In this paper, we study some fixed point theorems of a 2 × 2 block operator matrix defined on nonempty bounded closed convex subsets of Banach algebras, where the entries are nonlinear operators. Furthermore, we apply the obtained results to a coupled system of nonlinear equations. Copyright © 2012 John Wiley & Sons, Ltd.

Abstract. This paper is devoted to the study of a coupled system of nonlinear functional integral equations in suitable Banach algebras. This system is reduced to a fixed point problem for a 2 × 2 block operator matrix with nonlinear inputs. Hence, certain assumptions on its entries are given under a weak topology setting. These assumptions involve in particular the De Blasi measure of weak noncompactness in order to ensure the existence of solutions. Key Words and Phrases: integral equation, Banach algebra, weakly sequentially continuous, measure of weak noncompactness, fixed point theory. 2010 Mathematics Subject Classification: 47H10, 45G15. References

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This paper presents new fixed point theorems for 2 × 2 block operator matrix with countably condensing or countably 𝓓-set-contraction multi-valued inputs. Our theory will then be used to establish some new existence theorems for coupled system of functional differential inclusions in general Banach spaces under weak topology. Our results generalize, improve and complement a number of earlier works.