Existence Of Solutions And Algorithms For Generalized Variational-like Inequalities

It is well known that 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, which involve mathematical economics, finance, control theory, mechanics, physics and so on. 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. Therefore, the research for them has important learning value and certain degree of difficulty. This dissertation is devoted to study generalized variational-like inequalities in Banch spaces, especially, in reflexive Banach spaces, which is an unity and extension of a large number of known variational inequalities and vector variational inequalities including scalarization generalized variationl-like inequalities and generalized vector variational-like inequalities, 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 single-valued generalized nonlinear mixed variational-like inequalites in reflexive Banach spaces are studied. By applying minimax inequalities, the existence and uniqueness of solutions to a class of generalized nonlinear mixed variational-like inequalities are obtained. Also, a general algorithm is suggested exploiting auxiliary principle technique and the convergence of the iterative squences generated by the algorithm is proved. At the same time, the existence and uniqueness of solutions for a class of generalized strongly nonlinear quasivariational inequalities in Hilbert spaces are investigated and three-step perturbed iterative algorthm with errors is constructed and convergence of iterative sequence generated by the algorithm is discussed by projection technology;2. In chapter 3, the existence and algorithms for a class of set-valued generalized variational-like inclusions(inequalities) in Banach spaces is mainly discussed. Making use of the Jv—proximal mapping of t]—subdifferential of nonconvex, semicontinuous functional in reflexive Banach spaces, 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 squences generated by the two class of algorithms are proved. In addition, a class of three-step predictor-corrector iterative algorithms for a class of generalized set-valued mixed quasivariational inequalities in Hilbert spaces is suggested. Convergence rusults of algorithms are obtained without any monotonicity;3. In chapter 4, the existence of solutions for two classes of generalized vector variational-like inequalities is discussed. Applying certain pseudomonotonity and hemicontinuity of involving mappings and the known KKM theorem and Nadler lemma, the generalized vector Minty lemma (generalized linearization lemma) and the existence theorems for a class of generalized vector variational-like inequalities under the pseudomononity of the mappings are proved. Moreover, the existence result of the other class of generalized vector variational-like inequalities with compact-valued mappings using generalized Minty lemma and the existence result of the class of generalized vector variational-like inequalities is obtained;4. Chapter 5 is devoted to study a class of systems of vector variational-like inequalities based on Chapter 4. By means of the new concepts of Minty and Stampacchia T)—proper quasimonotonicity and r\—pseudomonotonicity and the known KKM theorem, the existence results of Stampacchia and Minty vector variational-like inequalities and relationship between the existence of solutions to the two vector variational-like inequalities are discussed under the certain hemicontinuity condition. Moreover, the existence of solutions to the systems of vector variational-like inequalities in reflexive Banach spaces is proved by the known Kakutani-Fan-Glicksberg fixed point theorem:5. Chaper 6 is devoted to study a class of systems of variational inequalities, which is a special case of the class of vector variational-like inequalities studied in chapter 5 and has important application in MPEC(mathematical program with equilibrium con-strains) problem. As result, several results on making use of systems of variational inequalities solving MPEC problem and algorithm scheme et.al. are given.