The Nonlinear Operator Fixed Points Of Iterative Approximation

The fixed point theory of nonlinear operator is important part of nonlinear functional analysis.Esipecially,the problem of approximating to fixed point of nonlinear operators becomes the topic that people study in the recently years.This paper discusses, the problem that several typical iteration sequences approximate to fixed point of nonexpansive mappings,asymptotically nonexpansive mappings and pseudocontractive mappings.In the first chapter,we consider the problems that Ishikawa iteration sequence approximate to fixed point of nonexpansive mappings and modified Ishikawa iteration approximate to fixed point of asymptotically nonexpansive mappings.On the one hand,it is shown that only the subsequence of iterative parameters satisfy some restrictions,the strong and weak convergence theorems for iterative process are proved.Meanwhile,the concept of approximate Picard iteration is put forward and its convergence theorems are proved.In the second chapter,we put forward two composite implicit iteration sequences, i.e.x_n = α_nx_n-1 + (1 -α_n)T_n(β_nx_n + (1 - β_n)T_nx_n)x_n = α_nx_n + (1 - α_n)T_n(β_nx_n + (1 -β_n)T_nx_n)where T_n = T_nmodN, and prove they can approximate to common fixed point of strictly pseudocontractive mappings using different ways.In the third chapter,we consider several iteration sequences approximating to especial fixed point of nonexpansive mapping, i.e. the following sequences:x_n+1 = α_nx + (1 - α_n)T(β_nx + (1 - β_n)Tx_n)y_n+1 = α_n(β_nx + (1 - β_n)Tx_n) + (1 - α_n)A_nx_n and furthermore prove the above sequences strongly convergence to Px ∈ F(T).where A_n = : C→ C and P is sunny nonexpansive retraction from C onto F(T).In the last of this paper,we make the appropriate modification tofamiliar Ishikawa iteration and discuss that it also approximate to especial fixed point of nonexpansive mapping.