Abstract
Abstract In this paper, we generalized the notion of proximal contractions of the first and second kinds by using Geraghty’s theorem and establish best proximity point theorems for proximal contractions. Our results improve and extend the recent results of Sadiq Basha and some others. MSC:47H09, 47H10.
Highlights
1 Introduction Several problems can be modeled as 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 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
The aim of this paper is to introduce the new classes of proximal contractions, which are more general than a class of proximal contractions of the first and second kinds, by giving the necessary condition to have best proximity points, and we give some illustrative example of our main results
Summary
We extend the result of Sadiq Basha [ ] and Banach’s fixed point theorem to the case of nonself-mappings satisfying Geraghty’s proximal contraction condition. Let S : A → B, T : B → A and g : A ∪ B → A ∪ B be the mappings satisfying the following conditions: (a) S and T are Geraghty’s proximal contractions of the first kind; (b) S(A ) ⊆ B , T(B ) ⊆ A ; (c) the pair (S, T) is a proximal cyclic contraction.
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