In this article, we propose a Krasnosel’skiǐ–Mann-type algorithm for finding a common fixed point of a countably infinite family of nonexpansive operators ${\left(T_n\right)}_{n\ge 0}$ in Hilbert spaces. We formulate an asymptotic property which the family ${\left(T_n\right)}_{n\ge 0}$ has to fulfill such that the sequence generated by the algorithm converges strongly to the element in ${\bigcap }_{n\ge 0}Fix{T}_n$ with minimum norm. Based on this, we derive a forward–backward algorithm that allows variable step sizes and generates a sequence of iterates that converge strongly to the zero with minimum norm of the sum of a maximally monotone operator and a cocoercive one. We demonstrate the superiority of the forward–backward algorithm with variable step sizes over the one with constant step size by means of numerical experiments on variational image reconstruction and split feasibility problems in infinite dimensional Hilbert spaces.

在本文中，我们提出了一种Krasnosel'skiǐ–Mann型算法，用于找到可数无限的非扩张算子族的公共不动点 ${（{Ť}_{ñ}）}_{ñ\ge 0}$在希尔伯特空间。我们制定了一个家庭的渐近性质${（{Ť}_{ñ}）}_{ñ\ge 0}$ 必须满足，以使算法生成的序列强烈收敛到 ${\bigcap }_{ñ\ge 0}使固定{Ť}_{ñ}$以最小的标准。基于此，我们推导了一种前向-后向算法，该算法允许可变步长，并生成一系列迭代，这些迭代以最大单调算子和一个矫顽算子之和的最小值为最小范数强烈收敛到零。我们通过对无穷维希尔伯特空间中的变分图像重建和分裂可行性问题进行了数值实验，证明了步长可变的正向算法比步长恒定的正向算法的优越性。