Strong Convergence Theorems For Asymptotically Nonexpansive Mappings In Banach Spaces

The theory of fixed points is an important research topic of functional analysis and it is widely used in many subjects such as differential equations, integral equations, numerical analysis, game theory, control and optimization theory.The theory of fixed points originated from Banach's principle of contraction mappings. By using the Banach's principle of contraction mappings, Browder proved the existence of fixed points of nonexpansive mappings. Browder's theorem was extended to uniformly smooth Banach spaces by Reich. Kirk gave the existence theorem of fixed points of nonexpansive self-mapping in Banach spaces with uniform normal structure. Goebel and Kirk showed that every asymptotically nonexpansive mapping has a fixed point on a nonempty bounded closed convex subset of uniformly convex Banach spaces. This result was extended to the case of Banach spaces with uniform normal structure by Kim and Xu. In 2002, Li and Sims showed that every asymptotically nonexpansive type mappings has at least a fixed point under certain appropriate conditions in Banach space with uniform normal structure: Let E be a Banach space with uniform normal structure and C be a nonempty bounded subset of E . let T :Câ†'C be an asymptotically nonexpansive type mapping such that T is continuous on C . Further, if there exists a nonempty closed convex subset K of C satisfies the following property: z∈K implies ( )ωw z ? K, then T has a fixed point in K . The method utilized there is to use the fixed points of contractions to approximate directly or approximate by iterations of the fixed points of nonexpansive mappings. In 1998, Shioji and Takahashi obtained the strong convergence theorems of the implicit viscosity iteration process for nonexpansive semigroups in Hilbert space. Shimizu and Takahashi studied the strong convergence of the explicit viscosity iteration process for nonexpansive semigroups in Hilbert space. In 2007, Chen and Song investigated the convergence of implicit iteration process and explicit iteration process for nonexpansive semigroups in uniformly convex Banach spaces with uniformly Gateaux differentiable norm.In this paper, under the framework of Banach space with uniformly Gateaux differentiable norm and uniform normal structure, we use the existence theorem of fixed points of Li and Sims to investigate the convergence of the implicit iteration process { }z n and the explicit iteration process { }xn for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroup.In chapter 2, we study the convergence of the following implicit viscosity iteration process { }z n and the explicit viscosity iteration process { }xn for asymptotically nonexpansive mappings in Banach space with uniformly Gateaux differentiable norm and uniform normal structure: And f∈ΠK, K is a nonempty closed convex subset of E , T :Kâ†'K is an asymptotically nonexpansive mapping with we show that { }z n and { }xn converge strongly to a fixed point, which is the unique solution to the variational inequalityIn chapter 3, we study the convergence of the following implicit viscosity iteration process { }z n and the explicit viscosity iteration process { }xn for asymptotically nonexpansive semigroup in Banach space with uniformly Gateaux differentiable norm and uniform normal structure:And f∈ΠK, K is a nonempty closed convex subset of E , ?= {T (t ), t≥0} is an asymptotically nonexpansive semigroup with converge strongly to a fixed point, which is the unique solution to the variational inequalityThe main results in this paper generalize and improve the main results in [9,10].