CONVERGENCE OF THE PATH AND ITS DISCRETIZATION TO THE MINIMUM-NORM FIXED POINT OF PSEUDOCONTRACTIONS

Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C -> C be a Lipschitz pseudocontractive mapping with Fix(T) not equal empty set. In this paper, we first show that as t -> 0+, the path x -> x(t), t is an element of (0, 1), in C, defined by x(t) = (1 - beta)P-C [(1 - t)x(t)]+beta Tx(t) converges strongly to the minimum-norm fixed point of T. Subsequently, by discreting the path, we suggest an explicit method x(n+1) = (1 - beta(n))P-C[(1 - alpha(n))x(n)] + beta(n)Tx(n). Under some assumptions, we prove the sequence {x(n)} also converges strongly to the minimum-norm fixed point of T.