Study On Some Important Properties Of Two Kinds Of Variational Inequalities

Along with the research on the properties of the solutions for variationalinequalities deepening, we study the two kinds of variational inequalities on gapfunctions and error bounds, generic stability and essential connected components,respectively.The research in this thesis contains two parts. We study how to use the image spaceanalysis to set up a class of gap functions and error bounds for variational inequalitieswith cone constrains in the first part. Then, we do some research on the stability and theexistence of essential components of the solution sets for generalized vectorquasi-variational-like inequalities.In the first part, we set up the Giannessi gap function and an error bound forvariational inequalities with cone constrains based on H and κ((?))linear separationwhose necessary and sufficient condition is built by saddle point of generalizedLagrange function. Then, we introduce regular weak separation functions and weakalternative theorem based on H and κ((?))separation(not only linear separation).Next,we get the necessary and sufficient condition and a class of gap functions forvariational inequalities with cone constrains by regular weak separation functions whichare u.s.c and satisfied some assumed conditions. Moreover, we can get the error bounds.Finally, we take some examples to see what kind of functions are the regular weakseparation functions which are u.s.c and satisfied some assumed conditions. And weprove that when the regular weak separation function takes regular Lagrange function, ifxsolves variational inequalities with cone constrains, then H andκ((?))admit linearseparation.In the second part, we construct a complete functional space M. Then, we prove theset-valued mapping Φ (u)from M into the solution sets K is usco(compact and uppersemicontinue). Next, we introduce the notion of essential solutions and essentialcomponents. Moreover, we prove that every solution sets mappingΦ (u)has at leastone essential connected component. Finally, we get some conclusions on the stability ofessential connected components for the solution sets.