A note on common fixed points for -weakly contractive mappings in generalized metric spaces
© Bilgili et al.; licensee Springer. 2013
Received: 20 May 2013
Accepted: 18 September 2013
Published: 8 November 2013
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).
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.
if and only if ,
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 semi-continuous.
Lakzian and Samet  proved the following fixed point theorem.
Theorem 1.1 
for all , where and . Then T has a unique fixed point in X.
Lemma 1.1 
Let X be a nonempty set and be a function. Then there exists a subset such that and is one-to-one.
Definition 1.2 Let X be a nonempty set, and let be self-mappings. 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 
Then T and F have a unique coincidence point in X. Moreover, if T and F are weakly compatible, then T and F have 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.
for all , where and . Then T has a unique fixed point in X.
by replacing and in inequality (4).
which implies that and then . Consequently, we have as .
which implies that . So, we conclude that and hence as .
As in Theorem 1.1 in , we notice that T has no periodic point.
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.
Then T has a unique fixed point in X.
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 2.3 Theorem 2.2 is a consequence of Theorem 2.1.
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 , we conclude the following result.
Theorem 2.4 Theorem 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 2.5 Theorem 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 thank anonymous referees for their remarkable comments, suggestions and ideas that helped to improve this paper.
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