Maximality of the Sum of a Maximally Monotone Linear Relation and a Maximally Monotone Operator

The most famous open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that Rockafellar’s constraint qualification holds. In this paper, we prove the maximal monotonicity of A?+?B provided that A, B are maximally monotone and A is a linear relation, as soon as Rockafellar’s constraint qualification holds: ${\operatorname{dom}}\,A\cap{\operatorname{int}}\,{\operatorname{dom}}\,B\neq\varnothing$ . Moreover, A?+?B is of type (FPV).