STRONG CONVERGENCE THEOREMS FOR STRICTLY PSEUDOCONTRACTIVE MAPPINGS OF BROWDER-PETRYSHYN TYPE

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 $T: K \rightarrow K$ be a strictly pseudocontractive mapping in the sense of F. E. Browder and W.V. Pstryshyn [1]. Let $\{u_n\}$ be a bounded sequence 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: $y_n = (1-\beta_n) x_n + \beta_n Tx_n$, $x_{n+1} = (1-\alpha_n) x_n + \alpha_n Ty_n + u_n$, $n \geq 1$. Sufficient and necessary conditions for the strong convergence $\{x_n\}$ to a fixed point of $T$ is established.