TYKHONOV WELL-POSEDNESS OF VARIATIONAL-HEMIVARIATIONAL INEQUALITIES AND MINIMIZATION PROBLEMS

In this paper, we deal with a Tykhonov well-posedness concept in the study of two nonlinear problems in reflexive Banach spaces: a general variational-hemivariational inequality problem ( VHVI) and a minimization problem (MP), constructed by using a gap function associated to VHVI. We prove the well-posedness of these problems with respect to specific Tykhonov triples, including a number of equivalence results which establish the link between the well-posedness and the unique solvability of each problem. We then show that, with a particular choice of the Tykhonov triples, it is possible to recover two well-posedness results obtained in the literature, for various classes of variational and hemivariational inequalities. Moreover, we use our results in the sensitivity analysis of VHVI, for which we prove the continuous dependence of the solution with respect to its data.