The purpose of this paper is to study a forward–backward algorithm for approximating a zero of the sum of maximally monotone mappings in the setting of Banach spaces. Under some mild conditions, we prove a new strong convergence theorem for the algorithm produced by the method in real reflexive Banach spaces. In addition, we give some applications to the minimization problems. Finally, we provide a numerical example, which supports our main result. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

中文翻译：

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Banach空间中最大单调映射的和为零的正向-向后方法的收敛结果

本文的目的是研究一种前向后算法，用于近似设置Banach空间中最大单调映射的和为零。在某些温和条件下，我们证明了该方法在实反自反Banach空间中产生的新的强收敛定理。另外，我们将一些应用程序应用于最小化问题。最后，我们提供一个数值示例，它支持我们的主要结果。我们的定理改进并统一了针对这一重要类别的非线性映射已证明的大多数结果。