# Nonlinear boundary value problems for mixed-type fractional equations and Ulam-Hyers stability

Huiwen Wang 1  and Fang Li 1
• 1 School of Mathematics, Yunnan Normal University, 650092, Kunming, People's Republic of China
Huiwen Wang
• School of Mathematics, Yunnan Normal University, Kunming, 650092, People's Republic of China
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and Fang Li
• Corresponding author
• School of Mathematics, Yunnan Normal University, Kunming, 650092, People's Republic of China
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## Abstract

In this article, we discuss the nonlinear boundary value problems involving both left Riemann-Liouville and right Caputo-type fractional derivatives. By using some new techniques and properties of the Mittag-Leffler functions, we introduce a formula of the solutions for the aforementioned problems, which can be regarded as a novelty item. Moreover, we obtain the existence result of solutions for the aforementioned problems and present the Ulam-Hyers stability of the fractional differential equation involving two different fractional derivatives. An example is given to illustrate our theoretical result.

## 1 Introduction

The fractional differential equations can model some engineering and scientific disciplines in the fields of physics, chemistry, electrodynamics of complex medium, polymer rheology, etc. In particular, the forward and backward fractional derivatives provide an excellent tool for the description of some physical phenomena such as the fractional oscillator equations and the fractional Euler-Lagrange equations [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. Recently, a linear boundary value problem (BVP) involving both the right Caputo and the left Riemann-Liouville fractional derivatives has been studied by some authors [10,11,12,13,14].

On the other hand, Ulam’s stability problem [15] has been attracted by many famous researchers (see [16,17] and references therein). Recently, studying the stability of Ulam-Hyers for fractional differential equations is gaining much importance and attention [18,19]. However, the Ulam-Hyers stabilities of differential equations involving with the forward and backward fractional derivatives have not yet been investigated.

In this article, we investigate the following BVP of the fractional differential equation with two different fractional derivatives:

$D1−βc(D0+αL+λ)u(t)=f(t,u(t)),t∈J≔(0,1],$
$limt→0+t1−αu(t)=u0,(I0+1−αu)(0)+(I0+qρu)(1)=0,$
where $α,β,α+β∈(0,1)$, $λ,ρ,q>0$, $α+ρ>1$. $D1−βc$ is the right Caputo fractional derivative of order $β$, $D0+αL$ is the left Riemann-Liouville fractional derivative of order $α$, $I0+1−α$ is the Riemann-Liouville fractional integral and $I0+qρ$ is the Katugampola fractional integral.

The rest of this article is organized as follows. In Section 2, we collect some concepts of fractional calculus. In Section 3, we prove some properties of classical and generalized Mittag-Leffler functions. In Section 4, we present the definition of solution to (1.1) and (1.2). In Section 5, we obtain the existence and uniqueness of solutions to problem (1.1) and (1.2). In Section 6, we present the Ulam-Hyers stability result for Eq. (1.1). An example is given in Section 7 to demonstrate the application of our result.

## 2 Preliminaries

In this section, we introduce some notations and definitions of fractional calculus. Throughout this article, we denote by $C(J,ℝ)$ the Banach space of all continuous functions from J to $ℝ$, $Lp(J,ℝ)$ the Banach space of all Lebesgue measurable functions $l:J→ℝ$ with the norm $∥l∥Lp=∫J|l(t)|pdt1p<∞$ and $AC([a,b],ℝ)$ the space of all the absolutely continuous functions defined on $[a,b]$.

Definition 2.1

[3,4] The left-sided and the right-sided fractional integrals of order γ for a function $x(t)∈L1$ are defined as

$(Ia+γx)(t)=1Γ(γ)∫at(t−s)γ−1x(s)ds,t>a,γ>0,(Ib−γx)(t)=1Γ(γ)∫tb(s−t)γ−1x(s)ds,t0,$
respectively. Here, $Γ(⋅)$ is the gamma function.

Definition 2.2

[3,4] If $x(t)∈AC([a,b],ℝ)$, then the left-sided and the right-sided Riemann-Liouville fractional derivatives $Da+γLx(t)$ and $Db−γLx(t)$ of order γ exist almost everywhere on $[a,b]$ and can be written as

$(Da+γLx)(t)=1Γ(1−γ)ddt∫at(t−s)−γx(s)ds=ddt(Ia+1−γx)(t),t>a,0<γ<1,(Db−γLx)(t)=1Γ(1−γ)−ddt∫tb(s−t)−γx(s)ds=−ddt(Ib−1−γx)(t),t
respectively.

Definition 2.3

[3,4] If $x(t)∈AC([a,b],ℝ)$, then the right-sided Caputo derivative $Db−γcx(t)$ of order γ exists almost everywhere on $[a,b]$ and can be written as

$(Db−γcx)(t)=(Db−γL[x(s)−x(b)])(t),t

Definition 2.4

[5] For $ρ,q>0$, then the Katugampola fractional integral of $y(t)$ can be defined as

$(Ia+qρy)(t)=ρ1−qΓ(q)∫at(tρ−τρ)q−1τρ−1y(τ)dτ,t>a,$
if the integral exists.

## 3 Properties of Mittag-Leffler functions

In this section, we prove some properties of the Mittag-Leffler functions.

Definition 3.1

[3,4] For $μ,ν>0,z∈ℝ$, the classical Mittag-Leffler functions $Eμ(z)$ and the generalized Mittag-Leffler functions $Eμ,ν(z)$ are defined by

$Eμ(z)=∑k=0∞zkΓ(μk+1),Eμ,ν(z)=∑k=0∞zkΓ(μk+ν).$
Clearly, $Eμ,1(z)=Eμ(z)$.

Lemma 3.2

[4,9] Let $α∈(0,1)$, $θ>α$ be arbitrary. The functions $Eα$, $Eα,α$ and $Eα,θ$ are nonnegative and have the following properties:

1. (i) For any $t∈J$, $Eα(−λtα)≤1$, $Eα,α(−λtα)≤1Γ(α)$, $Eα,θ(−λtα)≤1Γ(θ)$.
2. (ii) For any $t1,t2∈J$,
$|Eα(−λt2α)−Eα(−λt1α)|=O(|t2−t1|α),ast2→t1,|Eα,α(−λt2α)−Eα,α(−λt1α)|=O(|t2−t1|α),ast2→t1,|Eα,α+1(−λt2α)−Eα,α+1(−λt1α)|=O(|t2−t1|α),ast2→t1.$

Lemma 3.3

[4,9] For $γ,μ,ν,λ>0,t>0$, $0<α,β<1$ , the usual derivatives of $Eμ,ν$ and the fractional integrals and derivatives of $Eμ,ν$ are expressed by

1. (1) $ddtEμ(−λtμ)=−λtμ−1Eμ,μ(−λtμ);$
2. (2) $I0+γ(sν−1Eμ,ν(−λsμ))(t)=1Γ(γ)∫0t(t−s)γ−1sν−1Eμ,ν(−λsμ)ds=tγ+ν−1Eμ,γ+ν(−λtμ);$
3. (3) $[Da+νL(s−a)β−1Eα,β(−λ(s−a)α)](t)=(t−a)β−ν−1Eα,β−ν(−λ(t−a)α).$

Lemma 3.4

For $ν,λ>0,t>0$, $0<α,β<1$ , the fractional integrals and derivatives of $Eμ,ν$ are expressed by

1. (1) $[Da+αL(s−a)α−1Eα,α(−λ(s−a)α)](t)=−λ(t−a)α−1Eα,α(−λ(t−a)α);$
2. (2) $[D1−βcD0+αLsαEα,α+1(−λsα)](t)+λ[D1−βcsαEα,α+1(−λsα)](t)=0;$
3. (3) $[D1−βcD0+αLsα−1Eα,α(−λsα)](t)+λ[D1−βcsα−1Eα,α(−λsα)](t)=0;$
4. (4) $[I0+qρsν−1Eα,ν(−λsα)](t)=tqρ+ν−1ρqΓ(q)∫01sν−1ρ(1−s)q−1Eα,ν(−λtαsαρ)ds$, $ν≥α.$

Proof

1. (1)By using Lemma 3.3, we find
$[Da+αL(s−a)α−1Eα,α(−λ(s−a)α)](t)=1Γ(1−α)ddt∫at(t−s)−α(s−a)α−1Eα,α(−λ(s−a)α)ds=ddt[Eα(−λ(t−a)α)]=−λ(t−a)α−1Eα,α(−λ(t−a)α).$
Similarly, we have
$[Da+αL(s−a)αEα,α+1(−λ(s−a)α)](t)=Eα(−λ(t−a)α).$
2. (2)Using (3.1) and $(D1−βc1)(t)=0$, we arrive at
$[D1−βcD0+αLsαEα,α+1(−λsα)](t)+λ[D1−βcsαEα,α+1(−λsα)](t)=[D1−βc(Eα(−λsα)+λsαEα,α+1(−λsα))](t)=(D1−βc1)(t)=0.$
3. (3)The proof of (3) is similar to that of (2).
4. (4)Clearly, for $ν≥α$, the integral $∫0t(1−τ)q−1τν−1ρEα,ν(−λtαταρ)dτ$ exists, then we get
$[I0+qρsν−1Eα,ν(−λsα)](t)=ρ1−qΓ(q)∫0t(tρ−sρ)q−1sρ+ν−2Eα,ν(−λsα)ds=ρ1−qtρq+ν−1ρΓ(q)∫01(1−τ)q−1τν−1ρEα,ν(−λtαταρ)dτ=tρq+ν−1ρqΓ(q)∫01(1−τ)q−1τν−1ρEα,ν(−λtαταρ)dτ.$

## 4 Solutions for BVP

In this section, we present the formulas of solutions to problem (1.1) and (1.2).

Lemma 4.1

[4] For $θ>0$ , a general solution of the fractional differential equation $D1−θcu(t)=0$ is given by

$u(t)=∑i=0n−1Ci(1−t)i,$
where $Ci∈ℝ,i=0,1,2,…,n−1(n=[θ]+1)$ , and $[θ]$ denotes the integer part of the real number $θ$.

Lemma 4.2

For $α,β∈(0,1)$, $D1−βc(D0+αL+λ)u(t)=h(t),t∈J$ , then

$u(t)=C0tαEα,α+1(−λtα)+C1tα−1Eα,α(−λtα)+∫01K(t,τ)h(τ)dτ,t∈J,$
if the integral exists. Here,
$K(t,τ)=1Γ(β)∫0τ(t−s)α−1(τ−s)β−1Eα,α(−λ(t−s)α)ds,0<τ

Formally, by Lemma 4.1, for $C0∈ℝ$, we obtain $(D0+αL+λ)u(t)=C0+(I1−βh)(t)$. Based on the arguments of [4], we derive

$u(t)=C1tα−1Eα,α(−λtα)+∫0t(t−s)α−1Eα,α(−λ(t−s)α)(C0+(I1−βh)(s))ds=C0tαEα,α+1(−λtα)+C1tα−1Eα,α(−λtα)+1Γ(β)∫0t(t−s)α−1Eα,α(−λ(t−s)α)∫s1(τ−s)β−1h(τ)dτds=C0tαEα,α+1(−λtα)+C1tα−1Eα,α(−λtα)+1Γ(β)∫0th(τ)dτ∫0τ(t−s)α−1(τ−s)β−1Eα,α(−λ(t−s)α)ds+∫t1h(τ)dτ∫0t(t−s)α−1(τ−s)β−1Eα,α(−λ(t−s)α)ds=C0tαEα,α+1(−λtα)+C1tα−1Eα,α(−λtα)+∫01K(t,τ)h(τ)dτ.$
We define $C1−α([0,1],ℝ)={u∈C(J,ℝ):t1−αu(t)∈C([0,1],ℝ)}$ with the norm $∥u∥1−α=maxt∈[0,1]t1−α|u(t)|$ and we abbreviate $C1−α([0,1],ℝ)$ to $C1−α$.

To prove our results, we make the following assumptions.

1. (H1)Let $f:J×ℝ→ℝ$ be a function such that $f(⋅,u):J→ℝ$ is measurable for all $u∈ℝ$ and $f(t,⋅):ℝ→ℝ$ is continuous for a.e. $t∈J$, and there exists a function $φ∈L1p(J,ℝ+)(0) such that
$|f(t,u(t))−f(t,v(t))|≤φ(t)∥u−v∥1−α.$
2. (H2) $supt∈J|f(t,0)|<∞$.

For convenience of the following presentation, we set

$A(θ,t)=tqρ+θρqΓ(q)∫01sθρ(1−s)q−1Eα,θ+1(−λtαsαρ)ds;Nθ=Bθρ+1,qρqΓ(q)1Γ(θ+1);M1=1ρqΓ(q+1)Γ(α+1)A(α,1)+1;M2=1β1−pα−p1−p+1α1−pβ−p1−p∥φ∥L1pΓ(α)Γ(β);M3=2Γ(α+1)Γ(β+1),$
where $B(⋅,⋅)$ is the Beta function.

For $t∈J,y>p$, applying the Hölder inequality, there hold the following estimates:

$∫at(t−s)y−1φ(s)ds≤∫at(t−s)y−11−pds1−p∥φ∥L1p=1−py−p1−p(t−a)y−p∥φ∥L1p,t>a,$
$∫tb(s−t)y−1φ(s)ds≤∫tb(s−t)y−11−pds1−p∥φ∥L1p=1−py−p1−p(b−t)y−p∥φ∥L1p,t

Lemma 4.3

$∫0τ[(t1−s)α−1Eα,α(−λ(t1−s)α)−(t2−s)α−1Eα,α(−λ(t2−s)α)](τ−s)β−1ds≤O((t2−t1)α),0<τ
$∫0t1(t1−s)α−1−(t2−s)α−1(τ−s)β−1ds≤(τ−t2)β−1O((t2−t1)α),0

Proof

For $0, it follows from Lemma 3.2 and the mean value theorem that

$|t1α−1Eα,α(−λt1α)−t2α−1Eα,α(−λt2α)|≤|t1α−1−t2α−1|Eα,α(−λt1α)+|Eα,α(−λt1α)−Eα,α(−λt2α)|t2α−1≔O((t2−t1)α),$
which yields
$∫0τ[(t1−s)α−1Eα,α(−λ(t1−s)α)−(t2−s)α−1Eα,α(−λ(t2−s)α)](τ−s)β−1ds≤∫0τ(τ−s)β−1ds⋅O((t2−t1)α)≔O((t2−t1)α),for0<τ

For $0, by the mean value theorem, one can see that

$∫0t1[(t1−s)α−1−(t2−s)α−1](τ−s)β−1ds≤∫0t1[(t1−s)α−1−(t2−s)α−1]ds⋅(τ−t1)β−1≤1α[t2α−t1α+(t2−t1)α]⋅(τ−t1)β−1≤(τ−t2)β−1⋅O((t2−t1)α).$

Let us define

$k1(t,τ)=1Γ(β)∫0τ(t−s)α−1(τ−s)β−1Eα,α(−λ(t−s)α)ds,0<τ
clearly,
$∫01K(t,τ)dτ=∫0tk1(t,τ)dτ+∫t1k2(t,τ)dτ,$
one can obtain the following estimates.

Lemma 4.4

$|k1(t,τ)|≤(t−τ)α−1Γ(β+1)Γ(α),0<τ
$|k2(t,τ)|≤(τ−t)β−1Γ(α+1)Γ(β),0
$∫01|K(t2,τ)−K(t1,τ)|φ(τ)dτ=O((t2−t1)α−p)+O((t2−t1)β−p),$
$∫01|K(t2,τ)−K(t1,τ)|dτ=O((t2−t1)α)+O((t2−t1)β),$
$∫01K(t,τ)φ(τ)dτ≤M2,$
$∫01K(t,τ)dτ≤M3.$

Proof

Since

$∫0τ(t−s)α−1(τ−s)β−1ds≤∫0τ(t−τ)α−1(τ−s)β−1ds=τβ(t−τ)α−1β,0<τ
(4.5) and (4.6) hold. Moreover, (4.1) and (4.2) imply
$∫t1t2k1(t2,s)φ(s)ds≤∫t1t2(t2−s)α−1φ(s)Γ(β+1)Γ(α)ds≤1−pα−p1−p(t2−t1)α−pΓ(β+1)Γ(α)∥φ∥L1p,$
$∫t1t2k2(t1,s)φ(s)ds≤∫t1t2(s−t1)β−1φ(s)Γ(α+1)Γ(β)ds≤1−pβ−p1−p(t2−t1)β−pΓ(α+1)Γ(β)∥φ∥L1p.$
For $0<τ, by (4.3), we get
$|k1(t2,τ)−k1(t1,τ)|≤1Γ(β)∫0τ[(t2−s)α−1Eα,α(−λ(t2−s)α)−(t1−s)α−1Eα,α(−λ(t1−s)α)](τ−s)β−1ds=O((t2−t1)α),$
furthermore,
$∫0t1|k1(t2,τ)−k1(t1,τ)|φ(τ)dτ≤∥φ∥L1⋅O((t2−t1)α)≔O((t2−t1)α),$
$∫0t1|k1(t2,τ)−k1(t1,τ)|dτ≔O((t2−t1)α).$

For $0, by (4.4) and Lemma 3.2, we find

$|k2(t2,τ)−k2(t1,τ)|≤1Γ(β)1Γ(α)∫0t1[(t1−s)α−1−(t2−s)α−1](τ−s)β−1ds+∫0t1(t2−s)α−1(τ−s)β−1|Eα,α(−λ(t1−s)α)−Eα,α(−λ(t2−s)α)|ds+1Γ(α)∫t1t2(t2−s)α−1(τ−s)β−1ds≤(τ−t2)β−1⋅O((t2−t1)α)+(τ−t2)β−1Γ(α)Γ(β)∫t1t2(t2−s)α−1ds≔(τ−t2)β−1⋅O((t2−t1)α),$
then by (4.2), we arrive at
$∫t21|k2(t2,τ)−k2(t1,τ)|φ(τ)dτ≤∫t21(τ−t2)β−1φ(τ)dτ⋅O((t2−t1)α)≔O((t2−t1)α),$
$∫t21|k2(t2,τ)−k2(t1,τ)|dτ≔O((t2−t1)α).$

From (4.13), (4.15), (4.11) and (4.12), it follows that

$∫01|K(t2,τ)−K(t1,τ)|φ(τ)dτ≤∫0t1|k1(t2,s)−k1(t1,s)|φ(s)ds+∫t1t2k1(t2,s)φ(s)ds+∫t21|k2(t2,s)−k2(t1,s)|φ(s)ds+∫t1t2k2(t1,s)φ(s)ds=O((t2−t1)α−p)+O((t2−t1)β−p).$
Similarly, one can conclude from (4.14), (4.16), (4.5) and (4.6) that
$∫01|K(t2,τ)−K(t1,τ)|dτ≔O((t2−t1)α)+O((t2−t1)β).$
By (4.5), (4.6), (4.1) and (4.2), we observe that
$∫01K(t,τ)φ(τ)dτ=∫0tk1(t,τ)φ(τ)dτ+∫t1k2(t,τ)φ(τ)dτ≤M2,∫01K(t,τ)dτ≤M3,∫01K(t,τ)|f(τ,0)|dτ≤∫01K(t,τ)dτ⋅supt∈J|f(t,0)|≤M3supt∈J|f(t,0)|.□$

For the sake of convenience, we adopt the following notation:

$(Fu)(t)=∫01K(t,τ)f(τ,u(τ))dτ=∫0tk1(t,τ)f(τ,u(τ))dτ+∫t1k2(t,τ)f(τ,u(τ))dτ.$
Since
$|f(t,u(t))|≤|f(t,u(t))−f(t,0)|+|f(t,0)|≤φ(t)∥u∥1−α+|f(t,0)|,$
then by (4.9) and (4.10),
$|(Fu)(t))|≤∫01K(t,τ)φ(τ)dτ⋅∥u∥1−α+∫01K(t,τ)|f(τ,0)|dτ≤M2∥u∥1−α+M3supt∈J|f(t,0)|.$

Lemma 4.5

Assume that (H1) and (H2) hold. For $u∈C1−α$, $t∈J$, $(Fu)(t)$ satisfies the following relations:

1. (1) $(Fu)(t)∈AC(J,ℝ);$
2. (2) $[D0+αL(Fu)](t)=−λ(Fu)(t)+(I1−βf)(t);$
3. (3) $[D1−βcD0+αL(Fu)](t)+λ[D1−βc(Fu)](t)=f(t,u(t));$
4. (4) $[I0+qρ(Fu)](t)=tqρρqΓ(q)∫01(1−s)q−1(Fu)(ts1ρ)ds$.

Proof

For every finite collection ${(aj,bj)}1≤j≤n$ on J with $∑j=1n(bj−aj)→0$, noting that (4.7) and (4.8), we derive

$∑j=1n|(Fu)(bj)−(Fu)(aj)|=∑j=1n∫01K(bj,τ)f(τ,u(τ))dτ−∫01K(aj,τ)f(τ,u(τ))dτ≤∑j=1n∫01|K(bj,τ)−K(aj,τ)|φ(τ)dτ⋅∥u∥1−α+∑j=1n∫01|K(bj,τ)−K(aj,τ)|dτ⋅supt∈J|f(t,0)|→0.$
Hence, $(Fu)(t)$ is absolutely continuous on J. Furthermore, for almost all $t∈J$, $[LD0+α(Fu)(s)](t)$ exists and from the fact $(Fu)(s)=∫0s(s−τ)α−1Eα,α(−λ(s−τ)α)(I1−βf)(τ,u(τ))dτ$ and Lemma 3.3, we get
$[D0+αL(Fu)(s)](t)=1Γ(1−α)ddt∫0t∫0s(t−s)−α(s−τ)α−1Eα,α(−λ(s−τ)α)(I1−βf)(τ)dτds=1Γ(1−α)ddt∫0t(I1−βf)(τ)∫τt(t−s)−α(s−τ)α−1Eα,α(−λ(s−τ)α)dsdτ=ddt∫0t(I1−βf)(τ)Eα(−λ(t−τ)α)dτ=−λ(Fu)(t)+(I1−βf)(t).$
Thus,
$[D1−βcD0+αL(Fu)(s)](t)+λ[D1−βc(Fu)(s)](t)=[D1−βc(−λ(Fu)(s)+(I1−βf)(s))](t)+λ[D1−βc(Fu)(s)](t)=f(t,u(t)).$

It follows from (4.17) that $∫01(1−s)q−1(Fu)(ts1ρ)ds$ exists and

$[I0+qρ(Fu)](t)=ρ1−qΓ(q)∫0t(tρ−sρ)q−1sρ−1(Fu)(s)ds=tqρρqΓ(q)∫01(1−s)q−1(Fu)(ts1ρ)ds.□$

Lemma 4.6

Assume that (H1) and (H2) hold. A function u is a solution of the following fractional integral equation:

$u(t)=(Pu)(t)+(Qu)(t)$
if and only if u is a solution of problem (1.1 ) and ( 1.2), where
$(Pu)(t)=−u0Γ(α)(1+A(α−1,1))A(α,1)tαEα,α+1(−λtα)+u0Γ(α)tα−1Eα,α(−λtα),(Qu)(t)=−1ρqΓ(q)∫01(1−τ)q−1(Fu)(τ1ρ)dτA(α,1)tαEα,α+1(−λtα)+(Fu)(t).$

Proof

(Sufficiency) Let u be the solution of (1.1) and (1.2), Lemmas 3.3, 3.4, 4.2 and 4.5 imply

$u(t)=atαEα,α+1(−λtα)+btα−1Eα,α(−λtα)+(Fu)(t),t∈J,(I0+1−αu)(t)=atEα,2(−λtα)+bEα(−λtα)+[I0+1−α(Fu)](t),[I0+qρu](t)=aA(α,t)+bA(α−1,t)+tqρρqΓ(q)∫01(1−s)q−1(Fu)(ts1ρ)ds,$
where $a,b$ are constants. Using the boundary value condition (1.2), we derive that $b=u0Γ(α)$ and
$aA(α,1)+u0Γ(α)(1+A(α−1,1))+1ρqΓ(q)∫01(1−τ)q−1(Fu)(τ1ρ)dτ=0,$
which means
$a=−u0Γ(α)(1+A(α−1,1))+1ρqΓ(q)∫01(1−τ)q−1(Fu)(τ1ρ)dτA(α,1).$
Now we can see that (4.18) holds.

(Necessity) Let u satisfy (4.18). According to Lemma 3.4(2), (3) and Lemma 4.5(3), $[D1−βcD0+αLu](t)$ exists and $D1−βc(D0+αL+λ)u(t)=f(t,u(t))$ for $t∈J$. Clearly, the boundary value condition (1.2) holds and hence the necessity is proved.□

## 5 Existence and uniqueness result

In this section, we deal with the existence and uniqueness of solutions to problem (1.1) and (1.2).

Theorem 5.1

Assume that (H1) and (H2) are satisfied, then problem (1.1 ) and ( 1.2) has a unique solution $u∈C1−α$ if $M1M2<1$.

Proof

We consider an operator $ℱ:C1−α→C1−α$ defined by

$(ℱu)(t)=(Pu)(t)+(Qu)(t).$
Clearly, $ℱ$ is well defined and the fixed point of $ℱ$ is the solution of problem (1.1) and (1.2).

Let $Br={u∈C1−α:∥u∥1−α≤r}$ be a bounded set in $C1−α$, where

$r≥|u0|α1+Nα−1A(α,1)+α+M1M3supt∈J|f(t,0)|1−M1M2.$

For $u∈Br$, it follows from Lemmas 3.2(i) and (4.17) that

$∥Pu∥1−α≤|u0|α1+Nα−1A(α,1)+α,∥Qu∥1−α≤M1⋅[M2r+M3supt∈J|f(t,0)|]$
and hence
$∥(ℱu)∥1−α≤∥Pu∥1−α+∥Qu∥1−α≤|u0|α1+Nα−1A(α,1)+α+M1⋅[M2r+M3supt∈J|f(t,0)|]≤r.$
Then for $t∈J$, $u∈Br$, $ℱu∈Br$.

For any $u,v∈Br$, $t∈J$, by (4.9),

$|(Fu)(t)−(Fv)(t)|≤∫01K(t,τ)|f(τ,u(τ))−f(τ,v(τ))|dτ≤∫01K(t,τ)φ(τ)dτ⋅∥u−v∥1−α≤M2∥u−v∥1−α,$
and thus
$|(ℱu)(t)−(ℱv)(t)|≤∫01(1−τ)q−1(Fu)(τ1ρ)−(Fv)(τ1ρ)dτA(α,1)Γ(α+1)ρqΓ(q)+|(Fu)(t)−(Fv)(t)|≤M1M2∥u−v∥1−α.$
Furthermore, $∥ℱu−ℱv∥1−α<∥u−v∥1−α$. The proof now can be finished by using the Banach contraction mapping principle.□

## 6 Ulam-Hyers stability

Let $ϵ˜$ be a positive real number. We consider Eq. (1.1) with inequality

$|D1−βc(D0+αL+λ)v(t)−f(t,v(t))|≤ϵ˜,t∈J.$

Definition 6.1

Eq. (1.1) is Ulam-Hyers stable if there exists a real number $c>0$ such that for each solution $v(t)$ of inequality (6.1) there exists a solution u of Eq. (1.1) with

$|v(t)−u(t)|≤cϵ˜,t∈J.$

Remark 6.2

A function $v∈C1−α$ is a solution of inequality (6.1) if and only if there exists a function $g∈C1−α$ such that (i) $|g(t)|≤ϵ˜$; (ii) $D1−βc(D0+αL+λ)v(t)=f(t,v(t))+g(t)$.

Let

$v˜(t)=−u0Γ(α)(1+A(α−1,1))A(α,1)tαEα,α+1(−λtα)+u0Γ(α)tα−1Eα,α(−λtα)−1ρqΓ(q)∫01(1−τ)q−1(Fv)(τ1ρ)dτA(α,1)tαEα,α+1(−λtα)+(Fv)(t),$
where
$(Fv)(t)=∫01K(t,τ)f(τ,v(τ))dτ.$

Remark 6.3

Let $v∈C1−α$ be a solution of inequality (6.1) with $limt→0+t1−αv(t)=u0,(I0+1−αv)(0)+(I0+qρv)(1)=0$. Then, v is a solution of the inequality $|v(t)−v˜(t)|≤M1M3ϵ˜$.

Indeed, by Remark 6.2, we have

$D1−βc(D0+αL+λ)v(t)=f(t,v(t))+g(t),$
$limt→0+t1−αv(t)=u0,(I0+1−αv)(0)+(I0+qρv)(1)=0.$
Applying the same arguments as in the proof of Lemma 4.6, we obtain
$v(t)=−u0Γ(α)(1+A(α−1,1))A(α,1)tαEα,α+1(−λtα)+u0Γ(α)tα−1Eα,α(−λtα)−1ρqΓ(q)∫01(1−τ)q−1(Gv)(τ1ρ)dτA(α,1)tαEα,α+1(−λtα)+(Gv)(t),$
where
$(Gv)(t)=∫01K(t,τ)[f(τ,v(τ))+g(τ)]dτ.$
Therefore, by (4.10), we conclude that $|v(t)−v˜(t)|≤M1M3ϵ˜$.

Theorem 6.4

Assume that (H1) and (H2) are satisfied, then Eq. ( 1.1) is Ulam-Hyers stable if $M1M2<1$.

Proof

Let $v∈C1−α$ be a solution of inequality (6.1) with $limt→0+t1−αv(t)=u0,(I0+1−αv)(0)+(I0+qρv)(1)=0$. u denotes the unique solution of the following problem:

$D1−βc(D0+αL+λ)u(t)=f(t,u(t)),t∈J,limt→0+t1−αu(t)=u0,(I0+1−αu)(0)+(I0+qρu)(1)=0.$
It is easy to check that
$|v(t)−u(t)|≤|v(t)−v˜(t)|+|v˜(t)−u(t)|≤M1M3ϵ˜+∫01(1−τ)q−1|(Fv)(τ1ρ)−(Fu)(τ1ρ)|dτA(α,1)Γ(α+1)ρqΓ(q)+|(Fv)(t)−(Fu)(t)|≤M1M3ϵ˜+M1M2∥v−u∥1−α,$
then
$∥v−u∥1−α≤M1M3ϵ˜1−M1M2,$
that is, Eq. (1.1) is Ulam-Hyers stable.□

## 7 Example

As an example, we consider here the following BVP of the fractional differential equation with two different fractional derivatives:

$D1−15c(D0+35L+3)u(t)=130t10sin(t25u(t)+t34),t∈J≔(0,1],$
$limt→0+t25u(t)=u0,(I0+25u)(0)+(I0+32u)(1)=0.$
Corresponding to (1.1) and (1.2), we recognize that $α=35$, $β=15$, $λ=3$, $ρ=2$, $q=3$ and
$f(t,u(t))=130t10sin(t25u(t)+t34).$
The space $C1−α={u∈C(J,ℝ):t25u(t)∈C([0,1],ℝ)}$ with the norm $∥u∥25=maxt∈[0,1]t25|u(t)|$.

It is not difficult to obtain

$|f(t,u(t))−f(t,v(t))|≤φ(t)∥u−v∥25,$
where $φ(t)=130t10∈L1p[0,1](p=320)$ and $∥φ∥L1p=332030$. Moreover,
$|f(t,0)|=130t10|sint34|≤130,$
and thus $supt∈J|f(t,0)|≤130$.

By direct computation, we find

$A35,1=18Γ(3)∫01s310(1−s)2E35,85(−3s310)ds≈0.005,M1=1ρqΓ(q+1)Γ(α+1)A(α,1)+1=18Γ(4)Γ85A35,1+1≈5.6;M2=1β1−pα−p1−p+1α1−pβ−p1−p∥φ∥L1pΓ(α)Γ(β)=51791720+53(17)1720332030Γ35Γ15≈0.16,$
consequently, $M1M2≈0.9<1$, by Theorems 5.1 and 6.4, problem (7.1) and (7.2) has a unique solution and Eq. (7.1) is Ulam-Hyers stable.

## References

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Kenneth S. Miller and Bertram Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.

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Stefan G. Samko, Anatoly A. Kilbas, and Oleg I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.

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Igor Podlubny, Fractional Differential Equations, vol. 198, Mathematics in Science and Engineering, Academic Press, San Diego, 1999.

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Anatoly A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204, Elsevier, Amsterdam, 2006.

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Udita N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl. 6 (2014), no. 4, 1–15.

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Fang Li, Jin Liang, and Hong-Kun Xu, Existence of mild solutions for fractional integrodifferential equations of Sobolev type with nonlocal conditions, J. Math. Anal. Appl. 391 (2012), no. 2, 510–525, .

• Crossref
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Jin Liang, Yunyi Mu, and Ti-Jun Xiao, Solutions to fractional Sobolev-type integro-differential equations in Banach spaces with operator pairs and impulsive conditions, Banach J. Math. Anal. 13 (2019), no. 4, 745–768, .

• Crossref
• Export Citation
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Jin Liang, James H. Liu, and Ti-Jun Xiao, Condensing operators and periodic solutions of infinite delay impulsive evolution equations, Discrete Contin. Dyn. Syst. Ser. S 10 (2017), no. 3, 475–485, .

• Crossref
• Export Citation
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Yunsong Miao and Fang Li, Boundary value problems of the nonlinear multiple base points impulsive fractional differential equations with constant coefficients, Adv. Difference Equ. 2017 (2017), 190, .

• Crossref
• Export Citation
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Om P. Agrawal, Fractional variational calculus and transversality condition, J. Phys. A: Math. Gen. 39 (2006), no. 33, 10375–10384, .

• Crossref
• Export Citation
• [11]

Teodor M. Atanackovic and Bogoljub Stankovic, On a differential equation with left and right fractional derivatives, Fract. Calc. Appl. Anal. 10 (2007), no. 2, 139–150.

• [12]

Tomasz Blaszczyk and Mariusz Ciesielski, Fractional oscillator equation – transformation into integral equation and numerical solution, Appl. Math. Comput. 257 (2015), 428–435, .

• Crossref
• Export Citation
• [13]

Assia Guezane-Lakoud, Rabah Khaldi, and Adem Kiliçman, Solvability of a boundary value problem at resonance, SpringerPlus 5 (2016), 1504, .

• Crossref
• PubMed
• Export Citation
• [14]

Rabah Khaldi and Assia Guezane-Lakoud, Higher order fractional boundary value problems for mixed type derivatives, J. Nonlinear Funct. Anal. 2017 (2017), 30, .

• Crossref
• Export Citation
• [15]

Stanisław M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York, 1960.

• [16]

Donald H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. 27 (1941), no. 4, 222–224, .

• Crossref
• Export Citation
• [17]

Donald H. Hyers, George Isac, and Themistocles M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Basel, 1998.

• [18]

Wei Wei, Xuezhu Li, and Xia Li, New stability results for fractional integral equation, Comput. Math. Appl. 64 (2012), no. 10, 3468–3476, .

• Crossref
• Export Citation
• [19]

JinRong Wang and Xuezhu Li, Ulam-Hyers stability of fractional Langevin equations, Appl. Math. Comput. 258 (2015), 72–83, .

• Crossref
• Export Citation

If the inline PDF is not rendering correctly, you can download the PDF file here.

• [1]

Kenneth S. Miller and Bertram Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.

• [2]

Stefan G. Samko, Anatoly A. Kilbas, and Oleg I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.

• [3]

Igor Podlubny, Fractional Differential Equations, vol. 198, Mathematics in Science and Engineering, Academic Press, San Diego, 1999.

• [4]

Anatoly A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204, Elsevier, Amsterdam, 2006.

• [5]

Udita N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl. 6 (2014), no. 4, 1–15.

• [6]

Fang Li, Jin Liang, and Hong-Kun Xu, Existence of mild solutions for fractional integrodifferential equations of Sobolev type with nonlocal conditions, J. Math. Anal. Appl. 391 (2012), no. 2, 510–525, .

• Crossref
• Export Citation
• [7]

Jin Liang, Yunyi Mu, and Ti-Jun Xiao, Solutions to fractional Sobolev-type integro-differential equations in Banach spaces with operator pairs and impulsive conditions, Banach J. Math. Anal. 13 (2019), no. 4, 745–768, .

• Crossref
• Export Citation
• [8]

Jin Liang, James H. Liu, and Ti-Jun Xiao, Condensing operators and periodic solutions of infinite delay impulsive evolution equations, Discrete Contin. Dyn. Syst. Ser. S 10 (2017), no. 3, 475–485, .

• Crossref
• Export Citation
• [9]

Yunsong Miao and Fang Li, Boundary value problems of the nonlinear multiple base points impulsive fractional differential equations with constant coefficients, Adv. Difference Equ. 2017 (2017), 190, .

• Crossref
• Export Citation
• [10]

Om P. Agrawal, Fractional variational calculus and transversality condition, J. Phys. A: Math. Gen. 39 (2006), no. 33, 10375–10384, .

• Crossref
• Export Citation
• [11]

Teodor M. Atanackovic and Bogoljub Stankovic, On a differential equation with left and right fractional derivatives, Fract. Calc. Appl. Anal. 10 (2007), no. 2, 139–150.

• [12]

Tomasz Blaszczyk and Mariusz Ciesielski, Fractional oscillator equation – transformation into integral equation and numerical solution, Appl. Math. Comput. 257 (2015), 428–435, .

• Crossref
• Export Citation
• [13]

Assia Guezane-Lakoud, Rabah Khaldi, and Adem Kiliçman, Solvability of a boundary value problem at resonance, SpringerPlus 5 (2016), 1504, .

• Crossref
• PubMed
• Export Citation
• [14]

Rabah Khaldi and Assia Guezane-Lakoud, Higher order fractional boundary value problems for mixed type derivatives, J. Nonlinear Funct. Anal. 2017 (2017), 30, .

• Crossref
• Export Citation
• [15]

Stanisław M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York, 1960.

• [16]

Donald H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. 27 (1941), no. 4, 222–224, .

• Crossref
• Export Citation
• [17]

Donald H. Hyers, George Isac, and Themistocles M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Basel, 1998.

• [18]

Wei Wei, Xuezhu Li, and Xia Li, New stability results for fractional integral equation, Comput. Math. Appl. 64 (2012), no. 10, 3468–3476, .

• Crossref
• Export Citation
• [19]

JinRong Wang and Xuezhu Li, Ulam-Hyers stability of fractional Langevin equations, Appl. Math. Comput. 258 (2015), 72–83, .

• Crossref
• Export Citation
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