Abstract and Applied Analysis

Volume 2012 (2012), Article ID 232314, 14 pages

http://dx.doi.org/10.1155/2012/232314

## Existence Theorem for Integral and Functional Integral Equations with Discontinuous Kernels

Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 4 March 2012; Revised 9 May 2012; Accepted 9 May 2012

Academic Editor: Giovanni Galdi

Copyright © 2012 Ezzat R. Hassan. 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.

#### Abstract

Existence of extremal solutions of nonlinear discontinuous integral equations of Volterra type is proved. This result is extended herein to functional Volterra integral equations (FVIEs) and to a system of discontinuous VIEs as well.

#### 1. Introduction

In this work the existence of extremal solutions of nonlinear discontinuous integral as well as functional integral equations is proved by weakening all forms of Caratheodory’s condition. We consider the nonlinear Volterra integral equation (for short VIEs):

The main objective in this paper is to emphasize that the kernel

Lemma 1.1. *Suppose that *

*are Lebesgue measurable for each fixed* . In particular, for each , is Lebesgue measurable for each fixed .

Lemma 1.2. *Suppose that *

*The compositions* and are Lebesgue measurable for all any continuous , and, for almost all ,

The outline of the work is as follows. In Section 2 we present our existence theorem for (1.1) in

#### 2. Volterra Integral Equations

Theorem 2.1. *Let *

*)*

*(* is continuous.

*)*

*For each* , the function is Lebesgue measurable. For all and for almost all ,
where is a Lebesgue integrable function.(

*)*

*For each* ,
(

*)*

*Let* , where, . For every and all , the functions
are equicontinuous and tend to zero as .

Under the above assumptions VIE expressed by (1.1) has extremal solutions in the interval

*Proof. *We will prove the existence of a maximal solution the proof of the existence of a minimal solution is analogous and hence is omitted. The pattern of the proof consists of four steps. Similarly as it was done in [13, 14] we define the maximal solution as the limit of an appropriate sequence of approximations *Step 1. * Since *Step 2*. We claim that, for all

To prove these assertions we shall proceed inductively. Clearly,

If (

Suppose that we are in

If we are in *Step 3.* It follows, by (*C*4), that the constructed bounded nonincreasing sequence *Step 2*, for *Step 4*. Let

We proceed similarly to prove the existence of minimal solution; we first define recursively the functions

It is interesting to point out that “

*Example 2.2. *Let

#### 3. Functional Volterra Integral Equations

Our main concern in this section is to extend result established herein (Theorem 2.1) to a functional Volterra integral equation in deriving existence of extremal solutions for a class of FVIEs (1.2).

*Notations.*

Theorem 3.1. *Let *

*)*

*(* is continuous,

*)*

*For each* and is Lebesgue measurable, and for almost all ,
(

*)*

*For each* ,
(

*)*

*For each* , whenever with . (

*)*

*Let* , where, . Let be fixed, for every and all ; the functions

*are equicontinuous and tend to zero as* .

Under the previous assumptions FVIE expressed by (1.2) has extremal solutions in the interval

*Proof. * Since the proofs of existence of maximal and minimal solutions are similar, we concentrate our attention on showing the existence of the minimal solution.

For a fixed *F4*), that

#### 4. System of Volterra Integral Equations

The main obstacle to extending the results of the previous section for vector-valued functions is that the usual order in

Theorem 4.1. *Given *(

*)*

*(* is Lebesgue measurable for any continuous ,

*)*

*for each* and Lebesgue almost all , is nondecreasing in , and for all ,
(

*)*

*for each* and Lebesgue almost all , , where and is a Lebesgue integrable function,(

*)*

*let* , for every and all , the functions
are equicontinuous and tend to zero as , where and are interpreted componentwise.

Under the above assumptions VIE expressed by (1.1) (in

*Proof. *We shall only prove the existence of a maximal solution, since the same pattern could be followed to prove existence of a minimal solution. Note that, for *Step 1* of the proof of Theorem 2.1; *Step 4* of the proof of Theorem 2.1, one can show that

#### Acknowledgments

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. (017-3/430). The author, therefore, acknowledges with thanks DSR technical and financial support.

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