- Zead Mustafa
^{1}Email author and - Brailey Sims
^{2}

**2009**:917175

**DOI: **10.1155/2009/917175

© Z. Mustafa and B. Sims. 2009

**Received: **31 December 2008

**Accepted: **7 April 2009

**Published: **16 April 2009

## Abstract

## 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 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 -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 be a function satisfying the following axioms:

() (symmetry in all three variables),

() , for all , (rectangle inequality).

Then the function
is called a *generalized metric*, or, more specifically a
-*metric* on
, and the pair
is called a
-*metric space*.

Example 1.2 ([7]).

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

Definition 1.3.

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

Proposition 1.4.

Let be a -metric space then the following are equivalent.

Definition.

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

Proposition 1.6.

In a -metric space , the following are equivalent.

(2)For every there exists such that for all .

Definition 1.7.

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

Proposition 1.8.

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

Proposition 1.9.

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

Definition 1.10.

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

## 2. The Main Results

We begin with the following theorem.

Theorem 2.1.

where . Then has a unique fixed point (say ) and is -continuous at .

Proof.

and (2.13) leads to the following cases,

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

Remark 2.2.

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

Corollary 2.3.

where , then has a unique fixed point (say ), and is -continuous at .

Proof.

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

Theorem 2.4.

where , then has a unique fixed point (say ), and is -continuous at .

Proof.

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

Corollary 2.5.

where , then has a unique fixed point (say ), and is -continuous at .

Proof.

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

Theorem 2.6.

where , then has a unique fixed point, say , and is -continuous at .

Proof.

therefore, (2.38) implies two cases.

Case 1.

Case 2.

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

Corollary 2.7.

where , then has a unique fixed point, say , and is -continuous at .

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 , then has a unique fixed point, say , and is -continuous at .

Proof.

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

## Authors’ Affiliations

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## Copyright

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