Abstract
The purpose of this paper is to present a new method for the research of best proximity point theorems of nonlinear mappings in metric spaces. In this paper, the P-operator technique, which changes non-self-mapping to self-mapping, provides a new and simple method of proof. Best proximity point theorems for weakly contractive and weakly Kannan mappings, generalized best proximity point theorems for generalized contractions, and best proximity points for proximal cyclic contraction mappings have been proved by using this new method. Meanwhile, many recent results in this area have been improved.
Highlights
1 Introduction and preliminaries Several problems can be changed to equations of the form Tx = x, where T is a given self-mapping defined on a subset of a metric space, a normed linear space, a topological vector space or some suitable space
Best proximity point theorems serve as a natural generalization of fixed point theorems, for a best proximity point becomes a fixed point if the mapping under consideration is a self-mapping
Research on the best proximity point is an important topic in the nonlinear functional analysis and applications
Summary
If f : X → X is a weakly contractive mapping, f has a unique fixed point x∗ and the Picard sequence of iterates {f n(x)}n∈N converges, for every x ∈ X, to x∗. [ ] Let (X, d) be a complete metric space and let f : X → X be a mapping such that d f (x), f (y) ≤ α d x, f (x) + d y, f (y) for all x, y ∈ X and some α ∈ [ , ), f has a unique fixed point x∗ ∈ X.
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