Regularity and stability for a convex feasibility problem

Abstract: Let us consider two sequences of closed convex sets ${A_n}$ and ${B_n}$ converging with respect to the Attouch-Wets convergence to $A$ and $B$, respectively. 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 ${b_n}$ defined inductively by $b_n=P_{B_n}(a_{n-1})$ and $a_n=P_{A_n}(b_n)$. Suppose that $A\cap B$ (or a suitable substitute if $A \cap B=\emptyset$) is bounded, we prove that if the couple $(A,B)$ is (boundedly) regular then the couple $(A,B)$ is $d$-stable, i.e., for each ${a_n}$ and ${b_n}$ as above we have $\mathrm{dist}(a_n,A\cap B)\to 0$ and $\mathrm{dist}(b_n,A\cap B)\to 0$.