# Fixed Point Theorems for Contractive Mappings in Complete Open image in new window -Metric Spaces

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## Abstract

We prove some fixed point results for mappings satisfying various contractive conditions on Complete Open image in new window -metric Spaces. Also the Uniqueness of such fixed point are proved, as well as we showed these mappings are Open image in new window -continuous on such fixed points.

## Keywords

Basic Concept Differential Geometry Arbitrary Point Applied Science Point Theory## 1. Introduction

Metric spaces are playing an increasing role in mathematics and the applied sciences.

Over the past two decades the development of fixed point theory in metric spaces has attracted considerable attention due to numerous applications in areas such as variational and linear inequalities, optimization, and approximation theory.

Different generalizations of the notion of a metric space have been proposed by Gahler [1, 2] and by Dhage [3, 4]. However, HA et al. [5] have pointed out that the results obtained by Gahler for his Open image in new window metrics are independent, rather than generalizations, of the corresponding results in metric spaces, while in [6]the current authors have pointed out that Dhage's notion of a Open image in new window -metric space is fundamentally flawed and most of the results claimed by Dhage and others are invalid.

In 2003 we introduced a more appropriate and robust notion of a generalized metric space as follows.

Definition 1.1 ([7]).

Let X be a nonempty set, and let Open image in new window be a function satisfying the following axioms:

() Open image in new window if Open image in new window ,

() Open image in new window (symmetry in all three variables),

() Open image in new window , for all Open image in new window , (rectangle inequality).

Then the function Open image in new window is called a *generalized metric*, or, more specifically a Open image in new window -*metric* on Open image in new window , and the pair Open image in new window is called a Open image in new window -*metric space*.

Example 1.2 ([7]).

We now recall some of the basic concepts and results for Open image in new window -metric spaces that were introduced in ([7]).

Definition 1.3.

Let Open image in new window be a Open image in new window -metric space, let Open image in new window be a sequence of points of Open image in new window , we say that Open image in new window is Open image in new window -*convergent* to Open image in new window if Open image in new window ; that is, for any Open image in new window there exists Open image in new window such that Open image in new window , for all Open image in new window (throughout this paper we mean by Open image in new window the set of all natural numbers). We refer to Open image in new window as the *limit* of the sequence Open image in new window and write Open image in new window .

Proposition 1.4.

Let Open image in new window be a Open image in new window -metric space then the following are equivalent.

(1) Open image in new window is Open image in new window -convergent to Open image in new window .

(2) Open image in new window , as Open image in new window .

(3) Open image in new window , as Open image in new window .

Definition.

Let Open image in new window be a Open image in new window -metric space, a sequence Open image in new window is called Open image in new window -Cauchy if given Open image in new window , there is Open image in new window such that Open image in new window for all Open image in new window that is if Open image in new window as Open image in new window .

Proposition 1.6.

In a Open image in new window -metric space Open image in new window , the following are equivalent.

(1)The sequence Open image in new window is Open image in new window -Cauchy.

(2)For every Open image in new window there exists Open image in new window such that Open image in new window for all Open image in new window .

Definition 1.7.

Let Open image in new window and Open image in new window be Open image in new window -metric spaces and let Open image in new window be a function, then Open image in new window is said to be Open image in new window -*continuous at a point* Open image in new window if given Open image in new window , there exists Open image in new window such that Open image in new window ; Open image in new window implies Open image in new window . A function Open image in new window is Open image in new window -continuous on Open image in new window if and only if it is Open image in new window -continuous at all Open image in new window .

Proposition 1.8.

Let Open image in new window , Open image in new window be Open image in new window -metric spaces, then a function Open image in new window is Open image in new window -continuous at a point Open image in new window if and only if it is Open image in new window -sequentially continuous at Open image in new window ; that is, whenever Open image in new window is Open image in new window -convergent to Open image in new window , Open image in new window is Open image in new window -convergent to Open image in new window .

Proposition 1.9.

Let Open image in new window be a Open image in new window -metric space, then the function Open image in new window is jointly continuous in all three of its variables.

Definition 1.10.

A Open image in new window -metric space Open image in new window is said to be Open image in new window *-complete* (or a complete Open image in new window -metric space) if every Open image in new window -Cauchy sequence in Open image in new window is Open image in new window -convergent in Open image in new window .

## 2. The Main Results

We begin with the following theorem.

Theorem 2.1.

where Open image in new window . Then Open image in new window has a unique fixed point (say Open image in new window ) and Open image in new window is Open image in new window -continuous at Open image in new window .

Proof.

and (2.13) leads to the following cases,

(1) Open image in new window ,

In each case take the limit as Open image in new window to see that Open image in new window and so, by Proposition 1.4, we have that the sequence Open image in new window is Open image in new window -convergent to Open image in new window , therefor Proposition 1.8 implies that Open image in new window is Open image in new window -continuous at Open image in new window .

Remark 2.2.

If the Open image in new window -metric space is bounded (that is, for some Open image in new window we have Open image in new window for all Open image in new window ) then an argument similar to that used above establishes the result for Open image in new window .

Corollary 2.3.

where Open image in new window , then Open image in new window has a unique fixed point (say Open image in new window ), and Open image in new window is Open image in new window -continuous at Open image in new window .

Proof.

From the previous theorem, we have that Open image in new window has a unique fixed point (say u), that is, Open image in new window . But Open image in new window , so Open image in new window is another fixed point for Open image in new window and by uniqueness Open image in new window .

Theorem 2.4.

where Open image in new window , then Open image in new window has a unique fixed point (say Open image in new window ), and Open image in new window is Open image in new window -continuous at Open image in new window .

Proof.

Taking the limit of (2.27) as Open image in new window , we see that Open image in new window and so, by Proposition 1.8, we have Open image in new window which implies that Open image in new window is Open image in new window -continuous at Open image in new window .

Corollary 2.5.

where Open image in new window , then Open image in new window has a unique fixed point (say Open image in new window ), and Open image in new window is Open image in new window -continuous at Open image in new window .

Proof.

The proof follows from the previous theorem and the same argument used in Corollary 2.3.

Theorem 2.6.

where Open image in new window , then Open image in new window has a unique fixed point, say Open image in new window , and Open image in new window is Open image in new window -continuous at Open image in new window .

Proof.

therefore, (2.38) implies two cases.

Case 1.

Case 2.

But, by (G5) we have Open image in new window , so case 2 implies that Open image in new window In each case taking the limit as Open image in new window , we see that Open image in new window and so, by Proposition 1.8, we have Open image in new window which implies that Open image in new window is Open image in new window -continuous at Open image in new window .

Corollary 2.7.

where Open image in new window , then Open image in new window has a unique fixed point, say Open image in new window , and Open image in new window is Open image in new window -continuous at Open image in new window .

Proof.

The proof follows from the previous theorem and the same argument used in Corollary 2.3. The following theorem has been stated in [8] without proof, but this can be proved by using Theorem (2.6) as follows.

Theorem 2.8 ([8]).

where Open image in new window , then Open image in new window has a unique fixed point, say Open image in new window , and Open image in new window is Open image in new window -continuous at Open image in new window .

Proof.

Setting Open image in new window in condition (2.40), then it reduced to condition (2.29), and the proof follows from Theorem (2.6).

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