Vol. 53, No. 1, 1974

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Iterative techniques for approximation of fixed points of certain nonlinear mappings in Banach spaces

Robert L Thele

Vol. 53 (1974), No. 1, 259–266

Let D be a closed convex subset of a Banach space X, let T : D D be nonexpansive (that is, TxTyxy for every x,y D), and let Fλ = λT + (1 λ)I, where λ (0,1) and I denotes the identity on D. Several authors have found conditions under which the sequences of iterates {Tnx}, or the sequences {Fλnx}, converge strongly or weakly to fixed points of T for all x D. In this paper we establish conditions under which the sequences {F12nx} converge strongly to fixed points of T for all x in a neighborhood of the fixed point set of T; furthermore, our theorems hold for classes of mappings T more general than the class of nonexpansive mappings.

Mathematical Subject Classification 2000
Primary: 47H10
Received: 18 December 1972
Revised: 1 March 1974
Published: 1 July 1974
Robert L Thele