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Generalized I-nonexpansive maps and best approximations in Banach spaces. (English) Zbl 1095.41017

Let E=(E,·) be a Banach space and C a subset of E. Let T,I:EE. T is called I-nonexpansive on C if Tx-TyIx-Iy for all x,yC. The set of fixed points of T (resp. I) is denoted by F(T) (resp. F(I)). The set P C (x ^)={yC:y-x ˜=dist(x ^,C)} is called the set of best approximants to x ^X from C.

In the case that T and I are commuting (ITx=TIx) on P C (x ^), G. Jungck and S. Sessa [Math. Jap. 42, No. 2, 249–252 (1995; Zbl 0834.54026)] proved that, under certain conditions, P C (x ^)F(T)F(I). Later, N. Shahzad [Tamkang J. Math. 32, No. 1, 51–53 (2001; Zbl 0978.41020)] extended this result to a class of noncommuting maps. In this paper it is proved the validity of this result for generalized I-nonexpansive maps.


MSC:
41A50Best approximation, Chebyshev systems
47H10Fixed point theorems for nonlinear operators on topological linear spaces