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Fixed point theory for generalized Ćirić quasi-contraction maps in metric spaces
Fixed Point Theory and Applications volume 2013, Article number: 26 (2013)
Abstract
In this paper, we first give a new fixed point theorem for generalized Ćirić quasi-contraction maps in generalized metric spaces. Then we derive a common fixed point result for quasi-contractive type maps. Some examples are given to support our results. Our results extend and improve some fixed point and common fixed point theorems in the literature.
MSC:47H10.
1 Introduction and preliminaries
The well-known Banach fixed point theorem asserts that if is a complete metric space and is a map such that
where , then T has a unique fixed point and for any , the sequence converges to .
In recent years, a number of generalizations of the above Banach contraction principle have appeared. Of all these, the following generalization of Ćirić [1] stands at the top.
Theorem 1.1 Let be a complete metric space. Let be a Ćirić quasi-contraction map, that is, there exists such that
for any . Then T has a unique fixed point and for any , the sequence converges to .
For other generalizations of the above theorem, see [2] and the references therein.
2 Main results
Let X be a nonempty set and let be a mapping. If d satisfies all of the usual conditions of a metric except that the value of d may be infinity, we say that is a generalized metric space.
We now introduce the concept of a generalized Ćirić quasi-contraction map in generalized metric spaces.
Definition 2.1 Let be a generalized metric space. The self-map is said to be a generalized Ćirić quasi-contraction if
for any , where is a mapping.
As the following simple example due to Sastry and Naidu [3] shows, Theorem 1.1 is not true for generalized Ćirić quasi-contraction maps even if we suppose α is continuous and increasing.
Example 2.2 Let with the usual metric, be given by . Define by . Then, clearly, α is continuous and increasing, and
for each , but T has no fixed point.
Now, a natural question is what further conditions are to be imposed on T or α to guarantee the existence of a fixed point for T? For some partial answers to this question and application of quasi-contraction maps to variational inequalities, see [4] and the references therein.
Now, we are ready to state our main result.
Theorem 2.3 Let be a complete generalized metric space. Let be a generalized Ćirić quasi-contraction map such that α satisfies
Assume that there exists an with the bounded orbit, that is, the sequence is bounded. Furthermore, suppose that for each . Then T has a fixed point and . Moreover, if is a fixed point of T, then either or .
Proof If for some , , then for . Thus, is a fixed point of T, the sequence is convergent to , and we are finished (note that for each ). So, we may assume that for each . Now, we show that there exists such that
On the contrary, assume that
for some subsequence of . Since by our assumption the sequence is bounded, then the subsequence is bounded too, and so, by passing to subsequences if necessary, we may assume that it is convergent. Let . Then from (2.1), we have , a contradiction. Thus, (2.1) holds.
Now, we show that is a Cauchy sequence. To prove the claim, we first show by induction that for each ,
where K is a bound for the bounded sequence . If then, we get
Thus, (2.2) holds for . Suppose that (2.2) holds for each , and we show that it holds for . Since T is a generalized Ćirić quasi-contraction map, then we have
where
It is trivial that (2.2) holds if . Now, suppose that . In this case, we have
where
Again, it is trivial that (2.2) holds if or . If , then
By the assumption of induction,
Hence,
If , then
If , then
Therefore, by continuing this process, we see that (2.2) holds for each . From (2.2), we deduce that is a Cauchy sequence and since is a generalized complete metric space, then there exists an such that . Now, we show that is a fixed point of T. To show the claim, we first show that there exists such that for each . On the contrary, assume that for some subsequence . Since , then from the above, we get , a contradiction. Since T is a generalized Ćirić quasi-contraction, then we have
Then we have
which yields , and so (note that and by our assumptions). Now, let us assume that and are fixed points of T such that . Then
and so (note that ). □
The following example shows that in the statement of Theorem 2.3, the condition for each is necessary.
Example 2.4 Let , and let . Let be given by and . Then
for each , but T is fixed point free.
Example 2.5 Let , for each , for each and let . Then is a complete generalized metric space. Let be given by for each and . Define by for each and . Then we have
and for each . Thus, all of the assumptions of Theorem 2.3 are satisfied, and so T has a unique fixed point ( is a unique fixed point of T). But we cannot invoke the above mentioned theorem of Ćirić to show the existence of a fixed point for T.
To prove the following common fixed point result, we use the technique in [5].
Corollary 2.6 Let be a complete metric space and let the self-maps T and S satisfy the contractive condition
for each , where α satisfies for each . If and SX is a complete subset of X, then T and S have a unique coincidence point in X. Moreover, if T and S are weakly compatible (i.e., they commute at their coincidence points), then T and S have a unique common fixed point.
Proof It is well known that there exists such that and is one-to-one. Now, define a map by . Since S is one-to-one on E, U is well defined. Note that
for all . Since is complete, by using Theorem 2.3, there exists such that . Then , and so T and S have a coincidence point, which is also unique. Since and T and S commute, then we have
Thus, is also a coincidence point of T and S. By the uniqueness of a coincidence point of T and S, we get . □
References
Ćirić LB: A generalization of Banach’s contraction principle. Proc. Am. Math. Soc. 1974, 45(2):267–273.
Amini-Harandi A: Fixed point theory for set-valued quasi-contraction maps in metric spaces. Appl. Math. Lett. 2011, 24: 1791–1794. 10.1016/j.aml.2011.04.033
Sastry KPR, Naidu SVR: Fixed point theorems for generalized contraction mappings. Yokohama Math. J. 1980, 28: 15–19.
Ćirić L, Hussain N, Cakić N: Common fixed points for Ćirić type f -weak contraction with applications. Publ. Math. (Debr.) 2010, 76: 31–49.
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
Acknowledgements
The authors are grateful to the referees for their helpful comments leading to improvement of the presentation of the work. The second author acknowledge that this research was partially carried out at IPM-Isfahan Branch. The second author was partially supported by a grant from IPM (No. 91470412) and by the Center of Excellence for Mathematics, University of Shahrekord, Iran.
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Kiany, F., Amini-Harandi, A. Fixed point theory for generalized Ćirić quasi-contraction maps in metric spaces. Fixed Point Theory Appl 2013, 26 (2013). https://doi.org/10.1186/1687-1812-2013-26
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DOI: https://doi.org/10.1186/1687-1812-2013-26