Strong Convergence Of Approximants For Nonlipschitzian Semigroups In Banach Spaces

Nonlinear operator Semigroups theorem is an important branch of nonlinear functional analysis. The Study of it began in the middle of 1970's. Consequently,it got great development because it widely used in many problems, such as the numerical solutions of differentiablp equations, the existence theory of positive solution, control theory and optimization, In 1975,J.B.Baillon[l] introduced the first nonlinear nean ergodic convergence arid retraction theorem for nonexpansive mappings in the framework of Hilbert spafce: Let C be a nonexpansive mapping from C into itself. If the set F(T) of fixed points pf T is nonempty, then the Cesaro meanconverges weakly as n →∞ to a fixed point y of T for each x∈ C. In this case, putting y=Px for each x∈ C,P is a nonexpansive retraction of Conto F(T), satisfying PT = TP = P and Px∈ co{T~n x: n = 0,1,2, ? ? ?} ∨x∈ CIn later years, this theorem was extended to more general situations. Reich, Bruck and so on extended Baillon's theorem to uniformly convex Banach space with Frechet differentiable norm or Opial's condiction for the case of nonexpansive and asymptotically nonexpansive mappings. Li-Ma first proved the nonlinear ergodic theorems for non-lipschjtzian semigroup of type(γ) in a uniformly convex Banach space which has Frechet differentiable norm or Opial's condiction. In the paper, I will give the ergodic retraction theorem for the asymptotically nonexpansive type semigroup in uniformly convex Banach space. This ergodic retraction theorem makes an important role in the construction of a strong convergence sequence in the section 2.On the other hand,In 1967,Browder[3] first showed the convergence theorem of an approximated sequence for a nonexpansive mapping in Hilbert space: Let C be a closed, convex subset of a Hilbert space and let x be an element of C .Let T be a nonexpansivemapping from C into itself such that the set F(T) of fixed points of T is nonempty. For each t with 0 < t < 1 ,let xt be an element of C satisfyingx, = tx + (\-t)TxtThen {xt}convergences strongly to the element of F(T) which is nearest tox in F(T) as t -? 0. Reich [8] extended Browder's result to the case when E is a uniformly smooth Banach space .U^ing the idea of Browder [3],Shimizu and Takashi[ll] studied the convergence of another approximating sequence for an asymptotically nonexpansive mapping in Hilbert space. This result was extended to a Banach space by Shioji and Takashi[12]. Recently,I4 and Sims[6] gave an convergence theorem of an approximating sequence for an asymptotically nonexpasive type mapping in a uniformly convex Banach space wljich has uniformly Gateaux differentiable norm. On the base, I will extend this result to asymptotically nonexpasive type semiqroups in section 2, it contains a lot of results b(efore.