Existence and convergence of best proximity points. (English) Zbl 1105.54021

Eldred, A. Anthony; Veeramani, P.

J. Math. Anal. Appl. 323, No. 2, 1001-1006 (2006).

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 \(d(x, T_x) = \text{dist} (A,B)\).

Reviewer: Ioan A. Rus (Cluj-Napoca)

MSC:

54H25	Fixed-point and coincidence theorems (topological aspects)
47H10	Fixed-point theorems

Keywords:

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

Citations:

Zbl 1052.54032

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