WEAK AND STRONG CONVERGENCE FOR QUASI-NONEXPANSIVE MAPPINGS IN BANACH SPACES

In this paper, we first show that the iteration {x(n)} defined by x(n+1) = P((1 - alpha(n))x(n) + alpha nTP[beta(n)Tx(n) + (1 - beta(n))x(n)]) converges strongly to some fixed point of T when E is a real uniformly convex Banach space and T is a quasi-nonexpansive non-self mapping satisfying Condition A, which generalizes the result due to Shahzad [11]. Next, we show the strong convergence of the Mann iteration process with errors when E is a real uniformly convex Banach space and T is a quasi-nonexpansive self-mapping satisfying Condition A, which generalizes the result due to Senter-Dotson [10]. Finally, we show that the iteration {x(n)} defined by x(n+1) = alpha(n)Sx(n)+beta T-n[alpha'(n)Sx(n)+beta'(n)Tx(n)+gamma'(n)v(n)]+ gamma(n)u(n) converges strongly to a cornmon fixed point of T and S when E is a real uniformly convex Banach space and T,S are two quasi-nonexpansive self-mappings satisfying Condition D, which generalizes the result due to Ghosh-Debnath [3].