Let B be a uniformly convex and uniformly smooth real Banach space with dual space $B^{*}$. Let $F:B\to B^{*}$, $K:B^{*} \to B$ be maximal monotone mappings. An iterative algorithm is constructed and the sequence of the algorithm is proved to converge strongly to a solution of the Hammerstein equation $u+KFu=0$. This theorem is a significant improvement of some important recent results which were proved in real Hilbert spaces under the assumption that F and K are maximal monotone continuous and bounded. The continuity and boundedness restrictions on K and F have been dispensed with, using our new method, even in the more general setting considered in our theorems. Finally, numerical experiments are presented to illustrate the convergence of the sequence of our algorithm.

中文翻译：

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实Banach空间中Hammerstein方程解的迭代算法

令B为具有双空间$ B ^ {*} $的一致凸且一致光滑的实Banach空间。令$ F：B \至B ^ {*} $，$ K：B ^ {*} \至B $为最大单调映射。构造了一个迭代算法，证明了该算法的序列能很好地收敛于Hammerstein方程$ u + KFu = 0 $的解。该定理是对一些重要的最新结果的重大改进，这些最新结果在真实的希尔伯特空间中在F和K为最大单调连续且有界的情况下得到了证明。使用我们的新方法，即使在定理中考虑的更一般的情况下，也不再需要对K和F的连续性和有界性的限制。最后，通过数值实验说明了算法序列的收敛性。