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 pseudo-contraction 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.