科研之友
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摘要
Let X be a uniformly smooth Banach space and A be an m- accretive operator on X with A(-1)(0) not equal empty set. Assume that F : X -> X is delta-strongly accretive and lambda-strictly pseudocontractive with delta + lambda > 1. This article proposes hybrid viscosity approximation methods which combine viscosity approximation methods with hybrid steepest-descent methods. For each t is an element of (0, 1) and each integer n >= 0, let {x(t,n)} be defined by x(t,n) = tf (x(t,n)) + (1 - t)[J(rn) x(t,n) - theta(t) F (J(rn) x(t,n))] where f : X -> X is a contractive map, {r(n)} subset of [epsilon, infinity) for some epsilon > 0 and {theta(t) : t is an element of (0, 1)} subset of [0, 1) with lim(t -> 0) theta t/t = 0. We deduce that as t -> 0, {x(t,n)} converges strongly to a zero p of A, which is a unique solution of some variational inequality. On the other hand, given a point x(0) is an element of X and given sequences {lambda(n)}, {mu(n)} in [0, 1], {alpha(n)}, {beta(n)} in (0, 1], let the sequence {x(n)} be generated by {y(n) = alpha(n)x(n) + (1 - alpha(n)) J(rn) x(n), x(n+1) = beta(n)f (x(n)) + (1 - beta(n)) [J(rn) y(n) - lambda(n)mu F-n (J(rn) y(n))], for all(n) >= 0. It is proven that under appropriate conditions {x(n)} converges strongly to the same zero p of A. The results presented here extend, improve and develop some very recent theorems in the literature to a great extent.
