APPROXIMATION OF COMMON FIXED POINTS AND VARIATIONAL SOLUTIONS FOR ONE-PARAMETER FAMILY OF LIPSCHITZ PSEUDOCONTRACTIONS
Lu-Chuan Ceng, Adrian Petruşel, Silviu Gabriel Szentesi, Jen‐Chih Yao · 2010
Let X be a uniformly convex Banach space with a uniformly Gateaux dierentiable norm, let C be a nonempty closed convex subset of X and let T = {Tt : t 2 G} be a one-parameter family of Lipschitz pseudocontractions on C such that each Tt : C ! X satisfies the weakly inward condition. For any contraction f : C ! C, it is shown that the path t 7! xt, t 2 (0,1), in C, denoted by xt = tTtxt + (1 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 = {Tt : t 2 G} is a family of uniformly Lipschitz pseudocontractive self-mappings on C, it is also shown that the iteration process: x0 2 C, xn+1 = n( nTrnxn + (1 n)xn) + (1 n)f(xn), n 0, strongly converges to the common fixed point of T , which is the unique solution of the same varia- tional inequality.