In mathematics, Nemytskii operators are a class of nonlinear operators on Lp spaces with good continuity and boundedness properties. They take their name from the mathematician Viktor Vladimirovich Nemytskii.

General definition of Superposition operator

Let be non-empty sets, then — sets of mappings from \mathbb{X} with values in \mathbb{Y} and \mathbb{Z} respectively. The Nemytskii superposition operator is the mapping induced by the function, and such that for any function its image is given by the rule The function h is called the generator of the Nemytskii operator H.

Definition of Nemytskii operator

Let Ω be a domain (an open and connected set) in n-dimensional Euclidean space. A function f : Ω × Rm → R is said to satisfy the Carathéodory conditions if Given a function f satisfying the Carathéodory conditions and a function u : Ω → Rm, define a new function F(u) : Ω → R by The function F is called a Nemytskii operator.

Theorem on Lipschitzian Operators

Suppose that, and where operator H is defined as for any function and any x \in [a,b]. Under these conditions the operator H is Lipschitz continuous if and only if there exist functions such that

Boundedness theorem

Let Ω be a domain, let 1 < p < +∞ and let g ∈ Lq(Ω; R), with Suppose that f satisfies the Carathéodory conditions and that, for some constant C and all x and u, Then the Nemytskii operator F as defined above is a bounded and continuous map from Lp(Ω; Rm) into Lq(Ω; R).