The $g$-convergence of maximal monotone Nemytskii operators

We consider a sequence of superposition operators (Nemytskii operators) from the space of square-integrable functions on a line segment to a separable Hilbert space. Each term of the sequence is generated by a time-dependent family of maximal monotone operators in the Hilbert space. Under sufficiently general assumptions we show that every superposition operator is maximal monotone and study the $G$-convergence of the respective sequence of Nemytskii operators. The results can be used to study the parametric dependence of solutions to evolutionary inclusions with time-dependent maximal monotone operators.