Strong Convergence of Averaged Approximants for Lipschitz Pseudocontractive Maps

Let T be a Lipschitzian pseudocontractive self-mapping of a closed convex and bounded subset K of a Banach space E which is both uniformly convex and q-uniformly smooth such that the set FŽT. of fixed points of T is nonempty. Then FŽT. is a sunny nonexpansive retract of K. If U is the sunny nonexpansive retraction of K onto FŽT., is any point of K, and an4 n 0 a real sequence in Ž0, 1 , then the sequence xn4 n 0 in K defined by 1 n xn an Ž1 an.n 1 Ý Ž1 aj.I ajT xn j 0 for n 0, 1, 2, . . . , converges strongly to U . No compactness assumption is made on K.