On the Mann iterative process

Introduction. A self-mapping F of a Banach space F is said to be nonexpansive provided \-Ty\\ ^ \-y\\ for all x, y e E, and is said to be quasi-nonexpansive provided that if Tp=p then \-p\\ = \-p\\ for all x e E (i. e. , Fis nonexpansive about each of its fixed points). Nonexpansive mappings are clearly quasi-nonexpansive, and linear quasi-nonexpansive mappings are nonexpansive; but it is easily seen that there exist nonlinear continuous quasi-nonexpansive mappings which are not nonexpansive, e. g. Tx = (x\2) sin (l), F (0) = 0, on E1. The concept of quasi-nonexpansiveness is closely related to some ideas which have been investigated recently by J. B. Diaz and F. T. Metcalf 2. A mapping F is said to be quasi-nonexpansive on a subset C of E provided F maps C into C, and if// e C and Tp=p then ||Fjc-//|| = \-p\\ holds for all xeC. In this paper, an iterative process introduced by W. R. Mann 7 is applied to the approximation of fixed points of quasi-nonexpansive mappings in Hubert space and in uniformly convex and strictly convex Banach spaces. As corollaries, we obtain some results of M. A. Krasnosel'skiï 6, H. Schaefer 12, F. E. Browder and W. V. Petryshyn 1, and M. Edelstein 5. An affirmative answer is obtained for a recent conjecture of C. L. Outlaw and C. W. Groetsch 10, and a partial affirmative answer is obtained for a conjecture of H. Schaefer (for which Z. Opial 9 has recently obtained another partial affirmation). 2. The Mann iterative process. Suppose A = anj is an infinite real matrix satisfying (Al) anj⁰ for all n, j, and an;=0 for j>n; (A2) 2"=iam=l for all n; (A3) limn an, = 0 for all / Suppose F is a linear space, C is a convex subset of E, Fis a mapping of C into C, and xx e C. Then the Mann iterative process M (xly A, T) is defined by i>" = Z"=i onjx}, xn+1 = Tvn, «=1, 2, 3,. . . . W. R. Mann 7 introduced this process and proved that in case F is a Banach space, and C is closed, and F is continuous, then the convergence of either xn or vn to a point y implies the convergence of the other to y, and also implies Ty=y. Mann's proof is easily extended to a locally convex Hausdorff linear topological space F, by using the regularity of the matrix A together with properties of the continuous pseudo-norms which generate the topology of E. We state this as our first result. Theorem 1. Suppose E is a locally convex Hausdorff linear topological space, C is a closed convex subset ofE, T: C-> C is continuous, Xx e C, and A = anj satisfies