An answer to S. Simons’ question on the maximal monotonicity of the sum of a maximal monotone linear operator and a normal cone operator”, Set-Valued and Variational Analysis

The question whether or not the sum of two maximal monotone operators is maximal mono-tone under Rockafellar’s constraint qualification — that is, whether or not “the sum theorem ” is true — is themost famous open problem inMonotone Operator Theory. In his 2008monograph “From Hahn-Banach to Monotonicity”, Stephen Simons asked whether or not the sum theorem holds for the special case of a maximal monotone linear operator and a normal cone operator of a closed convex set provided that the interior of the set makes a nonempty intersection with the domain of the linear operator. In this note, we provide an affirmative answer to Simons ’ question. In fact, we show that the sum theorem is true for a maximal monotone linear relation and a normal cone operator. The proof relies on Rockafellar’s formula for the Fenchel conjugate of the sum as well as some results featuring the Fitzpatrick function. 2000 Mathematics Subject Classification