Nemytskii operator

Summary

A Nemytskii operator, also called a superposition operator, turns a function $g(x, u)$ of two variables into a map on functions. It sends $u$ to the function $x \mapsto g(x, u(x))$. It is the simplest kind of nonlinear operator on function spaces, since it acts point by point. Nonlinear integral equations such as the Hammerstein equation are compositions of a linear integral operator with a Nemytskii operator. Much of the theory is about when such an operator maps one $L^p$ space into another and is continuous.