- Research Article
- Open Access

# On a Nonlinear Integral Equation with Contractive Perturbation

- Huan Zhu
^{1}Email author

**Received:**19 December 2010**Accepted:**19 February 2011**Published:**13 March 2011

## Abstract

We get an existence result for solutions to a nonlinear integral equation with contractive perturbation by means of Krasnoselskii's fixed point theorem and especially the theory of measure of weak noncompactness.

## Keywords

- Banach Space
- Integral Equation
- Existence Result
- Fixed Point Theorem
- Nonlinear Integral Equation

## 1. Introduction

The integral equations have many applications in mechanics, physics, engineering, biology, economics, and so on. It is worthwhile mentioning that some problems considered in the theory of abstract differential equations also lead us to integral equations in Banach space, and some foundational work has been done in [1–8].

in the Banach space .

This equation has been studied when in [9] with and [10] with a perturbation term . Our paper extends their work to more general spaces and some modifications are also given on an error of [10].

Our paper is organized as follows.

In Section 2, some notations and auxiliary results will be given. We will introduce the main tools measure of weak noncompactness and Krasnoselskii's fixed point theorem in Section 3 and Section 4. The main theorem in our paper will be established in Section 5.

## 2. Preliminaries

First of all, we give out some notations to appear in the following.

denotes the set of real numbers and . Suppose that is a separable locally compact Banach space with norm in the whole paper. (Remark: the locally compactness of will be used in Lemma 2.2). Let be a Lebesgue measurable subset of and denote the Lebesgue measure of .

for .

Definition 2.1.

- (i)
is measurable in for any ;

- (ii)
is continuous in for almost all .

The following lemma which we will use in the proof of our main theorem explains the structure of functions satisfying Carathéodory conditions with the assumption that the space is separable and locally compact (see [11]).

Lemma 2.2.

Let be a bounded interval and be a functionsatisfying Carathéodory conditions. Then it possesses the Scorza-Dragoni property. That is each , there exists a closed subset of such that and is continuous.

The operator is called superposition operator or Nemytskij operator associated to . The following lemma on superposition operator is important in our theorem (see [12] and also in [13]).

Lemma 2.3.

for all and .

## 3. Measure of Weak Noncompactness

In this section we will recall the concept of measure of weak noncompactness which is the key point to complete our proof of main theorem in Section 5.

Let be a Banach space. and denote the collections of all nonempty bounded subsets and relatively weak compact subsets, respectively.

Definition 3.1.

- (1)
the family Ker is nonempty and Ker ;

- (2)
if , we have ;

- (3)
, where denotes the convex closed hull of ;

- (4)
for ;

- (5)
If is a decreasing sequence, that is, , every is weakly closed,

and , then is nonempty.

where denotes the closed ball in centered at 0 with radius .

for a nonempty and bounded subset of space .

for a nonempty and bounded subset of space .

It is easy to know that is a measure of weak noncompactness in space following the verification in [16].

## 4. Krasnoselskii's Fixed Point Theorem

The following is the Krasnoselskii's fixed point theorem which will be utilized to obtain the existence of solutions in the next section.

Theorem 4.1.

- (i)
;

- (ii)
is a contraction mapping;

- (iii)
is relatively compact and is continuous.

Then there exists such that .

Remark 4.2.

In [9], they proved the existence of solutions by means of Schauder fixed point theorem. With the presence of the Perturbation term in the integral equation, the Schauder fixed point theorem is invalid. To overcome this difficulty we will use the Kransnoselskii's fixed point theorem to obtain the existence of solutions.

Remark 4.3.

We will see in the following section that the important step is the construction of by means of measure of weak noncompactness. This is the biggest difference between our paper from [10].

Remark 4.4.

The Krasnoselskii's fixed point theorem was extended to general case in [17] (see also in [13]). In [10], they used the general Krasnoselskii's fixed point theorem to obtain the existence result. It can be seen in the next section of our paper that the classical Krasnoselskii's fixed point theorem is enough unless we need more general conditions on the perturbation term .

## 5. Main Theorem and Proof

Our main theorem in this paper is stated as follows.

Theorem 5.1.

Suppose that the following assumptions are satisfied.

for and .

transforms the space into itself.

(H4) The function such that where is an arbitrary subset of , and is bounded by for all .

(H5) , where denotes the norm of the linear Volterra operator .

(H6) .

Then the integral equation (1.1) has at least one solution .

Proof.

where is the linear Volterra integraloperator and is the superposition operator generated by the function .

The proof will be given in six steps.

Step 1.

There exists such that , where is a ball centered zero element with radius in .

where
by assumption *(H5)*.

Step 2.

for allbounded subset of .

Take a arbitrary numbers and such that .

It follows that by definition (3.2).

and then by definition (3.3).

From above, we then obtain for all bounded subset of .

Step 3.

We will construct a nonempty closed convex weakly compact set in on which we will apply fixed point theorem to prove the existence of solutions.

which yields that .

Denote , and then . By the definition of measure of weak noncompact we know that is nonempty. Moreover, .

is just nonempty closed convex weakly compact set which we need in the following steps.

Step 4.

is relatively compact in , where is just the set constructed in Step 3.

Considering the function on and on , we can find a closed subset of interval , such that , and such that and is continuous. Especially is uniformly continuous.

where denotes the modulusof continuity of the function on the set and . The last inequality of (5.12) is obtained since , where is just the one in the Step 1.

Taking into account the fact that the , we infer that the terms of the numerical sequence are arbitrarily small provided that the number is small enough.

Since is also arbitrarily small provided that the number is small enough, the right of (5.12) then tends to zero independent of as tends to zero. We then have is equicontinuous in the space .

From above, we then obtain that is equibounded in the space .

By assumption *(H1)*,we have the operator
is continuous. So
forms a relatively compact set in the space
.

Further observe that the above result does not depend on the choice of . Thus we can construct a sequence of closed subsets of the interval such that as and such that the sequence is relatively compact in every space . Passing to subsequence if necessary we can assume that is a cauchy sequence in each space .

for any , where .

for any .

which means that is a cauchy sequence in the space . Hence we conclude that is relatively compact in .

Step 5.

where we have made a transformation
in the above process. Since
by assumption *(H6)*, we then get the fact that the operator
is a contraction mapping.

Step 6.

- (1)
From Step 3, we know that , where is the set constructed in Step 3.

- (2)
From Step 5, we know that is a contraction mapping.

- (3)
From the Step 4 and assumptions

*(H1)*,*(H2)*, is relatively compact and is continuous.

We apply Theorem 4.1, and then obtain that (1.1) has at least one solution in .

Remark 5.2.

When , in [10] they said is weakly sequence compact in their Step 1 of main proof. From our proof, we know that their proof is not precise, since in Step 4, one of the crucial conditions to prove the relatively compactness of is that is weakly compact. We can only obtain that is weakly sequence compact as a mapping from to which is the weakly compact set defined in Step 3. The construction of set overcomes the fault in [10], and we obtain the existence result finally.

## Authors’ Affiliations

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