Abstract
Let C be a non-empty closed convex subset of a reflexive and strictly convex Banach space E which also has a weakly continuous duality map J(phi)(x) with the gauge phi. Let S and T be non-expansive mappings from C into itself such that F = F(S) boolean AND F(T) not equal 0. Let {alpha(n)} and (beta(n)} be sequences in (0, 1). Let {x(n)} be a sequence defined by
{x0 is an element of C, y(n) = beta(n)Sx(n) + (1 - beta(n))Tx(n), x(n+1) = alpha(n)u + (1 - alpha(n))y(n), n >= 0,
where u is an element of C is a given point. Assume that the following restrictions imposed on the control sequences are satisfied:
(a) Sigma(infinity)(n=0)a(n) = infinity, lim(n ->infinity) alpha(n) = 0;
(b) Sigma(infinity)(n=0)vertical bar a(n+1) - a(n)vertical bar < infinity, Sigma(infinity)(n=0)vertical bar beta(n+1) - beta(n)vertical bar < infinity;
(c) lim(n ->infinity) beta(n) = beta is an element of (0, 1).
Then the sequence {x} converges strongly to x* is an element of F, where x* = Q(u) and Q : C -> F is the unique sunny non-expansive retraction from C onto F.