In this paper, we study the problem of finding a common solution to variational inequality and fixed point problems for a countable family of Bregman weak relatively nonexpansive mappings in real reflexive Banach spaces. Two inertial-type algorithms with adaptive step size rules for solving the problem are presented and their strong convergence theorems are established. The usage of the Bregman distances and the Armijo line search technique (which avoids the need to know a priori the Lipschitz constant of the involved operators), enable great flexibility of the proposed scheme, and besides their theoretical extensions, it might also have a practical potential.

中文翻译：

---

求解 Banach 空间中变分不等式及相关问题的快速简单 Bregman 投影方法

在本文中，我们研究了在实自反 Banach 空间中为可数的 Bregman 弱相对非扩展映射族找到变分不等式和不动点问题的通用解的问题。提出了求解该问题的两种具有自适应步长规则的惯性型算法，并建立了它们的强收敛定理。Bregman 距离和 Armijo 线搜索技术的使用（避免了需要先验地知道所涉及算子的 Lipschitz 常数），使所提出的方案具有很大的灵活性，除了它们的理论扩展外，它还可能具有实际应用潜在的。