Abstract
AbstractIn this work, we discuss the existence of fixed points of monotone nonexpansive mappings defined on partially ordered Banach spaces. This work is a continuity of the previous works of Ran and Reurings, Nieto et al., and Jachimsky done for contraction mappings. As an application, we discuss the existence of solutions to an integral equations.
Highlights
1 Introduction Banach’s contraction principle [ ] is remarkable in its simplicity, yet it is perhaps the most widely applied fixed point theorem in all of analysis. This is because the contractive condition on the mapping is simple and easy to test, because it requires only a complete metric space for its setting, and because it finds almost canonical applications in the theory of differential and integral equations
A version of this theorem has been given in partially ordered metric spaces [, ] and in metric spaces with a graph [ ]
We discuss the case of nonexpansive mappings defined in partially ordered Banach spaces
Summary
Banach’s contraction principle [ ] is remarkable in its simplicity, yet it is perhaps the most widely applied fixed point theorem in all of analysis. We discuss the case of nonexpansive mappings defined in partially ordered Banach spaces. 2 Monotone nonexpansive mappings Let (X, · ) be a Banach vector space. Assume that we have a partial order defined on X such that order intervals are convex and τ -closed, where τ is a Hausdorff topology on X.
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