Some existence and uniqueness theorems on ordered metric spaces via generalized distances
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RESEARCH
Open Access
Some existence and uniqueness theorems on ordered metric spaces via generalized distances Fayyaz Rouzkard1 , Mohammad Imdad1 and Dhananjay Gopal2* *
Correspondence: gopal.dhananjay@rediffmail.com; [email protected] 2 Department of Applied Mathematics & Humanities, S. V. National Institute of Technology, Surat, 395007, 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 using w-distances together with the aid of altering functions. MSC: 54H25; 47H10 Keywords: partially ordered set; fixed point; complete metric space; altering functions; w-distance; nondecreasing; orbitally continuous; orbitally U-continuous
1 Introduction with preliminaries The concept of a w-distance was initiated by Kada, Suzuki and Takahashi [] and was primarily utilized to improve Caristi’s fixed point theorem [], Ekeland’s variational principle [] and nonconvex minimization theorems whose descriptions and details are available in Takahashi []. Proving existence results of fixed points on partially ordered metric spaces has been a relatively new development in metric fixed points theory. In [], an analogue of Banach’s fixed point theorem in a partially ordered metric space has been proved besides discussing some applications to matrix equations. Ran and Reurings have further weakened the usual contraction condition but merely up to monotone operators. Branciari [] established a fixed point result for an integral-type inequality, which is a generalization of the Banach contraction principle. Vijayaraju et al. [] obtained a general principle, which made it possible to prove many fixed point theorems for pairs of maps satisfying integral-type contraction conditions. Several fixed point and common fixed point theorems in metric and semi-metric spaces for compatible, weakly compatible and owc mappings satisfying contractive conditions of integral type were proved in [–] and in other papers. Later on, Suzuki [] proved that integral-type contractions are Meir-Keeler contractions. He also showed that MeirKeeler contractions of integral type are still Meir-Keeler contractions. Jachymski [] also proved that most contractive conditions of integral type given recently by many authors coincide with classical ones. But he gave a new contractive condition of integral type which is independent of classical ones. Recently Popa and Mocanu [] obtained integral-type contractions via an altering distance function and proved general common fixed point results for integral-type contractive conditions. In [], Razani et al. proved a fixed point theorem for (φ, ψ, p)-contractive mappings on X [i.e., for each x, y ∈ X , φp(T x, T y) ≤ ψφp(x, y)], which is a new version of the main © 2013 Rouzkard 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 u
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