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. 25:902-906, 2012).
MSC: 47H10, 54H25.
Keywords:fixed point; partial metric space; GP-metric space
1 Introduction and preliminaries
In 2012, Lakzian and Samet  proved a fixed point theorem of a self-mapping 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 , Isik and Turkoglu  reported a common fixed point result of two self-mappings 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  is a consequence of the main results of . The obtained results are inspired by the techniques and ideas of, e.g., [3-11].
Throughout the paper, we follow the notations used in . For the sake of completeness, we recall some basic definitions, notations and results.
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 semi-continuous.
Lakzian and Samet  proved the following fixed point theorem.
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 ).
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 semi-continuity.
Proof Let and for . Following the lines of the proof of Theorem 1.1 in , we conclude that there exists such that
We can easily derive that
and using the continuity of ψ and lower semi-continuity of ϕ, thus, we get
Next, we shall prove that
By using inequality (4), we derive that
Then, by using the continuity of ψ and lower semi-continuity of ϕ, we find
As in Theorem 1.1 in , we notice that T has no periodic point.
Again, repeating the steps of Theorem 1.1 in , we obtain that
Using the continuity of ψ and lower semi-continuity 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  and hence we omit the details. □
Inspired by Theorem 1.2, one can state the following theorem.
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
By regarding the techniques in , 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 one-to-one. Now, define a map by . Since F is one-to-one 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. □
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 Isik-Turkoglu  is a consequence of our improved result, Theorem 2.1.
The authors declare that there is no conflict of interests regarding the publication of this article.
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
The authors thank anonymous referees for their remarkable comments, suggestions and ideas that helped to improve this paper.
Lakzian, H, Samet, B: Fixed points -weakly contractive mappings in generalized metric spaces. Appl. Math. Lett.. 25, 902–906 (2012). Publisher Full Text
Haghi, RH, Rezapour, S, Shahzad, N: Some fixed point generalizations are not real generalizations. Nonlinear Anal.. 74, 1799–1803 (2011). Publisher Full Text
Di Bari, C, Vetro, P: Common fixed points in generalized metric spaces. Appl. Math. Comput. (2012). Publisher Full Text