Abstract
In this paper, we give new conditions for existence and uniqueness of a best proximity point for Geraghty- and Caristi-type mappings. The presented results are most valuable generalizations of the Geraghty and Caristi fixed point theorems.
Highlights
Introduction and PreliminariesThe Banach contraction principle (BCP) in metric spaces has been generalized and extended in various ways
The following interesting theorem for a cyclic map was given in [3]
If either A or B is boundedly compact, there exists x ∈ A ∪ B such that dðx, TxÞ = distðA, BÞ: A convenience attention has been recently devoted to the research on existence and uniqueness of best proximity points of self-mappings, as well as, to the investigation of associated relevant properties, for instance, stability of the iterations
Summary
The Banach contraction principle (BCP) in metric spaces has been generalized and extended in various ways. The following interesting theorem for a cyclic map was given in [3]. Eldred and Veeramani [5] extended Theorem 4 to include the case A ∩ B = ∅, by the following existence result of a best proximity point. Let A and B be nonempty closed subsets of a metric space X and let T : A ∪ B ⟶ A ∪ B be a cyclic contraction map. A convenience attention has been recently devoted to the research on existence and uniqueness of best proximity points of self-mappings, as well as, to the investigation of associated relevant properties, for instance, stability of the iterations. We ensure the existence of best proximity points for Geraghty and Caristi type contraction mappings
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