Several Classes Of Variational Inclusions With Generalized Resolvent Operator Technique

Variational inclusions are important generalizations of classical variational inequalities and thus, have wide applications to many fields, for example, mechanics, physics, optimization and control, nonlinear programming, economics and transportation equilibrium, and engineering sciences. For this reason, various variational inclusions have been intensively studied in recent years.Recently, sensitivity analysis of a solution set for variational inequalities and variational inclusions has been studied by many authors. Then, in year 2004, Ding [51]studied the behavior and sensitivity analysis of solutions for parametric completely generalized mixed implicit quasi-variational inclusions involving H-maximal monotone mappings; in year 2005, Peng and Long [52]studied the behavior and sensitivity analysis of solutions for parametric completely generalized strongly nonlinear implicit quasi-variational inclusions. Inspired and motivated by recent research works in this field, in the second chapter, by using (H,η)-resolvent operator technique and the property of fixed-point set of set-valued contractive mappings, we study the behavior and sensitivity analysis of solutions of a new class of parametric completely generalized strongly nonlinear mixed implicit quasi-variational inclusions with multi-valued and single-valued nonlinear mappings in Hilbert space.In year 2004, Agarwal, Huang and Tan [53]gave the sensitivity analysis of solution for a system of generalized nonlinear mixed quasi-variational inclusions in Hilbert spaces. In the third chapter, by using a resolvent operator technique of (H,η)-monotone mappings and the property of a fixed-point of contractive mappings, we study the behavior and sensitivity of the solutions of a new system of parametric variational inclusions with (H,η)-monotone operators which contains the system of parametric generalized nonlinear mixed quasi-variational inclusions.In year 2006, Verma [58] presented the sensitivity analysis for (A,η)-monotone quasivariational inclusions based on the generalized (A,η)-resolvent operator technique. In the forth chapter, by using a resolvent operator technique of (A,η)-monotone mappings and the property of a fixed-point of contractive mappings, we study the behavior and sensitivity of the solutions of parametric variational inclusions with (A,η)-monotone operators.