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 onto 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 \$Acap B\$ 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,Acap B) o 0\$ and \$mathrm{dist}(b_n,Acap B) o 0\$. Similar results are obtained also in the case \$A cap B=emptyset\$, considering the set of best approximation pairs instead of \$Acap B\$.

De Bernardi, C. A., Miglierina, E., Regularity and Stability for a Convex Feasibility Problem, <<SET-VALUED AND VARIATIONAL ANALYSIS>>, 2022; 30 (2): 521-542. [doi:10.1007/s11228-021-00602-3] [https://hdl.handle.net/10807/183435]

### 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 onto 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 \$Acap B\$ 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,Acap B) o 0\$ and \$mathrm{dist}(b_n,Acap B) o 0\$. Similar results are obtained also in the case \$A cap B=emptyset\$, considering the set of best approximation pairs instead of \$Acap B\$.
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De Bernardi, C. A., Miglierina, E., Regularity and Stability for a Convex Feasibility Problem, <<SET-VALUED AND VARIATIONAL ANALYSIS>>, 2022; 30 (2): 521-542. [doi:10.1007/s11228-021-00602-3] [https://hdl.handle.net/10807/183435]
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