By using our own approach, we study the strong convergence of an inexact proximal point algorithm with possible unbounded errors for a maximal monotone operator in a Banach space. We give a necessary and sufficient condition for the zero set of the operator to be nonempty and show that, in this case, this iterative sequence converges strongly to a zero of the operator. We present also some applications of our results to equilibrium problems and optimization. Our proximal point algorithm contains, as a special case, the one considered in Hilbert space by Djafari Rouhani and Moradi in (J Optim Theory Appl 172:222–235, 2017) and solves the open problem of extending it to a Banach space, which was stated in that paper and in Djafari Rouhani and Moradi in (J Optim Theory Appl 181:864–882, 2019) . Since the nonexpansiveness of the resolvent operator, which holds in Hilbert space, is not valid anymore in Banach space, our results require new methods of proofs, and significantly improve upon the previous results, both in theory and in applications.

中文翻译：

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Banach空间中不精确近邻点算法的强收敛性

通过使用我们自己的方法，我们研究了 Banach 空间中最大单调算子可能存在无限误差的不精确近端点算法的强收敛性。我们给出了算子的零集非空的充分必要条件，并表明，在这种情况下，这个迭代序列强烈收敛到算子的零。我们还介绍了我们的结果在平衡问题和优化中的一些应用。作为特例，我们的近点算法包含 Djafari Rouhani 和 Moradi 在 (J Optim Theory Appl 172:222–235, 2017) 中在 Hilbert 空间中考虑的算法，并解决了将其扩展到 Banach 空间的开放问题，其中该论文以及 Djafari Rouhani 和 Moradi 在 (J Optim Theory Appl 181:864–882, 2019) 中进行了陈述。由于解析算子的非膨胀性，