In this paper, using a Bregman distance technique, we introduce a new single projection process for approximating a common element in the set of solutions of variational inequalities involving a pseudo-monotone operator and the set of common fixed points of a finite family of Bregman quasi-nonexpansive mappings in a real reflexive Banach space. The stepsize of our algorithm is determined by a self-adaptive method, and we prove a strong convergence result under certain mild conditions. We further give some applications of our result to a generalized Nash equilibrium problem and bandwidth allocation problems. We also provide some numerical experiments to illustrate the performance of our proposed algorithm using various convex functions and compare this algorithm with other algorithms in the literature.

中文翻译：

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自反Banach空间中变分不等式和不动点问题的收敛性分析

在本文中，我们使用Bregman距离技术，介绍了一种新的单投影过程，用于逼近包含伪单调算子和Bregman拟有限元族的公共不动点集的变分不等式解集中的公共元素实反身Banach空间中的非扩张映射。我们的算法的步长是通过自适应方法确定的，我们证明了在某些温和条件下的强收敛性结果。我们进一步将我们的结果应用于广义纳什均衡问题和带宽分配问题。我们还提供了一些数值实验，以说明我们使用各种凸函数的算法的性能，并将该算法与文献中的其他算法进行比较。