g-reciprocal Continuity in Symmetric Spaces

Arvind Bhatt

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g-reciprocal Continuity in Symmetric Spaces

Arvind Bhatt

Applied Science Department, Bipin Tripahti Kumaon Institute of Technology, Almora, India

Abstract

In this paper, we obtain a common fixed point theorem by employing the notion of g-reciprocal continuity in symmetric spaces. We demonstrate that g-reciprocal continuity ensures the existence of common fixed point under strict contractive conditions, which otherwise do not ensure the existence of fixed points.

Cite this article:

  • Bhatt, Arvind. "g-reciprocal Continuity in Symmetric Spaces." American Journal of Mathematical Analysis 1.3 (2013): 39-41.
  • Bhatt, A. (2013). g-reciprocal Continuity in Symmetric Spaces. American Journal of Mathematical Analysis, 1(3), 39-41.
  • Bhatt, Arvind. "g-reciprocal Continuity in Symmetric Spaces." American Journal of Mathematical Analysis 1, no. 3 (2013): 39-41.

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1. Introduction

Fixed point theory of strict contractive conditions constitutes a very important class of mappings and includes contraction mappings as their subclass. It may be observed that strict contractive conditions do not ensure the existence of common fixed points unless some strong condition is assumed either on the space or on the mappings. In such cases either the space is taken to be compact or some sequence of iterates is assumed to be Cauchy sequence. The study of common fixed points of strict contractive conditions using noncompatibility was initiated by Pant [9]. Motivated by Pant [9] researchers [1, 3, 6, 7, 11, 12, 13] of this domain obtained common fixed point results for strict contractive conditions under generalized metric spaces. The significance of this paper lies in the fact that we can obtain fixed point theorems for g-reciprocally continuous mappings in generalized strict contractive conditions without assuming any strong conditions on the space or on the mappings.

2. Preliminaries

Definition 2.1 [14]. Let X be a non-empty set. A symmetric on a set X is a real valued function such that,

Let d be a symmetric on a set X and for and any x, let B(x,. A topology t(d) on X is given by Uif and only if for each x for some.A symmetric d is a semi-metric if for each x and each ,is a neighborhood of x in the topology t(d). There are several concepts of completeness in this setting. A sequence is a d-Cauchy if it satisfies the usual metric condition.

Definition 2.2. [14]. Let (X, d) be a symmetric (semi-metric) space.

(1). (X, d) is S-Complete if for every d-Cauchy sequence {xn } there exists x in X with lim.

(2). (X, d) is d-Cauchy Complete if for every d-Cauchy sequence {xn} there exists x in X with lim xn = x with respect to t(d).

(3). is d-Continuous if implies

(4). is t(d) continuous if with respect to t(d) implies with respect to t(d).

The following two axioms were given by Wilson [14].

Definition 2.3. Let (X, d) be a symmetric (semi-metric) space.

W1: Given {xn}, x and y in X, and.

W2: Given {xn}, {yn} and an x in X, and.

Definition 2.4. [4]. Two self maps and g of a metric space are called compatible if whenever is a sequence in such that for some in

The definition of compatibility implies that the mappings and will be noncompatible if there exists a sequence in such that for some in but is either non zero or nonexistent.

Definition 2.5. [1]. Two self maps and of a metric space are said to satisfy property (E. A.) if there exists a sequence in such that for some in .

Definition 2.6 .[5] Two self maps and of a metric space are called g-compatible if whenever is a sequence in such that for some in

Definition 2.7 [8]. Two selfmappings and of a metric space are called reciprocally continuous iff and whenever is a sequence such that for some in

Definition 2.8 [10]. Two selfmappings and of a metric space are called g-reciprocally continuous iff and whenever is a sequence such that for some in

It may be observed that if and are both continuous then they are obviously g-reciprocally continuous but the converse is not true. It may also be observed that g-reciprocal continuity is independent of the notion of reciprocal continuity. The following examples illustrate this fact.

Example 2.1. [9]. Let and be the symmetric (semi-metric) on Define as follows

Then and are g-reciprocally continuous but not reciprocally continuous. To see this let us consider the sequence Then and Thus and are g-reciprocally continuous but they are not reciprocally continuous.

Example 2.2. [9]. Let and be symmetric (semi-metric) on Define as follows

Then and are reciprocally continuous but not g-reciprocally continuous . To see this let us consider the sequence Then and Thus and are reciprocally continuous but they are not g-reciprocally continuous.

Examples 2.1 and 2.2 clearly show that reciprocal continuity and g-reciprocally continuous reciprocal continuity are independent of each other.

3. Main Results

Theorem 3.1. Let and be g-reciprocally continuous self mappings of a symmetric space satisfying

whenever the right hand side is nonzero. Suppose and satisfy property (E. A.). If and are g-compatible then and have a unique common fixed point.

Proof: Since and satisfy property (E. A.), there exists a sequence in such that for some in Suppose that and are g- compatible. Then g- reciprocally continuity of and implies that and The last two limits together imply Since g- compatibility implies commutativity at coincidence points, i.e., and, hence If then by using (i), we get

a contradiction. Hence and is a common fixed point of and This completes the proof of the theorem.

The next example illustrates the above theorem.

Example 3.1. Let with the symmetric (semi-metric) . Define as follows

Then and satisfy all the conditions of Theorem 3.1 and have a unique common fixed point at It can be verified in this example that and satisfy the contraction condition (i). Furthermore, and are g-reciprocally continuous g-compatible mappings. It is also obvious that and are not reciprocally continuous. Here and are not reciprocally continuous mappings.

Remark: In the above result we have not assumed strong conditions, e.g., completeness of the space, containment of the ranges of the mappings, closedness of the range of any one of the involved mappings and continuity of any mapping. We also proved a result using generalized strict contractive condition. It may be observed that strict contractive conditions do not ensure the existence of common fixed points unless the space is assumed compact or the strict contractive condition is replaced by some strong conditions, e.g., a Banach type contractive condition or a contractive condition or a Meir-Keeler type contractive condition. In the result established in this paper, we have not assumed any mapping to be continuous. Thus we provide more answers to the problem posed by Rhoades [2] regarding existence a contractive definition which is strong enough to generate a fixed point, but which does not force the map to be continuous at the fixed point.

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