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Volume 8, Issue 1

# A new method for converting boundary value problems for impulsive fractional differential equations to integral equations and its applications

Yuji Liu
• Corresponding author
• Department of Mathematics and Statics, Guangdong University of Finance and Economics, Guangzhou 510320, P. R. China
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Published Online: 2017-05-05 | DOI: https://doi.org/10.1515/anona-2016-0064

## Abstract

In this paper, we present a new method for converting boundary value problems of impulsive fractional differential equations to integral equations. Applications of this method are given to solve some types of anti-periodic boundary value problems for impulsive fractional differential equations. Firstly by using iterative method, we prove existence and uniqueness of solutions of Cauchy problems of differential equations involving Caputo fractional derivative, Riemann–Liouville and Hadamard fractional derivatives with order $q\in \left(0,1\right)$, see Theorem 2, Theorem 4, Theorem 6 and Theorem 8. Then we obtain exact expression of piecewise continuous solutions of these fractional differential equations see Theorem 1, Theorem 2, Theorem 3 and Theorem 4. Finally, four classes of integral type anti-periodic boundary value problems of singular fractional differential equations with impulse effects are proposed. Sufficient conditions are given for the existence of solutions of these problems. See Theorems 4.14.4. We allow the nonlinearity $p\left(t\right)f\left(t,x\right)$ in fractional differential equations to be singular at $t=0,1$ and be involved a super-linear and sub-linear term. The analysis relies on Schaefer’s fixed point theorem. In order to avoid misleading readers, we correct the results in [28] and [65]. We establish sufficient conditions for the existence of solutions of an anti-periodic boundary value problem for impulsive fractional differential equation. The results in [68] are complemented. The results in [81] are corrected. See Lemma 5.1, Lemma 5.7, Lemma 5.10 and Lemma 5.13.

MSC 2010: 34K37; 34K45; 34B37; 34B15; 34B10

## 1 Introduction

One knows that the fractional derivatives (Riemann–Liouville fractional derivative, Caputo fractional derivative and Hadamard fractional derivative and other types, see [33]) are actually nonlocal operators because integrals are nonlocal operators. Moreover, calculating time fractional derivatives of a function at some time requires all the past history and hence fractional derivatives can be used for modeling systems with memory.

Fractional order differential equations are generalizations of integer order differential equations. Using fractional order differential equations can help us to reduce the errors arising from the neglected parameters in modeling real life phenomena. Fractional differential equations have many applications, see [57, Chapter 10] and the books [34, 57, 60].

In recent years, there have been many results obtained on the existence and uniqueness of solutions of initial value problems or boundary value problems for nonlinear fractional differential equations, see [13, 15, 51, 55, 56, 59, 74, 79, 82].

Dynamics of many evolutionary processes from various fields such as population dynamics, control theory, physics, biology, and medicine, undergo abrupt changes at certain moments of time like earthquake, harvesting, shock, and so forth. These perturbations can be well approximated as instantaneous change of states or impulses. These processes are modeled by impulsive differential equations. In 1960, Milman and Myshkis introduced impulsive differential equations in their paper [52]. Based on their work, several monographs have been published by many authors like Samoilenko and Perestyuk [61], Lakshmikantham, Bainov and Simeonov [36], Bainov and Simeonov [10, 11], Bainov and Covachev [9], and Benchohra, Henderson and Ntonyas [16].

The fractional differential equation was extended to impulsive fractional differential equations, since Agarwal and Benchohra published the first paper on the topic [2] in 2008. Since then many authors [24, 25, 32, 46, 47, 42, 41, 54, 59, 75, 74, 78, 43, 48, 49] studied the existence or uniqueness of solutions of impulsive initial or boundary value problems for fractional differential equations. For examples, impulsive anti-periodic boundary value problems, see [3, 4, 2, 22, 44, 40, 63, 77], impulsive periodic boundary value problems, see [72, 67], impulsive initial value problems, see [18, 23, 53, 62], two-point, three-point or multi-point impulsive boundary value problems, see [8, 73, 80], and impulsive boundary value problems on infinite intervals, see [81].

Recently, in [28, 71, 83], the authors studied the existence and uniqueness of solutions of the following boundary value problem of impulsive fractional differential equation:

$\left\{\begin{array}{cc}\hfill {}^{C}D_{{0}^{+}}^{q}x\left(t\right)& =f\left(t,x\left(t\right)\right),t\in \left({t}_{i},{t}_{i+1}\right],i\in {ℕ}_{0},\hfill \\ \hfill {\mathrm{\Delta }x|}_{t={t}_{i}}& ={I}_{i}\left(x\left({t}_{i}^{-}\right)\right),i\in ℕ,ax\left(0\right)+bx\left(T\right)={x}_{0},\hfill \end{array}$(1.1)

where $q\in \left(0,1\right]$, ${}^{C}D_{{0}^{+}}^{q}$ is the standard Caputo fractional derivative of order q, ${ℕ}_{0}=\left\{0,1,\mathrm{\dots },m\right\}$ and $ℕ=\left\{1,2,\mathrm{\dots },m\right\}$, $f:\left[0,T\right]×ℝ\to ℝ$ is a jointly continuous function, ${I}_{k}:ℝ\to ℝ$ ($k\in ℕ$) are continuous functions, and $0={t}_{0}<{t}_{1}<\mathrm{\cdots }<{t}_{m}<{t}_{m+1}=T$,

${\mathrm{\Delta }x|}_{t={t}_{k}}=\underset{t\to {t}_{k}^{+}}{lim}x\left(t\right)-\underset{t\to {t}_{k}^{-}}{lim}x\left(t\right)=x\left({t}_{k}^{+}\right)-x\left({t}_{k}^{-}\right)$

and $x\left({t}_{k}^{+}\right),x\left({t}_{k}^{-}\right)$ represent the right and left limits of $x\left(t\right)$ at $t={t}_{k}$, respectively, $a,b,{x}_{0}$ are constants with $a+b\ne 0$. It was proved in [28] that if f is a jointly continuous function and there is a constant $\overline{\lambda }\in \left[0,1-\frac{1}{p}\right]$ for some $p\in \left(1,\frac{1}{1-q}\right]$ and $L>0$ such that $|f\left(t,x\right)|\le L\left(1+|x{|}^{\overline{\lambda }}\right]$ for each $t\in \left[0,T\right]$ and $x\in ℝ$. Then BVP (1.1) has at least one solution. We find that this result is wrong. For example, consider the following special problem of (1.1):

$\left\{\begin{array}{cc}\hfill {}^{C}D_{{0}^{+}}^{\alpha }x\left(t\right)& =t,t\in \left({t}_{i},{t}_{i+1}\right],i=0,1,\hfill \\ \hfill \mathrm{\Delta }x\left({t}_{1}\right)& =-2x\left({t}_{1}\right),x\left(0\right)+x\left(1\right)=0,\hfill \end{array}$(1.2)

where $\alpha \in \left(0,1\right)$, $0={t}_{0}<{t}_{1}<{t}_{2}=1$ with ${t}_{1}\ne {\left(\frac{1}{2}\right)}^{\frac{1}{\alpha +1}}$. It is easy to see that $f\left(t,x\right)=t$, ${I}_{1}\left(x\right)=-2x$, $a=b=1$ and ${x}_{0}=0$. One can see that f is a jointly continuous function and there is a constant $\overline{\lambda }\in \left[0,1-\frac{1}{p}\right]$ for some $p\in \left(1,\frac{1}{1-q}\right]$ and $L>0$ such that $|f\left(t,x\right)|\le L\left(1+|x{|}^{\overline{\lambda }}\right]$ for each $t\in \left[0,T\right]$ and $x\in ℝ$, $\left(\overline{\lambda }=0,L=1\right)$. But BVP (1.2) has no solution. In fact, if (1.2) has a solution x, then

$x\left(t\right)=\left\{\begin{array}{cccc}& {c}_{0}+\frac{{t}^{\alpha +1}}{\mathrm{\Gamma }\left(\alpha +1\right)},\hfill & & \hfill t\in \left(0,{t}_{1}\right],\\ & {c}_{0}+{c}_{1}+\frac{{t}^{\alpha +1}}{\mathrm{\Gamma }\left(\alpha +1\right)},\hfill & & \hfill t\in \left({t}_{1},1\right].\end{array}$

Thus we get

$2{c}_{0}+{c}_{1}+\frac{1}{\mathrm{\Gamma }\left(\alpha +1\right)}=0,{c}_{1}=-2\left({c}_{0}+\frac{{t}_{1}^{\alpha +1}}{\mathrm{\Gamma }\left(\alpha +1\right)}\right).$

So $2{t}_{1}^{\alpha +1}=1$, which contradicts to ${t}_{1}\ne {\left(\frac{1}{2}\right)}^{\frac{1}{\alpha +1}}$. Hence BVP (1.2) (which is called a anti-periodic boundary value problem) has no solution. It is needed to make a correction for (1.1), see Lemma 5.1.

In [71], it was proved that if f is a jointly continuous function and there exist ${C}_{f},{M}_{f}>0$ and ${q}_{1}\in \left[0,1\right]$ such that $|f\left(t,x\right)|\le {C}_{f}{|x|}^{{q}_{1}}+{M}_{f}$ for all $t\in \left[0,T\right]$ and $x\in ℝ$, there exist ${C}_{I},{M}_{I}>0$ and ${q}_{2}\in \left[0,1\right]$ such that $|{I}_{i}\left(x\right)|\le {C}_{I}{|x|}^{{q}_{2}}+{M}_{I}$ for all $i\in ℕ$ and $x\in ℝ$, there exist ${K}_{I}^{i}>0$ such that $|{I}_{i}\left(u\right)-{I}_{i}\left(v\right)|\le {K}_{I}^{i}|u-v|$ for all $u,v\in ℝ$ and $i\in ℕ$ with ${\sum }_{i=1}^{m}{K}_{I}^{i}<1$, then BVP (1.1) has at least one solution when

$\overline{K}=\frac{\mathrm{max}\left\{|a|,|b|\right\}}{|a+b|}\sum _{i=1}^{m}{K}_{I}^{i}<1.$

In [68], the authors actually studied the existence of solutions of the following boundary value problem for the impulsive fractional Langevin equations:

$\left\{\begin{array}{cc}\hfill {}^{C}D_{{0}^{+}}^{\beta }\left[{}^{C}D_{{0}^{+}}^{\alpha }x\left(t\right)+\lambda x\left(t\right)\right]& =f\left(t,x\left(t\right)\right),t\in \left[0,1\right]\setminus \left\{{t}_{1},{t}_{2},\mathrm{\dots },{t}_{m}\right\},\hfill \\ \hfill \mathrm{\Delta }x\left({t}_{i}\right)& ={I}_{i},i\in ℕ,x\left(0\right)=x\left({\eta }_{i}\right)=x\left(1\right)=0,i\in ℕ,\hfill \end{array}$(1.3)

where $\alpha ,\beta \in \left(0,1\right)$ with $\alpha +\beta <1$, $\lambda >0$, ${}^{C}D_{{0}^{+}}^{*}$ is the standard Caputo fractional derivative of order $*$, see [68, Definitions 2 and 3], $ℕ=\left\{1,\mathrm{\dots },m\right\}$, $f:\left[0,1\right]×ℝ\to ℝ$ is a given function, $0={t}_{0}<{t}_{1}<{t}_{2}<\mathrm{\cdots }<{t}_{m}<{t}_{m+1}=1$, ${I}_{i}\in ℝ$ ($i\in ℕ$) are constants, ${\eta }_{i}\in \left({t}_{i},{t}_{i+1}\right]$ ($i=0,1,2,\mathrm{\dots },m-1$),

$\mathrm{\Delta }Ix\left({t}_{i}\right)=x\left({t}_{i}^{+}\right)-x\left({t}_{i}^{-}\right)=\underset{t\to {t}_{k}^{+}}{lim}x\left(t\right)-\underset{t\to {t}_{k}^{-}}{lim}x\left(t\right).$

It is claimed in [68] that the presentation of solutions of

${}^{C}D_{t}^{\beta }\left[{}^{C}D_{{0}^{+}}^{\alpha }x\left(t\right)+\lambda x\left(t\right)\right]=f\left(t\right),t\in \left(ti,{t}_{i+1}\right],i\in {ℕ}_{0},$

is given by

$x\left(t\right)={𝐄}_{\alpha }\left(-\lambda {t}^{\alpha }\right){b}_{i}-\frac{1-{𝐄}_{\alpha }\left(-\lambda {t}^{\alpha }\right)}{\lambda }{a}_{i}+{\int }_{0}^{t}{\left(t-s\right)}^{\alpha +\beta -1}{𝐄}_{\alpha ,\alpha +\beta }\left(-\lambda {\left(t-s\right)}^{\alpha }\right)f\left(s\right)𝑑s,$

where ${a}_{i},{b}_{i}\in ℝ$ ($i\in {ℕ}_{0}$) are constants. We find that this claim is wrong, see Lemma 5.7.

The concept of “fractional relaxation differential equations” was introduced by Hilfer (see [30, (118), p. 115]) and [84]. That is, Hilfer considered the following generalized Hilfer fractional relaxation Cauchy problem

${}^{H}D_{{a}^{+}}^{\alpha ,\beta }x\left(t\right)=-Kx\left(t\right),t\in \left(a,b\right),{}^{\mathrm{RL}}I_{{a}^{+}}^{1-\gamma }x\left(a\right)={x}_{0},$

where $\alpha \in \left(0,1\right),\beta \in \left[0,1\right],\alpha \le \gamma =\alpha +\beta -\alpha \beta <1$, ${}^{H}D_{{a}^{+}}^{\alpha ,\beta }$ denotes the left-sided Hilfer fractional derivative of order α (see [30, Definition 3.3, p. 113] and type β, ${}^{\mathrm{RL}}I_{{a}^{+}}^{1-\gamma }$ denotes the left-sided Riemann–Liouville fractional integral (see [30, Definition 3.2, p. 112]), and K is called “fractional relaxation factor”. The solution of this problem can be written as (see [30, (124), p. 115]) $x\left(t\right)={x}_{0}{t}^{\gamma -1}{𝐄}_{\alpha ,\alpha +\beta \left(1-\gamma \right)}\left(-K{t}^{\alpha }\right)$. Obviously, the fractional relaxation factor K will generate the famous generalized Mittag-Leffler function ${𝐄}_{\alpha ,\alpha +\beta \left(1-\gamma \right)}\left(-K{t}^{\alpha }\right)$, which plays an important role in the study of various fractional relaxation differential equation [84].

In [65], the authors studied the following anti-periodic BVP for impulsive fractional differential equations with constant coefficients of the form:

$\left\{\begin{array}{cc}\hfill {}^{C}D_{{0}^{+}}^{\gamma }x\left(t\right)+\lambda x\left(t\right)& =f\left(t,x\left(t\right)\right),t\in \left[0,1\right]\setminus \left\{{t}_{1},{t}_{2},\mathrm{\dots },{t}_{m}\right\},\hfill \\ \hfill \mathrm{\Delta }x\left({t}_{i}\right)& =x\left({t}_{i}^{+}\right)-x\left({t}_{i}^{-}\right)={x}_{i},i\in ℕ,x\left(0\right)+x\left(1\right)=0,\hfill \end{array}$(1.5)

where $\lambda >0$, ${x}_{i}\in ℝ$ ($i\in ℕ$), ${}^{C}D_{{0}^{+}}^{\gamma }$ is the Caputo fractional derivative, see [65, Definition 2.2 and Definition 2.3], $f:\left[0,1\right]×ℝ\to ℝ$, $0={t}_{0}<{t}_{1}<\mathrm{\cdots }<{t}_{m}<{t}_{m+1}=1$. It is claimed (see [65, Lemma 3.1]) that

$\left\{\begin{array}{cc}\hfill {}^{C}D_{{0}^{+}}^{\gamma }x\left(t\right)+\lambda x\left(t\right)& =h\left(t\right),t\in \left[0,1\right]\setminus \left\{{t}_{1},{t}_{2},\mathrm{\dots },{t}_{m}\right\},\hfill \\ \hfill \mathrm{\Delta }x\left({t}_{i}\right)& =x\left({t}_{i}^{+}\right)-x\left({t}_{i}^{-}\right)={x}_{i},i\in ℕ,x\left(0\right)+x\left(1\right)=0,\hfill \end{array}$

has a unique solution

$x\left(t\right)=\frac{{𝐄}_{\gamma }\left(-\lambda {t}^{\gamma }\right)}{1+{𝐄}_{\gamma }\left(-\lambda \right)}\left[\sum _{i=1}^{m}\frac{{x}_{i}}{{𝐄}_{\gamma }\left(-\lambda {t}_{i}^{\gamma }\right)}-{\int }_{0}^{1}{\left(1-s\right)}^{\gamma -1}{𝐄}_{\gamma ,\gamma }\left(-\lambda {\left(1-s\right)}^{\gamma }\right)h\left(s\right)𝑑s\right]$$-{𝐄}_{\gamma }\left(-\lambda {t}^{\gamma }\right)\sum _{i=k+1}^{m}\frac{{x}_{i}}{{𝐄}_{\gamma }\left(-\lambda {t}_{i}^{\gamma }\right)}+{\int }_{0}^{t}{\left(t-s\right)}^{\gamma -1}{𝐄}_{\gamma ,\gamma }\left(-\lambda {\left(t-s\right)}^{\gamma }\right)h\left(s\right)𝑑s,t\in \left({t}_{k},{t}_{k+1}\right],k\in {ℕ}_{0}.$

This lemma is also wrong, see Lemma 5.10.

In [84], Zhang and Wang studied the following nonlocal Cauchy problems for implicit impulsive fractional relaxation differential systems:

$\left\{\begin{array}{cccc}\hfill {}^{C}D_{0,t}^{\gamma }x\left(t\right)& =-ax\left(t\right)+f\left(t,x\left(t\right),y\left(t\right)\right),\hfill & & \hfill t\in \left[0,1\right]\setminus \left\{{t}_{1},{t}_{2},\mathrm{\dots },{t}_{m}\right\},\\ \hfill {}^{C}D_{0,t}^{\gamma }y\left(t\right)& =-by\left(t\right)+g\left(t,x\left(t\right),y\left(t\right)\right),\hfill & & \hfill t\in \left[0,1\right]\setminus \left\{{t}_{1},{t}_{2},\mathrm{\dots },{t}_{m}\right\},\\ \hfill x\left(0\right)& =\sum _{i=1}^{m}{\alpha }_{i}x\left({\tau }_{i}\right),y\left(0\right)=\sum _{i=1}^{m}{\beta }_{i}y\left({\tau }_{i}\right),\hfill \\ \hfill \mathrm{\Delta }x\left({t}_{i}\right)& ={I}_{i}\left(x\left({t}_{i}\right)\right),\mathrm{\Delta }y\left({t}_{i}\right)={J}_{i}\left(y\left({t}_{i}\right)\right),i=1,2,\mathrm{\dots },m,\hfill \end{array}$(1.6)

where $a,b>0$ are fractional relaxation factors, ${}^{C}D_{0,t}^{\gamma }$ is the generalized Caputo fractional derivative of order $\gamma \in \left(0,1\right)$ with the lower limit zero, $f,g:\left[0,1\right]×ℝ×ℝ\to ℝ$ and ${\alpha }_{i},{\beta }_{i}$ are real numbers with $1+{\sum }_{i=1}^{m}{\alpha }_{i}\ne 0$, $1+{\sum }_{i=1}^{m}{\beta }_{i}\ne 0$. Next, ${I}_{i},{J}_{i}:ℝ\to ℝ$ and ${t}_{i},{\tau }_{i}$ are given points satisfying $0\le {t}_{1}\le {t}_{2}\le \mathrm{\cdots }\le {t}_{m}\le {t}_{m+1}=1$ and ${\tau }_{i}\in \left({t}_{i},{t}_{i+1}\right)$, $i=1,2,\mathrm{\dots },m$, $\mathrm{\Delta }x\left({t}_{i}\right)={lim}_{t\to {t}_{i}^{+}}x\left(t\right)-x\left({t}_{i}\right)$, $\mathrm{\Delta }y\left({t}_{i}\right)={lim}_{t\to {t}_{i}^{+}}y\left(t\right)-y\left({t}_{i}\right)$.

It is claimed [84] that one can make straightforward fractional calculations to show that the above system can be written as the following integral system:

$x\left(t\right)=-\alpha \sum _{i=1}^{m}{\alpha }_{i}\left[\sum _{0<{t}_{i}<{\tau }_{i}}{𝐄}_{\gamma }\left(-a{t}_{i}^{\gamma }\right){I}_{i}\left(x\left({t}_{i}\right)\right)+\sum _{0<{t}_{i}$+{\int }_{0}^{{\tau }_{i}}{\left({\tau }_{i}-s\right)}^{\gamma -1}{𝐄}_{\gamma ,\gamma }\left(-a{\left({\tau }_{i}-s\right)}^{\gamma }\right)f\left(s,x\left(s\right),y\left(s\right)\right)ds\right]$$+{\int }_{0}^{t}{\left(t-s\right)}^{\gamma -1}{𝐄}_{\gamma ,\gamma }\left(-a{\left(t-s\right)}^{\gamma }\right)f\left(s,x\left(s\right),y\left(s\right)\right)𝑑s,$$y\left(t\right)=-\beta \sum _{i=1}^{m}{\beta }_{i}\left[\sum _{0<{t}_{i}<{\tau }_{i}}{𝐄}_{\gamma }\left(-b{t}_{i}^{\gamma }\right){J}_{i}\left(y\left({t}_{i}\right)\right)+\sum _{0<{t}_{i}$+{\int }_{0}^{{\tau }_{i}}{\left({\tau }_{i}-s\right)}^{\gamma -1}{𝐄}_{\gamma ,\gamma }\left(-b{\left({\tau }_{i}-s\right)}^{\gamma }\right)g\left(s,x\left(s\right),y\left(s\right)\right)ds\right]$$+{\int }_{0}^{t}{\left(t-s\right)}^{\gamma -1}{𝐄}_{\gamma ,\gamma }\left(-b{\left(t-s\right)}^{\gamma }\right)g\left(s,x\left(s\right),y\left(s\right)\right)𝑑s,$

where

${𝐄}_{\gamma }\left(z\right)=\sum _{\chi =0}^{\mathrm{\infty }}\frac{{z}^{\chi }}{\mathrm{\Gamma }\left(\gamma \chi +1\right)},{𝐄}_{\gamma ,\gamma }\left(z\right)=\sum _{\chi =0}^{\mathrm{\infty }}\frac{{z}^{\chi }}{\mathrm{\Gamma }\left(\gamma \left(\chi +1\right)\right)},\alpha =1+\sum _{i=1}^{m}{\alpha }_{i}{𝐄}_{\gamma }\left(-a{\tau }_{i}^{\gamma }\right),\beta =1+\sum _{i=1}^{m}{\beta }_{i}{𝐄}_{\gamma }\left(-b{\tau }_{i}^{\gamma }\right).$

However, this claim is also wrong, see Lemma 5.13.

In a fractional differential equation, there exist two cases involving its fractional derivatives: the first case is ${D}^{\alpha }={D}_{{0}^{+}}^{\alpha }$ in which the fractional derivative has a unique start point. The second case is ${D}^{\alpha }={D}_{{t}_{i}^{+}}^{\alpha }$ in which ${D}^{\alpha }$ has multiple start points, i.e.,

${D}^{\alpha }={D}_{{t}_{i}^{+}}^{\alpha }.$

Concerning the second case, readers may find some discussions in [14, 72, 45, 77], and there is no confusion problem for this case.

In [26], Feckan and Zhou pointed out that the formula of solutions for impulsive fractional differential equations in [1, 12, 17] is incorrect and gave their correct formula. In [14, 70, 69], the authors established a general framework to find the solutions for impulsive fractional boundary value problems and obtained some sufficient conditions for the existence of the solutions to a kind of impulsive fractional differential equations, respectively. In [64], the authors illustrated their comprehensions for the counterexample in [26] and criticized the viewpoint in [26, 70, 69]. Next, in [27], Feckan, Zhou and Wang expounded for the counterexample in [26] and provided further five explanations in the paper. The concept of solution of an impulsive fractional differential equation is still an open question. This is the third motivation of this paper for studying the solvability of boundary value problems of impulsive fractional differential equations.

From the above discussion, it is of most importance and is interesting to search a new method for converting an impulsive boundary value problem for fractional differential equation to an equivalent integral equation. This is the motivation of this paper.

In this paper, we will study the existence of solutions of four classes of impulsive integral type boundary value problems of singular fractional differential systems. The first one is as follows:

$\left\{\begin{array}{cc}\hfill {}^{\mathrm{RL}}D_{{0}^{+}}^{\alpha }x\left(t\right)-\lambda x\left(t\right)& =p\left(t\right)f\left(t,x\left(t\right)\right),t\in \left({t}_{i},{t}_{i+1}\right],i\in {ℕ}_{0},\hfill \\ \hfill x\left(1\right)+\underset{t\to 0}{lim}{t}^{1-\alpha }x\left(t\right)& ={\int }_{0}^{1}\varphi \left(s\right)G\left(s,x\left(s\right)\right)𝑑s,\underset{t\to {t}_{i}^{+}}{lim}{\left(t-{t}_{i}\right)}^{1-\alpha }x\left(t\right)=I\left({t}_{i},x\left({t}_{i}\right)\right),i\in ℕ,\hfill \end{array}$(1.7)

where

• (a)

$0<\alpha <1$, $\lambda \in ℝ$, ${}^{\mathrm{RL}}D_{{0}^{+}}^{\alpha }$ is the Riemann–Liouville fractional derivative of order α,

• (b)

$0={t}_{0}<{t}_{1}<{t}_{2}<\mathrm{\cdots }<{t}_{m}<{t}_{m+1}=1$, ${ℕ}_{0}=\left\{0,1,2,\mathrm{\dots },m\right\}$ and $ℕ=\left\{1,2,\mathrm{\dots },m\right\}$,

• (c)

$\varphi :\left(0,1\right)\to ℝ$ satisfies ${\varphi |}_{\left({t}_{i},{t}_{i+1}\right)}$, $i\in {ℕ}_{0}$,

• (d)

$p:\left(0,1\right)\to ℝ$ is measurable and satisfies the growth conditions: there exist constants $k,l$ with $k>-1$ and $\mathrm{max}\left\{-\alpha ,-k-1\right\} such that $|p\left(t\right)|\le {t}^{k}{\left(1-t\right)}^{l}$, $t\in \left(0,1\right)$,

• (e)

$f,G$ defined on $\left(0,1\right]×ℝ$ are impulsive II-Carathéodory functions, $I:\left\{{t}_{i}:i\in ℕ\right\}×ℝ\to ℝ$ is a discrete II-Carathéodory function.

The second one is the following:

$\left\{\begin{array}{cc}\hfill {}^{C}D_{{0}^{+}}^{\alpha }x\left(t\right)-\lambda x\left(t\right)& =p\left(t\right)f\left(t,x\left(t\right)\right),t\in \left({t}_{i},{t}_{i+1}\right],i\in {ℕ}_{0},\hfill \\ \hfill x\left(1\right)+\underset{t\to {0}^{+}}{lim}x\left(t\right)& ={\int }_{0}^{1}\varphi \left(s\right)G\left(s,x\left(s\right)\right)𝑑s,\underset{t\to {t}_{i}^{+}}{lim}x\left(t\right)-x\left({t}_{i}\right)=I\left({t}_{i},x\left({t}_{i}\right)\right),i\in ℕ,\hfill \end{array}$(1.8)

where

• (f)

$0<\alpha <1$, $\lambda \in ℝ$, ${}^{C}D_{{0}^{+}}^{\alpha }$ is the Caputo fractional derivative of order α, ${t}_{i}$ satisfies (b), $\varphi :\left(0,1\right)\to ℝ$ satisfies (c), $p:\left(0,1\right)\to ℝ$ is measurable and satisfies that there exist constants $k,l$ with $k>-1$, $l\le 0,l\le 0$ with $\alpha +l>0$, $\alpha +k+l>0$ such that $|p\left(t\right)|\le {t}^{k}{\left(1-t\right)}^{l}$, $t\in \left(0,1\right)$,

• (g)

$f,G$ defined on $\left(0,1\right]×ℝ$ are impulsive I-Carathéodory functions, $I:\left\{{t}_{i}:i\in ℕ\right\}×ℝ\to ℝ$ is a discrete I-Carathéodory function.

We emphasize that much work on fractional boundary value problems involves either Riemann–Liouville or Caputo type fractional differential equations, see [5, 6, 7, 4]. Another kind of fractional derivatives that appears side by side to Riemann–Liouville and Caputo derivatives in the literature is the fractional derivative due to Hadamard introduced in 1892 [29], which differs from the preceding ones in the sense that the kernel of the integral (in the definition of Hadamard derivative) contains logarithmic function of arbitrary exponent. Recent studies can be seen in [19, 20, 21].

Thirdly we study the following impulsive integral type boundary value problems of singular fractional differential systems:

$\left\{\begin{array}{cc}\hfill {}^{\mathrm{RLH}}D_{{1}^{+}}^{\alpha }x\left(t\right)-\lambda x\left(t\right)& =p\left(t\right)f\left(t,x\left(t\right)\right),t\in \left({t}_{i},{t}_{i+1}\right],i\in {ℕ}_{0},\hfill \\ \hfill x\left(e\right)+\underset{t\to {1}^{+}}{lim}{\left(\mathrm{log}t\right)}^{1-\alpha }x\left(t\right)& ={\int }_{1}^{e}\varphi \left(s\right)G\left(s,x\left(s\right)\right)𝑑s,\underset{t\to {t}_{i}^{+}}{lim}{\left(\mathrm{log}\frac{t}{{t}_{i}}\right)}^{1-\alpha }x\left(t\right)=I\left({t}_{i},x\left({t}_{i}\right)\right),i\in ℕ,\hfill \end{array}$(1.9)

where

• (h)

$0<\alpha <1$, $\lambda \in ℝ$, ${}^{\mathrm{RLH}}D_{{1}^{+}}^{\alpha }$ is the Hadamard fractional derivative of order α,

• (i)

$1={t}_{0}<{t}_{1}<{t}_{2}<\mathrm{\cdots }<{t}_{m}<{t}_{m+1}=e$, $\varphi \in {L}^{1}\left(1,e\right)$, ${ℕ}_{0}=\left\{0,1,2,\mathrm{\dots },m\right\}$, $ℕ=\left\{1,2,\mathrm{\dots },m\right\}$, the function $p:\left(1,e\right)\to ℝ$ is measurable and satisfies the growth conditions: there exist constants k and l with $k>-1$ and $\mathrm{max}\left\{-\alpha ,-k-1\right\} such that $|p\left(t\right)|\le {\left(\mathrm{log}t\right)}^{k}{\left(1-\mathrm{log}t\right)}^{l}$, $t\in \left(1,e\right)$,

• (j)

$f,G$ defined on $\left(1,e\right]×ℝ$ are impulsive III-Carathéodory functions, $I:\left\{{t}_{i}:i\in ℕ\right\}×ℝ\to ℝ$ is a discrete III-Carathéodory function.

Finally, we study the following impulsive integral type boundary value problems of singular fractional differential systems:

$\left\{\begin{array}{cc}\hfill {}^{\mathrm{CH}}D_{{1}^{+}}^{\alpha }x\left(t\right)-\lambda x\left(t\right)& =p\left(t\right)f\left(t,x\left(t\right)\right),t\in \left({t}_{i},{t}_{i+1}\right],i\in {ℕ}_{0},\hfill \\ \hfill x\left(e\right)+\underset{t\to {1}^{+}}{lim}x\left(t\right)& ={\int }_{1}^{e}\varphi \left(s\right)G\left(s,x\left(s\right)\right)𝑑s,\underset{t\to {t}_{i}^{+}}{lim}x\left(t\right)-x\left({t}_{i}\right)=I\left({t}_{i},x\left({t}_{i}\right)\right),i\in ℕ,\hfill \end{array}$(1.10)

where

• (k)

$0<\alpha <1$, $\lambda \in ℝ$, ${}^{\mathrm{CH}}D_{{1}^{+}}^{\alpha }$ is the Caputo type Hadamard fractional derivative of order α,

• (l)

$1={t}_{0}<{t}_{1}<{t}_{2}<\mathrm{\cdots }<{t}_{m}<{t}_{m+1}=e$, ${ℕ}_{0}=\left\{0,1,2,\mathrm{\dots },m\right\}$ and $ℕ=\left\{1,2,\mathrm{\dots },m\right\}$, $\varphi \in {L}^{1}\left(1,e\right)$, and the function $p:\left(1,e\right)\to ℝ$ is measurable and satisfies that there exist constants k and l with $k>-1$ and $\mathrm{max}\left\{-\alpha ,-k-\alpha \right\} such that $|p\left(t\right)|\le {\left(\mathrm{log}t\right)}^{k}{\left(1-\mathrm{log}t\right)}^{l}$, $t\in \left(1,e\right)$,

• (m)

$f,G$ defined on $\left(1,e\right]×ℝ$ are impulsive I-Carathéodory functions, $I:\left\{{t}_{i}:i\in ℕ\right\}×ℝ\to ℝ$ is a discrete I-Carathéodory function.

We give an exact expression of continuous solutions of linear fractional differential equations by using the Picard iterative method. By using the mathematical induction method, we obtain an exact expression of piecewise continuous solutions of linear fractional differential equations. By using these results, we convert boundary value problems for impulsive fractional differential equations to integral equations technically.

In order to get solutions of a boundary value problem for impulsive fractional differential equations, we firstly define a Banach space X. Then we transform the boundary value problem into an integral equation and define a nonlinear operator T on X. Finally, we prove that T has a fixed point in X. This fixed point is just a solution of the boundary value problem. Three difficulties occur in known papers: one is how to transform the boundary value problem into a integral equation; the other one is how to define and prove a Banach space and the completely continuous property of the nonlinear operator defined; the third one is to choose a suitable fixed point theorem and impose suitable growth conditions on functions to get the fixed points of the operator.

To the best of the authors knowledge, no one has studied the existence of solutions of BVPs (1.7), (1.8), (1.9), (1.10). This paper fills this gap. Another purpose of this paper is to illustrate the similarity and difference of these three kinds of fractional differential equations. We obtain results on the existence of at least one solution for BVPs (1.7), (1.8), (1.9), (1.10), respectively. Some examples are given to illustrate the efficiency of the main theorems. For simplicity we only consider the left-sided operators here. The right-sided operators can be treated similarly.

The remainder of this paper is as follows: in Section 2, we present related definitions, especially the concept of solution of an impulsive fractional differential equation is proposed. In Section 3 some preliminary results are given. (By iterative method, we prove firstly the existence and uniqueness of solutions of the initial value problems of linear fractional differential equations, the results obtained generalize known ones. Then the exact piecewise continuous solutions of the impulsive linear fractional differential equations are obtained. Finally, we prove preliminary results for establishing existence of solutions of BVPs (1.7), (1.8), (1.9), (1.10) in the next section.) In Section 4, the main theorems (Theorems 4.14.4) and their proofs are given (we establish sufficient conditions for the existence of solutions (see Definitions 2.142.17) of BVPs (1.7), (1.8), (1.9), (1.10), respectively). In Section 5, we apply the methods proposed in this paper to solve five classes of boundary value problems more general than BVP (1.1), BVP (1.3), BVP (1.5) and BVP (1.6), respectively, see Theorems 5.35.15. The results in [68] are complemented. The results in [81] are corrected. See Lemmas 5.1, 5.7, 5.10 and 5.13.

## 2 Definitions

For the convenience of the readers, we firstly present the necessary definitions from the fractional calculus theory. These definitions and results can be found in the literature [34, 57, 60].

Let the Gamma function, Beta function and the classical Mittag-Leffler special function be defined by

$\mathrm{\Gamma }\left(\alpha \right)={\int }_{0}^{+\mathrm{\infty }}{x}^{\alpha -1}{e}^{-x}𝑑x,$$\alpha >0,$$𝐁\left(\alpha ,\beta \right)={\int }_{0}^{1}{x}^{\alpha -1}{\left(1-x\right)}^{\beta -1}𝑑x,$$\alpha >0,\beta >0,$${𝐄}_{\alpha ,\beta }\left(x\right)=\sum _{v=0}^{+\mathrm{\infty }}\frac{{x}^{v}}{\mathrm{\Gamma }\left(\alpha v+\beta \right)},$$\alpha >0,\beta >0,x\in ℝ.$

It is known that the Gamma function satisfies $\mathrm{\Gamma }\left(\alpha +1\right)=\alpha \mathrm{\Gamma }\left(\alpha \right)$, the Beta function satisfies $𝐁\left(\alpha ,\beta \right)=\frac{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(\beta \right)}{\mathrm{\Gamma }\left(\alpha +\beta \right)}$, and

$|{𝐄}_{\alpha ,\beta }\left(x\right)|\le {𝐄}_{\alpha ,\beta }\left(|x|\right)\le {𝐄}_{\alpha ,\beta }\left(|y|\right)$

for all $x,y\in ℝ$ with $|x|\le |y|$.

#### Definition 2.1 ([34]).

Let $c\in ℝ$ and $g:\left(c,+\mathrm{\infty }\right)\to ℝ$. The left Riemann–Liouville fractional integral of order $\alpha >0$ of g is given by

${I}_{{c}^{+}}^{\alpha }g\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{c}^{t}{\left(t-s\right)}^{\alpha -1}g\left(s\right)𝑑s,t>c,$

provided that the right-hand side exists.

#### Definition 2.2 ([34]).

Let $c\in ℝ$ and $g:\left(c,+\mathrm{\infty }\right)\to ℝ$. The left Riemann–Liouville fractional derivative of order $\alpha >0$ of g is given by

${}^{\mathrm{RL}}D_{{c}^{+}}^{\alpha }g\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(n-\alpha \right)}\frac{{d}^{n}}{d{t}^{n}}{\int }_{c}^{t}\frac{g\left(s\right)}{{\left(t-s\right)}^{\alpha -n+1}}𝑑s,t>c$

where $\alpha , i.e., $n=⌈\alpha ⌉$, provided that the right-hand side exists.

#### Definition 2.3 ([34]).

Let $c\in ℝ$ and $g:\left(c,+\mathrm{\infty }\right)\to ℝ$. The left Caputo fractional derivative of order $\alpha >0$ of g is given by

${}^{C}D_{{c}^{+}}^{\alpha }g\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(n-\alpha \right)}{\int }_{c}^{t}\frac{{g}^{\left(n\right)}\left(s\right)}{{\left(t-s\right)}^{\alpha -n+1}}𝑑s,t>c,$

where $\alpha , i.e., $n=⌈\alpha ⌉$, provided that the right-hand side exists.

#### Definition 2.4 ([34]).

Let $c>0$ and $g:\left(c,+\mathrm{\infty }\right)\to ℝ$. The left Hadamard fractional integral of order $\alpha >0$ of g is given by

${}^{H}I_{{c}^{+}}^{\alpha }g\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{c}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}g\left(s\right)\frac{ds}{s},t>c,$

provided that the right-hand side exists.

#### Definition 2.5 ([34]).

Let $c>0$ and $g:\left(c,+\mathrm{\infty }\right)\to ℝ$. The left Hadamard fractional derivative of order $\alpha >0$ of g is given by

${}^{\mathrm{RLH}}D_{{c}^{+}}^{\alpha }g\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(n-\alpha \right)}{\left(t\frac{d}{dt}\right)}^{n}{\int }_{c}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{n-\alpha -1}g\left(s\right)\frac{ds}{s},t>c,$

where $\alpha , i.e., $n=⌈\alpha ⌉$, provided that the right-hand side exists.

#### Definition 2.6 ([31]).

Let $c>0$ and $g:\left(c,+\mathrm{\infty }\right)\to ℝ$. The left Caputo type Hadamard fractional derivative of order $\alpha >0$ of g is given by

${}^{\mathrm{CH}}D_{{c}^{+}}^{\alpha }g\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(n-\alpha \right)}{\int }_{c}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{n-\alpha -1}{\left(s\frac{d}{ds}\right)}^{n}g\left(s\right)\frac{ds}{s},t>c,$

where $\alpha , i.e., $n=⌈\alpha ⌉$, provided that the right-hand side exists.

#### Definition 2.7.

We call $F:{\bigcup }_{i=0}^{m}\left({t}_{i},{t}_{i+1}\right)×ℝ\to ℝ$ an impulsive I-Carathéodory function if it satisfies

• (i)

$t↦F\left(t,u\right)$ is measurable on $\left({t}_{i},{t}_{i+1}\right)$ ($i\in {ℕ}_{0}$) for any $u\in ℝ$,

• (ii)

$u↦F\left(t,u\right)$ is continuous on $ℝ$ for almost all $t\in \left({t}_{i},{t}_{i+1}\right)$ ($i\in {ℕ}_{0}$),

• (iii)

for each $r>0$ there exists ${M}_{r}>0$ such that

$|F\left(t,u\right)|\le {M}_{r},t\in \left({t}_{i},{t}_{i+1}\right),|u|\le r,i\in {ℕ}_{0}.$

#### Definition 2.8.

We call $F:{\bigcup }_{i=0}^{m}\left({t}_{i},{t}_{i+1}\right)×ℝ\to ℝ$ an impulsive II-Carathéodory function if it satisfies

• (i)

$t↦F\left(t,{\left(t-{t}_{i}\right)}^{\alpha -1}u\right)$ is measurable on $\left({t}_{i},{t}_{i+1}\right)$ ($i=0,1$) for any $u\in ℝ$,

• (ii)

$u↦F\left(t,{\left(t-{t}_{i}\right)}^{\alpha -1}u\right)$ is continuous on $ℝ$ for almost all $t\in \left({t}_{i},{t}_{i+1}\right)$ ($i\in {ℕ}_{0}$),

• (iii)

for each $r>0$ there exists ${M}_{r}>0$ such that

$|F\left(t,{\left(t-{t}_{i}\right)}^{\alpha -1}u\right)|\le {M}_{r},t\in \left({t}_{i},{t}_{i+1}\right),|u|\le r,i\in {ℕ}_{0}.$

#### Definition 2.9.

We call $F:{\bigcup }_{i=0}^{m}\left({t}_{i},{t}_{i+1}\right)×ℝ\to ℝ$ an impulsive III-Carathéodory function if it satisfies

• (i)

$t↦F\left(t,{\left(\mathrm{log}\frac{t}{{t}_{i}}\right)}^{\alpha -1}u\right)$ is measurable on $\left({t}_{i},{t}_{i+1}\right)$ ($i\in {ℕ}_{0}$) for any $u\in ℝ$,

• (ii)

$u↦F\left(t,{\left(\mathrm{log}\frac{t}{{t}_{i}}\right)}^{\alpha -1}u\right)$ is continuous on $ℝ$ for almost all $t\in \left({t}_{i},{t}_{i+1}\right)$ ($i\in {ℕ}_{0}$),

• (iii)

for each $r>0$ there exists ${M}_{r}>0$ such that

$|F\left(t,{\left(\mathrm{log}\frac{t}{{t}_{i}}\right)}^{\alpha -1}u\right)|\le {M}_{r},t\in \left({t}_{i},{t}_{i+1}\right),|u|\le r,i\in {ℕ}_{0}.$

#### Definition 2.10.

We call $I:\left\{{t}_{i}:i\in ℕ\right\}×ℝ\to ℝ$ an discrete I-Carathéodory function if it satisfies

• (i)

$\left(u,v\right)↦I\left({t}_{i},u\right)$ is continuous on ${ℝ}^{2}$,

• (ii)

for each $r>0$ there exists ${M}_{r}>0$ such that $|I\left({t}_{i},u\right)|\le {M}_{r},|u|\le r$.

#### Definition 2.11.

We call $I:\left\{{t}_{i}:i\in {ℕ}_{0}\right\}×ℝ\to ℝ$ an discrete II-Carathéodory function if it satisfies

• (i)

$u↦I\left({t}_{i},{\left({t}_{i}-{t}_{i-1}\right)}^{\alpha -1}u\right)$ is continuous on $ℝ$,

• (ii)

for each $r>0$ there exists ${M}_{r}>0$ such that $|I\left({t}_{i},{\left({t}_{i}-{t}_{i-1}\right)}^{\alpha -1}u\right)|\le {M}_{r},|u|\le r$.

#### Definition 2.12.

We call $I:\left\{{t}_{i}:i\in {ℕ}_{0}\right\}×ℝ\to ℝ$ an discrete III-Carathéodory function if it satisfies

• (i)

$u↦I\left({t}_{i},{\left(\mathrm{log}\frac{{t}_{i}}{{t}_{i-1}}\right)}^{\alpha -1}u\right)$ is continuous on $ℝ$,

• (ii)

for each $r>0$ there exists ${M}_{r}>0$ such that $|I\left({t}_{i},{\left(\mathrm{log}\frac{{t}_{i}}{{t}_{i-1}}\right)}^{\alpha -1}u\right)|\le {M}_{r},|u|\le r$.

#### Definition 2.13 ([50]).

Let E and F be Banach spaces. A operator $T:E\to F$ is called a completely continuous operator if T is continuous and maps any bounded set into relatively compact set.

Let $a be constants. The following Banach spaces are used:

• (i)

$C\left(a,b\right]$ denotes the set of all continuous functions on $\left(a,b\right]$ with the limit ${lim}_{t\to {a}^{+}}x\left(t\right)$ existing, and the norm $\parallel x\parallel ={sup}_{t\in \left(a,b\right]}|x\left(t\right)|$.

• (ii)

${C}_{1-\alpha }\left(a,b\right]$ the set of all continuous functions on $\left(a,b\right]$ with the limit ${lim}_{t\to {a}^{+}}{\left(t-a\right)}^{1-\alpha }x\left(t\right)$ existing, the norm ${\parallel x\parallel }_{1-\alpha }={sup}_{t\in \left(a,b\right]}{\left(t-a\right)}^{1-\alpha }|x\left(t\right)|$.

• (iii)

$L{C}_{1-\alpha }\left(a,b\right]$ denotes the set of all continuous functions on $\left(a,b\right]$ with the limit ${lim}_{t\to {a}^{+}}{\left(\mathrm{log}\frac{t}{a}\right)}^{1-\alpha }x\left(t\right)$ existing, and the norm $\parallel x\parallel ={sup}_{t\in \left(a,b\right]}{\left(\mathrm{log}\frac{t}{a}\right)}^{1-\alpha }|x\left(t\right)|$.

Let m be a positive integer and ${ℕ}_{0}=\left\{0,1,2,\mathrm{\dots },m\right\}$, $0={t}_{0}<{t}_{1}<\mathrm{\cdots }<{t}_{m}<{t}_{m+1}=1$. The following Banach spaces are also used in this paper:

${P}_{m}{C}_{1-\alpha }\left(0,1\right]=\left\{x:\left(0,1\right]\to ℝ:{x|}_{\left({t}_{i},{t}_{i+1}\right]}\in {C}_{1-\alpha }\left({t}_{i},{t}_{i+1}\right],i\in {ℕ}_{0}\right\}$

with the norm

$\parallel x\parallel ={\parallel x\parallel }_{{P}_{m}{C}_{1-\alpha }}=\mathrm{max}\left\{\underset{t\in \left({t}_{i},{t}_{i+1}\right]}{sup}{\left(t-{t}_{i}\right)}^{1-\alpha }|x\left(t\right)|:i\in {ℕ}_{0}\right\},$

and

${P}_{m}C\left(0,1\right]=\left\{x:\left(0,1\right]\to ℝ:{x|}_{\left({t}_{i},{t}_{i+1}\right]}\in C\left({t}_{i},{t}_{i+1}\right],i\in {ℕ}_{0}\right\}$

with the norm

$\parallel x\parallel ={\parallel x\parallel }_{{P}_{m}C\left(0,1\right]}=\mathrm{max}\left\{\underset{t\in \left({t}_{i},{t}_{i+1}\right]}{sup}|x\left(t\right)|:i\in {ℕ}_{0}\right\}.$

Let $1={t}_{0}<{t}_{1}<\mathrm{\cdots }<{t}_{m}<{t}_{m+1}=e$. We also use the Banach spaces

$L{P}_{m}{C}_{1-\alpha }\left(1,e\right]=\left\{x:\left(1,e\right]\to ℝ:{x|}_{\left({t}_{i},{t}_{i+1}\right]}\in C\left({t}_{i},{t}_{i+1}\right],i\in {ℕ}_{0},\underset{t\to {t}_{i}^{+}}{lim}{\left(\mathrm{log}\frac{t}{{t}_{i}}\right)}^{1-\alpha }x\left(t\right),i\in {ℕ}_{0},\text{exist}\right\}$

with the norm

$\parallel x\parallel ={\parallel x\parallel }_{L{P}_{m}{C}_{1-\alpha }}=\mathrm{max}\left\{\underset{t\in \left({t}_{i},{t}_{i+1}\right]}{sup}{\left(\mathrm{log}\frac{t}{{t}_{i}}\right)}^{1-\alpha }|x\left(t\right)|:i\in {ℕ}_{0}\right\},$

and

${P}_{m}C\left(1,e\right]=\left\{x:\left(1,e\right]\to ℝ:{x|}_{\left({t}_{i},{t}_{i+1}\right]}\in C\left({t}_{i},{t}_{i+1}\right],i\in {ℕ}_{0}\right\}$

with the norm

$\parallel x\parallel ={\parallel x\parallel }_{{P}_{m}C}=\mathrm{max}\left\{\underset{t\in \left({t}_{i},{t}_{i+1}\right]}{sup}|x\left(t\right)|:i\in {ℕ}_{0}\right\}.$

Now, we propose new concepts of solution of an impulsive fractional differential equation with Caputo fractional derivative, Riemann–Liouville fractional derivative, Caputo type Hadamard fractional derivative and Riemann–Liouville type Hadamard fractional derivative respectively.

#### Definition 2.14.

A function $x:\left(0,1\right]\to ℝ$ is called a solution of BVP (1.7) if ${x|}_{\left({t}_{i},{t}_{i+1}\right]}$ is continuous on $\left({t}_{i},{t}_{i+1}\right]$ ($i\in {ℕ}_{0}$), the limits ${lim}_{t\to {t}_{i}^{+}}{\left(t-{t}_{i}\right)}^{1-\alpha }x\left(t\right)$ ($i\in {ℕ}_{0}$) exist, ${}^{\mathrm{RL}}D_{{0}^{+}}^{\alpha }x:\left(0,1\right]\to ℝ$ is measurable on $\left(0,1\right]$, and x satisfies all equations in (1.7).

#### Definition 2.15.

A function $x:\left(0,1\right]\to ℝ$ is called a solution of BVP (1.8) if ${x|}_{\left({t}_{i},{t}_{i+1}\right]}$ is continuous on $\left({t}_{i},{t}_{i+1}\right]$ ($i\in {ℕ}_{0}$), the limits ${lim}_{t\to {t}_{i}^{+}}x\left(t\right)$ ($i\in {ℕ}_{0}$) exist, ${}^{C}D_{{0}^{+}}^{\alpha }x:\left(0,1\right]\to ℝ$ is measurable on $\left(0,1\right]$ and x satisfies all equations in (1.8).

#### Definition 2.16.

A function $x:\left(1,e\right]\to ℝ$ is called a solution of BVP (1.9) if ${x|}_{\left({t}_{i},{t}_{i+1}\right]}$ is continuous on $\left({t}_{i},{t}_{i+1}\right]$ ($i\in {ℕ}_{0}$), the limits ${lim}_{t\to {t}_{i}^{+}}{\left(\mathrm{log}t-\mathrm{log}{t}_{i}\right)}^{1-\alpha }x\left(t\right)$ ($i\in {ℕ}_{0}$) exist, ${}^{\mathrm{RLH}}D_{{0}^{+}}^{\alpha }x:\left(1,e\right]\to ℝ$ is measurable on $\left(1,e\right]$, and x satisfies all equations in (1.9).

#### Definition 2.17.

A function $x:\left(1,e\right]\to ℝ$ is called a solution of BVP (1.10) if ${x|}_{\left({t}_{i},{t}_{i+1}\right]}$ is continuous on $\left({t}_{i},{t}_{i+1}\right]$ ($i\in {ℕ}_{0}$), the limits ${lim}_{t\to {t}_{i}^{+}}x\left(t\right)$ ($i\in {ℕ}_{0}$) exist, ${}^{\mathrm{CH}}D_{{0}^{+}}^{\alpha }x:\left(1,e\right]\to ℝ$ is measurable on $\left(1,e\right]$, and x satisfies all equations in (1.10).

## 3 Preliminary results

This section is divided into three subsections. By Picard iterative method, in Section 3.1, we prove the existence and uniqueness of solutions of the initial value problems for linear fractional differential equations, the results obtained generalize known ones. In Section 3.2, the exact piecewise continuous solutions of the impulsive linear fractional differential equations are obtained. In Sections 3.33.6, we prove preliminary results for establishing existence of solutions of BVPs (1.7), (1.8), (1.9), (1.10).

## 3.1 Picard iterative method for linear fractional differential equations

In [35, 37, 38, 39]. the basic theory of initial value problems for fractional differential equations involving Riemann–Liouville differential operators of order $q\in \left(0,1\right)$ were investigated. The existence and uniqueness of solutions of the following initial value problems (3.1.1)–(3.1.4) of fractional differential equations were discussed under the assumption that $f\in {C}_{r}\left[0,1\right]$. We will establish existence and uniqueness results for these problems under more weaker assumptions, see Assumptions A1A4 in this subsection.

Let $\eta \in ℝ$, and let $F,A:\left(0,1\right)\to ℝ$ and $B,G:\left(1,e\right)\to ℝ$ be continuous functions. We will consider the following four classes of initial value problems of nonhomogeneous linear fractional differential equations:

$\left\{\begin{array}{cc}\hfill {}^{C}D_{{0}^{+}}^{𝜶}x\left(t\right)& =A\left(t\right)x\left(t\right)+F\left(t\right),t\in \left(0,1\right),\hfill \\ \hfill \underset{t\to {0}^{+}}{lim}x\left(t\right)& =\eta ,\hfill \end{array}$(3.1.1)$\left\{\begin{array}{cc}\hfill {}^{\mathrm{RL}}D_{{0}^{+}}^{𝜶}x\left(t\right)& =A\left(t\right)x\left(t\right)+F\left(t\right),\in \left(0,1\right),\hfill \\ \hfill \underset{t\to {0}^{+}}{lim}{t}^{1-\alpha }x\left(t\right)& =\eta ,\hfill \end{array}$(3.1.2)$\left\{\begin{array}{cc}\hfill {}^{\mathrm{RLH}}D_{{0}^{+}}^{𝜶}x\left(t\right)& =B\left(t\right)x\left(t\right)+G\left(t\right),t\in \left(1,e\right),\hfill \\ \hfill \underset{t\to {1}^{+}}{lim}{\left(\mathrm{log}t\right)}^{1-\alpha }x\left(t\right)& =\eta ,\hfill \end{array}$(3.1.3)$\left\{\begin{array}{cc}\hfill {}^{\mathrm{CH}}D_{{0}^{+}}^{𝜶}x\left(t\right)& =B\left(t\right)x\left(t\right)+G\left(t\right),t\in \left(1,e\right),\hfill \\ \hfill \underset{t\to {1}^{+}}{lim}x\left(t\right)& =\eta ,\hfill \end{array}$(3.1.4)

To get solutions of (3.1.1), we need the following assumptions:

#### Assumption A1.

There exist constants ${k}_{i}>-1$ and ${l}_{i}\le 0$ with ${l}_{i}>\mathrm{max}\left\{-\alpha ,-\alpha -{k}_{i}\right\}$ ($i=1,2$), ${M}_{A}\ge 0$ and ${M}_{F}\ge 0$ such that $|A\left(t\right)|\le {M}_{A}{t}^{{k}_{1}}{\left(1-t\right)}^{{l}_{1}}$ and $|F\left(t\right)|\le {M}_{F}{t}^{{k}_{2}}{\left(1-t\right)}^{{l}_{2}}$ for all $t\in \left(0,1\right)$.

Choose Picard function sequence as

${\varphi }_{0}\left(t\right)=\eta +{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}F\left(s\right)𝑑s,$$t\in \left[0,1\right],$${\varphi }_{n}\left(t\right)={\varphi }_{0}\left(t\right)+{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}A\left(s\right){\varphi }_{n-1}\left(s\right)𝑑s,$$t\in \left[0,1\right],n=1,2,\mathrm{\dots }.$

#### Claim 3.1.

One has ${\varphi }_{n}\mathrm{\in }C\mathit{}\mathrm{\left[}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right]}$.

#### Proof.

One sees ${\varphi }_{0}\in C\left[0,1\right]$ by the assumption imposed on F in Assumption A1. Then ${\varphi }_{1}$ is continuous on $\left[0,1\right]$ by

$|{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}A\left(s\right){\varphi }_{0}\left(s\right)𝑑s|\le \parallel {\varphi }_{0}\parallel {\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}|A\left(s\right)|𝑑s$$\le \parallel {\varphi }_{0}\parallel {\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{M}_{A}|\eta |{s}^{{k}_{1}}{\left(1-s\right)}^{{l}_{1}}𝑑s$$\le {M}_{A}\parallel {\varphi }_{0}\parallel |\eta |{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha +{l}_{1}-1}}{\mathrm{\Gamma }\left(\alpha \right)}{s}^{{k}_{1}}𝑑s$$={M}_{A}|\eta |\parallel {\varphi }_{0}\parallel {t}^{\alpha +{k}_{1}+{l}_{1}}{\int }_{0}^{1}\frac{{\left(1-w\right)}^{\alpha +{l}_{1}-1}}{\mathrm{\Gamma }\left(\alpha \right)}{w}^{{k}_{1}}𝑑w$

By the mathematical induction method, we can prove that ${\varphi }_{n}\in C\left[0,1\right]$. ∎

#### Claim 3.2.

The sequence $\mathrm{\left\{}{\varphi }_{n}\mathrm{\right\}}$ is convergent uniformly on $\mathrm{\left[}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right]}$.

#### Proof.

In fact, we have for $t\in \left[0,1\right]$ that

$|{\varphi }_{1}\left(t\right)-{\varphi }_{0}\left(t\right)|=|{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}A\left(s\right){\varphi }_{0}\left(s\right)𝑑s|$$\le {M}_{A}|\eta |\parallel {\varphi }_{0}\parallel {\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{s}^{{k}_{1}}{\left(1-s\right)}^{{l}_{1}}𝑑s$$\le {M}_{A}|\eta |\parallel {\varphi }_{0}\parallel {\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha +{l}_{1}-1}}{\mathrm{\Gamma }\left(\alpha \right)}{s}^{{k}_{1}}𝑑s$$={M}_{A}|\eta |\parallel {\varphi }_{0}\parallel {t}^{\alpha +{k}_{1}+{l}_{1}}\frac{𝐁\left(\alpha +{l}_{1},{k}_{1}+1\right)}{\mathrm{\Gamma }\left(\alpha \right)}.$

So

$|{\varphi }_{2}\left(t\right)-{\varphi }_{1}\left(t\right)|=|{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}A\left(s\right)\left[{\varphi }_{1}\left(s\right)-{\varphi }_{0}\left(s\right)\right]𝑑s|$$\le {\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{M}_{A}{s}^{{k}_{1}}{\left(1-s\right)}^{{l}_{1}}\left({M}_{A}|\eta |\parallel {\varphi }_{0}\parallel {s}^{\alpha +{k}_{1}+{l}_{1}}\frac{𝐁\left(\alpha +{l}_{1},{k}_{1}+1\right)}{\mathrm{\Gamma }\left(\alpha \right)}\right)𝑑s$$\le |\eta |\parallel {\varphi }_{0}\parallel {M}_{A}^{2}{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha +{l}_{1}-1}}{\mathrm{\Gamma }\left(\alpha \right)}{s}^{\alpha +2{k}_{1}+{l}_{1}}\frac{𝐁\left(\alpha +{l}_{1},{k}_{1}+1\right)}{\mathrm{\Gamma }\left(\alpha \right)}𝑑s$$=|\eta |\parallel {\varphi }_{0}\parallel {M}_{A}^{2}{t}^{2\alpha +2{k}_{1}+2{l}_{1}}\frac{𝐁\left(\alpha +{l}_{1},{k}_{1}+1\right)}{\mathrm{\Gamma }\left(\alpha \right)}\frac{𝐁\left(\alpha +{l}_{1},\alpha +2{k}_{1}+{l}_{1}+1\right)}{\mathrm{\Gamma }\left(\alpha \right)}.$

Now suppose that

$|{\varphi }_{j}\left(t\right)-{\varphi }_{j-1}\left(t\right)|\le |\eta |\parallel {\varphi }_{0}\parallel {M}_{A}^{j}{t}^{j\alpha +j{k}_{1}+j{l}_{1}}\prod _{i=0}^{j-1}\frac{𝐁\left(\alpha +{l}_{1},i\alpha +\left(i+1\right){k}_{1}+i{l}_{1}+1\right)}{\mathrm{\Gamma }\left(\alpha \right)}.$

We get

$|{\varphi }_{j+1}\left(t\right)-{\varphi }_{j}\left(t\right)|=|{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}A\left(s\right)\left[{\varphi }_{j}\left(s\right)-{\varphi }_{j-1}\left(s\right)\right]𝑑s|$$\le {\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{M}_{A}\left(|\eta |\parallel {\varphi }_{0}\parallel {M}_{A}^{j}{s}^{j\alpha +j{k}_{1}+j{l}_{1}}\prod _{i=0}^{j-1}\frac{𝐁\left(\alpha +{l}_{1},i\alpha +\left(i+1\right){k}_{1}+i{l}_{1}+1\right)}{\mathrm{\Gamma }\left(\alpha \right)}\right){s}^{{k}_{1}}{\left(1-s\right)}^{{l}_{1}}𝑑s$$\le {\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha +{l}_{1}-1}}{\mathrm{\Gamma }\left(\alpha \right)}{M}_{A}\left(|\eta |\parallel {\varphi }_{0}\parallel {M}_{A}^{j}{s}^{j\alpha +j{k}_{1}+j{l}_{1}}\prod _{i=0}^{j-1}\frac{𝐁\left(\alpha +{l}_{1},i\alpha +\left(i+1\right){k}_{1}+i{l}_{1}+1\right)}{\mathrm{\Gamma }\left(\alpha \right)}\right){s}^{{k}_{1}}𝑑s$$\le |\eta |\parallel {\varphi }_{0}\parallel {M}_{A}^{j+1}{t}^{\left(j+1\right)\alpha +\left(j+1\right){k}_{1}+\left(j+1\right){l}_{1}}\prod _{i=0}^{j}\frac{𝐁\left(\alpha +{l}_{1},i\alpha +\left(i+1\right){k}_{1}+i{l}_{1}+1\right)}{\mathrm{\Gamma }\left(\alpha \right)}.$

From the mathematical induction method, we get for every $n=1,2,\mathrm{\dots }$ that

$|{\varphi }_{n+1}\left(t\right)-{\varphi }_{n}\left(t\right)|\le |\eta |\parallel {\varphi }_{0}\parallel {M}_{A}^{n+1}{t}^{\left(n+1\right)\alpha +\left(n+1\right){k}_{1}+\left(n+1\right){l}_{1}}\prod _{i=0}^{n}\frac{𝐁\left(\alpha +{l}_{1},i\alpha +\left(i+1\right){k}_{1}+i{l}_{1}+1\right)}{\mathrm{\Gamma }\left(\alpha \right)}$$\le |\eta |\parallel {\varphi }_{0}\parallel {M}_{A}^{n+1}\prod _{i=0}^{n}\frac{𝐁\left(\alpha +{l}_{1},i\alpha +\left(i+1\right){k}_{1}+i{l}_{1}+1\right)}{\mathrm{\Gamma }\left(\alpha \right)},t\in \left[0,1\right].$

Consider

$\sum _{n=1}^{+\mathrm{\infty }}{u}_{n}=\sum _{n=1}^{+\mathrm{\infty }}|\eta |\parallel {\varphi }_{0}\parallel {M}_{A}^{n+1}\prod _{i=0}^{n}\frac{𝐁\left(\alpha +{l}_{1},i\alpha +\left(i+1\right){k}_{1}+i{l}_{1}+1\right)}{\mathrm{\Gamma }\left(\alpha \right)}.$

One sees for sufficiently large n that

$\frac{{u}_{n+1}}{{u}_{n}}={M}_{A}\frac{𝐁\left(\alpha +{l}_{1},\left(n+1\right)\alpha +\left(n+1\right){k}_{1}+\left(n+1\right){l}_{1}\right)}{\mathrm{\Gamma }\left(\alpha \right)}$$={M}_{A}{\int }_{0}^{1}{\left(1-x\right)}^{\alpha +{l}_{1}-1}{x}^{\left(n+1\right)\alpha +\left(n+1\right){k}_{1}+\left(n+1\right){l}_{1}}𝑑x$$\le {M}_{A}{\int }_{0}^{\delta }{\left(1-x\right)}^{\alpha +{l}_{1}-1}𝑑x{\delta }^{\left(n+1\right)\alpha +\left(n+1\right){k}_{1}+\left(n+1\right){l}_{1}}+\frac{{M}_{A}}{\alpha +{l}_{1}}{\delta }^{\alpha +{l}_{1}}$$\le \frac{{M}_{A}}{\alpha +{l}_{1}}{\delta }^{\left(n+1\right)\alpha +\left(n+1\right){k}_{1}+\left(n+1\right){l}_{1}}+\frac{{M}_{A}}{\alpha +{l}_{1}}{\delta }^{\alpha +{l}_{1}}.$

For any $ϵ>0$, it is easy to see that there exists $\delta \in \left(0,1\right)$ such that

$\frac{{M}_{a}}{\alpha +{l}_{1}}{\delta }^{\alpha +{l}_{1}}<\frac{ϵ}{2}.$

For this δ, there exists an integer $N>0$ sufficiently large such that

$\frac{{M}_{a}}{\alpha +{l}_{1}}{\delta }^{\left(n+1\right)\alpha +\left(n+1\right){k}_{1}+\left(n+1\right){l}_{1}}<\frac{ϵ}{2}$

for all $n>N$. So $0<\frac{{u}_{n+1}}{{u}_{n}}<\frac{ϵ}{2}+\frac{ϵ}{2}=ϵ$ for all $n>N$. It follows that ${lim}_{n\to +\mathrm{\infty }}\frac{{u}_{n+1}}{{u}_{n}}=0$. Then ${\sum }_{n=1}^{+\mathrm{\infty }}{u}_{n}$ is convergent. Hence

${\varphi }_{0}\left(t\right)+\left[{\varphi }_{1}\left(t\right)-{\varphi }_{0}\left(t\right)\right]+\left[{\varphi }_{2}\left(t\right)-{\varphi }_{1}\left(t\right)\right]+\mathrm{\cdots }+\left[{\varphi }_{n}\left(t\right)-{\varphi }_{n-1}\left(t\right)\right]+\mathrm{\cdots },t\in \left[0,1\right]$

is uniformly convergent. Then $\left\{{\varphi }_{n}\left(t\right)\right\}$ is convergent uniformly on $\left[0,1\right]$. ∎

#### Claim 3.3.

The function ϕ defined on $\mathrm{\left[}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right]}$ by $\varphi \mathit{}\mathrm{\left(}t\mathrm{\right)}\mathrm{=}{\mathrm{lim}}_{n\mathrm{\to }\mathrm{+}\mathrm{\infty }}\mathit{}{\varphi }_{n}\mathit{}\mathrm{\left(}t\mathrm{\right)}$ is a unique continuous solution of the integral equation

$x\left(t\right)={\varphi }_{0}\left(t\right)+\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}A\left(s\right)x\left(s\right)𝑑s,t\in \left[0,1\right].$

#### Proof.

By $\varphi \left(t\right)={lim}_{n\to +\mathrm{\infty }}{\varphi }_{n}\left(t\right)$ and the uniformly convergence, we see $\varphi \left(t\right)$ is continuous on $\left[0,1\right]$ by defining ${x\left(t\right)|}_{t=0}={lim}_{t\to {0}^{+}}x\left(t\right)$. It follows that

$\varphi \left(t\right)=\underset{n\to +\mathrm{\infty }}{lim}{\varphi }_{n}\left(t\right)=\underset{n\to +\mathrm{\infty }}{lim}\left[{\varphi }_{0}\left(t\right)+{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}A\left(s\right){\varphi }_{n-1}\left(s\right)𝑑s\right]$$={\varphi }_{0}\left(t\right)+\underset{n\to +\mathrm{\infty }}{lim}{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}A\left(s\right){\varphi }_{n-1}\left(s\right)𝑑s$$={\varphi }_{0}\left(t\right)+{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}A\left(s\right)\underset{n\to +\mathrm{\infty }}{lim}{\varphi }_{n-1}\left(s\right)ds$$={\varphi }_{0}\left(t\right)+{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}A\left(s\right)\varphi \left(s\right)𝑑s.$

Then ϕ is a continuous solution of (3.1.5) defined on $\left[0,1\right]$.

Suppose that ψ defined on $\left[0,1\right]$ is also a solution of (3.1.5). Then

$\psi \left(t\right)={\varphi }_{0}\left(t\right)+{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}A\left(s\right)\psi \left(s\right)𝑑s,t\in \left(0,1\right].$

We need to prove that $\varphi \left(t\right)\equiv \psi \left(t\right)$ on $\left[0,1\right]$. Then

$|\psi \left(t\right)-{\varphi }_{0}\left(t\right)|=|{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}|A\left(s\right)\psi \left(s\right)|𝑑s|\le |\eta |\parallel \psi \parallel {M}_{A}{t}^{\alpha +{k}_{1}+{l}_{1}}\frac{𝐁\left(\alpha +{l}_{1},{k}_{1}+1\right)}{\mathrm{\Gamma }\left(\alpha \right)}.$

Furthermore, we have

$|\psi \left(t\right)-{\varphi }_{1}\left(t\right)|=|{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}A\left(s\right)\left[\psi \left(s\right)-{\varphi }_{0}\left(s\right)\right]𝑑s|$$\le |\eta |\parallel \psi \parallel {M}_{A}^{2}{t}^{2\alpha +2{k}_{1}+2{l}_{1}}\frac{𝐁\left(\alpha +{l}_{1},{k}_{1}+1\right)}{\mathrm{\Gamma }\left(\alpha \right)}\frac{𝐁\left(\alpha +{l}_{1},\alpha +2{k}_{1}+{l}_{1}+1\right)}{\mathrm{\Gamma }\left(\alpha \right)}.$

Now suppose that

$|\psi \left(t\right)-{\varphi }_{j-1}\left(t\right)|\le |\eta |\parallel \psi \parallel {M}_{A}^{j}{t}^{j\alpha +j{k}_{1}+j{l}_{1}}\prod _{i=0}^{j-1}\frac{𝐁\left(\alpha +{l}_{1},i\alpha +\left(i+1\right){k}_{1}+i{l}_{1}+1\right)}{\mathrm{\Gamma }\left(\alpha \right)}.$

Then

$|\psi \left(t\right)-{\varphi }_{j}\left(t\right)|=|{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}A\left(s\right)\left[\psi \left(s\right)-{\varphi }_{j-1}\left(s\right)\right]𝑑s|$$\le |\eta |\parallel \psi \parallel {M}_{A}^{j+1}{t}^{\left(j+1\right)\alpha +\left(j+1\right){k}_{1}+\left(j+1\right){l}_{1}}\prod _{i=0}^{j}\frac{𝐁\left(\alpha +{l}_{1},i\alpha +\left(i+1\right){k}_{1}+i{l}_{1}+1\right)}{\mathrm{\Gamma }\left(\alpha \right)}.$

Hence

$|\psi \left(t\right)-{\varphi }_{n}\left(t\right)|\le |\eta |\parallel \psi \parallel {M}_{A}^{n+1}{t}^{\left(n+1\right)\alpha +\left(n+1\right){k}_{1}+\left(n+1\right){l}_{1}}\prod _{i=0}^{n}\frac{𝐁\left(\alpha +{l}_{1},i\alpha +\left(i+1\right){k}_{1}+i{l}_{1}+1\right)}{\mathrm{\Gamma }\left(\alpha \right)}$

Similarly we have

$\underset{n\to +\mathrm{\infty }}{lim}|\eta |\parallel \psi \parallel {M}_{A}^{n+1}\prod _{i=0}^{n}\frac{𝐁\left(\alpha +{l}_{1},i\alpha +\left(i+1\right){k}_{1}+i{l}_{1}+1\right)}{\mathrm{\Gamma }\left(\alpha \right)}=0.$

Then ${lim}_{n\to +\mathrm{\infty }}{\varphi }_{n}\left(t\right)=\psi \left(t\right)$ uniformly on $\left[0,1\right]$. Thus $\varphi \left(t\right)\equiv \psi \left(t\right)$. Then (3.1.5) has a unique solution ϕ. The proof is complete. ∎

#### Theorem 1.

Suppose that Assumption A1 holds. Then $x$ is a solution of IVP (3.1.1) if and only if x is a solution of the integral equation (3.1.5).

#### Proof.

Suppose that $x\in C\left[0,1\right]$ is a solution of IVP (3.1.1). Then ${lim}_{t\to {0}^{+}}x\left(t\right)=\eta$ and $\parallel x\parallel =r<+\mathrm{\infty }$. From Assumption A1, we have

$|{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}A\left(s\right)x\left(s\right)𝑑s|\le \parallel x\parallel {\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}|A\left(s\right)|𝑑s$$\le {\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{M}_{A}r{s}^{{k}_{1}}{\left(1-s\right)}^{{l}_{1}}𝑑s$$\le {M}_{A}r{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha +{l}_{1}-1}}{\mathrm{\Gamma }\left(\alpha \right)}{s}^{{k}_{1}}𝑑s$$={M}_{A}r{t}^{\alpha +{k}_{1}+{l}_{1}}{\int }_{0}^{1}\frac{{\left(1-w\right)}^{\alpha +{l}_{1}-1}}{\mathrm{\Gamma }\left(\alpha \right)}{w}^{{k}_{1}}𝑑w$$={M}_{A}r{t}^{\alpha +{k}_{1}+{l}_{1}}\frac{𝐁\left(\alpha +{l}_{1},{k}_{1}+1\right)}{\mathrm{\Gamma }\left(\alpha \right)}.$

So $t↦{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}A\left(s\right)x\left(s\right)𝑑s$ is continuous on $\left[0,1\right]$. We have ${I}_{{0}^{+}}^{\alpha }{}^{C}D_{{0}^{+}}^{\alpha }x\left(t\right)={I}_{{0}^{+}}^{\alpha }\left[A\left(t\right)x\left(t\right)+F\left(t\right)\right]$. Thus

${\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\left[A\left(s\right)x\left(s\right)+F\left(s\right)\right]𝑑s$$={I}_{{0}^{+}}^{\alpha }\left[A\left(t\right)x\left(t\right)+F\left(t\right)\right]={I}_{{0}^{+}}^{\alpha }{}^{C}D_{{0}^{+}}^{\alpha }x\left(t\right)$$={\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\left(\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}{\int }_{0}^{s}{\left(s-w\right)}^{-\alpha }{x}^{\prime }\left(w\right)𝑑w\right)𝑑s\mathit{ }\text{(interchange the order of integrals)}$$={\int }_{0}^{t}{x}^{\prime }\left(w\right)𝑑w=x\left(t\right)-\underset{t\to {0}^{+}}{lim}x\left(t\right)=x\left(t\right)-\eta .$

Then $x\in C\left[0,1\right]$ is a solution of (3.1.5).

On the other hand, if x is a solution of (3.1.5), together with Claims 3.13.3, we have $x\in C\left[0,1\right]$ and ${lim}_{t\to {0}^{+}}x\left(t\right)=\eta$. So $x\in C\left[0,1\right]$. Furthermore,

${}^{C}D_{{0}^{+}}^{\alpha }x\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{-\alpha }{x}^{\prime }\left(s\right)𝑑s=\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{-\alpha }{\left(\eta +{\int }_{0}^{s}\frac{{\left(s-w\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\left[A\left(w\right)x\left(w\right)+F\left(w\right)\right]𝑑w\right)}^{\prime }𝑑s$$=\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{-\alpha }{\left({\int }_{0}^{s}\frac{{\left(s-w\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\left[A\left(w\right)x\left(w\right)+F\left(w\right)\right]𝑑w\right)}^{\prime }𝑑s$$={\left[\frac{1}{\mathrm{\Gamma }\left(2-\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{1-\alpha }\frac{1}{\mathrm{\Gamma }\left(1-\left(1-\alpha \right)\right)}{\left({\int }_{0}^{s}{\left(s-w\right)}^{-\left(1-\alpha \right)}\left[A\left(w\right)x\left(w\right)+F\left(w\right)\right]𝑑w\right)}^{\prime }𝑑s\right]}^{\prime }$$=\frac{1}{\mathrm{\Gamma }\left(2-\alpha \right)}{\left[.{\left(t-s\right)}^{1-\alpha }\frac{1}{\mathrm{\Gamma }\left(1-\left(1-\alpha \right)\right)}{\int }_{0}^{s}{\left(s-w\right)}^{-\left(1-\alpha \right)}\left[A\left(w\right)x\left(w\right)+F\left(w\right)\right]dw|}_{0}^{t}$$+\left(1-\alpha \right){\int }_{0}^{t}{\left(t-s\right)}^{-\alpha }\frac{1}{\mathrm{\Gamma }\left(1-\left(1-\alpha \right)\right)}{\int }_{0}^{s}{\left(s-w\right)}^{-\left(1-\alpha \right)}\left[A\left(w\right)x\left(w\right)+F\left(w\right)\right]dwds\right]{}^{\prime }$$=\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}{\left[{\int }_{0}^{t}{\left(t-s\right)}^{-\alpha }\frac{1}{\mathrm{\Gamma }\left(1-\left(1-\alpha \right)\right)}{\int }_{0}^{s}{\left(s-w\right)}^{-\left(1-\alpha \right)}\left[A\left(w\right)x\left(w\right)+F\left(w\right)\right]𝑑w𝑑s\right]}^{\prime }$$=\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\left[{\int }_{0}^{t}{\int }_{w}^{t}{\left(t-s\right)}^{-\alpha }{\left(s-w\right)}^{-\left(1-\alpha \right)}𝑑s\left[A\left(w\right)x\left(w\right)+F\left(w\right)\right]𝑑w\right]}^{\prime }$$=A\left(t\right)x\left(t\right)+F\left(t\right).$

So $x\in C\left[0,1\right]$ is a solution of IVP (3.1.1). ∎

#### Theorem 2.

Suppose that Assumption A1 holds. Then (3.1.1) has a unique solution. If there exist constants ${k}_{\mathrm{2}}\mathrm{>}\mathrm{-}\mathrm{1}$, ${l}_{\mathrm{2}}\mathrm{\le }\mathrm{0}$ with ${l}_{\mathrm{2}}\mathrm{>}\mathrm{max}\mathit{}\mathrm{\left\{}\mathrm{-}\alpha \mathrm{,}\mathrm{-}\alpha \mathrm{-}{k}_{\mathrm{2}}\mathrm{\right\}}$, ${M}_{F}\mathrm{\ge }\mathrm{0}$ such that $\mathrm{|}F\mathit{}\mathrm{\left(}t\mathrm{\right)}\mathrm{|}\mathrm{\le }{M}_{F}\mathit{}{t}^{{k}_{\mathrm{2}}}\mathit{}{\mathrm{\left(}\mathrm{1}\mathrm{-}t\mathrm{\right)}}^{{l}_{\mathrm{2}}}$ for all $t\mathrm{\in }\mathrm{\left(}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right)}$, then the special problem

$\left\{\begin{array}{cc}\hfill {}^{C}D_{{0}^{+}}^{𝜶}x\left(t\right)& =\lambda x\left(t\right)+F\left(t\right),t\in \left(0,1\right],\hfill \\ \hfill \underset{t\to {0}^{+}}{lim}x\left(t\right)& =\eta \hfill \end{array}$(3.1.7)

has a unique solution

$x\left(t\right)=\eta {E}_{\alpha ,1}\left(\lambda {t}^{\alpha }\right)+{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(\lambda {\left(t-s\right)}^{\alpha }\right)F\left(s\right)𝑑s,t\in \left(0,1\right].$

#### Proof.

From Claims 3.13.3, Theorem 1 implies that (3.1.1) has a unique solution. From the assumption and $A\left(t\right)\equiv \lambda$, it is easy to see that Assumption A1 holds with ${k}_{1}={l}_{1}=0$ and ${k}_{2},{l}_{2}$ mentioned. Thus (3.1.7) has a unique solution. We get from the Picard function sequence that

${\varphi }_{n}\left(t\right)=\eta +\lambda {\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{\varphi }_{n-1}\left(s\right)𝑑s+{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}F\left(s\right)𝑑s$$=\eta +\eta \lambda {\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}𝑑s+{\lambda }^{2}{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{s}\frac{{\left(s-w\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{\varphi }_{n-2}\left(w\right)𝑑w𝑑s$$+\lambda {\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\frac{{\left(s-w\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}F\left(w\right)𝑑w𝑑s+{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}F\left(s\right)𝑑s$$=\eta +\frac{\eta \lambda }{\mathrm{\Gamma }\left(\alpha +1\right)}{t}^{\alpha }+{\lambda }^{2}{\int }_{0}^{t}{\int }_{w}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\frac{{\left(s-w\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}𝑑s{\varphi }_{n-2}\left(w\right)𝑑w$$+\lambda {\int }_{0}^{t}{\int }_{w}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\frac{{\left(s-w\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}𝑑sF\left(w\right)𝑑w+{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}F\left(s\right)𝑑s$$=\eta +\frac{\eta \lambda }{\mathrm{\Gamma }\left(\alpha +1\right)}{t}^{\alpha }+{\lambda }^{2}{\int }_{0}^{t}{\left(t-w\right)}^{2\alpha -1}{\int }_{0}^{1}\frac{{\left(1-u\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\frac{{u}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}𝑑u{\varphi }_{n-2}\left(w\right)𝑑w$$+\lambda {\int }_{0}^{t}{\left(t-w\right)}^{2\alpha -1}{\int }_{0}^{1}\frac{{\left(1-u\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\frac{{u}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}𝑑uF\left(w\right)𝑑w+{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}F\left(s\right)𝑑s$$=\eta \left(1+\frac{\lambda {t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right)+{\lambda }^{2}{\int }_{0}^{t}\frac{{\left(t-w\right)}^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}{\varphi }_{n-2}\left(w\right)𝑑w+{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}\left(\frac{\lambda {\left(t-s\right)}^{\alpha }}{\mathrm{\Gamma }\left(2\alpha \right)}+\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\right)F\left(s\right)𝑑s$$\mathrm{⋮}$$=\eta \sum _{j=0}^{n-1}\frac{{\lambda }^{j}{t}^{j\alpha }}{\mathrm{\Gamma }\left(j\alpha +1\right)}+\eta {\lambda }^{n}{\int }_{0}^{t}\frac{{\left(t-w\right)}^{n\alpha -1}}{\mathrm{\Gamma }\left(n\alpha \right)}𝑑w+{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}\left(\sum _{j=0}^{n-1}\frac{{\lambda }^{j}{\left(t-s\right)}^{j\alpha }}{\mathrm{\Gamma }\left(\left(j+1\right)\alpha \right)}\right)F\left(s\right)𝑑s$$=\eta \sum _{j=0}^{n}\frac{{\lambda }^{j}{t}^{j\alpha }}{\mathrm{\Gamma }\left(j\alpha +1\right)}+{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}\left(\sum _{j=0}^{n}\frac{{\lambda }^{j}{\left(t-s\right)}^{j\alpha }}{\mathrm{\Gamma }\left(\left(j+1\right)\alpha \right)}\right)F\left(s\right)𝑑s$

and so

${\varphi }_{n}\left(t\right)\to \eta {E}_{\alpha ,1}\left(\lambda {t}^{\alpha }\right)+{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(\lambda {\left(t-s\right)}^{\alpha }\right)F\left(s\right)𝑑s.$

Then we get (3.1.8). The proof is complete. ∎

To get solutions of (3.1.2), we need the following assumptions.

#### Assumption A2.

There exist constants ${k}_{i}>-\alpha$ and ${l}_{i}\le 0$ with ${l}_{1}>\mathrm{max}\left\{-\alpha ,-1-{k}_{1}\right\}$, ${l}_{2}>\mathrm{max}\left\{-\alpha ,-1-{k}_{2}\right\}$, ${M}_{A}\ge 0$ and ${M}_{F}\ge 0$ such that $|A\left(t\right)|\le {M}_{A}{t}^{{k}_{1}}{\left(1-t\right)}^{{l}_{1}}$ and $|F\left(t\right)|\le {M}_{F}{t}^{{k}_{2}}{\left(1-t\right)}^{{l}_{2}}$ for all $t\in \left(0,1\right)$.

Choose Picard function sequence as

${\varphi }_{0}\left(t\right)=\eta {t}^{\alpha -1}+{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}F\left(s\right)𝑑s,$$t\in \left(0,1\right],$${\varphi }_{n}\left(t\right)={\varphi }_{0}\left(t\right)+{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}A\left(s\right){\varphi }_{n-1}\left(s\right)𝑑s,$$t\in \left(0,1\right],n=1,2,\mathrm{\dots }.$

#### Claim 3.4.

One has ${\varphi }_{n}\mathrm{\in }{C}_{\mathrm{1}\mathrm{-}\alpha }\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right]}$.

#### Proof.

Since ${\varphi }_{0}\in {C}_{1-\alpha }\left(0,1\right]$, we know that ${\varphi }_{1}$ is continuous on $\left(0,1\right]$, and together with

${t}^{1-\alpha }|{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}A\left(s\right){\varphi }_{0}\left(s\right)𝑑s|={t}^{1-\alpha }|{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}A\left(s\right){s}^{\alpha -1}{s}^{1-\alpha }{\varphi }_{0}\left(s\right)𝑑s|$$\le \parallel {\varphi }_{0}\parallel {t}^{1-\alpha }{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{M}_{A}|\eta |{s}^{\alpha -1}{s}^{{k}_{1}}{\left(1-s\right)}^{{l}_{1}}𝑑s$$\le \parallel {\varphi }_{0}\parallel {t}^{1-\alpha }{M}_{A}|\eta |{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha +{l}_{1}-1}}{\mathrm{\Gamma }\left(\alpha \right)}{s}^{\alpha +{k}_{1}-1}𝑑s$

we see that ${\varphi }_{1}\in {C}_{1-\alpha }\left(0,1\right]$. By the mathematical induction method, we obtain that ${\varphi }_{n}\in {C}_{1-\alpha }\left(0,1\right]$. ∎

#### Claim 3.5.

The sequence $\mathrm{\left\{}t\mathrm{↦}{t}^{\mathrm{1}\mathrm{-}\alpha }\mathit{}{\varphi }_{n}\mathit{}\mathrm{\left(}t\mathrm{\right)}\mathrm{\right\}}$ is convergent uniformly on $\mathrm{\left(}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right]}$.

#### Proof.

In fact, we have for $t\in \left(0,1\right]$ similarly to Claim 3.4 that

${t}^{1-\alpha }|{\varphi }_{1}\left(t\right)-{\varphi }_{0}\left(t\right)|=|{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}A\left(s\right){\varphi }_{0}\left(s\right)𝑑s|\le {M}_{A}|\eta |\parallel {\varphi }_{0}\parallel {t}^{1+{k}_{1}+{l}_{1}}\frac{𝐁\left(\alpha +{l}_{1},\alpha +{k}_{1}\right)}{\mathrm{\Gamma }\left(\alpha \right)}.$

It follows that

${t}^{1-\alpha }|{\varphi }_{2}\left(t\right)-{\varphi }_{1}\left(t\right)|=|{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}A\left(s\right)\left[{\varphi }_{1}\left(s\right)-{\varphi }_{0}\left(s\right)\right]𝑑s|$$\le {t}^{1-\alpha }{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{M}_{A}{s}^{{k}_{1}}{\left(1-s\right)}^{{l}_{1}}\left(|\eta |\parallel {\varphi }_{0}\parallel {M}_{A}{s}^{{k}_{1}+{l}_{1}+1}\frac{𝐁\left(\alpha +{l}_{1},{k}_{1}+1\right)}{\mathrm{\Gamma }\left(\alpha \right)}\right)𝑑s$$\le |\eta |\parallel {\varphi }_{0}\parallel {M}_{A}^{2}{t}^{1-\alpha }{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha +{l}_{1}-1}}{\mathrm{\Gamma }\left(\alpha \right)}{s}^{2{k}_{1}+{l}_{1}+1}\frac{𝐁\left(\alpha +{l}_{1},{k}_{1}+1\right)}{\mathrm{\Gamma }\left(\alpha \right)}𝑑s$$=|\eta |\parallel {\varphi }_{0}\parallel {M}_{A}^{2}{t}^{2{k}_{1}+2{l}_{1}+2}\frac{𝐁\left(\alpha +{l}_{1},{k}_{1}+1\right)}{\mathrm{\Gamma }\left(\alpha \right)}\frac{𝐁\left(\alpha +{l}_{1},2{k}_{1}+{l}_{1}+2\right)}{\mathrm{\Gamma }\left(\alpha \right)}.$

Again by the mathematical induction method, we get for every $n=1,2,\mathrm{\dots }$ that

${t}^{1-\alpha }|{\varphi }_{n}\left(t\right)-{\varphi }_{n-1}\left(t\right)|\le |\eta |M{\parallel {\varphi }_{0}\parallel }_{A}^{n}{t}^{n{k}_{1}+n{l}_{1}+n}\prod _{i=0}^{n-1}\frac{𝐁\left(\alpha +{l}_{1},\left(i+1\right){k}_{1}+i{l}_{1}+\left(i+1\right)\right)}{\mathrm{\Gamma }\left(\alpha \right)}$$\le |\eta |{M}_{A}^{n}\prod _{i=0}^{n-1}\frac{𝐁\left(\alpha +{l}_{1},\left(i+1\right){k}_{1}+i{l}_{1}+\left(i+1\right)\right)}{\mathrm{\Gamma }\left(\alpha \right)},t\in \left[0,1\right].$

Similarly we can prove that

$\sum _{n=1}^{+\mathrm{\infty }}{u}_{n}=\sum _{n=1}^{+\mathrm{\infty }}|\eta |{M}_{A}^{n}\prod _{i=0}^{n-1}\frac{𝐁\left(\alpha +{l}_{1},\left(i+1\right){k}_{1}+i{l}_{1}+\left(i+1\right)\right)}{\mathrm{\Gamma }\left(\alpha \right)}$

is convergent. Hence

${t}^{1-\alpha }{\varphi }_{0}\left(t\right)+{t}^{1-\alpha }\left[{\varphi }_{1}\left(t\right)-{\varphi }_{0}\left(t\right)\right]+{t}^{1-\alpha }\left[{\varphi }_{2}\left(t\right)-{\varphi }_{1}\left(t\right)\right]+\mathrm{\cdots }+{t}^{1-\alpha }\left[{\varphi }_{n}\left(t\right)-{\varphi }_{n-1}\left(t\right)\right]+\mathrm{\cdots },t\in \left[0,1\right],$

is uniformly convergent. Then $\left\{t↦{t}^{1-\alpha }{\varphi }_{n}\left(t\right)\right\}$ is convergent uniformly on $\left(0,1\right]$. ∎

#### Claim 3.6.

The function ϕ defined on $\mathrm{\left(}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right]}$ by $\varphi \mathit{}\mathrm{\left(}t\mathrm{\right)}\mathrm{=}{t}^{\alpha \mathrm{-}\mathrm{1}}\mathit{}{\mathrm{lim}}_{n\mathrm{\to }\mathrm{+}\mathrm{\infty }}\mathit{}{t}^{\mathrm{1}\mathrm{-}\alpha }\mathit{}{\varphi }_{n}\mathit{}\mathrm{\left(}t\mathrm{\right)}$ is a unique continuous solution of the integral equation

$x\left(t\right)={\varphi }_{0}\left(t\right)+{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}A\left(s\right)x\left(s\right)𝑑s,t\in \left(0,1\right].$

#### Proof.

By ${lim}_{n\to +\mathrm{\infty }}{t}^{1-\alpha }{\varphi }_{n}\left(t\right)={t}^{1-\alpha }\varphi \left(t\right)$ and the uniformly convergence, we see that $\varphi \left(t\right)$ is continuous on $\left(0,1\right]$. Then we know

$\varphi \left(t\right)={t}^{\alpha -1}\underset{n\to \mathrm{\infty }}{lim}{t}^{1-\alpha }{\varphi }_{n}\left(t\right)=\underset{n\to +\mathrm{\infty }}{lim}\left[{\varphi }_{0}\left(t\right)+{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}A\left(s\right){\varphi }_{n-1}\left(s\right)𝑑s\right]$$={\varphi }_{0}\left(t\right)+\underset{n\to +\mathrm{\infty }}{lim}{t}^{1-\alpha }{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}A\left(s\right){\varphi }_{n-1}\left(s\right)𝑑s={\varphi }_{0}\left(t\right)+{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}A\left(s\right)\varphi \left(s\right)𝑑s.$

Then ϕ is a continuous solution of (3.1.9) defined on $\left(0,1\right]$.

Suppose that ψ defined on $\left(0,1\right]$ is also a solution of (3.1.9). Then

$\psi \left(t\right)={\varphi }_{0}\left(t\right)+{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}A\left(s\right)\psi \left(s\right)𝑑s,t\in \left[0,1\right].$

We need to prove that $\varphi \left(t\right)\equiv \psi \left(t\right)$ on $\left(0,1\right]$. Then

${t}^{1-\alpha }|\psi \left(t\right)-{\varphi }_{0}\left(t\right)|={t}^{1-\alpha }|{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}|A\left(s\right)\psi \left(s\right)|𝑑s|$$\le |\eta |\parallel \psi \parallel {M}_{A}{t}^{{k}_{1}+{l}_{1}+1}\frac{𝐁\left(\alpha +{l}_{1},{k}_{1}+1\right)}{\mathrm{\Gamma }\left(\alpha \right)}+{M}_{F}{t}^{{k}_{2}+{l}_{2}+1}\frac{𝐁\left(\alpha +{l}_{2},{k}_{2}+1\right)}{\mathrm{\Gamma }\left(\alpha \right)}.$

Furthermore, we have

${t}^{1-\alpha }|\psi \left(t\right)-{\varphi }_{1}\left(t\right)|={t}^{1-\alpha }|{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}A\left(s\right)\left[\psi \left(s\right)-{\varphi }_{0}\left(s\right)\right]𝑑s|$$\le |\eta |\parallel \psi \parallel {M}_{A}^{2}{t}^{2{k}_{1}+2{l}_{1}+2}\frac{𝐁\left(\alpha +{l}_{1},{k}_{1}+1\right)}{\mathrm{\Gamma }\left(\alpha \right)}\frac{𝐁\left(\alpha +{l}_{1},2{k}_{1}+{l}_{1}+2\right)}{\mathrm{\Gamma }\left(\alpha \right)}.$

By the mathematical induction method, we can get that

${t}^{1-\alpha }|\psi \left(t\right)-{\varphi }_{n}\left(t\right)|={t}^{1-\alpha }|{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}A\left(s\right)\left[\psi \left(s\right)-{\varphi }_{n-1}\left(s\right)\right]𝑑s|$$\le |\eta |\parallel \psi \parallel {M}_{A}^{n}{t}^{n{k}_{1}+n{l}_{1}+n}\prod _{i=0}^{n-1}\frac{𝐁\left(\alpha +{l}_{1},\left(i+1\right){k}_{1}+i{l}_{1}+\left(i+1\right)\right)}{\mathrm{\Gamma }\left(\alpha \right)}.$

Hence

Similarly we have ${lim}_{n\to +\mathrm{\infty }}{t}^{1-\alpha }{\varphi }_{n}\left(t\right)={t}^{1-\alpha }\psi \left(t\right)$ uniformly on $\left(0,1\right]$. Thus $\varphi \left(t\right)\equiv \psi \left(t\right)$ on $\left(0,1\right]$. Then (3.1.9) has a unique solution ϕ. ∎

#### Theorem 3.

Suppose that Assumption A2 holds. Then $x\mathrm{\in }{C}_{\mathrm{1}\mathrm{-}\alpha }\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right]}$ is a solution of IVP (3.1.2) if and only if $x\mathrm{\in }{C}_{\mathrm{1}\mathrm{-}\alpha }\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right]}$ is a solution of the integral equation (3.1.9).

#### Proof.

Suppose that $x\in {C}_{1-\alpha }\left(0,1\right]$ is a solution of IVP (3.1.2). Then $t↦{t}^{1-\alpha }x\left(t\right)$is continuous on $\left(0,1\right]$ by defining ${{t}^{1-\alpha }x\left(t\right)|}_{t=0}={lim}_{t\to {0}^{+}}{t}^{1-\alpha }x\left(t\right)$ and $\parallel x\parallel =r<+\mathrm{\infty }$. So , we get

$\underset{s\to {0}^{+}}{lim}{\int }_{0}^{s}{\left(s-w\right)}^{-\alpha }x\left(w\right)𝑑w=\underset{s\to {0}^{+}}{lim}{\int }_{0}^{s}{\left(s-w\right)}^{-\alpha }{w}^{\alpha -1}{w}^{1-\alpha }x\left(w\right)𝑑w$$=\underset{s\to {0}^{+}}{lim}{\xi }^{1-\alpha }x\left(\xi \right){\int }_{0}^{s}{\left(s-w\right)}^{-\alpha }{w}^{\alpha -1}𝑑w$$=\underset{s\to {0}^{+}}{lim}{\xi }^{1-\alpha }x\left(\xi \right){\int }_{0}^{1}{\left(1-u\right)}^{-\alpha }{u}^{\alpha -1}𝑑u$$=\eta 𝐁\left(1-\alpha ,\alpha \right),$

by the mean value theorem for integrals, $\xi \in \left(0,s\right)$. From Assumption A2, we have

${t}^{1-\alpha }|{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\left[A\left(s\right)x\left(s\right)+F\left(s\right)\right]𝑑s|={t}^{1-\alpha }|{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\left[A\left(s\right){s}^{\alpha -1}{s}^{1-\alpha }x\left(s\right)+F\left(s\right)\right]𝑑s|$$\le {\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\left[{M}_{A}r{s}^{\alpha -1}{s}^{{k}_{1}}{\left(1-s\right)}^{{l}_{1}}+{M}_{F}{s}^{{k}_{2}}{\left(1-s\right)}^{{l}_{2}}\right]𝑑s$$\le {t}^{1-\alpha }{M}_{A}r{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha +{l}_{1}-1}}{\mathrm{\Gamma }\left(\alpha \right)}{s}^{\alpha +{k}_{1}-1}𝑑s+{t}^{1-\alpha }{M}_{F}{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha +{l}_{2}-1}}{\mathrm{\Gamma }\left(\alpha \right)}{s}^{{k}_{2}}𝑑s$$={M}_{A}r{t}^{\alpha +{k}_{1}+{l}_{1}}\frac{𝐁\left(\alpha +{l}_{1},\alpha +{k}_{1}\right)}{\mathrm{\Gamma }\left(\alpha \right)}+{M}_{F}{t}^{1+{k}_{2}+{l}_{2}}\frac{𝐁\left(\alpha +{l}_{2},{k}_{2}+1\right)}{\mathrm{\Gamma }\left(\alpha \right)}.$

So

$t↦{t}^{1-\alpha }{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\left[A\left(s\right)x\left(s\right)+F\left(s\right)\right]𝑑s$

is defined on $\left(0,1\right]$ and

$\underset{t\to {0}^{+}}{lim}{t}^{1-\alpha }{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\left[A\left(s\right)x\left(s\right)+F\left(s\right)\right]𝑑s=0.$

Furthermore, we have similarly to Theorem 1 that

$t↦{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\left[A\left(s\right)x\left(s\right)+F\left(s\right)\right]𝑑s$

is continuous on $\left(0,1\right]$. So

$t↦{t}^{1-\alpha }{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\left[A\left(s\right)x\left(s\right)+F\left(s\right)\right]𝑑s$

is continuous on $\left[0,1\right]$ by defining

${{t}^{1-\alpha }{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\left[A\left(s\right)x\left(s\right)+F\left(s\right)\right]𝑑s|}_{t=0}=\underset{t\to {0}^{+}}{lim}{t}^{1-\alpha }{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\left[A\left(s\right)x\left(s\right)+F\left(s\right)\right]𝑑s.$

We have ${I}_{{0}^{+}}^{\alpha }{}^{\mathrm{RL}}D_{{0}^{+}}^{\alpha }x\left(t\right)={I}_{{0}^{+}}^{\alpha }\left[A\left(t\right)x\left(t\right)+F\left(t\right)\right]$. So

${\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\left[A\left(s\right)x\left(s\right)+F\left(s\right)\right]𝑑s$$={I}_{{0}^{+}}^{\alpha }\left[A\left(t\right)x\left(t\right)+F\left(t\right)\right]={I}_{{0}^{+}}^{\alpha }{}^{\mathrm{RL}}D_{{0}^{+}}^{\alpha }x\left(t\right)$$={\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\left[\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}{\left({\int }_{0}^{s}{\left(s-w\right)}^{-\alpha }x\left(w\right)𝑑w\right)}^{\prime }\right]𝑑s$$={\left[{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}{\left({\int }_{0}^{s}{\left(s-w\right)}^{-\alpha }x\left(w\right)𝑑w\right)}^{\prime }𝑑s\right]}^{\prime }$$=\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)\mathrm{\Gamma }\left(\alpha +1\right)}{\left[{{\left(t-s\right)}^{\alpha }{\int }_{0}^{s}{\left(s-w\right)}^{-\alpha }x\left(w\right)𝑑w|}_{0}^{t}+\alpha {\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}{\int }_{0}^{s}{\left(s-w\right)}^{-\alpha }x\left(w\right)𝑑w𝑑s\right]}^{\prime }$$=\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)\mathrm{\Gamma }\left(\alpha +1\right)}{\left[-{t}^{\alpha }\underset{s\to {0}^{+}}{lim}{\int }_{0}^{s}{\left(s-w\right)}^{-\alpha }x\left(w\right)𝑑w+\alpha {\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}{\int }_{0}^{s}{\left(s-w\right)}^{-\alpha }x\left(w\right)𝑑w𝑑s\right]}^{\prime }$$=\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)\mathrm{\Gamma }\left(\alpha +1\right)}{\left[\alpha {\int }_{0}^{t}{\int }_{w}^{t}{\left(t-s\right)}^{\alpha -1}{\left(s-w\right)}^{-\alpha }𝑑sx\left(w\right)𝑑w\right]}^{\prime }-\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)\mathrm{\Gamma }\left(\alpha \right)}{t}^{\alpha -1}\underset{s\to {0}^{+}}{lim}{\int }_{0}^{s}{\left(s-w\right)}^{-\alpha }x\left(w\right)𝑑w$$=\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)\mathrm{\Gamma }\left(\alpha +1\right)}{\left[\alpha {\int }_{0}^{t}{\int }_{0}^{1}{\left(1-u\right)}^{\alpha -1}{u}^{-\alpha }𝑑ux\left(w\right)𝑑w\right]}^{\prime }-\frac{{t}^{\alpha -1}}{\mathrm{\Gamma }\left(1-\alpha \right)\mathrm{\Gamma }\left(\alpha \right)}\underset{s\to {0}^{+}}{lim}{\int }_{0}^{s}{\left(s-w\right)}^{-\alpha }x\left(w\right)𝑑w$$={\left[{\int }_{0}^{t}x\left(w\right)𝑑w\right]}^{\prime }$$=x\left(t\right)-\frac{{t}^{\alpha -1}}{\mathrm{\Gamma }\left(1-\alpha \right)\mathrm{\Gamma }\left(\alpha \right)}\eta 𝐁\left(1-\alpha ,\alpha \right)$$=x\left(t\right)-\eta {t}^{\alpha -1},$

by $\frac{s-w}{t-w}=u$. Then $x\in {C}_{1-\alpha }\left(0,1\right]$ is a solution of (3.1.9).

On the other hand, if $x\in {C}_{1-\alpha }\left(0,1\right]$ is a solution of (3.1.9), then this together with (3.1.10) implies that ${lim}_{t\to {0}^{+}}{t}^{1-\alpha }x\left(t\right)=\eta$. Furthermore, we have

${}^{\mathrm{RL}}D_{{0}^{+}}^{\alpha }x\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}\left({\int }_{0}^{t}{\left(t-s\right)}^{-\alpha }x\left(s\right)𝑑s\right)$$=\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}{\left({\int }_{0}^{t}{\left(t-s\right)}^{-\alpha }\left(\eta {s}^{\alpha -1}+{\int }_{0}^{s}\frac{{\left(s-w\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\left[A\left(w\right)x\left(w\right)+F\left(w\right)\right]𝑑w\right)𝑑s\right)}^{\prime }$$=\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}{\left(\eta {\int }_{0}^{t}{\left(t-s\right)}^{-\alpha }{s}^{\alpha -1}𝑑s\right)}^{\prime }+\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}{\left({\int }_{0}^{t}{\left(t-s\right)}^{-\alpha }{\int }_{0}^{s}\frac{{\left(s-w\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\left[A\left(w\right)x\left(w\right)+F\left(w\right)\right]𝑑w𝑑s\right)}^{\prime }$$=\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}{\left(\eta {\int }_{0}^{1}{\left(1-u\right)}^{-\alpha }{u}^{\alpha -1}𝑑u\right)}^{\prime }+\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}{\left({\int }_{0}^{t}{\int }_{w}^{t}{\left(t-s\right)}^{-\alpha }\frac{{\left(s-w\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}𝑑s\left[A\left(w\right)x\left(w\right)+F\left(w\right)\right]𝑑w\right)}^{\prime }$$=\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}{\left({\int }_{0}^{t}{\int }_{0}^{1}{\left(1-u\right)}^{-\alpha }\frac{{u}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}𝑑u\left[A\left(w\right)x\left(w\right)+F\left(w\right)\right]𝑑w\right)}^{\prime }$$=A\left(t\right)x\left(t\right)+F\left(t\right).$

So $x\in {C}_{1-\alpha }\left(0,1\right]$ is a solution of IVP (3.1.2). ∎

#### Theorem 4.

Suppose that Assumption A2 holds. Then (3.1.2) has a unique solution. If $A\mathit{}\mathrm{\left(}t\mathrm{\right)}\mathrm{\equiv }\lambda$ and there exist constants ${k}_{\mathrm{2}}\mathrm{>}\mathrm{-}\mathrm{1}$, ${l}_{\mathrm{2}}\mathrm{\le }\mathrm{0}$ with ${l}_{\mathrm{2}}\mathrm{>}\mathrm{max}\mathit{}\mathrm{\left\{}\mathrm{-}\alpha \mathrm{,}\mathrm{-}\mathrm{1}\mathrm{-}{k}_{\mathrm{1}}\mathrm{\right\}}$ and ${M}_{F}\mathrm{\ge }\mathrm{0}$ such that $\mathrm{|}F\mathit{}\mathrm{\left(}t\mathrm{\right)}\mathrm{|}\mathrm{\le }{M}_{F}\mathit{}{t}^{{k}_{\mathrm{2}}}\mathit{}{\mathrm{\left(}\mathrm{1}\mathrm{-}t\mathrm{\right)}}^{{l}_{\mathrm{2}}}$ for all $t\mathrm{\in }\mathrm{\left(}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right)}$, then the special problem

$\left\{\begin{array}{cc}\hfill {}^{\mathrm{RL}}D_{{0}^{+}}^{𝜶}x\left(t\right)& =\lambda x\left(t\right)+F\left(t\right),t\in \left(0,1\right],\hfill \\ \hfill \underset{t\to {0}^{+}}{lim}{t}^{1-\alpha }x\left(t\right)& =\eta \hfill \end{array}$(3.1.12)

has a unique solution

$x\left(t\right)=\eta \mathrm{\Gamma }\left(\alpha \right){t}^{\alpha -1}{E}_{\alpha ,\alpha }\left(\lambda {t}^{\alpha }\right)+{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(\lambda {\left(t-s\right)}^{\alpha }\right)F\left(s\right)𝑑s,t\in \left(0,1\right].$

#### Proof.

From Claims 3.43.6, IVP (3.1.2) and Theorem 3 has a unique solution. From the assumption and

$A\left(t\right)\equiv \lambda ,$

one sees that Assumption A2 holds with ${k}_{1}={l}_{1}=0$ and ${k}_{2},{l}_{2}$ mentioned. Thus (3.1.12) has a unique solution. We get from the Picard function sequence that

${\varphi }_{n}\left(t\right)=\eta {t}^{\alpha -1}+\lambda {\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{\varphi }_{n-1}\left(s\right)𝑑s+{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}F\left(s\right)𝑑s$$=\eta {t}^{\alpha -1}+\eta \lambda {\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{s}^{\alpha -1}𝑑s+{\lambda }^{2}{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{s}\frac{{\left(s-w\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}{\varphi }_{n-2}\left(w\right)𝑑w𝑑s$$+\lambda {\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\frac{{\left(s-w\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}F\left(w\right)𝑑w𝑑s+{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}F\left(s\right)𝑑s$$=\eta {t}^{\alpha -1}+\frac{\eta \lambda \mathrm{\Gamma }\left(\alpha \right){t}^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}+{\lambda }^{2}{\int }_{0}^{t}{\int }_{w}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\frac{{\left(s-w\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}𝑑s{\varphi }_{n-2}\left(w\right)𝑑w$$+\lambda {\int }_{0}^{t}{\int }_{w}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\frac{{\left(s-w\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}𝑑sF\left(w\right)𝑑w+{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}F\left(s\right)𝑑s$$\mathrm{⋮}$$=\eta \mathrm{\Gamma }\left(\alpha \right){t}^{\alpha -1}\sum _{j=0}^{n-1}\frac{{\lambda }^{j}{t}^{j\alpha }}{\mathrm{\Gamma }\left(\left(j+1\right)\alpha \right)}+\eta {\lambda }^{n}{\int }_{0}^{t}\frac{{\left(t-w\right)}^{n\alpha -1}}{\mathrm{\Gamma }\left(n\alpha \right)}𝑑w+{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}\left(\sum _{j=0}^{n-1}\frac{{\lambda }^{j}{\left(t-s\right)}^{j\alpha }}{\mathrm{\Gamma }\left(\left(j+1\right)\alpha \right)}\right)F\left(s\right)𝑑s$$=\eta \mathrm{\Gamma }\left(\alpha \right){t}^{\alpha -1}\sum _{j=0}^{n}\frac{{\lambda }^{j}{t}^{j\alpha }}{\mathrm{\Gamma }\left(\left(j+1\right)\alpha \right)}+{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}\left(\sum _{j=0}^{n}\frac{{\lambda }^{j}{\left(t-s\right)}^{j\alpha }}{\mathrm{\Gamma }\left(\left(j+1\right)\alpha \right)}\right)F\left(s\right)𝑑s$

and hence

${\varphi }_{n}\left(t\right)\to \eta \mathrm{\Gamma }\left(\alpha \right){t}^{\alpha -1}{E}_{\alpha ,\alpha }\left(\lambda {t}^{\alpha }\right)+{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(\lambda {\left(t-s\right)}^{\alpha }\right)F\left(s\right)𝑑s.$

Then we get (3.1.13). ∎

To get solutions of (3.1.3), we need the following assumptions:

#### Assumption A3.

There exist constants ${k}_{i}>-\alpha$, ${l}_{i}\le 0$ with ${l}_{1}>\mathrm{max}\left\{-\alpha ,-\alpha -{k}_{1}\right\}$, ${l}_{2}>\mathrm{max}\left\{-\alpha ,-1+{k}_{2}\right\}$, ${M}_{B}\ge 0$ and ${M}_{G}\ge 0$ such that $|B\left(t\right)|\le {M}_{B}{\left(\mathrm{log}t\right)}^{{k}_{1}}{\left(1-\mathrm{log}t\right)}^{{l}_{1}}$ and $|G\left(t\right)|\le {M}_{G}{\left(\mathrm{log}t\right)}^{{k}_{2}}{\left(1-\mathrm{log}t\right)}^{{l}_{2}}$ for all $t\in \left(1,e\right)$.

Choose Picard function sequence as

${\varphi }_{0}\left(t\right)=\eta {\left(\mathrm{log}t\right)}^{\alpha -1}+\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}G\left(s\right)\frac{ds}{s},$$t\in \left(1,e\right],$${\varphi }_{n}\left(t\right)={\varphi }_{0}\left(t\right)+\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}B\left(s\right){\varphi }_{n-1}\left(s\right)\frac{ds}{s},$$t\in \left(1,e\right],n=1,2,\mathrm{\dots }.$

#### Claim 3.7.

One has ${\varphi }_{n}\mathrm{\in }L\mathit{}{C}_{\mathrm{1}\mathrm{-}\alpha }\mathit{}\mathrm{\left(}\mathrm{1}\mathrm{,}e\mathrm{\right]}$.

#### Proof.

In fact, ${\varphi }_{0}\in L{C}_{1-\alpha }\left(1,e\right]$ and

${\left(\mathrm{log}t\right)}^{1-\alpha }|{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}B\left(s\right){\varphi }_{0}\left(s\right)\frac{ds}{s}|\le \parallel {\varphi }_{0}\parallel {\left(\mathrm{log}t\right)}^{1-\alpha }{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}{M}_{B}|\eta |{\left(\mathrm{log}s\right)}^{\alpha -1}{\left(\mathrm{log}s\right)}^{{k}_{1}}{\left(1-\mathrm{log}s\right)}^{{l}_{1}}\frac{ds}{s}$$\le \parallel {\varphi }_{0}\parallel {\left(\mathrm{log}t\right)}^{1-\alpha }{M}_{B}|\eta |{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha +{l}_{1}-1}{\left(\mathrm{log}s\right)}^{\alpha +{k}_{1}-1}\frac{ds}{s}$

We know that

$t↦\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}B\left(s\right){\varphi }_{0}\left(s\right)\frac{ds}{s}$

is continuous on $\left(1,e\right]$ and ${lim}_{t\to {0}^{+}}{\left(\mathrm{log}t\right)}^{1-\alpha }{\varphi }_{1}\left(t\right)$ exists. Then ${\varphi }_{1}\in L{C}_{1-\alpha }\left(1,e\right]$. By the mathematical induction method, we obtain that ${\varphi }_{n}\in L{C}_{1-\alpha }\left(1,e\right]$. ∎

#### Claim 3.8.

The sequence $\mathrm{\left\{}t\mathrm{↦}{\mathrm{\left(}\mathrm{log}\mathit{}t\mathrm{\right)}}^{\mathrm{1}\mathrm{-}\alpha }\mathit{}{\varphi }_{n}\mathit{}\mathrm{\left(}t\mathrm{\right)}\mathrm{\right\}}$ is convergent uniformly on $\mathrm{\left(}\mathrm{1}\mathrm{,}e\mathrm{\right]}$.

The proof is similar to that of Claim 3.5 and is omitted.

#### Claim 3.9.

The function ϕ defined on $\mathrm{\left(}\mathrm{1}\mathrm{,}e\mathrm{\right]}$ by $\varphi \mathit{}\mathrm{\left(}t\mathrm{\right)}\mathrm{=}{\mathrm{\left(}\mathrm{log}\mathit{}t\mathrm{\right)}}^{\alpha \mathrm{-}\mathrm{1}}\mathit{}{\mathrm{lim}}_{n\mathrm{\to }\mathrm{+}\mathrm{\infty }}\mathit{}{\mathrm{\left(}\mathrm{log}\mathit{}t\mathrm{\right)}}^{\mathrm{1}\mathrm{-}\alpha }\mathit{}{\varphi }_{n}\mathit{}\mathrm{\left(}t\mathrm{\right)}$ is a unique continuous solution of the integral equation

$x\left(t\right)=\eta {\left(\mathrm{log}t\right)}^{\alpha -1}+\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\left[B\left(s\right)x\left(s\right)+G\left(s\right)\right]\frac{ds}{s},t\in \left(1,e\right].$

#### Proof.

By

$\underset{n\to +\mathrm{\infty }}{lim}{\left(\mathrm{log}t\right)}^{1-\alpha }{\varphi }_{n}\left(t\right)={\left(\mathrm{log}t\right)}^{1-\alpha }\varphi \left(t\right)$

and the uniformly convergence, we see that $\varphi \left(t\right)$ is continuous on $\left(1,e\right]$. From

${\left(\mathrm{log}t\right)}^{1-\alpha }|{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\left[A\left(s\right){\varphi }_{n-1}\left(s\right)+F\left(s\right)\right]\frac{ds}{s}-{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\left[B\left(s\right){\varphi }_{m-1}\left(s\right)+G\left(s\right)\right]\frac{ds}{s}|$$\le {M}_{B}\parallel {\varphi }_{n-1}-{\varphi }_{m-1}\parallel {\left(\mathrm{log}t\right)}^{1-\alpha }{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha +{l}_{1}-1}{\left(\mathrm{log}s\right)}^{{k}_{1}}{\left(\mathrm{log}s\right)}^{\alpha -1}\frac{ds}{s}$$\le {M}_{B}\parallel {\varphi }_{n-1}-{\varphi }_{m-1}\parallel {\left(\mathrm{log}t\right)}^{1-\alpha }{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha +{l}_{1}-1}{\left(\mathrm{log}s\right)}^{\alpha +{k}_{1}-1}\frac{ds}{s}$$\le {M}_{B}\parallel {\varphi }_{n-1}-{\varphi }_{m-1}\parallel {\left(\mathrm{log}t\right)}^{\alpha +{k}_{1}+{l}_{1}}𝐁\left(\alpha +{l}_{1},\alpha +{k}_{1}\right)$

we know that

$\varphi \left(t\right)={\left(\mathrm{log}t\right)}^{\alpha -1}\underset{n\to +\mathrm{\infty }}{lim}{\left(\mathrm{log}t\right)}^{1-\alpha }{\varphi }_{n}\left(t\right)$$=\underset{n\to +\mathrm{\infty }}{lim}\left[\eta +{\left(\mathrm{log}t\right)}^{1-\alpha }\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\left[B\left(s\right){\varphi }_{n-1}\left(s\right)+G\left(s\right)\right]\frac{ds}{s}\right]$$=\eta {\left(\mathrm{log}t\right)}^{\alpha -1}+{\left(\mathrm{log}t\right)}^{\alpha -1}\underset{n\to +\mathrm{\infty }}{lim}{\left(\mathrm{log}t\right)}^{1-\alpha }{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\left[B\left(s\right){\varphi }_{n-1}\left(s\right)+G\left(s\right)\right]\frac{ds}{s}$$=\eta {\left(\mathrm{log}t\right)}^{\alpha -1}+\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\left[B\left(s\right)\varphi \left(s\right)+G\left(s\right)\right]\frac{ds}{s}.$

Then ϕ is a continuous solution of (3.1.14) defined on $\left(1,e\right]$.

Suppose that ψ defined on $\left(1,e\right]$ is also a solution of (3.1.14). Then

$\psi \left(t\right)=\eta {\left(\mathrm{log}t\right)}^{\alpha -1}+\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\left[B\left(s\right)\psi \left(s\right)+G\left(s\right)\right]\frac{ds}{s},t\in \left(1,e\right].$

We need to prove that $\varphi \left(t\right)\equiv \psi \left(t\right)$ on $\left(1,e\right]$. Then

${\left(\mathrm{log}t\right)}^{1-\alpha }|\psi \left(t\right)-{\varphi }_{0}\left(t\right)|={\left(\mathrm{log}t\right)}^{1-\alpha }|{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}|B\left(s\right)\psi \left(s\right)|\frac{ds}{s}|$$\le |\eta |\parallel \psi \parallel {M}_{B}{\left(\mathrm{log}t\right)}^{{k}_{1}+{l}_{1}+1}\frac{𝐁\left(\alpha +{l}_{1},{k}_{1}+1\right)}{\mathrm{\Gamma }\left(\alpha \right)}.$

By the mathematical induction method, we can get that

${\left(\mathrm{log}t\right)}^{1-\alpha }|\psi \left(t\right)-{\varphi }_{n}\left(t\right)|={\left(\mathrm{log}t\right)}^{1-\alpha }\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}|{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}B\left(s\right)\left[\psi \left(s\right)-{\varphi }_{n-1}\left(s\right)\right]𝑑s|$$\le |\eta |\parallel \psi \parallel {M}_{B}^{n}{\left(\mathrm{log}t\right)}^{n{k}_{1}+n{l}_{1}+n}\prod _{i=0}^{n-1}\frac{𝐁\left(\alpha +{l}_{1},\left(i+1\right){k}_{1}+i{l}_{1}+\left(i+1\right)\right)}{\mathrm{\Gamma }\left(\alpha \right)}$

Similarly we have ${lim}_{n\to +\mathrm{\infty }}{\left(\mathrm{log}t\right)}^{1-\alpha }{\varphi }_{n}\left(t\right)={\left(\mathrm{log}t\right)}^{1-\alpha }\psi \left(t\right)$ uniformly on $\left(1,e\right]$. Thus $\varphi \left(t\right)\equiv \psi \left(t\right)$ on $\left(1,e\right]$. Then (3.1.14) has a unique solution ϕ. ∎

#### Theorem 5.

Suppose that Assumption A3 holds. Then $x\mathrm{\in }L\mathit{}{C}_{\mathrm{1}\mathrm{-}\alpha }\mathit{}\mathrm{\left(}\mathrm{1}\mathrm{,}e\mathrm{\right]}$ is a solution of IVP (3.1.3) if and only if $x\mathrm{\in }L\mathit{}{C}_{\mathrm{1}\mathrm{-}\alpha }\mathit{}\mathrm{\left(}\mathrm{1}\mathrm{,}e\mathrm{\right]}$ is a solution of the integral equation (3.1.14).

#### Proof.

Suppose that $x\in {C}_{1-\alpha }\left(0,1\right]$ is a solution of IVP (3.1.3). Then $t↦{\left(\mathrm{log}t\right)}^{1-\alpha }x\left(t\right)$is continuous on $\left(1,e\right]$ by defining ${{\left(\mathrm{log}t\right)}^{1-\alpha }x\left(t\right)|}_{t=1}={lim}_{t\to {1}^{+}}{\left(\mathrm{log}t\right)}^{1-\alpha }x\left(t\right)$ and $\parallel x\parallel =r<+\mathrm{\infty }$. So

$\underset{s\to {1}^{+}}{lim}{\int }_{1}^{s}{\left(\mathrm{log}\frac{s}{w}\right)}^{-\alpha }x\left(w\right)\frac{dw}{w}=\underset{s\to {1}^{+}}{lim}{\int }_{1}^{s}{\left(\mathrm{log}\frac{s}{w}\right)}^{-\alpha }{\left(\mathrm{log}w\right)}^{\alpha -1}{\left(\mathrm{log}w\right)}^{1-\alpha }x\left(w\right)\frac{dw}{w}$$=\underset{s\to {1}^{+}}{lim}{\left(\mathrm{log}\xi \right)}^{1-\alpha }x\left(\xi \right){\int }_{1}^{s}{\left(\mathrm{log}\frac{s}{w}\right)}^{-\alpha }{\left(\mathrm{log}w\right)}^{\alpha -1}\frac{dw}{w}$$=\eta 𝐁\left(1-\alpha ,\alpha \right)$

by the mean value theorem for integrals, $\xi \in \left(1,s\right)$. From Assumption A3, we have

${\left(\mathrm{log}t\right)}^{1-\alpha }|{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\left[B\left(s\right)x\left(s\right)+G\left(s\right)\right]\frac{ds}{s}|$$\le {\left(\mathrm{log}t\right)}^{1-\alpha }{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\left[{M}_{B}r{\left(\mathrm{log}s\right)}^{\alpha -1}{\left(\mathrm{log}s\right)}^{{k}_{1}}{\left(1-\mathrm{log}s\right)}^{{l}_{1}}+{M}_{G}{\left(\mathrm{log}s\right)}^{{k}_{2}}{\left(1-\mathrm{log}s\right)}^{{l}_{2}}\right]\frac{ds}{s}$$\le {\left(\mathrm{log}t\right)}^{1-\alpha }{M}_{B}r{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha +{l}_{1}-1}{\left(\mathrm{log}s\right)}^{\alpha +{k}_{1}-1}\frac{ds}{s}+{\left(\mathrm{log}t\right)}^{1-\alpha }{M}_{G}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha +{l}_{2}-1}{\left(\mathrm{log}s\right)}^{{k}_{2}}\frac{ds}{s}$$={M}_{B}r{\left(\mathrm{log}t\right)}^{\alpha +{k}_{1}+{l}_{1}}𝐁\left(\alpha +{l}_{1},{k}_{1}+\alpha \right)+{M}_{G}{\left(\mathrm{log}t\right)}^{1+{k}_{1}+{l}_{1}}𝐁\left(\alpha +{l}_{2},{k}_{2}+1\right).$

It follows that

$t↦{\left(\mathrm{log}t\right)}^{1-\alpha }{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\left[B\left(s\right)x\left(s\right)+G\left(s\right)\right]\frac{ds}{s}$

is defined on $\left(1,e\right]$ and

$\underset{t\to {1}^{+}}{lim}{\left(\mathrm{log}t\right)}^{1-\alpha }{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\left[B\left(s\right)x\left(s\right)+G\left(s\right)\right]\frac{ds}{s}=0.$

Furthermore, we have similarly to Theorem 1 that

$t↦{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\left[B\left(s\right)x\left(s\right)+G\left(s\right)\right]\frac{ds}{s}$

is continuous on $\left(1,e\right]$. So

$t↦{\left(\mathrm{log}t\right)}^{1-\alpha }{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\left[B\left(s\right)x\left(s\right)+G\left(s\right)\right]\frac{ds}{s}$

is continuous on $\left[1,e\right]$ by defining

${{\left(\mathrm{log}t\right)}^{1-\alpha }{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\left[B\left(s\right)x\left(s\right)+G\left(s\right)\right]\frac{ds}{s}|}_{t=1}=\underset{t\to {1}^{+}}{lim}{\left(\mathrm{log}t\right)}^{1-\alpha }{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\left[B\left(s\right)x\left(s\right)+G\left(s\right)\right]\frac{ds}{s}.$

We have

${}^{H}I_{{1}^{+}}^{\alpha }{}^{\mathrm{RLH}}D_{{1}^{+}}^{\alpha }x\left(t\right)={}^{H}I_{{1}^{+}}^{\alpha }\left[B\left(t\right)x\left(t\right)+G\left(t\right)\right].$

Therefore

$\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\left[B\left(s\right)x\left(s\right)+G\left(s\right)\right]\frac{ds}{s}$$={}^{H}I_{{1}^{+}}^{\alpha }\left[B\left(t\right)x\left(t\right)+G\left(t\right)\right]={}^{H}I_{{1}^{+}}^{\alpha }{}^{\mathrm{RLH}}D_{{1}^{+}}^{\alpha }x\left(t\right)$$=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}s{\left({\int }_{1}^{s}{\left(\mathrm{log}\frac{s}{w}\right)}^{-\alpha }x\left(w\right)\frac{dw}{w}\right)}^{\prime }\frac{ds}{s}$$=\frac{1}{\mathrm{\Gamma }\left(\alpha +1\right)}\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}t{\left[{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha }{\left({\int }_{1}^{s}{\left(\mathrm{log}\frac{s}{w}\right)}^{-\alpha }x\left(w\right)\frac{dw}{w}\right)}^{\prime }𝑑s\right]}^{\prime }$$=\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)\mathrm{\Gamma }\left(\alpha +1\right)}t{\left[{{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha }{\int }_{1}^{s}{\left(\mathrm{log}\frac{s}{w}\right)}^{-\alpha }x\left(w\right)\frac{dw}{w}|}_{1}^{t}+\alpha {\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}{\int }_{1}^{s}{\left(\mathrm{log}\frac{s}{w}\right)}^{-\alpha }x\left(w\right)\frac{dw}{w}\frac{ds}{s}\right]}^{\prime }$$=\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)\mathrm{\Gamma }\left(\alpha +1\right)}t{\left[{\left(\mathrm{log}t\right)}^{\alpha }\underset{s\to {1}^{+}}{lim}{\int }_{1}^{s}{\left(\mathrm{log}\frac{s}{w}\right)}^{-\alpha }x\left(w\right)\frac{dw}{w}+\alpha {\int }_{1}^{t}{\int }_{w}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}{\left(\mathrm{log}\frac{s}{w}\right)}^{-\alpha }\frac{ds}{s}x\left(w\right)\frac{dw}{w}\right]}^{\prime }$$=\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)\mathrm{\Gamma }\left(\alpha +1\right)}t{\left[{\left(\mathrm{log}t\right)}^{\alpha }\eta 𝐁\left(1-\alpha ,\alpha \right)+\alpha {\int }_{1}^{t}𝐁\left(\alpha ,1-\alpha \right)x\left(w\right)\frac{dw}{w}\right]}^{\prime }$$=x\left(t\right)-\eta {\left(\mathrm{log}t\right)}^{\alpha -1}.$

Then $x\in L{C}_{1-\alpha }\left(1,e\right]$ is a solution of (3.1.14).

On the other hand, if x is a solution of (3.1.14), then we deduce from (3.1.15) that

$\underset{t\to {1}^{+}}{lim}{\left(\mathrm{log}t\right)}^{1-\alpha }x\left(t\right)=\eta .$

Then $x\in L{C}_{1-\alpha }\left(1,e\right]$. Furthermore, we have by Definition 2.5 that

${}^{\mathrm{RLH}}D_{{1}^{+}}^{\alpha }x\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}t{\left({\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{-\alpha }x\left(s\right)\frac{ds}{s}\right)}^{\prime }$$=\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}t{\left[{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{-\alpha }\left(\eta {\left(\mathrm{log}s\right)}^{\alpha -1}+\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{s}{\left(\mathrm{log}\frac{s}{w}\right)}^{\alpha -1}\left[A\left(w\right)x\left(w\right)+F\left(w\right)\right]\frac{dw}{w}\right)\frac{ds}{s}\right]}^{\prime }$$=\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}t{\left[\eta {\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{-\alpha }{\left(\mathrm{log}s\right)}^{\alpha -1}\frac{ds}{s}\right]}^{\prime }$$+\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}t{\left[\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{s}{\left(\mathrm{log}\frac{s}{w}\right)}^{\alpha -1}\left[A\left(w\right)x\left(w\right)+F\left(w\right)\right]\frac{dw}{w}\frac{ds}{s}\right]}^{\prime }$$=\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}t{\left[\eta 𝐁\left(1-\alpha ,\alpha \right)\right]}^{\prime }$$+\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}t{\left[\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\int }_{w}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{-\alpha }{\left(\mathrm{log}\frac{s}{w}\right)}^{\alpha -1}\frac{ds}{s}\left[A\left(w\right)x\left(w\right)+F\left(w\right)\right]\frac{dw}{w}\right]}^{\prime }$$=\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}t{\left[\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}𝐁\left(1-\alpha ,\alpha \right)\left[B\left(w\right)x\left(w\right)+G\left(w\right)\right]\frac{dw}{w}\right]}^{\prime }$$=B\left(t\right)x\left(t\right)+G\left(t\right).$

So $x\in L{C}_{1-\alpha }\left(1,e\right]$ is a solution of IVP (3.1.3). ∎

#### Theorem 6.

Suppose that Assumption A3 holds. Then (3.1.14) has a unique solution. If $B\mathit{}\mathrm{\left(}t\mathrm{\right)}\mathrm{\equiv }\lambda$ and there exist constants ${k}_{\mathrm{2}}\mathrm{>}\mathrm{-}\mathrm{1}$, ${l}_{\mathrm{2}}\mathrm{\le }\mathrm{0}$ with ${l}_{\mathrm{2}}\mathrm{>}\mathrm{max}\mathit{}\mathrm{\left\{}\mathrm{-}\alpha \mathrm{,}\mathrm{-}\mathrm{1}\mathrm{-}{k}_{\mathrm{2}}\mathrm{\right\}}$ and ${M}_{G}\mathrm{\ge }\mathrm{0}$ such that $\mathrm{|}G\mathit{}\mathrm{\left(}t\mathrm{\right)}\mathrm{|}\mathrm{\le }{M}_{G}\mathit{}{t}^{{k}_{\mathrm{2}}}\mathit{}{\mathrm{\left(}\mathrm{1}\mathrm{-}t\mathrm{\right)}}^{{l}_{\mathrm{2}}}$ for all $t\mathrm{\in }\mathrm{\left(}\mathrm{1}\mathrm{,}e\mathrm{\right)}$, then the special problem

$\left\{\begin{array}{cc}\hfill {}^{\mathrm{RLH}}D_{{1}^{+}}^{𝜶}x\left(t\right)& =\lambda x\left(t\right)+G\left(t\right),t\in \left(1,e\right],\hfill \\ \hfill \underset{t\to {0}^{+}}{lim}{\left(\mathrm{log}t\right)}^{1-\alpha }x\left(t\right)& =\eta \hfill \end{array}$(3.1.17)

has a unique solution

$x\left(t\right)=\eta \mathrm{\Gamma }\left(\alpha \right){\left(\mathrm{log}t\right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(\lambda {\left(\mathrm{log}t\right)}^{\alpha }\right)+{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(\lambda {\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha }\right)G\left(s\right)\frac{ds}{s},t\in \left(1,e\right].$

#### Proof.

From Claims 3.73.9, IVP (3.1.14) has a unique solution. From the assumption and $B\left(t\right)\equiv \lambda$, one sees that Assumption A3 holds with ${k}_{1}={l}_{1}=0$ and ${k}_{2},{l}_{2}$ mentioned in assumption. Thus (3.1.17) has a unique solution. We get from the Picard function sequence that

${\varphi }_{n}\left(t\right)=\eta {\left(\mathrm{log}t\right)}^{\alpha -1}+\lambda \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}{\varphi }_{n-1}\left(s\right)𝑑s+\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}G\left(s\right)\frac{ds}{s}$$=\eta {\left(\mathrm{log}t\right)}^{\alpha -1}+\frac{\eta \lambda }{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}{\left(\mathrm{log}s\right)}^{\alpha -1}\frac{ds}{s}$$+\frac{{\lambda }^{2}}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{s}{\left(\mathrm{log}\frac{s}{w}\right)}^{\alpha -1}{\varphi }_{n-2}\left(w\right)\frac{dw}{w}\frac{ds}{s}$$+\frac{\lambda }{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\left(\mathrm{log}\frac{s}{w}\right)}^{\alpha -1}G\left(w\right)\frac{dw}{w}\frac{ds}{s}+\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}G\left(s\right)\frac{ds}{s}$$=\eta {\left(\mathrm{log}t\right)}^{\alpha -1}+\frac{\eta \lambda {\left(\mathrm{log}t\right)}^{2\alpha -1}}{\mathrm{\Gamma }\left(2\alpha \right)}+\frac{{\lambda }^{2}}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\int }_{w}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\left(\mathrm{log}\frac{s}{w}\right)}^{\alpha -1}\frac{ds}{s}{\varphi }_{n-2}\left(w\right)\frac{dw}{w}$$+\frac{\lambda }{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\int }_{w}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}{\left(\mathrm{log}\frac{s}{w}\right)}^{\alpha -1}\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\frac{ds}{s}G\left(w\right)\frac{dw}{w}+\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}G\left(s\right)\frac{ds}{s}$$\mathrm{⋮}$$=\eta \mathrm{\Gamma }\left(\alpha \right){\left(\mathrm{log}t\right)}^{\alpha -1}\sum _{j=0}^{n-1}\frac{{\lambda }^{j}{\left(\mathrm{log}t\right)}^{j\alpha }}{\mathrm{\Gamma }\left(\left(j+1\right)\alpha \right)}+\frac{\eta {\lambda }^{n}}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\frac{dw}{w}$$+\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\left(\sum _{j=0}^{n-1}\frac{{\lambda }^{j}{\left(t-s\right)}^{j\alpha }}{\mathrm{\Gamma }\left(\left(j+1\right)\alpha \right)}\right)F\left(s\right)\frac{ds}{s}$$=\eta \mathrm{\Gamma }\left(\alpha \right){\left(\mathrm{log}t\right)}^{\alpha -1}\sum _{j=0}^{n}\frac{{\lambda }^{j}{\left(\mathrm{log}t\right)}^{j\alpha }}{\mathrm{\Gamma }\left(\left(j+1\right)\alpha \right)}+\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\left(\sum _{j=0}^{n}\frac{{\lambda }^{j}{\left(\frac{t}{s}\right)}^{j\alpha }}{\mathrm{\Gamma }\left(\left(j+1\right)\alpha \right)}\right)G\left(s\right)\frac{ds}{s}$

and so

${\varphi }_{n}\left(t\right)\to \eta \mathrm{\Gamma }\left(\alpha \right){\left(\mathrm{log}t\right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(\lambda {\left(\mathrm{log}t\right)}^{\alpha }\right)+{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(\lambda {\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha }\right)G\left(s\right)\frac{ds}{s}.$

Then we get (3.1.18). The proof is complete. ∎

To get solutions of (3.1.4), we need the following assumptions:

#### Assumption A4.

There exist constants ${k}_{i}>-1$ and ${l}_{i}\le 0$ with ${l}_{i}>\mathrm{max}\left\{-\alpha ,-\alpha -{k}_{i}\right\}$, ${M}_{B}\ge 0$ and ${M}_{G}\ge 0$ such that $|B\left(t\right)|\le {M}_{B}{\left(\mathrm{log}t\right)}^{{k}_{1}}{\left(1-\mathrm{log}t\right)}^{{l}_{1}}$ and $|G\left(t\right)|\le {M}_{G}{\left(\mathrm{log}t\right)}^{{k}_{2}}{\left(1-\mathrm{log}t\right)}^{{l}_{2}}$ for all $t\in \left(1,e\right)$.

Choose Picard function sequence as

${\varphi }_{0}\left(t\right)=\eta +\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}G\left(s\right)\frac{ds}{s},$$t\in \left(1,e\right],$${\varphi }_{n}\left(t\right)={\varphi }_{0}\left(t\right)+\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}B\left(s\right){\varphi }_{n-1}\left(s\right)\frac{ds}{s},$$t\in \left(1,e\right],n=1,2,\mathrm{\dots }.$

#### Claim 3.10.

One has ${\varphi }_{n}\mathrm{\in }C\mathit{}\mathrm{\left[}\mathrm{1}\mathrm{,}e\mathrm{\right]}$.

The proof is similar to that of Claim 3.1 and is omitted.

#### Claim 3.11.

The sequence ${\varphi }_{n}$ is convergent uniformly on $\mathrm{\left[}\mathrm{1}\mathrm{,}e\mathrm{\right]}$.

The proof is similar to that of Claim 3.2 and is omitted.

#### Claim 3.12.

The function ϕ defined on $\mathrm{\left(}\mathrm{1}\mathrm{,}e\mathrm{\right]}$ by $\varphi \mathit{}\mathrm{\left(}t\mathrm{\right)}\mathrm{=}{\mathrm{lim}}_{n\mathrm{\to }\mathrm{+}\mathrm{\infty }}\mathit{}{\varphi }_{n}\mathit{}\mathrm{\left(}t\mathrm{\right)}$ is a unique continuous solution of the integral equation

$x\left(t\right)=\eta +\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\left[B\left(s\right)x\left(s\right)+G\left(s\right)\right]\frac{ds}{s}.$

The proof is similar to that of Claim 3.3 and is omitted.

#### Lemma 7.

Suppose that Assumption A4 holds. Then $x\mathrm{\in }C\mathit{}\mathrm{\left(}\mathrm{1}\mathrm{,}e\mathrm{\right]}$ is a solution of IVP (3.1.4) if and only if $x\mathrm{\in }C\mathit{}\mathrm{\left(}\mathrm{1}\mathrm{,}e\mathrm{\right]}$ is a solution of the integral equation (3.1.19).

#### Proof.

Suppose that $x\in C\left(1,e\right]$ is a solution of IVP (3.1.4). Then the function $t↦x\left(t\right)$ is continuous on $\left[1,e\right]$ and $\parallel x\parallel =r<+\mathrm{\infty }$. One can see that

$={\left(\mathrm{log}t\right)}^{\alpha +{k}_{1}+{l}_{1}}{\int }_{0}^{1}{\left(1-u\right)}^{\alpha +{l}_{1}-1}{u}^{{k}_{1}}𝑑u$$\le {\left(\mathrm{log}t\right)}^{\alpha +{k}_{1}+{l}_{1}}{\int }_{0}^{1}{\left(1-u\right)}^{\alpha +{l}_{1}-1}{u}^{{k}_{1}}𝑑u$$={\left(\mathrm{log}t\right)}^{\alpha +{k}_{1}+{l}_{1}}𝐁\left(\alpha +{l}_{1},{k}_{1}+1\right).$

From Assumption A4, we have for $t\in \left(1,e\right]$ that

$|{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\left[B\left(s\right)x\left(s\right)+G\left(s\right)\right]\frac{ds}{s}|$$\le {\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\left[{M}_{B}r{\left(\mathrm{log}s\right)}^{{k}_{1}}{\left(1-\mathrm{log}s\right)}^{{l}_{1}}+{M}_{G}{\left(\mathrm{log}s\right)}^{{k}_{2}}{\left(1-\mathrm{log}s\right)}^{{l}_{2}}\right]\frac{ds}{s}$$\le {M}_{B}r{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}{\left(\mathrm{log}s\right)}^{{k}_{1}}{\left(1-\mathrm{log}s\right)}^{{l}_{1}}\frac{ds}{s}+{M}_{G}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}{\left(\mathrm{log}s\right)}^{{k}_{2}}{\left(1-\mathrm{log}s\right)}^{{l}_{2}}\frac{ds}{s}$$={M}_{B}r{\left(\mathrm{log}t\right)}^{\alpha +{k}_{1}+{l}_{1}}𝐁\left(\alpha +{l}_{1},{k}_{1}+1\right)+{M}_{G}{\left(\mathrm{log}t\right)}^{\alpha +{k}_{2}+{l}_{2}}𝐁\left(\alpha +{l}_{2},{k}_{2}+1\right).$

It follows that

$t↦{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\left[B\left(s\right)x\left(s\right)+G\left(s\right)\right]\frac{ds}{s}$

is defined on $\left(1,e\right]$ and

$\underset{t\to {1}^{+}}{lim}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\left[B\left(s\right)x\left(s\right)+G\left(s\right)\right]\frac{ds}{s}=0.$(3.1.20)

Furthermore, we have similarly to Theorem 1 that

$t↦{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\left[B\left(s\right)x\left(s\right)+G\left(s\right)\right]\frac{ds}{s}$

is continuous on $\left(1,e\right]$. So

$t↦{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\left[B\left(s\right)x\left(s\right)+G\left(s\right)\right]\frac{ds}{s}$

is continuous on $\left[1,e\right]$ by defining

${{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\left[B\left(s\right)x\left(s\right)+G\left(s\right)\right]\frac{ds}{s}|}_{t=1}=0.$(3.1.21)

One sees that

${\int }_{w}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}{\left(\mathrm{log}\frac{s}{w}\right)}^{-\alpha }\frac{ds}{s}={\int }_{0}^{1}{\left(1-u\right)}^{\alpha -1}{u}^{-\alpha }𝑑u=\mathrm{\Gamma }\left(1-\alpha \right)\mathrm{\Gamma }\left(\alpha \right)$

by $\frac{\mathrm{log}s-\mathrm{log}w}{\mathrm{log}t-\mathrm{log}w}=u$. We have by Definition 2.6 and ${}^{H}I_{{1}^{+}}^{\alpha }{}^{\mathrm{CH}}D_{{1}^{+}}^{\alpha }x\left(t\right)={}^{H}I_{{1}^{+}}^{\alpha }\left[B\left(t\right)x\left(t\right)+G\left(t\right)\right]$ that

${\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\left[A\left(s\right)x\left(s\right)+F\left(s\right)\right]\frac{ds}{s}={}^{H}I_{{1}^{+}}^{\alpha }\left[B\left(t\right)x\left(t\right)+G\left(t\right)\right]={}^{H}I_{{1}^{+}}^{\alpha }{}^{\mathrm{CH}}D_{{1}^{+}}^{\alpha }x\left(t\right)$$=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\left[\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}{\int }_{1}^{s}{\left(\mathrm{log}\frac{s}{w}\right)}^{-\alpha }w{x}^{\prime }\left(w\right)\frac{dw}{w}\right]\frac{ds}{s}$$=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\left[\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}{\int }_{0}^{s}{\left(\mathrm{log}\frac{s}{w}\right)}^{-\alpha }{x}^{\prime }\left(w\right)𝑑w\right]\frac{ds}{s}$$=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}{\int }_{1}^{t}{\int }_{w}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}{\left(\mathrm{log}\frac{s}{w}\right)}^{-\alpha }\frac{ds}{s}{x}^{\prime }\left(w\right)𝑑w$$={\int }_{1}^{t}{x}^{\prime }\left(w\right)𝑑w=x\left(t\right)-\underset{t\to {1}^{+}}{lim}x\left(t\right)=x\left(t\right)-\eta .$

Then $x\in C\left(1,e\right]$ is a solution of (3.1.19).

On the other hand, if $x\in C\left(1,e\right]$ is a solution of (3.1.19), then this together with (3.1.20) implies ${lim}_{t\to {1}^{+}}\left(t\right)=\eta$. Furthermore, we have that

${}^{\mathrm{CH}}D_{{1}^{+}}^{\alpha }x\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{-\alpha }s{x}^{\prime }\left(s\right)\frac{ds}{s}$$=\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{-\alpha }{\left(\eta +\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{s}{\left(\mathrm{log}\frac{s}{w}\right)}^{\alpha -1}\left[B\left(w\right)x\left(w\right)+G\left(w\right)\right]\frac{dw}{w}\right)}^{\prime }𝑑s$$=\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{-\alpha }{\left(\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{s}{\left(\mathrm{log}\frac{s}{w}\right)}^{\alpha -1}\left[B\left(w\right)x\left(w\right)+G\left(w\right)\right]\frac{dw}{w}\right)}^{\prime }𝑑s$$=\frac{t}{\mathrm{\Gamma }\left(2-\alpha \right)}{\left({\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{1-\alpha }{\left(\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{s}{\left(\mathrm{log}\frac{s}{w}\right)}^{\alpha -1}\left[B\left(w\right)x\left(w\right)+G\left(w\right)\right]\frac{dw}{w}\right)}^{\prime }𝑑s\right)}^{\prime }$$=\frac{t}{\mathrm{\Gamma }\left(2-\alpha \right)}{\left[{\left(\mathrm{log}\frac{t}{s}\right)}^{1-\alpha }\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{s}{\left(\mathrm{log}\frac{s}{w}\right)}^{\alpha -1}\left[B\left(w\right)x\left(w\right)+G\left(w\right)\right]\frac{dw}{w}|}_{1}^{t}$$+\left(1-\alpha \right)\frac{1}{s}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{-\alpha }\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{s}{\left(\mathrm{log}\frac{s}{w}\right)}^{\alpha -1}\left[B\left(w\right)x\left(w\right)+G\left(w\right)\right]\frac{dw}{w}ds\right]{}^{\prime }$$=\frac{t}{\mathrm{\Gamma }\left(1-\alpha \right)}\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\left[\frac{1}{s}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{-\alpha }{\int }_{1}^{s}{\left(\mathrm{log}\frac{s}{w}\right)}^{\alpha -1}\left[B\left(w\right)x\left(w\right)+G\left(w\right)\right]\frac{dw}{w}𝑑s\right]}^{\prime }$$=B\left(t\right)x\left(t\right)+G\left(t\right).$

So $x\in C\left(1,e\right]$ is a solution of IVP (3.1.4). ∎

#### Theorem 8.

Suppose that Assumption A4 holds. Then (3.1.4) has a unique solution. If $B\mathit{}\mathrm{\left(}t\mathrm{\right)}\mathrm{\equiv }\lambda$ and there exist constants ${k}_{\mathrm{2}}\mathrm{>}\mathrm{-}\mathrm{1}$, ${l}_{\mathrm{2}}\mathrm{\le }\mathrm{0}$ with ${l}_{\mathrm{2}}\mathrm{>}\mathrm{max}\mathit{}\mathrm{\left\{}\mathrm{-}\alpha \mathrm{,}\mathrm{-}\alpha \mathrm{-}{k}_{\mathrm{2}}\mathrm{\right\}}$ and ${M}_{G}\mathrm{\ge }\mathrm{0}$ such that $\mathrm{|}G\mathit{}\mathrm{\left(}t\mathrm{\right)}\mathrm{|}\mathrm{\le }{M}_{G}\mathit{}{t}^{{k}_{\mathrm{2}}}\mathit{}{\mathrm{\left(}\mathrm{1}\mathrm{-}t\mathrm{\right)}}^{{l}_{\mathrm{2}}}$ for all $t\mathrm{\in }\mathrm{\left(}\mathrm{1}\mathrm{,}e\mathrm{\right)}$, then the special problem

$\left\{\begin{array}{cc}\hfill {}^{\mathrm{CH}}D_{{0}^{+}}^{𝜶}x\left(t\right)& =\lambda x\left(t\right)+G\left(t\right),t\in \left(1,e\right],\hfill \\ \hfill \underset{t\to {1}^{+}}{lim}x\left(t\right)& =\eta \hfill \end{array}$(3.1.22)

has a unique solution

$x\left(t\right)=\eta {E}_{\alpha ,1}\left(\lambda {\left(\mathrm{log}t\right)}^{\alpha }\right)+{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(\lambda {\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha }\right)G\left(s\right)\frac{ds}{s},t\in \left(1,e\right].$

#### Proof.

From Claims 3.103.12 and Lemma 7, IVP (3.1.4) has a unique solution. From the assumption and

$A\left(t\right)\equiv \lambda ,$

one sees that Assumption A4 holds with ${k}_{1}={l}_{1}=0$ and ${k}_{2},{l}_{2}$ mentioned. Thus (3.1.22) has a unique solution. We get from the Picard function sequence that

${\varphi }_{n}\left(t\right)=\eta +\lambda \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}{\varphi }_{n-1}\left(s\right)\frac{ds}{s}+\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}G\left(s\right)\frac{ds}{s}$$=\eta +\eta \lambda \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\frac{ds}{s}+{\lambda }^{2}\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}{\int }_{1}^{s}{\left(\mathrm{log}\frac{s}{w}\right)}^{\alpha -1}{\varphi }_{n-2}\left(w\right)\frac{dw}{w}\frac{ds}{s}$$+\lambda \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}{\int }_{1}^{s}{\left(\mathrm{log}\frac{s}{w}\right)}^{\alpha -1}G\left(w\right)\frac{dw}{w}\frac{ds}{s}+\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}G\left(s\right)\frac{ds}{s}$$=\eta +\frac{\eta \lambda {\left(\mathrm{log}t\right)}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}+{\lambda }^{2}\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\int }_{w}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}{\left(\mathrm{log}\frac{s}{w}\right)}^{\alpha -1}\frac{ds}{s}{\varphi }_{n-2}\left(w\right)\frac{dw}{w}$$+\lambda \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{t}{\int }_{w}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}{\left(\mathrm{log}\frac{s}{w}\right)}^{\alpha -1}\frac{ds}{s}F\left(w\right)\frac{dw}{w}+\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}F\left(s\right)\frac{ds}{s}$$=\eta \left(1+\frac{\lambda {\left(\mathrm{log}t\right)}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right)+{\lambda }^{2}\frac{1}{\mathrm{\Gamma }\left(2\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{2\alpha -1}{\varphi }_{n-2}\left(w\right)\frac{dw}{w}$$+\lambda \frac{1}{\mathrm{\Gamma }\left(2\alpha \right)}{\int }_{0}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{2\alpha -1}G\left(w\right)\frac{dw}{w}+\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}G\left(s\right)\frac{ds}{s}$$\mathrm{⋮}$$=\eta \sum _{j=0}^{n}\frac{{\lambda }^{j}{\left(\mathrm{log}t\right)}^{j\alpha }}{\mathrm{\Gamma }\left(j\alpha +1\right)}+{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}\left(\sum _{j=0}^{n}\frac{{\lambda }^{j}{\left(\mathrm{log}\frac{t}{s}\right)}^{j\alpha }}{\mathrm{\Gamma }\left(\left(j+1\right)\alpha \right)}\right)G\left(s\right)𝑑s$

and so

${\varphi }_{n}\left(t\right)\to \eta {E}_{\alpha ,1}\left(\lambda {\left(\mathrm{log}t\right)}^{\alpha }\right)+{\int }_{1}^{t}{\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(\lambda {\left(\mathrm{log}\frac{t}{s}\right)}^{\alpha }\right)G\left(s\right)\frac{ds}{s}.$

Then we get (3.1.23). The proof is complete. ∎

#### Theorem 9 (Schaefer’s fixed point theorem).

Let E be a Banach spaces and let $T\mathrm{:}E\mathrm{\to }E$ be a completely continuous operator. If the set is bounded, then T has at least a fixed point in E.

## 3.2 Exact piecewise continuous solutions of impulsive FDEs

In this subsection, we present exact piecewise continuous solutions of the following fractional differential equations, respectively

${}^{C}D_{{0}^{+}}^{𝜶}x\left(t\right)=\lambda x\left(t\right)+F\left(t\right),$$t\in \left({t}_{i},{t}_{i+1}\right],i\in {ℕ}_{0},$(3.2.1)${}^{\mathrm{RL}}D_{{0}^{+}}^{𝜶}x\left(t\right)=\lambda x\left(t\right)+F\left(t\right),$$t\in \left({t}_{i},{t}_{i+1}\right],i\in {ℕ}_{0},$(3.2.2)${}^{\mathrm{RLH}}D_{{0}^{+}}^{𝜶}x\left(t\right)=\lambda x\left(t\right)+G\left(t\right),$$t\in \left({t}_{i},{t}_{i+1}\right],i\in {ℕ}_{0},$(3.2.3)${}^{\mathrm{CH}}D_{{0}^{+}}^{𝜶}x\left(t\right)=\lambda x\left(t\right)+G\left(t\right),$$t\in \left({t}_{i},{t}_{i+1}\right],i\in {ℕ}_{0},$(3.2.4)

where $\lambda \in ℝ$, $0={t}_{0}<{t}_{1}<\mathrm{\cdots }<{t}_{m}<{t}_{m+1}=1$ in (3.2.1) and (3.2.2) and $1={t}_{0}<{t}_{1}<\mathrm{\cdots }<{t}_{m}<{t}_{m+1}=e$ in (3.2.3) and (3.2.4). We say that $x:\left(0,1\right]\to R$ is a piecewise solution of (3.2.1) (or (3.2.2) if $x\in {P}_{m}C\left(0,1\right]$ (or ${P}_{m}{C}_{1-\alpha }\left(0,1\right]$ and satisfies (3.2.1) or (3.2.2). We say that $x\left(1,e\right]\to ℝ$ is a piecewise continuous solutions of (3.2.3) (or (3.2.4)) if $x\in L{P}_{m}{C}_{1-\alpha }\left(1,e\right]$ (or $L{P}_{m}C\left(1,e\right]$) and x satisfies all equations in (3.2.3) (or (3.2.4)).

#### Theorem 1.

Suppose that F is continuous on $\mathrm{\left(}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right)}$ and there exist constants $k\mathrm{>}\mathrm{-}\mathrm{1}$ and $l\mathrm{\in }\mathrm{\left(}\mathrm{-}\alpha \mathrm{,}\mathrm{-}\alpha \mathrm{-}k\mathrm{,}\mathrm{0}\mathrm{\right]}$ such that $\mathrm{|}F\mathit{}\mathrm{\left(}t\mathrm{\right)}\mathrm{|}\mathrm{\le }{t}^{k}\mathit{}{\mathrm{\left(}\mathrm{1}\mathrm{-}t\mathrm{\right)}}^{l}$ for all $t\mathrm{\in }\mathrm{\left(}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right)}$. Then x is a piecewise solution of (3.2.1) if and only if x and there exists constants ${c}_{i}\mathrm{\in }\mathrm{R}$ ($i\in {ℕ}_{0}$) such that

$x\left(t\right)=\sum _{v=0}^{j}{c}_{v}{𝐄}_{\alpha ,1}\left(\lambda {\left(t-{t}_{v}\right)}^{\alpha }\right)+{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {\left(t-s\right)}^{\alpha }\right)F\left(s\right)𝑑s,t\in \left({t}_{j},{t}_{j+1}\right],j\in {ℕ}_{0}.$(3.2.5)

#### Proof.

Firstly, we have for $t\in \left({t}_{i},{t}_{i+1}\right]$ that

$|{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {\left(t-s\right)}^{\alpha }\right)F\left(s\right)𝑑s|\le {\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {\left(t-s\right)}^{\alpha }\right)|F\left(s\right)|𝑑s$$\le {\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {\left(t-s\right)}^{\alpha }\right){s}^{k}{\left(1-s\right)}^{l}𝑑s$$=\sum _{j=0}^{+\mathrm{\infty }}\frac{{\lambda }^{j}}{𝚪\left(\left(j+1\right)\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}{\left(t-s\right)}^{\alpha j}{s}^{k}{\left(1-s\right)}^{l}𝑑s$$\le \sum _{j=0}^{+\mathrm{\infty }}\frac{{\lambda }^{j}}{𝚪\left(\left(j+1\right)\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha +l-1}{\left(t-s\right)}^{\alpha j}{s}^{k}𝑑s$$=\sum _{j=0}^{+\mathrm{\infty }}\frac{{\lambda }^{j}}{𝚪\left(\left(j+1\right)\alpha \right)}{t}^{\alpha +\alpha j+k+l}{\int }_{0}^{1}{\left(1-w\right)}^{\alpha +\alpha j+l-1}{w}^{k}𝑑w$$\le \sum _{j=0}^{+\mathrm{\infty }}\frac{{\lambda }^{j}{t}^{\alpha j}}{𝚪\left(\left(j+1\right)\alpha \right)}{t}^{\alpha +k+l}{\int }_{0}^{1}{\left(1-w\right)}^{\alpha +l-1}{w}^{k}𝑑w$$={t}^{\alpha +k+l}{𝐄}_{\alpha ,\alpha }\left(\lambda {t}^{\alpha }\right)𝐁\left(\alpha +l,k+1\right).$

Then ${\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {\left(t-s\right)}^{\alpha }\right)F\left(s\right)𝑑s$ is convergent and is continuous on $\left[0,1\right]$. If x is a piecewise continuous solution of (3.2.5), then we know that $x\in {P}_{m}C\left(0,1\right]$ and ${lim}_{t\to {t}_{i}^{+}}x\left(t\right)$ ($i\in {ℕ}_{0}$) exist. Now we prove that x satisfies differential equation in (3.2.1). In fact, for $t\in \left({t}_{i},{t}_{i+1}\right]$ ($i\in {ℕ}_{0}$), we have

${}^{C}D_{{0}^{+}}^{\alpha }x\left(t\right)=\frac{1}{𝚪\left(1-\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{-\alpha }{x}^{\prime }\left(s\right)𝑑s$$=\frac{1}{𝚪\left(1-\alpha \right)}\left[\sum _{j=0}^{i-1}{\int }_{{t}_{j}}^{{t}_{j+1}}{\left(t-s\right)}^{-\alpha }{x}^{\prime }\left(s\right)𝑑s+{\int }_{{t}_{i}}^{t}{\left(t-s\right)}^{-\alpha }{x}^{\prime }\left(s\right)𝑑s\right]$$=\frac{1}{𝚪\left(1-\alpha \right)}\left[\sum _{j=0}^{i-1}{\int }_{{t}_{j}}^{{t}_{j+1}}{\left(t-s\right)}^{-\alpha }{\left(\sum _{\kappa =0}^{j}{c}_{\kappa }{𝐄}_{\alpha ,1}\left(\lambda {\left(s-{t}_{\kappa }\right)}^{\alpha }\right)+{\int }_{0}^{s}{\left(s-v\right)}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {\left(s-v\right)}^{\alpha }\right)F\left(v\right)dv\right)}^{\prime }ds$$+{\int }_{{t}_{i}}^{t}{\left(t-s\right)}^{-\alpha }{\left(\sum _{\kappa =0}^{i}{c}_{\kappa }{𝐄}_{\alpha ,1}\left(\lambda {\left(s-{t}_{\kappa }\right)}^{\alpha }\right)+{\int }_{0}^{s}{\left(s-v\right)}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {\left(s-v\right)}^{\alpha }\right)F\left(v\right)dv\right)}^{\prime }ds\right]$$=\frac{1}{𝚪\left(1-\alpha \right)}\sum _{j=0}^{i-1}{\int }_{{t}_{j}}^{{t}_{j+1}}{\left(t-s\right)}^{-\alpha }{\left[\sum _{\kappa =0}^{j}{c}_{\kappa }{𝐄}_{\alpha ,1}\left(\lambda {\left(s-{t}_{\kappa }\right)}^{\alpha }\right)\right]}^{\prime }𝑑s$$+\frac{1}{𝚪\left(1-\alpha \right)}{\int }_{{t}_{i}}^{t}{\left(t-s\right)}^{-\alpha }{\left[\sum _{\kappa =0}^{i}{c}_{\kappa }{𝐄}_{\alpha ,1}\left(\lambda {\left(s-{t}_{\kappa }\right)}^{\alpha }\right)\right]}^{\prime }𝑑s$$+\frac{1}{𝚪\left(1-\alpha \right)}{\int }_{{t}_{0}}^{t}{\left(t-s\right)}^{-\alpha }{\left[{\int }_{0}^{s}{\left(s-v\right)}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {\left(s-v\right)}^{\alpha }\right)F\left(v\right)𝑑v\right]}^{\prime }𝑑s$$=\frac{1}{𝚪\left(1-\alpha \right)}\sum _{j=0}^{i-1}\sum _{\kappa =0}^{j}{c}_{\kappa }{\int }_{{t}_{j}}^{{t}_{j+1}}{\left(t-s\right)}^{-\alpha }{\left[\sum _{m=0}^{+\mathrm{\infty }}\frac{{\lambda }^{m}{\left(s-{t}_{\kappa }\right)}^{m\alpha }}{𝚪\left(m\alpha +1\right)}\right]}^{\prime }𝑑s$$+\frac{1}{𝚪\left(1-\alpha \right)}\sum _{\kappa =0}^{i}{c}_{\kappa }{\int }_{{t}_{i}}^{t}{\left(t-s\right)}^{-\alpha }{\left[\sum _{m=0}^{+\mathrm{\infty }}\frac{{\lambda }^{m}{\left(s-{t}_{\kappa }\right)}^{m\alpha }}{𝚪\left(m\alpha +1\right)}\right]}^{\prime }𝑑s$$+\frac{1}{𝚪\left(1-\alpha \right)}\sum _{m=0}^{+\mathrm{\infty }}\frac{{\lambda }^{m}}{𝚪\left(\alpha \left(m+1\right)\right)}{\int }_{0}^{t}{\left(t-s\right)}^{-\alpha }{\left[{\int }_{0}^{s}{\left(s-v\right)}^{\alpha +m\alpha -1}F\left(v\right)𝑑v\right]}^{\prime }𝑑s$$=\frac{1}{𝚪\left(1-\alpha \right)}\sum _{j=0}^{i-1}\sum _{\kappa =0}^{j}{c}_{\kappa }\sum _{m=1}^{+\mathrm{\infty }}\frac{{\lambda }^{m}m\alpha }{𝚪\left(m\alpha +1\right)}{\int }_{{t}_{j}}^{{t}_{j+1}}{\left(t-s\right)}^{-\alpha }{\left(s-{t}_{\kappa }\right)}^{m\alpha -1}𝑑s$$+\frac{1}{𝚪\left(1-\alpha \right)}\sum _{m=1}^{+\mathrm{\infty }}\frac{m\alpha {\lambda }^{m}}{𝚪\left(m\alpha +1\right)}\sum _{\kappa =0}^{i}{c}_{\kappa }{\int }_{{t}_{i}}^{t}{\left(t-s\right)}^{-\alpha }{\left(s-{t}_{\kappa }\right)}^{m\alpha -1}𝑑s+\sum _{m=0}^{+\mathrm{\infty }}{\lambda }^{m}{D}_{{0}^{+}}^{\alpha }{I}_{{0}^{+}}^{\alpha \left(m+1\right)}F\left(t\right)$$=\frac{1}{𝚪\left(1-\alpha \right)}\sum _{m=1}^{+\mathrm{\infty }}\frac{{\lambda }^{m}m\alpha }{𝚪\left(m\alpha +1\right)}\sum _{j=0}^{i-1}\sum _{\kappa =0}^{j}{c}_{\kappa }{\left(t-{t}_{\kappa }\right)}^{m\alpha -\alpha }{\int }_{\frac{{t}_{j}-{t}_{\kappa }}{t-{t}_{\kappa }}}^{\frac{{t}_{j+1}-{t}_{\kappa }}{t-{t}_{\kappa }}}{\left(1-w\right)}^{-\alpha }{w}^{m\alpha -1}𝑑w$$+\frac{1}{𝚪\left(1-\alpha \right)}\sum _{m=1}^{+\mathrm{\infty }}\frac{m\alpha {\lambda }^{m}}{𝚪\left(m\alpha +1\right)}\sum _{\kappa =0}^{i}{c}_{\kappa }{\left(t-{t}_{\kappa }\right)}^{\alpha m-\alpha }{\int }_{\frac{{t}_{i}-{y}_{\kappa }}{t-{t}_{\kappa }}}^{1}{\left(1-w\right)}^{-\alpha }{w}^{m\alpha -1}𝑑w+\sum _{m=0}^{+\mathrm{\infty }}{\lambda }^{m}{I}_{{0}^{+}}^{\alpha m}F\left(t\right)$$=\frac{1}{𝚪\left(1-\alpha \right)}\sum _{m=1}^{+\mathrm{\infty }}\frac{{\lambda }^{m}m\alpha }{𝚪\left(m\alpha +1\right)}\sum _{\kappa =0}^{i-1}{c}_{\kappa }{\left(t-{t}_{\kappa }\right)}^{m\alpha -\alpha }\sum _{j=\kappa }^{i-1}{\int }_{\frac{{t}_{j}-{t}_{\kappa }}{t-{t}_{\kappa }}}^{\frac{{t}_{j+1}-{t}_{\kappa }}{t-{t}_{\kappa }}}{\left(1-w\right)}^{-\alpha }{w}^{m\alpha -1}𝑑w$$+\frac{1}{𝚪\left(1-\alpha \right)}\sum _{m=1}^{+\mathrm{\infty }}\frac{m\alpha {\lambda }^{m}}{𝚪\left(m\alpha +1\right)}\sum _{\kappa =0}^{i-1}{c}_{\kappa }{\left(t-{t}_{\kappa }\right)}^{\alpha m-\alpha }{\int }_{\frac{{t}_{i}-{y}_{\kappa }}{t-{t}_{\kappa }}}^{1}{\left(1-w\right)}^{-\alpha }{w}^{m\alpha -1}𝑑w$$+\frac{{c}_{i}}{𝚪\left(1-\alpha \right)}\sum _{m=1}^{+\mathrm{\infty }}\frac{m\alpha {\lambda }^{m}}{𝚪\left(m\alpha +1\right)}{\int }_{0}^{1}{\left(1-w\right)}^{-\alpha }{w}^{m\alpha -1}𝑑w+f\left(t\right)$$+\lambda {\int }_{0}^{t}\sum _{m=1}^{+\mathrm{\infty }}{\left(t-s\right)}^{\alpha -1}\frac{{\lambda }^{m-1}{\left(t-s\right)}^{\alpha \left(m-1\right)}}{𝚪\left(\alpha m\right)}F\left(s\right)ds$$=\frac{1}{𝚪\left(1-\alpha \right)}\sum _{m=1}^{+\mathrm{\infty }}\frac{{\lambda }^{m}m\alpha }{𝚪\left(m\alpha +1\right)}\sum _{\kappa =0}^{i-1}{c}_{\kappa }{\left(t-{t}_{\kappa }\right)}^{m\alpha -\alpha }{\int }_{0}^{1}{\left(1-w\right)}^{-\alpha }{w}^{m\alpha -1}𝑑w$$+\frac{{c}_{i}}{𝚪\left(1-\alpha \right)}\sum _{m=1}^{+\mathrm{\infty }}\frac{m\alpha {\lambda }^{m}}{𝚪\left(m\alpha +1\right)}{\int }_{0}^{1}{\left(1-w\right)}^{-\alpha }{w}^{m\alpha -1}𝑑w+f\left(t\right)$$+\lambda {\int }_{0}^{t}\sum _{m=1}^{+\mathrm{\infty }}{\left(t-s\right)}^{\alpha -1}\frac{{\lambda }^{m-1}{\left(t-s\right)}^{\alpha \left(m-1\right)}}{𝚪\left(\alpha m\right)}F\left(s\right)ds$$=\lambda x\left(t\right)+F\left(t\right).$

We have done that x satisfies (3.2.1) if x satisfies (3.2.5).

Now, we suppose that x is a solution of (3.2.1). We will prove that x satisfies (3.2.5) by the mathematical induction method. Since x is continuous on $\left({t}_{i},{t}_{i+1}\right]$ and the limit ${lim}_{t\to {t}_{i}^{+}}x\left(t\right)$ ($i\in {ℕ}_{0}$) exists, it follows that $x\in {P}_{m}C\left(0,1\right]$. For $t\in \left({t}_{0},{t}_{1}\right]$, we know from Theorem 2 that there exists ${c}_{0}\in ℝ$ such that

$x\left(t\right)={c}_{0}{𝐄}_{\alpha ,1}\left(\lambda {t}^{\alpha }\right)+{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {\left(t-s\right)}^{\alpha }\right)F\left(s\right)𝑑s,t\in \left({t}_{0},{t}_{1}\right].$

Then (3.2.5) holds for $j=0$. We suppose that (3.2.5) holds for all $j=0,1,\mathrm{\dots },i$. We derive the expression of x on $\left({t}_{i+1},{t}_{i+2}\right]$. Suppose that

$x\left(t\right)=\mathrm{\Phi }\left(t\right)+\sum _{j=0}^{i}{c}_{j}{𝐄}_{\alpha ,1}\left(\lambda {\left(t-{t}_{j}\right)}^{\alpha }\right)+{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {\left(t-s\right)}^{\alpha }\right)F\left(s\right)𝑑s,t\in \left({t}_{i+1},{t}_{i+2}\right].$

By

${}^{C}D_{{0}^{+}}^{\alpha }x\left(t\right)-\lambda x\left(t\right)=f\left(t\right),t\in \left({t}_{i+1},{t}_{i+2}\right],$

we get

$F\left(t\right)+\lambda x\left(t\right)={}^{C}D_{{0}^{+}}^{\alpha }x\left(t\right)$$=\frac{1}{𝚪\left(1-\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{-\alpha }{x}^{\prime }\left(s\right)𝑑s$$=\frac{1}{𝚪\left(1-\alpha \right)}\left(\sum _{j=0}^{i}{\int }_{{t}_{j}}^{{t}_{j+1}}{\left(t-s\right)}^{-\alpha }{x}^{\prime }\left(s\right)𝑑s+{\int }_{{t}_{i+1}}^{t}{\left(t-s\right)}^{-\alpha }{x}^{\prime }\left(s\right)𝑑s\right)$$=\frac{1}{𝚪\left(1-\alpha \right)}\left[\sum _{j=0}^{i}{\int }_{{t}_{j}}^{{t}_{j+1}}{\left(t-s\right)}^{-\alpha }{\left(\sum _{v=0}^{j}{c}_{v}{𝐄}_{\alpha ,1}\left(\lambda {\left(s-{t}_{v}\right)}^{\alpha }\right)+{\int }_{0}^{s}{\left(s-u\right)}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {\left(s-u\right)}^{\alpha }\right)F\left(u\right)du\right)}^{\prime }ds$$+{\int }_{{t}_{i+1}}^{t}{\left(t-s\right)}^{-\alpha }{\left(\mathrm{\Phi }\left(s\right)+\sum _{v=0}^{i}{c}_{v}{𝐄}_{\alpha ,1}\left(\lambda {\left(s-{t}_{v}\right)}^{\alpha }\right)+{\int }_{0}^{s}{\left(s-u\right)}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {\left(s-u\right)}^{\alpha }\right)F\left(u\right)du\right)}^{\prime }ds\right]$$={}^{C}D_{{t}_{i+1}^{+}}^{\alpha }\mathrm{\Phi }\left(t\right)+\frac{1}{𝚪\left(1-\alpha \right)}\left[\sum _{j=0}^{i}{\int }_{{t}_{j}}^{{t}_{j+1}}{\left(t-s\right)}^{-\alpha }{\left(\sum _{v=0}^{j}{c}_{v}{𝐄}_{\alpha ,1}\left(\lambda {\left(s-{t}_{v}\right)}^{\alpha }\right)\right)}^{\prime }ds$$+{\int }_{{t}_{i+1}}^{t}{\left(t-s\right)}^{-\alpha }{\left(\sum _{v=0}^{i}{c}_{v}{𝐄}_{\alpha ,1}\left(\lambda {\left(s-{t}_{v}\right)}^{\alpha }\right)\right)}^{\prime }𝑑s$$+{\int }_{0}^{t}{\left(t-s\right)}^{-\alpha }{\left({\int }_{0}^{s}{\left(s-u\right)}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {\left(s-u\right)}^{\alpha }\right)F\left(u\right)du\right)}^{\prime }ds\right]$$={}^{C}D_{{t}_{i+1}^{+}}^{\alpha }\mathrm{\Phi }\left(t\right)+\frac{1}{𝚪\left(1-\alpha \right)}\left[\sum _{j=0}^{i}\sum _{v=0}^{j}{c}_{v}{\int }_{{t}_{j}}^{{t}_{j+1}}{\left(t-s\right)}^{-\alpha }{\left(\sum _{\iota =0}^{+\mathrm{\infty }}\frac{{\lambda }^{\iota }{\left(s-{t}_{v}\right)}^{\iota \alpha }}{\mathrm{\Gamma }\left(\alpha \iota +1\right)}\right)}^{\prime }ds$$+\sum _{v=0}^{i}{c}_{v}{\int }_{{t}_{i+1}}^{t}{\left(t-s\right)}^{-\alpha }{\left(\sum _{\iota =0}^{+\mathrm{\infty }}\frac{{\lambda }^{\iota }{\left(s-{t}_{v}\right)}^{\iota \alpha }}{\mathrm{\Gamma }\left(\alpha \iota +1\right)}\right)}^{\prime }ds\right]$$+\frac{1}{\mathrm{\Gamma }\left(2-\alpha \right)}{\left[{\int }_{0}^{t}{\left(t-s\right)}^{1-\alpha }{\left({\int }_{0}^{s}{\left(s-u\right)}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {\left(s-u\right)}^{\alpha }\right)F\left(u\right)𝑑u\right)}^{\prime }𝑑s\right]}^{\prime }$$={}^{C}D_{{t}_{i+1}^{+}}^{\alpha }\mathrm{\Phi }\left(t\right)+\frac{1}{𝚪\left(1-\alpha \right)}\left[\sum _{j=0}^{i}\sum _{v=0}^{j}{c}_{v}{\int }_{{t}_{j}}^{{t}_{j+1}}{\left(t-s\right)}^{-\alpha }\left(\sum _{\iota =1}^{+\mathrm{\infty }}\frac{\left(\iota \alpha \right){\lambda }^{\iota }{\left(s-{t}_{v}\right)}^{\iota \alpha -1}}{\mathrm{\Gamma }\left(\alpha \iota +1\right)}\right)ds$$+\sum _{v=0}^{i}{c}_{v}{\int }_{{t}_{i+1}}^{t}{\left(t-s\right)}^{-\alpha }\left(\sum _{\iota =1}^{+\mathrm{\infty }}\frac{\left(\alpha \iota \right){\lambda }^{\iota }{\left(s-{t}_{v}\right)}^{\iota \alpha -1}}{\mathrm{\Gamma }\left(\alpha \iota +1\right)}\right)ds\right]$$+\frac{1}{\mathrm{\Gamma }\left(2-\alpha \right)}{\left[{\left(t-s\right)}^{1-\alpha }\left({\int }_{0}^{s}{\left(s-u\right)}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {\left(s-u\right)}^{\alpha }\right)F\left(u\right)du\right)|}_{0}^{t}$$+\left(1-\alpha \right){\int }_{0}^{t}{\left(t-s\right)}^{-\alpha }{\int }_{0}^{s}{\left(s-u\right)}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {\left(s-u\right)}^{\alpha }\right)F\left(u\right)duds\right]{}^{\prime }$$={}^{C}D_{{t}_{i+1}^{+}}^{\alpha }\mathrm{\Phi }\left(t\right)+\frac{1}{𝚪\left(1-\alpha \right)}\left[\sum _{j=0}^{i}\sum _{v=0}^{j}{c}_{v}\sum _{\iota =1}^{+\mathrm{\infty }}\frac{{\lambda }^{\iota }}{\mathrm{\Gamma }\left(\alpha \iota \right)}{\int }_{{t}_{j}}^{{t}_{j+1}}{\left(t-s\right)}^{-\alpha }{\left(s-{t}_{v}\right)}^{\iota \alpha -1}ds$$+\sum _{v=0}^{i}{c}_{v}\sum _{\iota =1}^{+\mathrm{\infty }}\frac{{\lambda }^{\iota }}{\mathrm{\Gamma }\left(\alpha \iota \right)}{\int }_{{t}_{i+1}}^{t}{\left(t-s\right)}^{-\alpha }{\left(s-{t}_{v}\right)}^{\iota \alpha -1}ds\right]$$+\frac{1}{\mathrm{\Gamma }\left(2-\alpha \right)}{\left[\left(1-\alpha \right)\sum _{\iota =0}^{+\mathrm{\infty }}\frac{{\lambda }^{\iota }}{\mathrm{\Gamma }\left(\alpha \left(j+1\right)\right)}{\int }_{0}^{t}{\int }_{u}^{t}{\left(t-s\right)}^{-\alpha }{\left(s-u\right)}^{\alpha j+\alpha -1}𝑑sF\left(u\right)𝑑u\right]}^{\prime }$$={}^{C}D_{{t}_{i+1}^{+}}^{\alpha }\mathrm{\Phi }\left(t\right)+\frac{1}{𝚪\left(1-\alpha \right)}\left[\sum _{v=0}^{i}{c}_{v}\sum _{j=v}^{i}\sum _{\iota =1}^{+\mathrm{\infty }}\frac{{\lambda }^{\iota }{\left(t-tv\right)}^{\alpha \left(\iota -1\right)}}{\mathrm{\Gamma }\left(\alpha \iota \right)}{\int }_{\frac{{t}_{j}-{t}_{v}}{t-{t}_{v}}}^{\frac{{t}_{j+1}-{t}_{v}}{t-{t}_{v}}}{\left(1-w\right)}^{-\alpha }{w}^{\iota \alpha -1}dw$$+\sum _{v=0}^{i}{c}_{v}\sum _{\iota =1}^{+\mathrm{\infty }}\frac{{\lambda }^{\iota }{\left(t-{t}_{v}\right)}^{\alpha \left(\iota -1\right)}}{\mathrm{\Gamma }\left(\alpha \iota \right)}{\int }_{\frac{{t}_{i+1}-{t}_{v}}{t-{t}_{v}}}^{1}{\left(1-w\right)}^{-\alpha }{w}^{\iota \alpha -1}dw\right]$$+\frac{1}{\mathrm{\Gamma }\left(2-\alpha \right)}{\left[\left(1-\alpha \right)\sum _{\iota =0}^{+\mathrm{\infty }}\frac{{\lambda }^{\iota }}{\mathrm{\Gamma }\left(\alpha \left(j+1\right)\right)}{\int }_{0}^{t}{\left(t-u\right)}^{\alpha j}{\int }_{0}^{1}{\left(1-w\right)}^{-\alpha }{w}^{\alpha j+\alpha -1}𝑑wF\left(u\right)𝑑u\right]}^{\prime }$$={}^{C}D_{{t}_{i+1}^{+}}^{\alpha }\mathrm{\Phi }\left(t\right)+\frac{1}{𝚪\left(1-\alpha \right)}\left[\sum _{v=0}^{i}{c}_{v}\sum _{\iota =1}^{+\mathrm{\infty }}\frac{{\lambda }^{\iota }{\left(t-tv\right)}^{\alpha \left(\iota -1\right)}}{\mathrm{\Gamma }\left(\alpha \iota \right)}{\int }_{0}^{\frac{{t}_{i+1}-{t}_{v}}{t-{t}_{v}}}{\left(1-w\right)}^{-\alpha }{w}^{\iota \alpha -1}dw$$+\sum _{v=0}^{i}{c}_{v}\sum _{\iota =1}^{+\mathrm{\infty }}\frac{{\lambda }^{\iota }{\left(t-{t}_{v}\right)}^{\alpha \left(\iota -1\right)}}{\mathrm{\Gamma }\left(\alpha \iota \right)}{\int }_{\frac{{t}_{i+1}-{t}_{v}}{t-{t}_{v}}}^{1}{\left(1-w\right)}^{-\alpha }{w}^{\iota \alpha -1}dw\right]$$+{\left[\sum _{\iota =0}^{+\mathrm{\infty }}\frac{{\lambda }^{\iota }}{\mathrm{\Gamma }\left(\alpha j+1\right)}{\int }_{0}^{t}{\left(t-u\right)}^{\alpha j}F\left(u\right)𝑑u\right]}^{\prime }$$={}^{C}D_{{t}_{i+1}^{+}}^{\alpha }\mathrm{\Phi }\left(t\right)+\frac{1}{𝚪\left(1-\alpha \right)}\left[\sum _{v=0}^{i}{c}_{v}\sum _{\iota =1}^{+\mathrm{\infty }}\frac{{\lambda }^{\iota }{\left(t-tv\right)}^{\alpha \left(\iota -1\right)}}{\mathrm{\Gamma }\left(\alpha \iota \right)}{\int }_{0}^{1}{\left(1-w\right)}^{-\alpha }{w}^{\iota \alpha -1}𝑑w\right]$$+{\left[\sum _{\iota =0}^{+\mathrm{\infty }}\frac{{\lambda }^{\iota }}{\mathrm{\Gamma }\left(\alpha j+1\right)}{\int }_{0}^{t}{\left(t-u\right)}^{\alpha j}F\left(u\right)𝑑u\right]}^{\prime }$$={}^{C}D_{{t}_{i+1}^{+}}^{\alpha }\mathrm{\Phi }\left(t\right)+\sum _{v=0}^{i}{c}_{v}\sum _{\iota =1}^{+\mathrm{\infty }}\frac{{\lambda }^{\iota }{\left(t-tv\right)}^{\alpha \left(\iota -1\right)}}{\mathrm{\Gamma }\left(\alpha \left(\iota -1\right)+1\right)}+\sum _{\iota =1}^{+\mathrm{\infty }}\frac{{\lambda }^{\iota }}{\mathrm{\Gamma }\left(\alpha j\right)}{\int }_{0}^{t}{\left(t-u\right)}^{\alpha j-1}F\left(u\right)𝑑u$$=F\left(t\right)+\lambda x\left(t\right)+{}^{c}D_{{t}_{i+1}^{+}}^{\alpha }\mathrm{\Phi }\left(t\right)-\lambda \mathrm{\Phi }\left(t\right).$

Thus

${}^{C}D_{{t}_{i+1}^{+}}^{\alpha }\mathrm{\Phi }\left(t\right)-\lambda \mathrm{\Phi }\left(t\right)=0$

for all $t\in \left({t}_{i+1},{t}_{i+2}\right]$. By Theorem 2, we know that there exists a constant ${c}_{i+1}\in ℝ$ such that

$\mathrm{\Phi }\left(t\right)={c}_{i+1}{𝐄}_{\alpha ,1}\left(\lambda {\left(t-{t}_{i+1}\right)}^{\alpha }\right)$

for $t\in \left({t}_{i+1},{t}_{i+2}\right]$. Substituting Φ into (3.2.6), we get that (3.2.5) holds for $j=i+1$. Now suppose that (3.2.5) holds for all $j\in {ℕ}_{0}$. By the mathematical induction method, we know that x satisfies (3.2.5) and ${x|}_{\left({t}_{i},{t}_{i+1}\right]}$ is continuous and ${lim}_{t\to {t}_{i}^{+}}x\left(t\right)$ exists. ∎

#### Theorem 2.

Suppose that F is continuous on $\mathrm{\left(}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right)}$ and there exist constants $k\mathrm{>}\mathrm{-}\mathrm{1}$ and $l\mathrm{\in }\mathrm{\left(}\mathrm{-}\alpha \mathrm{,}\mathrm{-}\mathrm{1}\mathrm{-}k\mathrm{,}\mathrm{0}\mathrm{\right]}$ such that $\mathrm{|}F\mathit{}\mathrm{\left(}t\mathrm{\right)}\mathrm{|}\mathrm{\le }{t}^{k}\mathit{}{\mathrm{\left(}\mathrm{1}\mathrm{-}t\mathrm{\right)}}^{l}$ for all $t\mathrm{\in }\mathrm{\left(}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right)}$. Then x is a solution of (3.2.2) if and only if there exists constants ${c}_{i}\mathrm{\in }\mathrm{R}$ ($i\in {ℕ}_{0}$) such that

$x\left(t\right)=\sum _{v=0}^{j}{c}_{v}{\left(t-{t}_{v}\right)}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {\left(t-{t}_{v}\right)}^{\alpha }\right)+{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {\left(t-s\right)}^{\alpha }\right)F\left(s\right)𝑑s,t\in \left({t}_{j},{t}_{j+1}\right],j\in {ℕ}_{0}.$(3.2.7)

#### Proof.

For $t\in \left({t}_{j},{t}_{j+1}\right]$ ($j\in {ℕ}_{0}$), similarly to the beginning of the proof of Theorem 1 we know that

${t}^{1-\alpha }|{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {\left(t-s\right)}^{\alpha }\right)F\left(s\right)𝑑s|\le {\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {\left(t-s\right)}^{\alpha }\right)|F\left(s\right)|𝑑s$$\le {t}^{1-\alpha }{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {\left(t-s\right)}^{\alpha }\right){s}^{k}{\left(1-s\right)}^{l}𝑑s$$={t}^{1-\alpha }\sum _{j=0}^{+\mathrm{\infty }}\frac{{\lambda }^{j}}{𝚪\left(\left(j+1\right)\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}{\left(t-s\right)}^{\alpha j}{s}^{k}{\left(1-s\right)}^{l}𝑑s$$\le {t}^{1-\alpha }\sum _{j=0}^{+\mathrm{\infty }}\frac{{\lambda }^{j}}{𝚪\left(\left(j+1\right)\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha +l-1}{\left(t-s\right)}^{\alpha j}{s}^{k}𝑑s$$={t}^{1-\alpha }\sum _{j=0}^{+\mathrm{\infty }}\frac{{\lambda }^{j}}{𝚪\left(\left(j+1\right)\alpha \right)}{t}^{\alpha +\alpha j+k+l}{\int }_{0}^{1}{\left(1-w\right)}^{\alpha +\alpha j+l-1}{w}^{k}𝑑w$$\le {t}^{1-\alpha }\sum _{j=0}^{+\mathrm{\infty }}\frac{{\lambda }^{j}{t}^{\alpha j}}{𝚪\left(\left(j+1\right)\alpha \right)}{t}^{\alpha +k+l}{\int }_{0}^{1}{\left(1-w\right)}^{\alpha +l-1}{w}^{k}𝑑w$$={t}^{1+k+l}{𝐄}_{\alpha ,\alpha }\left(\lambda {t}^{\alpha }\right)𝐁\left(\alpha +l,k+1\right).$

So ${t}^{1-\alpha }{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {\left(t-s\right)}^{\alpha }\right)F\left(s\right)𝑑s$ is convergent and is continuous on $\left[0,1\right]$.

If x is a solution of (3.2.7), we have

$x\in {P}_{m}{C}_{1-\alpha }\left(0,1\right].$

It follows for $t\in \left({t}_{i},{t}_{i+1}\right]$ that

${}^{\mathrm{RL}}D_{{0}^{+}}^{\alpha }x\left(t\right)=\frac{1}{𝚪\left(1-\alpha \right)}{\left[{\int }_{0}^{t}{\left(t-s\right)}^{-\alpha }x\left(s\right)𝑑s\right]}^{\prime }$$=\frac{1}{𝚪\left(1-\alpha \right)}{\left[\sum _{j=0}^{i-1}{\int }_{{t}_{j}}^{{t}_{j+1}}{\left(t-s\right)}^{-\alpha }\left(\sum _{\kappa =0}^{j}{c}_{\kappa }{\left(s-{t}_{\kappa }\right)}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {\left(s-{t}_{\kappa }\right)}^{\alpha }\right)+{\int }_{0}^{s}{\left(s-v\right)}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {\left(s-v\right)}^{\alpha }\right)f\left(v\right)𝑑v\right)𝑑s\right]}^{\prime }$$+\frac{1}{𝚪\left(1-\alpha \right)}\left[{\int }_{{t}_{i}}^{t}{\left(t-s\right)}^{-\alpha }\left(\sum _{\kappa =0}^{i}{c}_{\kappa }{\left(t-{t}_{\kappa }\right)}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {\left(s-{t}_{\kappa }\right)}^{\alpha }\right)$$+{\int }_{0}^{s}{\left(s-v\right)}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {\left(s-v\right)}^{\alpha }\right)F\left(v\right)dv\right)ds\right]{}^{\prime }$$=\frac{1}{𝚪\left(1-\alpha \right)}{\left[\sum _{j=0}^{i-1}\sum _{\kappa =0}^{j}{c}_{\kappa }{\int }_{{t}_{j}}^{{t}_{j+1}}{\left(t-s\right)}^{-\alpha }{\left(s-{t}_{\kappa }\right)}^{\alpha -1}\sum _{m=0}^{+\mathrm{\infty }}\frac{{\lambda }^{m}{\left(s-{t}_{\kappa }\right)}^{\alpha m}}{𝚪\left(\alpha \left(m+1\right)\right)}ds\right]}^{\prime }$$+\frac{1}{𝚪\left(1-\alpha \right)}{\left[\sum _{\kappa =0}^{i}{c}_{\kappa }{\int }_{{t}_{i}}^{t}{\left(t-s\right)}^{-\alpha }{\left(t-{t}_{\kappa }\right)}^{\alpha -1}\sum _{m=0}^{+\mathrm{\infty }}\frac{{\lambda }^{m}{\left(s-{t}_{\kappa }\right)}^{\alpha m}}{𝚪\left(\alpha \left(m+1\right)\right)}ds\right]}^{\prime }$$+\frac{1}{𝚪\left(1-\alpha \right)}{\left[{\int }_{0}^{t}{\left(t-s\right)}^{-\alpha }{\int }_{0}^{s}{\left(s-v\right)}^{\alpha -1}\sum _{m=0}^{+\mathrm{\infty }}\frac{{\lambda }^{m}{\left(s-v\right)}^{\alpha m}}{𝚪\left(\alpha \left(m+1\right)\right)}F\left(v\right)dvds\right]}^{\prime }$$=\frac{1}{𝚪\left(1-\alpha \right)}{\left[\sum _{m=0}^{+\mathrm{\infty }}\frac{{\lambda }^{m}}{𝚪\left(\alpha \left(m+1\right)\right)}\sum _{j=0}^{i-1}\sum _{\kappa =0}^{j}{c}_{\kappa }{\int }_{{t}_{j}}^{{t}_{j+1}}{\left(t-s\right)}^{-\alpha }{\left(s-{t}_{\kappa }\right)}^{\alpha +\alpha m-1}𝑑s\right]}^{\prime }$$+\frac{1}{𝚪\left(1-\alpha \right)}{\left[\sum _{\kappa =0}^{i}{c}_{\kappa }\sum _{m=0}^{+\mathrm{\infty }}\frac{{\lambda }^{m}}{𝚪\left(\alpha \left(m+1\right)\right)}{\int }_{{t}_{i}}^{t}{\left(t-s\right)}^{-\alpha }{\left(t-{t}_{\kappa }\right)}^{\alpha +\alpha m-1}𝑑s\right]}^{\prime }$$+\frac{1}{𝚪\left(1-\alpha \right)}{\left[\sum _{m=0}^{+\mathrm{\infty }}{\lambda }^{m}{\int }_{0}^{t}{\left(t-s\right)}^{-\alpha }{\int }_{0}^{s}\frac{{\left(s-v\right)}^{\alpha +\alpha m-1}}{𝚪\left(\alpha \left(m+1\right)\right)}F\left(v\right)𝑑v𝑑s\right]}^{\prime }$$=\frac{1}{𝚪\left(1-\alpha \right)}{\left[\sum _{m=0}^{+\mathrm{\infty }}\frac{{\lambda }^{m}}{𝚪\left(\alpha \left(m+1\right)\right)}\sum _{\kappa =0}^{i-1}{c}_{\kappa }{\left(t-{t}_{\kappa }\right)}^{\alpha m}\sum _{j=\kappa }^{i-1}{\int }_{\frac{{t}_{j}-{t}_{\kappa }}{t-{t}_{\kappa }}}^{\frac{{t}_{j+1}-{t}_{\kappa }}{t-{t}_{\kappa }}}{\left(1-w\right)}^{-\alpha }{w}^{\alpha +\alpha m-1}𝑑w\right]}^{\prime }$$+\frac{1}{𝚪\left(1-\alpha \right)}{\left[\sum _{m=0}^{+\mathrm{\infty }}\sum _{\kappa =0}^{i}{c}_{\kappa }{\left(t-{t}_{\kappa }\right)}^{\alpha m}\frac{{\lambda }^{m}}{𝚪\left(\alpha \left(m+1\right)\right)}{\int }_{\frac{{t}_{i}-{t}_{\kappa }}{t-{t}_{\kappa }}}^{1}{\left(1-w\right)}^{-\alpha }{w}^{\alpha +\alpha m-1}𝑑w\right]}^{\prime }$$+\frac{1}{𝚪\left(1-\alpha \right)}{\left[\sum _{m=0}^{+\mathrm{\infty }}{\lambda }^{m}{\int }_{0}^{t}{\int }_{v}^{t}{\left(t-s\right)}^{-\alpha }\frac{{\left(s-v\right)}^{\alpha +\alpha m-1}}{𝚪\left(\alpha \left(m+1\right)\right)}𝑑sF\left(v\right)𝑑v\right]}^{\prime }$$=\frac{1}{𝚪\left(1-\alpha \right)}{\left[\sum _{m=0}^{+\mathrm{\infty }}\frac{{\lambda }^{m}}{𝚪\left(\alpha \left(m+1\right)\right)}\sum _{\kappa =0}^{i-1}{c}_{\kappa }{\left(t-{t}_{\kappa }\right)}^{\alpha m}{\int }_{0}^{\frac{{t}_{i}-{t}_{\kappa }}{t-{t}_{\kappa }}}{\left(1-w\right)}^{-\alpha }{w}^{\alpha +\alpha m-1}𝑑w\right]}^{\prime }$$+\frac{1}{𝚪\left(1-\alpha \right)}\left[\sum _{m=0}^{+\mathrm{\infty }}\sum _{\kappa =0}^{i-1}{c}_{\kappa }{\left(t-{t}_{\kappa }\right)}^{\alpha m}\frac{{\lambda }^{m}}{𝚪\left(\alpha \left(m+1\right)\right)}{\int }_{\frac{{t}_{i}-{t}_{\kappa }}{t-{t}_{\kappa }}}^{1}{\left(1-w\right)}^{-\alpha }{w}^{\alpha +\alpha m-1}dw$$+\sum _{m=0}^{+\mathrm{\infty }}{c}_{i}{\left(t-{t}_{i}\right)}^{\alpha m}\frac{{\lambda }^{m}}{𝚪\left(\alpha \left(m+1\right)\right)}{\int }_{0}^{1}{\left(1-w\right)}^{-\alpha }{w}^{\alpha +\alpha m-1}dw\right]{}^{\prime }$$+\frac{1}{𝚪\left(1-\alpha \right)}{\left[\sum _{m=0}^{+\mathrm{\infty }}{\lambda }^{m}{\int }_{0}^{t}{\left(t-v\right)}^{\alpha m}{\int }_{0}^{1}{\left(1-w\right)}^{-\alpha }\frac{{w}^{\alpha +\alpha m-1}}{𝚪\left(\alpha \left(m+1\right)\right)}𝑑wF\left(v\right)𝑑v\right]}^{\prime }$$=\frac{1}{𝚪\left(1-\alpha \right)}{\left[\sum _{\kappa =0}^{i-1}{c}_{\kappa }\sum _{m=0}^{+\mathrm{\infty }}{\left(t-{t}_{\kappa }\right)}^{\alpha m}\frac{{\lambda }^{m}}{𝚪\left(\alpha \left(m+1\right)\right)}{\int }_{0}^{1}{\left(1-w\right)}^{-\alpha }{w}^{\alpha +\alpha m-1}𝑑w\right]}^{\prime }$$+\frac{1}{𝚪\left(1-\alpha \right)}{\left[{c}_{i}\sum _{m=0}^{+\mathrm{\infty }}{\left(t-{t}_{i}\right)}^{\alpha m}\frac{{\lambda }^{m}}{𝚪\left(\alpha \left(m+1\right)\right)}{\int }_{0}^{1}{\left(1-w\right)}^{-\alpha }{w}^{\alpha +\alpha m-1}𝑑w\right]}^{\prime }$$+\frac{1}{𝚪\left(1-\alpha \right)}{\left[\sum _{m=0}^{+\mathrm{\infty }}{\lambda }^{m}{\int }_{0}^{t}{\left(t-v\right)}^{\alpha m}{\int }_{0}^{1}{\left(1-w\right)}^{-\alpha }\frac{{w}^{\alpha +\alpha m-1}}{𝚪\left(\alpha \left(m+1\right)\right)}𝑑wF\left(v\right)𝑑v\right]}^{\prime }$$=\frac{1}{𝚪\left(1-\alpha \right)}{\left[\sum _{\kappa =0}^{i-1}{c}_{\kappa }\sum _{m=0}^{+\mathrm{\infty }}{\left(t-{t}_{\kappa }\right)}^{\alpha m}\frac{{\lambda }^{m}}{𝚪\left(\alpha \left(m+1\right)\right)}\frac{\mathrm{\Gamma }\left(1-\alpha \right)\mathrm{\Gamma }\left(\alpha \left(m+1\right)\right)}{\mathrm{\Gamma }\left(\alpha m\right)}\right]}^{\prime }$$+\frac{1}{𝚪\left(1-\alpha \right)}{\left[{c}_{i}\sum _{m=0}^{+\mathrm{\infty }}{\left(t-{t}_{i}\right)}^{\alpha m}\frac{{\lambda }^{m}}{𝚪\left(\alpha \left(m+1\right)\right)}\frac{\mathrm{\Gamma }\left(1-\alpha \right)\mathrm{\Gamma }\left(\alpha \left(m+1\right)\right)}{\mathrm{\Gamma }\left(\alpha m\right)}\right]}^{\prime }$$+\frac{1}{𝚪\left(1-\alpha \right)}{\left[\sum _{m=0}^{+\mathrm{\infty }}\frac{{\lambda }^{m}}{𝚪\left(\alpha \left(m+1\right)\right)}\frac{\mathrm{\Gamma }\left(1-\alpha \right)\mathrm{\Gamma }\left(\alpha \left(m+1\right)\right)}{\mathrm{\Gamma }\left(\alpha m\right)}{\int }_{0}^{t}{\left(t-v\right)}^{\alpha m}F\left(v\right)𝑑v\right]}^{\prime }$$=\sum _{\kappa =0}^{i-1}{c}_{\kappa }\sum _{m=1}{}^{+\mathrm{\infty }}\left(t-{t}_{\kappa }\right)^{\alpha m-1}\frac{\left(\alpha m\right){\lambda }^{m}}{𝚪\left(\alpha m\right)}+{c}_{i}\sum _{m=1}^{+\mathrm{\infty }}{\left(t-{t}_{i}\right)}^{\alpha m-1}\frac{\left(\alpha m\right){\lambda }^{m}}{𝚪\left(\alpha m\right)}$$+\sum _{m=1}^{+\mathrm{\infty }}\frac{\left(\alpha m\right){\lambda }^{m}}{𝚪\left(\alpha m\right)}{\int }_{0}^{t}{\left(t-v\right)}^{\alpha m-1}F\left(v\right)𝑑v+F\left(t\right)$$=F\left(t\right)+\lambda \sum _{\kappa =0}^{i}{c}_{\kappa }{\left(t-{t}_{\kappa }\right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(\lambda {\left(t-{t}_{\kappa }\right)}^{\alpha }\right)+\sum _{m=1}^{+\mathrm{\infty }}\frac{\left(\alpha m\right){\lambda }^{m}}{𝚪\left(\alpha \left(m+1\right)\right)}{\int }_{0}^{t}{\left(t-v\right)}^{\alpha m-1}F\left(v\right)𝑑v$$=F\left(t\right)+\lambda \sum _{\kappa =0}^{i}{c}_{\kappa }{\left(t-{t}_{\kappa }\right)}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {\left(t-{t}_{\kappa }\right)}^{\alpha }\right)+\lambda {\int }_{0}^{t}{\left(t-v\right)}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {\left(t-v\right)}^{\alpha }\right)F\left(v\right)𝑑v$$=\lambda x\left(t\right)+F\left(t\right),t\in \left({t}_{i},{t}_{i+1}\right].$

It follows that x is a solution of (3.2.2).

Now we prove that if x is a solution of (3.2.2), then x satisfies (3.2.7) and $x\in {P}_{m}{C}_{1-\alpha }\left(0,1\right]$ by the mathematical induction method. By Theorem 4 we know that there exists a constant ${c}_{0}\in ℝ$ such that

$x\left(t\right)={c}_{0}{t}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {t}^{\alpha }\right)+{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {\left(t-s\right)}^{\alpha }\right)F\left(s\right)𝑑s,t\in \left({t}_{0},{t}_{1}\right].$

Hence (3.2.7) holds for $j=0$. Assume that (3.2.7) holds for $j=0,1,2,\mathrm{\dots },i\le m$, we will prove that (3.2.7) holds for $j=i+1$. Suppose that

$x\left(t\right)=\mathrm{\Phi }\left(t\right)+\sum _{j=0}^{i}{c}_{j}{\left(t-{t}_{j}\right)}^{\alpha -1}{𝐄}_{\alpha ,\alpha }\left(\lambda {\left(t-{t}_{j}\right)}^{\alpha }\right)+$