The Approximation Problem Of The Fixed Point Of The Asymptotically Nonexpansive Mappings

The whole thesis falls into six chapters:Chapter 1 acts as the general introduction to the significance and curren situation inth study of the fixed points of asymptotically nonexpansive mapping.Chapter 2 gives a necessary and sufficient conditions for {x_n} converges stronglyto a fixed point of T. By using viscosity approximation methods for the interative sequence of asymptotical nonexpansive mappings in Banach space, we study thesequence {x_n}:(?),f be acontractive mapping in Banach space, T be an asymptotically nonexpansive mapping. A strong convergence theorem for viscosity approximation sequence of asymptotically nonexpansive mappings in Banach space is obtained.Chapter 3 suppose D is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction, let T:D→E be an asymptotically nonexpansive noneself-mapping. we study thesequence {x_n}:(?) and give anecessary and sufficient condition for {x_n} converges strongly to a fixed point of T.Chapter 4 suppose D is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction, let T:D→E be an asymptotically nonexpansive noneself-mapping. By using viscosity approximation methods for the interative sequence of asymptotical nonexpansive mappings in Banach space, we study the covergence of the sequence {x_n} defined by(?). A strong convergencetheorem for viscosity approximation sequence of asymptotically nonexpansivemappings in Banach space is obtained.Chapter 5 study the viscosity approximation process for a asymptotically nonexpansive semigroup and prove another strong convergence therom for a asymptotically nonexpansive semigroup in Banach space, which is defined by x_n=α_nf(x_n)+(1-α_n)T~n(t_n)(x_n),n∈N,where E be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E~*, and D be a nonempty closed convex subet of E, {T~n(t):t∈R_+} be a asymptotically nonexpansive semigroup on D such that F :=(?), f:D→D be a fixd contractive mapping.Chapter 6 comes to a conclusion. In the meanwhile, it put forward some problems for further study.In this thesis, main results gather in the 2-th, 3-th, 4-th, 5-th chaper.