- Research Article
- Open Access
© The Author(s). 2010
- Received: 8 July 2009
- Accepted: 12 January 2010
- Published: 20 January 2010
- Banach Space
- Natural Number
- Differential Geometry
- Normed Space
- Nonempty Subset
As a generalization of Banach contraction principle, Kirk et al. proved, in 2003, the following fixed point result; see .
Let and be nonempty closed subsets of a metric space and a strictly increasing map. We say that is a cyclic -contraction map  whenever , and
In 2005, Petru el proved some periodic point results for cyclic contraction maps . Then, Eldered and Veeramani proved some results on best proximity points of cyclic contraction maps in 2006 . They raised a question about the existence of a best proximity point for a cyclic contraction map in a reflexive Banach space. In 2009, Al-Thagafi and Shahzad gave a positive answer to the question . More precisely, they proved some results on the existence and convergence of best proximity points of cyclic contraction maps defined on reflexive (and strictly convex) Banach spaces [2, Theorems 9, 10, 11, and 12]. They also introduced cyclic -contraction maps and raised the following question in .
In this paper, we provide a positive answer to the above question. For obtaining the answer, we use some results of .
First, we give the following extension of [4, Proposition 3.3] for cyclic -contraction maps, where is unbounded.
Since the proof of last result was classic, we presented it separately. Here, we provide our key result via a special proof which is a general case of Theorem 2.1.
Now by using this key result, we provide our main results which give positive answer to the question. Their proofs are basically due to Al-Thagafi and Shahzad . However, the crucial role is played by our key result. Weak convergence of to is denoted by .
Let be a strictly increasing map. Also, let and be nonempty subsets of a reflexive Banach space such that is weakly closed and a cyclic -contraction map. Then, there exists such that provided that one of the following conditions is satisfied
If , the result follows from [2, Theorem 1]. So, we assume that . For , define for all . By Theorem 2.2, the sequence is bounded. Since is reflexive and is weakly closed, the sequence has a subsequence such that as .
(b)By [2, Theorem 3], we have
Let be a strictly increasing map. Also, let and be nonempty closed and convex subsets of a reflexive and strictly convex Banach space and a cyclic -contraction map. If , then there exists a unique such that and .
Let be a strictly increasing map. Also, let and be nonempty subsets of a reflexive and strictly convex Banach space such that is closed and convex and a cyclic -contraction map. Then, there exists a unique such that and provided that one of the following conditions is satisfied
which is a contradiction. The uniqueness of follows as in the proof of [2, Theorem 8].
The authors express their gratitude to the referees for their helpful suggestions concerning the final version of this paper.
- Kirk WA, Srinivasan PS, Veeramani P: Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory 2003,4(1):79–89.MathSciNetMATHGoogle Scholar
- Al-Thagafi MA, Shahzad N: Convergence and existence results for best proximity points. Nonlinear Analysis. Theory, Methods & Applications 2009,70(10):3665–3671. 10.1016/j.na.2008.07.022MathSciNetView ArticleMATHGoogle Scholar
- Petruşel G: Cyclic representations and periodic points. Universitatis Babeş-Bolyai. Studia. Mathematica 2005,50(3):107–112.MathSciNetMATHGoogle Scholar
- Eldred AA, Veeramani P: Existence and convergence of best proximity points. Journal of Mathematical Analysis and Applications 2006,323(2):1001–1006. 10.1016/j.jmaa.2005.10.081MathSciNetView ArticleMATHGoogle Scholar
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