Asian Research Thesis Index

Abstract

A novel generalization of variational inequalities, which is called strongly mixed variational inequalities, is introduced and investigated. It is shown that the minimum of a sum of differentiable convex functions and nondifferentiable strongly convex functions can be characterized by strongly mixed variational inequalities. Auxiliary principle technique is used to investigate the existence of the unique solution of strongly mixed variational inequalities. This technique suggests us to propose some iterative schemes for solving strongly mixed variational inequalities. It is shown that strongly mixed variational inequalities are equivalent to fixed point problem and the resolvent equations under some conditions. These equivalent formulations can be used to examine the existence of a solution of the strongly mixed variational inequalities as well as used to develop new iterative schemes for solving strongly mixed variational inequalities. Convergence criteria of proposed methods is analyzed under some conditions. Dynamical systems related to strongly mixed variational inequalities is introduced. This approach is also used to examine the existence of the solution of strongly mixed variational inequalities. The error bounds for a solution of the strongly mixed variational inequalities using the merit function technique are derived. Moreover various special cases have also been discussed. System of strongly mixed variational inequalities involving two different operators is considered.

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