Approximation of fixed points of strongly pseudocontractive mappings

Abstract:

Let $E$ be a real Banach space with a uniformly convex dual, and let $K$ be a nonempty closed convex and bounded subset of $E$. Let $T:K \to K$ be a continuous strongly pseudocontractive mapping of $K$ into itself. Let $\{ {c_n}\} _{n = 1}^\infty$ be a real sequence satisfying: (i) $0 < {c_n} < 1$ for all $n \geqslant 1$; (ii) $\sum \nolimits _{n = 1}^\infty {{c_n} = \infty }$; and (iii) $\sum \nolimits _{n = 1}^\infty {{c_n}b({c_n}) < \infty }$, where $b:[0,\infty ) \to [0,\infty )$ is some continuous nondecreasing function satisfying $b(0) = 0, b(ct) \leqslant cb(t)$ for all $c \geqslant 1$. Then the sequence $\{ {x_n}\} _{n = 1}^\infty$ generated by ${x_1} \in K$, \[ {x_{n + 1}} = (1 - {c_n}){x_n} + {c_n}T{x_n},\qquad n \geqslant 1,\] converges strongly to the unique fixed point of $T$. A related result deals with the Ishikawa iteration scheme when $T$ is Lipschitzian and strongly pseudocontractive.