Fractional Differential Equations with Fractional Impulsive and Nonseparated Boundary Conditions

Abstract

This paper studies the existence results for nonseparated boundary value problems of fractional differential equations with fractional impulsive conditions. By means of Schaefer fixed point theorem, Banach fixed point theorem, and nonlinear alternative of Leray-Schauder type, some existence results are obtained. Examples are given to illustrate the results.

1. Introduction

The subject of fractional differential equations has recently evolved as an interesting and popular field of research. In fact, fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. More and more researchers have found that fractional differential equations play important roles in many research areas, such as physics, chemical technology, population dynamics, biotechnology, and economics (see [1–4]). Hence, there are many literatures devoted to solving fractional differential equations through theoretical analysis or numerical methods (e.g., see [5–9]).

The theory of impulsive differential equations of integer order has found its extensive applications in realistic mathematical modeling of a wide variety of practical situations and has emerged as an important area of investigation in recent years. For the general theory and applications of impulsive differential equations, see [10–15] and so forth. However, impulsive differential equations of fractional order have not been much studied and many aspects of these equations are yet to be explored. For some recent work on impulsive fractional differential equations, we can refer to [16–24] and the references therein.

In this paper, we investigate the existence and uniqueness of solutions for fractional differential equations with fractional impulsive conditions ^𝑐𝐷^𝛼𝑥(𝑡)((𝑡,[]=𝑓𝑡,𝑥))𝑡∈𝐽∶=0,𝑇,𝑡≠𝑡_𝑘𝑡,𝑘=1,2,…,𝑚,Δ𝑥_𝑘=𝐼_𝑘𝑥𝑡⁻_𝑘,Δ^𝑐𝐷^𝛾𝑥𝑡_𝑘=𝐼^∗_𝑘𝑥𝑡⁻_𝑘,𝑘=1,2,…,𝑚, (1) and nonseparated boundary conditions 𝑎₁𝑥(0)+𝑏₁𝑥(𝑇)=𝑐₁,𝑎₂𝑥^′(0)+𝑏₂𝑥^′(𝑇)=𝑐₂,𝑎₁𝑥(0)+𝑏₁𝑥(𝑇)=𝑐₁,𝑎₂^𝑐𝐷^𝛾𝑥(0)+𝑏₂^𝑐𝐷^𝛾𝑥(𝑇)=𝑐₂, (2)(3) where ^𝑐𝐷^𝛼 is the Caputo fractional derivative of order 𝛼∈(1,2), 0<𝛾<1, 𝑓∈𝐶(𝐽×ℝ,ℝ), 𝐼_𝑘,𝐼^∗_𝑘∈𝐶(ℝ,ℝ), 0=𝑡₀<𝑡₁<⋯<𝑡_𝑚<𝑡_𝑚+1=𝑇, Δ𝑥(𝑡_𝑘)=𝑥(𝑡⁺_𝑘)−𝑥(𝑡⁻_𝑘) with 𝑥(𝑡⁺_𝑘)=lim_𝜖→0⁺𝑥(𝑡_𝑘+𝜖), 𝑥(𝑡⁻_𝑘)=lim_𝜖→0⁻𝑥(𝑡_𝑘+𝜖) representing the right and left limits of 𝑥(𝑡) at 𝑡=𝑡_𝑘, Δ(^𝑐𝐷^𝛾𝑥(𝑡_𝑘)) has a similar meaning for ^𝑐𝐷^𝛾𝑥(𝑡_𝑘), and 𝑎_𝑖,𝑏_𝑖,𝑐_𝑖,𝑖=1,2, are real constants such that (𝑎₁+𝑏₁)(𝑎₂+𝑏₂)≠0 for (2) and (𝑎₁+𝑏₁)𝑏₂≠0 for (3).

We note that, as pointed out in the papers [22–24], the concept of piecewise continuous solutions used in some already published works to handle the impulsive fractional differential equations is not appropriate (see counterexamples given in Lemma 3.1 of [22], Section 1 in [23], and Section 3 in [24]). The papers on this topic cited above except [22] all deal with the Caputo derivative and the impulsive conditions only involve integer order derivatives. Here we study the fractional differential equations with fractional impulsive conditions and nonseparated boundary conditions [25–27].

If we let 𝑎₁=𝑏₁=1, 𝑎₂=𝑏₂=1, and 𝑐₁=𝑐₂=0 in boundary conditions (2) and (3), then they reduce to antiperiodic boundary conditions 𝑥(0)=−𝑥(𝑇), 𝑥^′(0)=−𝑥^′(𝑇) and 𝑥(0)=−𝑥(𝑇),^𝑐𝐷^𝛾𝑥(0)=−^𝑐𝐷^𝛾𝑥(𝑇).

The rest of the paper is organized as follows. In Section 2 we introduce some notations and definitions needed in the following sections and give the appropriate formula of solutions for our problems. In Section 3 we present the existence results for the problems (1), (2) and (1), (3). Two examples are presented in Section 4 to illustrate the results. Concluding remarks are provided in Section 5.

2. Preliminaries

Let us set 𝐽₀=[0,𝑡₁], 𝐽₁=(𝑡₁,𝑡₂],…,𝐽_𝑚−1=(𝑡_𝑚−1,𝑡_𝑚], 𝐽_𝑚=(𝑡_𝑚,𝑡_𝑚+1], and 𝐽^′∶=𝐽⧵{𝑡₁,𝑡₂,…,𝑡_𝑚} and introduce the space 𝑃𝐶(𝐽,ℝ)∶={𝑢∶𝐽→ℝ∣𝑢∈𝐶(𝐽_𝑘,ℝ),𝑘=0,1,2,…,𝑚, and there exist 𝑢(𝑡⁺_𝑘) and 𝑢(𝑡⁻_𝑘), 𝑘=1,2,…,𝑚, with 𝑢(𝑡⁻_𝑘)=𝑢(𝑡_𝑘)}. It is clear that 𝑃𝐶(𝐽,ℝ) is a Banach space with the norm ‖𝑢‖=sup{|𝑢(𝑡)|∶𝑡∈𝐽}.

Definition 1 (see [4]). The Riemann-Liouville fractional integral of order 𝑞 for a function 𝑓 is defined as 𝐼^𝑞𝑓(𝑡)=1Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1𝑓(𝑠)𝑑𝑠,𝑞>0, (4) provided the integral exists.

Definition 2 (see [4]). For a continuous function 𝑓, the Caputo derivative of order 𝑞 is defined as ^𝑐𝐷^𝑞𝑓(𝑡)=1Γ𝑛−𝑞^𝑡₀(𝑡−𝑠)^{𝑛−𝑞−1}𝑓^(𝑛)(𝑠)𝑞𝑑𝑠,𝑛−1<𝑞<𝑛,𝑛=+1, (5) where [𝑞] denotes the integer part of the real number 𝑞.

Lemma 3 (see [4]). Let 𝛼>0; then the differential equation ^𝑐𝐷^𝛼ℎ(𝑡)=0 (6) has solutions ℎ(𝑡)=𝑐₀+𝑐₁𝑡+𝑐₂𝑡²+⋯+𝑐_𝑛−1𝑡^𝑛−1 and 𝐼^𝛼𝑐𝐷^𝛼ℎ(𝑡)=ℎ(𝑡)+𝑐₀+𝑐₁𝑡+𝑐₂𝑡²+⋯+𝑐_𝑛−1𝑡^𝑛−1, (7) where 𝑐_𝑖∈ℝ, 𝑖=0,1,2,…,𝑛−1, 𝑛=[𝛼]+1.

Definition 4. A function 𝑥∈𝑃𝐶(𝐽,ℝ) with its 𝛼-derivative existing on 𝐽^′ is said to be a solution of the problems (1), (2) (or the problems (1), (3)), if 𝑥 satisfies the equation ^𝑐𝐷^𝛼𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)) on 𝐽^′, the impulsive conditions Δ𝑥𝑡_𝑘=𝐼_𝑘𝑥𝑡⁻_𝑘,Δ^𝑐𝐷^𝛾𝑥𝑡_𝑘=𝐼^∗_𝑘𝑥𝑡⁻_𝑘,𝑘=1,2,…,𝑚, (8) and the nonseparated boundary conditions (2) (or (3)).

Lemma 5. Let 𝑦∈𝑃𝐶(𝐽,ℝ). A function 𝑥 is a solution of the fractional integral equation 𝑥(𝑡)=^𝑡0(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝑑₀+𝑒₀𝑡,𝑡∈𝐽₀;_𝑡0(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝑑₀+𝐼₁𝑥𝑡⁻₁−Γ2−𝛾𝑡^𝛾₁𝐼^∗₁𝑥𝑡⁻₁+𝑒₀𝑡+Γ2−𝛾𝑡₁^𝛾−1𝐼^∗₁𝑥𝑡⁻₁𝑡,𝑡∈𝐽₁;_𝑡0⋮(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝑑₀+^𝑘_𝑖=1𝐼_𝑖𝑥𝑡⁻_𝑖−Γ2−𝛾^𝑘_𝑖=1𝑡^𝛾_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖+𝑒₀𝑡+Γ2−𝛾𝑡^𝑘_𝑖=1𝑡_𝑖^𝛾−1𝐼^∗_𝑖𝑥𝑡⁻_𝑖,𝑡∈𝐽_𝑘,𝑘=1,2,…,𝑚, (9) where 𝑑₀=𝑐₁−𝑏₁∫^𝑇₀(𝑇−𝑠)^𝛼−1/Γ(𝛼)𝑦(𝑠)𝑑𝑠−𝑏₁𝐴𝑎₁+𝑏₁−𝑏₁𝑐₂𝑇+𝑎₂𝑏₁𝑇𝐵−𝑏₁𝑏₂𝑇^𝑇₀(𝑇−𝑠)^𝛼−2Γ()𝑦(𝑠)×𝑎𝛼−1𝑑𝑠₁+𝑏₁𝑎₂+𝑏₂⁻¹,𝑒₀=𝑐₂−𝑏₂∫^𝑇₀(𝑇−𝑠)^𝛼−2/Γ(𝛼−1)𝑦(𝑠)𝑑𝑠−𝑏₂𝐵𝑎₂+𝑏₂,𝐴=^𝑚_𝑖=1𝐼_𝑖𝑥𝑡⁻_𝑖−Γ2−𝛾^𝑚_𝑖=1𝑡^𝛾_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖,𝐵=Γ2−𝛾^𝑚_𝑖=1𝑡_𝑖^𝛾−1𝐼^∗_𝑖𝑥𝑡⁻_𝑖 (10)(11) if and only if 𝑥 is a solution of the impulsive fractional boundary value problem ^𝑐𝐷^𝛼𝑥(𝑡)=𝑦(𝑡),𝑡∈𝐽^′𝑡,1<𝛼<2,Δ𝑥_𝑘=𝐼_𝑘𝑥𝑡⁻_𝑘,Δ^𝑐𝐷^𝛾𝑥𝑡_𝑘=𝐼^∗_𝑘𝑥𝑡⁻_𝑘,𝑎𝑘=1,2,…,𝑚,₁𝑥(0)+𝑏₁𝑥(𝑇)=𝑐₁,𝑎₂𝑥^′(0)+𝑏₂𝑥^′(𝑇)=𝑐₂. (12)

Proof. For 1<𝛼<2, by Lemma 3, we know that a general solution of the equation ^𝑐𝐷^𝛼𝑥(𝑡)=𝑦(𝑡) on each interval 𝐽_𝑘 (𝑘=0,1,2,…,𝑚) is given by 𝑥(𝑡)=𝐼^𝛼𝑦(𝑡)+𝑑_𝑘+𝑒_𝑘𝑡=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝑑_𝑘+𝑒_𝑘𝑡,𝑡∈𝐽_𝑘, (13) where 𝑑_𝑘 and 𝑒_𝑘∈ℝ are arbitrary constants. Then we have 𝑥^′(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−2Γ()𝑦(𝑠)𝛼−1𝑑𝑠+𝑒_𝑘,𝑡∈𝐽_𝑘. (14)
Since ^𝑐𝐷^𝛾𝐶=0 (𝐶 is a constant), ^𝑐𝐷^𝛾𝑡=𝑡^1−𝛾/Γ(2−𝛾), and ^𝑐𝐷^𝛾𝐼^𝛼𝑦(𝑡)=𝐼^𝛼−𝛾𝑦(𝑡) (see [4]), then from (13), we have ^𝑐𝐷^𝛾𝑥(𝑡)=^𝑡₀(𝑡−𝑠)^{𝛼−𝛾−1}Γ𝑦(𝑠)𝑒𝛼−𝛾𝑑𝑠+^𝑘𝑡^1−𝛾Γ2−𝛾,𝑡∈𝐽_𝑘. (15)
Applying the boundary conditions of (12), we get (since 0∈𝐽₀ and 𝑇∈𝐽_𝑚) 𝑎₁×𝑑₀+𝑏₁×^𝑇₀(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝑑_𝑚+𝑒_𝑚𝑇=𝑐₁,𝑎₂×𝑒₀+𝑏₂×^𝑇₀(𝑇−𝑠)^𝛼−2Γ()𝑦(𝑠)𝛼−1𝑑𝑠+𝑒_𝑚=𝑐₂. (16)
Next using the impulsive conditions in (12), we obtain that, for 𝑘=1,2,…,𝑚, 𝑑_𝑘−𝑑_𝑘−1+𝑒_𝑘−𝑒_𝑘−1𝑡_𝑘=𝐼_𝑘𝑥𝑡⁻_𝑘,𝑒_𝑘−𝑒_𝑘−1𝑡_𝑘^1−𝛾Γ2−𝛾=𝐼^∗_𝑘𝑥𝑡⁻_𝑘. (17)
That is to say, we have, for 𝑘=1,2,…,𝑚, 𝑑_𝑘=𝑑₀+^𝑘_𝑖=1𝐼_𝑖𝑥𝑡⁻_𝑖−Γ2−𝛾^𝑘_𝑖=1𝑡^𝛾_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖,𝑒_𝑘=𝑒₀+Γ2−𝛾^𝑘_𝑖=1𝑡_𝑖^𝛾−1𝐼^∗_𝑖𝑥𝑡⁻_𝑖. (18)
Now the above two equations together with (16) imply that 𝑑₀=𝑐₁−𝑏₁∫^𝑇₀(𝑇−𝑠)^𝛼−1/Γ(𝛼)𝑦(𝑠)𝑑𝑠−𝑏₁𝐴𝑎₁+𝑏₁−𝑏₁𝑐₂𝑇+𝑎₂𝑏₁𝑇𝐵−𝑏₁𝑏₂𝑇^𝑇₀(𝑇−𝑠)^𝛼−2Γ()𝑦(𝑠)×𝑎𝛼−1𝑑𝑠₁+𝑏₁𝑎₂+𝑏₂⁻¹,𝑒₀=𝑐₂−𝑏₂∫^𝑇₀(𝑇−𝑠)^𝛼−2/Γ(𝛼−1)𝑦(𝑠)𝑑𝑠−𝑏₂𝐵𝑎₂+𝑏₂, (19) where 𝐴=^𝑚_𝑖=1𝐼_𝑖𝑥𝑡⁻_𝑖−Γ2−𝛾^𝑚_𝑖=1𝑡^𝛾_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖,𝐵=Γ2−𝛾^𝑚_𝑖=1𝑡_𝑖^𝛾−1𝐼^∗_𝑖𝑥𝑡⁻_𝑖. (20)
Hence, for 𝑘=0,1,2,…,𝑚, we have 𝑑_𝑘+𝑒_𝑘𝑡=𝑑₀+^𝑘_𝑖=1𝐼_𝑖𝑥𝑡⁻_𝑖−Γ2−𝛾^𝑘_𝑖=1𝑡^𝛾_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖+𝑒₀𝑡+Γ2−𝛾𝑡^𝑘_𝑖=1𝑡_𝑖^𝛾−1𝐼^∗_𝑖𝑥𝑡⁻_𝑖. (21)
Substituting the value of 𝑑₀ and 𝑒₀ in (21), and by (13), we get the fractional integral equation (9).
Conversely, assume that 𝑥 satisfies the fractional integral equation (9); that is, for 𝑡∈𝐽_𝑘, 𝑘=0,1,2,…,𝑚, we have 𝑥(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝑑₀+^𝑘_𝑖=1𝐼_𝑖𝑥𝑡⁻_𝑖−Γ2−𝛾^𝑘_𝑖=1𝑡^𝛾_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖+𝑒₀𝑡+Γ2−𝛾𝑡^𝑘_𝑖=1𝑡_𝑖^𝛾−1𝐼^∗_𝑖𝑥𝑡⁻_𝑖,𝑥^′(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−2Γ()𝑦(𝑠)𝛼−1𝑑𝑠+𝑒₀+Γ2−𝛾^𝑘_𝑖=1𝑡_𝑖^𝛾−1𝐼^∗_𝑖𝑥𝑡⁻_𝑖. (22)(23)
Since 1<𝛼<2, we have ^𝑐𝐷^𝛼𝐶=0 (𝐶 is a constant) and ^𝑐𝐷^𝛼𝑡=0. Using the fact that ^𝑐𝐷^𝛼 is the left inverse of 𝐼^𝛼, we get ^𝑐𝐷^𝛼𝑥(𝑡)=𝑦(𝑡),𝑡∈𝐽^′, (24) which means that 𝑥 satisfies the first equation of the impulsive fractional boundary value problem (12). Next we will verify that 𝑥 satisfies the impulsive conditions. Taking fractional derivative ^𝑐𝐷^𝛾 of (22), we have, for 𝑡∈𝐽_𝑘, ^𝑐𝐷^𝛾𝑥(𝑡)=^𝑡₀(𝑡−𝑠)^{𝛼−𝛾−1}Γ𝑦(𝑠)+𝑒𝛼−𝛾𝑑𝑠₀𝑡^1−𝛾Γ2−𝛾+𝑡^𝑘1−𝛾_𝑖=1𝑡_𝑖^𝛾−1𝐼^∗_𝑖𝑥𝑡⁻_𝑖. (25)
From (22), we obtain 𝑥𝑡⁺_𝑘=^𝑡𝑘₀𝑡_𝑘−𝑠^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝑑₀+^𝑘_𝑖=1𝐼_𝑖𝑥𝑡⁻_𝑖−Γ2−𝛾^𝑘_𝑖=1𝑡^𝛾_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖+𝑒₀𝑡_𝑘+Γ2−𝛾𝑡_𝑘^𝑘_𝑖=1𝑡_𝑖^𝛾−1𝐼^∗_𝑖𝑥𝑡⁻_𝑖,𝑥𝑡⁻_𝑘=^𝑡𝑘₀𝑡_𝑘−𝑠^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝑑₀+^𝑘−1_𝑖=1𝐼_𝑖𝑥𝑡⁻_𝑖−Γ2−𝛾^𝑘−1_𝑖=1𝑡^𝛾_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖+𝑒₀𝑡_𝑘+Γ2−𝛾𝑡_𝑘^𝑘−1_𝑖=1𝑡_𝑖^𝛾−1𝐼^∗_𝑖𝑥𝑡⁻_𝑖. (26)
The two equations above imply that, for 𝑘=1,2,…,𝑚, Δ𝑥𝑡_𝑘=𝐼_𝑘𝑥𝑡⁻_𝑘−Γ2−𝛾𝑡^𝛾_𝑘𝐼^∗_𝑘𝑥𝑡⁻_𝑘+Γ2−𝛾𝑡_𝑘𝑡_𝑘^𝛾−1𝐼^∗_𝑘𝑥𝑡⁻_𝑘=𝐼_𝑘𝑥𝑡⁻_𝑘. (27)
Similarly, from (25), we have ^𝑐𝐷^𝛾𝑥𝑡⁺_𝑘=^𝑡𝑘₀𝑡_𝑘−𝑠^{𝛼−𝛾−1}Γ𝑦(𝑠)𝑒𝛼−𝛾𝑑𝑠+⁰𝑡_𝑘^1−𝛾Γ2−𝛾+𝑡_𝑘^𝑘1−𝛾_𝑖=1𝑡_𝑖^𝛾−1𝐼^∗_𝑖𝑥𝑡⁻_𝑖,_𝑐𝐷^𝛾𝑥𝑡⁻_𝑘=^𝑡𝑘₀𝑡_𝑘−𝑠^{𝛼−𝛾−1}Γ𝑦(𝑠)+𝑒𝛼−𝛾𝑑𝑠₀𝑡_𝑘^1−𝛾Γ2−𝛾+𝑡_𝑘^{1−𝛾𝑘−1}_𝑖=1𝑡_𝑖^𝛾−1𝐼^∗_𝑖𝑥𝑡⁻_𝑖. (28)
Therefore, for 𝑘=1,2,…,𝑚, we have Δ^𝑐𝐷^𝛾𝑥𝑡_𝑘=𝑡_𝑘^1−𝛾𝑡_𝑘^𝛾−1𝐼^∗_𝑘𝑥𝑡⁻_𝑘=𝐼^∗_𝑘𝑥𝑡⁻_𝑘. (29)
Finally, it follows from (22) and (23) that (since 0∈𝐽₀, 𝑇∈𝐽_𝑚) 𝑥(0)=𝑑₀,𝑥(𝑇)=^𝑇₀(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝑑₀+𝐴+𝑒₀𝑥𝑇+𝑇𝐵,_′(0)=𝑒₀,𝑥^′(𝑇)=^𝑇₀(𝑇−𝑠)^𝛼−2Γ()𝑦(𝑠)𝛼−1𝑑𝑠+𝑒₀+𝐵. (30)
Hence, we get 𝑎₁𝑥(0)+𝑏₁𝑥(𝑇)=𝑎₁𝑑₀+𝑏₁^𝑇₀(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝑑₀+𝐴+𝑒₀𝑇+𝑇𝐵=𝑐₁+𝑏₁𝑇𝐵+𝑏₁𝑒₀𝑇−𝑏₁𝑐₂𝑇+𝑎₂𝑏₁𝑇𝐵−𝑏₁𝑏₂𝑇^𝑇₀(𝑇−𝑠)^𝛼−2Γ()𝑦(𝑠)×𝑎𝛼−1𝑑𝑠₂+𝑏₂⁻¹=𝑐₁,𝑎₂𝑥^′(0)+𝑏₂𝑥^′(𝑇)=𝑎₂𝑒₀+𝑏₂^𝑇₀(𝑇−𝑠)^𝛼−2Γ()𝑦(𝑠)𝛼−1𝑑𝑠+𝑒₀+𝐵=𝑐₂. (31)
Therefore, 𝑥 given by (9) satisfies the impulsive fractional boundary value problem (12). The proof is complete.

Lemma 6. Let 𝑦∈𝑃𝐶(𝐽,ℝ). A function 𝑥 is a solution of the fractional integral equation 𝑥(𝑡)=^𝑡0(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝑑₀+𝑒₀𝑡,𝑡∈𝐽₀;_𝑡0(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝑑₀+𝐼₁𝑥𝑡⁻₁−Γ2−𝛾𝑡^𝛾₁𝐼^∗₁𝑥𝑡⁻₁+𝑒₀𝑡+Γ2−𝛾𝑡₁^𝛾−1𝐼^∗₁𝑥𝑡⁻₁𝑡,𝑡∈𝐽₁;_𝑡0⋮(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝑑₀+^𝑘_𝑖=1𝐼_𝑖𝑥𝑡⁻_𝑖−Γ2−𝛾^𝑘_𝑖=1𝑡^𝛾_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖+𝑒₀𝑡+Γ2−𝛾𝑡^𝑘_𝑖=1𝑡_𝑖^𝛾−1𝐼^∗_𝑖𝑥𝑡⁻_𝑖,𝑡∈𝐽_𝑘,𝑘=1,2,…,𝑚, (32) where 𝑑₀𝑏=−¹𝑎₁+𝑏₁()+𝑐𝐶+𝐴¹𝑏₂−𝑏₁𝑐₂𝑇^𝛾Γ2−𝛾𝑎₁+𝑏₁𝑏₂,𝑒₀Γ=−2−𝛾𝑇^1−𝛾𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑦(𝑠)𝑐𝛼−𝛾𝑑𝑠−²𝑏₂−𝐵 (33) with 𝐴 and 𝐵 defined by (11) and 𝐶=^𝑇₀(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠−𝑇^𝛾Γ2−𝛾^𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑦(𝑠)𝛼−𝛾𝑑𝑠, (34) if and only if 𝑥 is a solution of the impulsive fractional boundary value problem ^𝑐𝐷^𝛼𝑥(𝑡)=𝑦(𝑡),𝑡∈𝐽^′𝑡,1<𝛼<2,Δ𝑥_𝑘=𝐼_𝑘𝑥𝑡⁻_𝑘,Δ^𝑐𝐷^𝛾𝑥𝑡_𝑘=𝐼^∗_𝑘𝑥𝑡⁻_𝑘,𝑎𝑘=1,2,…,𝑚,₁𝑥(0)+𝑏₁𝑥(𝑇)=𝑐₁,𝑎₂^𝑐𝐷^𝛾𝑥(0)+𝑏₂^𝑐𝐷^𝛾𝑥(𝑇)=𝑐₂. (35)

Proof. The proof is similar to that of Lemma 5. Let the notations be given as in the proof of Lemma 5. Applying the boundary conditions in (35), from the relations (13), (15), and (17), we have 𝑎₁×𝑑₀+𝑏₁×^𝑇₀(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝑑_𝑚+𝑒_𝑚𝑇=𝑐₁,𝑎₂×0+𝑏₂×^𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑦(𝑠)𝑒𝛼−𝛾𝑑𝑠+^𝑚𝑇^1−𝛾Γ2−𝛾=𝑐₂. (36)
After a direct computation, we obtain that 𝑑₀𝑏=−¹𝑎₁+𝑏₁()+𝑐𝐶+𝐴¹𝑏₂−𝑏₁𝑐₂𝑇^𝛾Γ2−𝛾𝑎₁+𝑏₁𝑏₂,𝑒₀Γ=−2−𝛾𝑇^1−𝛾𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑦(𝑠)𝑐𝛼−𝛾𝑑𝑠−²𝑏₂−𝐵 (37) with 𝐴 and 𝐵 defined by (11) and 𝐶=^𝑇₀(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠−𝑇^𝛾Γ2−𝛾^𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑦(𝑠)𝛼−𝛾𝑑𝑠. (38)
The remaining part of proof is the same as that of Lemma 5.

Remark 7. We note that the solution expression (32) of the problem (35) does not depend on the parameter 𝑎₂ appearing in the boundary conditions. Thus, by Lemma 6, we conclude that the parameter 𝑎₂ is of arbitrary nature of the problem (35).

Remark 8. Our approach for the construction of solutions for impulsive fractional differential equations is general, which provides an effective way to deal with such problems. Taking the problem (1) for example, on each interval 𝐽_𝑘, we use 𝑥(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑓(𝑠,𝑥(𝑠))𝑑𝑠+𝑑_𝑘+𝑒_𝑘𝑡 (I) instead of 𝑥(𝑡)=^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑓(𝑠,𝑥(𝑠))𝑑𝑠+𝑑_𝑘+𝑒_𝑘𝑡−𝑡_𝑘. (II)
From the counterexamples given in Lemma 3.1 of [22], Section 1 in [23], and Section 3 in [24], we know that if the case (II) was chosen to construct solutions, the solution formula obtained in the fractional integral equation form is not equivalent to the original impulsive fractional differential equation. This is the main difference between impulsive fractional differential equations and impulsive ordinary differential equations. Observe that fractional calculus has memory property.

Theorem 9 (nonlinear alternative of Leray-Schauder type [28]). Let 𝑋 be a Banach space, 𝐶 a nonempty convex subset of 𝑋, and 𝑈 a nonempty open subset of 𝐶 with 0∈𝑈. Suppose that 𝑃∶𝑈→𝐶 is a continuous and compact map. Then either (a)  𝑃 has a fixed point in 𝑈 or (b) there exist a 𝑥∈𝜕𝑈 (the boundary of 𝑈) and 𝜆∈(0,1) with 𝑥=𝜆𝑃(𝑥).

Theorem 10 (Schaefer fixed point theorem [29]). Let 𝑋 be a normed space and 𝑃 a continuous mapping of 𝑋 into 𝑋 which is compact on each bounded subset 𝐵 of 𝑋. Then either (I) the equation 𝑥=𝜆𝑃𝑥 has a solution for 𝜆=1 or (II) the set of all such solutions 𝑥, for 0<𝜆<1, is unbounded.

3. Main Results

This section deals with the existence and uniqueness of solutions for the problem (1), (2) and the problem (1), (3).

First we study the existence results for the problem (1), (2). In view of Lemma 5, we define an operator 𝐹∶𝑃𝐶(𝐽,ℝ)→𝑃𝐶(𝐽,ℝ) by (𝐹𝑥)(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑓(𝑠,𝑥(𝑠))𝑑𝑠+𝑑^𝑥₀+^𝑘_𝑖=1𝐼_𝑖𝑥𝑡⁻_𝑖−Γ2−𝛾^𝑘_𝑖=1𝑡^𝛾_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖+𝑒^𝑥₀𝑡+Γ2−𝛾𝑡^𝑘_𝑖=1𝑡_𝑖^𝛾−1𝐼^∗_𝑖𝑥𝑡⁻_𝑖,𝑡∈𝐽_𝑘,𝑘=0,1,2,…,𝑚, (39) where 𝑑^𝑥₀=𝑐₁−𝑏₁∫^𝑇₀(𝑇−𝑠)^𝛼−1/Γ(𝛼)𝑓(𝑠,𝑥(𝑠))𝑑𝑠−𝑏₁𝐴^𝑥𝑎₁+𝑏₁−𝑏₁𝑐₂𝑇+𝑎₂𝑏₁𝑇𝐵^𝑥−𝑏₁𝑏₂𝑇×^𝑇₀(𝑇−𝑠)^𝛼−2Γ()𝑓((𝑠×𝑎𝛼−1𝑠,𝑥))𝑑𝑠₁+𝑏₁𝑎₂+𝑏₂⁻¹,𝑒^𝑥₀=𝑐₂−𝑏₂∫^𝑇₀(𝑇−𝑠)^𝛼−2/Γ(𝛼−1)𝑓(𝑠,𝑥(𝑠))𝑑𝑠−𝑏₂𝐵^𝑥𝑎₂+𝑏₂,𝐴^𝑥=^𝑚_𝑖=1𝐼_𝑖𝑥𝑡⁻_𝑖−Γ2−𝛾^𝑚_𝑖=1𝑡^𝛾_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖,𝐵^𝑥=Γ2−𝛾^𝑚_𝑖=1𝑡_𝑖^𝛾−1𝐼^∗_𝑖𝑥𝑡⁻_𝑖. (40) Here 𝑑^𝑥₀, 𝑒^𝑥₀, 𝐴^𝑥, and 𝐵^𝑥 mean that 𝑑₀, 𝑒₀, 𝐴, and 𝐵 defined in Lemma 5 are related to 𝑥∈𝑃𝐶(𝐽,ℝ). It is obvious that 𝐹 is well defined because of the continuity of 𝑓, 𝐼_𝑘, and 𝐼^∗_𝑘 and that the problem (1), (2) has solutions if and only if the operator equation 𝐹𝑥=𝑥 has fixed points.

For the sake of convenience, we set Δ=𝑎₁+𝑏₁𝑎₂+𝑏₂. (41)

Lemma 11. The operator 𝐹∶𝑃𝐶(𝐽,ℝ)→𝑃𝐶(𝐽,ℝ) defined by (39) is completely continuous.

Proof. Since 𝑓, 𝐼_𝑘, and 𝐼^∗_𝑘 are continuous, it is easy to check that 𝐹 is continuous on 𝑃𝐶(𝐽,ℝ).
Let 𝐵⊆𝑃𝐶(𝐽,ℝ) be bounded; then there exist positive constants 𝑁_𝑖, 𝑖=1,2,3, such that |𝑓(𝑡,𝑥(𝑡))|≤𝑁₁, |𝐼_𝑘(𝑥(𝑡⁻_𝑘))|≤𝑁₂, and |𝐼^∗_𝑘(𝑥(𝑡⁻_𝑘))|≤𝑁₃, for all 𝑡∈𝐽,𝑥∈𝐵, 𝑘=1,2,…,𝑚. Thus, for 𝑥∈𝐵 and 𝑡∈𝐽, we have 𝑡≤𝑁|(𝐹𝑥)()|¹𝑇^𝛼Γ()+𝑑𝛼+1^𝑥0+𝑚𝑁₂+Γ2−𝛾𝑁₃^𝑚_𝑖=1𝑡^𝛾_𝑖+𝑒^𝑥₀𝑇+Γ2−𝛾𝑇𝑁₃^𝑚_𝑖=1𝑡_𝑖^𝛾−1,𝑑^𝑥₀≤𝑐₁𝑎₁+𝑏₁+𝑏₁𝑎₁+𝑏₁×𝑁₁𝑇^𝛼Γ(𝛼+1)+𝑚𝑁₂+Γ2−𝛾𝑁₃^𝑚_𝑖=1𝑡^𝛾_𝑖+𝑏₁𝑐₂𝑇|Δ|+𝑎₂𝑏₁𝑇|Δ|Γ2−𝛾𝑁₃^𝑚_𝑖=1𝑡_𝑖^𝛾−1+𝑏₁𝑏₂𝑁₁𝑇^𝛼|Δ|Γ(𝛼),𝑒^𝑥₀≤𝑐₂𝑎₂+𝑏₂+𝑏₂𝑎₂+𝑏₂𝑁₁𝑇^𝛼−1Γ(𝛼)+Γ2−𝛾𝑁₃^𝑚_𝑖=1𝑡_𝑖^𝛾−1. (42)(43) Therefore, we can deduce that, for all 𝑥∈𝐵 and 𝑡∈𝐽, 𝑡≤𝑁|(𝐹𝑥)()|¹𝑇^𝛼Γ()𝑏𝛼+11+¹𝑎₁+𝑏₁+𝑁₁𝑇^𝛼Γ(𝛼)𝑏₁𝑏₂|Δ|+𝑏₂𝑎₂+𝑏₂+𝑚𝑁₂𝑏1+¹𝑎₁+𝑏₁+Γ2−𝛾𝑁₃^𝑚_𝑖=1𝑡^𝛾_𝑖𝑏1+¹𝑎₁+𝑏₁+Γ2−𝛾𝑇𝑁₃^𝑚_𝑖=1𝑡_𝑖^𝛾−1𝑎1+²𝑏₁|Δ|+𝑏₂𝑎₂+𝑏₂+𝑐₁𝑎₁+𝑏₁+𝑏₁𝑐₂𝑇|Δ|+𝑐₂𝑇𝑎₂+𝑏₂, (44) which implies that the operator 𝐹 is uniformly bounded on 𝐵.
On the other hand, let 𝑥∈𝐵 and, for any 𝜏₁,𝜏₂∈𝐽_𝑘, 𝑘=0,1,2,…,𝑚, with 𝜏₁<𝜏₂, we have (𝐹𝑥)𝜏₂−(𝐹𝑥)𝜏₁≤^𝜏2₀𝜏₂−𝑠^𝛼−1Γ(𝛼)𝑓((𝑠−𝑠,𝑥))𝑑𝑠_𝜏10𝜏₁−𝑠^𝛼−1Γ(𝛼)𝑓((𝑠+𝑒𝑠,𝑥))𝑑𝑠_𝑥0𝜏₂−𝜏₁+Γ2−𝛾^𝑘_𝑖=1𝑡_𝑖^𝛾−1𝐼^∗_𝑖𝑥𝑡⁻_𝑖𝜏₂−𝜏₁≤𝑁₁𝜏^𝛼₂−𝜏^𝛼₁Γ()+𝑒𝛼+1^𝑥0𝜏₂−𝜏₁+Γ2−𝛾𝑁₃^𝑘_𝑖=1𝑡_𝑖^𝛾−1𝜏₂−𝜏₁. (45) From (43) and the above inequality, we deduce that |(𝐹𝑥)(𝜏₂)−(𝐹𝑥)(𝜏₁)| tends to zero as 𝜏₂→𝜏₁. This implies that 𝐹 is equicontinuous on the interval 𝐽_𝑘. Hence, by PC-type Arzela-Ascoli Theorem (see Theorem 2.1 [15]), the operator 𝐹∶𝑃𝐶(𝐽,ℝ)→𝑃𝐶(𝐽,ℝ) is completely continuous.(H1):(1) there exist ℎ∈𝐿^∞(𝐽,ℝ⁺) and 𝜑∶[0,∞)→(0,∞) continuous nondecreasing such that |𝑓(𝑡,𝑥)|≤ℎ(𝑡)𝜑(|𝑥|) for (𝑡,𝑥)∈𝐽×ℝ; (2) there exist 𝜓 and 𝜓^∗∶[0,∞)→(0,∞) continuous nondecreasing such that |𝐼_𝑘(𝑥)|≤𝜓(|𝑥|) and |𝐼^∗_𝑘(𝑥)|≤𝜓^∗(|𝑥|) for all 𝑥∈ℝ and 𝑘=1,2,…,𝑚.

Theorem 12. Let (H1) hold and there exists a constant 𝑀>0 such that 𝑀‖ℎ‖_𝐿^∞𝜑(𝑀)𝑃+Γ2−𝛾𝜓^∗(𝑀)𝑅+𝜓(𝑀)𝑄+𝐻>1, (46) where 𝑇𝑃=^𝛼Γ()𝑏𝛼+11+¹𝑎₁+𝑏₁+𝑇^𝛼Γ(𝛼)𝑏₁𝑏₂|Δ|+𝑏₂𝑎₂+𝑏₂,𝑏𝑄=𝑚1+¹𝑎₁+𝑏₁,𝑐𝐻=¹𝑎₁+𝑏₁+𝑏₁𝑐₂𝑇|Δ|+𝑐₂𝑇𝑎₂+𝑏₂,𝑏𝑅=1+¹𝑎₁+𝑏₁^𝑚_𝑖=1𝑡^𝛾_𝑖𝑎+𝑇1+²𝑏₁|Δ|+𝑏₂𝑎₂+𝑏₂^𝑚_𝑖=1𝑡_𝑖^𝛾−1. (47) Then the impulsive fractional boundary value problem (1), (2) has at least one solution on 𝐽.

Proof. We will show that the operator 𝐹 defined by (39) satisfies the assumptions of the nonlinear alternative of Leray-Schauder type.
From Lemma 11, the operator 𝐹∶𝑃𝐶(𝐽,ℝ)→𝑃𝐶(𝐽,ℝ) is continuous and completely continuous.
Let 𝑥∈𝑃𝐶(𝐽,ℝ) be such that 𝑥(𝑡)=𝜆(𝐹𝑥)(𝑡) for some 𝜆∈(0,1). Then using the computations in the proof that 𝐹 maps bounded sets into bounded sets in Lemma 11, we have |𝑥(𝑡)|≤‖ℎ‖_𝐿^∞𝜑𝑥𝑇(‖‖)^𝛼Γ()𝑏𝛼+11+¹𝑎₁+𝑏₁+𝑇^𝛼Γ(𝛼)𝑏₁𝑏₂|Δ|+𝑏₂𝑎₂+𝑏₂𝑥𝑏+𝑚𝜓(‖‖)1+¹𝑎₁+𝑏₁+Γ2−𝛾𝜓^∗𝑥𝑏(‖‖)1+¹𝑎₁+𝑏₁^𝑚_𝑖=1𝑡^𝛾_𝑖+Γ2−𝛾𝜓^∗𝑥𝑎(‖‖)×𝑇1+²𝑏₁|Δ|+𝑏₂𝑎₂+𝑏₂^𝑚_𝑖=1𝑡_𝑖^𝛾−1+𝑐₁𝑎₁+𝑏₁+𝑏₁𝑐₂𝑇|Δ|+𝑐₂𝑇𝑎₂+𝑏₂≤‖ℎ‖_𝐿^∞𝜑𝑥𝑥𝑄𝜓(‖‖)𝑃+𝜓(‖‖)+Γ2−𝛾^∗(‖𝑥‖)𝑅+𝐻. (48) Consequently, we have ‖𝑥‖‖ℎ‖_𝐿^∞𝜑(‖𝑥‖)𝑃+Γ2−𝛾𝜓^∗(‖𝑥‖)𝑅+𝜓(‖𝑥‖)𝑄+𝐻≤1. (49) Then, in view of condition (46), there exists 𝑀>0 such that ‖𝑥‖≠𝑀. Let us set 𝑈={𝑥∈𝑃𝐶(𝐽,ℝ)∶‖𝑥‖<𝑀}. (50) The operator 𝐹∶𝑈→𝑃𝐶(𝐽,ℝ) is continuous and compact. From the choice of the set 𝑈, there is no 𝑥∈𝜕𝑈 such that 𝑥=𝜆𝐹𝑥 for some 𝜆∈(0,1). Therefore, by the nonlinear alternative of Leray-Schauder type (see Theorem 9), we deduce that 𝐹 has a fixed point 𝑥 in 𝑈 which is a solution of the impulsive fractional boundary value problem (1), (2). The proof is complete.(H2):there exist ℎ∈𝐿^∞(𝐽,ℝ⁺) and positive constants 𝐻₁ and 𝐻₂ such that, for 𝑡∈𝐽, 𝑥∈ℝ, 𝑘=1,2,…,𝑚, 𝑓(𝑡,𝑥)≤ℎ(𝑡),𝐼_𝑘(𝑥)≤𝐻₁,𝐼^∗_𝑘(𝑥)≤𝐻₂. (51)

Theorem 13. If (H2) holds, then the impulsive fractional boundary value problem (1), (2) has at least one solution on 𝐽.

Proof. Lemma 11 tells us that the operator 𝐹∶𝑃𝐶(𝐽,ℝ)→𝑃𝐶(𝐽,ℝ) defined by (39) is continuous and compact on each bounded subset 𝐵 of 𝑃𝐶(𝐽,ℝ).
Now let us show that the set 𝑉={𝑢∈𝑃𝐶(𝐽,ℝ)∶𝑢=𝜆𝐹𝑢,0<𝜆<1} is bounded. Let 𝑥∈𝑉; then 𝑥=𝜆𝐹𝑥 for some 0<𝜆<1. For each 𝑡∈𝐽, using the similar estimations given in Theorem 12, we have |𝑥(𝑡)|≤‖ℎ‖_𝐿^∞𝑇^𝛼Γ()𝑏𝛼+11+¹𝑎₁+𝑏₁+𝑇^𝛼Γ(𝛼)𝑏₁𝑏₂|Δ|+𝑏₂𝑎₂+𝑏₂+𝑚𝐻₁𝑏1+¹𝑎₁+𝑏₁+Γ2−𝛾𝐻₂𝑏1+¹𝑎₁+𝑏₁^𝑚_𝑖=1𝑡^𝛾_𝑖+Γ2−𝛾𝐻₂𝑇𝑎1+²𝑏₁|Δ|+𝑏₂𝑎₂+𝑏₂^𝑚_𝑖=1𝑡_𝑖^𝛾−1+𝑐₁𝑎₁+𝑏₁+𝑏₁𝑐₂𝑇|Δ|+𝑐₂𝑇𝑎₂+𝑏₂. (52) This implies that there exists some 𝐶>0 such that ‖𝑥‖≤𝐶 for all 𝑥∈𝑉; that is, 𝑉 is bounded. Thus, by Theorem 10, the operator 𝐹 has at least one fixed point. Hence the problem (1), (2) has at least one solution. The proof is completed.(H3):there exist ℎ∈𝐿^∞(𝐽,ℝ⁺) and positive constants 𝐿 and 𝐿^∗ such that, for 𝑡∈𝐽, 𝑥,𝑦∈ℝ, 𝑘=1,2,…,𝑚, 𝑓()(𝑡,𝐼𝑡,𝑥−𝑓𝑡,𝑦≤ℎ)𝑥−𝑦_𝑘(𝑥)−𝐼_𝑘𝑦,𝐼≤𝐿𝑥−𝑦_∗𝑘(𝑥)−𝐼^∗_𝑘𝑦≤𝐿^∗𝑥−𝑦. (53)

Theorem 14. Let (H3) hold. Moreover, ‖ℎ‖_𝐿^∞𝑇^𝛼Γ()𝐿𝛼+1+𝑚𝐿+Γ2−𝛾^∗𝑚_𝑖=1𝑡^𝛾_𝑖𝑏1+¹𝑎₁+𝑏₁+Γ2−𝛾𝑇𝐿^∗×𝑎1+²𝑏₁|Δ|+𝑏₂𝑎₂+𝑏₂^𝑚_𝑖=1𝑡_𝑖^𝛾−1+𝑏₁𝑏₂|Δ|+𝑏₂𝑎₂+𝑏₂‖ℎ‖_𝐿^∞𝑇^𝛼Γ(𝛼)<1. (54) Then the impulsive fractional boundary value problem (1), (2) has a unique solution on 𝐽.

Proof. Let 𝑥,𝑦∈𝑃𝐶(𝐽,ℝ). Then, for each 𝑡∈𝐽, we have (𝐹𝑥)(𝑡)−𝐹𝑦(𝑡)≤^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑓(𝑠,𝑥(𝑠))−𝑓𝑠,𝑦(𝑠)𝑑𝑠+𝑑^𝑥₀−𝑑^𝑦₀+^𝑚_𝑖=1𝐼_𝑖𝑥𝑡⁻_𝑖−𝐼_𝑖𝑦𝑡⁻_𝑖+Γ2−𝛾^𝑚_𝑖=1𝑡^𝛾_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖−𝐼^∗_𝑖𝑦𝑡⁻_𝑖+𝑒^𝑥₀−𝑒^𝑦₀𝑇+Γ2−𝛾𝑇^𝑚_𝑖=1𝑡_𝑖^𝛾−1𝐼^∗_𝑖𝑥𝑡⁻_𝑖−𝐼^∗_𝑖𝑦𝑡⁻_𝑖. (55) Since 𝑑^𝑥₀−𝑑^𝑦₀≤𝑏₁𝑎₁+𝑏₁×‖ℎ‖_𝐿^∞𝑇^𝛼Γ()𝐿𝛼+1+𝑚𝐿+Γ2−𝛾^∗𝑚_𝑖=1𝑡^𝛾_𝑖+𝑎𝑥−𝑦₂𝑏₁|Δ|Γ2−𝛾𝑇𝐿^∗𝑚_𝑖=1𝑡_𝑖^𝛾−1+𝑏₁𝑏₂|Δ|‖ℎ‖_𝐿^∞𝑇^𝛼Γ(𝛼),𝑒𝑥−𝑦_𝑥0−𝑒^𝑦₀𝑏𝑇≤²𝑎₂+𝑏₂×‖ℎ‖_𝐿^∞𝑇^𝛼Γ(𝛼)+Γ2−𝛾𝑇𝐿^∗𝑚_𝑖=1𝑡_𝑖^𝛾−1𝑥−𝑦, (56) then combining these two estimations with (55), we obtain ≤‖ℎ‖𝐹𝑥−𝐹𝑦_𝐿∞𝑇^𝛼Γ()𝐿𝛼+1+𝑚𝐿+Γ2−𝛾^∗𝑚_𝑖=1𝑡^𝛾_𝑖×𝑏1+¹𝑎₁+𝑏₁+Γ2−𝛾𝑇𝐿^∗𝑎1+²𝑏₁|Δ|+𝑏₂𝑎₂+𝑏₂^𝑚_𝑖=1𝑡_𝑖^𝛾−1+𝑏₁𝑏₂|Δ|+𝑏₂𝑎₂+𝑏₂‖ℎ‖_𝐿^∞𝑇^𝛼Γ(𝛼)𝑥−𝑦. (57) Therefore, by (54), the operator 𝐹 is a contraction mapping on 𝑃𝐶(𝐽,ℝ). Then it follows from Banach’s fixed point theorem that the problems (1), (2) has a unique solution on 𝐽. This completes the proof.

Next we will state some existence results for the problems (1), (3) without proofs since these are similar to the ones obtained for the problems (1), (2) above.

Theorem 15. Let (H1) hold and there exists a constant 𝑀>0 such that 𝑀‖ℎ‖_𝐿^∞𝜑(𝑀)𝑃₁+Γ2−𝛾𝜓^∗(𝑀)𝑅₁+𝜓(𝑀)𝑄₁+𝐻₁>1, (58) where 𝑃₁=𝑏1+¹𝑎₁+𝑏₁𝑇^𝛼Γ()+Γ𝑇𝛼+12−𝛾^𝛼Γ,𝑄𝛼−𝛾+1₁𝑏=𝑚1+¹𝑎₁+𝑏₁,𝑅₁=𝑏1+¹𝑎₁+𝑏₁^𝑚_𝑖=1𝑡^𝛾_𝑖+2𝑇^𝑚_𝑖=1𝑡_𝑖^𝛾−1,𝐻₁=𝑐₁𝑏₂−𝑏₁𝑐₂𝑇^𝛾Γ2−𝛾𝑎₁+𝑏₁𝑏₂+Γ2−𝛾𝑐₂𝑇^𝛾𝑏₂. (59)
Then the impulsive fractional boundary value problem (1), (3) has at least one solution on 𝐽.

Theorem 16. Let (H2) hold; then the impulsive fractional boundary value problem (1), (3) has at least one solution on 𝐽.

Theorem 17. Assume that (H3) holds. Moreover, 𝑏1+¹𝑎₁+𝑏₁‖ℎ‖_𝐿^∞𝑇^𝛼Γ()+‖ℎ‖𝛼+1^𝐿∞Γ2−𝛾𝑇^𝛼Γ𝐿𝛼−𝛾+1+𝑚𝐿+Γ2−𝛾^∗𝑚_𝑖=1𝑡^𝛾_𝑖+2Γ2−𝛾𝑇𝐿^∗𝑚_𝑖=1𝑡_𝑖^𝛾−1<1. (60) Then the impulsive fractional boundary value problem (1), (3) has a unique solution on 𝐽.

Remark 18. The results obtained in this paper can easily be generalized to the boundary value problems of impulsive fractional differential equations (1) with nonseparated integral boundary conditions 𝑎₁𝑥(0)+𝑏₁𝑥(𝑇)=𝑐₁^𝑇₀𝑔((𝑠𝑎𝑠,𝑥))𝑑𝑠,₂𝑥^′(0)+𝑏₂𝑥^′(𝑇)=𝑐₂^𝑇₀ℎ((𝑠𝑎𝑠,𝑥))𝑑𝑠,₁𝑥(0)+𝑏₁𝑥(𝑇)=𝑐₁^𝑇₀𝑔((𝑠𝑎𝑠,𝑥))𝑑𝑠,_2𝑐𝐷^𝛾𝑥(0)+𝑏₂^𝑐𝐷^𝛾𝑥(𝑇)=𝑐₂^𝑇₀ℎ(𝑠,𝑥(𝑠))𝑑𝑠, (61) where 𝑔 and ℎ∶[0,𝑇]×ℝ→ℝ are given continuous functions.

4. Examples

In this section, we give two simple examples to show the applicability of our results.

Example 1. Consider the following impulsive fractional BVP: ^𝑐𝐷^7/4𝑥(𝑡)=cos𝑡(𝑡+6)²(𝑥(𝑡)(𝑡,[]1+arctan𝑥))𝑡∈0,1,𝑡≠2,1Δ𝑥2=|𝑥(1/2⁻)|(17+|𝑥(1/2⁻,Δ)|)_𝑐𝐷^1/4𝑥12=|𝑥(1/2⁻)|(20+|𝑥(1/2⁻,(0)+1)|)3𝑥3𝑥(1)2=1,_𝑐𝐷^1/4𝑥(0)+3^𝑐𝐷^1/4𝑥(1)=14. (62)

Here 𝛼=7/4, 𝛾=1/4, 𝑇=1, 𝑚=1, 𝑎₁=3, 𝑏₁=1/3, 𝑐₁=1, 𝑎₂=2, 𝑏₂=3, and 𝑐₂=1/4. Clearly, we can take ℎ(𝑡)=2cos𝑡/(𝑡+6)², 𝐿=1/17, and 𝐿^∗=1/20 such that the relations (53) hold. Moreover, 𝑏1+¹𝑎₁+𝑏₁‖ℎ‖_𝐿^∞𝑇^𝛼Γ()+‖ℎ‖𝛼+1^𝐿∞Γ2−𝛾𝑇^𝛼Γ𝐿𝛼−𝛾+1+𝑚𝐿+Γ2−𝛾^∗𝑚_𝑖=1𝑡^𝛾_𝑖+2Γ2−𝛾𝑇𝐿^∗𝑚_𝑖=1𝑡_𝑖^𝛾−1≈11×101.3131+11817+0.0386+0.1546=0.3420<1. (63) Thus, all the assumptions of Theorem 17 are satisfied. Hence, by the conclusion of Theorem 17, the impulsive fractional BVP (62) has a unique solution on [0,1].

Example 2. Consider the following impulsive fractional BVP: ^𝑐𝐷^3/2𝑥(𝑡)=5𝑡²+𝑒^{−|𝑥(𝑡)|}(𝑡),[]1+sin𝑥𝑡∈0,1,𝑡≠4,1Δ𝑥4=2|𝑥(1/4⁻)|(1+|𝑥(1/4⁻,Δ)|)_𝑐𝐷^1/2𝑥141=cos𝑥4⁻(0)(1)+3,3𝑥+2𝑥=1,𝑥^′(0)+3𝑥^′(1)=2.5. (64)

Here 𝛼=3/2, 𝛾=1/2, 𝑇=1, 𝑚=1, 𝑎₁=3, 𝑏₁=2, 𝑐₁=1, 𝑎₂=1, 𝑏₂=3, and 𝑐₂=2.5. In the context of this problem, we have 𝑓(𝑡,𝑥)=5𝑡²+𝑒^−|𝑥|[]𝐼+sin𝑥≤7,𝑡∈0,1,𝑥∈ℝ,_𝑘(𝑥)≤2,𝐼^∗_𝑘(𝑥)≤4,𝑥∈ℝ. (65)

Put ℎ(𝑡)≡7, 𝐻₁=2, and 𝐻₂=4; we know that the condition (51) holds. Then from Theorem 13, the impulsive fractional BVP (64) has at least one solution on [0,1].

5. Conclusions

In this paper, we investigated the existence and uniqueness of solutions for a class of fractional differential equations with fractional impulsive conditions and nonseparated boundary conditions. We obtained the results by using Schaefer fixed point theorem, Banach fixed point theorem, and nonlinear alternative of Leray-Schauder type. And two examples are provided in Section 4 to illustrate the results.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the Doctor Priming Fund Project of University of South China (Grant no. 2013XQD16).