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. 25:902-906, 2012).
MSC:47H10, 54H25.
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1 Introduction and preliminaries
In 2012, Lakzian and Samet [1] 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 [1], Isik and Turkoglu [2] 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 [2] is a consequence of the main results of [1]. The obtained results are inspired by the techniques and ideas of, e.g., [3–11].
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:
-
(RM1)
if and only if ,
-
(RM2)
,
-
(RM3)
.
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 [1] proved the following fixed point theorem.
Theorem 1.1 [1]
Let be a Hausdorff and complete RMS, and let be a self-map satisfying
for all , where and . Then T has a unique fixed point in X.
Lemma 1.1 [3]
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 [2]
Let be a Hausdorff and complete RMS, and let be self-mappings such that , and is a closed subspace of X, and that the following condition holds:
for all , where , , , and these mappings satisfy the condition
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 [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 semi-continuity.
Theorem 2.1 Let be a Hausdorff and complete RMS, and let be a self-map satisfying
for all , where and . Then T has a unique fixed point in X.
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 semi-continuity 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 semi-continuity 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 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 [1] and hence we omit the details. □
Inspired by Theorem 1.2, one can state the following theorem.
Theorem 2.2 Let be a Hausdorff and complete RMS, and let be self-mappings such that
for all , where , , and these mappings satisfy the condition
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.
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
Define by , . Then 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 2.4 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. □
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 Isik-Turkoglu [2] is a consequence of our improved result, Theorem 2.1.
References
Lakzian H, Samet B:Fixed points -weakly contractive mappings in generalized metric spaces. Appl. Math. Lett. 2012, 25: 902–906. 10.1016/j.aml.2011.10.047
Isik H, Turkoglu D:Common fixed points for -weakly contractive mappings in generalized metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 131
Haghi RH, Rezapour S, Shahzad N: Some fixed point generalizations are not real generalizations. Nonlinear Anal. 2011, 74: 1799–1803. 10.1016/j.na.2010.10.052
Aydi H, Karapınar E, Samet B: Remarks on some recent fixed point theorems. Fixed Point Theory Appl. 2012., 2012: Article ID 76
Kirk WA, Shahzad N: Generalized metrics and Caristi’s theorem. Fixed Point Theory Appl. 2013., 2013: Article ID 129
Arshad M, Ahmad J, Karapınar E: Some common fixed point results in rectangular metric spaces. Int. J. Anal. 2013., 2013: Article ID 307234
Sarma IJ, Rao JM, Rao SS: Contractions over generalized metric spaces. J. Nonlinear Sci. Appl. 2009, 2(3):180–182.
Arshad M, Ahmad J, Vetro C: On a theorem of Khan in a generalized metric space. Int. J. Anal. 2013., 2013: Article ID 852727
Chen C-M, Sun WY:Periodic points and fixed points for the weaker -contractive mappings in complete generalized metric spaces. J. Appl. Math. 2012., 2012: Article ID 856974
Chen C-M: Common fixed point theorems in complete generalized metric spaces. J. Appl. Math. 2012., 2012: Article ID 945915
Di Bari C, Vetro P: Common fixed points in generalized metric spaces. Appl. Math. Comput. 2012. 10.1016/j.amc.2012.01.010
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The authors thank anonymous referees for their remarkable comments, suggestions and ideas that helped to improve this paper.
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Bilgili, N., Karapınar, E. & Turkoglu, D. A note on common fixed points for -weakly contractive mappings in generalized metric spaces. Fixed Point Theory Appl 2013, 287 (2013). https://doi.org/10.1186/1687-1812-2013-287
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DOI: https://doi.org/10.1186/1687-1812-2013-287