Common Fixed Points for JH-Operators and Occasionally Weakly g-Biased Pairs under Relaxed Condition on Probabilistic Metric Space

Arvind Bhatt,Harish Chandra

doi:10.1155/2013/846315

Abstract

We obtain some fixed point theorems for JH-operators and occasionally weakly g-biased maps on a setXtogether with the functionF:X×X→Δwithout using the triangle inequality and without using the symmetric condition. Our results extend the results of Bhatt et al. (2010).

Highlights

Fixed point theory in probabilistic metric spaces can be considered as a part of probabilistic analysis, which is a one of the emerging areas of interdisciplinary mathematical research with many diverse applications
The theory of probabilistic metric spaces was introduced by Menger [1] in connection with some measurements in Physics
The first effort in this direction was made by Sehgal [4], who, in his doctoral dissertation, initiated the study of contraction mapping theorems in probabilistic metric spaces

Summary

Introduction

Fixed point theory in probabilistic metric spaces can be considered as a part of probabilistic analysis, which is a one of the emerging areas of interdisciplinary mathematical research with many diverse applications. Sessa [6] initiated the tradition of improving commutativity in fixed point theorems by introducing the notion of weakly commuting maps in metric spaces. The results obtained in the metric fixed point theory by using the notion of non-compatible maps or the (E.A) property are very interesting. Bhatt et al [12] have given application of occasionally weakly compatible mappings in dynamical system. We obtain some fixed point theorems for JHoperators and occasionally weakly biased pairs under relaxed condition on F. We extend the concept of JH-operators and occasionally weakly g-biased pairs and the space (X, F) satisfying condition (2). Two selfmaps f and g of a space (X, F) are called occasionally weakly g-biased, if and only if there exists some x ∈ X such that fx = gx and Fgfx,gx(t) ≥ Ffgx,fx(t). An occasionally weakly compatible and a nontrivial weakly g-biased pair (f, g) are occasionally weakly g-biased pairs, but the converse does not hold

Section II

Section III

Application to Dynamic Programming