Maximal Monotonicity of a Nemytskii Operator

A family of maximally monotone operators on a separable Hilbert space is considered. The domains of these operators depend on time ranging over an interval of the real line. The space of square-integrable functions on this interval taking values in the same Hilbert space is also considered. On the space of square-integrable functions a superposition (Nemytskii) operator is constructed based on a family of maximally monotone operators. Under fairly general assumptions, the maximal monotonicity of the Nemytskii operator is proved. This result is applied to the family of maximally monotone operators endowed with a pseudodistance in the sense of A. A. Vladimirov, to the family of subdifferential operators generated by a proper convex lower semicontinuous function depending on time, and to the family of normal cones of a moving closed convex set.