Iterative Algorithms For The Solutions Of Variational Inequalities And Equilibrium Problems

Variational inequalities and equilibrium problems are important parts of nonlinear functional analysis.Many famous scholars have studied extensively them.At the same time,the development of fixed point theory stimulates the study of many fundamental mathematical fields such as variational inequality theory and equilibrium theory.In this dissertation,the problems of fixed points of nonexpansive mappings,variational inequalities and equilibrium have been explored extensively and we constructed effective iterative schemes to approximate the common element of the three different sets which extend and improve the domestic and international previous known results.First,we suggest a general iterative algorithm by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. Our results improve and extend the corresponding results of S.Takahashi and W.Takahashi(2007),G.Marino and H.K.Xu(2006),P.L.Combettes and S.A. Hirstoaga(2005)and some others.Second,we construct a composite iterative algorithm by viscosity approximation method for finding a common element of the set of fixed points of a nonexpansive mapping,the set of solutions of variational inequalities for an inversestrongly monotone mapping and the set of solutions of equilibrium problems in a Hilbert space.We show that the strong convergence of the iterative sequence. The results improve and extend the corresponding results announced by H.Iiduka and W.Takahashi(2005),S.Takahashi and W.Takahashi(2007).Third,we prove that the general variational inequalities are equivalent to the general Wiener-Hopf equations and use this alterative equivalence to suggest and analyze a new iterative method for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the general variational inequality involving multivalued relaxed monotone operators.Fourth,we suggest a new iterative algorithm for finding the common element of the set of a generalized system for nonlinear variational inequalities and the set of fixed points of quasi-nonexpansive mappings in Hilbert spaces.Our results include some recent results as some special cases,improve and extend the corresponding results of R.U.Verma(2004),S.S.Chang,H.W.Joseph Lee and C.K.Chan(2007),Z.Huang and M.A.Noor(2007)and some others.