The Iterative Analysis For A Common Solution On Three Class Problems In Banach Spaces

In this paper,we studied strong convergence problems of generalized nonlinear vari-ational inclusions(or variational inequalities) and hybrid pseudo-viscosity approximation schemes with strongly positive bounded linear operators,using the projection operator technique、the new generalized resolvent operator technique and viscosity approximation method etc.our results extend and improve the existing relevant results.Specifically stated as follows.In the Chapter1, we discuss the issues of domestic and foreign scholars about this research,some scholars research results in recent two or three years,some explored back-grounds and advantages of the article,which thus highlights the application value and the practical significance of this topic.In the Chapter2, we study convergence problems of a new system of generalized non-linear set-valued variational inclusions involving (A,η, m)-accretive operator in Banach spaces.By Nadler’s lemma,the new generalized resolvent operator technique associated with (A,η,m)-accretive operator,we construct some new iterative algorithms, under suitable conditions, some strong convergence theorems are proved.Our results extend and improve the recent ones announced by many others.More details please read the Chapter2.In the Chapter3, a new system of generalized mixed nonlinear variational inequali-ties is introduced and considered in Banach spaces,which includes some known systems of variational equalities (introduced by Alber and so on) and the classical variational inequali-ties as special cases.Using the projection operator technique,we suggest some new iterative algorithms for solving the system of generalized mixed variational inequalities, and prove the convergence of the proposed iterative methods under suitable conditions.The results in this Chapter extend and improve some well-known results in the literature.More details please read the Chapter3.In the Chapter4, Mainly studied a common element on three class problems,by constructing a new iterative algorithm, iterative solution of a generalized mixed equilib-rium problems is found,and this solution is the set of fixed points of a family of infinitely non-expansive mappings.We prove the strong convergence of the sequences generated by hybrid pseudo-viscosity approximation schemes with strongly positive bounded linear op-erators to a solution of an equilibrium problem which is also a common fixed point of infinitely many non-expansive mappings.Meanwhile, these results improve and generalize many recent corresponding results obtained by Ceng L.C.,Ansari Q.H. and Yao J.C. and the other authors.More details please read the Chapter4.