Existence And Boundedness Of Solutions To The Hemivariational Inequalities Of Hartmann-Stampacchia Type

This paper consists of three parts. In the first part, we focus on the existence and boundedness of solution for variational-hemivariational inequali-ties of Hartman-Stampacchia type; in the second part, we discuss the sufficient condition and the necessary condition for the existence and boundedness of the solution set of variational inclusion, only provided that the mapping is maximal monotone; in the third part, we discuss the properties of generalized differentials of the perturbed minimal time function.In the first part, we consider the existence and boundedness of solution, when the mapping has a certain kind of monotonicity and when the mapping is only upper semicontinuous, respectively.On the one hand, assuming that the mapping is set-valued, lower hemi-continuous and stably quasimonotone with respect to a certain set, when the constraint set is bounded, we prove the existence of solution; when the con-straint set is not bounded, we derive a sufficient condition for the existence and boundedness of solution.On the other hand, assuming that the mapping is set-valued and upper semicontinuous and the space is Euclidean space Rn, when the set K is bounded, we prove the existence and boundedness of solutions to this problem; when the set K is unbounded, we give a sufficient condition for the existence of solutions and a sufficient condition for the existence and boundedness of solutions.In the second part, we prove the solution set of variational inclusion being nonempty is equivalent to a coercivity condition, only provided that the mapping is maximal monotone. We also derive a sufficient condition and a necessary condition for the existence and boundedness of the solution set.In the third part, using the sudifferential of a perturbed function and the level sets of the support function of a bounded closed convex set, we show that the Frechet subdifferential and proximal subdifferential of a perturbed minimal time function. Some results is a special case of this result.