Existence And Algorithm Of Solutions For Generalized Mixed Quasi-variational-like Inequalities

For many years, the variational inequality problem which was introduced and studied by the G. Stampacchia, P. Hartman and J. L. Lions has developed into a highly important branch of mathematics. In the scientific theory and application of science, the variational principle as a powerful tool can explain the math and finance and the basic principles of physics and so on. Theory of variational inequalities as the main promotional variational principle, it can describe the mathematics, physics, economics and engineering aspects of many problems. In recent years, the classical theory of variational inequalities have been extensively used to study the production in applied mathematics, optimization and control theory, mechanical and thermal, linear and nonlinear programming, economics and finance, traffic and transportation equilibrium, differential and integral equations, game theory and other areas. In the theory of variational inequalities, the most interesting and most important issue is to develop an effective approach for seeking iterative algorithm.This paper systematically introduces the theory of variational inequalities historical background of the development, current situation and the main work done by many scholars. Inspired by recent research achievement, this made the discussion of the following aspects:(1) We introduce and study a class of single values of generalized quasi-variational-like inequalities in reflexive Banach space, which involve a coercive continuous bilinear form a(u,v) and non-linear form b(u,v). First of all, we give a single value of generalized quasi-variational-like inequalities existence and uniqueness theorem in the relatively weak assumptions through using the application of minimax inequality and literature in the lemma. Second, we proposed the variational inequality iterative algorithm and the algorithm of the iterative sequence generated by the convergence criteria through using the application of KKM theorem and the auxiliary principle technique,(2) We introduce and study a class of set values of generalized quasi-variational-like inequalities in reflexive Banach space, which involve a coercive continuous bilinear form a(u,v) and non-linear form b(u,v). First, we use the auxiliary principle technique to prove a kind of vanational inequalities existence of solutions. Second, we proposed such variational inequalities iterative algorithm for approximate solutions and discuss the convergence of this algorithm.(3) We introduce and study a class of parametric generalized quasi-variational-like inequalities in reflexive Banach space, which involve a coercive continuous bilinear form a(u,v) and non-linear form b(u,v). Unlike in the past on the methodology used, this will all familiar auxiliary principle technique applied to the variational inequalities sensitivity analysis.