Abstract
We prove some common fixed point results for two mappings satisfying generalized contractive condition in -metric space. Note that -metric of main results in this work are not necessarily continuous. So our results extend and improve several previous works. We also present one example that shows the applicability and usefulness of our results.
1. Introduction
In 1998, Czerwik [1] introduced the concept of -metric space. Since then, several papers deal with fixed point theory for single-valued and multivalued operators in -metric spaces (see also [1–19]). For example, Pacurar [16] proved results on sequences of almost contractions and fixed points in -metric spaces. Also, Hussain and Shah [11] obtained results on KKM mappings in cone -metric spaces. Furthermore, Khamsi ([12, 13]) also showed that each cone metric space has a -metric structure.
The aim of this paper is to present some common fixed point results for two mappings under generalized contractive condition in -metric space, where the -metric is not necessarily continuous. Because many of the authors in their works have used the -metric spaces in which the -metric is continuous, the techniques used in this paper can be used for many of the results on the context of -metric space. From this point of view the results obtained in this paper generalize and extend several ones obtained earlier in a lot of papers concerning -metric space.
2. Preliminaries
Throughout this paper, we denote by , , and the sets of positive integers, nonnegative real numbers, and real numbers, respectively.
Consistent with [1] and [18, page 264], the following definition and results will be needed in the sequel.
Definition 1 (see [1]). Let be a nonempty set and let be a given real number. A function is a -metric if and only if, for all , the following conditions are satisfied: (b1) if and only if ;(b2);(b3).The pair is called a -metric space with coefficient .
It should be noted that the class of -metric spaces is effectively larger than that of metric spaces, since a -metric is a metric when .
We present an example which shows that a -metric on need not be a metric on (see also [18, page 264]).
Example 2. Let be a metric space and , where is a real number. We show that is a -metric with .
Obviously conditions (b1) and (b2) in Definition 1 are satisfied. Now we show that condition (b3) holds for .
It is easy to see that if , then the convexity of the function , where , implies and henceTherefore, for each , we obtain that So condition (b3) in Definition 1 holds and then is a -metric coefficient .
It should be noted that in Example 2, if is a metric space, then is not necessarily a metric space (see Example 3).
Example 3. For example, if and the usual Euclidean metric is defined by for all , then is a -metric on with but is not a metric on , because the triangle inequality does not hold.
Before stating and proving our results, we present some definition and proposition in -metric space. We recall first the notions of convergence, closedness, and completeness in a -metric space.
Definition 4 (see [6]). Let be a -metric space. Then a sequence in is called (i)convergent if and only if there exists such that as , and in this case, we write ,(ii)Cauchy if and only if as .
Proposition 5 (Remark 2.1 in [6]). In a -metric space the following assertions hold: (i)a convergent sequence has a unique limit;(ii)each convergent sequence is Cauchy;(iii)in general, a -metric is not continuous.
Definition 6 (see [6]). The -metric space is complete if every Cauchy sequence in converges.
It should be noted that in general a -metric function for is not jointly continuous in all the two of its variables.
Since in general a -metric is not continuous, we need the following simple lemma about the -convergent sequences.
Lemma 7 (see [20]). Let be a -metric space with coefficient , and suppose that and are -convergent to , respectively; then one has In particular, if , then one has . Moreover, for each one has
3. Common Fixed Point Results
Definition 8. Let and be mappings from a -metric space into itself. The mappings and are said to be weakly commuting if for each in .
Definition 9. Let and be mappings from a -metric space into itself. The mappings and are said to be -weakly commuting if there exists some positive real number such that for each in .
Remark 10. Weak commutativity implies -weak commutativity in -metric space. However, -weak commutativity implies weak commutativity only when .
Example 11. Let and defined as follows: for all . Then is a -metric space. Define and . Then Therefore, for , and are -weakly commuting. But and are not weakly commuting.
Theorem 12. Let be a complete -metric space and let and be -weakly commuting self-mappings on satisfying the following conditions: (a);(b) or is continuous;(c) for all , where is a continuous and nondecreasing function such that for each and .Then and have a unique common fixed point.
Proof. Let be an arbitrary point in . By , choose a point in such that . In general choose such that for all . Now we observe that for each , we haveThis implies that is nonincreasing sequence in . Therefore, it tends to a limit . Next, we claim that . Suppose that . Making in the inequality (11), we get which is a contradiction. Hence ; that is,
Now, we prove that is a Cauchy sequence in . Suppose that is not a Cauchy sequence in . For convenience, let for . Then there is an such that, for each integer , there exist integers and with such thatWe may assume thatby choosing as the smallest number exceeding for which (15) holds. Using (11), we haveHence, . Also notice for each . Thus, as in the above inequality we haveand thus which is a contradiction. Thus, is Cauchy sequence in and by the completeness of , converges to in . Also converges to in . Let us suppose that the mapping is continuous. Then and . Further we have that since and are -weakly commuting for all . Taking the upper limit as in the above inequality, we get Similarly,and hence we get . We now prove that . Suppose that and then . By , we have for each . On taking the upper limit as in the above inequality we get which is a contradiction. Therefore, . Since we can find in such that . Now, we have for all . Taking limit as we get since , which implies that ; that is, . Also, which again implies that . Thus is a common fixed point of and .
Now to prove uniqueness let if possible be another common fixed point of and . Then and so which is a contradiction. Therefore, , that is, is a unique common fixed point of and . This completes the proof.
Now we give an example to support Theorem 12.
Example 13. Let and defined by for all . Then is a -metric space for . Define and on . It is evident that and is continuous. Now we observe that for all in and defined by for . Moreover, it is easy to see that and are -weakly commuting. Thus all the conditions of Theorem 12 are satisfied and is a common fixed point of and .
Corollary 14. Let be a complete -metric space and let be a self-mapping on satisfying the following condition: for all , where is a continuous and nondecreasing function such that for each and . Then has a unique fixed point.
Proof. If we take as identity mapping on , then Theorem 12 follows that has a unique fixed point.
Corollary 15. Let be a complete metric space and let and be -weakly commuting self-mappings of satisfying the following conditions: (a);(b) or is continuous;(c) for all , where is a continuous and nondecreasing function such that for each and .Then and have a unique common fixed point.
Proof. If we take , then Theorem 12 follows that and have a unique common fixed point.
Conflict of Interests
The authors declare that they have no conflict of interests.
Authors’ Contribution
All authors contributed equally and significantly in writing this paper. All authors read and approved the paper.
Acknowledgment
Poom Kumam and Wutiphol Sintunavarat would like to thank the Thailand Research Fund and Thammasat University under Grant no. TRG5780013 for financial support during the preparation of this paper.