Abstract
The aim of the paper is to introduce new Suzuki and convex type contractions and prove new best proximity results for these contractions in the setting of a metric space. As applications, we deduce similar results for such type of contractions in partially ordered metric spaces and derive new Suzuki type fixed point results. An illustrative example is provided here to highlight our findings.
Highlights
1 Introduction and preliminaries The background literature on best proximity theory and associated fixed point theory in metric spaces, Banach spaces and fuzzy metric spaces is very abundant in the literature; see, for instance, [ – ] and references therein
We shall denote the set of best proximity points of T by Bpp(T)
We introduce new concepts of proximal mappings, for more details see [ ]
Summary
Let A and B be nonempty closed subsets of a complete partially ordered metric space (X, d, ) such that A is nonempty and T : A → B be a Suzuki type ordered ψ proximal map satisfying the following assertions: (i) T(A ) ⊆ B and (A, B) satisfies the weak P-property, (ii) T is proximally ordered-preserving, (iii) there are x and x in A such that d(x , Tx ) = d(A, B) and x x , (iv) T is continuous, or (v) if {xn} is a increasing sequence in A with xn → x ∈ A as n → ∞, xn n ∈ N. T : A → B is an ordered convex proximal contractive mapping of the first type (or the second type) satisfying T(A ) ⊆ B and the conditions (ii)-(iv) of Theorem .
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