In this paper, three new algorithms are proposed for solving a pseudomonotone equilibrium problem with a Lipschitz-type condition in a 2-uniformly convex and uniformly smooth Banach space. The algorithms are constructed around the $$\phi $$ -proximal mapping associated with cost bifunction. The first algorithm is designed with the prior knowledge of the Lipschitz-type constant of bifunction. This means that the Lipschitz-type constant is an input parameter of the algorithm while the next two algorithms are modified such that they can work without any information of the Lipschitz-type constant, and then they can be implemented more easily. Some convergence theorems are proved under mild conditions. Our results extend and enrich existing algorithms for solving equilibrium problem in Banach spaces. The numerical behavior of the new algorithms is also illustrated via several experiments.

中文翻译：

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求解 Banach 空间中平衡问题的新的超梯度方法

在本文中，提出了三种新算法，用于在 2-均匀凸和均匀光滑的 Banach 空间中解决具有 Lipschitz 类型条件的伪单调平衡问题。算法是围绕与成本双函数相关的 $$\phi $$ -近端映射构建的。第一个算法是利用双函数的 Lipschitz 型常数的先验知识设计的。这意味着 Lipschitz 型常数是算法的输入参数，而接下来的两个算法经过修改，以便它们可以在没有 Lipschitz 型常数的任何信息的情况下工作，然后它们可以更容易地实现。在温和条件下证明了一些收敛定理。我们的结果扩展并丰富了解决巴拿赫空间平衡问题的现有算法。