For the purpose of this article, we introduce a modified form of a generalized system of variational inclusions, called the generalized system of modified variational inclusion problems (GSMVIP). This problem reduces to the classical variational inclusion and variational inequalities problems. Motivated by several recent results related to the subgradient extragradient method, we propose a new subgradient extragradient method for finding a common element of the set of solutions of GSMVIP and the set of a finite family of variational inequalities problems. Under suitable assumptions, strong convergence theorems have been proved in the framework of a Hilbert space. In addition, some numerical results indicate that the proposed method is effective.

中文翻译：

---

实希尔伯特空间中变分包含问题和变分不等式问题的有限族系统的改进次梯度超梯度方法

出于本文的目的，我们介绍了变分包含的广义系统的一种改进形式，称为变分包含问题的广义系统（GSMVIP）。该问题简化为经典的变分包含和变分不等式问题。受最近与次梯度超梯度方法有关的一些结果的启发，我们提出了一种新的次梯度超梯度方法，用于寻找GSMVIP解集和变分不等式问题的有限族集的共同元素。在适当的假设下，已在希尔伯特空间的框架中证明了强收敛定理。另外，一些数值结果表明该方法是有效的。