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Existence and convergence of best proximity points. (English) Zbl 1105.54021
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 $d(x, T_x) = \text{dist} (A,B)$.

54H25Fixed-point and coincidence theorems in topological spaces
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
Full Text: DOI
[1] Kirk, W. A.; Reich, S.; Veeramani, P.: Proximinal retracts and best proximity pair theorems. Numer. funct. Anal. optim. 24, 851-862 (2003) · Zbl 1054.47040
[2] Kirk, W. A.; Srinivasan, P. S.; Veeramani, P.: Fixed points for mappings satisfying cyclical contractive conditions. Fixed point theory 4, 79-89 (2003) · Zbl 1052.54032