Abstract
Very recently, Isik and Turkoglu (Fixed Point Theory Appl. 2013:131, 2013) proved a common fixed point theorem in a rectangular metric space by using three auxiliary distance functions. In this paper, we note that this result can be derived from the recent paper of Lakzian and Samet (Appl. Math. Lett. 25:902906, 2012).
MSC: 47H10, 54H25.
Keywords:
fixed point; partial metric space; GPmetric space1 Introduction and preliminaries
In 2012, Lakzian and Samet [1] proved a fixed point theorem of a selfmapping with certain conditions in the context of a rectangular metric space via two auxiliary functions. Very recently, as a generalization of the main result of [1], Isik and Turkoglu [2] reported a common fixed point result of two selfmappings in the setting of a rectangular metric space by using three auxiliary functions. In this paper, unexpectedly, we conclude that the main result of Isik and Turkoglu [2] is a consequence of the main results of [1]. The obtained results are inspired by the techniques and ideas of, e.g., [311].
Throughout the paper, we follow the notations used in [2]. For the sake of completeness, we recall some basic definitions, notations and results.
Definition 1.1 Let X be a nonempty set, and let satisfy the following conditions for all and all distinct , each of which is different from x and y:
Then the map d is called a rectangular metric and the pair is called a rectangular metric space (or, for short, RMS).
We note that a rectangular metric space is also known as a generalized metric space (g.m.s.) in some sources.
We first recall the definitions of the following auxiliary functions: Let ℱ be the set of functions satisfying the condition if and only if . We denote by Ψ the set of functions such that ψ is continuous and nondecreasing. We reserve Φ for the set of functions such that α is continuous. Finally, by Γ we denote the set of functions satisfying the following condition: β is lower semicontinuous.
Lakzian and Samet [1] proved the following fixed point theorem.
Theorem 1.1[1]
Letbe a Hausdorff and completeRMS, and letbe a selfmap satisfying
for all, whereand. ThenThas a unique fixed point inX.
Lemma 1.1[3]
LetXbe a nonempty set andbe a function. Then there exists a subsetsuch thatandis onetoone.
Definition 1.2 Let X be a nonempty set, and let be selfmappings. The mappings are said to be weakly compatible if they commute at their coincidence points, that is, if for some implies that .
Theorem 1.2[2]
Letbe a Hausdorff and completeRMS, and letbe selfmappings such that, andis a closed subspace ofX, and that the following condition holds:
for all, where, , , and these mappings satisfy the condition
ThenTandFhave a unique coincidence point inX. Moreover, ifTandFare weakly compatible, thenTandFhave a unique common fixed point.
Remark 1.1 Let be RMS. Then d is continuous (see, e.g., Proposition 2 in [5]).
2 Main results
We start this section with the following theorem which is a slightly improved version of Theorem 1.1, obtained by replacing the continuity condition of ϕ with a lower semicontinuity.
Theorem 2.1Letbe a Hausdorff and completeRMS, and letbe a selfmap satisfying
for all, whereand. ThenThas a unique fixed point inX.
Proof Let and for . Following the lines of the proof of Theorem 1.1 in [1], we conclude that there exists such that
We can easily derive that
by replacing and in inequality (4).
Taking lim sup in inequality (6) as , we find that
and using the continuity of ψ and lower semicontinuity of ϕ, thus, we get
which implies that and then . Consequently, we have as .
Next, we shall prove that
By using inequality (4), we derive that
From the monotone property of the function ψ, it follows that is monotone decreasing. Thus, there exists such that
Taking lim sup of inequality (10) as , we derive that
Then, by using the continuity of ψ and lower semicontinuity of ϕ, we find
which implies that . So, we conclude that and hence as .
As in Theorem 1.1 in [1], we notice that T has no periodic point.
We assert that is a Cauchy sequence. Suppose, on the contrary, that there exists for which we can obtain subsequences and of with such that
Again, repeating the steps of Theorem 1.1 in [1], we obtain that
Now, letting lim sup in inequality (15) as , we observe that
Using the continuity of ψ and lower semicontinuity of ϕ, we get
which implies that and then , a contradiction with . Hence, is a Cauchy sequence. The rest of the proof is the mimic of the proof of Theorem 1.1 in [1] and hence we omit the details. □
Inspired by Theorem 1.2, one can state the following theorem.
Theorem 2.2Letbe a Hausdorff and completeRMS, and letbe selfmappings such that
for all, where, , and these mappings satisfy the condition
ThenThas a unique fixed point inX.
Since the proof is the mimic of the proof of Theorem 1.2, we omit it.
We first prove that the above theorem is equivalent to Theorem 2.1.
Theorem 2.3Theorem 2.2 is a consequence of Theorem 2.1.
Proof Taking in Theorem 2.2, we obtain immediately Theorem 2.1. Now, we shall prove that Theorem 2.2 can be deduced from Theorem 2.1. Indeed, let be a mapping satisfying (18) with , , , and let these mappings satisfy condition (19). From (18), for all , we have
for all . Due to the definition of θ, we observe that . Now, Theorem 2.2 follows immediately from Theorem 2.1. □
By regarding the techniques in [3], we conclude the following result.
Theorem 2.4Theorem 1.2 is a consequence of Theorem 2.2.
Proof By Lemma 1.1, there exists such that and is onetoone. Now, define a map by . Since F is onetoone on E, h is well defined. Note that for all . Since is complete, by using Theorem 2.2, there exists such that . Hence, T and F have a point of coincidence, which is also unique. It is clear that T and F have a unique common fixed point whenever T and F are weakly compatible. □
Theorem 2.5Theorem 1.2 is a consequence of Theorem 2.1.
Proof It is evident from Theorem 2.3 and Theorem 2.4. □
3 Conclusion
In this paper, we first slightly improve the main result of Lakzian and Samet, Theorem 1.1. Then, we conclude that the main result (Theorem 1.2) of IsikTurkoglu [2] is a consequence of our improved result, Theorem 2.1.
Competing interests
The authors declare that there is no conflict of interests regarding the publication of this article.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Acknowledgements
The authors thank anonymous referees for their remarkable comments, suggestions and ideas that helped to improve this paper.
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