Let H be a real Hilbert space. Consider on H a nonexpansive mapping T with a fixed point, a contraction f with coefficient 0 < α < 1, and a strongly positive linear bounded operator A with coefficientγ > 0. Let 0 < γ <γ /α. It is proved that the sequence {x n } generated by the iterative method x n+1 = (I − α n A)T x n + α n γf (x n ) converges strongly to a fixed pointx ∈ Fix(T ) which solves the variational inequality (γf − A)x, x −x 0 for x ∈ Fix(T ).

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.

No abstract

We introduce an iterative method for finding a common element of the set of solutions of an equilibrium problem and of the set of fixed points of a finite family of nonexpansive mappings in a Hilbert space. We prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality, which is the optimality condition for a minimization problem.

No abstract