Leray–Schauder Fixed Point Theorems for Block Operator Matrix with an Application

Farid, Mohamed Amine; Marhrani, El Miloudi; Aamri, Mohamed

doi:https://doi.org/10.1155/2021/9985817

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Volume 2021 | Article ID 9985817 | https://doi.org/10.1155/2021/9985817

Leray–Schauder Fixed Point Theorems for Block Operator Matrix with an Application

Mohamed Amine Farid,¹El Miloudi Marhrani,¹and Mohamed Aamri¹

Academic Editor: Huseyin Isik

Received10 Mar 2021

Revised16 Jun 2021

Accepted10 Jul 2021

Published27 Jul 2021

Abstract

In this paper, we establish some new variants of Leray–Schauder-type fixed point theorems for a block operator matrix defined on nonempty, closed, and convex subsets of Banach spaces. Note here that need not be bounded. These results are formulated in terms of weak sequential continuity and the technique of De Blasi measure of weak noncompactness on countably subsets. We will also prove the existence of solutions for a coupled system of nonlinear equations with an example.

1. Introduction

With the development of new problems in diverse fields of sciences as well as in physical, biological, and social sciences, the theory of fixed point and its applications are very diverse and continuously growing. Also, the theory of block operator matrix is a subject of great interest thanks to the useful applications for studying some systems of integral equations as well as systems of partial or ordinary differential equations. Recent work has employed the fixed point technique for the operator matrix with nonlinear entries acting on Banach spaces or Banach algebras for studying the existence of solutions for several classes of systems of nonlinear integral equations, see, for example, [1–5]. These operators are defined by a block operator matrix:

Based on new generalized Schauder and Krasnoselskii fixed point theorems for the block operator matrix (1), Ben Amar et al., in [6], have established some results for a coupled system of differential equations on for , under abstract boundary conditions of Rotenberg’s model type. These last equations were proposed by M. Rotenberg and model the evolution of a cell population [7]. Due to the lack of compactness on spaces, the study in [6] did not cover the case . Later, Jeribi et al. [1] proposed to extend the results of Amar et al. [6] to the case by establishing new variants of fixed point theorems for (1), and their analysis was carried out via arguments of weak topology and, particularly, the technique of measures of weak noncompactness. In the above quoted works, the assumptions that or are invertible play a fundamental role in the arguments. Jeribi et al., in [8], were interested in studying the case when is not injective and established some fixed point theorems for operator (1), involving multivalued maps acting on Banach spaces. This way, their results were formulated in terms of weak sequential continuity and the technique of De Blasi measure of weak noncompactness. The results obtained are then applied to the two-dimensional nonlinear functional integral equation:for all , where with and ; here, is a Banach space. And, , , , and are suitably defined functions.

The main purpose of this paper is to obtain some new variants of Leray–Schauder-type fixed point theorem for operator (1) on a Banach space. From application, we discuss the existence of solutions to problem (2) in a suitable Banach space and an example of a nonlinear integral equation in the Banach space .

Note that system (2) can be written as a fixed point problem:where

The present paper is built up as follows. In Section 2, we introduce the necessary definitions and preliminary concepts. Section 3 is devoted to present some new variants of Leray–Schauder-type fixed point theorems for a block operator matrix maps acting on Banach spaces. Finally, in Section 4, we apply Corollary 1 in order to discuss the existence of solutions for problem (2).

2. Basic Definitions and Preliminary Concepts

In this section, we give some essential definitions, properties, and theorems of fixed point theory, which should be used in the present paper. Throughout this paper, unless otherwise mentioned, denotes a Banach space endowed with the norm and with the zero element , denotes the closed ball in centered at with radius , and denotes the collection of all nonempty bounded subsets of . Moreover, we write and to denote, respectively, the strong convergence and the weak convergence of a sequence to . We say that a map is weakly sequentially continuous if, for every sequence with , we have .

The De Blasi measure of weak noncompactness [9] is the map defined by

The De Blasi measure of weak noncompactness satisfies the following properties. For a proof, we refer the reader to [9, 10].

Lemma 1. Let and be in , and we have(i)If , then (ii) if and only if is relatively weakly compact(iii), where is the weak closure of (iv), for all (v), where is the closed convex hull of (vi)(vii), for all

In [10], Appell and De Pascale proved that, in spaces, the maps has the following form:

For all bounded subsets of , where is a finite dimensional Banach space and denotes the Lebesgue measure, we recall the following definitions.

Definition 1. Let be a subset of a Banach space , and . Let be a mapping, we say that1. is -contractive if for any bounded set (2) is condensing if for any bounded set with (3) is countably -contractive if for any countable bounded set (4) is countably condensing if for any countable bounded set with Clearly, every contractive is countably contractive. Now, is said to be Lipschitzian if with . If , is called a contraction.

Definition 2. A mapping is said to be weakly compact if is relatively weakly compact for every nonempty bounded subset .
Following [11], we recall the next definition.

Definition 3. A mapping is called to be a separate contraction if there exist two functions , satisfying(1) is strictly increasing and (2)(3) for any Note that every contraction mapping is a separate contraction mapping.

Definition 4 (see [12]). A mapping is a nonlinear contraction mapping if there exists a continuous nondecreasing function satisfying(1) for any (2)

Definition 5. An operator is said to be -expansive if there exists a function such that(1)(2), for any (3) is either continuous or nondecreasing(4) for all In particular, for with , then is an expansive mapping (see [13]). The following results are the nonlinear alternatives of Leray–Schauder-type states in [14].

Theorem 1. Let be a nonempty closed and convex subset of a Banach space and be a weakly open subset of with such that is a weakly compact subset of and is a weakly sequentially continuous mapping.
Then, either(A1) has a fixed point or(A2)there is a point and with , where denotes the weak boundary of in .

Remark 1. In Theorem 1, the condition “ is a weakly compact” can be replaced by “ is relatively weakly compact,” for the proof, see Remark 3.2 in [14].

Theorem 2. Let be a nonempty closed and convex subset of a Banach space and be a weakly open subset of with . Assume that is a weakly sequentially continuous and condensing map with bounded. Then, either(A1) has a fixed point or(A2)there is a point and with , where denotes the weak boundary of in .

Amar et al., in [15], showed that the condition “condensing” in Theorem 2 can be relaxed by the assumption “countably condensing.”

Theorem 3. Let be a nonempty closed and convex subset of a Banach space and be a weakly open subset of with . Assume that is a weakly sequentially continuous and countably condensing map with bounded. Then, either(A1) has a fixed point or(A2)there is a point and with , where denotes the weak boundary of in .

The following results are crucial for our purposes.

Lemma 2 (see [16]). Let be a subset of a Banach space and let be a Lipschitzian map. Assume that is a sequentially weakly continuous map. Then, for each bounded subset of ; here, stands for the De Blasi measure of weak noncompactness.

Lemma 3 (see [17]). Let be a Hausdorff compact space and be a Banach space. A bounded sequence converges weakly to if and only if, for every , the sequence converges weakly (in ) to .

3. Main Result

In this section, we give some new variants of Leray–Schauder for the operator (1). The first result is formulated as follows.

Theorem 4. Let be a nonempty closed and convex subset of a Banach space and be a weakly open subset of with . Let and be four operators such that(i) is linear and bounded, and there is such that is a separate contraction(ii) and are weakly sequentially continuous and is relatively weakly compact(iii) is linear and bounded, and there is such that is a separate contraction(iv), for all Then, either(A1)the block operator matrix (1) has a fixed point or(A2)there exists and such thatwhere denotes the weak boundary of in .

Proof. If we refer to Lemma 1.2 in [11] and to page 39 in [18], we can prove that and exist and are weakly continuous. Hence, we can define the mapping byIn view of Theorem 1, it suffices to establish that is weakly sequentially continuous and is relatively weakly compact.
We have and , and and are weakly sequentially continuous; then, is weakly sequentially continuous, and because is relatively weakly compact, we get is relatively weakly compact. Hence, either(1) has a fixed point or(2)there is a point and with .In the first case, the vector solves the problem, whereas in the second case we use the vector to achieve the proof.

Remark 2. (1)Theorem 4 remains true if we suppose that or is a nonlinear contraction(2)Theorem 4 is a generalization of Theorem 4 and Theorem 3.2 in [19]We also have the following result.

Theorem 5. Let be a nonempty closed and convex subset of a Banach space and be a weakly open subset of with . Let and be four operators such that(i) and are relatively weakly compact(ii), , and are weakly sequentially continuous(iii) is linear and bounded, and there is such that is a separate contraction(iv), for all Then, either(A1)the block operator matrix (1) has a fixed point or(A2)there exists and such thatwhere denotes the weak boundary of in .

Proof. The reasoning in the proof of Theorem 4 yields that exists and is weakly continuous. Hence, we can define the mapping byWe can see that is weakly sequentially continuous because , , , and are weakly sequentially continuous. The assumption proves that is relatively weakly compact. Hence, by Theorem 1, we have(1) has a fixed point or(2)There is a point and with In the first case, the vector solves the problem, whereas in the second case we use the vector to achieve the proof.

Remark 3. (1)In Theorem 5, we can replace the assumption “, for all ,” by “, for all implies ”(2)Theorem 5 is a generalization of Theorem 3.3 in [19]Thus, we give the following result

Theorem 6. Let be a nonempty closed and convex subset of a Banach space and be a weakly open subset of with . Let and be four operators such that(i) and are relatively weakly compact(ii), , and are weakly sequentially continuous(iii) is linear and bounded, and there is such that is a separate contraction(iv), for all Then, either(A1)the block operator matrix (1) has a fixed point or(A2)there exists and such thatwhere denotes the weak boundary of in .

Proof. Reasoning as in the proof of Theorem 5, we obtain(1) has a fixed point or(2)There is a point and with In the first case, the vector solves the problem, whereas in the second case, we use the vector to achieve the proof.

Remark 4. Theorem 6 is a generalization of Theorem 3.4 in [19].

Theorem 7. Let be a nonempty closed and convex subset of Banach space and be a weakly open subset of with . Let and be four weakly sequentially continuous operators such that(i) and are relatively weakly compact(ii) is a contraction with (iii), for all Then, either(A1)the block operator matrix (1) has a fixed point or(A2)there exists and such thatwhere denotes the weak boundary of in

Proof. By assumption , we can see that exists and continuous; hence, from Theorem 5, we deduce the desired result.
Inspired by [20], we deduce the following result.

Corollary 1. Let be a nonempty closed and convex subset of a Banach space and be a weakly open subset of with . Let and be four weakly sequentially continuous operators such that(i) and are relatively weakly compact(ii) is a contraction with (iii) for all In addition, assume thatfor all and . Then, the set of fixed points of the block operator matrix (1) in is nonempty.

In the next results, we explore the case where and are nonlinear.

Theorem 8. Let be a nonempty closed and convex subset of a Banach space and be a weakly open subset of with . Let and be four weakly sequentially continuous operators such that(i) exists on (ii) is relatively weakly compact and is countably contractive(iii) is countably contractive and is countably contractive with (iv) is a bounded subset of Then, either(A1)the block operator matrix (1) has a fixed point or(A2)there exists and such thatwhere denotes the weak boundary of in .

Proof. We defend the operator byWe show that is weakly sequentially continuous and countably condensing.
Let be a sequence in such that ; since is weakly sequentially continuous, then the set is relatively weakly compact. Note thatWe have is countably contractive; then,which implies that ; then, is relatively weakly compact. Consequently, there exists a subsequence of such that . From (16) and because and are weakly sequentially continuous, we get ; then, ; consequently, . Now, we show that . Suppose the contrary; then, there exists a subsequence of and a weak neighborhood of such that for all . Since , then arguing as before, we may extract a subsequence of such that , which is a contraction. As a result, is weakly sequentially continuous. Because and are weakly sequentially continuous, we get is weakly sequentially continuous.
Now, we show that is countably condensing. Let be a countably subset of such that ; by (16), we haveThen,Using the subadditivity of the De Blasi measure of weak noncompactness, we obtainHence, is countably condensing.
Then, by Theorem 3, we obtain(1) has a fixed point or(2)There is a point and with In the first case, the vector solves the problem, whereas in the second case, we use the vector to achieve the proof.

Remark 5. In Theorem 8, we can replace the De Blasi measure of weak noncompactness by any subadditive measures of weak noncompactness on .
In Theorem 8, condition is difficult to verify; in the following, we will change it under weaker conditions.

Theorem 9. Let be a nonempty closed and convex subset of a Banach space and be a weakly open subset of with . Let and be four weakly sequentially continuous operators such that(i) is expansive and (ii) is relatively weakly compact and is countably contractive(iii) is countably contractive and is countably contractive with (iv) is a bounded subset of Then, either(A1)the block operator matrix (1) has a fixed point or(A2)there exists and such thatwhere denotes the weak boundary of in .

Proof. Let such that ; since is expansive, we obtainThus, is bijective, and because , we get is well defined on . Now, reasoning as in the proof of Theorem 8, we get the desired result.

Theorem 10. Let be a nonempty closed and convex subset of a Banach space and be a weakly open subset of with . Let and be four weakly sequentially continuous operators such that(i) is countably contractive and is relatively weakly compact(ii) is expansive for some and each , where for (iii) is a contraction with constant and is countably contractive with (iv) is a bounded subset of Then, either(A1)the block operator matrix (1) has a fixed point, or(A2)there exists and such thatwhere denotes the weak boundary of in .

Proof. By assumptions and with Lemma 2.3 in [21], we get is well defend. And, by assumption and Lemma 2, we have that is countably contractive. Now, reasoning as in the proof of Theorem 8, we get the desired result.

Remark 6. Theorem 10 remains true if we suppose that is contractive.

4. Application

Let be a Banach space. As usual, we will denote by the Banach space of all valued continuous functions defined on . We equip the space with its standard norm:

The goal of this section is to apply Corollary 1 to study the existence of continuous solutions to the nonlinear functional integral equations (2).

Let us now introduce the following assumptions.(H0)(i)The function is continuous and nondecreasing(ii)The function is continuous(iii)The function is continuous, and (H1)The operator is such that(i)For all , the operator is weakly sequentially continuous and weakly compact.(ii)For each , the operator is continuous.(iii)There exists a continuous function with bound such that(H2)The operator is continuous such that, for each and , and the operator is continuous uniformly.(H3)The operator is such that(i)For each , the operator is weakly sequentially continuous.(ii)For each , the operator is continuous.(iii)There exists a constant such that(H4)The operators is such that(i)For each , the operator is measurable.(ii)For each , the operator is weakly sequentially continuous.(iii)There exists a constant and a function such that(H5)Assume that there exists such that , , and .

Let us define the subset of by

We can see that is a nonempty closed and convex subset of . Let be a weakly open subset of such that . Notice that (2) is equivalent to the system:where the operators , , , and defined by

Now, we have come to a place where we give the main result of this section.

Theorem 11. Assume that , , , , , and hold. In addition, assume thatfor all and .

Then, system (2) has, at least, one solution in .

Proof. In order to apply Corollary 1, we divided the proof into four steps. Step 1: in this step, we prove in , , and , respectively, that the operators , , and are well defined and weakly sequentially continuous.(a)Let , and let be a sequence in such that . We have Since and taking into account the assumptions , we obtain . Accordingly, . Now, we show that is weakly sequentially continuous. To see this, let be a sequence in such that . Hence, is bounded; then, by Lemma 3, we have , for all ; consequently, Using again Lemma 3, we get because is bounded by ; hence, is weakly sequentially continuous.(b)Let , and let be a sequence in such that . We have By using assumption with the dominated convergence theorem, we obtain Hence, This implies that the function is continuous. Now, let be a sequence in such that . For all , we have , where Because is a real bounded sequence, we deduce that there is a renamed subsequence such that . Since is bounded, we obtain ; hence, is weakly sequentially continuous.(c)Let , and let be a sequence in such that . We have where If we refer to assumption , we deduce is continuous. Now, let be a sequence in such that ; using assumption , we have Using the dominated convergence theorem, we obtain Since is bounded, then we can apply Lemma 3, and we get ; then, is weakly sequentially continuous. Step 2: in this step, in and , respectively, we prove that and are relatively weakly compact.(a)Let be any sequence in . By , we have For all , this proves that is a uniformly bounded sequence in . As a result, is sequentially relatively weakly compact. Now, we proceed to show that it is also weakly equicontinuous. If we take , , and (without loss of generality assume that ), then we have where Using assumption , we have Since and , as , we obtain Applying the Arzelà-Ascoli’s theorem [22], we get is sequentially relatively weakly compact in , and an application of Eberlein–Smulian’s theorem [23] shows that is relatively weakly compact.(b)We have ; this subset is nothing else than Hence, by hypothesis , we can see that is relatively weakly compact. Step 3: in this step in , we show that , and in , we prove that , for all .(a)Let , we defined the mapping by . Because is a contraction, we can see that is a contraction; then, an application of Banach’s fixed point theorem yields there is a unique point such that , and this implies that . Since , then there is such that This implies that Then, .(b)Let and such that or, equivalently, for all ,We haveThis implies thatThus, all the hypotheses of Corollary 1 are satisfied, and therefore, system (2) has, at least, one solution in .

5. Example

Consider the Banach space of all continuous real-valued functions on , with norm . In this case, , and is a reflexive Banach space. We consider the following coupled nonlinear integral equation in :

To show that (54) has a solution in , we will verify that all conditions of Theorem 11 are satisfied.

Here, for all and , we have

For each , the operator is continuous (then weakly sequentially continuous), and for each , the operator is continuous. Now, let and , and we havewhere the function is continuous with bound .

Next, we have is continuous, and for each and , the operator is continuous uniformly.

Moreover, for all and , we have

Thus, .

Next, we havewhere , for all .

Choose and , we obtain , , and . Then, the inequalities , , and are verified.

Now, let , we have . Suppose that there exists and such that . We have . And, which is a contraction. Hence, for all and , we have . Hence, all conditions of Theorem 11 are verified.

6. Conclusion

In recent years, some works were devoted to the investigation of fixed point theorems for operator matrices with entries acting on Banach spaces and Banach algebras. The aim of the present paper was to establish some new variants of Leray–Schauder-type fixed point theorems for a block operator matrix. The second aim of this study was to use our results to prove the existence of solutions for a coupled system of nonlinear equations. An example to illustrate our theory is included.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2021 Mohamed Amine Farid et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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