## Rong Hu, Mircea Sofonea, On the well-posedness of variational-hemivariational inequalities and associated fixed point problems

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DOI: 10.23952/jnva.6.2022.5.09

Volume 6, Issue 5, 1 October 2022, Pages 567-584

Abstract. We consider an elliptic variational-hemivariational inequality $P$ in a $p$-uniformly smooth Banach space. We prove that the inequality is governed by a multivalued maximal monotone operator, and, for each $\lambda>0$, we use the resolvent of this operator to construct an auxiliary fixed point problem, denoted $P_\lambda$. Next, we perform a parallel study of problems $P$ and $P_\lambda$ based on their intrinsic equivalence. In this way, we prove existence, uniqueness, and well-posedness results with respect to specific Tykhonov triples. The existence of a unique common solution to problems $P$ and $P_\lambda$ is proved by using the Banach contraction principle in the study of Problem $P_\lambda$. In contrast, the well-posedness of the problems is obtained by using a monotonicity argument in the study of Problem $P$. Finally, the properties of Problem $P_\lambda$ allow us to deduce a convergence criterion in the study of Problem $P$.