Existence of Solutions for Integrodifferential Equations of Fractional Order with Antiperiodic Boundary Conditions

Abstract

We discuss the existence of solutions for a nonlinear antiperiodic boundary value problem of integrodifferential equations of fractional order 𝑞∈(1,2]. The contraction mapping principle and Krasnoselskii's fixed point theorem are applied to establish the results.

1. Introduction

Recently, the subject of fractional differential equations has emerged as an important area of investigation. Fractional differential equations arise in many engineering and scientific disciplines as the fractional derivatives describe numerous events and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, and so forth. For some recent development on the subject, see [1–15] and the references therein.

Integrodifferential equations arise in many engineering and scientific disciplines, often as approximation to partial differential equations, which represent much of the continuum phenomena. Many forms of these equations are possible. For details, see [16–20] and the references therein.

Antiperiodic boundary value problems have recently received considerable attention as antiperiodic boundary conditions appear in numerous situations, for instance, see [21–25].

In this paper, we prove some existence and uniqueness results for the following antiperiodic fractional boundary value problem: ^𝑐𝐷^𝑞𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡),(𝜒𝑥)(𝑡)),𝑡∈[0,𝑇],𝑇>0,1<𝑞≤2,𝑥(0)=−𝑥(𝑇),𝑥^′(0)=−𝑥^′(𝑇), (1.1) where ^𝑐𝐷^𝑞 denotes the Caputo fractional derivative of order 𝑞, 𝑓∶[0,𝑇]×𝑋×𝑋→𝑋, and for 𝛾∶[0,𝑇]×[0,𝑇]→[0,∞), (𝜒𝑥)(𝑡)=^𝑡₀𝛾(𝑡,𝑠)𝑥(𝑠)𝑑𝑠, (1.2)

with 𝛾₀=max{∫^𝑡₀𝛾(𝑡,𝑠)𝑑𝑠∶(𝑡,𝑠)∈[0,𝑇]×[0,𝑇]}. Here, (𝑋,‖⋅‖) is a Banach space and 𝐶=𝐶([0,𝑇],𝑋) denotes the Banach space of all continuous functions from [0,𝑇]→𝑋 endowed with a topology of uniform convergence with the norm denoted by ‖⋅‖.

2. Preliminaries

First of all, we recall some basic definitions [26–28].

Definition 2.1. For a function 𝑔∶[0,∞)→ℝ, the Caputo derivative of fractional order 𝑞 is defined as ^𝑐𝐷^𝑞1𝑔(𝑡)=Γ(𝑛−𝑞)^𝑡₀(𝑡−𝑠)^{𝑛−𝑞−1}𝑔^(𝑛)(𝑠)𝑑𝑠,𝑛−1<𝑞<𝑛,𝑛=[𝑞]+1, (2.1)where [𝑞] denotes the integer part of the real number 𝑞.

Definition 2.2. The Riemann-Liouville fractional integral of order 𝑞 is defined as 𝐼^𝑞1𝑔(𝑡)=Γ(𝑞)^𝑡₀𝑔(𝑠)(𝑡−𝑠)^1−𝑞𝑑𝑠,𝑞>0, (2.2)provided that the integral exists.

Definition 2.3. The Riemann-Liouville fractional derivative of order 𝑞 for a function 𝑔(𝑡) is defined by 𝐷^𝑞1𝑔(𝑡)=(𝑑Γ(𝑛−𝑞))𝑑𝑡^𝑛𝑡₀𝑔(𝑠)(𝑡−𝑠)^{𝑞−𝑛+1}𝑑𝑠,𝑛=[𝑞]+1, (2.3)provided that the right-hand side is pointwise defined on (0,∞).

Lemma 2.4 (see [8]). For 𝑞>0, the general solution of the fractional differential equation ^𝑐𝐷^𝑞𝑥(𝑡)=0 is given by 𝑥(𝑡)=𝑐₀+𝑐₁𝑡+𝑐₂𝑡²+⋯+𝑐_𝑛−1𝑡^𝑛−1, (2.4)where 𝑐_𝑖∈ℝ, 𝑖=0,1,2,…,𝑛−1 (𝑛=[𝑞]+1).

In view of Lemma 2.4, it follows that 𝐼^𝑞𝑐𝐷^𝑞𝑥(𝑡)=𝑥(𝑡)+𝑐₀+𝑐₁𝑡+𝑐₂𝑡²+⋯+𝑐_𝑛−1𝑡^𝑛−1, (2.5) for some 𝑐_𝑖∈ℝ, 𝑖=0,1,2,…,𝑛−1 (𝑛=[𝑞]+1).

Now, we state a known result due to Krasnoselskii [29] which is needed to prove the existence of at least one solution of (1.1).

Theorem 2.5. Let 𝑀 be a closed convex and nonempty subset of a Banach space 𝑋. Let 𝐴,𝐵 be the operators such that (i)𝐴𝑥+𝐵𝑦∈𝑀 whenever 𝑥,𝑦∈𝑀; (ii)𝐴 is compact and continuous; (iii)𝐵 is a contraction mapping. Then there exists 𝑧∈𝑀 such that 𝑧=𝐴𝑧+𝐵𝑧.

Lemma 2.6. For any 𝜔∈𝐶[0,𝑇], the unique solution of the boundary value problem ^𝑐𝐷^𝑞𝑥(𝑡)=𝜔(𝑡),0<𝑡<𝑇,1<𝑞≤2,𝑥(0)=−𝑥(𝑇),𝑥^′(0)=−𝑥^′(𝑇) (2.6) is given by 𝑥(𝑡)=^𝑇₀𝐺(𝑡,𝑠)𝜔(𝑠)𝑑𝑠, (2.7)where 𝐺(𝑡,𝑠) is the Green's function given by 𝐺(𝑡,𝑠)=(𝑡−𝑠)^𝑞−1−(1/2)(𝑇−𝑠)^𝑞−1+Γ(𝑞)(𝑇−2𝑡)(𝑇−𝑠)^𝑞−2−4Γ(𝑞−1),𝑠≤𝑡,(𝑇−𝑠)^𝑞−1+2Γ(𝑞)(𝑇−2𝑡)(𝑇−𝑠)^𝑞−24Γ(𝑞−1),𝑡≤𝑠. (2.8)

Proof. Using (2.5), for some constants 𝑏₁,𝑏₂∈ℝ, we have 𝑥(𝑡)=𝐼^𝑞𝜔(𝑡)−𝑏₁−𝑏₂𝑡=^𝑡₀(𝑡−𝑠)^𝑞−1Γ(𝑞)𝜔(𝑠)𝑑𝑠−𝑏₁−𝑏₂𝑡. (2.9)In view of the relations ^𝑐𝐷^𝑞𝐼^𝑞𝑥(𝑡)=𝑥(𝑡) and 𝐼^𝑞𝐼^𝑝𝑥(𝑡)=𝐼^𝑞+𝑝𝑥(𝑡) for 𝑞,𝑝>0,𝑥∈𝐿(0,1), we obtain 𝑥^′(𝑡)=^𝑡₀(𝑡−𝑠)^𝑞−2Γ(𝑞−1)𝜔(𝑠)𝑑𝑠−𝑏₂. (2.10)Applying the boundary conditions 𝑥(0)=−𝑥(𝑇),𝑥^′(0)=−𝑥^′(𝑇), we find that 𝑏₁=12Γ(𝑞)^𝑇₀(𝑇−𝑠)^𝑞−1𝑇𝜔(𝑠)𝑑𝑠−4Γ(𝑞−1)^𝑇₀(𝑇−𝑠)^𝑞−2𝑏𝜔(𝑠)𝑑𝑠,₂=12Γ(𝑞−1)^𝑇₀(𝑇−𝑠)^𝑞−2𝜔(𝑠)𝑑𝑠. (2.11) Thus, the unique solution of (2.6) is 𝑥(𝑡)=^𝑡₀(𝑡−𝑠)^𝑞−11Γ(𝑞)𝜔(𝑠)𝑑𝑠−2^𝑇₀(𝑇−𝑠)^𝑞−1+1Γ(𝑞)𝜔(𝑠)𝑑𝑠4(𝑇−2𝑡)^𝑇₀(𝑇−𝑠)^𝑞−2Γ(𝑞−1)𝜔(𝑠)𝑑𝑠=^𝑇₀𝐺(𝑡,𝑠)𝜔(𝑠)𝑑𝑠, (2.12)where 𝐺(𝑡,𝑠) is given by (2.8). This completes the proof.

3. Main Results

To prove the main results, we need the following assumptions:

(𝐴₁)‖𝑓(𝑡,𝑥(𝑡),(𝜒𝑥)(𝑡))−𝑓(𝑡,𝑦(𝑡),(𝜒𝑦)(𝑡))‖≤𝐿₁‖𝑥−𝑦‖+𝐿₁‖𝜒𝑥−𝜒𝑦‖, for all 𝑡∈[0,𝑇], 𝑥,𝑦∈𝑋;(𝐴₂)‖𝑓(𝑡,𝑥(𝑡),(𝜒𝑥)(𝑡))‖≤𝜇(𝑡), for all (𝑡,𝑥,𝜒𝑥)∈[0,𝑇]×𝑋×𝑋 and 𝜇∈𝐿¹([0,𝑇],𝑅⁺).

Theorem 3.1. Let 𝑓∶[0,𝑇]×𝑋×𝑋→𝑋 be a jointly continuous function satisfying the assumption (𝐴₁) with (𝐿₁+𝛾₀𝐿₁)≤Γ(𝑞+1)/𝑇^𝑞(3+𝑞/2). Then the antiperiodic boundary value problem (1.1) has a unique solution.

Proof. Define Θ∶𝐶→𝐶 by (Θ𝑥)(𝑡)=^𝑡₀(𝑡−𝑠)^𝑞−1−1Γ(𝑞)𝑓(𝑠,𝑥(𝑠),(𝜒𝑥)(𝑠))𝑑𝑠2^𝑇₀(𝑇−𝑠)^𝑞−1+1Γ(𝑞)𝑓(𝑠,𝑥(𝑠),(𝜒𝑥)(𝑠))𝑑𝑠4(𝑇−2𝑡)^𝑇₀(𝑇−𝑠)^𝑞−2Γ(𝑞−1)𝑓(𝑠,𝑥(𝑠),(𝜒𝑥)(𝑠))𝑑𝑠,𝑡∈[0,𝑇]. (3.1) Setting sup_{𝑡∈[0,𝑇]}‖𝑓(𝑡,0,0)‖=𝑀 and choosing 𝑟≥(𝑀𝑇^𝑞/Γ(𝑞+1))(3+𝑞/2), we show that Θ𝐵_𝑟⊂𝐵_𝑟, where 𝐵_𝑟={𝑥∈𝐶∶‖𝑥‖≤𝑟}. For 𝑥∈𝐵_𝑟, we have‖(Θ𝑥)(𝑡)‖≤^𝑡₀(𝑡−𝑠)^𝑞−1+1Γ(𝑞)‖𝑓(𝑠,𝑥(𝑠),(𝜒𝑥)(𝑠))‖𝑑𝑠2^𝑇₀(𝑇−𝑠)^𝑞−1+1Γ(𝑞)‖𝑓(𝑠,𝑥(𝑠),(𝜒𝑥)(𝑠))‖𝑑𝑠4|𝑇−2𝑡|^𝑇₀(𝑇−𝑠)^𝑞−2≤Γ(𝑞−1)‖𝑓(𝑠,𝑥(𝑠),(𝜒𝑥)(𝑠))‖𝑑𝑠_𝑡0(𝑡−𝑠)^𝑞−1+1Γ(𝑞)[‖𝑓(𝑠,𝑥(𝑠),(𝜒𝑥)(𝑠))−𝑓(𝑠,0,0)‖+‖𝑓(𝑠,0,0)‖]𝑑𝑠2^𝑇₀(𝑇−𝑠)^𝑞−1+1Γ(𝑞)[‖𝑓(𝑠,𝑥(𝑠),(𝜒𝑥)(𝑠))−𝑓(𝑠,0,0)‖+‖𝑓(𝑠,0,0)‖]𝑑𝑠4|𝑇−2𝑡|^𝑇₀(𝑇−𝑠)^𝑞−2Γ(𝑞−1)[‖𝑓(𝑠,𝑥(𝑠),(𝜒𝑥)(𝑠))−𝑓(𝑠,0,0)‖+‖𝑓(𝑠,0,0)‖]𝑑𝑠≤((𝐿₁+𝛾₀𝐿₁1)𝑟+𝑀)[Γ(𝑞)^𝑡₀(𝑡−𝑠)^𝑞−11𝑑𝑠+2Γ(𝑞)^𝑇₀(𝑇−𝑠)^𝑞−1+1𝑑𝑠4Γ(𝑞−1)|𝑇−2𝑡|^𝑇₀(𝑇−𝑠)^𝑞−2𝑑𝑠]≤((𝐿₁+𝛾₀𝐿₁𝑇)𝑟+𝑀)[^𝑞𝑞2Γ(𝑞+1)(3+2)]≤𝑟. (3.2)Now, for 𝑥,𝑦∈𝐶 and for each 𝑡∈[0,𝑇], we obtain ‖(Θ𝑥)(𝑡)−(Θ𝑦)(𝑡)‖≤^𝑡₀(𝑡−𝑠)^𝑞−1+1Γ(𝑞)‖𝑓(𝑠,𝑥(𝑠),(𝜒𝑥)(𝑠))−𝑓(𝑠,𝑦(𝑠),(𝜒𝑦)(𝑠))‖𝑑𝑠2^𝑇₀(𝑇−𝑠)^𝑞−1+1Γ(𝑞)‖𝑓(𝑠,𝑥(𝑠),(𝜒𝑥)(𝑠))−𝑓(𝑠,𝑦(𝑠),(𝜒𝑦)(𝑠))‖𝑑𝑠4|𝑇−2𝑡|^𝑇₀(𝑇−𝑠)^𝑞−2Γ(𝑞−1)‖𝑓(𝑠,𝑥(𝑠),(𝜒𝑥)(𝑠))−𝑓(𝑠,𝑦(𝑠),(𝜒𝑦)(𝑠))‖𝑑𝑠≤(𝐿₁+𝛾₀𝐿₁1)‖𝑥−𝑦‖[Γ(𝑞)^𝑡₀(𝑡−𝑠)^𝑞−11𝑑𝑠+2Γ(𝑞)^𝑇₀(𝑇−𝑠)^𝑞−1+1𝑑𝑠4Γ(𝑞−1)|𝑇−2𝑡|^𝑇₀(𝑇−𝑠)^𝑞−2≤𝑑𝑠](𝐿₁+𝛾₀𝐿₁)𝑇^𝑞𝑞2Γ(𝑞+1)(3+2)‖𝑥−𝑦‖≤Λ_{𝐿1,𝐿1,𝛾0,𝑇,𝑞}‖𝑥−𝑦‖, (3.3)where Λ_{𝐿1,𝐿1,𝛾0,𝑇,𝑞}=((𝐿₁+𝛾₀𝐿₁)𝑇^𝑞/2Γ(𝑞+1))(3+𝑞/2), which depends only on the parameters involved in the problem. As Λ_{𝐿1,𝐿1,𝛾0,𝑇,𝑞}<1, therefore Θ is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle (Banach fixed point theorem).

Theorem 3.2. Let 𝑓∶[0,𝑇]×𝑋×𝑋→𝑋 be a jointly continuous function mapping bounded subsets of [0,1]×𝑋×𝑋 into relatively compact subsets of 𝑋, and the assumptions (𝐴₁)-(𝐴₂) hold with ((𝐿₁+𝛾₀𝐿₁)𝑇^𝑞/2Γ(𝑞+1))(1+𝑞/2)<1. Then the antiperiodic boundary value problem (1.1) has at least one solution on [0,𝑇].

Proof. Let us fix 𝑟≥‖𝜇‖_𝐿1𝑇^𝑞𝑞2Γ(𝑞+1)(3+2), (3.4)and consider 𝐵_𝑟={𝑥∈𝐶∶‖𝑥‖≤𝑟}. We define the operators Φ and Ψ on 𝐵_𝑟 as 1(Φ𝑥)(𝑡)=Γ(𝑞)^𝑡₀(𝑡−𝑠)^𝑞−11𝑓(𝑠,𝑥(𝑠),(𝜒𝑥)(𝑠))𝑑𝑠,(Ψ𝑥)(𝑡)=−2^𝑇₀(𝑇−𝑠)^𝑞−1+1Γ(𝑞)𝑓(𝑠,𝑥(𝑠),(𝜒𝑥)(𝑠))𝑑𝑠4(𝑇−2𝑡)^𝑇₀(𝑇−𝑠)^𝑞−2Γ(𝑞−1)𝑓(𝑠,𝑥(𝑠),(𝜒𝑥)(𝑠))𝑑𝑠. (3.5)For 𝑥,𝑦∈𝐵_𝑟, we find that ‖Φ𝑥+Ψ𝑦‖≤‖𝜇‖_𝐿1𝑇^𝑞𝑞2Γ(𝑞+1)(3+2)≤𝑟. (3.6)Thus, Φ𝑥+Ψ𝑦∈𝐵_𝑟. It follows from the assumption (𝐴₁) that Ψ is a contraction mapping for ((𝐿₁+𝛾₀𝐿₁)𝑇^𝑞/2Γ(𝑞+1))(1+𝑞/2)<1. Continuity of 𝑓 implies that the operator Φ is continuous. Also, Φ is uniformly bounded on 𝐵_𝑟 as ‖Φ𝑥‖≤‖𝜇‖_𝐿1𝑇^𝑞.Γ(𝑞+1) (3.7)Now we prove the compactness of the operator Φ. In view of (𝐴₁), we define sup_{(𝑡,𝑥,𝜒𝑥)∈Ω}‖𝑓(𝑡,𝑥,(𝜒𝑥))‖=𝑓_max, Ω=[0,𝑇]×𝐵_𝑟×𝐵_𝑟, and consequently we have ‖(Φ𝑥)(𝑡₁)−(Φ𝑥)(𝑡₂1)‖=‖Γ(𝑞)^𝑡1₀[(𝑡₂−𝑠)^𝑞−1−(𝑡₁−𝑠)^𝑞−1+]𝑓(𝑠,𝑥(𝑠),(𝜒𝑥)(𝑠))𝑑𝑠_𝑡2𝑡1(𝑡₂−𝑠)^𝑞−1≤𝑓𝑓(𝑠,𝑥(𝑠),(𝜒𝑥)(𝑠))𝑑𝑠‖_maxΓ(𝑞+1)|2(𝑡₂−𝑡₁)^𝑞+𝑡^𝑞₁−𝑡^𝑞₂|, (3.8)which is independent of 𝑥. So Φ is relatively compact on 𝐵_𝑟. Hence, by Arzela Ascoli theorem, Φ is compact on 𝐵_𝑟. Thus all the assumptions of Theorem 2.5 are satisfied and the conclusion of Theorem 2.5 implies that the antiperiodic boundary value problem (1.1) has at least one solution on [0,𝑇].

Example 3.3. Consider the following antiperiodic boundary value problem: ^𝑐𝐷^𝑞1𝑥(𝑡)=(𝑡+4)²‖𝑥‖+1+‖𝑥‖^𝑡₀𝑒^{−(𝑠−𝑡)}16𝑥(𝑠)𝑑𝑠,𝑡∈[0,1],1<𝑞≤2,𝑥(0)=−𝑥(1),𝑥^′(0)=−𝑥^′(1). (3.9) Here, 𝑓(𝑡,𝑥)=(1/(𝑡+4)²)(‖𝑥‖/(1+‖𝑥‖)),𝛾(𝑡,𝑠)=𝑒^{−(𝑠−𝑡)}/16, 𝑇=1. Clearly, 1‖𝑓(𝑡,𝑥,𝜒𝑥)−𝑓(𝑡,𝑦,𝜒𝑦)‖≤16(‖𝑥−𝑦‖+‖𝜒𝑥−𝜒𝑦‖). (3.10)So (𝐴₁) is satisfied with 𝐿₁=𝐿₁=1/16. Further (𝐿₁+𝛾₀𝐿₁)𝑇^𝑞𝑞Γ(𝑞+1)(3+2)<1⇔(3+𝑞/2)<Γ(𝑞+1)16𝑒. (3.11)Thus, by Theorem 3.1, the boundary value problem (3.9) has a unique solution on [0,1].