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

In this paper, we establish some new variants of Leray–Schauder-type ﬁxed point theorems for a 2 × 2 block operator matrix deﬁned on nonempty, closed, and convex subsets Ω of Banach spaces. Note here that Ω 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.


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][2][3][4][5]. ese operators are defined by a 2 × 2 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 L p × L p for p ∈ (1, ∞), under abstract boundary conditions of Rotenberg's model type. ese last equations were proposed by M. Rotenberg and model the evolution of a cell population [7]. Due to the lack of compactness on L 1 spaces, the study in [6] did not cover the case p � 1. Later, Jeribi et al. [1] proposed to extend the results of Amar et al. [6] to the case p � 1 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 I − A or I − D are invertible play a fundamental role in the arguments. Jeribi et al., in [8], were interested in studying the case when I − A is not injective and established some fixed point theorems for operator (1), involving multivalued maps acting on Banach spaces. is way, their results were formulated in terms of weak sequential continuity and the technique of De Blasi measure of weak noncompactness. e results obtained are then applied to the two-dimensional nonlinear functional integral equation: here, X is a Banach space. And, σ: J ⟶ J, g: J × J × X ⟶ X, k, w: J × J ⟶ R, and b: J ⟶ X are suitably defined functions. e 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 C([0, 1], R).
Note that system (2) can be written as a fixed point problem: where e 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 2 × 2 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).

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. roughout this paper, unless otherwise mentioned, X denotes a Banach space endowed with the norm ‖ · ‖ and with the zero element θ, B r denotes the closed ball in X centered at θ with radius r > 0, and P bd (X) denotes the collection of all nonempty bounded subsets of X. Moreover, we write x n ⟶ x and x n ⇀x to denote, respectively, the strong convergence and the weak convergence of a sequence x n n to x. We say that a map T: X ⟶ X is weakly sequentially continuous if, for every sequence x n n ⊂ X with x n ⇀x, we have Tx n ⇀Tx.
e De Blasi measure of weak noncompactness [9] is the map ω: e De Blasi measure of weak noncompactness satisfies the following properties. For a proof, we refer the reader to [9,10].

Lemma 1.
Let M 1 and M 2 be in P bd (X), and we have In [10], Appell and De Pascale proved that, in L 1 − spaces, the maps ω(·) has the following form: For all bounded subsets M of L 1 (Ω, X), where X is a finite dimensional Banach space and meas(·) denotes the Lebesgue measure, we recall the following definitions.
Definition 2. A mapping T: X ⟶ X is said to be weakly compact if T(M) is relatively weakly compact for every nonempty bounded subset M⊆X.

Journal of Mathematics
Note that every contraction mapping is a separate contraction mapping.
Definition 4 (see [12]). A mapping T: X ⟶ X is a nonlinear contraction mapping if there exists a continuous nondecreasing function φ: In particular, for ψ(t) � ht with h > 1, then T is an expansive mapping (see [13]). e following results are the nonlinear alternatives of Leray-Schauder-type states in [14].

Theorem 1.
Let Ω be a nonempty closed and convex subset of a Banach space X and U ⊂ Ω be a weakly open subset of Ω with θ ∈ U such that U w is a weakly compact subset of Ω and F: U w ⟶ Ω is a weakly sequentially continuous mapping. en, either

Remark 1.
In eorem 1, the condition "U w is a weakly compact" can be replaced by "F(U w ) is relatively weakly compact," for the proof, see Remark 3.2 in [14].

Theorem 2.
Let Ω be a nonempty closed and convex subset of a Banach space X and U ⊂ Ω be a weakly open subset of Ω with θ ∈ U. Assume that F: U w ⟶ Ω is a weakly sequentially continuous and condensing map with F(U w ) bounded. en, either Amar et al., in [15], showed that the condition "condensing" in eorem 2 can be relaxed by the assumption "countably condensing." Theorem 3. Let Ω be a nonempty closed and convex subset of a Banach space X and U ⊂ Ω be a weakly open subset of Ω with θ ∈ U. Assume that F: U w ⟶ Ω is a weakly sequentially continuous and countably condensing map with F(U w ) bounded. en, either e following results are crucial for our purposes.
Lemma 2 (see [16]). Let Ω be a subset of a Banach space X and let T: Ω ⟶ X be a k− Lipschitzian map. Assume that T is a sequentially weakly continuous map. en, ω(T(M)) ≤ kω(M) for each bounded subset M of Ω; here, ω(·) stands for the De Blasi measure of weak noncompactness.
Lemma 3 (see [17]). Let K be a Hausdorff compact space and X be a Banach space. A bounded sequence f n n ⊂ C(K, X) converges weakly to f ∈ C(K, X) if and only if, for every t ∈ K, the sequence f n (t) n converges weakly (in X) to f(t).

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

Theorem 4.
Let Ω be a nonempty closed and convex subset of a Banach space X and U ⊂ Ω be a weakly open subset of Ω with θ ∈ U. Let A, C: U w ⟶ X and B, D: X ⟶ X be four operators such that (i) A is linear and bounded, and there is p ∈ N * such that A p is a separate contraction (ii) B and C are weakly sequentially continuous and C(U w ) is relatively weakly compact (iii) D is linear and bounded, and there is p ∈ N * such that D p is a separate contraction where z Ω U denotes the weak boundary of U in Ω.
Proof. If we refer to Lemma 1.2 in [11] and to page 39 in [18], we can prove that (I − A) − 1 and (I − D) − 1 exist and are weakly continuous. Hence, we can define the mapping F: U w ⟶ Ω by In view of eorem 1, it suffices to establish that F is weakly sequentially continuous and F(U w ) is relatively weakly compact. We have (I − A) − 1 and (I − D) − 1 , and B and C are weakly sequentially continuous; then, F is weakly Journal of Mathematics sequentially continuous, and because C(U w ) is relatively weakly compact, we get F(U w ) is relatively weakly compact. Hence, either (1) F has a fixed point or (2) there is a point u ∈ z Ω U and λ ∈ (0, 1) with u � λFu.
In the first case, the vector y � (I − D) − 1 Cx solves the problem, whereas in the second case we use the vector v � (I − D) − 1 Cu to achieve the proof.

Remark 2
(1) eorem 4 remains true if we suppose that A p or D p is a nonlinear contraction (2) eorem 4 is a generalization of eorem 4 and eorem 3.2 in [19] We also have the following result.
where z Ω U denotes the weak boundary of U in Ω.
Proof. e reasoning in the proof of eorem 4 yields that (I − D) − 1 exists and is weakly continuous. Hence, we can define the mapping F: U w ⟶ Ω by We can see that F is weakly sequentially continuous because A, B, C, and (I − D) − 1 are weakly sequentially continuous. e assumption (i) proves that F(U w ) is relatively weakly compact. Hence, by eorem 1, we have (1) F has a fixed point or (2) ere is a point u ∈ z Ω U and λ ∈ (0, 1) with u � λFu In the first case, the vector y � (I − D) − 1 x solves the problem, whereas in the second case we use the vector v � (I − D) − 1 Cu to achieve the proof.
Proof. Reasoning as in the proof of eorem 5, we obtain (1) F has a fixed point or (2) ere is a point u ∈ z Ω U and λ ∈ (0, 1) with u � λFu In the first case, the vector y � (I − D) − 1 x solves the problem, whereas in the second case, we use the vector v � λ(I − D) − 1 Cu to achieve the proof.
where z Ω U denotes the weak boundary of U in Ω.
Proof. By assumption (ii), we can see that (I − D) − 1 exists and continuous; hence, from eorem 5, we deduce the desired result.
Inspired by [20], we deduce the following result.
In addition, assume that λAu + λBv ≠ u, (13) for all (u, v) ∈ z Ω U × X and λ ∈ (0, 1). en, the set of fixed points of the block operator matrix (1) in U w × X is nonempty.
In the next results, we explore the case where A and D are nonlinear.

exists on C(Ω) (ii) A(U w ) is relatively weakly compact and B is countably β− contractive (iii) C is countably c− contractive and D is countably
where z Ω U denotes the weak boundary of U in Ω.
Proof. We defend the operator F: U w ⟶ Ω by We show that F is weakly sequentially continuous and countably condensing. Let x n n be a sequence in Ω such that x n ⇀x; since C is weakly sequentially continuous, then the set Cx n : n ∈ N is relatively weakly compact. Note that We have D is countably δ− contractive; then, Using the subadditivity of the De Blasi measure of weak noncompactness, we obtain

(20)
Hence, F is countably condensing. en, by eorem 3, we obtain (1) F has a fixed point or (2) ere is a point u ∈ z Ω U and λ ∈ (0, 1) with u � λFu In the first case, the vector y � (I − D) − 1 Cx solves the problem, whereas in the second case, we use the vector v � (I − D) − 1 Cu to achieve the proof. □ Remark 5. In eorem 8, we can replace the De Blasi measure of weak noncompactness ω by any subadditive measures of weak noncompactness on X.
In eorem 8, condition (i) is difficult to verify; in the following, we will change it under weaker conditions.
where z Ω U denotes the weak boundary of U in Ω.
Proof. Let x, y ∈ X such that x ≠ y; since (I − D) is ψ− expansive, we obtain

i) B is countably β− contractive and A(U w ) is relatively
weakly compact (ii) T p y is expansive for some p ∈ N and each y ∈ X, where T y x � Dx + y for x ∈ X (iii) D is a contraction with constant 0 < δ < 1 and C is countably where z Ω U denotes the weak boundary of U in Ω.

Application
Let X be a Banach space. As usual, we will denote by E � C(J, X) the Banach space of all X− valued continuous functions defined on J � [0, 1]. We equip the space E with its standard norm: e 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) e function σ: J ⟶ J is continuous and nondecreasing (ii) e function q: J ⟶ R is continuous (iii) e function a: J ⟶ X is continuous, and ‖a‖ ∞ < 1 (H1) e operator k: J × X ⟶ X is such that for all x, y ∈ X and t ∈ J.
(25) (H2) e operator g: J × J × X ⟶ R is continuous such that, for each s ∈ J and x ∈ X, and the operator g(·, s, t): J ⟶ R is continuous uniformly. (H3) e operator ϕ: J × X ⟶ R is such that (i) For each t ∈ J, the operator ϕ(t, ·): X ⟶ R is weakly sequentially continuous. (ii) For each x ∈ X, the operator ϕ(·, x): J ⟶ R is continuous. (iii) ere exists a constant L such that for all x, y ∈ X and t ∈ J.
(26) (H4) e operatorsw: J × X ⟶ R is such that (i) For each x ∈ X, the operator w(·, x): J ⟶ R is measurable. (H5) Assume that there exists r 0 > 1 such that Let us define the subset Ω of C(J, X) by We can see that Ω is a nonempty closed and convex subset of E. Let U be a weakly open subset of Ω such that 0 ∈ U. Notice that (2) is equivalent to the system: where the operators A, B, C, and D defined by s, x(s))ds · u, for all t ∈ J, t 0 t t + s w(s, x(s))ds · v, for all t ∈ J, Dy(t) � a(t)y(t), for all t ∈ J.
(30) Now, we have come to a place where we give the main result of this section.
Proof. In order to apply Corollary 1, we divided the proof into four steps.
Step 1: in this step, we prove in (a), (b), and (c), respectively, that the operators A, B, and C are well defined and weakly sequentially continuous.

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. e aim of the present paper was to establish some new variants of Leray-Schauder-type fixed point theorems for a 2 × 2 block operator matrix. e 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
e authors declare that they have no conflicts of interest.