## Compactness Conditions in the Theory of Nonlinear Differential and Integral Equations

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Shengli Xie, "Existence of Solutions for Nonlinear Mixed Type Integrodifferential Functional Evolution Equations with Nonlocal Conditions", *Abstract and Applied Analysis*, vol. 2012, Article ID 913809, 11 pages, 2012. https://doi.org/10.1155/2012/913809

# Existence of Solutions for Nonlinear Mixed Type Integrodifferential Functional Evolution Equations with Nonlocal Conditions

**Academic Editor:**Beata Rzepka

#### Abstract

Using Mönch fixed point theorem, this paper proves the existence and controllability of mild solutions for nonlinear mixed type integrodifferential functional evolution equations with nonlocal conditions in Banach spaces, some restricted conditions on a priori estimation and measure of noncompactness estimation have been deleted, our results extend and improve many known results. As an application, we have given a controllability result of the system.

#### 1. Introduction

This paper related to the existence and controllability of mild solutions for the following nonlinear mixed type integrodifferential functional evolution equations with nonlocal conditions in Banach space : where is closed linear operator on with a dense domain which is independent of , , and defined by for and , , , , are given functions.

For the existence and controllability of solutions of nonlinear integrodifferential functional evolution equations in abstract spaces, there are many research results, see [1–13] and their references. However, in order to obtain existence and controllability of mild solutions in these study papers, usually, some restricted conditions on a priori estimation and compactness conditions of evolution operator are used. Recently, Xu [6] studied existence of mild solutions of the following nonlinear integrodifferential evolution system with equicontinuous semigroup: Some restricted conditions on a priori estimation and measure of noncompactness estimation: are used, and some similar restricted conditions are used in [14, 15]. But estimations (3.15) and (3.21) in [15] seem to be incorrect. Since spectral radius of linear Volterra integral operator , in order to obtain the existence of solutions for nonlinear Volterra integrodifferential equations in abstract spaces by using fixed point theory, usually, some restricted conditions on a priori estimation and measure of noncompactness estimation will not be used even if the infinitesimal generator .

In this paper, using Mönch fixed point theorem, we investigate the existence and controllability of mild solution of nonlinear Volterra-Fredholm integrodifferential system (1.1), some restricted conditions on a priori estimation and measure of noncompactness estimation have been deleted, our results extend and improve the corresponding results in papers [2–20].

#### 2. Preliminaries

Let be a real Banach space and let be a Banach space of all continuous -valued functions defined on with norm for . denotes the Banach space of bounded linear operators from into itself.

*Definition 2.1. *The family of linear bounded operators on is said an evolution system, if the following properties are satisfied:(i), where is the identity operator in ;(ii) for ;(iii) the space of bounded linear operator on , where for every and for each , the mapping is continuous.

The evolution system is said to be equicontinuous if for all bounded set , is equicontinuous for . is said to be a mild solution of the nonlocal problem (1.1), if for , and, for , it satisfies the following integral equation:

The following lemma is obvious.

Lemma 2.2. * Let the evolution system be equicontinuous. If there exists a such that for a.e. , then the set is equicontinuous. *

Lemma 2.3 (see [21]). *Let . If there exists such that for any and a.e. , then and
*

Lemma 2.4 (see [22]). * Let be an equicontinuous bounded subset. Then , . *

Lemma 2.5 (see [23]). * Let be a Banach space, a closed convex subset in , and . Suppose that the operator is continuous and has the following property:
**
Then has a fixed point in . *

Let , and denote the Kuratowski measure of noncompactness in and , respectively. For details on properties of noncompact measure, see [22].

#### 3. Existence Result

We make the following hypotheses for convenience. is continuous, compact and there exists a constant such that . satisfies the Carathodory conditions, that is, is measurable for each , is continuous for a.e. . There is a bounded measure function such that For each are measurable and is continuous for a.e. . For each , there are nonnegative measure functions on such that and are bounded on . For any bounded set , there is bounded measure function such that The resolvent operator is equicontinuous and there are positive numbers and such that , where , .

Theorem 3.1. *Let conditions be satisfied. Then the nonlocal problem (1.1) has at least one mild solution. *

*Proof. * Define an operator by
We have by and ,
Consequently,
where . Taking , let
Then is a closed convex subset in and . Similar to the proof in [14, 24], it is easy to verify that is a continuous operator from into . For , and imply
We can show that from , (3.9) and Lemma 2.2 that is an equicontinuous in .

Let be a countable set and
From equicontinuity of and (3.10), we know that is an equicontinuous subset in . By , it is easy to see that . By properties of noncompact measure, and Lemma 2.3, we have
Consequently,
Equations (3.10), (3.12), and Lemma 2.4 imply
where . Hence and is relative compact in . Lemma 2.5 implies that has a fixed point in , then the system (1.1), (1.2) has at least one mild solution. The proof is completed.

#### 4. An Example

In this section, we give an example to illustrate Theorem 3.1.

Let . Consider the following functional integrodifferential equation with nonlocal condition: where functions is continuous on and uniformly Hölder continuous in , is bounded measure on , , and are continuous, respectively. Taking , The operator defined by with the domain Then generates an evolution system, and can be deduced from the evolution systems so that is equicontinuous and for some constants and (see [24, 25]). The system (4.1) can be regarded as a form of the system (1.1), (1.2). We have by (4.2) for , and is continuous and compact (see the example in [7]). and can be chosen such that . In addition, for any bounded set , we can show that by the diagonal method It is easy to verify that all conditions of Theorem 3.1 are satisfied, so the system (5.1) has at least one mild solution.

#### 5. An Application

As an application of Theorem 3.1, we shall consider the following system with control parameter: where is a bounded linear operator from a Banach space to and . Then the mild solution of systems (5.1) is given by where the resolvent operator , , and satisfy the conditions stated in Section 3.

*Definition 5.1. *The system (5.2) is said to be controllable on , if for every initial function and there is a control such that the mild solution of the system (5.1) satisfies .

To obtain the controllability result, we need the following additional hypotheses. The resolvent operator is equicontinuous and , , for and positive number where are as before. The linear operator from into , defined by has an inverse operator , which takes values in and there exists a positive constant such that .

Theorem 5.2. * Let the conditions and be satisfied. Then the nonlocal problem (1.1), (1.2) is controllable. *

*Proof. * Using hypothesis , for an arbitrary , define the control
Define the operator by
Now we show that, when using this control, has a fixed point. Then this fixed point is a solution of the system (5.1). Substituting in (5.6), we get
Clearly, , which means that the control steers the system (5.1) from the given initial function to the origin in time , provided we can obtain a fixed point of nonlinear operator . The remaining part of the proof is similar to Theorem 3.1, we omit it.

*Remark 5.3. *Since the spectral radius of linear Fredholm type integral operator may be greater than 1, in order to obtain the existence of solutions for nonlinear Volterra-Fredholm type integrodifferential equations in abstract spaces by using fixed point theory, some restricted conditions on a priori estimation and measure of noncompactness estimation will not be used even if the generator . But, these restrictive conditions are not being used in Theorems 3.1 and 5.2.

#### Acknowledgments

The work was supported by Natural Science Foundation of Anhui Province (11040606M01) and Education Department of Anhui (KJ2011A061, KJ2011Z057), China.

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#### Copyright

Copyright © 2012 Shengli Xie. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.