Study on the existence of solutions for a generalized functional integral equation in spaces
© Yang et al.; licensee Springer. 2013
Received: 4 November 2012
Accepted: 24 April 2013
Published: 8 May 2013
Using a nonlinear alternative theorem of Krasnosel’skii type proved recently by Smaïl Djebali and Zahira Sahnoun, we investigate, in this paper, the existence of solutions for a generalized mixed-type functional integral equation in space. We also present some examples of the integral equation to confirm the efficiency of our results.
The functional integral equations describe many physical phenomena in various areas of natural science, mathematical physics, mechanics and population dynamics [1–4]. The theory of integral equations is developing rapidly with the help of tools in functional analysis, topology and fixed-point theory (see, for instance, [5–8]) and it serves as a useful tool, in turn, for other branches of mathematics, for example, for differential equations (see [9, 10]). A fixed point theorem, frequently used to solve integral equations, is a theorem proved by Krasnosel’skii in 1958 (see, for instance, [11, 12]). The Krasnosel’skii theorem asserts that has a fixed point in a closed, convex nonempty subset M of X if A, B satisfy the following conditions:
A is compact and continuous;
B is a strict contraction;
However, the Krasnosel’skii fixed point theorem sometimes turns out to be restrictive for some equations due to the weak topology of the problem. In order to use this result and its variant, one has to find a self-mapped closed convex set M so that maps M into itself or the weaker one: . From the application point of view, this condition is also generally strict and is hard to achieve. To relax these conditions, a new effort is made in  by establishing a new variant of nonlinear Krasnosel’skii type fixed point theorem for nonself maps.
Let us first recall the nonlinear alternative Krasnosel’skii fixed point theorem established in , which plays a central role in our discussion.
Theorem 1 Let be an open subset of a Banach space X and let be the closure of S. Let and be two mappings satisfying:
A is continuous, is relatively weakly compact, and A verifies the condition H1.
B is a contraction and verifies the condition H2.
Then either the equation admits a solution in , or there exists an element (∂S denotes the boundary of S) such that for some , where conditions H1 and H2 are given in Section 2.
The advantage of Theorem 1 lies in that in applying Theorem 1, one does not need to verify that the involved operator maps a closed convex subset onto itself.
The outline of this paper is as follows. In Section 2, we introduce some basic facts and use them to obtain our aims in Section 3. In the last section, we present some examples that verify the application of this kind of nonlinear integral equation.
2.1 The weak MNC
for each .
The following Lemma 1 comes from .
if and only if is relatively weakly compact.
(where is the weak closure of ).
for all .
( refers to the convex hull of ).
The map is called the De Blasi measure of weak noncompactness.
for all bounded , where represents the Lebesgue measure, X is a finite-dimensional Banach space.
Let J be a nonlinear operator from X into itself. In what follows, we need the following two conditions:
H1. If is a weakly convergent sequence in X, then has a strongly convergent subsequence in X;
H2. If is a weakly convergent sequence in X, then has a weakly convergent subsequence in X.
2.2 The superposition operator
for any , the map is measurable from Ω to Y;
for almost all , the map is continuous from X to Y.
Definition 1 (Nemytskii’s operator)
Let be a Carathéodory function, Nemytskii’s operator associated with f, is defined by , .
The superposition operator enjoys several nice properties. Specifically, we have the following results.
Lemma 2 
where stands for the positive cone of the space .
Lemma 3 
Let X, Y be two finite-dimensional Banach spaces and let Ω be a bounded domain of . If is a Carathéodory function and maps into , then satisfies the condition H2.
We give a fixed point lemma for bilinear forms.
Lemma 4 Let X be a Banach space and let be a bilinear map. Let denote the norm in X. If for all , . Then for all satisfying , the equation has a solution satisfying and uniquely defined by the condition .
Remark 1 The proof of this lemma also shows that , where the approximate solutions are defined by and . Moreover, for all k.
3 Main results
In this section, we investigate the solvability of the nonlinear functional integral Eq. (1) in the space by applying Theorem 1.
The function is a measurable function, and g is a contraction with respect to the second variable, i.e., there exists an such that for almost all and all .
, satisfy Carathéodory conditions and , , act from into itself continuously.
The operators T and A are linear and bounded on .
The Urysohn operator U defined as before maps continuously into .
- (e)for and , where ξ belongs to , μ is a nonnegative constant and is a measurable function such that its associated integral operator K defined by(4)
- (f)There exists a constant independent of such that any solution of the integral equation
Remark 2 It is deserved to mention that though the Urysohn operator U maps into itself, it does not have to be continuous. Sufficient conditions showing that U maps into itself and is continuous can be found in .
Before going on, we give crucial Lemma 5.
Lemma 5 Let X be a finite-dimensional Banach space and let Ω be a compact subset of . If the conditions (b)-(e) are satisfied, then the operator satisfies the condition H2.
for any bounded subset S of .
Next, let be a weakly convergent sequence of . Owing to (6), we infer that . This shows that the set is relatively weakly compact in . This completes the proof. □
This shows that the operator ℬ is continuous and maps a bounded set of into a bounded set of . According to Lemma 3, we obtain ℬ satisfies the condition H2.
Now we are in a position to state our main result.
Theorem 2 Let X be a finite-dimensional Banach space and let Ω be a bounded domain of . Assume that the conditions (a)-(f) hold true. Then Eq. (1) admits at least one solution in .
So, ℬ is a strict contraction mapping on , and from Remark 3, ℬ satisfies the condition H2.
Claim 2. Clearly, by the assumptions (b)-(d), is continuous on . Now we check that satisfies the condition H1. To do this, let be a weakly convergent sequence of . By Lemma 5, has a weakly convergent subsequence, say . Furthermore, the continuity of the linear operator T implies its weak continuity on for almost all . Thus, the sequence , i.e., converges pointwisely for almost all . By the Vitali convergence theorem [, p.150], we conclude that converges strongly in . Hence, satisfies the condition H1.
Owing to (5), we deduce that , and hence is relatively weakly compact.
Finally, thanks to the assumption (f), the second situation of Theorem 1 does not occur. Now, applying Theorem 1, we get that has a fixed point in , that is to say, Eq. (1) has a solution in . This completes the proof. □
Remark 4 The requirement that X should be a finite-dimensional Banach space comes from the usage of the relation (3) proved in  for bounded subsets in the space of Lebesgue integrable functions with values in a finite-dimensional Banach space.
By Theorem 2, we can get a special existence criterion for Eq. (1).
- (i)There exists a continuous function such that whenever and
where and L is the constant in the assumption (a). Then Eq. (1) has a solution in .
, where the constants , , μ are these in Lemma 2 and the hypothesis (f), denotes the norm of the operator K defined in (4).
Then Eq. (1) has a solution in .
which is a contradiction. So, the hypothesis (g) is satisfied and the result then follows from Theorem 2. This completes the proof. □
In this section, we provide some examples of a classical integral and functional equation considered in nonlinear analysis which are a particular case of Eq. (1).
is also a special case of Eq. (1) with and .
We can suppose T to be an arbitrary linear bounded operator on . It is easy to see that the function g satisfies the assumption (a) with , the function with and .
The work is supported by the National Natural Science Foundation of China (No. 10861014, No. 11161057).
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