We prove a common fixed-point theorem generalizing results of Dotson and Habiniak. Using this result, we extend, generalize, and unify well known results on fixed points and common fixed points of best approximation.
1996 Academic Press, Inc.
1. Introduction
Let X be a normed linear space, D and M subsets of X, I and T selfmaps of X, F(I, T) the set of common fixed points of I and T, and F(I) the set of fixed points of I. I and T commute on
&Tx&Ty& &x&y&] for every x, y # D. T is I-contraction on D if &Tx&Ty& k&Ix&Iy& for every x, y # D and some k # [0, 1). D is q-starshaped if there exists q # D such that (1&k) q+kx # D for all x # D and all k # (0, 1). D is convex if it is q-starshaped for every q # D. Let B M ( p) :=[x # M: &x& p&=$( p, M)] be the set of best M-approximants to p # X, where $( p, M)=inf z # M &z& p&, and let
Brosowski [1] proved that if T is nonexpansive with p # F(T ), T(M)/M and B M ( p) is nonempty, compact and convex, then T has a fixed point in B M ( p). Subrahmanyam [11] replaced the requirement that B M ( p) is nonempty by the assumption that M is a finite-dimensional subspace of X. Singh [8] noted that Brosowski's result remains true if B M ( p) is only q-starshaped. Singh [9] noted that the nonexpansiveness of T on B M ( p) _ [ p] is enough for his earlier result. Hicks and Humphries [4] noted that Singh's first result remains true if T(M)/M is replaced by T( M)/M, where M is the boundary of M in X. Smoluk [10] noted that the finite-dimensionality of M in Subrahmanyam's result can be replaced by the assumptions that T is linear and that Cl(T(D)) is compact for every bounded subset D of M. Habiniak [3] removed the linearity of T from Smoluk's result. Sahab, Khan and Sessa [7] unified and generalized the result of Hicks and Humphries and the results of Singh by the following: article no. 0045 318