| In this thesis, we shall introduce the concepts of h-quasi-monotone, quasi-monotone, bi-quasi-monotone, h-quasi-semi-monotone, quasi-semi-monotone, quasi-nonexpansive, semi-nonexpansive, lower hemi-continuous, upper hemi-continuous, weakly lower (respectively, upper) demi-continuous, strongly lower (respectively, upper) demi-continuous, strong h-pseudo-monotone, strong pseudo-monotone, h-pseudo-monotone, pseudo-monotone, h-demi-monotone, and demi-monotone operators. We shall first obtain some generalizations of Ky Fan's minimax inequality. As applications, we shall obtain results on fixed point theorems, generalized variational, quasi-variational and bi-quasi-variational inequalities, and complementarity and bi-complementarity problems.; In Chapter 2, we shall first obtain a minimax inequality which generalizes Ky Fan's minimax inequality in several respects. Then, we shall obtain a Knaster-Kuratowski-Mazurkiewicz (in short KKM) type lemma which will be more general than KKM Lemma in all of its practical applications. By applying our KKM type lemma, we shall obtain a generalization of Brezis-Nirenberg-Stampacchia's generalization of Ky Fan's minimax inequality.; In Chapter 3, as applications of the minimax inequalities of Chapter 2, and as applications of most of the above mentioned operators, we shall obtain several existence theorems for compact and non-compact generalized variational inequalities and for non-compact generalized complementarity problems in topological vector spaces.; Finally, in Chapter 4, as applications of the generalized variational inequalities of Chapter 3, we shall first obtain some fixed point theorems in Hilbert spaces for some of the operators introduced in this thesis. Applying the minimax inequalities of Chapter 2 or some minimax inequalities in the literature and/or some of the operators introduced in this thesis, we shall obtain several existence theorems for non-compact generalized quasi-variational inequalities as well as for both compact and non-compact generalized bi-quasi-variational inequalities, and non-compact bi-complementarity problems in locally convex Hausdorff topological vector spaces. |
