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In this paper we establish existence, uniqueness, and boundedness results for an elliptic variational inequality coupled with a nonlinear ordinary differential equation. Under the general framework, we present a new application modelling the antiplane shear deformation of a static frictional adhesive contact problem. The adhesion process has been extensively studied, but it is usual to assume a priori that the intensity of adhesion is bounded by introducing truncation operators. The aim of this article is to remove this restriction. The proof is based on an iterative approximation scheme showing that the problem has a unique solution. A key ingredient is finding uniform a priori bounds for each iterate. These are obtained by adapting versions of the Moser iteration to our system of equations., Comment: 39 pages, 1 figure

In this paper, we study the well-posedness of a class of evolutionary variational-hemivariational inequalities coupled with a nonlinear ordinary differential equation in Banach spaces. The proof is based on an iterative approximation scheme showing that the problem has a unique mild solution. In addition, we established the continuity of the flow map with respect to the initial data. Under the general framework, we consider two new applications for modelling of frictional contact for viscoelastic materials. In the first application, we consider Coulomb friction with normal compliance, and in the second, normal damped response. The structure of the friction coefficient $\mu$ is new with motivation from geophysical applications in earth sciences with dependence on an external state variable $\alpha$ and the slip rate $|\dot{u}_\tau|$., Comment: 49 pages, 3 figures