Approximation to Fixed Points of Generalized Nonexpansive Mappings

Let K be a convex subset of a uniformly convex Banach space. It is proved that if K is compact, then the fixed points of a continuous generalized nonexpansive self-mapping T on K can be approximated by the iterates of T, with t B (0,1), T,(x) = (1 t)x + tT(x), x e K; T, is asymptotically regular if T has a fixed point. Let (X, d) be a (nonempty) metric space. A function a of X X X into [0, oo) is symmetric if a(x,y) = a(y,x) for all x,y in X. Let Abe a self-mapping on X. T is generalized nonexpansive if there exist symmetric functions a,-, i = 1, 2, ..., 5, of X X X into [0, oo) such that (0 sup l 2 0Lt(x,y): x,y E x\ < 1 N=l -* and for all x, >> in X, d(T(x), T(y)) < a, d(x,y) + a2c/(x, A(^)) + a3d(y, T(x)) (2) + a4<7(x, T(x)) + a5d(y, T(y)), where a, = at(x,y). It is clear that T is generalized nonexpansive if it is nonexpansive (d(T(x), T(y)) < d(x,y), x,y £ X. R. Kannan first considered those T which satisfy (2) with ax — a2 = a3 = 0 and a4 = a5 < \ [5]-[9]. S. Reich considered those A which satisfy (2) with a2 = a3 = 0 and with constants ax, a4, a5 [11]—[13]. Recently, G. Hardy and T. Rogers considered those T which satisfy (2) with constants a,'s [4]. In [3], K. Goebel, W. A. Kirk and Tawfik N. Shimi proved that A has a fixed point if A is a weakly compact convex subset of a uniformly convex Banach space and if 77 satisfies (2) with constant coefficients. Other related work can be found in [ 15]—[19]. In this paper, we shall investigate the theory of approximations to fixed points of generalized nonexpansive mappings. 1. Asymptotic regular mappings. Let A be a self-mapping on a metric space Received by the editors July 23, 1973 and, in revised form, November 1, 1974. AMS (MOS) subject classifications (1970). Primary 47H10; Secondary 54H25. 1 This research was partially supported by the National Research Council of Canada Grant A8518. © American Mathematical Society 1976 93 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use