Let E be a real normed space. A new notion of quasi-boundedness for operators \(A:E\rightarrow 2^E\) is introduced and the following general important result for accretive operators is proved: an accretive operator with zero in the interior of its domain is quasi-bounded. Using this result, a new strong convergence theorem for approximating a zero of an m-accretive operator is proved in a uniformly smooth real Banach space. This result complements the celebrated proximal point algorithm for approximating solutions of \(0\in Au\) in a real Hilbert space where A is a maximal monotone operator. Furthermore, as an application of our theorem, a new strong convergence theorem for approximating a solution of a Hammerstein equation is proved. Finally, several numerical experiments are presented to illustrate the strong convergence of the sequence generated by our algorithm and the results obtained are compared with those obtained using some recent important algorithms.

中文翻译：

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近点算法的强收敛性及其在哈默斯坦方程中的应用

令E为一个实数范空间。引入了算子\（A：E \ rightarrow 2 ^ E \）的拟有界性的新概念，并证明了增生算子的以下一般重要结果：在其域内部具有零的增生算子是拟有界的。利用这一结果，在均匀光滑的实Banach空间中证明了一个新的强收敛定理，用于逼近m-增生算子的零。该结果补充了著名的近点算法，用于逼近真实希尔伯特空间中\（0 \ in Au \）的解，其中A是最大单调算符。此外，作为我们定理的应用，证明了一种新的强收敛定理，用于近似Hammerstein方程的解。最后，提出了几个数值实验来说明我们的算法生成的序列的强收敛性，并将所获得的结果与使用一些最近的重要算法所获得的结果进行比较。