The 2-sets convex feasibility problem aims at finding a point in the nonempty intersection of two closed convex sets A and B in a Hilbert space H. The method of alternating projections is the simplest iterative procedure for finding a solution and it goes back to von Neumann. In the present paper, we study some stability properties for this method in the following sense: we consider two sequences of closed convex sets \(\{A_n\}\) and \(\{B_n\}\), each of them converging, with respect to the Attouch-Wets variational convergence, respectively, to A and B. Given a starting point \(a_0\), we consider the sequences of points obtained by projecting on the “perturbed” sets, i.e., the sequences \(\{a_n\}\) and \(\{b_n\}\) given by \(b_n=P_{B_n}(a_{n-1})\) and \(a_n=P_{A_n}(b_n)\). Under appropriate geometrical and topological assumptions on the intersection of the limit sets, we ensure that the sequences \(\{a_n\}\) and \(\{b_n\}\) converge in norm to a point in the intersection of A and B. In particular, we consider both when the intersection \(A\cap B\) reduces to a singleton and when the interior of \(A \cap B\) is nonempty. Finally we consider the case in which the limit sets A and B are subspaces.

2集凸可行性问题的目的是在希尔伯特空间H中找到两个闭合凸集A和B的非空交点。交替投影的方法是找到解决方案的最简单的迭代过程，它可以追溯到冯·诺伊曼（von Neumann）。在本文中，我们从以下意义上研究此方法的一些稳定性：我们考虑闭合凸集\（\ {A_n \} \）和\（\ {B_n \} \）的两个序列，它们每个都收敛，相对于所述Attouch-才子变收敛，分别阿和乙。给定一个起点\（a_0 \），我们考虑通过投影在“扰动”集上获得的点序列，即\（b_n = P_ {B_n}给出的序列\（\ {a_n \} \）和\（\ {b_n \} \）（a_ {n-1}）\）和\（a_n = P_ {A_n}（b_n）\）。在极限集的交集的适当几何和拓扑假设下，我们确保序列\（\ {a_n \} \）和\（\ {b_n \} \）在范数内收敛到A和A的交点。乙。特别是，我们考虑了当交点\（A \ cap B \）减少为单例时以及当\（A \ cap B \）的内部时，我们都考虑是非空的。最后，我们考虑极限集A和B为子空间的情况。