Nonlinear Operator Approximation Fixed Point Problems

Approximation for fixed points of nonexpansive mappings method is an important part ofnonlinear analysi’s thoery and attract many scholars attenlions.The purpose of the paper is todiscuss the approximation for fixed points of nonexpansive mappings and generalized equilib?rium problem、nonexpansive nonself-mapping、nonexpansive mappings and monotone map?pings. The results presented in this paper improve and extend some corresponding results.This paper includes four chapters.Now wc will describe them briefly.In chapter1, we recall the history and introduce the attentions of the research on approxima?tion for fixed points of nonexpansive mappings method and application and the main work of thisthesis.In chapter2, by using Fan-KKM Theorem, we prove the existence and uniqueness of solu?tion of the auxiliary problem for a generalized equilibrium problem.A common clement of thesolution set of ihe generalized equilibrium problem and the fixed point set for an inlinile family ofnonexpansive mappings can be achieved by an iterative sequence.In chapter3，we study the following iterative sequencewhere E is a real strictly convex and relfexive Banach space with a uniformly Gateaux differen-tiable norm and K is a nonempty closed convex subset of E which is also a sunny nonexpansiveretract of E. Wn is the W-mapping generated by and An, An Ai for anyn》1,where is a family of countable infinite nonexpansive nonself-inappingssuch that the common ifxed points set0and be a sequence of nonneg-ativc numbers in is a contractive mapping, P is a sunny nonexpansive retractionof E onto K. It is shown that under very mild conditions on the parameters, the sequenceconverges strongly to p G F to the following variational inequalityIn chapter4’ we study a general iterative process to ifnd a common element of the set of fixedpoints of nonexpansive mappings and the set of solutions of variations inequality for a stronglymonotone mappings.