Approximation of fixed points of strongly pseudocontractive maps without Lipschitz assumption

Abstract:

In the present paper, the following result is shown: Let $X$ be a real Banach space with a uniformly convex dual $X^*$, and let $K$ be a nonempty closed convex and bounded subset of $X$. Assume that $T:\,K\rightarrow K$ is a continuous strong pseudocontraction. Let $\{\alpha _n\}^{\infty }_{n=1}$ and $\{\beta _n\}^{\infty }_{n=1}$ be two real sequences satisfying (i) $0<\alpha _n,\,\beta _n<1$ for all $n\ge 1$; (ii) $\sum _{n=1}^{\infty }\alpha _n=\infty$; and (iii) $\alpha _n \rightarrow 0,\, \beta _n \rightarrow 0$ as $n\rightarrow \infty .$ Then the Ishikawa iterative sequence $\{x_n\}_{n=1}^{\infty }$ generated by \begin{equation*} \mathrm {(I)} \quad \left \{ \begin {array}{l} x_1\in K,\\ x_{n+1}=(1-\alpha _n)x_n+\alpha _nTy_n,\\ y_n=(1-\beta _n)x_n+\beta _nTx_n,\,n\geq 1, \end{array} \right . \end{equation*} converges strongly to the unique fixed point of $T$.