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A fixed point theorem for nonautonomous type superposition operators and integrable solutions of a general nonlinear functional integral equation
Journal of Inequalities and Applications volume 2014, Article number: 487 (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.
It is well known that the class of nonlinear operator equations of various types has many useful applications in describing numerous problems of the real world. A number of equations which include several given operators have arisen in many branches of science such as the theory of optimal control, economics, biological, mathematical physics and engineering. The present paper is concerned with the solvability of the following quite general nonlinear functional integral equation:
in , the space of Lebesgue integrable functions on . Here, and are two given functions, while k is a given real function defined on .
Among nonlinear operators, there is a distinguished class called superposition operators. The solvability of Eq. (1.1) is closely related to the fixed points of the nonautonomous type superposition operator, which is asked to prove that there exists satisfying the following operator equation:
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.
Definition 2.1 Let I be an interval in ℝ. A function is said to be a Carathéodory function if:
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])
Let I be a (bounded or unbounded) interval in ℝ. The superposition operator maps into if and only if there exist a function and a constant such that
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 .
Definition 2.4 Let X be a Banach space; let M, and be in . A function is said to be a measure of weak noncompactness if it satisfies the following conditions:
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 first important example of a measure of weak noncompactness has been defined by De Blasi  as follows:
The De Blasi measure of weak noncompactness has some interesting properties. It plays a significant role in nonlinear analysis and has many applications.
Nevertheless, it is rather difficult to express the De Blasi measure of weak noncompactness with the help of a convenient formula in a concrete Banach space. Such a formula is only known in the case of the space of , where I is a bounded subinterval of ℝ. In , Appell and De Pascale gave to ω the following simple form in spaces:
for all bounded subsets M of , where denotes the Lebesgue measure.
For a nonempty and bounded subset M of the space , Banas and Knap  constructed a useful measure of weak noncompactness as follows:
Finally, let us put
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 .
Theorem 2.5 A bounded set N is relatively weakly compact in if and only if the following two conditions are satisfied:
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.
Definition 2.6 Let be a subset of the Banach space X. An operator is said to be nonlinear contractive (or a φ-nonlinear contraction) if there exists a continuous and nondecreasing function with for such that
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.
For any and a sequence which converges to , we have
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. □
3 Fixed point theorem of nonautonomous type superposition operators
Theorem 3.1 Let X and Y be two Banach spaces, and let be a nonempty subset of X. Suppose that the operators and satisfy the following:
A and F is continuous,
for any , the operator is a φ-nonlinear contraction,
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 .
By assumption (3), for any , we infer that there is such that . This shows that . Since A and J are all continuous, the composite operator is continuous on . Now applying the Schauder fixed point theorem, we conclude that has at least one fixed point such that , which implies that
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]).
As another example, let and be a closed convex and bounded subset of the Banach algebra X in Theorem 3.1, where is completely continuous, and satisfy
where are two continuous nondecreasing functions satisfying and
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].
4 The solvability of general nonlinear integral equations in space
In this section, we study the existence of integrable solutions for Eq. (1.1). A number of functional integral equations, such as the following:
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;
() for an arbitrary fixed with , where satisfies
there exists a continuous and nondecreasing function with for such that
Remark 4.2 First notice that Eq. (1.1) may be written in an abstract form by Eq. (1.2), where F is the superposition operator associated to the function f (, the superposition operator of double variables type was proposed by ):
and appears as the composition of the superposition operator associated to u with the linear Volterra integral operator defined by
Our aim is now to prove that the nonautonomous type superposition operator has a fixed point in . Before starting to study this problem, we give some remarks to illustrate that the operators A and F are well defined as follows.
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 .
Theorem 4.3 If Assumption 4.1 is verified, then the equation
that is, Eq. (1.1) has at least a solution .
Proof It is clear that the solutions of the operator equation satisfy Eq. (1.1). We will use Theorem 3.1 to prove the present theorem, thus the assumptions of Theorem 3.1 have to be checked. Our proving is divided into several steps.
By Remark 4.2, the operators , are well defined and continuous, and then the assumption (1) of Theorem 3.1 is fulfilled.
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.
If there is such that for , then by () we have
It follows that
since by (). This shows that the nonautonomous type superposition operator maps into itself.
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 .
On the other hand, for each and a nonempty measurable subset D of such that , we know that if there exist and such that
which implies that
Taking into account the fact that the set consisting of one element is weakly compact, the use of Theorem 2.5 leads to
As a result,
where by ().
In the sequel let us fix arbitrarily the number . Then, for and with
which implies that
and the use of Theorem 2.5 leads to
As a result,
Thus, combining estimates (4.4) and (4.5), we obtain that
Further, from for , we obtain that
Setting , we see that is nonempty and weakly compact by Definition 2.4. Moreover, we infer that for any if holds, then .
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 .
Since is weakly compact, for an arbitrary fixed , by Theorem 2.5 there exists such that
for all .
Moreover, for the sequence in (4.6), let
According to Theorem 2.3, there exists a closed subset of such that the functions k and u are continuous on and , respectively, where . By taking with , we have
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 ).
On the other hand, we have
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 .
In what follows, by virtue of the fact that is weakly compact, let us choose a number such that for each closed subset of the interval with satisfies
Since is a Cauchy sequence in each space , there is such that and for
Consequently, (4.9) and (4.10) imply that for we have
Now, combining (4.7) and (4.11) for , we obtain that
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.
Example 4.5 Consider the following nonlinear integral equation:
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.
Define the functions as follows:
It obvious that u, k and f are all Carathéodory functions. Taking and , we have
So, u satisfies (). Taking , and , we have
It follows that f satisfies (). By a simple calculation, we obtain that
It follows that
which shows that () and () are satisfied.
From the inequality
it follows that
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).
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The author is grateful to the referees for the careful reading of the manuscript. The remarks motivated the author to make some valuable improvements.
The author declares that they have no competing interests.
This paper is completed by the sole author.
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Wang, F. A fixed point theorem for nonautonomous type superposition operators and integrable solutions of a general nonlinear functional integral equation. J Inequal Appl 2014, 487 (2014). https://doi.org/10.1186/1029-242X-2014-487
- superposition operators
- fixed points
- measure of weak noncompactness
- nonlinear functional integral equations