Strong convergence of three-step iteration methods for a countable family of generalized strict pseudocontractions in Hilbert spaces

Abstract

Abstract In this paper, we introduce a new class of generalized strict pseudocontractions in a real Hilbert space, and we consider a three-step Ishikawa-type iteration method { z n = ( 1 − γ n ) x n + γ n T n x n , y n = ( 1 − β n ) x n + β n T n z n , x n + 1 = ( 1 − α n ) x n + α n T n y n , for finding a common fixed point of a countable family { T n } of uniformly Lipschitz generalized λ n -strict pseudocontractions. Under mild conditions imposed on the parameter sequences { α n } , { β n } and { γ n } , we prove the strong convergence of { x n } to a common fixed point of a countable family { T n } of uniformly Lipschitz generalized strict pseudocontractions. On the other hand, we also introduce three-step hybrid viscosity approximation method for finding a common fixed point of a countable family { T n } of uniformly Lipschitz generalized λ n -strict pseudocontractions with λ n = 0 , i.e., a countable family { T n } of uniformly Lipschitz pseudocontractions. Under appropriate conditions we derive the strong convergence results for this method. The results presented in this paper improve and extend the corresponding results in the earlier and recent literature. MSC:47H06, 47H09, 47J20, 47J30.