Several Generalized Variational Inequalities Problems

The theory of variational inequality has been well developing since 1960s andbecome one of very e?cient mathematical methods. Nowadays it plays an im-portant role in the study of a wide class of problems arising in pure and appliedsciences including mechanics, optimization and optimal control, operation researchand engineering sciences. Because the theory of variational inequality can providepeople with a simple and natural framework for the research of unrelated linear andnonlinear problems.In this thesis we further studied several mixed type variational inequalities andvariational inclusions problems. Firstly, by using auxiliary principle technique, weproved the existence and uniqueness theorem of solution for the generalized stronglynonlinear set-valued variational-like type inequalities problem in a Hilbert spaceand proposed iterative algorithm for computing approximate solutions. Secondly,by defining a new auxiliary variational inequality, we established a more generaliterative algorithm for solving the mixed quasi-variational-like inclusions problemin a re?exive Banach space and gave its convergence analysis. Thirdly, we studiedthe generalized mixed implicit quasi-η-variational inequalities problem in a Hilbertspace. By suggesting a new auxiliary problem, we construct and analyze an algo-rithm for solving generalized mixed implicit quasi-η-variational inequalities prob-lem. Finally, utilizing the alternative equivalent formulation between general mixedquasi-variational inequalities problem and implicit fixed-point problem, we extendedNoor's predictor-corrector iterative algorithm to develop the new modified iterativealgorithm for solving generalized mixed quasi-variational inequalities problem in areal Hilbert space and discussed the convergence criteria of the iterative sequencegenerated by our algorithm. Our results improved and generalized some previouslyknown results.