Abstract The 2-sets convex feasibility problem aims at finding a point in the nonempty intersection of two closed convex setsAandBin a Hilbert spaceH. 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 $$\{A_n\}$$ {An} and $$\{B_n\}$$ {Bn} , each of them converging, with respect to the Attouch-Wets variational convergence, respectively, toAandB. Given a starting point $$a_0$$ a0 , we consider the sequences of points obtained by projecting on the “perturbed” sets, i.e., the sequences $$\{a_n\}$$ {an} and $$\{b_n\}$$ {bn} given by $$b_n=P_{B_n}(a_{n-1})$$ bn=PBn(an-1) and $$a_n=P_{A_n}(b_n)$$ an=PAn(bn) . Under appropriate geometrical and topological assumptions on the intersection of the limit sets, we ensure that the sequences $$\{a_n\}$$ {an} and $$\{b_n\}$$ {bn} converge in norm to a point in the intersection ofAandB. In particular, we consider both when the intersection $$A\cap B$$ A∩B reduces to a singleton and when the interior of $$A \cap B$$ A∩B is nonempty. Finally we consider the case in which the limit setsAandBare subspaces.

Abstract The 2-sets convex feasibility problem aims at finding a point in the intersection of two closed convex sets A and B in a normed space X. More generally, we can consider the problem of finding (if possible) two points in A and B, respectively, which minimize the distance between the sets. In the present paper, we study some stability properties for the convex feasibility problem: we consider two sequences of sets, each of them converging, with respect to a suitable notion of set convergence, respectively, to A and B. Under appropriate assumptions on the original problem, we ensure that the solutions of the perturbed problems converge to a solution of the original problem. We consider both the finite-dimensional and the infinite-dimensional case. Moreover, we provide several examples that point out the role of our assumptions in the obtained results.