Let X be a uniformly convex and uniformly smooth real Banach space with dual space $X^{*}$ . In this paper, a Mann-type iterative algorithm that approximates the zero of a generalized-Φ-strongly monotone map is constructed. A strong convergence theorem for a sequence generated by the algorithm is proved. Furthermore, the theorem is applied to approximate the solution of a convex optimization problem, a Hammerstein integral equation, and a variational inequality problem. This theorem generalizes, improves, and complements some recent results. Finally, examples of generalized-Φ-strongly monotone maps are constructed and numerical experiments which illustrate the convergence of the sequence generated by our algorithm are presented.

中文翻译：

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广义Φ强单调映射的一个强收敛定理及其应用。

令X是具有对偶空间$ X ^ {*} $的一致凸且一致光滑的实Banach空间。在本文中，构造了一种Mann型迭代算法，该算法逼近广义Φ强单调映射的零。证明了该算法生成的序列的强收敛定理。此外，该定理适用于逼近凸优化问题，Hammerstein积分方程和变分不等式问题的解。该定理可以概括，改进和补充最近的一些结果。最后，构造了广义Φ-强单调映射的例子，并进行了数值实验，说明了我们算法生成的序列的收敛性。