Show Summary Details
In This Section

# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

1 Issue per year

IMPACT FACTOR 2016 (Open Mathematics): 0.682
IMPACT FACTOR 2016 (Central European Journal of Mathematics): 0.489

CiteScore 2016: 0.62

SCImago Journal Rank (SJR) 2015: 0.521
Source Normalized Impact per Paper (SNIP) 2015: 1.233

Mathematical Citation Quotient (MCQ) 2015: 0.39

Open Access
Online
ISSN
2391-5455
See all formats and pricing
In This Section

# Existence theory for sequential fractional differential equations with anti-periodic type boundary conditions

Mohammed H. Aqlan
• Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Arabia
/ Ahmed Alsaedi
• Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Arabia
• Corresponding author
• Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Arabia
• Email:
/ Juan J. Nieto
• Analisis Matematico, Facultad de Matematicas, University of Santiago de Compostela, Santiago de Compostela, 15782, Spain
Published Online: 2016-10-16 | DOI: https://doi.org/10.1515/math-2016-0064

## Abstract

We develop the existence theory for sequential fractional differential equations involving Liouville-Caputo fractional derivative equipped with anti-periodic type (non-separated) and nonlocal integral boundary conditions. Several existence criteria depending on the nonlinearity involved in the problems are presented by means of a variety of tools of the fixed point theory. The applicability of the results is shown with the aid of examples. Our results are not only new in the given configuration but also yield some new special cases for specific choices of parameters involved in the problems.

MSC 2010: 34A08; 34B10; 34B15

## 1 Introduction

Recently, there has been an utterly great interest in developing theoretical analysis for boundary value problems of nonlinear fractional-order differential equations supplemented with a variety of boundary conditions. It has been mainly due to nonlocal nature of fractional-order differential operators which take into account memory and hereditary properties of some important and useful materials and processes. Fractional calculus has played a key role in improving the mathematical modelling of several phenomena occurring in engineering and scientific disciplines, such as blood flow problems, control theory, aerodynamics, nonlinear oscillation of earthquake, the fluid-dynamic traffic model, polymer rheology, regular variation in thermodynamics etc. For more details and explanation, see, for instance [13]. Some recent results on fractional-order boundary value problem can be found in a series of papers [412] and the references cited therein. Sequential fractional differential equations have also received considerable attention, for instance see [1317].

Anti-periodic boundary conditions are found to be quite significant and important in the mathematical modeling of certain physical processes and phenomena, for example trigonometric polynomials in the study of interpolation problems, wavelets, physics etc. For more details, see [18] and the references cited therein. For some recent works on fractional-order anti-periodic boundary value problems, we refer the reader to [1923]. However, the study of sequential fractional differential equations equipped with anti-periodic boundary conditions is yet to be initiated.

In this paper, we study new boundary value problems of Liouville-Caputo type sequential fractional differential equation: $(cDα+k cDα−1)u(t)=f(t, u(t)),1<α≤2, 00,$(1)

subject to anti-periodic type (non-separated) boundary conditions of the form: $α1u(0)+ρ1u(T)=β1, α2u′(0)+ρ2u′(T)=β2,$(2)

and anti-periodic type (non-separated) nonlocal integral boundary conditions: $α1u(0)+ρ1u(T)=λ1∫0nu(s)ds+λ2, α2u′(0)+ρ2u′(T)=μ1∫ξTu(s)ds+μ2,$(3)

where c Dα denotes the Liouville-Caputo fractional derivative of order α, k ∈ ℝ+, 0 < η < ξ < T, α1, α2, ρ1, ρ2, β1, β2, λ1, λ2, μ1μ2 ∈ ℝ with α1 + ρ1 ≠ 0, α2 + ρ2e-kT ≠ 0 and f : [0, T] × ℝ → ℝ is a given continuous function. Instead of writing the so-called “Caputo” derivative, we will call it “Liouville-Caputo” derivative as it was introduced by Liouville many decades ago.

The rest of the paper is organized as follows. In Section 2, we recall some basic concepts of fractional calculus and obtain the integral solution for the linear variants of the given problems. Section 3 contains the existence results for problem (1)-(2) obtained by applying Schaefer’s fixed point theorem, Leray-Schauder’s nonlinear alternative, Leray-Schauder’s degree theory, Banach’s contraction mapping principle and Krasnoselskii’s fixed point theorem. In Section 4, we provide the outline for the existence results of problem (1)-(3).

## 2 Preliminaries and auxiliary results

This section is devoted to some basic definitions of fractional calculus [1, 2] and auxiliary lemmas.

The fractional integral of order q with the lower limit zero for a function f is defined as $Iqf(t)=1Γ(q)∫0tf(s)(t−s)1−qds, t > 0, q > 0,$

provided the right hand-side is point-wise defined on [0,∞), where $\text{Γ}\left(\cdot \right)$ is the gamma function, which is defined by $\text{Γ}\left(q\right)={\int }_{0}^{\infty }{t}^{q-1}{e}^{-t}dt$.

The Riemann-Liouville fractional derivative of order q > 0, n — 1 < q < n, nN, is defined as $D0+qf(t)=1Γ(n−q)(ddt)n∫0t(t−s)n−q−1f(s)ds,$

where the function f(t) has absolutely continuous derivative up to order (n — 1).

The Liouville-Caputo derivative of order q for a function f : [0,∞) → ℝ can be written as $cDqf(t)=Dq(f(t)−∑k=0n−1tkk!f(k)(0)), t>0, n−1

If $f\left(t\right)\in {C}^{n}\left[0,\infty \right)$ then $cDqf(t)=1Γ(n−q)∫0tf(n)(s)(t−s)q+1−nds=In−qf(n)(t), t>0, n−1

To define the fixed point problems associated with problems (1)-(2) and (1)-(3), we consider the following lemmas dealing with the linear variant of equation (1).

Let hC([0, T], ℝ). Then the problem consisting of the equation $(cDα+kcDα−1)u(t)=h(t)), 1<α≤2, 00,$(4)

and the boundary conditions (2) is equivalent to the integral equation $u(t)=v1(t)+∫0te−k(t−s)(∫0s(s−x)α−2Γ(α−1)h(x)dx)ds+v2(t)∫0T(T−s)α−2Γ(α−1)h(s)ds +v3(t)∫0Te−k(T−s)(∫0s(s−x)α−2Γ(α−1)h(x)dx)ds,$(5)

where $v1(t)=β1(α1+ρ1)+((α1+ρ1e−kT)−(α1+ρ1)e−kT)β2k(α1+ρ1)(α2+ρ2e−kT), v2(t)=ρ2((α1+ρ1)e−kT−(α1+ρ1e−kT))k(α1+ρ1)(α2+ρ2e−kT), v3(t)=α1ρ2−α2ρ1−ρ2(α1+ρ1)e−kt(α1+ρ1)(α2+ρ2e−kT).$(6)

Let hC([0, T], ℝ). Then the problem consisting of linear equation (4) equipped with boundary conditions (3) is equivalent to the integral equation $u(t)=B1(t){λ1∫0n(∫0se−k(s−x)Iα−1h(x)dx)ds−ρ1∫0Te−k(T−s)Iα−1h(s)ds+λ2} +B2(t){μ1∫ξT(∫0se−k(s−x)Iα−1h(x)dx)ds+kρ2∫0Te−k(T−s)Iα−1h(s)ds −ρ2Iα−1h(T)+μ2}+∫0te−k(t−s)Iα−1h(s)ds,$(11)

where $B1(t)=(ϵ2e−kt+δ2)Δ, B2(t)=(ϵ1e−kt−δ1)Δ, Δ=δ1ϵ2+δ2ϵ1, δ1=α1+ρ1e−kT+λ1k(e−kn−1), ϵ1=(α1+ρ1−λ1η), δ2=−kα2−kρ2e−kT+μ1k(e−kT−e−kξ), ϵ2=μ1(T−ξ).$(12)

## 3 Existence results for the problem (1)-(2)

In view of Lemma 2.5, we introduce a fixed point problem associated with the problem (1)-(2) as follows: $u=Hu,$(13)

where the operator $\mathcal{H}:\mathcal{E}\to \mathcal{E}$ is $(Hu)(t)=v1(t)+∫0te−k(t−s)∫0s(s−x)α−2Γ(α−1)f(x, u(x))dxds+v2(t)∫0T(T−s)α−2Γ(α−1)f(s, u(s))ds +v3(t)∫0Te−k(T−s)∫0s(s−x)α−2Γ(α−1)f(x, u(x))dxds.$(14)

Here $\mathcal{E}=C\left(\left[0,T\right],ℝ\right)$ denotes the Banach space of all continuous functions from [0, T] → ℝ endowed with the norm defined by $‖u‖=\mathrm{sup}\left\{|u\left(t\right)|,t\in \left[0,T\right]\right\}$.

Observe that that problem (1)-(2) has solutions if the operator equation (13) has fixed points. For computational convenience, we set the notation: $Q=supt∈[0,T]{1−e−ktkΓ(α)t1−α+|v2(t)|Γ(α)T1−α+|v3(t)|(1−e−kT)kΓ(α)T1−α}.$(15)

Now we are in a position to present our main results for the problem (1)-(2). The first one deals with Schaefer’s fixed point theorem [24].

Let X be a Banach space. Assume that $\mathcal{T}:X\to X$ is a completely continuous operator and the set $Y=\left\{u\in X|u=\mu \mathcal{T}u,0<\mu <1\right\}$ is bounded. Then $\mathcal{T}$ has a fixed point in X.

Assume that there exists a positive constant L1 such that $|f\left(t,u\left(t\right)\right)|\le {L}_{1}$ for $t\in \left[0,T\right],u\in ℝ$. Then the boundary value problem (1)-(2) has at least one solution on [0, T].

In the first step, we show that the operator $\mathcal{H}$ defined by (14) is completely continuous. Observe that continuity of $\mathcal{H}$ follows from the continuity of f. For a positive constant r, let ${B}_{r}=\left\{u\in \mathcal{E}:‖u‖\le r\right\}$ be a bounded ball in $\mathcal{E}$. Then for t ∈ [0, T], we have $|(Hu)(t)|≤|v1(t)|+∫0te−k(t−s)(∫0s(s−x)α−2Γ(α−1)|f(x, u(x))|dx)ds+|v2(t)|∫0T(T−s)α−2Γ(α−1)|f(s, u(s))|ds +|v3(t)|∫0Te−k(T−s)(∫0s(s−x)α−2Γ(α−1)|f(x, u(x))|dx)ds. ≤|v1(t)|+L1{∫0te−k(t−s)(∫0s(s−x)α−2Γ(α−1)dx)ds+|v2(t)|∫0T(T−s)α−2Γ(α−1)ds +|v3(t)|∫0Te−k(T−s)(∫0s(s−x)α−2Γ(α−1)dx)ds}≤L1Q+‖v1‖,$

which consequently implies that $(Hu)≤L1Q+v1,$

where Q is defined by (15).

Next we show that the operator $\mathcal{H}$ maps bounded sets into equicontinuous sets of $\mathcal{E}$. Let τ1, τ2 ∈ [0, T] with τ1 < τ2 and uBr. Then we have $(Hu)(τ2)−(Hu)(τ1)≤v1(τ2)−v1(τ1) +L1e−kτ2−e−kτ1∫0τ1eks∫0s(s−x)α−2Γ(α−1)dxds+∫τ1τ2e−k(τ2−s)∫0s(s−x)α−2Γ(α−1)dxds +v2(τ2)−v2(τ1)∫0T(T−s)α−2Γ(α−1)ds+v3(τ2)−v3(τ1)∫0Te−k(T−s)∫0s(s−x)α−2Γ(α−1)dxds.$

As τ1τ2 → 0, the right-hand side of the above inequality tends to zero independently of uBr. Therefore, by the Arzelá-Ascoli theorem, the operator $\mathcal{H}:\mathcal{E}\to \mathcal{E}$ is completely continuous.

Finally, we consider the set $V=\left\{u\in \mathcal{E}:u=\mu \mathcal{H}u,\text{\hspace{0.17em}}0<\mu <1\right\}$ and show that V is bounded. For uV and t ∈ [0, T], we get $‖u‖≤L1Q+‖v1‖.$

Therefore, V is bounded. Hence, by Lemma 3.1, the problem (1)-(2) has at least one solution on [0, T].

Our next existence result is based on Leray-Schauder’s nonlinear alternative.

(Nonlinear alternative for single valued maps [25]). Let E be a Banach space, C a closed, convex subset of E, U an open subset of C and 0 ∈ U. Suppose that $\mathcal{A}:\overline{U}\to C$ is a continuous, compact (that is, $\mathcal{A}\left(\overline{U}\right)$ is a relatively compact subset of C ) map. Then either

• (i)

$\mathcal{A}$ has a fixed point in $\overline{U}$, or

• (ii)

there is a x ∈ ∂U (the boundary of U in C) and λ ∈ (0, 1) with $x=\text{λ}\mathcal{A}\left(x\right)$.

Assume that

• (E1)

there exists a continuous nondecreasing function $\chi :\left[0,\infty \right)\to \left(0,\infty \right)$ and a function pC([0, T], ℝ+) such that $|f(t, u)|≤p(t)χ(‖u‖) for each (t, u)∈[0,T]×ℝ;$

• (E2)

there exists a constant N > 0 such that $Nχ(N)‖p‖Q+‖v1‖>1,$(16)

where Q is given by (15).

Then the boundary value problem (1)-(2) has at least one solution on [0, T].

The next existence result is based on Leray-Schauder’s degree theory [25].

Let f : [0, T] × ℝ → ℝ be a continuous function. Suppose that (E3) there exist constants 0 ≤ ω < Q-1, and M1 > 0 such that $|f(t, u)|≤ω|u|+M1 for all (t, u)∈[0,T]×ℝ,$

where Q is given by (15).

Then the boundary value problem (1)-(2) has at least one solution on [0, T].

Next we show the existence of a unique solution of the problem (1)-(2) by applying Banach’s contraction mapping principle.

Assume that f : [0, T] × ℝ → ℝ is a continuous function satisfying the Lipschitz condition: (E4) there exists a positive number ℓ such that $|f\left(t,u\right)-f\left(t,\upsilon \right)|\le \ell |u-\upsilon |,\forall t\in \left[0,\text{\hspace{0.17em}}T\right],\text{\hspace{0.17em}}u,\text{\hspace{0.17em}}\upsilon \in ℝ$.

Then the boundary value problem (1)-(2) has a unique solution on [0, T] if Q < 1/, where Q is given by (15).

Consider a set ${B}_{r}=\left\{u\in \mathcal{E}:‖u‖\le r\right\}$ with $r\ge \frac{QM+‖{v}_{1}‖}{1-\ell Q}$, where $M={\mathrm{sup}}_{t\in \left[0,T\right]}|f\left(t,0\right)|$ and Q is given by (15). In the first step, we show that $\mathcal{H}{B}_{r}\subset {B}_{r}$, where the operator $\mathcal{H}$ is defined by (14). For any uBr, t ∈ [0, T], observe that $|f(t, u(t))|=|f(t, u(t))−f(t, 0)+f(t, 0)|≤|f(t, u(t))−f(t, 0)|+|f(t, 0)|≤ℓ‖u‖+M≤ℓr+M,$

where we have used the assumption (E4). Then, for uBr, we obtain $‖(Hu)‖≤supt∈[0,T]{|v1(t)+|∫0te−k(t−s)(∫0s(s−x)α−2Γ(α−1)|f(x, u(x))|dx)ds+|v2(t)|∫0T(T−s)α−2Γ(α−1)|f(s, u(s))|ds+|v3(t)|∫0Te−k(T−s)(∫0s(s−x)α−2Γ(α−1)|f(x, u(x))|dx)ds.}≤(ℓr+M)supt∈[0,T]{∫0te−k(t−s)(∫0s(s−x)α−2Γ(α−1)dx)ds+|v2(t)|∫0T(T−s)α−2Γ(α−1)ds+|v3(t)|∫0Te−k(T−s)(∫0s(s−x)α−2Γ(α−1)dx)ds}+‖v1‖≤(ℓr+M)Q+‖v1‖≤r,$

which implies that $\mathcal{H}u\in {B}_{r}$. Thus $\mathcal{H}{B}_{r}\subset {B}_{r}$. Next we show that the operator $\mathcal{H}$ is a contraction. Using the assumption (E4) and (15), we get $‖Hu−Hv‖≤supt∈[0,T]{∫0te−k(t−s)(∫0s(s−x)α−2Γ(α−1)|f(x,u(x))−f(x,v(x))|dx)ds+|v2(t)|∫0T(T−s)α−2Γ(α−1)|f(s,u(s))−f(s,v(s))|ds+|v3(t)|∫0Te−k(T−s)(∫0s(s−x)α−2Γ(α−1)|f(x,u(x))−f(x,v(x))|dx)ds}≤ℓ‖u−v‖supt∈[0,T]{∫0te−k(t−s)(∫0s(s−x)α−2Γ(α−1)dx)ds+|v2(t)|∫0T(T−s)α−2Γ(α−1)ds+|v3(t)|∫0Te−k(T−s)(∫0s(s−x)α−2Γ(α−1)dx)ds}≤ℓQ‖u−v‖.$

In view of the assumption: Q < 1/ℓ, it follows that the operator $\mathcal{H}$ is a contraction. Thus, by Banach’s contraction mapping principle, we deduce that the operator $\mathcal{H}$ has a fixed point, which in turn implies that there exists a unique solution for the problem (1)-(2) on [0, T].

In the following theorem, we show the existence of solutions for the problem (1)-(2) by applying Krasnoselskii’s fixed point theorem.

(Krasnoselskii’s fixed point theorem [24]). Let Y be a closed bounded, convex and nonempty subset of a Banach space X. Let B1, B2 be the operators such that (i) B1y1 + B2y2 Y whenever y1,y2 Y; (ii) B1 is compact and continuous and (iii) B2 is a contraction mapping. Then there exists z Y such that z = B1z + B2z.

Let f : [0,T] × ℝ → ℝ be a continuous function satisfying the condition (E4) and that $|f\left(t,x\right)|\le g\left(t\right)$ (t,x) ∈ [0, T] × ℝ with gC([0, T], ℝ+) and ${\mathrm{sup}}_{t\in \left[0,T\right]}|g\left(t\right)|=‖g‖$ . In addition, it is assumed that ${Q}_{1}<1/\ell$, where$Q1=supt∈[0,T]{|v2(t)Γ(α)T1−α+|v3(t)|(1−e−kT)kΓ(α)T1−α}.$(19)

Then problem (1)-(2) has at least one solution on [0, T].

Consider the following anti-periodic fractional boundary value problem:${(cD3/2+1.5cD1/2)u(t)=f(t,u(t)), t∈[0,2],1.25u(0)+4u(2)=−1, 0.5u′(0)−2u′(2)=2.5.$(20)

Here T = 2, k = 1.5, α1 = 1.25, ρ1 = 4, β1 = —1, α2 = 0.5, ρ2 = —2, β2 = 2.5. With the given values, we find that Q ≈ 7.742915 (Q is given by (15)).

(a) Let$f(t,u)=e−u2(cos2(3u+2)t2+9+tsint1+u2+2t+3).$(21)

Clearly $|f\left(t,u\left(t\right)\right)|\le 3={L}_{1}$ for all t ∈ [0, 2], u ∈ ℝ. Thus, by Theorem 3.2, the problem (20) with f(t,u) given by (21) has at least one solution on [0, 2].

(b) Letting$f(t,u)=e−t27(|u|31+|u|3+|u|1+|u|+1t+1),$(22)

we have $|f\left(t,u\right)|\le {e}^{-t}/9=p\left(t\right)\chi \left(‖u‖\right)$ . Selecting $\chi \left(‖u‖\right)=1$ and $p\left(t\right)={e}^{-t}/9\left(‖p‖=1/9\right)$, we find that the assumption (E2) holds true for N > 4.064141. As all the conditions of Theorem 3.4 are satisfied, there exists at least one solution of the problem (20) with f(t, u) given by (22) on [0, 2].

(c) Let us take$f(t,u)=1t2+100sinu+1t+2.$(23)

Then $|f\left(t,u\right)|\le \left(1/10\right)u+1/2$ implies that ω = 1/10, M1 = 1/2. Clearly ω < 1/Q(Q ≈ 7.742915). In consequence, the conclusion of Theorem 3.5 applies and the problem (20) with f(t, u) given by (23) has a solution on [0, T].

(d) Let us choose$f(t,u)=110tan−1u(t)+cost.$(24)

Clearly $\ell =1/10$ as $|f\left(t,u\right)-f\left(t,\upsilon \right)|\le \frac{1}{10}|u-\upsilon |$ and $\ell Q\approx 0.774292<1$ . Thus all the conditions of Theorem 3.6 are satisfied. Hence we deduce by the conclusion of Theorem 3.6 that there exists a unique solution for the problem (20) with f(t, u) given by (24).

For the applicability of Theorem 3.8, we find that $|f\left(t,u\right)|\le g\left(t\right)=\pi /20+\mathrm{cos}t$ with $‖g‖=\left(20+\pi \right)/20$ and Q1 ≈ 6.732035 (Q1 is given by (19)). Obviously $\ell {Q}_{1}\approx 0.673203<1$ . Thus all the conditions of Theorem 3.8 are satisfied. Hence the conclusion of Theorem 3.8 implies that the problem (20) with f(t, u) given by (24) has at least one solution on [0, 2].

By fixing the parameters involved in the boundary conditions (2), we can obtain some new special results for different problems arising from the problem (1)-(2). For instance, for α1 = α2 = ρ1 = ρ2 = 1, β1 = β2 = 0, we obtain the existence results for sequential fractional differential equation (1) with anti-periodic boundary condition: u(0) + u(T) = 0, u′(0) + u′(T) = 0. Our results correspond to the ones obtα1ned in [14] for u(0) = a, u′(0) = u′(1) by taking α1 = 1 = α2, ρ1 = 0, ρ2 = —1, β1 = a, β2 = 0.

## 4 Existence results for the problem (1)-(3)

In this section, we present some existence results for the problem (1)-(3). We omit the proofs as the method of proof is similar to the one employed in the previous section. First of all, by Lemma 2.6, we define a fixed point operator $\mathcal{G}:\mathcal{E}\to \mathcal{E}$ associated with the problem (1)-(3) as $(Gu)(t)=B1(t){λ1∫0η(∫0se−k(s−x)Iα−1h(x)dx)ds−ρ1∫0Te−k(T−s)Iα−1h(s)ds+λ2}+B2(t){μ1∫ξT(∫0se−k(s−x)Iα−1h(x)dx)ds+kρ2∫0Te−k(T−s)Iα−1h(s)ds−ρ2Iα−1h(T)+μ2}+∫0te−k(t−s)Iα−1h(s)ds$(25)

where B1(t) and B2(t) are given by (12).

Using the operator (25) and the method of proof for the results obtained in the last section, we can establish the following results for the problem (1)-(3).

Let f : [0,T] × ℝ → ℝ be a continuous function satisfying the assumption (E4). Then the boundary value problem (1)-(3) has a unique solution on [0, T] if $\overline{Q}<1/\ell$, where$Q¯=supt∈[0,T]{|B1(t)[λ1kΓ(α)(ηαα+ηα−1(e−kη−1)k)−ρ1Tα−1(1−e−kT)kΓ(α)]|+|B2(t)[μ1kΓ(α)(Tα−ξαα+Tα−1(e−kT−e−kξ)k)−ρ2Tα−1Γ(α)]|+(1−e−kt)tα−1kΓ(α)}.$(26)

Let f : [0,T] × ℝ → ℝ be a continuous functions satisfying the condition (E4) and that $|f\left(t,x\right)|\le \overline{g}\left(t\right)$, $\forall \left(t,x\right)\in \left[0,T\right]×ℝ$ with $\overline{g}\in C\left(\left[0,T\right],{ℝ}^{+}\right)$ .In addition, it is assumed that $\overline{Q}<1/\ell$, where$Q¯1=supt∈[0,T]{|B1(t)[λ1kΓ(α)(ηαα+ηα−1(e−kη−1)k)−ρ1Tα−1(1−e−kT)kΓ(α)]|+|B2(t)[μ1kΓ(α)(Tα−ξαα+Tα−1(e−kT−e−kξ)k)−ρ2Tα−1Γ(α)]|}.$(27)

Then problem (1)-(3) has at least one solution on [0,T].

Let f : [0,T] × ℝ → ℝ be a jointly continuous function. Assume that (E5) there exists a continuous nondecreasing function ψ : [0,) → (0,) and a function φ ∈ C([0, T], ℝ+) such that$|f(t,u)|≤φ(t)ψ(‖u‖) for each (t,u)∈[0,T]×ℝ;$

(E6) there exists a constant N1 > 0 such that$N1ψ(N1)‖φ‖Q¯+Q^>1,$(28)

where $\overline{Q}$ is given by (26),$Q^=supt∈[0,T]{|λ2B1(t)+μ2B2(t)|}.$(29)

Then the boundary value problem (1)-(3) has at least one solution on [0,T].

Let f : [0,T] × ℝ → ℝ be a continuous function. Suppose that (E7) there exist constants $0\le \omega <1/\overline{Q}$, and M2 > 0 such that$|f(t,u)|≤ω1|u|+M2 for all (t,u)∈[0,T]×ℝ,$

where $\overline{Q}$ is given by (26).

Then the boundary value problem (1)-(3) has at least one solution on [0, T].

Let f : [0,T] × ℝ → ℝ be a continuous function. Assume that there exists a positive constant L2 suchthat $|f\left(t,u\left(t\right)\right)|\le {L}_{2}$ for t ∈ [0,T], u ∈ ℝ. Then the boundary value problem (1)-(3) has at least one solution on [0, T].

Consider the following anti-periodic fractional boundary value problem:${(cD3/2+1.5cD1/2)u(t)=1t+25((5−t)sinu30+e−tcos t), t∈[0,2],1.25u(0)+4u(2)=−∫01u(s)ds−1, 0.5.u′(0)−2u′(2)=2∫1.252u(s)ds+2.5,$(30)

where $f\left(t,u\left(t\right)\right)=\frac{1}{\sqrt{t+25}}\left(\frac{\left(5-t\right)\mathrm{sin}u}{15}+{e}^{-t}\mathrm{cos}t\right)$, T = 2, η = 1, ξ = 1.25, k = 1.5, α1 = 1.25, ρ1 = 4, λ1 = —1, λ2 = —1, λ2 = 0.5, ρ2 = —2, μ1 = 2, μ2 = 2.5.

With the given values, we find that $\overline{Q}\approx 14.422595$ ( $\overline{Q}$ is given by (26)) and $\ell =1/30$ as $|f\left(t,u\right)-f\left(t,v\right)|\le \frac{1}{30}|u-\upsilon |$ . Clearly, $\overline{Q}<1/\ell$ .Thus all the conditions of Theorem 4.1 are satisfied. Hence we deduce by the conclusion of Theorem 4.1 that there exists a unique solution for the problem (30).

For the applicability of Theorem 4.2, we find that $|f\left(t,u\right)|\le \overline{g}\left(t\right)=\frac{1}{\sqrt{t+25}}\left(\frac{5-t}{30}+1\right)$ with $‖\overline{g}‖=\frac{7}{30}$ and ${\overline{Q}}_{1}\approx 13.411715$ ( ${\overline{Q}}_{1}$ is given by (27)). Obviously ${\overline{Q}}_{1}<1/\ell$ .Thus all the conditions of Theorem 4.2 are satisfied. Hence the conclusion of Theorem 4.2 applies to the problem (30).

To illustrate Theorem 4.3, we take $|f\left(t,u\right)|\le \phi \left(t\right)\psi \left(‖u‖\right)\frac{1}{\sqrt{t+25}}\left(\frac{5-t}{30}+1\right)$, $\phi \left(t\right)=\frac{1}{\sqrt{t+25}}\left(\frac{5-t}{30}+1\right)$, $\psi \left(‖u‖\right)=1$ with $‖\phi ‖=\frac{7}{30}$ and $N>\psi \left({N}_{\right)}‖\phi ‖\overline{Q}+\stackrel{^}{Q}=9.333636$ ( $\overline{Q}$ and $\stackrel{^}{Q}$ are given by (26) and (29) respectively). Thus all the conditions of Theorem 4.3 are satisfied. Hence the conclusion of Theorem 4.3 implies that the problem (30) has at least one solution on [0, T].

Several special cases of the existence results for the problem (1)-(3) follow by fixing the values of the parameters involved in the problem. For example, by taking α1= 1 = α2, ρ1 = 0 = ρ2, λ1 = 1 = μ1, λ2 = 0 = μ2, the results of this section correspond to the conditions: $u\left(0\right)={\int }_{0}^{n}u\left(s\right)ds$, ${u}^{\prime }\left(0\right)={\int }_{\xi }^{T}u\left(s\right)ds$ . In case we take α1 = 0 = α2, ρ1 = 1 = ρ2, λ1 = 1 = μ1, λ2 = 0 = μ2, we obtain the results for terminal-point conditions: $u\left(T\right)={\int }_{0}^{n}u\left(s\right)ds$, ${u}^{\prime }\left(T\right)={\int }_{\xi }^{T}u\left(s\right)ds$ . Letting α1 = 1 = α2, ρ1 = 1 = ρ2, λ1 = 1/eta, μ1 = 1/((Tξ), λ2 = 0 = μ2, we get the results for the average valued (integral) conditions: $u\left(0\right)+u\left(T\right)=\left(1/\eta \right){\int }_{0}^{n}u\left(s\right)ds$, ${u}^{\prime }\left(0\right)+{u}^{\prime }\left(T\right)=1/\left(T-\xi \right){\int }_{\xi }^{T}u\left(s\right)ds$. By taking α = 2, our results correspond to the equation: (D2 + kD)u(t) = f (t, u(t)), 0 < t < T, T > 0, which are also new.

## Acknowledgement

The research of J.J. Nieto was partially supported by the Ministerio de Economia y Competi-tividad of Spain under grant and MTM2013-43014-P, and Xunta de Galicia under grant GRC 2015/004.

## References

• [1]

Podlubny I., Fractional differential equations, Academic Press, San Diego, 1999 Google Scholar

• [2]

Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006 Google Scholar

• [3]

Klafter J., Lim S.C., Metzler R. (Editors), Fractional dynamics in physics, World Scientific, Singapore, 2011 Google Scholar

• [4]

Agarwal R.P., O'Regan D., Hristova S., Stability of Caputo fractional differential equations by Lyapunov functions, Appl. Math., 2015, 60, 653-676 Google Scholar

• [5]

Area I., Losada J., Nieto J.J., A note on the fractional logistic equation, Phys. A, 2016, 444, 182-187 Google Scholar

• [6]

Bai C., Impulsive periodic boundary value problems for fractional differential equation involving Riemann-Liouville sequential fractional derivative, J. Math. Anal. Appl., 2011, 384, 211-231Google Scholar

• [7]

Henderson J., Luca R., Positive solutions for a system of nonlocal fractional boundary value problems, Fract. Calc. Appl. Anal., 2013, 16, 985-1008Google Scholar

• [8]

Liu X., Liu Z., Fu X., Relaxation in nonconvex optimal control problems described by fractional differential equations, J. Math. Anal. Appl., 2014, 409, 446-458Google Scholar

• [9]

O'Regan D., Stanek S., Fractional boundary value problems with singularities in space variables, Nonlinear Dynam., 2013, 71, 641-652Google Scholar

• [10]

Punzo F., Terrone G., On the Cauchy problem for a general fractional porous medium equation with variable density, Nonlinear Anal., 2014, 98, 27-47Google Scholar

• [11]

Wang J.R., Wei W., Feckan M., Nonlocal Cauchy problems for fractional evolution equations involving Volterra-Fredholm type integral operators, Miskolc Math. Notes, 2012, 13, 127-147Google Scholar

• [12]

Wang J.R., Zhou Y., Feckan M., On the nonlocal Cauchy problem for semilinear fractional order evolution equations, Cent. Eur. J. Math., 2104, 12, 911-922 Google Scholar

• [13]

Ahmad B., Nieto J.J., Sequential fractional differential equations with three-point boundary conditions, Comput. Math. Appl., 2012, 64, 3046-3052. Google Scholar

• [14]

Ahmad B., Nieto J.J., Boundary value problems for a class of sequential integrodifferential equations of fractional order, J. Funct. Spaces Appl., 2013, Art. ID 149659, 8pp Google Scholar

• [15]

Ahmad B., Ntouyas S.K., Existence results for a coupled system of Caputo type sequential fractional differential equations with nonlocal integral boundary conditions, Appl. Math. Comput., 2015, 266, 615-622Google Scholar

• [16]

Klimek M., Sequential fractional differential equations with Hadamard derivative, Commun. Nonlinear Sci. Numer. Simul., 2011, 16, 4689-4697Google Scholar

• [17]

Ye H., Huang, R., On the nonlinear fractional differential equations with Caputo sequential fractional derivative, Adv. Math. Phys., 2015, Art. ID 174156, 9 pp Google Scholar

• [18]

Alsaedi A., Sivasundaram S., Ahmad B., On the generalization of second order nonlinear anti-periodic boundary value problems, Nonlinear Stud., 2009, 16, 415-420 Google Scholar

• [19]

Agarwal R.P., Ahmad B., Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions, Comput. Math. Appl., 2011, 62, 1200-1214 Google Scholar

• [20]

Ahmad B., Nieto J.J., Anti-periodic fractional boundary value problems, Comput. Math. Appl., 2011, 62, 1150-1156.Google Scholar

• [21]

Ahmad B., Losada J., Nieto J.J., On antiperiodic nonlocal three-point boundary value problems for nonlinear fractional differential equations, Discrete Dyn. Nat. Soc., 2015, Art. ID 973783, 7 ppGoogle Scholar

• [22]

Cao J., Yang Q., Huang Z., Existence of anti-periodic mild solutions for a class of semilinear fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 2012, 17, 277-283

• [23]

Jiang J., Solvability of anti-periodic boundary value problem for coupled system of fractional p-Laplacian equation, Adv. Difference Equ., 2015, 2015:305, 11 pp Google Scholar

• [24]

Smart D.R., Fixed point theorems, Cambridge University Press, 1980 Google Scholar

• [25]

Granas A., Dugundji J., Fixed point theory, Springer-Verlag, New York, 2003 Google Scholar

Accepted: 2016-07-27

Published Online: 2016-10-16

Published in Print: 2016-01-01

Citation Information: Open Mathematics, ISSN (Online) 2391-5455,

Export Citation