Best proximity point theorems for generalized contractions in partially ordered metric spaces
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Best proximity point theorems for generalized contractions in partially ordered metric spaces Jingling Zhang, Yongfu Su* and Qingqing Cheng *
Correspondence: [email protected]; [email protected] Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, P.R. China
Abstract The purpose of this paper is to obtain four best proximity point theorems for generalized contractions in partially ordered metric spaces. Further, our P-operator technique, which changes a non-self mapping to a self-mapping, plays an important role. Some recent results in this area have been improved. MSC: 47H05; 47H09; 47H10 Keywords: generalized contraction; fixed point; best proximity point; weak P-monotone property
1 Introduction and preliminaries Let A and B be nonempty subsets of a metric space (X, d). An operator T : A → B is said to be contractive if there exists k ∈ [, ) such that d(Tx, Ty) ≤ kd(x, y) for any x, y ∈ A. The well-known Banach contraction principle is as follows: Let (X, d) be a complete metric space, and let T : X → X be a contraction of X into itself. Then T has a unique fixed point in X. In the sequel, we denote by Γ the functions β : [, ∞) → [, ) satisfying the following condition: β(tn ) →
⇒
tn → .
In , Geraghty introduced the Geraghty-contraction and obtained Theorem . as follows. Definition . [] Let (X, d) be a metric space. A mapping T : X → X is said to be a Geraghty-contraction if there exists β ∈ Γ such that for any x, y ∈ X, d(Tx, Ty) ≤ β d(x, y) d(x, y). Theorem . [] Let (X, d) be a complete metric space, and let T : X → X be an operator. Suppose that there exists β ∈ Γ such that for any x, y ∈ X, d(Tx, Ty) ≤ β d(x, y) d(x, y). Then T has a unique fixed point. © 2013 Zhang et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Zhang et al. Fixed Point Theory and Applications 2013, 2013:83 http://www.fixedpointtheoryandapplications.com/content/2013/1/83
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Obviously, Theorem . is an extensive version of the Banach contraction principle. Recently, the generalized contraction principle has been studied by many authors in metric spaces or more generalized metric spaces. Some results have been got in partially ordered metric spaces as follows. Theorem . [] Let (X, ≤) be a partially ordered set, and suppose that there exists a metric d such that (X, d) is a complete metric space. Let f : X → X be an increasing mapping such that there exists an element x ∈ X with x ≤ f (x ). Suppose that there exists β ∈ Γ such that d f (x), f (y) ≤ β d(x, y) d(x, y),
∀y ≥ x.
Assume that either f is continuous or X is such that if an increasing sequence xn → x ∈ X, then xn ≤ x, ∀n. Besides, if for each x, y ∈ X, there exists z ∈ X which is comparable to x and y, then f has a unique fixed point.
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