Variational Inclusions with General Over-relaxed Proximal Point and Variational-like Inequalities with Densely Pseudomonotonicity

This dissertation focuses on the existence and uniqueness of the solutions of variational inclusion and variational inequality problems and then attempts to develop efficient algorithms to estimate numerical solutions for the problems. The dissertation consists a total of five chapters. Chapter 1 is an introduction to variational inequality problems, variational inclusion problems, monotone operators, and some basic definitions and preliminaries from convex analysis. Chapter 2 is a study of a general class of nonlinear implicit inclusion problems. The objective of this study is to explore how to omit the Lipschitz continuity condition by using an alternating approach to the proximal point algorithm to estimate the numerical solution of the implicit inclusion problems. In chapter 3 we introduce generalized densely relaxed ƞ - α pseudomonotone operators and generalized relaxed ƞ - α proper quasimonotone operators as well as relaxed ƞ - α quasimonotone operators. Using these generalized monotonicity notions, we establish the existence results for the generalized variational-like inequality in the general setting of Banach spaces. In chapter 4, we use the auxiliary principle technique to introduce a general algorithm for solutions of the densely relaxed pseudomonotone variational-like inequalities. Chapter 5 is the chapter concluding remarks and scope for future work