Fixed-Point Iteration Methods For Variational Inequalities With Optimization Problems

Variational inequality theory is a very powerful research tool in today’s mathematical technology.It plays an important role in the research of pure and applied sciences,and has attracted the attention of many scholars.More and more scholars use different methods to conduct inq-depth research under appropriate conditions.Variational inclusion is another important generalization of variational inequality,it is widely used in optimization and cybernetics,traffic balance theory,economics,engineering science theory and other fields.And traffic balance system,spatial balance system,Nash balance system and general balance programming system all take the system of variational inequalities as its mathematical model.Therefore,the study of variational inequality system(Abbreviated as SVI)is of great academic value to us.In this paper,We introduce a composite viscosity implicit method for solving the VI and CFPP with the SVI constraint in the framework of uniformly convex and q-uniformly smooth Banach space where(1 < ≤ 2).Moreover,we prove the strong convergence of the sequences generated by the proposed implicit method to a solution of a certain hierarchical variational inequality(HVI).In addition,by the means of modified extragradient methods,we dicussed a modified viscosity implicit rule for finding a common element of the set of solutions of variational inequalities for two inverseq-strongly monotone operators and the set of fixed points of an asymptotically nonexpansive mapping in Hilbert spaces.The outline of this paper is organized as follows:In Chapter 1,we introduce the research background of the variational inequality system and fixed point system,composite viscosity implicit method and the development status at home and abroad.And we specify the main research work of this paper.In Chapter 2,the basic definitions,propositions and related lemmas are presented.In Chapter 3,we introduce a composite viscosity implicit method for solving the VI and CFPP with the SVI constraint in the framework of uniformly convex and q-uniformly smooth Banach space where(1 < ≤ 2).Moreover,we prove the strong convergence of the sequences generated by the proposed implicit method to a solution of a certain hierarchical variational inequality(HVI).In addion,our results are also applied for solving the fixed point problem(FPP)of nonexpansive mapping,variational inequality problem,convex minimization problem and split feasibility problem in Hilbert spaces.In Chapter 4,With the help of the modified extragradient methods,we use the technique of a modified viscosity implicit rule for finding a common element of the set of solutions of variational inequalities for two inverseq-strongly monotone operators and the set of fixed points of an asymptotically nonexpansive mapping in Hilbert spaces.And we obtained some strong convergence theorems under some suitable assumptions imposed on the parameters.We also give an algorithm to apply the theorems to solve fixed point problems for nonexpansive mappings FPP,VIP,and equilibrium problems in Hilbert spaces.In Chapter 5,we give the conclusion and outlook of this paper.