Abstract

We present a fixed point result in quasi -metric spaces. Our result generalizes recent fixed point results obtained by Aleksić et al., Dung and Hang, Jovanović et al., Sarwar, and Rahman and classical results obtained by Hardy, Rogers, and Ćirić. Also, we obtain a common fixed point result in -metric spaces. As a special case, we get a result of Ćirić and Wong.

1. Introduction

The notion of a generalized contraction was presented by Ćirić in his dissertation [1]. In [1], Ćirić proved the first fixed point result for this class of mappings, which was published in [2]. Ćirić also published several papers on generalized contractions, such as for multivalued mappings in [3], on common fixed point of not necessarily commuting mappings in [4], for probabilistic metric spaces in [5, 6] and fixed point result of Meir-Keeler type in [7]. For further historical remarks of the papers of Ćirić, see [8].

In 1973, Hardy and Rogers [9] proved a result of fixed point on metric space, which was extended to common fixed point result by Wong [10].

The results of common fixed points of Wong [10] and Ćirić [4] are independent. More concepts of common fixed points can be seen in [11, 12].

Also, Fréchet in the paper [13] introduced a class of metric spaces which are included in the class -metric spaces. First, fixed point result in a -metric space was presented by Bakhtin [14] and Czerwik [15] (for more on -metric spaces see [16–23]). In the last few decades, many generalizations of a metric space appeared in literature. For some historical aspects of various generalizations of a metric space, the reader may refer to [24].

In this paper, we present a fixed point theorem for a mapping defined on a quasi -metric space which generalizes recent fixed point results obtained by Aleksić et al. [16], Dung and Hang [18], Jovanović et al. [25], and Sarwar and Rahman [22]. Further, we obtain a result of common fixed point on a -metric space. Our result generalizes the classical results presented by Ćirić [4] and Wong [10].

2. The Quasi -Metric Spaces

We start with definition of quasi -metric spaces, which was introduced by Shah and Hussain [23].

Definition 1. Let be a nonempty set, and . Then, is a quasi -metric space if (1) if and only if (2), for all

Clearly, is a quasi metric space.

Remark 2. Let be a quasi -metric space and for all . Then, is a -metric space.

Lemma 3. Let be a quasi -metric space. Then, .

Proof. Let . Then, . So, .☐

Remark 4. Let be a sequence of nonnegative real numbers such that and . A quasi -metric space is a topological space with , as a base of neighborhood filter of the point where

Definition 5. Let be a quasi -metric space and a sequence . (1)Sequence is a left Cauchy sequence, if as (2)A quasi -metric space is left complete if every left Cauchy sequence converges to some

Definition 6. Let be a quasi -metric space and the sequences , in be such that and A mapping is sequentially continuous if .

We will use the following lemma in our main results.

Lemma 7 (see [26]). Let be a quasi -metric space and . If there exists such that for all , then is a left Cauchy sequence.

3. A Fixed Point Theorem in Quasi -Metric Spaces

Let and be a given mapping. Then, is a fixed point of mapping if . Let , and consider the sequence defined by , i.e., is a sequence of Picard iterates of mapping at point .

Now, we present our first result, which generalizes recent fixed point results obtained in [16, 18, 22, 25] for generalized contractive mappings defined on -metric spaces.

Theorem 8. Let be a left complete quasi -metric space and a mapping . If there exist such that , and for any , then for any sequence of Picard iterates defined by mapping at is left Cauchy sequence. Moreover, if is sequentially continuous or is sequentially continuous, then, has unique fixed point and as .

Proof. Let be arbitrary and sequence of Picard iterates defined by at . Then If , then, which implies So, which implies Hence, we get that where So, by Lemma 7, we obtain that is left Cauchy sequence. It is convergent because is left complete. Thus, exists such that .☐

Case 9. Let a mapping be a sequentially continuous.
Then

Case 10. Let be a sequentially continuous.
Then which implies So, we get that Hence, It follows that because . Finally, suppose that there are two fixed points of mapping , i.e., , . Then, we get which implies that .

Corollary 11. Let be a left complete quasi -metric space and a mapping . If there exist such that for any , then for any sequence of Picard iterates defined by mapping at is left Cauchy sequence. Moreover, if is sequentially continuous or is sequentially continuous, then, has unique fixed point and as .

Example 12. Let and mapping defined by Let defined by Since, and , for all , we obtain that for holds for all . Also, if and only if . So, is a quasi -metric space. Note that does not hold in the general case. In this case, all the conditions of Corollary 11 are valid, and we conclude that the mapping has a fixed point.

4. A Common Fixed Point Theorem in -Metric Spaces

Now we obtain a common fixed point result for mappings defined on -metric spaces. Our result improves the classical results presented by Ćirić [4] and Wong [10].

Theorem 13. Let be a complete -metric space and the mappings . If there exist such that and for any , then for any sequence of Picard iterates defined by at is left Cauchy sequence. If and are sequentially continuous or is sequentially continuous then and has unique fixed point which is unique limit of all Picard sequences defined by .

Proof. Let be arbitrary and sequence defined by and . Then If then So, we get that therefore, which implies Hence, we get that So, we obtained where Further, we have If then which implies therefore, which implies that Therefore, we obtain It follows So we obtain, where Hence, for each positive integer . So, by Lemma 7, we obtain that is a Cauchy sequence. It is convergent because is complete. Therefore, there exists such that .☐