An iterative method generates a sequence associated with an equation, that, under appropriate conditions, converges to a solution of that equation. A perturbation of the equation produces also a perturbation of the sequence. In this paper, we study the Ulam stability (the behavior of the solutions of the perturbed equation with respect to the solutions of the exact equation) of an operatorial equation of the form \(x_{n+1}=T_nx_n+a_n\), where \(T_n:X \rightarrow X\), \(n \in \mathbb {N}\), are linear and bounded operators acting on a Banach space X. As applications we obtain some stability results for the case of Volterra, Fredholm and Gram–Schmidt operators. In this way, we improve and complement some results on this topic.

迭代方法生成与方程关联的序列，该序列在适当的条件下收敛到该方程的解。方程的扰动也会产生序列的扰动。在本文中，我们研究形式为（（x_ {n + 1} = T_nx_n + a_n \）的算子方程的Ulam稳定性（扰动方程的解相对于精确方程的解的行为），其中\（T_n：X \ rightarrow X \），\（n \ in \ mathbb {N} \）是作用在Banach空间X上的线性有界算子。作为应用，对于Volterra，Fredholm和Gram–Schmidt算子，我们获得了一些稳定性结果。通过这种方式，我们改进并补充了有关该主题的一些结果。