Best Proximity Points of Cyclic -Contractions on Reflexive Banach Spaces
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Research Article Best Proximity Points of Cyclic ϕ-Contractions on Reflexive Banach Spaces Sh. Rezapour,1 M. Derafshpour,1 and N. Shahzad2 1 2
Department of Mathematics, Azarbaidjan University of Tarbiat Moallem, Azarshahr, Tabriz, Iran Department of Mathematics, King AbdulAziz University, P.O. Box 80203, Jeddah 21859, Saudi Arabia
Correspondence should be addressed to N. Shahzad, [email protected] Received 8 July 2009; Revised 4 January 2010; Accepted 12 January 2010 Academic Editor: Juan Jose Nieto Copyright q 2010 Sh. Rezapour et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We provide a positive answer to a question raised by Al-Thagafi and Shahzad Nonlinear Analysis, 70 2009, 3665-3671 about best proximity points of cyclic ϕ-contractions on reflexive Banach spaces.
1. Introduction As a generalization of Banach contraction principle, Kirk et al. proved, in 2003, the following fixed point result; see 1. Theorem 1.1. Let A and B be nonempty closed subsets of a complete metric space X, d. Suppose that T : A ∪ B → A ∪ B is a map satisfying T A ⊆ B, T B ⊆ A and there exists k ∈ 0, 1 such that dT x, T y ≤ kdx, y for all x ∈ A and y ∈ B. Then, T has a unique fixed point in A ∩ B. Let A and B be nonempty closed subsets of a metric space X, d and ϕ : 0, ∞ → 0, ∞ a strictly increasing map. We say that T : A ∪ B → A ∪ B is a cyclic ϕ-contraction map 2 whenever T A ⊆ B, T B ⊆ A and d T x, T y ≤ d x, y − ϕ d x, y ϕdA, B
1.1
for all x ∈ A and y ∈ B, where dA, B : inf{dx, y : x ∈ A, y ∈ B}. Also, x ∈ A ∪ B is called a best proximity point if dx, T x dA, B. As a special case, when ϕt 1 − αt, in which α ∈ 0, 1 is a constant, T is called cyclic contraction. In 2005, Petrusel proved some periodic point results for cyclic contraction maps 3. Then, Eldered and Veeramani proved some results on best proximity points of cyclic
2
Fixed Point Theory and Applications
contraction maps in 2006 4. 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 2. 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 ϕ-contraction maps and raised the following question in 2. Question 1. It is interesting to ask whether Theorems 9 and 10 resp., Theorems 11 and 12 hold for cyclic ϕ-contraction maps where the space is only reflexive resp., reflexive and strictly convex Banach space. In this paper, we provide a positive answer to the above question. For obtaining the answer, we use some results of 2.
2. Main Results First, we give the following extension of 4,
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