Existence Results for Fractional Differential Equations with Separated Boundary Conditions and Fractional Impulsive Conditions

Abstract

This paper is concerned with the fractional separated boundary value problem of fractional differential equations with fractional impulsive conditions. By means of the 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

Recently, much attention has been paid to study fractional differential equations due to the fact that they have been proven to be valuable tools in the mathematical modeling of many phenomena in physics, biology, mechanics, and so forth, (see [1–3]).

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 [4–10] and so forth. However, impulsive fractional differential equations 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 [11–26] and the references therein.

In this paper, we consider the existence and uniqueness of solutions for the following fractional separated boundary value problem 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,𝑎𝑘=1,2,…,𝑚,₁𝑥(0)+𝑏₁^𝑐𝐷^𝛾𝑥(0)=𝑐₁,𝑎₂𝑥(𝑇)+𝑏₂^𝑐𝐷^𝛾𝑥(𝑇)=𝑐₂, (1) where ^𝑐𝐷^𝛼 is the Caputo fractional derivative of order 𝛼∈(1,2) with the lower limit zero, 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 with 𝑎₁≠0 and 𝑎₂𝑇^𝛾Γ(2−𝛾)≠−𝑏₂.

We note that the papers on this topic cited above except [24] 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 fractional separated boundary conditions.

In [24], the author considered the following two impulsive problems: ^𝑐𝐷^𝛿𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)),𝑡∈(0,1]⧵𝑡₁,𝑡₂,…,𝑡_𝑚,_𝑐𝐷^𝛾𝑥𝑡⁺_𝑘−^𝑐𝐷^𝛾𝑥𝑡⁻_𝑘=𝐽_𝑘𝑥𝑡_𝑘,𝑥(0)𝑘=1,2,…,𝑚,=𝑥₀,𝑥^′(0)=𝑥₁, (2) where ^𝑐𝐷^𝛿 is the Caputo fractional derivative of order 𝛿∈(1,2) with the lower limit zero, 0<𝛾<1, and ^𝐿𝐷^𝛿𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)),𝑡∈(0,1]⧵𝑡₁,𝑡₂,…,𝑡_𝑚,_𝐿𝐷^𝛾𝑥𝑡⁺_𝑘−^𝐿𝐷^𝛾𝑥𝑡⁻_𝑘=𝐽_𝑘𝑥𝑡_𝑘,𝐼𝑘=1,2,…,𝑚,_1−𝛼𝑥(0)=𝑥₀, (3) where ^𝐿𝐷^𝛿 is the Riemann-Liouville fractional derivative of order 𝛿∈(0,1) with the lower limit zero and 0<𝛾<𝛿.

In [25], Fečkan et al. studied the impulsive problem of the following form: ^𝑐𝐷^𝛿𝑥(𝑡)((𝑡,(]⧵𝑡=𝑓𝑡,𝑥))𝑡∈0,𝑇₁,𝑡₂,…,𝑡_𝑚(),𝑡,𝛿∈0,1Δ𝑥_𝑘=𝐼_𝑘𝑥𝑡⁻_𝑘𝑥(0),𝑘=1,2,…,𝑚,=𝑥₀, (4) where 𝑓∶[0,𝑇]×ℝ→ℝ is jointly continuous, 𝐼_𝑘∶ℝ→ℝ and 𝑡_𝑘 satisfy 0=𝑡₀<𝑡₁<⋯<𝑡_𝑚+1=𝑇, 𝑥(𝑡⁺_𝑘)=lim_𝜖→0⁺𝑥(𝑡_𝑘+𝜖), and 𝑥(𝑡⁻_𝑘)=lim_𝜖→0⁻𝑥(𝑡_𝑘+𝜖).

Furthermore, Wang et al. [26] considered the impulsive fractional differential equations with boundary conditions as follows: ^𝑐𝐷^𝛿𝑢(𝑡)=ℎ(𝑡),𝑡∈𝐽^′(),𝑡,𝛿∈1,2Δ𝑢_𝑘=𝑦_𝑘,Δ𝑢^′𝑡_𝑘=𝑦_𝑘𝑢(0),𝑘=1,2,…,𝑚,=0,𝑢^′(1)=0, (5) where 𝑦_𝑘,𝑦_𝑘∈ℝ.

To the best of our knowledge, there are few papers concerning fractional differential equations with separated boundary conditions [27, 28].

The rest of the paper is organized as follows. In Section 2 we introduce some preliminary results needed in the sequel. In Section 3 we present the existence results for the problem (1). Two examples are given in Section 4 to illustrate the results.

2. Preliminaries

Let us set 𝐽₀=[0,𝑡₁], 𝐽₁=(𝑡₁,𝑡₂],…, 𝐽_𝑚−1=(𝑡_𝑚−1,𝑡_𝑚], and 𝐽_𝑚=(𝑡_𝑚,𝑡_𝑚+1], 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 [3]). The Riemann-Liouville fractional integral of order 𝑞 for a continuous function 𝑓∶[0,∞)→ℝ is defined as 𝐼^𝑞𝑓(𝑡)=1Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1𝑓(𝑠)𝑑𝑠,𝑞>0, (6) which provided that the integral exists.

Definition 2 (see [3]). For 𝑛−1 times an absolutely continuous function 𝑓∶[0,∞)→ℝ, 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, (7) where [𝑞] denotes the integer part of the real number 𝑞.

Lemma 3 (see [3]). Let 𝛼>0. Then the differential equation ^𝑐𝐷^𝛼ℎ(𝑡)=0 (8) 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 (9) which hold for almost all points on the interval [0,∞), here 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 problem (1) if 𝑥 satisfies the equation ^𝑐𝐷^𝛼𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)) on 𝐽^′ and the conditions Δ𝑥𝑡_𝑘=𝐼_𝑘𝑥𝑡⁻_𝑘,Δ^𝑐𝐷^𝛾𝑥𝑡_𝑘=𝐼^∗_𝑘𝑥𝑡⁻_𝑘,𝑎𝑘=1,2,…,𝑚,₁𝑥(0)+𝑏₁^𝑐𝐷^𝛾𝑥(0)=𝑐₁,𝑎₂𝑥(𝑇)+𝑏₂^𝑐𝐷^𝛾𝑥(𝑇)=𝑐₂ (10)
are satisfied.

By using a similar discussion of [25], we have the following lemma.

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)𝑐𝑑𝑠+¹𝑎₁−Λ𝑡𝑣−𝑎₂Π𝑡𝑣−𝑎₂𝑐₁𝑡𝑣𝑎₁+𝑐₂𝑡𝑣−Γ2−𝛾𝑡^𝑚_𝑖=1𝐼^∗_𝑖𝑥𝑡⁻_𝑖𝑡_𝑖^1−𝛾,𝑡∈𝐽₀;_𝑡0(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑐𝑑𝑠+¹𝑎₁+𝐼₁𝑥𝑡⁻₁−Γ2−𝛾𝑡^𝛾₁𝐼^∗₁𝑥𝑡⁻₁−Λ𝑡𝑣−𝑎₂Π𝑡𝑣−𝑎₂𝑐₁𝑡𝑣𝑎₁+𝑐₂𝑡𝑣−Γ2−𝛾𝑡^𝑚_𝑖=2𝐼^∗_𝑖𝑥𝑡⁻_𝑖𝑡_𝑖^1−𝛾,𝑡∈𝐽₁;_𝑡0(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑐𝑑𝑠+¹𝑎₁+^𝑘_𝑖=1𝐼_𝑖𝑥𝑡⁻_𝑖−Γ2−𝛾^𝑘_𝑖=1𝑡^𝛾_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖−Λ𝑡𝑣−𝑎₂Π𝑡𝑣−𝑎₂𝑐₁𝑡𝑣𝑎₁+𝑐₂𝑡𝑣−Γ2−𝛾𝑡^𝑚_𝑖=𝑘+1𝐼^∗_𝑖𝑥𝑡⁻_𝑖𝑡_𝑖^1−𝛾,𝑡∈𝐽_𝑘,𝑘=2,…,𝑚, (11) where 𝑎𝑣=²𝑇Γ2−𝛾+𝑏₂𝑇^1−𝛾Γ,2−𝛾Λ=𝑎₂^𝑇₀(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝑏₂^𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑦(𝑠)𝛼−𝛾𝑑𝑠,Π=^𝑚_𝑖=1𝐼_𝑖𝑥𝑡⁻_𝑖−Γ2−𝛾^𝑚_𝑖=1𝑡^𝛾_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖, (12)(13) if and only if 𝑥 is a solution of the impulsive fractional BVP: ^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)+𝑏₁^𝑐𝐷^𝛾𝑥(0)=𝑐₁,𝑎₂𝑥(𝑇)+𝑏₂^𝑐𝐷^𝛾𝑥(𝑇)=𝑐₂. (14)

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, (15) where 𝑑_𝑘,𝑒_𝑘∈ℝ are arbitrary constants. 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 [3]), then from (15), we have ^𝑐𝐷^𝛾𝑥(𝑡)=𝐼^𝛼−𝛾𝑦(𝑡)+𝑒_𝑘𝑡^1−𝛾Γ=2−𝛾_𝑡0(𝑡−𝑠)^{𝛼−𝛾−1}Γ𝑦(𝑠)𝑒𝛼−𝛾𝑑𝑠+^𝑘𝑡^1−𝛾Γ,2−𝛾 (16) for 𝑡∈𝐽_𝑘. Applying the boundary conditions of (14), we get 𝑎₁×𝑑₀+𝑏₁×0=𝑐₁,𝑎₂×^𝑇₀(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝑑_𝑚+𝑒_𝑚𝑇+𝑏₂×^𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑦(𝑠)𝑒𝛼−𝛾𝑑𝑠+^𝑚𝑇^1−𝛾Γ2−𝛾=𝑐₂. (17) Next, using the impulsive conditions in (14), we obtain that for 𝑘=1,2,…,𝑚𝑑_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. (18) Now we can derive the values of 𝑑_𝑘,𝑒_𝑘, 𝑘=0,1,2,…,𝑚 from formulae (17)-(18). That is, 𝑑₀=𝑐₁𝑎₁,𝑑_𝑘=𝑑₀+^𝑘_𝑖=1𝐼_𝑖𝑥𝑡⁻_𝑖−Γ2−𝛾^𝑘_𝑖=1𝑡^𝛾_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖 (19) for 𝑘=1,2,…,𝑚 and 𝑒_𝑚1=−𝑣𝑎₂^𝑇₀(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝑏₂^𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑦(𝑠)−𝑎𝛼−𝛾𝑑𝑠²𝑑_𝑚𝑣+𝑐₂𝑣,𝑒_𝑘=𝑒_𝑚−Γ2−𝛾^𝑚_𝑖=𝑘+1𝐼^∗_𝑖𝑥𝑡⁻_𝑖𝑡_𝑖^1−𝛾,for𝑘=0,1,2,…,𝑚−1. (20) Hence for 𝑘=0,1,2,…,𝑚, we have 𝑑_𝑘+𝑒_𝑘𝑡=𝑐₁𝑎₁+^𝑘_𝑖=1𝐼_𝑖𝑥𝑡⁻_𝑖−Γ2−𝛾^𝑘_𝑖=1𝑡^𝛾_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖−𝑡𝑣×𝑎₂^𝑇₀(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝑏₂^𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑦(𝑠)−𝑎𝛼−𝛾𝑑𝑠₂𝑡𝑣^𝑚_𝑖=1𝐼_𝑖𝑥𝑡⁻_𝑖−Γ2−𝛾^𝑚_𝑖=1𝑡^𝛾_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖−𝑎₂𝑐₁𝑡𝑣𝑎₁+𝑐₂𝑡𝑣−Γ2−𝛾𝑡^𝑚_𝑖=𝑘+1𝐼^∗_𝑖𝑥𝑡⁻_𝑖𝑡_𝑖^1−𝛾. (21) Now it is clear that a solution of the problem (14) has the form of (11).
Conversely, assume that 𝑥 satisfies the fractional integral equation (11). That is, for 𝑡∈𝐽_𝑘, 𝑘=0,1,2,…,𝑚, we have 𝑥(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑐𝑑𝑠+¹𝑎₁+^𝑘_𝑖=1𝐼_𝑖𝑥𝑡⁻_𝑖−Γ2−𝛾^𝑘_𝑖=1𝑡^𝛾_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖−Λ𝑣+𝑎₂Π𝑣+𝑎₂𝑐₁𝑣𝑎₁−𝑐₂𝑣𝑡−Γ2−𝛾𝑡^𝑚_𝑖=𝑘+1𝐼^∗_𝑖𝑥𝑡⁻_𝑖𝑡_𝑖^1−𝛾. (22) 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^′, (23) which means that 𝑥 satisfies the first equation of the impulsive fractional BVP (14). 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)−Λ𝛼−𝛾𝑑𝑠𝑣+𝑎₂Π𝑣+𝑎₂𝑐₁𝑣𝑎₁−𝑐₂𝑣𝑡^1−𝛾Γ2−𝛾−𝑡^𝑚1−𝛾_𝑖=𝑘+1𝐼^∗_𝑖𝑥𝑡⁻_𝑖𝑡_𝑖^1−𝛾. (24) From (22), we obtain 𝑥𝑡⁺_𝑘=^𝑡𝑘₀𝑡_𝑘−𝑠^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+^𝑘_𝑖=1𝐼_𝑖𝑥𝑡⁻_𝑖−Γ2−𝛾^𝑘_𝑖=1𝑡^𝛾_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖−Λ𝑣+𝑎₂Π𝑣+𝑎₂𝑐₁𝑣𝑎₁−𝑐₂𝑣𝑡_𝑘−Γ2−𝛾𝑡_𝑘^𝑚_𝑖=𝑘+1𝐼^∗_𝑖𝑥𝑡⁻_𝑖𝑡_𝑖^1−𝛾+𝑐₁𝑎₁,𝑥𝑡⁻_𝑘=^𝑡𝑘₀𝑡_𝑘−𝑠^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+^𝑘−1_𝑖=1𝐼_𝑖𝑥𝑡⁻_𝑖−Γ2−𝛾^𝑘−1_𝑖=1𝑡^𝛾_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖−Λ𝑣+𝑎₂Π𝑣+𝑎₂𝑐₁𝑣𝑎₁−𝑐₂𝑣𝑡_𝑘−Γ2−𝛾𝑡_𝑘^𝑚_𝑖=𝑘𝐼^∗_𝑖𝑥𝑡⁻_𝑖𝑡_𝑖^1−𝛾+𝑐₁𝑎₁. (25)(26) Hence we have, for 𝑘=1,2,…,𝑚, Δ𝑥𝑡_𝑘=𝐼_𝑘𝑥𝑡⁻_𝑘−Γ2−𝛾𝑡^𝛾_𝑘𝐼^∗_𝑘𝑥𝑡⁻_𝑘+Γ2−𝛾𝑡_𝑘𝐼^∗_𝑘𝑥𝑡⁻_𝑘𝑡_𝑘^1−𝛾=𝐼_𝑘𝑥𝑡⁻_𝑘. (27) Similarly, from (24), we can obtain that, for 𝑘=1,2,…,𝑚, Δ^𝑐𝐷^𝛾𝑥𝑡_𝑘=𝑡_𝑘^1−𝛾𝐼^∗_𝑘𝑥𝑡⁻_𝑘𝑡_𝑘^1−𝛾=𝐼^∗_𝑘𝑥𝑡⁻_𝑘. (28) Finally, it follows from (22) and (24) that (since 0∈𝐽₀, 𝑇∈𝐽_𝑚) 𝑥(0)=𝑐₁/𝑎₁, ^𝑐𝐷^𝛾𝑥(0)=0, and 𝑥(𝑇)=^𝑇₀(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑐𝑑𝑠+¹𝑎₁Λ+Π−𝑣+𝑎₂Π𝑣+𝑎₂𝑐₁𝑣𝑎₁−𝑐₂𝑣𝑇,_𝑐𝐷^𝛾𝑥(𝑇)=^𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑦(𝑠)−Λ𝛼−𝛾𝑑𝑠𝑣+𝑎₂Π𝑣+𝑎₂𝑐₁𝑣𝑎₁−𝑐₂𝑣𝑇^1−𝛾Γ.2−𝛾 (29) Now we get 𝑎₁𝑥(0)+𝑏₁^𝑐𝐷^𝛾𝑥(0)=𝑐₁,𝑎₂𝑥(𝑇)+𝑏₂^𝑐𝐷^𝛾𝑥(𝑇)𝑎=Λ+²𝑐₁𝑎₁+𝑎₂Π−𝑎₂𝑇Λ𝑣+𝑎₂Π𝑣+𝑎₂𝑐₁𝑣𝑎₁−𝑐₂𝑣−𝑏₂𝑇^1−𝛾ΓΛ2−𝛾𝑣+𝑎₂Π𝑣+𝑎₂𝑐₁𝑣𝑎₁−𝑐₂𝑣=𝑐₂. (30) Therefore 𝑥 given by (11) satisfies the impulsive fractional boundary value problem (14). The proof is complete.

Remark 6. We notice that the expression of (11) does not depend on the parameter 𝑏₁ appearing in the boundary conditions of the problem (14). Thus by Lemma 5, we conclude that the parameter 𝑏₁ is of arbitrary nature of the problem (14).

Let 𝑋,𝑌 be Banach spaces and 𝑓∶𝑋→𝑌, and we say that 𝑓 is a compact if the image of each bounded set in 𝑋 under 𝑓 is relatively compact. The following are two fixed point theorems which will be used in the sequel.

Theorem 7 (nonlinear alternative of Leray-Schauder type [29]). 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 8 (Schaefer fixed point theorem [30]). 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).

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((𝑠𝑐𝑠,𝑥))𝑑𝑠+¹𝑎₁+^𝑘_𝑖=1𝐼_𝑖𝑥𝑡⁻_𝑖−Γ2−𝛾^𝑘_𝑖=1𝑡^𝛾_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖−Λ_𝑥𝑡𝑣−𝑎₂Π_𝑥𝑡𝑣−𝑎₂𝑐₁𝑡𝑣𝑎₁+𝑐₂𝑡𝑣−Γ2−𝛾𝑡^𝑚_𝑖=𝑘+1𝐼^∗_𝑖𝑥𝑡⁻_𝑖𝑡_𝑖^1−𝛾,𝑡∈𝐽_𝑘,𝑘=0,1,2,…,𝑚, (31) with Λ_𝑥=𝑎₂^𝑇₀(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑓(𝑠,𝑥(𝑠))𝑑𝑠+𝑏₂×^𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑓((𝑠Π𝛼−𝛾𝑠,𝑥))𝑑𝑠,_𝑥=^𝑚_𝑖=1𝐼_𝑖𝑥𝑡⁻_𝑖−Γ2−𝛾^𝑚_𝑖=1𝑡^𝛾_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖. (32) Here Λ_𝑥, Π_𝑥 mean that Λ, Π 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. Observe that the problem (1) has solutions if and only if the operator equation 𝐹𝑥=𝑥 has fixed points.

Lemma 9. The operator 𝐹∶𝑃𝐶(𝐽,ℝ)→𝑃𝐶(𝐽,ℝ) defined by (31) is completely continuous.

Proof. Since 𝑓, 𝐼_𝑘, and 𝐼^∗_𝑘 are continuous, it is easy to show 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()+𝑐𝛼+1𝑎¹₁+𝑚𝑁₂+Γ2−𝛾𝑁₃^𝑚_𝑖=1𝑡^𝛾_𝑖+Λ_𝑥𝑇|𝑣|+𝑎₂Π_𝑥𝑇|𝑣|+𝑎₂𝑐₁𝑇𝑣𝑎₁+𝑐₂𝑇|𝑣|+Γ2−𝛾𝑇𝑁₃^𝑚_𝑖=1𝑡_𝑖^𝛾−1,Λ_𝑥≤𝑎₂𝑁₁𝑇^𝛼Γ()+𝑏𝛼+1²𝑁₁𝑇^𝛼−𝛾Γ,Π𝛼−𝛾+1_𝑥≤𝑚𝑁₂+Γ2−𝛾𝑁₃^𝑚_𝑖=1𝑡^𝛾_𝑖. (33)(34) Now we can obtain that, for all 𝑥∈𝐵, and 𝑡∈𝐽, 𝑡≤𝑁|(𝐹𝑥)()|₁𝑇^𝛼Γ()+𝑇𝛼+1|𝑣|𝑎₂𝑁₁𝑇^𝛼Γ()+𝑏𝛼+1²𝑁₁𝑇^𝛼−𝛾Γ+𝑎𝛼−𝛾+11+²𝑇|𝑣|𝑚𝑁₂+Γ2−𝛾𝑁₃×𝑎1+²𝑇|𝑣|^𝑚_𝑖=1𝑡^𝛾_𝑖+𝑇^𝑚_𝑖=1𝑡_𝑖^𝛾−1+𝑎₂𝑐₁𝑇𝑣𝑎₁+𝑐₂𝑇|𝑣|+𝑐₁𝑎₁, (35) 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+Λ𝑠,𝑥))𝑑𝑠_𝑥|𝑣|+𝑎₂Π_𝑥|𝑣|+𝑎₂𝑐₁𝑣𝑎₁+𝑐₂|𝑣|𝜏₂−𝜏₁+Γ2−𝛾^𝑚_𝑖=𝑘+1𝑡_𝑖^𝛾−1𝐼^∗_𝑖𝑥𝑡⁻_𝑖𝜏₂−𝜏₁≤𝑁₁𝜏^𝛼₂−𝜏^𝛼₁Γ()+Λ𝛼+1|𝑣^𝑥|+𝑎₂Π_𝑥|𝑣|+𝑎₂𝑐₁𝑣𝑎₁+𝑐₂|𝑣|×𝜏₂−𝜏₁+Γ2−𝛾𝑁₃^𝑚_𝑖=𝑘+1𝑡_𝑖^𝛾−1𝜏₂−𝜏₁. (36) By (34) and the above inequality, we deduce that (𝐹𝑥)𝜏₂−(𝐹𝑥)𝜏₁⟶0as𝜏₂⟶𝜏₁. (37) This implies that 𝐹 is equicontinuous on the interval 𝐽_𝑘. Hence by PC-type Arzela-Ascoli theorem (see Theorem 2.1 [10]), the operator 𝐹∶PC(𝐽,ℝ)→PC(𝐽,ℝ) is completely continuous.

Theorem 10. Assume that (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,𝜓^∗∶[0,∞)→(0,∞) continuous, nondecreasing such that |𝐼_𝑘(𝑥)|≤𝜓(|𝑥|), |𝐼^∗_𝑘(𝑥)|≤𝜓^∗(|𝑥|) for all 𝑥∈ℝ and 𝑘=1,2,…,𝑚; (3) there exists a constant 𝑀>0 such that 𝑀𝑃𝜑(𝑀)‖ℎ‖_𝐿^∞+𝑄𝜓(𝑀)+𝑅𝜓^∗(𝑀)+𝐻>1, (38) where 𝑇𝑃=^𝛼Γ()+𝑇𝛼+1|𝑣|𝑎₂𝑇^𝛼Γ()+𝑏𝛼+1²𝑇^𝛼−𝛾Γ,𝑎𝛼−𝛾+1𝑄=𝑚1+²𝑇|𝑣|,𝑎𝑅=Γ2−𝛾1+²𝑇|𝑣|^𝑚_𝑖=1𝑡^𝛾_𝑖+𝑇^𝑚_𝑖=1𝑡_𝑖^𝛾−1,𝑎𝐻=²𝑐₁𝑇𝑣𝑎₁+𝑐₂𝑇|𝑣|+𝑐₁𝑎₁. (39) Then, BVP (1) has at least one solution.

Proof. We will show that the operator 𝐹 defined by (31) satisfies the assumptions of the nonlinear alternative of Leray-Schauder type.
From Lemma 9, the operator 𝐹∶PC(𝐽,ℝ)→PC(𝐽,ℝ) is continuous and completely continuous.
Let 𝑥∈PC(𝐽,ℝ) such that 𝑥(𝑡)=𝜆(𝐹𝑥)(𝑡) for some 𝜆∈(0,1). Then using the computations in proving that 𝐹 maps bounded sets into bounded sets in Lemma 9, we have |𝑥(𝑡)|≤‖ℎ‖_𝐿^∞𝜑𝑥×𝑇(‖‖)_𝛼Γ()+𝑇𝛼+1|𝑣|𝑎₂𝑇^𝛼Γ()+𝑏𝛼+1²𝑇^𝛼−𝛾Γ𝜓𝛼−𝛾+1+Γ2−𝛾^∗𝑥𝑎(‖‖)1+²𝑇|𝑣|^𝑚_𝑖=1𝑡^𝛾_𝑖+𝑇^𝑚_𝑖=1𝑡_𝑖^𝛾−1+𝑎1+²𝑇|𝑣|𝑥+𝑎𝑚𝜓(‖‖)²𝑐₁𝑇𝑣𝑎₁+𝑐₂𝑇|𝑣|+𝑐₁𝑎₁. (40) Consequently, we have ‖𝑥‖𝑃‖ℎ‖_𝐿^∞𝜑(‖𝑥‖)+𝑄𝜓(‖𝑥‖)+𝑅𝜓^∗(‖𝑥‖)+𝐻≤1. (41) Then by condition (38), ‖𝑥‖≠𝑀. Let us set 𝑈={𝑥∈PC(𝐽,ℝ)∶‖𝑥‖<𝑀}. (42) 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 7), we deduce that 𝐹 has a fixed point 𝑥 in 𝑈 which is a solution of the problem (1). The proof is complete.

Theorem 11. Assume that there exist ℎ∈𝐿^∞(𝐽,ℝ⁺) and positive constants 𝐻₁,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. (43) Then, BVP (1) has at least one solution on [0,𝑇].

Proof. Lemma 9 tells us that the operator 𝐹∶PC(𝐽,ℝ)→PC(𝐽,ℝ) defined by (31) is continuous and compact on each bounded subset 𝐵 of PC(𝐽,ℝ).
Let 𝑉={𝑢∈PC(𝐽,ℝ)∶𝑢=𝜆𝐹𝑢,0<𝜆<1}. Since, for each 𝑡∈𝐽, |𝑥(𝑡=|𝜆(𝑡≤‖ℎ‖)|𝐹𝑥)()|_𝐿∞×𝑇^𝛼Γ()+𝑇𝛼+1|𝑣|𝑎₂𝑇^𝛼Γ()+𝑏𝛼+1²𝑇^𝛼−𝛾Γ𝐻𝛼−𝛾+1+Γ2−𝛾₂𝑎1+²𝑇|𝑣|^𝑚_𝑖=1𝑡^𝛾_𝑖+𝑇^𝑚_𝑖=1𝑡_𝑖^𝛾−1+𝑎1+²𝑇|𝑣|𝑚𝐻₁+𝑎₂𝑐₁𝑇𝑣𝑎₁+𝑐₂𝑇|𝑣|+𝑐₁𝑎₁, (44) we know that 𝑉 is bounded. Thus, by Theorem 8, the operator 𝐹 has at least one fixed point. Hence the problem (1) has at least one solution. The proof is completed.

Theorem 12. Assume that there exist ℎ∈𝐿^∞(𝐽,ℝ⁺) and positive constants 𝐿,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. (45) Moreover ‖ℎ‖_𝐿^∞𝑇^𝛼Γ()+𝑇‖ℎ‖𝛼+1|𝑣|^𝐿∞𝑎₂𝑇^𝛼Γ()+𝑏𝛼+1²𝑇^𝛼−𝛾Γ+𝑎𝛼−𝛾+11+²𝑇|𝑣|𝑚𝐿+Γ2−𝛾𝐿^∗×𝑎1+²𝑇|𝑣|^𝑚_𝑖∑₌₁𝑡^𝛾_𝑖+𝑇∑^𝑚_𝑖=1𝑡_𝑖^𝛾−1<1. (46) Then, BVP (1) 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(𝑠)+𝑠,𝑥))−𝑓𝑠,𝑦𝑑𝑠_𝑚𝑖=1𝐼_𝑖𝑥𝑡⁻_𝑖−𝐼_𝑖𝑦𝑡⁻_𝑖+Γ2−𝛾^𝑚_𝑖=1𝑡^𝛾_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖−𝐼^∗_𝑖𝑦𝑡⁻_𝑖+𝑇|𝑣|Λ_𝑥−Λ_𝑦+𝑎₂𝑇|𝑣|Π_𝑥−Π_𝑦𝑇×+Γ2−𝛾_𝑚𝑖=1𝑡_𝑖^𝛾−1𝐼^∗_𝑖𝑥𝑡⁻_𝑖−𝐼^∗_𝑖𝑦𝑡⁻_𝑖. (47) Since Λ_𝑥−Λ_𝑦≤𝑎₂^𝑇₀(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑓((𝑠(𝑠)+𝑏𝑠,𝑥))−𝑓𝑠,𝑦𝑑𝑠_2𝑇0(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑓((𝑠(𝑠)≤‖ℎ‖𝛼−𝛾𝑠,𝑥))−𝑓𝑠,𝑦𝑑𝑠_𝐿∞𝑎₂𝑇^𝛼Γ()+‖ℎ‖𝛼+1𝑥−𝑦^𝐿∞𝑏₂𝑇^𝛼−𝛾Γ,Π𝛼−𝛾+1𝑥−𝑦_𝑥−Π_𝑦≤^𝑚_𝑖=1𝐼_𝑖𝑥𝑡⁻_𝑖−𝐼_𝑖𝑦𝑡⁻_𝑖+Γ2−𝛾^𝑚_𝑖=1𝑡^𝛾_𝑖𝐼^∗_𝑖𝑥𝑡⁻_𝑖−𝐼^∗_𝑖𝑦𝑡⁻_𝑖≤𝑚𝐿𝑥−𝑦+Γ2−𝛾𝐿^∗𝑚_𝑖=1𝑡^𝛾_𝑖𝑥−𝑦, (48) then combining these two estimations with (47), we obtain ≤𝑇‖ℎ‖𝐹𝑥−𝐹𝑦_𝐿∞|𝑣|𝑎₂𝑇^𝛼Γ()+𝑏𝛼+1²𝑇^𝛼−𝛾Γ𝐿𝛼−𝛾+1+Γ2−𝛾^∗𝑎1+²𝑇|𝑣|^𝑚_𝑖=1𝑡^𝛾_𝑖+𝑇^𝑚_𝑖=1𝑡_𝑖^𝛾−1+‖ℎ‖_𝐿^∞𝑇^𝛼Γ()+𝑎𝛼+11+²𝑇|𝑣|𝑚𝐿𝑥−𝑦. (49) Therefore, by (46), the operator 𝐹 is a contraction mapping on PC(𝐽,ℝ). Then it follows Banach's fixed point theorem that the problem (1) has a unique solution on 𝐽. This completes the proof.

4. Examples

Finally we give two simple examples to show the applicability of our results.

Example 1. Consider the following impulsive fractional separated 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)⁻,𝑥(0))|)+2^𝑐𝐷^1/4𝑥(0)=12,12𝑥(1)+13^𝑐𝐷^1/4𝑥(1)=2. (50)
Here 𝛼=7/4, 𝛾=1/4, 𝑇=1, and 𝑚=1. Clearly, we can take ℎ(𝑡)=2cos𝑡/(𝑡+6)², 𝐿=1/17 and 𝐿^∗=1/20 such that the relations (45) hold. Moreover ‖ℎ‖_𝐿^∞𝑇^𝛼Γ()+𝑇‖ℎ‖𝛼+1|𝑣|^𝐿∞𝑎₂𝑇^𝛼Γ()+𝑏𝛼+1²𝑇^𝛼−𝛾Γ+𝑎𝛼−𝛾+11+²𝑇|𝑣|𝑚𝐿+Γ2−𝛾𝐿^∗×𝑎1+²𝑇|𝑣|^𝑚_𝑖=1𝑡^𝛾_𝑖+𝑇^𝑚_𝑖=11𝑡_𝑖^1−𝛾≈1118×1.2728+117×1.5796+20×2.7666<1. (51) Thus, all the assumptions of Theorem 12 are satisfied. Hence, by the conclusion of Theorem 12, the impulsive fractional BVP (50) has a unique solution on [0,1].

Example 2. Consider the following impulsive fractional separated 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+3,3𝑥2^𝑐𝐷^1/2𝑥(0)(1)=1,2𝑥+3^𝑐𝐷^1/2𝑥(1)=2.5. (52)
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. (53) Put ℎ(𝑡)≡7, 𝐻₁=2, and 𝐻₂=4. Then from Theorem 11, the impulsive fractional BVP (52) has at least one solution on [0,1].