ISHIKAWA ITERATION WITH ERRORS FOR APPROXIMATING FIXED POINTS OF STRICTLY PSEUDOCONTRACTIVE MAPPINGS OF BROWDER-PETRYSHYN TYPE

Abstract

Let $q \gt 1$ and $E$ be a real $q−$uniformly smooth Banach space. Let $K$ be a nonempty closed convex subset of $E$ and $T : K \to K$ be a strictly pseudocontractive mapping in the sense of F. E. Browder and W. V. Petryshyn [1]. Let $\{u_n\}$ be a bounded sequence in $K$ and $\{\alpha_n\}, \{\beta_n\}, \{\gamma_n\}$ be real sequences in $[0,1]$ satisfying some restrictions. Let $\{x_n\}$ be the bounded sequence in $K$ generated from any given $x_1 \in K$ by the Ishikawa iteration method with errors: $y_n = (1 − \beta_n) x_n + \beta_n Tx_n$, $x_{n+1} = (1 − \alpha_n − \gamma_n) x_n + \alpha_n Ty_n + \gamma_n u_n$, $n \geq 1$. It is shown in this paper that if $T$ is compact or demicompact, then $\{x_n\}$ converges strongly to a fixed point of $T$.