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A best proximity point theorem for weakly contractive non-self-mappings. (English) Zbl 1228.54046

Let A,B be nonempty subsets of a metric space (X,d). A mapping T:AB is called weakly contractive if there exists a continuous nondecreasing function ψ:[0,)[0,) with ψ(0)=0,ψ(t)>0,t>0, and lim t ψ(t)=0, such that d(x,y)d(x,y)-ψ(d(x,y)) for all (x,y)A×B. If B=A and ψ(t)=(1-k)t for some 0<k<1, then T is a k-contraction on A. It was proved by Y. Alber and S. Guerre-Delabriere [Oper. Theory, Adv. Appl. 98, 7–22 (1997; Zbl 0897.47044)] that, if Y is a closed convex set of a Hilbert space, then every weakly contractive mapping T:YY 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 (X,d).

The author extends the above result of Rhoades to weakly contractive mappings T:AB. A best proximity point for T is a point x * A such that d(x * ,Tx * )=d(A,B)(=inf{d(x,y):xA,yB}). Recall that Ky Fan [Math. Z. 112, 234–240 (1969; Zbl 0185.39503)] proved that, if A is a nonempty compact convex subset of a normed space X, then for every continuous mapping T:AX there exists xA such that x-Tx=d(Tx,A)(=inf{Tx-a:aA}). One says that the pair A,B of sets has the property P if d(x 1 ,y 1 )=d(x 2 ,y 2 )=d(A,B) implies d(x 1 ,x 2 )=d(y 1 ,y 2 ) for all x 1 ,x 2 A and y 1 ,y 2 B. Denote A 0 ={xA:yB,d(x,y)=d(A,B)} and B 0 ={yB:xA,d(x,y)=d(A,B)}.

The main result of the paper (Theorem 3.1) asserts that, if A,B are subsets of a complete metric space (X,d) satisfying the property P and such that A 0 , then for every weakly contractive mapping T:AB satisfying T(A 0 )B 0 there exists a unique point x * A such that d(x * ,Tx * )=d(A,B).

54H25Fixed-point and coincidence theorems in topological spaces
54E40Special maps on metric spaces
54E50Complete metric spaces
41A65Abstract approximation theory
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