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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 W. A. Kirk, P. S. Srinivasan and P. Veeramani [Fixed Point Theory 4, No. 1, 79–89 (2003; Zbl 1052.54032)] and 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 $$d(x, T_x) = \text{dist} (A,B)$$.

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects) 47H10 Fixed-point theorems
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##### References:
 [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
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