Common fixed point theorems in symmetric spaces employing a new implicit function and common property (E.A). — Bulletin of the Belgian Mathematical Society — Simon Stevin | HighBeam Research

7 мая 2014 | Author: | No comments yet »
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Common fixed point theorems in symmetric spaces employing a new implicit function and common property (E.A).

1 Introduction

Banach contraction principle has been extended and generalized in several ways which include the noted article due to Jungck [15]. The paper due to Jungck [15] has inspired vigorous research activity around it since its appearance. Sessa [32] initiated the tradition of weakening the commutativity condition in such common fixed point theorems by introducing the notion of weak commutativity. After the appearance of this notion, several authors introduced the similar conditions of weak commutativity such as: R-weak commutativity, compatible mappings, compatible mappings of type (A), type (B), type (C), type (P) and weak compatibility whose systematic comparisons and illustrations are available in Murthy [22].

In the setting of metric as well as symmetric spaces, contrastive conditions do not ensure the existence of fixed points unless the space is compact (cf. [4]) or the contrastive condition is replaced by a relatively stronger condition. In recent years, noncompatible mappings have made it possible to prove results on strict contractions beyond compact metric space. The study of common fixed points of noncompatible mappings is a subject of investigation in the recent past and still continues to be an interesting aspect for further investigation. In this regard, the results contained in Pant [23] deserve special mention wherein author has shown the existence of common fixed points of an strict contraction when the underlying space is not essentially compact.

Rhoades [30,31] carried out an exhaustive comparative study of contraction conditions wherein he introduced some contraction conditions and also established the equivalence of several contraction conditions. In recent years, Popa [29] utilized implicit functions instead of contraction conditions to prove common fixed point theorems. Implicit functions are proving fruitful due to their unifying power besides admitting new contraction conditions. Imdad and Ali [12] also proved some results on common fixed points of self mappings using implicit function. In this paper, we define a new class of implicit function and utilize the same to prove our results because of their versatility of deducing several known and unknown contraction conditions in one go. One of the most striking feature of our implicit function (to be introduced in the next section) lies in the nonrequirement of triangular inequality in the course of the proofs of our results in this paper and this is why we opt to prove our results in symmetric spaces instead of metric spaces.

Let X be a nonempty set. A symmetric d is a nonnegative real function defined on X x X such that

(a) d (x, y) = 0 if and only if x = y,

(b) d(x,y) = d(y,x) [for all] x,y [member of] X.


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As expected by (X, d), we denote a nonempty set X equipped with a symmetric d on X and call it a symmetric space. The spaces (X, d) in which limiting points are defined in the usual way is also sometime called an E-space. The idea of E-spaces is due to Frechet and Menger. For more details, one can see [2,9,11,13,35].

Most recently, Aamri and Moutawakil [1] introduced the notion of property (E.A) which is a generalization of compatible (nontrivial) as well as noncompatible mappings and utilize the same to prove some common fixed point theorems for strict contractions in metric spaces. In this continuation, Imdad and Ali [12] also shown that the property (E.A) relaxes the required containment of ranges of involved mappings up to a pair of mappings. Only recently, Liu et al. [20] introduced the notion of common property (E.A) which is in fact an extension of property (E.A) to two pairs of mappings and utilize the same to prove common fixed points for strict contractions.

Definition 1.1.[1] A pair (S, T) of self mappings of a symmetric space (X, d) is said to satisfy the property (E.A) if there exists a sequence in X such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Remark 1.1. Recall that a pair (S, T) of self mappings of a

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