| Fixed point theory is an important part of functional analysis. Since the 1920s, from the Banach classical theorem of contractive mapping to Ishikawa iterative sequence and Mann iterative sequence approximation , the fixed point of several mappings theory have formed a very abundant and sound system.In the purpose of this paper is to study the fixed point theory of two more general mappings . The paper is organized asIn chapter 1, The significance and current situation in the study of the fixed point theory is introduced.In chapter 2,we present some basic concept and Lemma.In chapter 3,we study the Iterative Approximation Problem of fixed points for generalized asymptotically quasi-nonexpansive type mappings. Based on [26],the purpose of this paper is to study the Iterative Approximation Problem of Fixed Points for generalized asymptotically quasi-nonexpansive type mappings in the the non-empty closed convex subset in Banach spaces,and to give some necessary and sufficient conditions for the modified Ishikawa iterative sequence with errors to converge strongly to a fixed point of generalized asymptotically quasi-nonexpansive type mappings: Let E be a real Banach space ,C be a nonempty closed conex subset of E and T:C→C be generalized asymptotically quasi-nonexpansive type mappings. {Kn}is asymptotic coefficient of the generalized asymptotically quasi-nonexpansive type mappings satisfying .Let {x_n} be the modified Ishikawa iterative sequence with errors defined by where {u_n} and {v_n} are two se quences in E and {α_n } , {β_n } , {γ_n},{ξ_n } , {η_n } ,{δ_n} are the sequences in [0,1] satisfying the follow conditions: 1) ,where L is some constant; 2) 3)If T is uniformly continuous in the point of F(T),then the sequences {xn} converges to some fixed point of T. After that,we give a necessary and sufficient conditions for the modified N-step iterative sequence with errors to converges strongly to a common fixed point of a finite family of generalized asymptotically quasi-nonexpansive type mappings.The result in this paper extendes a great deal of the achievement now existed.In chapter 4,we discuss the Iterative Approximation Problem of Fixed Points for total asymptotically nonexpansive mappings. In this paper,we research into the Iterative Approximation Problem of Fixed Points for total asymptotically nonexpansive mappings and receive some necessary and sufficient conditions for the modified Ishikawa iterative sequence with errors to converge strongly to a fixed point of total asymptotically nonexpansive mappings: Let E be a real Banach space ,C be a nonempty closed conex subset of E and T:C→C be generalized asymptotically quasi-nonexpansive type mappings.suppose ,whereμnand l nare two sequences appeared in the definition of total asymptotically nonexpansive mappings,and suppose that there exist M,M*>0 such thatφ(λ)≤M~*λforλ≥M.Let {x_n} be given by (1) , where {un} and {vn} are two sequences in E and{α_n } , {β_n } , {γ_n},{ξ_n } , {η_n } ,{δ_n}are the sequences in [0,1] satisfying the follow conditions: 1) ,where L is some constant; 2) 3)If T is uniformly continuous in the point of F(T),then the sequences {xn} converges to some fixed point of T.Meanwhile, we give a necessary and sufficient conditions for the modified N-step iterative sequence with errors to converges strongly to a common fixed point of a finite family of total asymptotically nonexpansive mappings.The result presented in this paper extendes and improves the current results.In chapter 5,we aggregate this paper,and Propose some Problem for studing in future. |
