Some Existence Results for Impulsive Nonlinear Fractional Differential Equations with Closed Boundary Conditions

Abstract

We investigate some existence results for the solutions to impulsive fractional differential equations having closed boundary conditions. Our results are based on contracting mapping principle and Burton-Kirk fixed point theorem.

1. Introduction

This paper considers the existence and uniqueness of the solutions to the closed boundary value problem (BVP), for the following impulsive fractional differential equation: ^𝐶𝐷^𝛼𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)),𝑡∈𝐽∶=[0,𝑇],𝑡≠𝑡_𝑘𝑡,1<𝛼≤2,Δ𝑥_𝑘=𝐼_𝑘𝑥𝑡⁻_𝑘,Δ𝑥^′𝑡_𝑘=𝐼^∗_𝑘𝑥𝑡⁻_𝑘𝑥(𝑇)(0),𝑘=1,2,…,𝑝,=𝑎𝑥+𝑏𝑇𝑥^′(0),𝑇𝑥^′(𝑇)=𝑐𝑥(0)+𝑑𝑇𝑥^′(0), (1.1) where ^𝐶𝐷^𝛼 is Caputo fractional derivative, 𝑓∈𝐶(𝐽×𝑅,𝑅), 𝐼_𝑘,𝐼^∗_𝑘∈𝐶(𝑅,𝑅), Δ𝑥𝑡_𝑘=𝑥𝑡⁺_𝑘−𝑥𝑡⁻_𝑘, (1.2) with 𝑥𝑡⁺_𝑘=lim_ℎ→0+𝑥𝑡_𝑘+ℎ,𝑥𝑡⁻_𝑘=lim_ℎ→0−𝑥𝑡_𝑘+ℎ, (1.3) and Δ𝑥^′(𝑡_𝑘) has a similar meaning for 𝑥^′(𝑡), where 0=𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑝<𝑡_𝑝+1=𝑇, (1.4)𝑎, 𝑏, 𝑐, and 𝑑 are real constants with Δ∶=𝑐(1−𝑏)+(1−𝑎)(1−𝑑)≠0.

The boundary value problems for nonlinear fractional differential equations have been addressed by several researchers during last decades. That is why, the fractional derivatives serve an excellent tool for the description of hereditary properties of various materials and processes. Actually, fractional differential equations arise in many engineering and scientific disciplines such as, physics, chemistry, biology, electrochemistry, electromagnetic, control theory, economics, signal and image processing, aerodynamics, and porous media (see [1–7]). For some recent development, see, for example, [8–14].

On the other hand, theory of impulsive differential equations for integer order has become important and found its extensive applications in mathematical modeling of phenomena and practical situations in both physical and social sciences in recent years. One can see a noticeable development in impulsive theory. For instance, for the general theory and applications of impulsive differential equations we refer the readers to [15–17].

Moreover, boundary value problems for impulsive fractional differential equations have been studied by some authors (see [18–20] and references therein). However, to the best of our knowledge, there is no study considering closed boundary value problems for impulsive fractional differential equations.

Here, we notice that the closed boundary conditions in (1.1) include quasi-periodic boundary conditions (𝑏=𝑐=0) and interpolate between periodic (𝑎=𝑑=1,𝑏=𝑐=0) and antiperiodic (𝑎=𝑑=−1,𝑏=𝑐=0) boundary conditions.

Motivated by the mentioned recent work above, in this study, we investigate the existence and uniqueness of solutions to the closed boundary value problem for impulsive fractional differential equation (1.1). Throughout this paper, in Section 2, we present some notations and preliminary results about fractional calculus and differential equations to be used in the following sections. In Section 3, we discuss some existence and uniqueness results for solutions of BVP (1.1), that is, the first one is based on Banach’s fixed point theorem, the second one is based on the Burton-Kirk fixed point theorem. At the end, we give an illustrative example for our results.

2. Preliminaries

Let us set 𝐽₀=[0,𝑡₁], 𝐽₁=(𝑡₁,𝑡₂],…,𝐽_𝑘−1=(𝑡_𝑘−1,𝑡_𝑘], 𝐽_𝑘=(𝑡_𝑘,𝑡_𝑘+1], 𝐽^′∶=[0,𝑇]⧵{𝑡₁,𝑡₂,…,𝑡_𝑝} and introduce the set of functions:

𝑃𝐶(𝐽,𝑅)={𝑥∶𝐽→𝑅∶𝑥∈𝐶((𝑡_𝑘,𝑡_𝑘+1],𝑅), 𝑘=0,1,2,…,𝑝 and there exist 𝑥(𝑡⁺_𝑘) and 𝑥(𝑡⁻_𝑘), 𝑘=1,2,…,𝑝 with 𝑥(𝑡⁻_𝑘)=𝑥(𝑡_𝑘)} and

𝑃𝐶¹(𝐽,𝑅)={𝑥∈𝑃𝐶(𝐽,𝑅), 𝑥^′∈𝐶((𝑡_𝑘,𝑡_𝑘+1],𝑅), 𝑘=0,1,2,…,𝑝 and there exist 𝑥^′(𝑡⁺_𝑘) and 𝑥^′(𝑡⁻_𝑘), 𝑘=1,2,…,𝑝 with 𝑥^′(𝑡⁻_𝑘)=𝑥^′(𝑡_𝑘)} which is a Banach space with the norm ‖𝑥‖=sup_𝑡∈𝐽{‖𝑥‖_𝑃𝐶,‖𝑥^′‖_𝑃𝐶} where ‖𝑥‖_𝑃𝐶∶=sup{|𝑥(𝑡)|∶𝑡∈𝐽}.

The following definitions and lemmas were given in [4].

Definition 2.1. The fractional (arbitrary) order integral of the function ℎ∈𝐿¹(𝐽,𝑅₊) of order 𝛼∈𝑅₊ is defined by 𝐼^𝛼₀ℎ(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1ℎ(𝑠)𝑑𝑠, (2.1) where Γ(⋅) is the Euler gamma function.

Definition 2.2. For a function ℎ given on the interval 𝐽, Caputo fractional derivative of order 𝛼>0 is defined by ^𝐶𝐷^𝛼₀⁺ℎ(𝑡)=1Γ(𝑛−𝛼)^𝑡₀(𝑡−𝑠)^{𝑛−𝛼−1}ℎ^(𝑛)(𝑠)𝑑𝑠,𝑛=[𝛼]+1, (2.2) where the function ℎ(𝑡) has absolutely continuous derivatives up to order (𝑛−1).

Lemma 2.3. Let 𝛼>0, then the differential equation ^𝐶𝐷^𝛼ℎ(𝑡)=0 (2.3) has solutions ℎ(𝑡)=𝑐₀+𝑐₁𝑡+𝑐₂𝑡²+⋯+𝑐_𝑛−1𝑡^𝑛−1,𝑐_𝑖∈𝑅,𝑖=0,1,2,…,𝑛−1,𝑛=[𝛼]+1. (2.4)

The following lemma was given in [4, 10].

Lemma 2.4. Let 𝛼>0, then 𝐼^𝛼𝐶𝐷^𝛼ℎ(𝑡)=ℎ(𝑡)+𝑐₀+𝑐₁𝑡+𝑐₂𝑡²+⋯+𝑐_𝑛−1𝑡^𝑛−1, (2.5) for some 𝑐_𝑖∈𝑅, 𝑖=0,1,2,…,𝑛−1, 𝑛=[𝛼]+1.

The following theorem is known as Burton-Kirk fixed point theorem and proved in [21].

Theorem 2.5. Let 𝑋 be a Banach space and 𝐴, 𝐷∶𝑋→𝑋 two operators satisfying: (a)𝐴 is a contraction, and (b)𝐷 is completely continuous. Then either (i) the operator equation 𝑥=𝐴(𝑥)+𝐷(𝑥) has a solution, or(ii) the set 𝜀={𝑥∈𝑋∶𝑥=𝜆𝐴(𝑥/𝜆)+𝜆𝐷(𝑥)} is unbounded for 𝜆∈(0,1).

Theorem 2.6 (see [22], Banach’s fixed point theorem). Let 𝑆 be a nonempty closed subset of a Banach space 𝑋, then any contraction mapping 𝑇 of 𝑆 into itself has a unique fixed point.

Next we prove the following lemma.

Lemma 2.7. Let 1<𝛼≤2 and let ℎ∶𝐽→𝑅 be continuous. A function 𝑥(𝑡) is a solution of the fractional integral equation: 𝑥(𝑡)=∫^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)ℎ(𝑠)𝑑𝑠+Ω₁(𝑡)Λ₁+Ω₂(𝑡)Λ₂,𝑡∈𝐽₀∫^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1Γ(𝛼)ℎ(𝑠)𝑑𝑠+1+Ω₁(𝑡)∑^𝑘_𝑖=1∫^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1Γ(𝛼)ℎ(𝑠)+𝑑𝑠𝑇−𝑡_𝑘Ω₁(𝑡)+Ω₂(𝑡)+𝑡−𝑡∑_𝑘^𝑘_𝑖=1∫^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()ℎ(𝑠)+𝛼−1𝑑𝑠1+Ω₁(𝑡)∑^𝑘−1_𝑖=1𝑡_𝑘−𝑡_𝑖∫^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()ℎ(𝑠)𝛼−1𝑑𝑠+Ω₁(𝑡)∫^𝑇_𝑡𝑘(𝑇−𝑠)^𝛼−1Γ(𝛼)ℎ(𝑠)𝑑𝑠+Ω₂(𝑡)∫^𝑇_𝑡𝑘(𝑇−𝑠)^𝛼−2Γ()ℎ(𝑠)𝛼−1𝑑𝑠+1+Ω₁(𝑡)∑^𝑘_𝑖=1𝐼_𝑖𝑥𝑡⁻_𝑖+𝐼^∗_𝑘𝑥𝑡⁻_𝑘+Ω₁(𝑡)∑^𝑘−1_𝑖=1𝑇−𝑡_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖+Ω₂(𝑡)∑^𝑘−1_𝑖=1𝐼^∗_𝑖𝑥𝑡⁻_𝑖+∑^𝑘−1_𝑖=1𝑡−𝑡_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖,𝑡∈𝐽_𝑘,𝑘=1,2,…,𝑝 (2.6) if and only if 𝑥(𝑡) is a solution of the fractional BVP ^𝐶𝐷^𝛼𝑥(𝑡)=ℎ(𝑡),𝑡∈𝐽^′,Δ𝑥𝑡_𝑘=𝐼_𝑘𝑥𝑡⁻_𝑘,Δ𝑥^′𝑡_𝑘=𝐼^∗_𝑘𝑥𝑡⁻_𝑘,𝑥(𝑇)=𝑎𝑥(0)+𝑏𝑇𝑥^′(0),𝑇𝑥^′(𝑇)=𝑐𝑥(0)+𝑑𝑇𝑥^′(0), (2.7) whereΛ₁∶=^𝑇_𝑡𝑘(𝑇−𝑠)^𝛼−1Γ(𝛼)ℎ(𝑠)𝑑𝑠+^𝑘_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1Γ(𝛼)ℎ(𝑠)+𝑑𝑠_𝑘𝑖=1𝑇−𝑡_𝑘^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()ℎ(𝑠)𝛼−1𝑑𝑠+^𝑘−1_𝑖=1𝑡_𝑘−𝑡_𝑖^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()ℎ(𝑠)+𝛼−1𝑑𝑠_𝑘𝑖=1𝐼_𝑖𝑥𝑡⁻_𝑖+^𝑘−1_𝑖=1𝑇−𝑡_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖+𝐼^∗_𝑘𝑥𝑡⁻_𝑘,Λ₂∶=^𝑇_𝑡𝑘(𝑇−𝑠)^𝛼−2Γ()ℎ(𝑠)𝛼−1𝑑𝑠+^𝑘_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()ℎ(𝑠)𝛼−1𝑑𝑠+^𝑘−1_𝑖=1𝐼^∗_𝑖𝑥𝑡⁻_𝑖,Ω₁(𝑡)()∶=−1−𝑑Δ−𝑐𝑡,Ω𝑇Δ₂(𝑡)()𝑇∶=1−𝑏Δ−(1−𝑎)𝑡Δ. (2.8)

Proof. Let 𝑥 be the solution of (2.7). If 𝑡∈𝐽₀, then Lemma 2.4 implies that 𝑥(𝑡)=𝐼^𝛼ℎ(𝑡)−𝑐₀−𝑐₁𝑡=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)ℎ(𝑠)𝑑𝑠−𝑐₀−𝑐₁𝑥𝑡,_′(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−2Γ()ℎ(𝑠)𝛼−1𝑑𝑠−𝑐₁, (2.9) for some 𝑐₀,𝑐₁∈𝑅.
If 𝑡∈𝐽₁, then Lemma 2.4 implies that 𝑥(𝑡)=^𝑡_𝑡1(𝑡−𝑠)^𝛼−1Γ(𝛼)ℎ(𝑠)𝑑𝑠−𝑑₀−𝑑₁𝑡−𝑡₁,𝑥^′(𝑡)=^𝑡_𝑡1(𝑡−𝑠)^𝛼−2Γ()ℎ(𝑠)𝛼−1𝑑𝑠−𝑑₁, (2.10) for some 𝑑₀,𝑑₁∈𝑅. Thus we have 𝑥𝑡⁻₁=^𝑡1_𝑡0𝑡₁−𝑠^𝛼−1Γ(𝛼)ℎ(𝑠)𝑑𝑠−𝑐₀−𝑐₁𝑡₁,𝑥𝑡⁺₁=−𝑑₀,𝑥^′𝑡⁻₁=^𝑡1_𝑡0𝑡₁−𝑠^𝛼−2Γ()ℎ(𝑠)𝛼−1𝑑𝑠−𝑐₁,𝑥^′𝑡⁺₁=−𝑑₁. (2.11) Observing that Δ𝑥𝑡₁=𝑥𝑡⁺₁−𝑥𝑡⁻₁=𝐼₁𝑥𝑡⁻₁,Δ𝑥^′𝑡₁=𝑥^′𝑡⁺₁−𝑥^′𝑡⁻₁=𝐼^∗₁𝑥𝑡⁻₁, (2.12) then we have −𝑑₀=^𝑡1_𝑡0𝑡₁−𝑠^𝛼−1Γ(𝛼)ℎ(𝑠)𝑑𝑠−𝑐₀−𝑐₁𝑡₁+𝐼₁𝑥𝑡⁻₁,−𝑑₁=^𝑡1_𝑡0𝑡₁−𝑠^𝛼−2Γ()ℎ(𝑠)𝛼−1𝑑𝑠−𝑐₁+𝐼^∗₁𝑥𝑡⁻₁, (2.13) hence, for 𝑡∈(𝑡₁,𝑡₂], 𝑥(𝑡)=^𝑡_𝑡1(𝑡−𝑠)^𝛼−1Γ(𝛼)ℎ(𝑠)𝑑𝑠+^𝑡1_𝑡0𝑡₁−𝑠^𝛼−1Γ(𝛼)ℎ(𝑠)+𝑑𝑠𝑡−𝑡₁^𝑡1_𝑡0𝑡₁−𝑠^𝛼−2Γ()ℎ(𝑠)𝛼−1𝑑𝑠+𝐼₁𝑥𝑡⁻₁+𝑡−𝑡₁𝐼^∗₁𝑥𝑡⁻₁−𝑐₀−𝑐₁𝑥𝑡,_′(𝑡)=^𝑡_𝑡1(𝑡−𝑠)^𝛼−2Γ()ℎ(𝑠)𝛼−1𝑑𝑠+^𝑡1_𝑡0𝑡₁−𝑠^𝛼−2Γ()ℎ(𝑠)𝛼−1𝑑𝑠+𝐼^∗₁𝑥𝑡⁻₁−𝑐₁. (2.14) If 𝑡∈𝐽₂, then Lemma 2.4 implies that 𝑥(𝑡)=^𝑡_𝑡2(𝑡−𝑠)^𝛼−1Γ(𝛼)ℎ(𝑠)𝑑𝑠−𝑒₀−𝑒₁𝑡−𝑡₂,𝑥^′(𝑡)=^𝑡_𝑡2(𝑡−𝑠)^𝛼−2Γ()ℎ(𝑠)𝛼−1𝑑𝑠−𝑒₁, (2.15) for some 𝑒₀,𝑒₁∈𝑅. Thus we have 𝑥𝑡⁻₂=^𝑡2_𝑡1𝑡₂−𝑠^𝛼−1Γ(𝛼)ℎ(𝑠)𝑑𝑠+^𝑡1_𝑡0𝑡₁−𝑠^𝛼−1Γ(𝛼)ℎ(𝑠)+𝑡𝑑𝑠₂−𝑡₁^𝑡1_𝑡0𝑡₁−𝑠^𝛼−2Γ()ℎ(𝑠)𝛼−1𝑑𝑠+𝐼₁𝑥𝑡⁻₁+𝑡₂−𝑡₁𝐼^∗₁𝑥𝑡⁻₁−𝑐₀−𝑐₁𝑡₂,𝑥𝑡⁺₂=−𝑒₀,𝑥^′𝑡⁻₂=^𝑡2_𝑡1𝑡₂−𝑠^𝛼−2Γ()ℎ(𝑠)𝛼−1𝑑𝑠+^𝑡1_𝑡0𝑡₁−𝑠^𝛼−2Γ()ℎ(𝑠)𝛼−1𝑑𝑠+𝐼^∗₁𝑥𝑡⁻₁−𝑐₁,𝑥^′𝑡⁺₂=−𝑒₁. (2.16) Similarly we observe that Δ𝑥𝑡₂=𝑥𝑡⁺₂−𝑥𝑡⁻₂=𝐼₂𝑥𝑡⁻₂,Δ𝑥^′𝑡₂=𝑥^′𝑡⁺₂−𝑥^′𝑡⁻₂=𝐼^∗₂𝑥𝑡⁻₂, (2.17) thus we have −𝑒₀=𝑥𝑡⁻₂+𝐼₂𝑥𝑡⁻₂,−𝑒₁=𝑥^′𝑡⁻₂+𝐼^∗₂𝑥𝑡⁻₂. (2.18) Hence, for 𝑡∈(𝑡₂,𝑡₃], 𝑥(𝑡)=^𝑡_𝑡2(𝑡−𝑠)^𝛼−1Γ(𝛼)ℎ(𝑠)𝑑𝑠+^𝑡1_𝑡0𝑡₁−𝑠^𝛼−1Γ(𝛼)ℎ(𝑠)𝑑𝑠+^𝑡2_𝑡1𝑡₂−𝑠^𝛼−1Γ(𝛼)ℎ(𝑠)+𝑡𝑑𝑠₂−𝑡₁^𝑡1_𝑡0𝑡₁−𝑠^𝛼−2Γ()ℎ(𝑠)+𝛼−1𝑑𝑠𝑡−𝑡₂^𝑡1_𝑡0𝑡₁−𝑠^𝛼−1Γ()ℎ(𝑠)𝛼−1𝑑𝑠+^𝑡2_𝑡1𝑡₂−𝑠^𝛼−1Γ()ℎ(𝑠)𝛼−1𝑑𝑠+𝐼₁𝑥𝑡⁻₁+𝐼₂𝑥𝑡⁻₂+𝑡−𝑡₁𝐼^∗₁𝑥𝑡⁻₁+𝐼^∗₂𝑥𝑡⁻₂−𝑐₀−𝑐₁𝑡. (2.19) By a similar process, if 𝑡∈𝐽_𝑘, then again from Lemma 2.4 we get 𝑥(𝑡)=∫^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1Γ(𝛼)ℎ(𝑠)𝑑𝑠+∑^𝑘_𝑖=1∫^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1Γ(𝛼)ℎ(𝑠)+𝑑𝑠_𝑘𝑖∑₌₁𝑡−𝑡_𝑘∫^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()ℎ(𝑠)𝛼−1𝑑𝑠+∑^𝑘−1_𝑖=1𝑡_𝑘−𝑡_𝑖∫^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()ℎ(𝑠)+𝛼−1𝑑𝑠_𝑘𝑖∑₌₁𝐼_𝑖𝑥𝑡⁻_𝑖+∑^𝑘−1_𝑖=1𝑡−𝑡_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖+𝐼^∗_𝑘𝑥𝑡⁻_𝑘−𝑐₀−𝑐₁𝑥𝑡,_′(𝑡)=∫^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−2Γ()ℎ(𝑠)𝛼−1𝑑𝑠+∑^𝑘_𝑖=1∫^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()ℎ(𝑠)𝛼−1𝑑𝑠+∑^𝑘−1_𝑖=1𝐼^∗_𝑖𝑥𝑡⁻_𝑖−𝑐₁. (2.20) Now if we apply the conditions: 𝑥(𝑇)=𝑎𝑥(0)+𝑏𝑇𝑥^′(0),𝑇𝑥^′(𝑇)=𝑐𝑥(0)+𝑑𝑇𝑥^′(0), (2.21) we have Λ₁=(1−𝑎)𝑐₀+𝑇(1−𝑏)𝑐₁,Λ₂𝑐=−𝑇𝑐₀+(1−𝑑)𝑐₁,−𝑐₀()Λ=−1−𝑑₁Δ+(1−𝑏)𝑇Λ₂Δ,−𝑐₁=−𝑐Λ₁−()Λ𝑇Δ1−𝑎₂Δ. (2.22) In view of the relations (2.8), when the values of −𝑐₀ and −𝑐₁ are replaced in (2.9) and (2.20), the integral equation (2.7) is obtained.
Conversely, assume that 𝑥 satisfies the impulsive fractional integral equation (2.6), then by direct computation, it can be seen that the solution given by (2.6) satisfies (2.7). The proof is complete.

3. Main Results

Definition 3.1. A function 𝑥∈𝑃𝐶¹(𝐽,𝑅) with its 𝛼-derivative existing on 𝐽^′ is said to be a solution of (1.1), if 𝑥 satisfies the equation ^𝐶𝐷^𝛼𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)) on 𝐽^′ and satisfies the conditions: Δ𝑥𝑡_𝑘=𝐼_𝑘𝑥𝑡⁻_𝑘,Δ𝑥^′𝑡_𝑘=𝐼^∗_𝑘𝑥𝑡⁻_𝑘,𝑥(𝑇)=𝑎𝑥(0)+𝑏𝑇𝑥^′(0),𝑇𝑥^′(𝑇)=𝑐𝑥(0)+𝑑𝑇𝑥^′(0). (3.1)

For the sake of convenience, we define Ω^∗₁=sup_𝑡∈𝐽Ω₁(𝑡),Ω^∗₂=sup_𝑡∈𝐽Ω₂(𝑡),Ω₁^∗∗=sup_𝑡∈𝐽Ω^′₁(𝑡),Ω₂^∗∗=sup_𝑡∈𝐽Ω^′₂(𝑡). (3.2) The followings are main results of this paper.

Theorem 3.2. Assume that (A1) the function 𝑓∶𝐽×𝑅→𝑅 is continuous and there exists a constant 𝐿₁>0 such that ‖𝑓(𝑡,𝑢)−𝑓(𝑡,𝑣)‖≤𝐿₁‖𝑢−𝑣‖, for all 𝑡∈𝐽, and 𝑢,𝑣∈𝑅,(A2)𝐼_𝑘, 𝐼^∗_𝑘∶𝑅→𝑅 are continuous, and there exist constants 𝐿₂>0 and 𝐿₃>0 such that ‖𝐼_𝑘(𝑢)−𝐼_𝑘(𝑣)‖≤𝐿₂‖𝑢−𝑣‖, ‖𝐼^∗_𝑘(𝑢)−𝐼^∗_𝑘(𝑣)‖≤𝐿₃‖𝑢−𝑣‖ for each 𝑢,𝑣∈𝑅 and 𝑘=1,2,…,𝑝.Moreover, consider the following: 𝐿₁𝑇^𝛼Γ(𝛼+1)1+Ω^∗₁+𝐿1+𝑝+2𝑝𝛼¹𝑇^𝛼−1Γ(𝛼)Ω^∗₂+1+𝑝1+Ω^∗₁𝑝𝐿₂+𝐿₃+Ω^∗₁𝑇+Ω^∗₂+𝑇𝑝𝐿₃<1. (3.3) Then, BVP (1.1) has a unique solution on 𝐽.

Proof. Define an operator 𝐹∶𝑃𝐶¹(𝐽,𝑅)→𝑃𝐶¹(𝐽,𝑅) by (𝐹𝑥)(𝑡)=^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑓(𝑠,𝑥(𝑠))𝑑𝑠+1+Ω₁(𝑡)^𝑘_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1Γ(𝛼)𝑓((𝑠+𝑠,𝑥))𝑑𝑠𝑇−𝑡_𝑘Ω₁(𝑡)+Ω₂(𝑡)+𝑡−𝑡_𝑘^𝑘_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠+𝛼−1𝑠,𝑥))𝑑𝑠1+Ω₁(𝑡)^𝑘−1_𝑖=1𝑡_𝑘−𝑡_𝑖^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠𝛼−1𝑠,𝑥))𝑑𝑠+Ω₁(𝑡)^𝑇_𝑡𝑘(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑓(𝑠,𝑥(𝑠))𝑑𝑠+Ω₂(𝑡)^𝑇_𝑡𝑘(𝑇−𝑠)^𝛼−2Γ()𝑓((𝑠𝛼−1𝑠,𝑥))𝑑𝑠+1+Ω₁(𝑡)^𝑘_𝑖=1𝐼_𝑖𝑥𝑡⁻_𝑖+𝐼^∗_𝑘𝑥𝑡⁻_𝑘+Ω₁(𝑡)^𝑘−1_𝑖=1𝑇−𝑡_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖+Ω₂(𝑡)^𝑘−1_𝑖=1𝐼^∗_𝑖𝑥𝑡⁻_𝑖+^𝑘−1_𝑖=1𝑡−𝑡_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖. (3.4) Now, for 𝑥,𝑦∈𝑃𝐶(𝐽,𝑅) and for each 𝑡∈𝐽, we obtain (𝐹𝑥)(𝑡)−𝐹𝑦(𝑡)≤^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑓((𝑠(𝑠)+𝑠,𝑥))−𝑓𝑠,𝑦𝑑𝑠1+Ω₁(𝑡)^𝑘_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1Γ(𝛼)𝑓((𝑠(𝑠)+𝑠,𝑥))−𝑓𝑠,𝑦𝑑𝑠𝑇−𝑡_𝑘Ω₁(𝑡)+Ω₂(𝑡)+𝑡−𝑡_𝑘×^𝑘_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠(𝑠+𝛼−1𝑠,𝑥))−𝑓𝑠,𝑦)𝑑𝑠1+Ω₁(𝑡)^𝑘−1_𝑖=1𝑡_𝑘−𝑡_𝑖^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠(𝑠)+Ω𝛼−1𝑠,𝑥))−𝑓𝑠,𝑦𝑑𝑠₁(𝑡)^𝑇_𝑡𝑘(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑓((𝑠(𝑠)+Ω𝑠,𝑥))−𝑓𝑠,𝑦𝑑𝑠₂(𝑡)^𝑇_𝑡𝑘(𝑇−𝑠)^𝛼−2Γ()𝑓((𝑠(𝑠)+𝛼−1𝑠,𝑥))−𝑓𝑠,𝑦𝑑𝑠1+Ω₁(𝑡)^𝑘_𝑖=1𝐼_𝑖𝑥𝑡⁻_𝑖−𝐼_𝑖𝑦𝑡⁻_𝑖+𝐼^∗_𝑘𝑥𝑡⁻_𝑘−𝐼^∗_𝑘𝑦𝑡⁻_𝑘+Ω₁(𝑡)^𝑘−1_𝑖=1𝑇−𝑡_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖−𝐼^∗_𝑖𝑦𝑡⁻_𝑖+Ω₂(𝑡)^𝑘−1_𝑖=1𝐼^∗_𝑖𝑥𝑡⁻_𝑖−𝐼^∗_𝑖𝑦𝑡⁻_𝑖+^𝑘−1_𝑖=1𝑡−𝑡_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖−𝐼^∗_𝑖𝑦𝑡⁻_𝑖,𝑡)−(𝑡≤𝐿(𝐹𝑥)(𝐹𝑦)¹𝑇^𝛼Γ(𝛼+1)1+Ω^∗₁+𝐿1+𝑝+2𝑝𝛼¹𝑇^𝛼−1Γ(𝛼)Ω^∗₂+1+𝑝1+Ω^∗₁𝑝𝐿₂+𝐿₃+Ω^∗₁𝑇+Ω^∗₂+𝑇𝑝𝐿₃𝑥(𝑠)−𝑦(𝑠). (3.5) Therefore, by (3.3), the operator 𝐹 is a contraction mapping. In a consequence of Banach’s fixed theorem, the BVP (1.1) has a unique solution. Now, our second result relies on the Burton-Kirk fixed point theorem.

Theorem 3.3. Assume that (A1)-(A2) hold, and(A3) there exist constants 𝑀₁>0, 𝑀₂>0, 𝑀₃>0 such that ‖𝑓(𝑡,𝑢)‖≤𝑀₁, ‖𝐼_𝑘(𝑢)‖≤𝑀₂, ‖𝐼^∗_𝑘(𝑢)‖≤𝑀₃ for each 𝑢,𝑣∈𝑅 and 𝑘=1,2,…,𝑝.
Then the BVP (1.1) has at least one solution on 𝐽.

Proof. We define the operators 𝐴,𝐷∶𝑃𝐶¹(𝐽,𝑅)→𝑃𝐶¹(𝐽,𝑅) by (𝐴𝑥)(𝑡)=1+Ω₁(𝑡)^𝑘_𝑖=1𝐼_𝑖𝑥𝑡⁻_𝑖+𝐼^∗_𝑘𝑥𝑡⁻_𝑘+Ω₁(𝑡)^𝑘−1_𝑖=1𝑇−𝑡_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖+Ω₂(𝑡)^𝑘−1_𝑖=1𝐼^∗_𝑖𝑥𝑡⁻_𝑖+^𝑘−1_𝑖=1𝑡−𝑡_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖,(𝐷𝑥)(𝑡)=^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑓(𝑠,𝑥(𝑠))𝑑𝑠+1+Ω₁(𝑡)^𝑘_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1Γ(𝛼)𝑓((𝑠+𝑠,𝑥))𝑑𝑠𝑇−𝑡_𝑘Ω₁(𝑡)+Ω₂(𝑡)+𝑡−𝑡_𝑘^𝑘_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠+𝛼−1𝑠,𝑥))𝑑𝑠1+Ω₁(𝑡)^𝑘−1_𝑖=1𝑡_𝑘−𝑡_𝑖^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠𝛼−1𝑠,𝑥))𝑑𝑠+Ω₁(𝑡)^𝑇_𝑡𝑘(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑓(𝑠,𝑥(𝑠))𝑑𝑠+Ω₂(𝑡)^𝑇_𝑡𝑘(𝑇−𝑠)^𝛼−2Γ()𝑓((𝑠𝛼−1𝑠,𝑥))𝑑𝑠. (3.6) It is obvious that 𝐴 is contraction mapping for 1+Ω^∗₁𝑝𝐿₂+𝐿₃+Ω^∗₁𝑇+Ω^∗₂+𝑇𝑝𝐿₃<1. (3.7) Now, in order to check that 𝐷 is completely continuous, let us follow the sequence of the following steps.
Step 1 (𝐷 is continuous). Let {𝑥_𝑛} be a sequence such that 𝑥_𝑛→𝑥 in 𝑃𝐶(𝐽,𝑅). Then for 𝑡∈𝐽, we have 𝐷𝑥_𝑛(𝑡)−(𝐷𝑥)(𝑡)≤^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑓𝑠,𝑥_𝑛(𝑠)((𝑠+−𝑓𝑠,𝑥))𝑑𝑠1+Ω₁(𝑡)^𝑘_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1Γ(𝛼)𝑓𝑠,𝑥_𝑛(𝑠)((𝑠+−𝑓𝑠,𝑥))𝑑𝑠𝑇−𝑡_𝑘Ω₁(𝑡)+Ω₂(𝑡)+𝑡−𝑡_𝑘×^𝑘_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓𝛼−1𝑠,𝑥_𝑛(𝑠((𝑠+)−𝑓𝑠,𝑥))𝑑𝑠1+Ω₁(𝑡)^𝑘−1_𝑖=1𝑡_𝑘−𝑡_𝑖^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓𝛼−1𝑠,𝑥_𝑛(𝑠)((𝑠+Ω−𝑓𝑠,𝑥))𝑑𝑠₁(𝑡)^𝑇_𝑡𝑘(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑓𝑠,𝑥_𝑛(𝑠)((𝑠+Ω−𝑓𝑠,𝑥))𝑑𝑠₂(𝑡)^𝑇_𝑡𝑘(𝑇−𝑠)^𝛼−2Γ()𝑓𝛼−1𝑠,𝑥_𝑛(𝑠)−𝑓(𝑠,𝑥(𝑠))𝑑𝑠. (3.8) Since 𝑓 is continuous function, we get 𝐷𝑥_𝑛(𝑡)−(𝐷𝑥)(𝑡)→0as𝑛→∞. (3.9)
Step 2 (𝐷 maps bounded sets into bounded sets in 𝑃𝐶(𝐽,𝑅)). Indeed, it is enough to show that for any 𝑟>0, there exists a positive constant 𝑙 such that for each 𝑥∈𝐵_𝑟={𝑥∈𝑃𝐶(𝐽,𝑅)∶‖𝑥‖≤𝑟}, we have ‖𝐷(𝑥)‖≤𝑙. By (A3), we have for each 𝑡∈𝐽, |(𝐷𝑥)(𝑡)|≤^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑓((𝑠+𝑠,𝑥))𝑑𝑠1+Ω₁(𝑡)^𝑘_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1Γ(𝛼)𝑓((𝑠+𝑠,𝑥))𝑑𝑠𝑇−𝑡_𝑘Ω₁(𝑡)+Ω₂(𝑡)+𝑡−𝑡_𝑘^𝑘_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠+𝛼−1𝑠,𝑥))𝑑𝑠1+Ω₁(𝑡)^𝑘−1_𝑖=1𝑡_𝑘−𝑡_𝑖^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠+Ω𝛼−1𝑠,𝑥))𝑑𝑠₁(𝑡)^𝑇_𝑡𝑘(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑓(𝑠,𝑥(𝑠))𝑑𝑠+Ω₂(𝑡)^𝑇_𝑡𝑘(𝑇−𝑠)^𝛼−2Γ()𝑓((𝑠‖𝐷(𝑥≤𝑀𝛼−1𝑠,𝑥))𝑑𝑠,)‖¹𝑇^𝛼Γ(𝛼+1)1+Ω^∗₁+𝑀1+𝑝+2𝑝𝛼¹𝑇^𝛼−1Γ(𝛼)Ω^∗₂1+𝑝∶=𝑙. (3.10)
Step 3 (𝐷 maps bounded sets into equicontinuous sets in 𝑃𝐶¹(𝐽,𝑅)). Let 𝜏₁,𝜏₂∈𝐽_𝑘,0≤𝑘≤𝑝 with 𝜏₁<𝜏₂ and let 𝐵_𝑟 be a bounded set of 𝑃𝐶¹(𝐽,𝑅) as in Step 2, and let 𝑥∈𝐵_𝑟. Then 𝐷𝑦𝜏₂−(𝐷)𝜏₁≤^𝜏2_𝜏1𝐷𝑦^′(𝑠)𝑑𝑠≤𝐿𝜏₂−𝜏₁, (3.11) where (𝐷𝑥)^′(𝑡)≤^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−2Γ()𝑓((𝑠Ω𝛼−1𝑠,𝑥))𝑑𝑠+^′₁(𝑡)^𝑘_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1Γ(𝛼)𝑓((𝑠+𝑠,𝑥))𝑑𝑠𝑇−𝑡_𝑘Ω^′₁(𝑡)+Ω^′₂(𝑡)+1^𝑘_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠+Ω𝛼−1𝑠,𝑥))𝑑𝑠_′1(𝑡)^𝑘−1_𝑖=1𝑡_𝑘−𝑡_𝑖^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠+Ω𝛼−1𝑠,𝑥))𝑑𝑠_′1(𝑡)^𝑇_𝑡𝑘(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑓(𝑠,𝑥(𝑠))𝑑𝑠+Ω^′₂(𝑡)^𝑇_𝑡𝑘(𝑇−𝑠)^𝛼−2Γ()𝑓((𝑠≤𝑀𝛼−1𝑠,𝑥))𝑑𝑠,₁𝑇^𝛼Γ()Ω𝛼+1^∗1+𝑀1+𝑝+2𝑝𝛼¹𝑇^𝛼−1Γ(𝛼)1+Ω^∗₂1+𝑝∶=𝐿. (3.12) This implies that 𝐴 is equicontinuous on all the subintervals 𝐽_𝑘, 𝑘=0,1,2,…,𝑝. Therefore, by the Arzela-Ascoli Theorem, the operator 𝐷∶𝑃𝐶¹(𝐽,𝑅)→𝑃𝐶¹(𝐽,𝑅) is completely continuous.

To conclude the existence of a fixed point of the operator 𝐴+𝐷, it remains to show that the set 𝑥𝜀=𝑥∈𝑋∶𝑥=𝜆𝐴𝜆+𝜆𝐷(𝑥)forsome𝜆∈(0,1) (3.13) is bounded.

Let 𝑥∈𝜀, for each 𝑡∈𝐽, 𝑥(𝑡)(𝑥𝑡)𝑥=𝜆𝐷)(+𝜆𝐴𝜆(𝑡). (3.14) Hence, from (A3), we have |𝑥(𝑡)|≤𝜆^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑓(𝑠,𝑥(𝑠))𝑑𝑠+𝜆1+Ω₁(𝑡)^𝑘_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1Γ(𝛼)𝑓(𝑠,𝑥(𝑠))𝑑𝑠+𝜆𝑇−𝑡_𝑘Ω₁(𝑡)+Ω₂(𝑡)+𝑡−𝑡_𝑘^𝑘_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠𝛼−1𝑠,𝑥))𝑑𝑠+𝜆1+Ω₁(𝑡)^𝑘−1_𝑖=1𝑡_𝑘−𝑡_𝑖^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠Ω𝛼−1𝑠,𝑥))𝑑𝑠+𝜆₁(𝑡)^𝑇_𝑡𝑘(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑓(𝑠,𝑥(𝑠))𝑑𝑠+𝜆Ω₂(𝑡)^𝑇_𝑡𝑘(𝑇−𝑠)^𝛼−2Γ()𝑓((𝑠𝛼−1𝑠,𝑥))𝑑𝑠+𝜆1+Ω₁(𝑡)^𝑘_𝑖=1𝐼_𝑖𝑥𝑡⁻_𝑖𝜆+𝐼^∗_𝑘𝑥𝑡⁻_𝑘𝜆+𝜆Ω₁(𝑡)^𝑘−1_𝑖=1𝑇−𝑡_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖𝜆+𝜆Ω₂(𝑡)^𝑘−1_𝑖=1𝐼^∗_𝑖𝑥𝑡⁻_𝑖𝜆+𝜆^𝑘−1_𝑖=1𝑡−𝑡_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖𝜆,‖𝑥(𝑡≤𝑀)‖¹𝑇^𝛼Γ(𝛼+1)1+Ω^∗₁+𝑀1+𝑝+2𝑝𝛼¹𝑇^𝛼−1Γ(𝛼)Ω^∗₂+1+𝑝1+Ω^∗₁𝑝𝑀₂+𝑀₃+Ω^∗₁𝑇+Ω^∗₂+𝑇𝑝𝑀₃. (3.15) Consequently, we conclude the result of our theorem based on the Burton-Kirk fixed point theorem.

4. An Example

Consider the following impulsive fractional boundary value problem: ^𝐶𝐷^3/2𝑥(𝑡)=sin2𝑡|𝑥(𝑡)|(𝑡+5)²(|𝑥(𝑡[]11+)|),𝑡∈0,1,𝑡≠3,1Δ𝑥3=|𝑥(1⁻/3)|15+|𝑥(1⁻/3)|,Δ𝑥^′13=𝑥^′(1⁻/3)10+𝑥^′(1⁻,𝑥(1)(0)/3)=5𝑥−2𝑥^′(0),𝑥^′(1)=𝑥(0)+4𝑥^′(0). (4.1) Here, 𝑎=5, 𝑏=−2, 𝑐=1, 𝑑=4, 𝛼=3/2, 𝑇=1, 𝑝=1. Obviously, 𝐿₁=1/25, 𝐿₂=1/15, 𝐿₃=1/10, Ω^∗₁=4/15, Ω^∗₂=7/15. Further, 𝐿₁𝑇^𝛼Γ(𝛼+1)1+Ω^∗₁1+𝑝1+𝐿₂+2𝑝𝛼+𝐿₃+𝐿₁𝑇^𝛼−1Γ(𝛼)Ω^∗₂1+𝑝+Ω^∗₁𝑇+Ω^∗₂+𝑇𝑝𝐿₃=4641125√𝜋+173450<1. (4.2) Since the assumptions of Theorem 3.2 are satisfied, the closed boundary value problem (4.1) has a unique solution on [0,1]. Moreover, it is easy to check the conclusion of Theorem 3.3.

Acknowledgments

The authors would like to express their sincere thanks and gratitude to the reviewer(s) for their valuable comments and suggestions for the improvement of this paper. The second author also gratefully acknowledges that this research was partially supported by the University Putra Malaysia under the Research University Grant Scheme  05-01-09-0720RU.