On a Nonlinear Integral Equation with Contractive Perturbation
© Huan Zhu. 2011
Received: 19 December 2010
Accepted: 19 February 2011
Published: 13 March 2011
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.
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].
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.
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 .
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 ).
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.
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.
It is easy to know that is a measure of weak noncompactness in space following the verification in .
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.
In , 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.
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 .
The Krasnoselskii's fixed point theorem was extended to general case in  (see also in ). In , 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.
Suppose that the following assumptions are satisfied.
Then the integral equation (1.1) has at least one solution .
The proof will be given in six steps.
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.
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 .
When , in  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 , and we obtain the existence result finally.
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