A Note on Approximating Fixed Points of Pseudocontractive Mappings

Abstract:

      Let $K$ be a nonempty bounded closed convex subset of a real reflexive Banach space $E$ with a uniformly Gateaux differentiable norm. Let $T:K\rightarrow K$ be a uniformly continuous pseudocontractive mapping. Suppose every closed convex and bounded subset of $K$ has the fixed point property for nonexpansive mappings. Let $\{\lambda_n\}\subset (0,\frac{1}{2}]$ be a sequence satisfying the conditions: (i) $\lim_{n\rightarrow \infty}\lambda_n=0$; (ii) $\sum_{n=0}^{\infty}\lambda_n=\infty$. Let the sequence $\{x_n\}$ be generated from arbitrary $x_1\in K$ by $x_{n+1}=(1-\lambda_n)x_n+\lambda_nTx_n-\lambda_n(x_n-x_1)$, $n\geq 1$. Suppose $\lim_{n\rightarrow \infty}\|x_n-Tx_n\|=0$. Then $\{x_n\}$ converges strongly to a fixed point of $T$.