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Zbl 1105.54021
Eldred, A.Anthony; Veeramani, P.
Existence and convergence of best proximity points.
(English)
[J] J. Math. Anal. Appl. 323, No. 2, 1001-1006 (2006). ISSN 0022-247X

Let $A$ and $B$ be nonempty closed subsets of a complete metric space $(X,d)$ and $T:A \cup B \to A \cup B$ satisfying $T(A)\subset B$ and $T(B) \subset A$. Fixed point theorems for such mappings satisfying cyclic contractive conditions were given by {\it W. A. Kirk, P. S. Srinivasan} and {\it P. Veeramani} [Fixed Point Theory 4, No. 1, 79--89 (2003; Zbl 1052.54032)] and {\it I. A. Rus} [Ann. T. Popoviciu Seminar of Funct. Eq., Approx. and Convexity 3, 171--178 (2005)]. In this paper the authors extend some of the above results to the case when $A \bigcap B = \emptyset$ and for the best proximity points, i.e., $x \in A \cup B$ such that $d(x, T_x) = \text{dist} (A,B)$.
[Ioan A. Rus (Cluj-Napoca)]
MSC 2000:
*54H25 Fixed-point theorems in topological spaces
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces

Keywords: cyclic contraction; best proximity point; uniformly convex Banach space; strict convexity

Citations: Zbl 1052.54032

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