APPROXIMATION OF COMMON FIXED POINTS AND VARIATIONAL SOLUTIONS FOR ONE-PARAMETER FAMILY OF LIPSCHITZ PSEUDOCONTRACTIONS

Let X be a uniformly convex Banach space with a uniformly Gateaux differentiable norm, let C be a nonempty closed convex subset of X and let T= {T(t) : t is an element of G} be a one-parameter family of Lipschitz pseudocontractions on C such that each T(t) : C -> X satisfies the weakly inward condition. For any contraction f : C -> C, it is shown that the path t <-> x(t), t is an element of [0,1), in C, denoted by x(t) = alpha(t)T(t)x(t) + (1 - alpha t) f(xt) is continuous and strongly converges to a common fixed point of T, which is the unique solution of some variational inequality. On the other hand, if T= {T(t) : t is an element of G} is a family of uniformly Lipschitz pseudocontractive self-mappings on C, it is also shown that the iteration process: xo is an element of C, x(n+1) = beta(n)(alpha nTr(n) xn+ (1-alpha(n))x(n)) + (1-beta(n))f (x(n)), n >= 0, strongly converges to the common fixed point of T, which is the unique solution of the same variational inequality.