Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces

Abstract

Let C be a closed convex subset of a real Hilbert space H and assume that T is a κ-strict pseudocontraction on C with a fixed point, for some 0 κ < 1. Given an initial guess x 0 ∈ C and given also a real sequence {α n } in (0, 1). The Mann's algorithm generates a sequence {x n } by the formula: x n+1 = α n x n + (1 − α n )T x n , n 0. It is proved that if the control sequence {α n } is chosen so that κ < α n < 1 and ∞ n=0 (α n − κ)(1 − α n ) = ∞, then {x n } converges weakly to a fixed point of T . However this convergence is in general not strong. We then modify Mann's algorithm by applying projections onto suitably constructed closed convex sets to get an algorithm which generates a strong convergent sequence. This result extends a recent result of Nakajo and Takahashi [K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372-379] from nonexpansive mappings to strict pseudo-contractions.

Viscosity approximation methods for the multiple-set split equality common fixed-point problems of demicontractive mappings

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