In this article, we present an inertial subgradient extragradient-type method that uses a non-monotone step size rule to find a numerical solution to equilibrium problems in real Hilbert spaces. The presented iterative scheme is based on an extragradient subgradient method and an inertial-type scheme. In fact, the proposed iterative scheme is effective in terms of performance, and the key advantage derives directly from the use of the variable step size rule, which is revised by each iteration on the basis of the Lipschitz-type constants as well as certain prior iterations. We obtain a weak convergence theorem for a new method by using mild conditions on a bifunction. Applications of the main results are given to solve various nonlinear problems. Several numerical findings are given in order to illustrate the numerical behaviour of the proposed method and to compare it to others.

在本文中，我们提出了一种惯性次梯度外梯度型方法，该方法使用非单调步长规则来找到真实希尔伯特空间中平衡问题的数值解。所提出的迭代方案是基于外梯度次梯度方法和惯性型方案。事实上，所提出的迭代方案在性能方面是有效的，其关键优势直接来自于可变步长规则的使用，该规则在每次迭代的基础上根据 Lipschitz 型常数以及某些先验迭代。我们通过在双函数上使用温和条件获得了一种新方法的弱收敛定理。给出了主要结果的应用，以解决各种非线性问题。