On the Mann iterative process
W. G. Dotson · Transactions of the American Mathematical Society · 1970
Introduction.A self-mapping F of a Banach space F is said to be nonexpansive provided \\Tx-Ty\\ ^ \\x-y\\ for all x, y e E, and is said to be quasi-nonexpansive provided that if Tp=p then \\Tx-p\\ = \\x-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\x), 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-//|| = \\x-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) anj^0 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