Fixed point theorems in ordered metric spaces via w -distances
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RESEARCH
Open Access
Fixed point theorems in ordered metric spaces via w-distances Mohammad Imdad1* and Fayyaz Rouzkard1,2 *
Correspondence: [email protected] 1 Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India Full list of author information is available at the end of the article
Abstract The purpose of this paper is to prove some fixed point theorems in a complete metric space equipped with a partial ordering employing generalized distances together with altering distance functions. MSC: 54H25; 47H10 Keywords: ordered set; fixed point; complete metric space; altering functions; w-distance; nondecreasing map; orbitally continuous; orbitally U-continuous
1 Introduction The existing literature of metric fixed point theory contains numerous noted generalizations of the Banach contraction mapping principle (e.g., [] and []). One variety of such generalizations is the contractive fixed point theorems contained in Khan et al. [] wherein the authors utilized altering functions to alter the distance between two points in a metric space. Such altering functions are also sometimes referred to as control functions. The following altering distance function is instrumental in our forthcoming results. Definition A (cf. []) A map φ : [, ∞) → [, ∞) is said to be an altering distance function if (a) φ is continuous and nondecreasing and (b) φ(t) = if and only if t = . Using the function φ, Khan et al. [] proved the following result. Theorem A (cf. []) Let T be a self-mapping defined on a complete metric space (X , d) satisfying the condition φ d(T x, T y) ≤ c · φ d(x, y) for x, y ∈ X and < c < , where φ is the earlier described altering distance function. Then T has a unique fixed point. In the recent past, the idea of altering function has been utilized by many researchers (e.g., [–]). Quite recently, Alber and Guerre-Delabriere [] initiated the study of weakly contractive mappings which were initially confined to Hilbert spaces. Rhoades [] utilized this idea in the context of complete metric spaces and proved the following interesting theorem. © 2012 Imdad and Rouzkard; 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.
Imdad and Rouzkard Fixed Point Theory and Applications 2012, 2012:222 http://www.fixedpointtheoryandapplications.com/content/2012/1/222
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Theorem B (cf. []) Let T be a self-mapping defined on a complete metric space (X , d) satisfying the condition d(T x, T y) ≤ d(x, y) – φ d(x, y) for x, y ∈ X , where φ is the earlier described altering distance function. Then T has a unique fixed point. In fact, Alber and Guerre-Delabriere assumed the additional assumption limt→∞ φ(t) = ∞ (on φ). But Rhoades [] proved his theorem without this requirement on φ. In [], Dutta and Choudhury presented a generalization of Th
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