Existence Of Solutions And Gap Functions For Vector Variational-Like Inequalities

Vector variational-like inequalities are generalization of variational inequalities. Thisdissertation is devoted to studying existence results of generalized mixed vector variational-likeinequalities with a variable ordering relation. Moreover, the gap functions and weak sharpnessfor the vector variational-like inequalities with set-valued mapping are discussed.The main results obtained here may be summarized as follows:On the one hand, the existence of solutions for generalized mixed vector variational-likeinequalities with a variable ordering relation (for short, GMVVLI) is discussed. Firstly, byapplying the known KKM–Fan theorem and Nadler’s lemma, the existence of GMVVLIwithmonotone vector multifunction are proved. Secondly, by utilizing the Brouwer’s fixed pointtheorem, some existence theorems of GMVVLI without monotonicity assumptions under somequite mild conditions are obtained.On the other hand, it is devoted to studying several kinds of set-valued scalarvariational-like inequalities (SVLI and WVLI) and vector variational-like inequalities withset-valued mappings(SVVLI and WVVLI). Moreover, the equivalence of the variational-likeinequalities is established under certain conditions by using the scalarization approach of Konnov.In addition, the gap functions for SVVLI WVVLI, SVLI, WVLI are given respectively, and therelations among these gap functions are shown. Moreover, it is shown that weak sharp minimafor the sets of VVLI holds under the assumption of α η l strong pseudomonotonicity.