Nonlinear ergodic theorems for nonexpansive mappings in Hilbert spaces
Norimichi Hirano, Wataru Takahashi · Kodai Mathematical Journal · 1979
Let C be a closed convex subset of a Hubert space H and T be a mapping of C into itself.T is said to be asymptotically nonexpansive if for each x, y^C, \\T % x-T x y\\^(X+a % )\\x-y\\ for i=l,2,-, where limα^O.In particular if a t =0, i=l, 2, •••, T is said to be nonexpansive.τ In [1], Baillon proved the first nonlinear ergodic theorem: Let C be a closed convex subset of a real Hubert space H and T be a nonexpansive mapping of C into itself.If T has a fixed point in C, then for each x in C, converges weakly to a fixed point of T. Brezis and Browder [3] extended this theorem to general averaging processes B n (x)= Σ α n .kT*x(0ύa n .k , Σ a n .k = l).The argument there was very simple and elegant.In this paper, at first, we extend Baillon's theorem to asymptotically nonexpansive mappings and we prove that the converse of Baillon's theorem is also true if for each x in C, A n (x) converges weakly to a point in C, then T has a fixed point in C.Moreover, we obtain nonlinear ergodic theorems for a family {T t : 0^ί<oo} of mappings on C satisfying some conditions.Finally, a nonlinear ergodic theorem for a commutative semigroup of nonexpansive mappings on C is given by using the asymptotic center defined in Lim's paper [7].The authors wish to express their hearty thanks to Professor Hisaharu Umegaki for many kind suggestions and advice.§ 2. Ergodic theorems for nonlinear mappings.Let H be a real Hubert space and C be a closed convex subset of H. Let