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The 2-sets convex feasibility problem aims at finding a point in the nonemptyintersection of two closed convex sets $A$ and $B$ in a Hilbert space $X$. Themethod of alternating projections is the simplest iterative procedure forfinding a solution and it goes back to von Neumann. In the present paper, westudy some stability properties for this method in the following sense: weconsider two sequences of sets, each of them converging, with respect to theAttouch-Wets variational convergence, respectively, to $A$ and $B$. Given astarting point $a_0$, we consider the sequences of points obtained byprojecting 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 appropriategeometrical and topological assumptions on the intersection of the limit sets,we ensure that the sequences $\{a_n\}$ and $\{b_n\}$ converge in norm to apoint in the intersection of $A$ and $B$. In particular, we consider both whenthe 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.

A variational approach to the alternating projections method