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:() if , (), (), () (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]). Let be a usual metric space, then and are -metric space, where 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. (1) is -convergent to .(2), as .(3), as .

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. (1)The sequence is -Cauchy.(2)For everythere existssuch 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. Let be a complete -metric space and let be a mapping which satisfies the following condition, for all , where . Then has a unique fixed point (say ) and is -continuous at .

Proof. Suppose that satisfies condition (2.1), let be an arbitrary point, and define the sequence by , then by (2.1), we have so, But, by (G5), we have So, (2.3)becomes So, it must be the case that which implies Let , then and by repeated application of (2.7), we have Then, for all we have by repeated use of the rectangle inequality and (2.8) that Then, , as , since , as . For (G5) implies that , taking limit as , we get . So is -Cauchy a sequence. By completeness of , there exists such that is -converges to . Suppose that , then taking the limit as , and using the fact that the function is continuous on its variables, we have , which is a contradiction since . So, . To prove uniqueness, suppose that is such that , then (2.1) implies that , thus again by the same argument we will find , thus which implies that , since . To see that is -continuous at , let be a sequence such that , then and we deduce that but (G5) implies that and (2.13) leads to the following cases,(1),(2)(3) 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. Let be a complete -metric spaces and let be a mapping which satisfies the following condition for some and for all : 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. Let be a complete -metric space, and let be a mapping which satisfies the following condition for all where , then has a unique fixed point (say ), and is -continuous at .

Proof. Suppose that satisfies the condition (2.16), let be an arbitrary point, and define the sequence by , then by (2.16) we get since , then it must be the case that but from (G5), we have so (2.18) implies that let , then and by repeated application of (2.20), we have Then, for all we have, by repeated use of the rectangle inequality, So, , as and is -Cauchy sequence. By the completeness of , there exists such tha is -convergent to .Suppose that , then Taking the limit as , and using the fact that the function is continuous in its variables, we get since , this contradiction implies that .To prove uniqueness, suppose that such that , then so we deduce that . This implies that and by repeated use of the same argument we will find . Therefor we get , since , this contradiction implies that . To show that is -continuous at let be a sequence such that in , then Thus, (2.25) becomes but by (G5) we have , therefor (2.26) implies that and we deduce that 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. Let be a complete -metric space, and let be a mapping which satisfies the following condition for some and for all 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. Let be a complete -metric space, and let be a mapping which satisfies the following condition, for all where , then has a unique fixed point, say , and is -continuous at .

Proof. Suppose that satisfies the condition (2.29). Let be an arbitrary point, and define the sequence by , then by (2.29), we have thus and so But by (G5) we have Let , then since and from (2.31) we deduce that Continuing this procedure we get Then, for all