On the Mann Iterative Process

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. This content downloaded from 138.77.91.58 on Wed, 1. Introduction. A self-mapping T of a Banach space E is said to be nonexpansive provided 1Tx-Tyl ? 1x--yl for all x, y E E, and is said to be quasi-nonexpansive provided that if Tp =p then ITx-p ?| -11x-p || for all x E E (i.e., T is nonexpansive about each of its fixed points). Nonexpansive mappings are clearly quasi-non-expansive, 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 (1 Ix), T(O) =0, on El. 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 T is said to be quasi-nonexpansive on a subset C of E provided T maps C into C, and if p E C and Tp=p then 11Tx-pl -< ? lx-p 1 holds for all x E C. 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 Hilbert space and in uniformly convex and strictly convex Banach spaces. As corollaries, we obtain some results of M. A. Krasnosel'skil [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).