Abstract
In this paper, we propose a new methodology to study evolutionary variational-hemivariational inequalities based on the theory of evolution equations governed by maximal monotone operators. More precisely, the proposed approach, based on a hidden maximal monotonicity, is used to explore the well-posedness for a class of evolutionary variational-hemivariational inequalities involving history-dependent operators and related problems with periodic and antiperiodic boundary conditions. The applicability of our theoretical results is illustrated through applications to a fractional evolution inclusion and a dynamic semipermeability problem.
Highlights
Variational and hemivariational inequalities serve as theoretical models for various problems arising in mechanics, physics, and engineering sciences
It should be mentioned that the study of evolutionary variational-hemivariational inequalities has been performed typically through surjectivity results for pseudomonotone operators and fixed point theorems for nonlinear operators
This paper aims to propose a new approach to study evolutionary variational-hemivariational inequalities based on the theory of evolution problems governed by maximal monotone operators
Summary
Variational and hemivariational inequalities serve as theoretical models for various problems arising in mechanics, physics, and engineering sciences. It should be mentioned that the study of evolutionary variational-hemivariational inequalities has been performed typically through surjectivity results for pseudomonotone operators and fixed point theorems for nonlinear operators (see, e.g., [21] and the references therein). This paper aims to propose a new approach to study evolutionary variational-hemivariational inequalities based on the theory of evolution problems governed by maximal monotone operators. A key assumption to apply the surjectivity result is the so-called relaxed monotonicity for the subdifferential in the sense of Clarke (see Definition 2 below), which is a weaker notion than monotonicity, but which still permits to obtain the existence of solutions We characterize this notion in terms of the convexity of an associated function (see Section 3).
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