Abstract
We discuss some extensions of Caristi’s fixed point theorem for mappings defined on a metric space endowed with a graph. This work should be seen as a generalization of the classical Caristi’s fixed point theorem where the assumptions in Caristi’s theorem can, a priori, be weakened. It extends some recent works on Caristi’s fixed point theorem for mappings defined on metric spaces with a graph. MSC:47H09, 46B20, 47H10, 47E10.
Highlights
1 Introduction This work was motivated by some recent work on Caristi’s fixed point theorem for mappings defined on metric spaces with a graph [ ]. It seems that the terminology of graph theory instead of partial ordering gives clearer pictures and yields generalized fixed point theorems
Caristi’s fixed point theorem may be one of the most beautiful extensions of the Banach contraction principle [, ]. Recall that this theorem states that any map T : M → M has a fixed point provided that M is a complete metric space, and there exists a lower semicontinuous map φ mapping M into the nonnegative numbers such that d(x, Tx) ≤ φ(x) – φ(Tx) for every x ∈ M
In this work we present a characterization to the existence of minimal elements in partially ordered sets in terms of fixed point of multivalued maps, see [ ]
Summary
Introduction This work was motivated by some recent work on Caristi’s fixed point theorem for mappings defined on metric spaces with a graph [ ]. It seems that the terminology of graph theory instead of partial ordering gives clearer pictures and yields generalized fixed point theorems. Recall that this theorem states that any map T : M → M has a fixed point provided that M is a complete metric space, and there exists a lower semicontinuous map φ mapping M into the nonnegative numbers such that d(x, Tx) ≤ φ(x) – φ(Tx) for every x ∈ M.
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