Solvability and Stability Results for a Class of History-Dependent Quasi-Variational Hemivariational Inequalities

What the paper says

(Communicated by Xinmin Yang) In this paper, we study a history-dependent quasi-variational hemivariational inequality as follows. Problem 1. Find $u \in C(I;\Lambda)$ such that, for all $t \in I$, $u(t) \in K(u(t))$ and $Au(t),v-u(t)\rangle + \varphi(Su(t),u(t),v)- \varphi(Su(t),u(t),u(t))$ $+ j^\circ(\gamma u(t),\gamma u(t);\gamma v-\gamma u(t)) \geq \langle f(t),v-u(t)\rangle, ~~\forall v \in K(u(t)).$ Here $I=[0,+\infty)$ is an infinite time interval, $V$ and $X$ are reflexive Banach spaces, $Y$ is a normed space, $\Lambda$ is a nonempty subset of $V$, $C(I;\Lambda)$ stands for a set of all continuous functions defined on $I$ with values on $\Lambda$, $A: V \to V^*$ represents a nonlinear operator, $\gamma: V \to X$ is a linear continuous operator, $S: C(I;V) \to C(I;Y)$ is a history-dependent operator, $\varphi: Y \times V \times V \to \mathbb{R}$ is convex, lower semicontinuous with respect to its last argument, $j:X \times X \to \mathbb{R}$ is a locally Lipschitz functional with respect to its last argument, $j^\circ$ is the generalized directional derivative (in the sense of Clarke) of $j$, $K: \Lambda \to 2^\Lambda$ is a set-valued mapping and $f \in C(I;V^*)$ is given. By applying a fixed point argument about history-dependent operators and the Gronwall inequality, we obtain a unique solvability result to the history-dependent quasi-variational hemivariational inequality in a space of continuous functions. In addition, when the data of the history-dependent quasi-variational hemivariational inequality are perturbed, sufficient conditions are given to guarantee that the solution sequence of the perturbed problem converges to the unique solution of the original problem.