General Cesaro mean approximation methods for nonexpansive mappings in Hilbert spaces

Let C be a nonempty closed convex subset of a real Hilbert space H, f a contraction on C and A a strongly bounded linear operator on H with coefficient (gamma) over bar > 0. Consider a general Cesaro mean iterative method x(0) is an element of C, x(n+1) = alpha(n)gamma f(x(n)) + beta(n)x(n) + ((1 - beta(n)) I + alpha(n)A) 1/n +1 Sigma(n)(j=0) T(j)x(n), n >= 0, where {alpha(n)}, {beta(n)} are the sequences in [0,1] satisfying certain conditions, and T is a nonexpansive mapping of C into itself. It is proved that the sequence {x(n)} generated by above method, converges strongly to a point (x) over tilde is an element of F(T) which solves the variational inequality <(A - gamma f)(x) over tilde,(x) over tilde - x > <= 0, x is an element of F(T). The results presented in this paper generalize, extend and improve the corresponding results of Shimizu and Takahashi [1.1], Matsushita and Kuroiwa [8] and many others.