Impulsive fractional boundary-value problems with fractional integral jump conditions

In this paper we establish the existence and uniqueness of solutions for impulsive fractional boundary-value problems with fractional integral jump conditions. By using a variety of fixed-point theorems, some new existence and uniqueness results are obtained. Illustrative examples of our results are also presented.

MSC:34A08, 34B37, 34B15, 34B10.

In this paper, we investigate the following boundary-value problem for impulsive fractional differential equations with fractional integral jump conditions:

where {}^{c}D^{\alpha } is the Caputo fractional derivative of order α, , f:[0,T]\times \mathbb{R}\to \mathbb{R} is a continuous function, J_k:\mathbb{R}\to \mathbb{R}, \mathrm{\Delta }x(t_k)=x(t_k^{+})-x(t_k^{-}) with x(t_k^{+})={lim}_{h\to 0^{+}}x(t_k+h), x(t_k^{-})={lim}_{h\to 0^{-}}x(t_k+h), d_{k,j} are constants, I^{{\beta }_{k,j}} is the Riemann-Liouville fractional integral of order {\beta }_{k,j}>0 for j=1,2\dots ,k and k=1,2,\dots ,m, , a, b, c are given constants such that a+b\ne 0.

The integral jump conditions are very general and include many conditions as special cases. In particular, if d_{k,j}=d_j and {\beta }_{k,j}={\beta }_j, then the impulsive fractional integral of equation (1.1) reduces to

Recently, much attention has been paid to the existence of solutions for fractional differential equations due to its wide application in engineering, economics and other fields. A variety of results on initial- and boundary-value problems of fractional differential equations and inclusions can easily be found in the literature on the topic. For some recent results, we can refer to [1–13] and references cited therein.

On the other hand, integer order impulsive differential equations have become important in recent years as mathematical models of phenomena in both the physical and social sciences. There has a significant development in impulsive theory especially in the area of impulsive differential equations with fixed moments, see for instance [14–24].

In this paper we prove some new existence and uniqueness results by using a variety of fixed-point theorems. In Theorem 3.1 we prove an existence and uniqueness result by using Banach’s contraction principle, in Theorem 3.2 we prove an existence and uniqueness result by using Banach’s contraction principle and Hölder’s inequality, in Theorem 3.3 we prove the existence of a solution by using Krasnoselskii’s fixed-point theorem, while in Theorem 3.4 we prove the existence of a solution via Leray-Schauder’s nonlinear alternative. Leray-Schauder’s degree theory is used in proving the existence result in Theorem 3.5.

The rest of the paper is organized as follows: In Section 2 we recall some preliminaries and present a basic lemma which is used to convert the impulsive fractional boundary-value problem (1.1) into an equivalent integral equation. The main results are presented in Section 3, while illustrative examples are contained in Section 4.

Let PC([0,T],\mathbb{R}) = {x:[0,T]\to \mathbb{R}:x(t) is continuous everywhere except for some t_k at which x(t_k^{+}) and x(t_k^{-}) exist and x(t_k^{-})=x(t_k), k=1,2,\dots ,m}. PC([0,T],\mathbb{R}) is a Banach space endowed with the norm defined by \parallel x\parallel ={sup}_{t\in [0,T]}|x(t)|. Next, we introduce some notations, definitions of fractional calculus [25–27], and we present a preliminary result needed in our proofs later.

Definition 2.1 The Riemann-Liouville fractional integral of order \alpha >0 of a function g\in L^1((0,T),\mathbb{R}) is defined by

where Γ is the Gamma function.

Definition 2.2 The Riemann-Liouville fractional derivative of order \alpha >0 of a continuous function g:(0,\mathrm{\infty })\to \mathbb{R} is defined by

where n=[\alpha ]+1, [\alpha ] denotes the integral part of real number α, provided the right-hand side is point-wise defined on (0,\mathrm{\infty }).

Definition 2.3 For a continuous function g:(0,\mathrm{\infty })\to \mathbb{R}, the Caputo derivative of fractional order α is defined as

where n=[\alpha ]+1, [\alpha ] denotes the integral part of real number α, provided g^{(n)}(t) exists.

Lemma 2.1 ([28])

Let \alpha \in (0,1) and h:[0,T]\to \mathbb{R} be continuous. A function x\in C([0,T],\mathbb{R}) is a solution of the fractional Cauchy problem

if and only if x is a solution of the following integral equation:

Lemma 2.2 Let and a+b\ne 0. The unique solution of the impulsive fractional boundary-value problem (1.1) is given by

Proof For t\in [t_0,t_1], Riemann-Liouville fractional integrating of order α, from 0 to t, for the first equation of (1.1), we have

Substituting t=t_1 into (2.2), we get

For t\in (t_1,t_2], by Lemma 2.1 with the second equation of (1.1), we obtain

If t\in (t_2,t_3] then again from Lemma 2.1, we have

If t\in (t_k,t_{k+1}] then again from Lemma 2.1, we get

In particular, for t=T, we have

From the third equation of (1.1) and (2.3), we get

Therefore, we have

This completes the proof. □

As in Lemma 2.2, we define an operator A:PC([0,T],\mathbb{R})\to PC([0,T],\mathbb{R}) by

with a+b\ne 0. It should be noticed that problem (1.1) has solutions if and only if the operator A has fixed points.

We are in a position to establish our main results. In the following subsections we prove existence as well as existence and uniqueness results for the impulsive fractional BVP (1.1) by using a variety of fixed-point theorems.

3.1 Existence and uniqueness results via Banach’s fixed-point theorem

In this subsection we give first an existence and uniqueness result for the impulsive fractional BVP (1.1) by using Banach’s fixed-point theorem.

For convenience, we set

Theorem 3.1 Assume the following.

(H₁) There exists a constant L_1>0 such that |f(t,x)-f(t,y)|\le L_1|x-y|, for each t\in [0,T] and x,y\in \mathbb{R}.

(H₂) There exists a constant L_2>0 such that |J_k(x)-J_k(y)|\le L_2|x-y| for each x,y\in \mathbb{R}, k=1,2,\dots ,m.

If

then impulsive fractional boundary-value problem (1.1) has a unique solution in [0,T].

Proof We transform the problem (1.1) into a fixed-point problem, x=Ax, where the operator A is defined by equation (2.4). Using Banach’s contraction principle, we shall show that A has a fixed point.

Setting , and choosing r\ge \frac{1}{1-\epsilon }(M\mathrm{\Omega }+\mathrm{\Psi }), we show that A{B}_r\subset B_r, where B_r=\{x\in PC([0,T],\mathbb{R}):\parallel x\parallel \le r\}. For x\in B_r, we have

which proves that A{B}_r\subset B_r.

Now let x,y\in PC([0,T],\mathbb{R}). Then, for t\in [0,T], we have

Therefore,

As follows from equation (3.4), A is a contraction. As a consequence of Banach’s fixed-point theorem, we have A has a fixed point which is a unique solution of the impulsive fractional boundary-value problem (1.1). This completes the proof. □

Now we give another existence and uniqueness result for impulsive fractional BVP (1.1) by using Banach’s fixed-point theorem and Hölder’s inequality. In addition, for \sigma \in (0,1), we set

Theorem 3.2 Assume that the following conditions hold:

(H₃) |f(t,x)-f(t,y)|\le \xi (t)|x-y|, for each t\in [0,T], x,y\in \mathbb{R}, where \xi \in L^{\frac{1}{\gamma }}([0,T],{\mathbb{R}}^{+}), \gamma \in (0,\alpha ).

(H₄) |J_i(x)-J_i(y)|\le {\eta }_i|x-y|, for each x,y\in \mathbb{R}, with constants {\eta }_i>0, i=1,2,\dots ,m.

Denote \parallel \xi \parallel ={({\int }_0^T|\xi (s){|}^{\frac{1}{\gamma }}ds)}^{\gamma }.

If

then the impulsive fractional boundary-value problem (1.1) has a unique solution.

Proof For x,y\in PC([0,T],\mathbb{R}) and for each t\in [0,T], by Hölder’s inequality, we get

Therefore,

It follows that A is a contraction mapping. Hence Banach’s fixed-point theorem implies that A has a unique fixed point, which is the unique solution of the impulsive fractional boundary-value problem (1.1). This completes the proof. □

3.2 Existence result via Krasnoselskii’s fixed-point theorem

Lemma 3.1 (Krasnoselskii’s fixed point theorem) [29]

Let M be a closed, bounded, convex and nonempty subset of a Banach space X. Let A, B be the operators such that (a) Ax+By\in M whenever x,y\in M; (b) A is compact and continuous; (c) B is a contraction mapping. Then there exists z\in M such that z=Az+Bz.

Theorem 3.3 Let f:[0,T]\times \mathbb{R}\to \mathbb{R} be a continuous function and let (H₂) holds. In addition, we assume that:

(H₅) |f(t,x)|\le \mu (t), \mathrm{\forall }(t,x)\in [0,T]\times \mathbb{R}, and \mu \in C([0,T],{\mathbb{R}}^{+}).

(H₆) There exists a constant N>0 such that |J_k(x)|\le N, \mathrm{\forall }x\in \mathbb{R}, for k=1,2,\dots ,m.

Then the impulsive fractional boundary-value problem (1.1) has at least one solution on [0,T] if

where Φ is defined by equation (3.2).

Proof We define {sup}_{t\in [0,T]}|\mu (t)|=\parallel \mu \parallel and choose a suitable constant \overline{r} as

where Ω and Ψ are defined by equations (3.1) and (3.3), respectively. We define the operators \mathcal{P} and \mathcal{Q} on B_{\overline{r}}=\{x\in PC([0,T],\mathbb{R}):\parallel x\parallel \le \overline{r}\} as

For x,y\in B_{\overline{r}}, we find that

Thus, \mathcal{P}x+\mathcal{Q}y\in B_{\overline{r}}. It follows from the assumption (H₂) together with (3.8) that \mathcal{Q} is a contraction mapping. Continuity of f implies that the operator \mathcal{P} is continuous. Also, \mathcal{P} is uniformly bounded on B_{\overline{r}} as

Now we prove the compactness of the operator \mathcal{P}.

We define , {\tau }_1,{\tau }_2\in [0,T] with and consequently we have

which is independent of x and tends to zero as {\tau }_2-{\tau }_1\to 0. Thus, \mathcal{P} is equicontinuous. So \mathcal{P} is relatively compact on B_{\overline{r}}. Hence, by the Arzelá-Ascoli theorem, \mathcal{P} is compact on B_{\overline{r}}. Thus all the assumptions of Lemma 3.1 are satisfied. So the conclusion of Lemma 3.1 implies that the impulsive fractional boundary-value problem (1.1) has at least one solution on [0,T]. The proof is completed. □

3.3 Existence result via Leray-Schauder’s Nonlinear Alternative

Lemma 3.2 (Nonlinear alternative for single valued maps) [30]

Let E be a Banach space, C a closed, convex subset of E, U an open subset of C and 0\in U. Suppose that F:\overline{U}\to C is a continuous, compact (that is, F(\overline{U}) is a relatively compact subset of C) map. Then either

1. (i)

   F has a fixed point in \overline{U}, or

2. (ii)

   there is a u\in \partial U (the boundary of U in C) and \lambda \in (0,1) with u=\lambda F(u).

Theorem 3.4 Assume the following.

(H₇) There exist a continuous nondecreasing function \psi :[0,\mathrm{\infty })\to (0,\mathrm{\infty }) and a function p\in L^1([0,T],{\mathbb{R}}^{+}) such that

(H₈) There exists a continuous nondecreasing function \phi :[0,\mathrm{\infty })\to (0,\mathrm{\infty }) such that

(H₉) There exists a constant M^{\ast }>0 such that

Then the impulsive fractional boundary-value problem (1.1) has at least one solution on [0,T].

Proof We show that A maps bounded sets (balls) into bounded sets in PC([0,T],\mathbb{R}). For a positive number r, let B_r=\{x\in C([0,T],\mathbb{R}):\parallel x\parallel \le r\} be a bounded ball in PC([0,T],\mathbb{R}). Then for t\in [0,T] we have

Consequently

Next we show that A maps bounded sets into equicontinuous sets of PC([0,T],\mathbb{R}). Let , {\tau }_1,{\tau }_2\in [0,T] with {\tau }_1\in (t_u,t_{u+1}], {\tau }_2\in (t_v,t_{v+1}], u\le v, u,v\in \{1,2,\dots ,m\} and x\in B_r. Then we have

Obviously the right-hand side of the above inequality tends to zero independently of x\in B_r as {\tau }_2-{\tau }_1\to 0. As A satisfies the above assumptions, therefore it follows by the Arzelá-Ascoli theorem that A:PC([0,T],\mathbb{R})\to PC([0,T],\mathbb{R}) is completely continuous.

Let x be a solution. Then, for t\in [0,T], and following the similar computations as in the first step, we have

Consequently, we have

In view of (H₉), there exists M^{\ast } such that \parallel x\parallel \ne M^{\ast }. Let us set