Let be nonempty subsets of a metric space . A mapping is called weakly contractive if there exists a continuous nondecreasing function with and , such that for all . If and for some , then is a -contraction on . It was proved by Y. Alber and S. Guerre-Delabriere [Oper. Theory, Adv. Appl. 98, 7–22 (1997; Zbl 0897.47044)] that, if is a closed convex set of a Hilbert space, then every weakly contractive mapping has a unique fixed point. The result was extended by B. E. Rhoades [Nonlinear Anal., Theory Methods Appl. 47, No. 4, 2683–2693 (2001; Zbl 1042.47521)] to weakly contractive mappings on an arbitrary complete metric space .
The author extends the above result of Rhoades to weakly contractive mappings . A best proximity point for is a point such that . Recall that Ky Fan [Math. Z. 112, 234–240 (1969; Zbl 0185.39503)] proved that, if is a nonempty compact convex subset of a normed space , then for every continuous mapping there exists such that . One says that the pair of sets has the property if implies for all and . Denote and .
The main result of the paper (Theorem 3.1) asserts that, if are subsets of a complete metric space satisfying the property and such that , then for every weakly contractive mapping satisfying there exists a unique point such that .