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Best proximity pair theorems for relatively nonexpansive mappings. (English) Zbl 1213.47062
Let A,B be nonempty bounded closed convex subsets of a uniformly convex Banach space and T:ABAB be a map such that T(A)B, T(B)A, and Tx-Tyx-y for all xA and yB. The authors discuss the problem of finding an element x 0 A such that x 0 -Tx 0 =inf{a-b:aA,bB}. The proof of the existence of such a point x 0 is given without using Zorn’s lemma, as was the case in A. A. Eldred, W. A. Kirk and P. Veeramani’s result in [Stud. Math. 171, No. 3, 283–293 (2005; Zbl 1078.47013)].
47H10Fixed point theorems for nonlinear operators on topological linear spaces