Abstract We consider an elliptic variational-hemivariational inequality 𝓟 in a reflexive Banach space, governed by a set of constraints K, a nonlinear operator A, and an element f. We associate to this inequality a sequence {𝓟n} of variational-hemivariational inequalities such that, for each n ∈ ℕ, inequality 𝓟n is obtained by perturbing the data K and A and, moreover, it contains an additional term governed by a small parameter εn. The unique solvability of 𝓟 and, for each n ∈ ℕ, the solvability of its perturbed version 𝓟n, are guaranteed by an existence and uniqueness result obtained in literature. Denote by u the solution of Problem 𝓟 and, for each n ∈ ℕ, let un be a solution of Problem 𝓟n. The main result of this paper states the strong convergence of un → u in X, as n → ∞. We show that the main result extends a number of results previously obtained in the study of Problem 𝓟. Finally, we illustrate the use of our abstract results in the study of a mathematical model which describes the contact of an elastic body with a rigid-deformable foundation and provide the corresponding mechanical interpretations.

中文翻译：

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椭圆变分-半变分不等式的收敛结果

摘要 我们考虑自反 Banach 空间中的椭圆变分半变分不等式 𝓟，由一组约束 K、一个非线性算子 A 和一个元素 f 控制。我们将这个不等式与一个变分半变分不等式的序列 {𝓟n} 相关联，这样，对于每个 n ∈ ℕ，不等式 𝓟n 是通过扰动数据 K 和 A 获得的，此外，它包含一个由小参数 εn 控制的附加项. 𝓟 的唯一可解性以及对于每个 n ∈ ℕ，其扰动版本 𝓟n 的可解性由文献中获得的存在性和唯一性结果保证。用u 表示问题𝓟 的解，对于每个n ∈ ℕ，令un 成为问题𝓟n 的解。本文的主要结果表明 un → u 在 X 中的强收敛性，即 n → ∞。我们表明，主要结果扩展了先前在问题 𝓟 研究中获得的许多结果。最后，我们说明了我们的抽象结果在数学模型研究中的使用，该模型描述了弹性体与刚性可变形基础的接触，并提供了相应的力学解释。