viscosity approximation methods with weak contraction for l-lipschitzian pseudocontractive self-mapping

in this paper, under the framework of real reflexive banach space which admits a weakly sequentially continuous duality mapping from e to e∗, we study the strong convergence of an implicit and an explicit composite viscosity approximation algorithm (i) and (ii), respectively for a pseudocontractive mapping t by using the weakly contractive mapping f as follows:

x_t,s = tf (x_t,s) + (1 − t)y_t,s

y_t,s = sx_t,s + (1 − s)tx_t,s (i) and

x_n+1 = α_nf (x_n) + (1 − α_n)y_n,

y_n = β_nx_n + (1 − β_n)tx_n, n ≥ 0. (ii)

our results unify, improve and complement several recent ones existing in the current literature.