The aim the paper is to study a large class of variational-hemivariational inequalities involving constraints in a Banach space. First, we establish a general existence theorem for this class. Second, we introduce a sequence of penalized problems without constraints. Under the suitable assumptions, we prove that the Kuratowski upper limit with respect to the weak topology of the sets of solutions to penalized problems, $w\text{--}{lim\; sup}_{n\to \infty }{\mathbb{S}}_n,$ is nonempty and is contained in the set of solutions to original inequality problem. Also, we prove the identity, $w\text{--}{lim\; sup}_{n\to \infty }{\mathbb{S}}_n=s\text{--}{lim\; sup}_{n\to \infty }{\mathbb{S}}_n,$ when operator A satisfies ${\left(S\right)}_{+}$-property. Finally, we illustrate the applicability of the theoretical results and we explore two complicated partial differential systems of elliptic type, which are an elliptic mixed boundary value problem involving a nonlinear nonhomogeneous differential operator with an obstacle effect, and a nonlinear elastic contact problem in mechanics with unilateral constraints, respectively.

本文的目的是研究涉及Banach空间中约束的一大类变分半变分不等式。首先，我们为此类建立一个普遍存在定理。其次，我们引入了一系列无约束的惩罚性问题。在适当的假设下，我们证明了针对惩罚性问题的解集的弱拓扑的Kuratowski上限，$w\text{-}{lim\; sup}_{ñ\to \infty }{\mathbb{小号}}_{ñ}，$是非空的，包含在原始不平等问题的解决方案集中。另外，我们证明身份，$w\text{-}{lim\; sup}_{ñ\to \infty }{\mathbb{小号}}_{ñ}=s\text{-}{lim\; sup}_{ñ\to \infty }{\mathbb{小号}}_{ñ}，$当运算符A满足时${（\mathrm{小号}）}_{+}$-属性。最后，我们说明了理论结果的适用性，并探索了两个复杂的椭圆型偏微分系统，这是一个椭圆形混合边值问题，涉及具有障碍作用的非线性非齐次微分算子，以及一个具有弹性的力学非线性接触问题。单方面的约束。