STRONG CONVERGENCE THEOREMS OF ISHIKAWA ITERATION PROCESS WITH ERRORS FOR FIXED POINTS OF LIPSCHITZ CONTINUOUS MAPPINGS IN BANACH SPACES

Abstract

Let $q \gt 1$ and $E$ be a real $q$-uniformly smooth Banach space, $K$ be a nonempty closed convex subset of $E$ and $T : K \to K$ be a Lipschitz continuous mapping. Let $\{u_n\}$ and $\{v_n\}$ be bounded sequences in $K$ and $\{\alpha_n\}$ and $\{\beta_n\}$ be real sequences in $[0,1]$ satisfying some restrictions. Let $\{x_n\}$ be the sequence generated from an arbitrary $x_1 \in K$ by the Ishikawa iteration process with errors: $y_n = (1 − \beta n) x_n + \beta_n Tx_n + v_n$, $x_{n+1} = (1 − \alpha_n) x_n + \alpha_n Ty_n + u_n$, $n \geq 1$. Sufficient and necessary conditions for the strong convergence $\{x_n\}$ to a fixed point of $T$ is established.