A Number Of Algorithms For Generalized Variational Inequalities

In recent years,variational inequality theory has been become very effective and powerful tools for studying a wide class of nonlinear problems arising in many diverse fields of pure mathematics and applied sciences,such as mathematical programming,optimization,mechanics, elasticity,transportation,economic equilibrium,fluid flow through porous media and many other branches of mathematical and engineering science.Variational inequalities have been extended and generalized in different directions by using novel and innovative techniques both for own sake and for its application.We arrange this dissertation as follows:1.The historic background of variational inequality theory is recalled briefly.2.We recall some basic concepts and theories.3.A new class of completely generalized quasi-variational-like inclusions in reflexive Banach spaces is introduced and studied.Using the J~η—proximal mapping,two iterative algorithms to compute approximate solutions for this class of completely generalized quasi-variational-like inclusions are suggested and analyzed,and the convergence of the iterative sequences generated by the algorithms is also proved.4.We study the strong convergence for the viscosity iterative sequences {x_t} and {z_m} defined by(4.1.1.1) and(4.2.1.1),respectively.We prove that {x_t} and {z_m} converges strongly to some p∈F(T),where p is a unique solution to the variational inequality(4.1.1.2).5.In real Hilbert spaces,a new system of completely generalized strongly nonlinear mixed variational-like inequalities(SCGSNMVLI) is introduced.We establish an existence and uniqueness theorem of solutions to the auxiliary variational inequality problems for the SCGSNMVLI. Based on the auxiliary problems,we construct an iterative algorithm to compute the approximate solutions of the SCGSNMVLI.And also we give the convergence analysis of the iterative sequences generalized by the algorithm.6.We first introduce and study a new system of multi-valued variational inclusions involving (H,η)-accretive operators in Banach spaces.Using the resolvent operator associated with (H,η)-accretive operators,we construct an algorithm of this system and prove the convergence of the iterative sequences generated by the algorithm.Then,we introduce a new concept of generalized (A,η)-accretive mappings,study some properties of generalized(A,η)-accretive mappings and define resolvent operators associated with generalized(A,η)-accretive mappings.In terms of the new resolvent operator technique,we construct an algorithm for a class of multi-valued nonlinear variational inclusions involving generalized(A,η)-accretive mappings and prove the convergence of the iterative sequences generated by the algorithm.