# Impulsive Integrodifferential Equations Involving Nonlocal Initial Conditions

- Rong-Nian Wang
^{1}Email author and - Jun Xia
^{1}

**2011**:634701

https://doi.org/10.1155/2011/634701

© R.-N. Wang and J. Xia. 2011

**Received: **23 November 2010

**Accepted: **7 March 2011

**Published: **15 March 2011

## Abstract

We focus on a Cauchy problem for impulsive integrodifferential equations involving nonlocal initial conditions, where the linear part is a generator of a solution operator on a complex Banach space. A suitable mild solution for the Cauchy problem is introduced. The existence and uniqueness of mild solutions for the Cauchy problem, under various criterions, are proved. In the last part of the paper, we construct an example to illustrate the feasibility of our results.

## 1. Introduction

Let denote a complex Banach space and denote by the space of all bounded linear operators from into with the usual operator norm . Let us recall the following definitions.

Definition 1.1 (see [1]).

is called the Riemann-Liouville integral of order .

Definition 1.2 (see [2]).

Remark 1.3.

It is to be noted that in the border case , the family corresponds to a classical strongly continuous semigroup, whereas in the case a solution operator corresponds to the concept of a cosine family. Moreover, according to [3], one can find that solution operators are a particular case of -regularized families and a solution operator corresponds to a -regularized family.

Remark 1.4.

Note that solution operator does not satisfy the semigroup property.

Remark 1.5.

Various solution operators are usually key tools in dealing with the abstract Cauchy problems and related issues. For more information, please see, for example, [4–11] and references therein.

Starting from some speculations of Leibniz and Euler, the fractional calculus (such as the Riemann-Liouville fractional integral) which allows us to consider integration and differentiation of any order, not necessarily integer, have been the object of extensive study for analyzing not only stochastic processes driven by fractional Brownian motion, but also nonrandom fractional phenomena in physics and optimal control (cf. e.g., [1, 12, 13]). One of the emerging branches of the study is the Cauchy problems of abstract differential equations involving fractional integration or fractional differentiation (see, e.g., [1, 14–17]). Let us point out that many phenomena in engineering, physics, economy, chemistry, aerodynamics, and electrodynamics of complex medium can be modeled by this class of equations.

where , is a generator of a solution operator , , and stand for the right and left limits of at , respectively, and , are appropriate functions to be specified later. As can be seen, the convolution integral in (1.3) is the Riemann-Liouville fractional integral, and the function constitutes a nonlocal condition.

As usual, the solution with the points of discontinuity at the moments follows that , that is, at which it is continuous from the left.

We mention that in recent years, the theory of various integrodifferential equations in Banach spaces has been studied deeply due to their important values in sciences and technologies, and many significant results have been established (see, e.g., [2, 18–23] and references therein).

where , is a -almost sectorial operator (not necessarily densely defined).

In this work, motivated by the above contributions, we shall combine these earlier work and extend the study to the Cauchy problem (1.3). New existence and uniqueness results in the case when is a generator of a solution operator, under various criterions, are proved. In the last part of paper, we construct an example to illustrate the feasibility of our results.

## 2. Preliminaries

with , , and let be the restriction of a function to .

It is easy to see is a Banach space.

Motivated by the above consideration, we give the following definition.

Definition 2.1.

where is the solution operator generated by .

We list the following basic assumptions of this paper.

is continuous in on and there exists a constant such that

is continuous and there exists a function such that

is completely continuous and there exists a continuous nondecreasing function such that for each ,

is Lipschitz continuous with Lipschitz constant .

For , is Lipschitz continuous with Lipschitz constant .

For , is completely continuous and there exists a continuous nondecreasing function such that for each ,

The following fixed-point theorem plays a key role in the proof of our main results.

Lemma 2.2 (see [34]).

Let be a convex, bounded, and closed subset of a Banach space and let be a condensing map. Then, has a fixed point in .

## 3. Main Results

To set the framework for our main existence results, we will make use of the following lemma.

Lemma 3.1.

Proof.

Assume that is a mild solution of (2.14) in the sense of Definition 2.1. Obviously, if , then one sees from Definition 2.1, that the assertion of theorem remains true. Thus, the rest proof of the theorem is done under .

This proves, for the case , that the conclusion of theorem holds.

here and are given by (3.2) and (3.3) with , respectively. A continuation of the same process shows that for any , the assertion of theorem holds.

In this work, we adopt the following concept of mild solution for the problem (1.3).

Definition 3.2.

here , is called a mild solution of the Cauchy problem (1.3).

Remark 3.3.

Note that if there is no discontinuity, that is, if , , then Definition 2.1 is equivalent to Definition 3.2.

Now we present and prove our main results.

Theorem 3.4.

Proof.

Then it is clear that is well defined.

To prove the theorem, it is sufficient to prove that has a fixed point in .

which contradicts (3.12).

maps into , here is a positive number yet to be determined, as the cases for other subintervals are the same.

This is a contradiction to (3.12). Thus, we prove that there exists an integer such that .

which means that is a contraction due to (3.12).

Thus, is a condensing map on . Then, it follows from Lemma 2.2 that the Cauchy problem (1.3) admits at least one mild solution. This completes the proof.

Theorem 3.5.

Proof.

This is a contradiction to (3.25).

Next, we will verify that for each , is a completely continuous operator, while, is a contraction. Obviously, by assumptions , it easily seen that is a completely continuous operator. Moreover, by a similar proof with that in Theorem 3.4, we can prove that is a contraction.

As a consequence of the above discussion and Lemma 2.2, we can conclude that the problem (1.3) admits at least one mild solution. The proof is completed.

Theorem 3.6.

Proof.

which implies is a contractive mapping on due to (3.29). Thus has a unique fixed point , this means that is a mild solution of (1.3). This completes the proof of the theorem.

## 4. Example

In this section, we present an example to illustrate the abstract results of this paper, which do not aim at generality but indicate how our theorems can be applied to concrete problems.

Clearly is densely defined in and is sectorial of type . Hence is a generator of a solution operator satisfying the estimate (2.7) on . Here, without lost of generality, we take .

which implies that one can choose large enough such that the first inequality of (3.29) is satisfied. Hence, according to Theorem 3.6, the Cauchy problem (4.1) has a unique mild solution.

## Declarations

### Acknowledgments

This research was supported in part by the NSF of JiangXi Province of China (2009GQS0018) and the Youth Foundation of JiangXi Provincial Education Department of China (GJJ10051).

## Authors’ Affiliations

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