- Open Access
A fixed point theorem for nonautonomous type superposition operators and integrable solutions of a general nonlinear functional integral equation
© Wang; licensee Springer. 2014
- Received: 24 April 2014
- Accepted: 25 November 2014
- Published: 8 December 2014
We first establish a new fixed point theorem for nonautonomous type superposition operators. After that, we prove the existence of integrable solutions for a general nonlinear functional integral equation in an space on an unbounded interval by using our theorem. Our main tool is the measure of weak noncompactness.
- superposition operators
- fixed points
- measure of weak noncompactness
- nonlinear functional integral equations
in , the space of Lebesgue integrable functions on . Here, and are two given functions, while k is a given real function defined on .
for two given operators and , where X and Y are two Banach spaces. Our goal in this paper is to study under what conditions Eq. (1.1) is solvable in an space. To this end, we establish a fixed point theorem for the solvability of Eq. (1.2) in advance.
The organization of this paper is as follows. In Section 2, we gather some notions and preliminary facts, including the concepts and properties of the measure of weak noncompactness, which will be needed in our further considerations. In Section 3, we establish a new fixed point theorem for Eq. (1.2). In Section 4, we prove the existence of integrable solutions for Eq. (1.1) by virtue of the measure of weak noncompactness.
for each fixed , the function is Lebesgue measurable in I;
for almost everywhere (a.e., for short) fixed , the function is continuous.
Let be a set of all measurable functions . If f is a Carathéodory function, then f defines a mapping by . This mapping is called the superposition operator (or Nemytskii operator) associated to f. The theory concerning superposition operators is presented in .
For a given measurable function , the composite operator which maps I into ℝ is said to be a nonautonomous type superposition operator. By generalizing this concept, the solvability of Eq. (1.2) may be thought the existence of fixed points of the nonautonomous type superposition operator on .
Theorem 2.2 (see [, Theorem 3.1, pp.93])
where denotes a positive cone of the space .
In this case, the operator is continuous and bounded in the sense that it maps bounded sets in bounded sets.
The following Scorza-Dragoni theorem explains the structure of the Carathéodory functions on a bounded interval.
Theorem 2.3 (see [, Theorem 3])
Let I be a bounded interval of ℝ, and let be a Carathéodory function. Then, for each , there exists a closed subset of the interval I such that and is continuous.
Next, we gather together some notations and preliminary facts of some weak topology feature which will be needed in our further considerations. Let be a collection of all nonempty bounded subsets of a Banach space X, and let be a subset of consisting of all weakly compact subsets of X. Also, let denote a closed ball in X centered in 0 and with radius r.
In what follows, we accept the following definition .
the family is nonempty and is contained in the set of relatively weakly compact sets of X;
, where refers to the closed convex hull of M;
if is a decreasing sequence of nonempty, bounded and weakly closed subsets of X with , then is nonempty.
The family described in (1) is called the kernel of the measure of weak noncompactness μ. Note that the intersection set from (5) belongs to since for every and .
The De Blasi measure of weak noncompactness has some interesting properties. It plays a significant role in nonlinear analysis and has many applications.
for all bounded subsets M of , where denotes the Lebesgue measure.
Based on the following criterion for weak noncompactness due to Dieudonné , it was shown that the function μ is a measure of weak noncompactness in the space .
for any , there exists such that if then for all ,
for any , there exists such that for all .
The nonlinear contractive property of the operators plays some important roles in our subsequent considerations.
for all .
Remark 2.7 If we take with , then such a φ-nonlinear contraction is also said to be λ-contraction.
Lemma 2.8 Let X and Y be two Banach spaces, and let be a subset of Y. If is continuous, and for any the operator is a φ-nonlinear contraction, then there exists a continuous map such that for any .
Proof For arbitrary fixed , the mapping defined by is a φ-nonlinear contraction and maps X into X, so it has a unique fixed point by [, Theorem 1]. Let us denote by the map which assigns to each the unique point in X such that . Thus, the map J is well defined.
Let . Since the operator F is continuous, we obtain as . The properties of the function φ show that , that is, . The continuity of J is proved. □
A and F is continuous,
for any , the operator is a φ-nonlinear contraction,
- (3)there exists a nonempty, compact and convex subset of such that
Then there is a point x in such that .
Proof Let us denote by the map which assigns to each the unique point in X such that . From Lemma 2.8, the map J is well defined and continuous on .
This completes the proof. □
Remark 3.2 There are some fixed point theorems, which involve several operators such as the operators in a Banach space, or in Banach algebras etc., and they may be formulated by Theorem 3.1 in a coincident form.
For example, let and be a nonempty, convex and closed set of a Banach space X in Theorem 3.1, where is compact and continuous, is a contraction mapping and for all . Then we immediately obtain the celebrated Krasnosel’skii fixed point theorem (see [, Theorem 4.4.1, pp.31]), which implies that Theorem 3.1 is a generalization of the Krasnosel’skii fixed point theorem. In fact, if we take , then satisfies the assumption (3) of Theorem 3.1 (see the proof of [, Lemma 4.4.2, pp.32]).
It is obvious that is a φ-nonlinear contraction for any with , and if we take , it is also easily proved that satisfies the assumption (3) of Theorem 3.1. Thus, we obtain the fixed point theorem for the operator in Banach algebras, which implies that Theorem 3.1 is a generalization of [, Theorem 1.5].
may all be illustrated as special examples of Eq. (1.1).
Solutions to Eq. (1.1) will be sought in , the space of Lebesgue integrable functions on with values in ℝ, endowed with the standard norm . Here are some hypotheses on the nonlinear functions involved in Eq. (1.1).
Assumption 4.1 Assume that
() is a Carathéodory function, and there exist a function and a constant such that ;
() is a Carathéodory function, and ;
() is a Carathéodory function; there exist two positive numbers α, β and a function such that for a.e. ;
() if , otherwise , where denotes the norm of the linear Volterra integral operator K generated by the function k;
- (1)It should be noted that assumption () leads to the estimate
which shows that the linear Volterra integral operator K is continuous from an space into itself, and .
Assumption () shows that the superposition operator is continuous and maps bounded sets of into bounded sets of by Theorem 2.2.
Note that being an equivalent norm of in , according to the Lucchetti-Patrone theorem (see  or [, Theorem 1]), assumption () shows that the superposition operator is continuous and maps bounded sets of into bounded sets of .
that is, Eq. (1.1) has at least a solution .
By Remark 4.2, the operators , are well defined and continuous, and then the assumption (1) of Theorem 3.1 is fulfilled.
- (2)By (), for arbitrary fixed with , there exists a continuous and nondecreasing function such that
for any , and then the assumption (2) of Theorem 3.1 is fulfilled.
- (3)If there is such that for , then by () we haveIt follows thatthat is,
since by (). This shows that the nonautonomous type superposition operator maps into itself.
- (4)Let , and let
Then () are all nonempty closed convex, and then they are weakly closed. Moreover, we have from step (3), and by the induction we may infer that for all .
where by ().
- (5)In this final step, let us prove that is compact. To this end, for any sequence with(4.6)
we shall show that possesses a convergent subsequence in .
for all .
where , and denotes the modulus of continuity of the function k on the set .
Since is relatively weakly compact and the set consisting of one element is also weakly compact, by Theorem 2.5 we infer that the terms and in (4.8) may all be arbitrarily small provided that the number is small enough. Thus, we obtain that the sequence is equicontinuous on (the space of all continuous functions defined on ).
which implies the sequence is uniformly bounded on .
Since the map J, which signs each the unique point such that , is well defined and uniformly continuous on , by Lemma 2.8 we infer that the sequence with is uniformly bounded and equicontinuous on . Hence, by applying the Arzéla-Ascoli theorem, we obtain that the set forms a relatively compact set in .
Note that our reasoning does not depend on the choice of ε. Thus we can construct a sequence of closed subsets of the interval such that as , and the sequence is relatively compact in every space . Passing to subsequences if necessary, we can assume that is a Cauchy sequence in each space for .
which shows that is a Cauchy sequence in an space. Thus, the sequence has a convergent subsequence, which implies that the closed set is compact.
This shows that the assumptions of Theorem 3.1 are all fulfilled, which completes the proof. □
Remark 4.4 The techniques of the proof of Theorem 4.3 based on Carathéodory conditions and the Scorza-Dragoni theorem were already used in [16–19] for proving the solvability of Eq. (4.1), (4.2), etc.
Finally, we provide an example, which is not included in Eq. (4.1)-(4.3), and which may be treated by our Theorem 4.3.
for . In order to show that such an equation admits a solution in , we are going to check that the conditions of Theorem 4.3 are satisfied.
which shows that () and () are satisfied.
for all . So () is satisfied for .
Since the assumptions ()-() are all satisfied, we apply Theorem 4.3 to derive the existence of a solution to Eq. (4.12).
The author is grateful to the referees for the careful reading of the manuscript. The remarks motivated the author to make some valuable improvements.
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