Existence Theorems for Fractional Semilinear Integrodifferential Equations with Noninstantaneous Impulses and Delay

Abstract

In this paper, we consider a class of fractional semilinear integrodifferential equations with noninstantaneous impulses and delay. By the semigroup theory and fixed point theorems, we establish various theorems for the existence of mild solutions for the problem. An example involving partial differential equations with noninstantaneous impulses is given to show the application of our main results.

1. Introduction

Over the last couple of decades, fractional differential equations have been applied successfully to model many phenomena in physics, engineering, chemistry, financial, biology, etc. Consequently, the subject of fractional differential equations has attracted more and more attention worldwide and for more details, see for example [1–19] and the references therein.

Meanwhile, differential equations with impulsive effects have been used widely as mathematical models for the study of many phenomena in physical, biology, optimal control model of economics, etc. Much attention has been paid to the existence of solutions for the differential equations with impulses in abstract space. For details, see [7, 20–28].

In [26], the author initially studied the differential equations with noninstantaneous impulsive effects as follows:where is the generator of a -semigroup of bounded operators defined on a Banach space .

In [20], the author studied the following integer order integrodifferential equations with instantaneous impulses in a Banach space :where for any , the linear operator is the infinitesimal generator of a compact, analytic semigroup, and the nonlinear term is Lipschitz continuous. The existence of mild solutions has been proved.

In this paper, we investigate the following fractional semilinear integrodifferential equations with noninstantaneous impulses and delay:where , is the Caputo’s fractional derivative of order , is a closed and linear operator with domain defined on a Banach space , and the fixed points and satisfying are prefixed numbers. and are to be specified later.

Inspired by the results mentioned above, by the semigroup theory and fixed point theorems, we consider the existence of mild solutions for the fractional semilinear integrodifferential equations with noninstantaneous impulses and delay (3). In [7], the authors discussed the existence of solutions for the fractional ordinary differential equation with a generalized impulsive term. In [20–23], the authors discussed the integer or fractional differential equations with instantaneous impulses and the linear operator is independent of . In [26, 29, 30], the authors discussed the integer-order differential equations with noninstantaneous impulses and the linear operator is independent of . In [31–33], the authors discussed the fractional differential equations with noninstantaneous impulses and the linear operator is also independent of . In this paper, we consider the fractional semilinear integrodifferential equations with noninstantaneous impulses and delay, and the linear operator is assumed to be dependent on . Therefore, the mentioned results above are special cases of the problem investigated in this paper. Our results improve and generalize the results in References [7, 20–23, 26, 29–33].

The rest of this paper is organized as follows. In Section 2, we present the basic notation and preliminary results. In Section 3, we prove the existence of mild solutions. In Section 4, we give an illustrative example, followed by the conclusion of this paper in Section 5.

2. Preliminaries

Let be a Banach space, , , with the PC-norm .

Definition 1 [34]. The Riemann-Liouville fractional integral of order of a function is given byprovided that the right-hand side is pointwise defined on .

Definition 2 [34]. The Caputo fractional derivative of order of a function is given bywhere denotes the Gamma function, is a fractional number, , provided that the right-hand side is pointwise defined on .

Definition 3 [35, 36]. Let be a closed and linear operator with domain defined on a Banach space and . Let be the resolvent set of , we call the generator of an -resolvent family if there exist and a strongly continuous function such that and

In this case, is called the -resolvent family generated by .

Definition 4. A function is called a mild solution of the problem (3), if , and

Lemma 5. (Sadovskii fixed point theorem). Let be a convex, closed, and bounded subset of a Banach space and be a condensing map. Then, has one fixed point in .

3. Main Results

For convenience in presentation, we give here the basic assumptions to be used later throughout the paper.

(H₁) The function is continuous, and for all , there exist nonnegative Lebesgue integrable functions such that for any , we have

(H₂) The function is continuous, and for all , there exists a nonnegative Lebesgue integrable function such that for any , we have

(H₃) are continuous such that for all .

(H₄) The functions and are continuous, and for all , there exist nonnegative constants such that for any , we have

(H₄’) There is a function such that for every and for every . Let .

(H₅) The function satisfies the Carathéodory condition, i.e., is measurable for all , and is continuous for a.e. , and for all , there exists such that for .

(H₆) The function is continuous, and for all , there exists such that for .

Theorem 6. Assume and conditions (H₁)–(H₄) hold, where , ,

Then, there exists a unique mild solution of the problem (3) on and , where

Proof. Define an operator byLet . We will prove . Let , for , we haveFor all , we haveFor all , we haveFrom the above inequalities, we have that Next, we prove that is a contraction map on . For all and , we haveFor all , we haveFor all , we haveFrom the above results, for all , we haveTherefore, is a contraction map and there exists a unique mild solution of the problem (3) in and .

Theorem 7. Assume that the conditions (H₃), (H₄), (H₄’), (H₅), and (H₆) hold, is compact for and

Then, the problem (3) has a mild solution and , where

Proof. We introduce the decomposition , where are defined byLet , we then prove that the operator is a condensing map on . It is easy to see that is a closed, bounded, and convex subset of .

Step 1. We prove . Let . For , we have

For all , we have

For all , we have

Hence, .

Step 2. We prove that is a contraction on . For any and , we haveFor all , we haveFor all , we have

Thus, since , we get that is a contraction on .

Step 3. We prove that is completely continuous on . Firstly, we prove that the operator is continuous. Let , and . For any , by , we haveBy (H₅),By the Lebesgue dominated convergence theorem, we haveas Therefore, as For , the proof is similar to that for . Hence, is continuous. By the proof of Step 1, for any , it is easy to see that maps a bounded set into a bounded set in .

Secondly, we prove that is equicontinuous.

Case 1. For ,By (H₅), we have

Further, the compactness of for implies that the for is continuous in the sense of uniformly operator topology. Thus, as . It is easy to see that as since the functions and . Therefore, as .

Case 2. For