Fixed points by a new iteration method

Abstract

The following result is shown. If T T is a lipschitzian pseudo-contractive map of a compact convex subset E E of a Hilbert space into itself and x 1 {x_1} is any point in E E , then a certain mean value sequence defined by x n + 1 = α n T [ β n T x n + ( 1 − β n ) x n ] + ( 1 − α n ) x n {x_{n + 1}} = {\alpha _n}T[{\beta _n}T{x_n} + (1 - {\beta _n}){x_n}] + (1 - {\alpha _n}){x_n} converges strongly to a fixed point of T T , where { α n } \{ {\alpha _n}\} and { β n } \{ {\beta _n}\} are sequences of positive numbers that satisfy some conditions.