Research Article | Open Access
Existence of Solutions for Fractional Impulsive Integrodifferential Equations in Banach Spaces
We investigate the existence of solutions for a class of impulsive fractional evolution equations with nonlocal conditions in Banach space by using some fixed point theorems combined with the technique of measure of noncompactness. Our results improve and generalize some known results corresponding to those obtained by others. Finally, two applications are given to illustrate that our results are valuable.
Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modelling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, and polymer rheology and have been emerging as an important area of investigation in the last few decades; see [1–4]. However, the theory of impulsive fractional evolution equations was still in the initial stages and many aspects of this theory need to be explored.
The theory of impulsive differential equations is a new and important branch of differential equation theory, which has an extensive physical, population dynamics, ecology, chemical, biological systems, and engineering background. Therefore, it has been an object of intensive investigation in recent years; some basic results on impulsive differential equations have been obtained and applications to different areas have been considered by many authors; see [4–8].
The study of nonlocal Cauchy problem for abstract evolution differential equations has been initiated by Byszewski . The existence of solutions for fractional abstract differential equations with nonlocal initial condition was recently investigated by N’Guérékata  and Balachandran and Park . In , Balachandran et al. were concerned with the existence of solutions of first-order nonlinear impulsive fractional integrodifferential equations in Banach spaces:the results are obtained by using fixed point principles.
Shu and Wang  studied the existence of mild solutions for fractional differential equation with nonlocal conditions in a Banach space :By using the contraction mapping principle and Krasnoselskii’s fixed point theorem, they obtained the existence of solutions for the equation.
In , Gou and Li investigated local and global existence of mild solution for an impulsive fractional functional integrodifferential equation with noncompact semigroup in Banach spaces :and they establish a general framework to find the mild solutions for impulsive fractional integrodifferential equations, which will provide an effective way to deal with such problems.
Motivated by this consideration, we investigate the existence of solutions for a class of impulsive fractional evolution equations with nonlocal conditions in Banach space : by using some fixed point theorems combined with the technique of measure of noncompactness, where is the Caputo fractional derivative of order , is a closed linear operator, and generates a -semigroup in ; , , where is a constant, ; is an impulsive function, and ; and , is an -valued function to be given later and, , , , and denotes the jump of at ; that is, , and represent the right and left limits of at , respectively.
The paper is organized as follows. In Section 2, we recall some concepts and facts about the Kuratowski measure of noncompactness and some fixed point theorems. In Section 3, we obtain the existence solutions of problem (4). In Section 4, we give two examples to illustrate our results.
In this section, we briefly recall some definitions and the fixed point theorems which will be used in the sequel. Throughout this paper, let be a Banach space; we assume that is a closed linear operator and generates a uniformly bounded -semigroup on a Banach space and , where stands for the Banach space of all linear and bounded operators in .
denote the Banach space of all continuous -value functions on interval and , and there exist and , with . Obviously, is a Banach space with the supnorm .
Definition 1. The fractional integral of order with the lower limit zero for a function is defined aswhere is the Gamma function.
Definition 2. The Caputo fractional derivative of order with the lower limit zero for a function is defined aswhere the function has absolutely continuous derivatives up to order .
Definition 4. By a mild solution of the initial value problemon , we mean that a continuous function defined from into satisfying
whereare the functions of Wright type defined on which satisfies
Lemma 5. The operators and have the following properties:(i)For any fixed and are linear and bounded operators; that is, for any ,(ii)For every , and are continuous functions from into .(iii)The operators and are strongly continuous, which means that, for and , one has (iv)If the semigroup is continuous by operator norm for every , then and are continuous in by the operator norm.
Let denote the Kuratowski measure of noncompactness of the bounded set. For the details of the definition and properties of the measure of noncompactness, see .
The following lemmas are to be used in proving our main results.
Lemma 6 (see ). Let be a Banach space, and let be equicontinuous and bounded; then is continuous on , and
Lemma 7 (see ). Let be bounded. Then there exists a countable set , such that
Lemma 8 (see ). Let be a Banach space, and let be a bounded and countable set. Then is the Lebesgue integral on , and
In the following, we introduce the definition of -contractive mapping.
Definition 9 (see ). Let be a metric space with distance and a given mapping. We say that is an -contractive mapping if there exist two functions and such that for all
We will show some fixed point theorems about condensing operator and -contractive mapping, which play a key role in the proof of our main results.
Lemma 10 (see ). Let be a Banach space. Assume that is a bounded closed and convex set on , is condensing. Then has at least one fixed point in .
Lemma 11 (see ). Let be a distance on a complete metric space and let be an -contractive mapping. Suppose that the following conditions hold:(i) is an -admissible mapping;(ii)there exists a point such that ;(iii)either is continuous or, for any sequence in , if for all and as , then for all . Then there exists a point such that Moreover, if , then
3. Main Results
In this section, we will establish the existence theorems of solutions for the nonlocal problem (4). For convenience, we give some notations.
For , let and denote . Let , . First of all, let us start by defining what we mean by a solution of problem (4).
Definition 12. A function is said to be a mild solution of problem (4) if satisfies the equation
To prove our main results, we state the following basic assumptions of this paper.(H1)There exists a positive constant such that and for , , there exists a constant such that(H2)For any , there exist a Lebesgue-integrable function and nondecreasing continuous function such that for all , .(H3) is a continuous and compact mapping; furthermore, there exists positive number such that and for any .(H4)The functions are continuous and there exists a constant and such that (H5)There exist constants such that for any bounded and equicontinuous sets and ,
Lemma 13 (see ). Let be a continuous function and let be the generator of a -semigroup . If is a mild solution of (4) in the sense of Definition 12, then for any ,is a solution of (4). In other words is a mild solution of (4).
Theorem 14. Let be a Banach space, let be a closed linear operator, and generates an equicontinuous -semigroup of uniformly bounded operators in . Suppose that the conditions (H1)–(H5) are satisfied. Then for every there exists a , such that problem (4) has a solution .
Proof. Since we are interested here only in local solutions, we may assume that . By using our assumption (H1)–(H4), let be such that , for and , and let us chooseSet ; then is a closed ball in with center and radius . Consider the operator defined by It is easy to see that the fixed points of are the solutions of the nonlocal problem (4); we shall prove that has a fixed point by using Lemma 10. For any and , by Lemma 5(i), we have where . So . Similarly, we prove .
And by (19), we haveTherefore, . Now we show that is continuous from into . To show this, we first observe that since is continuous in , it follows that for any and for a fixed there exists such that for any and let . Then for any , and choose Then we have Thus, we proved that is a continuous operator.
Now, we demonstrate that the operator is equicontinuous. For any and , we get that whereHere we calculate Therefore, we inspect that tend to , when ,
For , by Lemma 5(iii), as
For , by Lemma 5(iii), we have For , by Lemma 5(i), we have For , by Lemma 5(i), we have For , by Lemma 5(iii), we have For , by Lemma 5(iii), we have In conclusion, tends to as , which implies that is equicontinuous.
Let . Then it is easy to verify that maps into itself and is equicontinuous. Now, we prove that is a condensing operator. For any , by Lemma 7, there exists a countable set , such that By the equicontinuity of , we know that is also equicontinuous.
By the fact that we have Thus, by (H1), (H2), (39), and Lemma 13, we have Since is bounded and equicontinuous, we know from Lemma 6 that Therefore, from (37), (40), and (41), we know thatThus, is a condensing operator. It follows from Lemma 10 that has at least one fixed point , which is just a solution of problem (4) on the interval . This completes the proof.
Corollary 15. Let be a Banach space. be a closed linear operator and generates an equicontinuous -semigroup of uniformly bounded operators in . Suppose that the conditions (H1)–(H5) are satisfied. Then for every there exists a such that the nonlocal problemhas a solution .
Theorem 16. Let be a given function. Assume that the following conditions hold:(A)there exists such that for all and for all with ;(B)there exists such that , for all , where a mapping is defined by(C)for each , and implies that ;(D)for each , if is a sequence in such that in and for all , then for all
Proof. First of all, let . It is easy to see that is a solution of (4) if and only if is a solution of the integral equationThen problem (4) is equivalent to finding which is a fixed point of
Now, let such that for all By condition (A), we have