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

Authors: Mohamed Amine Farid; El Miloudi Marhrani; Mohamed Aamri
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we establish some new variants of Leray–Schauder-type fixed point theorems for a2×2 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.


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## Body

## 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 2×2 block operator matrix:(1)ABCD.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 Lp×Lp for p∈1,∞, 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 L1 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. 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:(2)xt=kt,xt+qt+∫0σtgt,s,ysds⋅u,yt=ϕt,∫0ttt+sws,xsds⋅v+atyt,for all t∈J=0,1, where u,v∈X with u≠0 and v≠0; 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.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 C0,1,ℝ.Note that system (2) can be written as a fixed point problem:(3)xtyt=kt,⋅Gt,⋅φt,⋅atxtyt,where(4)Gt,yt=qt+∫0σtgt,s,ysds⋅u,φt,xt=ϕt,∫0ttt+sws,xsds⋅v.The present paper is built up as follows. In Section2, 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).


## 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,X denotes a Banach space endowed with the norm ⋅ and with the zero element θ, Br denotes the closed ball in X centered at θ with radius r>0, and PbdX denotes the collection of all nonempty bounded subsets of X. Moreover, we write xn⟶x and xn⇀x to denote, respectively, the strong convergence and the weak convergence of a sequence xnn to x. We say that a map T:X⟶X is weakly sequentially continuous if, for every sequence xnn⊂X with xn⇀x, we have Txn⇀Tx.The De Blasi measure of weak noncompactness [9] is the map ω:PbdX⟶0,∞ defined by(5)ωM=infr>0,there exists a weakly compact setKsuch thatM⊆K+Br.The De Blasi measure of weak noncompactness satisfies the following properties. For a proof, we refer the reader to [9, 10].Lemma 1.
LetM1 and M2 be in PbdX, and we have(i)
IfM1⊂M2, then ωM1≤ωM2(ii)
ωM1=0 if and only if M1 is relatively weakly compact(iii)
ωM1w¯=ωM1, where M1w¯ is the weak closure of M1(iv)
ωλM1=λωM1, for all λ∈ℝ(v)
ωco¯M1=ωM1, where co¯M1 is the closed convex hull of M1(vi)
ωM1+M2≤ωM1+ωM2(vii)
ωM1∪x=ωM1, for all x∈XIn [10], Appell and De Pascale proved that, in L1−spaces, the maps ω⋅ has the following form:(6)ωM=lim supε⟶0supψ∈M∫DψtXdt:measD≤ε.For all bounded subsetsM of L1Ω,X, where X is a finite dimensional Banach space and meas⋅ denotes the Lebesgue measure, we recall the following definitions.Definition 1.
LetΩ be a subset of a Banach space X, and k∈0,1. Let T:Ω⟶X be a mapping, we say that1.
T is k-contractive if ωTM≤kωM for any bounded set M⊂Ω(2)
T is condensing if ωTM<ωM for any bounded set M⊂Ω with ωM>0(3)
T is countably k-contractive if ωTM≤kωM for any countable bounded set M⊂Ω(4)
T is countably condensing if ωTM<ωM for any countable bounded set M⊂Ω with ωM>0
Clearly, everyk−contractive is countably k−contractive. Now, T is said to be k−Lipschitzian if Tx−Ty≤kx−y with k∈0,∞. If k∈0,1, T is called a contraction.Definition 2.
A mappingT:X⟶X is said to be weakly compact if TM is relatively weakly compact for every nonempty bounded subset M⊆X.
Following [11], we recall the next definition.Definition 3.
A mappingT:X⟶X is called to be a separate contraction if there exist two functions φ,ψ:0,∞⟶0,∞, satisfying(1)
ψ is strictly increasing and ψ0=0(2)
Tx−Ty≤φx−yfor allx,y∈X(3)
ψr+φr≤r for any r>0
Note that every contraction mapping is a separate contraction mapping.Definition 4 (see [12]).
A mappingT:X⟶X is a nonlinear contraction mapping if there exists a continuous nondecreasing function φ:0,∞⟶0,∞ satisfying(1)
φr<r for any r>0(2)
Tx−Ty≤φx−yfor allx,y∈XDefinition 5.
An operatorT:X⟶X is said to be ψ-expansive if there exists a function ψ:0,∞⟶0,∞ such that(1)
ψ0=0(2)
ψr>r, for any r>0(3)
ψ is either continuous or nondecreasing(4)
Tx−Ty≥ψx−y for all x,y∈X
In particular, forψt=ht with h>1, then T is an expansive mapping (see [13]). The following results are the nonlinear alternatives of Leray–Schauder-type states in [14].Theorem 1.
LetΩ 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 Uw¯ is a weakly compact subset of Ω and F:Uw¯⟶Ω is a weakly sequentially continuous mapping.
Then, either(A1)
F has a fixed point or(A2)
there is a pointx∈∂ΩU and λ∈0,1 with x=λFx, where ∂ΩU denotes the weak boundary of U in Ω.Remark 1.
In Theorem1, the condition “Uw¯ is a weakly compact” can be replaced by “FUw¯ is relatively weakly compact,” for the proof, see Remark 3.2 in [14].Theorem 2.
LetΩ 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:Uw¯⟶Ω is a weakly sequentially continuous and condensing map with FUw¯ bounded. Then, either(A1)
F has a fixed point or(A2)
there is a pointx∈∂ΩU and λ∈0,1 with x=λFx, where ∂ΩU denotes the weak boundary of U in Ω.Amar et al., in [15], showed that the condition “condensing” in Theorem 2 can be relaxed by the assumption “countably condensing.”Theorem 3.
LetΩ 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:Uw¯⟶Ω is a weakly sequentially continuous and countably condensing map with FUw¯ bounded. Then, either(A1)
F has a fixed point or(A2)
there is a pointx∈∂ΩU and λ∈0,1 with x=λFx, where ∂ΩU denotes the weak boundary of U in Ω.The following results are crucial for our purposes.Lemma 2 (see [16]).
LetΩ 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. Then, ωTM≤kωM for each bounded subset M of Ω; here, ω⋅ stands for the De Blasi measure of weak noncompactness.Lemma 3 (see [17]).
LetK be a Hausdorff compact space and X be a Banach space. A bounded sequence fnn⊂CK,X converges weakly to f∈CK,X if and only if, for every t∈K, the sequence fntn converges weakly (in X) to ft.


## 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.
LetΩ 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:Uw¯⟶X and B,D:X⟶X be four operators such that(i)
A is linear and bounded, and there is p∈ℕ∗ such that Ap is a separate contraction(ii)
B and C are weakly sequentially continuous and CUw¯ is relatively weakly compact(iii)
D is linear and bounded, and there is p∈ℕ∗ such that Dp is a separate contraction(iv)
I−A−1BI−D−1Cx∈Ω, for all x∈Uw¯
Then, either(A1)
the block operator matrix (1) has a fixed point or(A2)
there existsu,v∈∂ΩU×X and λ∈0,1 such that(7)AλBCDuv=uv,where ∂ΩU 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:Uw¯⟶Ω by(8)Fx=I−A−1BI−D−1Cx.
In view of Theorem1, it suffices to establish that F is weakly sequentially continuous and FU¯w is relatively weakly compact.
We haveI−A−1 and I−D−1, and B and C are weakly sequentially continuous; then, F is weakly sequentially continuous, and because CUw¯ is relatively weakly compact, we get FUw¯ is relatively weakly compact. Hence, either(1)
F has a fixed point or(2)
there is a pointu∈∂ΩU and λ∈0,1 with u=λFu.
In the first case, the vectory=I−D−1Cx solves the problem, whereas in the second case we use the vector v=I−D−1Cu to achieve the proof.Remark 2.
(1)
Theorem4 remains true if we suppose that Ap or Dp is a nonlinear contraction(2)
Theorem4 is a generalization of Theorem 4 and Theorem 3.2 in [19]
We also have the following result.Theorem 5.
LetΩ 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:Uw¯⟶X and B,D:X⟶X be four operators such that(i)
AUw¯ and CUw¯ are relatively weakly compact(ii)
A, B, and C are weakly sequentially continuous(iii)
D is linear and bounded, and there is p∈ℕ∗ such that Dp is a separate contraction(iv)
Ax+BI−D−1Cx∈Ω, for all x∈Uw¯
Then, either(A1)
the block operator matrix (1) has a fixed point or(A2)
there existsu,v∈∂ΩU×X and λ∈0,1 such that(9)λAλBCDuv=uv,where ∂ΩU denotes the weak boundary of U in Ω.Proof.
The reasoning in the proof of Theorem4 yields that I−D−1 exists and is weakly continuous. Hence, we can define the mapping F:Uw¯⟶Ω by(10)Fx=Ax+BI−D−1Cx.
We can see thatF is weakly sequentially continuous because A, B, C, and I−D−1 are weakly sequentially continuous. The assumption i proves that FUw¯ is relatively weakly compact. Hence, by Theorem 1, we have(1)
F has a fixed point or(2)
There is a pointu∈∂ΩU and λ∈0,1 with u=λFu
In the first case, the vectory=I−D−1x solves the problem, whereas in the second case we use the vector v=I−D−1Cu to achieve the proof.Remark 3.
(1)
In Theorem5, we can replace the assumption “Ax+BI−D−1Cx∈Ω, for all x∈Uw¯,” by “x=Ax+BI−D−1Cy∈Ω, for all y∈Uw¯ implies x∈Ω”(2)
Theorem5 is a generalization of Theorem 3.3 in [19]
Thus, we give the following resultTheorem 6.
LetΩ 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:Uw¯⟶X and B,D:X⟶X be four operators such that(i)
AUw¯ and CUw¯ are relatively weakly compact(ii)
A, B, and C are weakly sequentially continuous(iii)
D is linear and bounded, and there is p∈ℕ∗ such that Dp is a separate contraction(iv)
Ax+BI−D−1Cx∈Ω, for all x∈Uw¯
Then, either(A1)
the block operator matrix (1) has a fixed point or(A2)
there existsu,v∈∂ΩU×X and λ∈0,1 such that(11)λAλBλCλDuvλ=uv,where ∂ΩU denotes the weak boundary of U in Ω.Proof.
Reasoning as in the proof of Theorem5, we obtain(1)
F has a fixed point or(2)
There is a pointu∈∂ΩU and λ∈0,1 with u=λFu
In the first case, the vectory=I−D−1x solves the problem, whereas in the second case, we use the vector v=λI−D−1Cu to achieve the proof.Remark 4.
Theorem6 is a generalization of Theorem 3.4 in [19].Theorem 7.
LetΩ be a nonempty closed and convex subset of Banach space X and U⊂Ω be a weakly open subset of Ω with θ∈U. Let A,C:Uw¯⟶X and B,D:X⟶X be four weakly sequentially continuous operators such that(i)
AUw¯ and CUw¯ are relatively weakly compact(ii)
D is a contraction with CUw¯⊂I−DΩ(iii)
Ax+BI−D−1Cx∈Ω, for all x∈Uw¯
Then, either(A1)
the block operator matrix (1) has a fixed point or(A2)
there existsu,v∈∂ΩU×X and λ∈0,1 such that(12)λAλBCDuv=uv,where ∂ΩU denotes the weak boundary of U in Ω.Proof.
By assumptionii, we can see that I−D−1 exists and continuous; hence, from Theorem 5, we deduce the desired result.
Inspired by [20], we deduce the following result.Corollary 1.
LetΩ 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:Uw¯⟶X and B,D:X⟶X be four weakly sequentially continuous operators such that(i)
AUw¯ and CUw¯ are relatively weakly compact(ii)
D is a contraction with CUw¯⊂I−DΩ(iii)
Ax+BI−D−1Cx∈Ω for all x∈Uw¯
In addition, assume that(13)λAu+λBv≠u,for all u,v∈∂ΩU×X and λ∈0,1. Then, the set of fixed points of the block operator matrix (1) in Uw¯×X is nonempty.In the next results, we explore the case whereA and D are nonlinear.Theorem 8.
LetΩ 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:Uw¯⟶X and B,D:X⟶X be four weakly sequentially continuous operators such that(i)
I−D−1 exists on CΩ(ii)
AUw¯ is relatively weakly compact and B is countably β−contractive(iii)
C is countably γ−contractive and D is countably δ−contractive with γ+δ<1(iv)
A+BI−D−1CUw¯ is a bounded subset of Ω
Then, either(A1)
the block operator matrix (1) has a fixed point or(A2)
there existsu,v∈∂ΩU×X and λ∈0,1 such that(14)λAλBCDuv=uv,where ∂ΩU denotes the weak boundary of U in Ω.Proof.
We defend the operatorF:Uw¯⟶Ω by(15)Fx=Ax+BI−D−1Cx.
We show thatF is weakly sequentially continuous and countably condensing.
Letxnn be a sequence in Ω such that xn⇀x; since C is weakly sequentially continuous, then the set Cxn:n∈ℕ is relatively weakly compact. Note that(16)I−D−1C=C+DI−D−1C.
We haveD is countably δ−contractive; then,(17)ωI−D−1Cxn:n∈ℕ≤ωCxn:n∈ℕ+ωDI−D−1Cxn:n∈ℕ≤ωDI−D−1Cxn:n∈ℕ≤δωI−D−1Cxn:n∈ℕ<ωI−D−1Cxn:n∈ℕ,which implies that ωI−D−1Cxn:n∈ℕ=0; then, I−D−1Cxn:n∈ℕ is relatively weakly compact. Consequently, there exists a subsequence xσnn∈ℕ of xnn∈ℕ such that I−D−1Cxσn⇀y. From (16) and because C and D are weakly sequentially continuous, we get Cx+Dy=y; then, I−D−1Cx=y; consequently, I−D−1Cxσn⇀I−D−1Cx. Now, we show that I−D−1Cxn⇀I−D−1Cx. Suppose the contrary; then, there exists a subsequence xφnn∈ℕ of xnn∈ℕ and a weak neighborhood V of I−D−1Cx such that I−D−1Cxφn∉V for all n∈ℕ. Since xφn⇀x, then arguing as before, we may extract a subsequence xϕnn∈ℕ of xφnn∈ℕ such that I−D−1Cxϕn⇀I−D−1Cx, which is a contraction. As a result, I−D−1C is weakly sequentially continuous. Because A and B are weakly sequentially continuous, we get A+BI−D−1C is weakly sequentially continuous.
Now, we show thatA+BI−D−1C is countably condensing. Let M be a countably subset of Uw¯ such that ωM>0; by (16), we have(18)ωI−D−1CM≤ωCM+ωDI−D−1CM≤γωM+δωI−D−1CM.
Then,(19)ωI−D−1CM≤γ1−δωM≤ωM.
Using the subadditivity of the De Blasi measure of weak noncompactness, we obtain(20)ωFM≤ωAM+ωBI−D−1CM≤βωM<ωM.
Hence,F is countably condensing.
Then, by Theorem3, we obtain(1)
F has a fixed point or(2)
There is a pointu∈∂ΩU and λ∈0,1 with u=λFu
In the first case, the vectory=I−D−1Cx solves the problem, whereas in the second case, we use the vector v=I−D−1Cu to achieve the proof.Remark 5.
In Theorem8, we can replace the De Blasi measure of weak noncompactness ω by any subadditive measures of weak noncompactness on X.
In Theorem8, condition i is difficult to verify; in the following, we will change it under weaker conditions.Theorem 9.
LetΩ 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:Uw¯⟶X and B,D:X⟶X be four weakly sequentially continuous operators such that(i)
I−D is ψ−expansive and CΩ⊂I−DX(ii)
AUw¯ is relatively weakly compact and B is countably β−contractive(iii)
C is countably γ−contractive and D is countably δ−contractive with γ+δ<1(iv)
A+BI−D−1CUw¯ is a bounded subset of Ω
Then, either(A1)
the block operator matrix (1) has a fixed point or(A2)
there existsu,v∈∂ΩU×X and λ∈0,1 such that(21)λAλBCDuv=uv,where ∂ΩU denotes the weak boundary of U in Ω.Proof.
Letx,y∈X such that x≠y; since I−D is ψ−expansive, we obtain(22)I−Dx−I−Dy≥ψx−y>0,
Thus,I−D:X⟶I−DX is bijective, and because CΩ⊂I−DX, we get I−D−1 is well defined on CΩ. Now, reasoning as in the proof of Theorem 8, we get the desired result.Theorem 10.
LetΩ 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:Uw¯⟶X and B,D:X⟶X be four weakly sequentially continuous operators such that(i)
B is countably β−contractive and AUw¯ is relatively weakly compact(ii)
Typ is expansive for some p∈ℕ and each y∈X, where Tyx=Dx+y for x∈X(iii)
D is a contraction with constant 0<δ<1 and C is countably γ−contractive with γ+δ<1(iv)
A+BI−D−1CUw¯ is a bounded subset of Ω
Then, either(A1)
the block operator matrix (1) has a fixed point, or(A2)
there existsu,v∈∂ΩU×X and λ∈0,1 such that(23)λAλBCDuv=uv,where ∂ΩU denotes the weak boundary of U in Ω.Proof.
By assumptionsi and ii with Lemma 2.3 in [21], we get I−D−1C is well defend. And, by assumption i and Lemma 2, we have that D is countably δ−contractive. Now, reasoning as in the proof of Theorem 8, we get the desired result.Remark 6.
Theorem10 remains true if we suppose that Typ is contractive.


## 4. Application

LetX be a Banach space. As usual, we will denote by E=CJ,X the Banach space of all X−valued continuous functions defined on J=0,1. We equip the space E with its standard norm:(24)x∞=supt∈Jxt.The goal of this section is to apply Corollary1 to study the existence of continuous solutions to the nonlinear functional integral equations (2).Let us now introduce the following assumptions.(H0)
(i)
The functionσ:J⟶J is continuous and nondecreasing(ii)
The functionq:J⟶ℝ is continuous(iii)
The functiona:J⟶X is continuous, and a∞<1(H1)
The operatork:J×X⟶X is such that(i)
For allt∈J, the operator kt,⋅:X⟶X is weakly sequentially continuous and weakly compact.(ii)
For eachx∈X, the operator k⋅,x:J⟶X is continuous.(iii)
There exists a continuous functionδ:J⟶0,+∞ with bound Δ=δ∞ such that(25)kt,x−kt,y≤δtx−y,for allx,y∈Xandt∈J.(H2)
The operatorg: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)
The operatorϕ:J×X⟶ℝ is such that(i)
For eacht∈J, the operator ϕt,⋅:X⟶ℝ is weakly sequentially continuous.(ii)
For eachx∈X, the operator ϕ⋅,x:J⟶ℝ is continuous.(iii)
There exists a constantL such that(26)ϕt,x−ϕt,y≤Lx−y,for allx,y∈Xandt∈J.(H4)
The operatorsw:J×X⟶ℝ is such that(i)
For eachx∈X, the operator w⋅,x:J⟶ℝ is measurable.(ii)
For eacht∈J, the operator wt,⋅:X⟶ℝ is weakly sequentially continuous.(iii)
There exists a constantb and a function m∈L10,1 such that(27)wt,x≤mtand∫011t+smsds≤b.(H5)
Assume that there existsr0>1 such that g⋅,⋅,⋅≤r0−q∞, supt∈Jwt,0≤1−a∞r0−LmL1v∞/v∞, and supt∈Jkt,0≤r01−Δ−u∞.Let us define the subsetΩ of CJ,X by(28)Ω=x∈CJ,X:x≤r0.We can see thatΩ 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:(29)xt=Axt+Byt,yt=Cxt+Dyt,where the operators A, B, C, and D defined by(30)Axt=kt,xt,for allt∈J,Bxt=qt+∫0σtgt,s,xsds⋅u,for allt∈J,Cxt=ϕt,∫0ttt+sws,xsds⋅v,for allt∈J,Dyt=atyt,for allt∈J.Now, we have come to a place where we give the main result of this section.Theorem 11.
Assume thatH0, H1, H2, H3, H4, and H5 hold. In addition, assume that(31)λAu1+λBu2≠u1,for all u1,u2∈∂ΩU×X and λ∈0,1.Then, system (2) has, at least, one solution in CJ,X×CJ,X.Proof.
In order to apply Corollary1, we divided the proof into four steps.Step 1: in this step, we prove ina, b, and c, respectively, that the operators A, B, and C are well defined and weakly sequentially continuous.(a)
Letx∈CJ,X, and let tnn be a sequence in J such that tn⟶t∈J. We have(32)Axtntn−Axt=ktn,xtn−kt,xt≤ktn,xtn−ktn,xt+ktn,xt−kt,xt≤δtnxtn−xt+ktn,xt−kt,xt≤Δxtn−xt+ktn,xt−kt,xt.Sincetn⟶t and taking into account the assumptions H1ii, we obtain Axtn−Axt⟶0. Accordingly, Ax∈E. Now, we show that A is weakly sequentially continuous. To see this, let xnn be a sequence in Ω such that xn⇀x∈Ω. Hence, xnn is bounded; then, by Lemma 3, we have xnt⇀xt, for all t∈J; consequently,(33)Axnt=kt,xnt⇀kt,xt=Axt.Using again Lemma3, we get Axn⇀Ax because Axnn is bounded by Δr0+supt∈Jkt,0; hence, A is weakly sequentially continuous.(b)
Letx∈CJ,X, and let tnn be a sequence in J such that tn⟶t∈J. We have(34)Bxtn−Bxt≤∫0σtngtn,s,xsds−∫0σtgt,s,xsdsu∞+qtn−qtu∞≤∫0σtngtn,s,xs−gt,s,xsdsu∞+∫σtnσtgt,s,xsds+qtn−qtu∞≤∫01gtn,s,xs−gt,s,xsdsu∞+r0σtn−σt+qtn−qtu∞.By using assumptionH2 with the dominated convergence theorem, we obtain(35)∫01gtn,s,xs−gt,s,xsds⟶0.Hence,(36)Bxtn−Bxt⟶0,This implies that the functionBx is continuous. Now, let xnn be a sequence in E such that xn⇀x∈E. For all t∈J, we have Bxnt=rnt⋅u, where(37)rnt=qt+∫0σtgt,s,xnsds.Becausernt,n∈ℕ is a real bounded sequence, we deduce that there is a renamed subsequence such that rnt⟶rt. Since Bxn,n∈ℕ is bounded, we obtain Bxn⇀Bx; hence, B is weakly sequentially continuous.(c)
Letx∈CJ,X, and let tnn be a sequence in J such that tn⟶t∈J. We have(38)Cxtn−Cxt≤ϕtn,∫0tntntn+sws,xsds−ϕt,∫0ttt+sws,xsdsv∞≤ϕtn,∫0tntntn+sws,xsds−ϕtn,∫0ttt+sws,xsdsv∞+ϕtn,∫0ttt+sws,xsds−ϕt,∫0ttt+sws,xsdsv∞≤L∫0tntntn+sws,xsds−∫0ttt+sws,xsdsv∞+ϕtn,∫0ttt+sws,xsds−ϕt,∫0ttt+sws,xsdsv∞≤L∫0tnstn−ttn+st+sws,xsds−∫tnttt+sws,xsdsv∞+ϕtn,∫0ttt+sws,xsds−ϕt,∫0ttt+sws,xsdsv∞≤2Lbtn−tv∞+ϕtn,z−ϕt,zv∞,where(39)zt=∫0ttt+sws,xsds,for allt∈J.If we refer to assumptionH3ii, we deduce Cx is continuous.Now, letxnn be a sequence in E such that xn⇀x∈E; using assumption H4ii, we have(40)tt+sws,xns⇀tt+sws,xs,inℝ.Using the dominated convergence theorem, we obtain(41)Cxnt⇀Cxt,inX.SinceCxn,n∈ℕ is bounded, then we can apply Lemma 3, and we get Cxn⇀Cx; then, C is weakly sequentially continuous.Step 2: in this step, ina and b, respectively, we prove that CUw¯ and AUw¯ are relatively weakly compact.(a)
Letxnn be any sequence in Ω. By H3iii, we have(42)Cxnt=ϕt,∫0ttt+sws,xnsds−ϕt,0+ϕt,0v∞≤L∫0ttt+sws,xnsds+supt∈Jwt,0v∞≤LmL1+supt∈Jwt,0v∞.For allt∈0,1, this proves that Cxn,n∈ℕ is a uniformly bounded sequence in CΩ. As a result, CΩt is sequentially relatively weakly compact. Now, we proceed to show that it is also weakly equicontinuous. If we take ε>0, x∈Ω, and t,t′∈0,1 (without loss of generality assume that t<t′), then we have(43)x∗Cxnt−Cxnt′≤LMε+Φεx∗v,where(44)Mε=∫0ttt+sws,xnsds−∫0t′t′t′+sws,xnsds,t−t′<ε,and,Φε=ϕt,∫0t′t′t′+sws,xsds−ϕt′,∫0t′t′t′+sws,xsds,t−t′<ε.Using assumptionH4, we have(45)Mε≤t−t′∫011t+sws,xnsds+∫t′tt′t′+sws,xnsds,≤at−t′+∫t′tmsds.SinceMε⟶0 and Φε⟶0, as ε⟶0, we obtain(46)x∗Cxnt−Cxnt′⟶0,ast⟶t′.Applying the Arzelà-Ascoli’s theorem [22], we get CΩ is sequentially relatively weakly compact in X, and an application of Eberlein–Smulian’s theorem [23] shows that CΩ is relatively weakly compact.(b)
We haveAΩ=Ax:x≤r0; this subset is nothing else than(47)k⋅,Ω=k⋅,x:x≤r0,Hence, by hypothesisH1i, we can see that AΩ is relatively weakly compact.Step 3: in this step ina, we show that CUw¯⊂I−DΩ, and in b, we prove that Ax+BI−D−1Cx∈Ω, for all x∈Uw¯.(a)
Letx∈Uw¯, we defined the mapping ϕx:CJ,X⟶CJ,X by ϕxy=Cx+Dy. Because D is a contraction, we can see that ϕx is a contraction; then, an application of Banach’s fixed point theorem yields there is a unique point y∈CJ,X such that y=Cx+Dy, and this implies that CUw¯⊂I−DCJ,X. Since y∈CJ,X, then there is t∗∈J such that(48)y∞=yt∗=Cxt∗+Dyt∗≤ϕt∗,∫0t∗t∗t∗+sws,xnsdsv∞+a∞yt∗.This implies that(49)y∞≤LmL1+supt∈Jwt,0v∞1−a∞≤r0.Then,CUw¯⊂I−DΩ.(b)
Lety∈CJ,X and x∈Uw¯ such that(50)yt=Axt+BI−D−1Cxt.or, equivalently, for allt∈J,(51)yt=Axt+BI−D−1Cxt.
We have(52)yt=Axt+BI−D−1Cxt≤Axt+BI−D−1Cxt≤Δxt+supt∈Jkt,0+r0u∞≤Δr0+supt∈Jkt,0+r0u∞≤r0.
This implies that(53)Ax+BI−D−1Cx∈Ω,for allx∈Uw¯.
Thus, all the hypotheses of Corollary1 are satisfied, and therefore, system (2) has, at least, one solution in CJ,X×CJ,X.


## 5. Example

Consider the Banach spaceE=C0,1,ℝ of all continuous real-valued functions on J=0,1, with norm x∞=supt∈0,1xt. In this case, X=ℝ, and E is a reflexive Banach space. We consider the following coupled nonlinear integral equation in E:(54)xt=14t3cosxt2+1814+∫0tys1+ysds,yt=t∫0ttsxst+sexsds+13sintyt.To show that (54) has a solution in E, we will verify that all conditions of Theorem 11 are satisfied.Here, for allt,s∈J and x,y∈E, we have(55)σt=t,qt=14,at=13sint,u=18,v=1,ϕt,xt=txt,kt,xt=14t3cosxt2,gt,s,ys=ys1+ys,ws,xs=sxs+12exs+1.For eacht∈J, the operator kt,⋅:X⟶X is continuous (then weakly sequentially continuous), and for each x∈X, the operator k⋅,x:J⟶X is continuous. Now, let x,y∈E and t∈J, and we have(56)kt,xt−kt,yt≤18t3xt−yt,where the function δ:t⟶1/8t3 is continuous with bound Δ=1/8.Next, we haveg is continuous, and for each s∈J and x∈ℝ, the operator g⋅,s,t:J⟶ℝ is continuous uniformly.Moreover, for allx,y∈ℝ and t∈J, we have(57)ϕt,xt−ϕt,yt≤xt−yt.Thus,L=1.Next, we have(58)wt,xt≤t2,and∫011t+smsds≤1.where mt=t, for all t∈J.Chooseb=1 and r0=2, we obtain r0−q∞=7/4, 1−a∞r0−LmL1v∞/v∞=5/6, and r01−Δ−u∞=3/2. Then, the inequalities g⋅,⋅,⋅≤7/4, supt∈Jwt,0≤5/6, and supt∈Jkt,0≤3/2 are verified.Now, letU=x∈C0,1,ℝ:x<2, we have ∂U=x∈C0,1,ℝ:x=2. Suppose that there exists u1×u2∈∂U×X and λ∈0,1 such that λAu1+λBu2=u1. We have λAu1+λBu2<13/32. And, u1=2 which is a contraction. Hence, for all u1×u2∈∂U×X and λ∈0,1, we have λAu1+λBu2≠u1. 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 a2×2 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.



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*Source: 9985817-2021-07-27.xml*