General Variational Inequalities And Wiener-Hopf Equation

Extended general variational inequalities are the generalization on the basis of general variational inequalities. This new class of variatioanl inequalities was fist introduced and studied by M.A.Noor in2009. Theory of extended variational inequalities is quite a new one, ideas and techniques used to analyze general variational inequalities and other results can be used to research the new variational inequalities for the quick development of the theory.In this paper, we first introduce the development of variational inequalities theory, mainly including the background of the theory, the present state of the research and the main works that have been done by other authors. Inspired by Professor Noor’s and other scholars’publications, we do research into the following two aspects:First, we study the equivalence between the general variational inequality problem and solving the Wiener-Hopf equations. Then use the equivalence to suggest and analyze several iterative algorithms to solve general variational inequalities, at last we discuss and prove the convergence criteria for the algorithms.Second, inspired by the research of the last part, we study and prove that solving the Wiener-Hopf equations are still equivalent to the extended general variational inequalities problem. Similarly, the next step we suggest and analyze the iterative algorithms and prove the convergence criteria for the algorithms at last.Third, we studied the equivalence between general Wiener-Hopf equations and extended Verma general invariational inequalities, then we also suggest and analyze the iterative algorithms and prove the convergence criteria for the algorithms at last.