The goal of this paper is to study a comprehensive system called differential variational–hemivariational inequality which is composed of a nonlinear evolution equation and a time-dependent variational–hemivariational inequality in Banach spaces. Under the general functional framework, a generalized existence theorem for differential variational–hemivariational inequality is established by employing KKM principle, Minty’s technique, theory of multivalued analysis, the properties of Clarke’s subgradient. Furthermore, we explore a well-posedness result for the system, including the existence, uniqueness, and stability of the solution in mild sense. Finally, using penalty methods to the inequality, we consider a penalized problem-associated differential variational–hemivariational inequality, and examine the convergence result that the solution to the original problem can be approached, as a parameter converges to zero, by the solution of the penalized problem.

中文翻译：

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微分变异-半变异不等式：存在，唯一性，稳定性和收敛性

本文的目的是研究一个综合系统，该系统称为微分变分-半偏差不等式，它由非线性演化方程和Banach空间中随时间变化的半变分-不等式不等式组成。在通用功能框架下，利用KKM原理，Minty技术，多值分析理论，Clarke次梯度的性质，建立了微分变分-半变分不等式的广义存在性定理。此外，我们探索了该系统的适定性结果，包括温和意义上解的存在性，唯一性和稳定性。最后，使用对不等式的惩罚方法，我们考虑了与问题相关的惩罚性变分-半变异不等式，