Nonlinear Operator Subsemigroups Fixed Point Theorem And The Ergodic Theory Of Banach Spaces

This thesis is devoted to the study of the fixed point theory and the ergodic theory of nonlinear operators in Banach spaces. It has two main chapters.It is well known that the fixed point theory is an important and widely used branch of nonlinear analysis. Recently, Suzuki and Takahashi [3], Takahashi and Zembayashi [8] proved the fixed point theorems of the nonexpansive semigroup and the asymptotically nonexpansive semigroup: Let C be a compact convex subset of Banach space X and (?) = {T(t): t > 0} be an asymptotically nonexpansive semigroup on C. Then the set of fixed points F(3) of 3 is nonempty. The first chapter presents the fixed point theorem of the asymptotically nonexpansive type semigroup in a general Banach space: Let C be a compact convex subset of Banach space X and (?) = {T(t): t ≥ 0} be an asymptoti -cally nonexpansive type semigroup on C. Then the set of fixed points F(3) of 3 is nonempty. Using the fixed point theorem, we also prove that Mann's type sequence for the asymptotically nonexpansive type semigroup is strongly convergent. This is a generalization of the results of Suzuki and Takahashi[3], Takahashi and Zembayashi [8] for nonexpansive semigroup and asymptotically nonexpansive semigroup. We also generalize our results to the case of right reversible semigroups of asymptotically nonex -pansive mappings and the case of the general semitopological semigroup of asymptoti -cally nonexpansive mappings. The main results are following theorems: Theorem 4.2 Let C be a compact convex subset of Banach space X and G a right reversible semigroup. Let (?) = {T(t) :t ∈ G} be an asymptotically nonexpansive type semigroup on C. Then the set of fixed points F(3) of 3 is nonempty. Theorem 5.2 : Let C be a compact convex subset of Banach space X and G a semitopological semigroup. Let 3 = {T(t) :t∈G} be an asymptotically nonexpansive type semigroup on C. Then the set of fixed points F(3) of 3 is nonempty.The research of nonlinear ergodic theory began in the mid-seventies. Consequently,it got great development because it was widely used in many problems such as the numerical solution of differential equation, the existence theory of positive solution, control theory and optimization. Baillon proved the first nonlinear ergodic theorem for nonexpansive mappings in the framework of Hilbert space [9]. Baillon's theorem was extended to the case of uniformly convex Banach space with the Frechet differentiable norm by Brack, Hirano and Reich [10,11,12]. Hirano-Kido-Takahashi, Oka and Park-Jenong proved the ergodic theorem for commutative semigroups of nonexpansive mappings and asymptotically nonexpansive mappings in the uniformly convex Banach space with the Frechet differentiable norm [13,16,17,18]. Li Gang and Ma Jipu proved the ergodic theorem for the almost-orbits of non-Lipschitzian commutative semigroups of type γ in the uniformly convex Banach space which has Frechet differentiable norm or Opial's condition. In the setting of right reversible semigroups [20], Li Gang proved the ergodic theorem for non-Lipschitzian semigroups of type γ in the uniformly convex Banach space with a Frechet differentiable norm [21]. However, it remains open for a few years whether Baillon's theorem is valid for the case of right reversible semigroups of asymptotically nonexpansive mappings in the uniformly convex Banach space without Frechet differentiable norm or Opial's condition. In chapter 2, by using technique of product nets and other new ideas, we succeed in proving the nonlinear ergodic convergence theorem for the almost-orbits of right reversible semigroups of asymptotically nonexpansive mappings in the uniformly convex Banach space with Opial's condition. Further, we give the weak convergence theorem for the almost-orbits of right reversible semigroups of asymptotically nonexpansive mappings in the uniformly convex Banach space with Opial's condition or its dual space has KK property. We should notice that the condition X~* has the KK property is strictly weaker than the condition X has the Frechet differentiable norm.