Existence Of A Solution Of The Variational Inequality And Its Well-posedness

In this dissertation, we study the existence of a solution of the vari-ational inequality and its well-posedness. In chapter2, we prove the existence ofsolutions of generalized mixed variational inequalities under a rather weak coerciv-ity condition. Then we give the Tikhonov regularization result for generalized mixedvariational inequalities. In chapter3, the concept of Levitin-Polyak perturbation α-well-posedness of a mixed variational inequality is introduced. Let MVI<F,, KA bea mixed variational inequality. It is shown that the Levitin-Polyak perturbation α-well-posedness of MVI<F,, KA and its Levitin-Polyak α-well-posedness are equivalent.Then a metric characterization of the Levitin-Polyak perturbation α-well-posedness forMVI<F,, KA is given, even F is not a monotone mapping. Moreover, we also givesome conditions under which a mixed variational inequality is (generalized) Levitin-Polyak perturbation α-well-posed. Finally, although one can transform the Levitin-Polyak well-posedness of a mixed variational inequality into the Levitin-Polyak well-posedness of the gap function, we prove directly the relations between Levitin-Polyakwell-posedness and Hadamard well-posedness of a mixed variational inequality.