Variational-Like Inequalities And Equilibrium Problems

This paper mainly further studies several focus problems in nonlinear functional analysis, unites and generalizes some known results in recent literature. At first, using auxiliary variational principle, we prove the existence of a class of nonlinear mixed variational-like inequalities, compute the approximate solutions of it by the iterative algorithms and show the convergence of iterative sequence in reflexive Banach space. At the same time, by applying the continuous selection theorem and fixed point theorem, we establish the equilibrium points'existence theorems of a class of quasi-equilibrium problems and generalized quasi-equilibrium problems for multimaps, and generalize some corresponding results. At last, by applying the generalized R-KKM type theorem in topological spaces and the KKM type theorems in FC-spaces, we establish equilibrium point's existence theorems of a class of abstract generalized vector equilibrium problems in FC-spaces.