Existence Of Solutions And Algorithms For Several Classes Of Generalized Variational Inequalities And Variational Inclusions

It is well known that variational inequality theory is very powerful tools of the current mathematical technology. Variational inequalities have many important applications in operation research, computer science, system science, engineering technology, transportation, economics and management et. al. In the last 20 years of the twentith century, they have been paid close attention by many scholars. Generalized variational-like inequalities are generalization of variational inequalities, they become an important foundation and tool for studying multiobjective and multilevel programs and one of focal point problems paid close attention by scholars in the field of applied mathematics, research on which touch upon such mathematical branches as convex, linear analysis and nonlinear analysis, nonsmooth analysis, set-valued analysis. Generalized variational inclusions are generalization of variational inequalities, they have wide applications in fields of optimization and control, economics and transportation equilibrium, engineering science. Recently, some new and interesting problems, which are called to be system of variational inequality problems were studied. Some scholars considered the traffic equilibrium problem, the spatial equilibrium problem, the Nash equilibrium, and the general equilibrium programming problem can be modeled as a system of variational inequalities. Therefore, the research for them has important learning value . This dissertation is devoted to study bilinear completely generalized variational-like inequalities in Banach spaces, especially, in reflexive Banach spaces and generalized variational inclusions,where, generalized variational inclusions including: a system of variational inclusions with (H,η)-monotone operators in Hilbert space , variational inclusions with H-accretive operators in Banach space, which is an unity and extension of a large number of known variational inequalities and variational inclusions, from theory and algorithm. The main results, obtained in this dissertation, may be summarized as follows:1. In chapter 2, the existence of the solution and algorithms for a class of bilinear completely generalized variational-like inequalities in reflexive Banach spaces are studied. By applying minimax inequalities, the existence and uniqueness of solutions to a class of bilinear completely generalized variational-like inequalities are obtained. Also, a general algorithm is suggested exploiting auxiliary principle technique and the convergence of the iterative sequences generated by the algorithm is proved; 2. In chapter 3, the existence and algorithms for a new system of variational inclusions with (H,η)-monotone operators is mainly discussed. Making use of the resolvent operator method associated with (H,η)-monotone operators, we prove the existence and uniqueness of solutions and the convergence of some new three-step iterative algorithms for this system of variational inclusions.3. In chapter 4 , the existence and algorithms for a class of set-valued generalized variational-like inclusions (inequalities) in Banach spaces is mainly discussed. By using the resolvent operator method associated with H-accretive, the equivalence between the class of generalized set-valued variational-like inclusions and a class of Wiener-Hopf equations is established. Furthermore, two classes of new and general iterative algorithms are suggested and the existence of solutions to the class of generalized set-valued variational-like inclusions and the convergence of sequences generated by the two class of algorithms are proved.