Fixed Point Theorems on Ordered Metric Spaces through a Rational Contraction

Poom Kumam,Fayyaz Rouzkard,Mohammad Imdad,Dhananjay Gopal

doi:10.1155/2013/206515

Abstract

Ran and Reurings (2004) established an interesting analogue of Banach Contraction Principle in a complete metric space equipped with a partial ordering and also utilized the same oneto discuss the existence of solutions to matrix equations. Motivated by this paper, we prove results on coincidence points for a pair of weakly increasing mappings satisfying a nonlinear contraction condition described by a rational expression on an ordered complete metric space. The uniqueness of common fixed point is also discussed. Some examples are furnished to demonstrate the validity of the hypotheses of our results. As an application, we derive an existence theorem for the solution of an integral equation.

Highlights

Introduction with PreliminariesA variety of generalizations of the Classical Banach Contraction Principle [1] are available in the existing literature of metric fixed point theory
Let (X, ⪯) be a partially ordered set equipped with a metric d on X such that (X, d) is a complete metric space
Let (X, ⪯) be a partially ordered set on which there exists a metric d on X such that (X, d) is a complete metric space

Summary

Introduction with Preliminaries

A variety of generalizations of the Classical Banach Contraction Principle [1] are available in the existing literature of metric fixed point theory. (b) a mapping T : X → X is called nondecreasing with respect to “⪯” if x ⪯ y implies Tx ⪯ Ty. Let X be a nonempty set and R : X → X be a given mapping. In order to show that the mapping T is weakly increasing with respect to mapping R, we distinguish three cases. A pair (R, T) of self-mappings of a metric space (X, d) is said to be reciprocally continuous if and only if limn → ∞RTxn = Rz and limn → ∞TRxn = Tz for every sequence {xn} in X satisfying nl→im∞Rxn = nl→im∞Txn = z (7). A pair (R, T) of self mappings of a metric space (X, d) is said to be weakly reciprocally continuous if and only if limn → ∞RTxn = Rz, for every sequence {xn} in X satisfying nl→im∞Rxn = nl→im∞Txn = z (8). Every pair of reciprocally continuous mappings is always weakly reciprocally but not as demonstrated in Pant et al [19]

Results

Uniqueness Results

An Application