- Open Access
Fixed point theory for generalized Ćirić quasi-contraction maps in metric spaces
© Kiany and Amini-Harandi; licensee Springer 2013
- Received: 16 June 2012
- Accepted: 13 January 2013
- Published: 6 February 2013
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.
1 Introduction and preliminaries
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  stands at the top.
for any . Then T has a unique fixed point and for any , the sequence converges to .
For other generalizations of the above theorem, see  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.
for any , where is a mapping.
As the following simple example due to Sastry and Naidu  shows, Theorem 1.1 is not true for generalized Ćirić quasi-contraction maps even if we suppose α is continuous and increasing.
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  and the references therein.
Now, we are ready to state our main result.
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 .
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.
and so (note that ). □
The following example shows that in the statement of Theorem 2.3, the condition for each is necessary.
for each , but T is fixed point free.
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 .
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.
Thus, is also a coincidence point of T and S. By the uniqueness of a coincidence point of T and S, we get . □
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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