Solvability and optimization for a class of mixed variational problems

TL;DR

This paper investigates the solvability and optimization of a class of nonlinear mixed variational problems in Banach spaces, establishing conditions for unique solutions, continuous dependence, and applications to elastic contact models.

Contribution

It introduces new conditions for the unique solvability and continuous dependence of solutions in nonlinear mixed variational problems, and applies these results to elastic contact modeling.

Findings

Proved unique solvability under generalized monotonicity.

Established continuous dependence of solutions on data.

Applied abstract results to nonlinear elastic contact problems.

Abstract

We consider an abstract mixed variational problem governed by a nonlinear operator $A$ and a bifunctional $J$, in a real reflexive Banach space $X$. The operator $A$ is assumed to be continuous, Lipschitz continuous on each bounded subset of $X,$ and generalized monotone. First, we pay attention to the unique solvability of the problem. Next, we prove a continuous dependence result of the solution with respect to the data. Based on this result we prove the existence of at least one solution for an associated optimization problem. Finally, we apply our abstract results to the well-posedness and the optimization of an antiplane frictional contact model for nonlinearly elastic materials of Hencky-type.