Abstract
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.
Highlights
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
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: AB
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 applied to the two-dimensional nonlinear functional integral equation:
Summary
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. 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]. 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.
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