Asian Research Thesis Index

Abstract

It is known that the general variational inequalities are equivalent to the fixed point problem and the Wiener-Hopf equations. We use this equivalent formulation to develop some new self-adaptive methods for solving the general variational inequalities. It is shown that the convergence of these new self-adaptive methods requires only the pseudomonotonicity, which is weaker condition than monotonicity. Relationship of these new methods with previous known methods is considered. Several examples are given to illustrate the efficiency and implementation of these methods. It is shown that the new self-adaptive methods perform better than the previous ones. A new class of variational inequalities is introduced and studied which is called the extended general mixed variational inequality. We establish the equivalence between the extended general mixed variational inequalities and the fixed point problems. This alternative equivalent formulation is used to suggest and analyze some new iterative methods for solving the extended general mixed variational inequalities. The convergence analysis of these methods is considered under suitable mild conditions. A new class of resolvent equation is introduced. It is shown that the extended general mixed variational inequalities are equivalent to the resolvent equation. This equivalence is used to suggest some iterative methods for solving the extended general mixed variational inequalities. The convergence of these iterative methods is discussed. Since the extended general mixed variational inequalities include extended general variational inequalities and related optimization as special cases, results obtained in this thesis continue to hold for these problems.

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