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 E (0, 1), 4(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, cx) is symmetric if a(x,y) = a(y, x) for all x,y in X. Let T be 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 (1) sup{ ai(x,y): x,y E X}< and for all x, y in X, (2) d(T(x), T(y)) < a, d(x,y) + a2 d(x, T(y)) + a3 d(y, T(x)) + a4d(x, T(x)) + a5d(y, T(y)), where ai = ai(x,y). It is clear that T is generalized nonexpansive if it is nonexpansive (d(T(x), T(y)) < d(x,y), x,y E X. R. Kannan first considered those T which satisfy (2) with a, = a2 = a3 = 0 and a4 = a5 < 2 [5]-[9]. S. Reich considered those T which satisfy (2) with a2 = a3 = 0 and with constants a,, a4, a5 [1 1]-[13]. Recently, G. Hardy and T. Rogers considered those T which satisfy (2) with constants ai's [4]. In [3], K. Goebel, W. A. Kirk and Tawfik N. Shimi proved that T has a fixed point if X is a weakly compact convex subset of a uniformly convex Banach space and if T 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 T 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. I This research was partially supported by the National Research Council of Canada Grant A8518. ' American Mathematical Society 1976