Nemytskii operator

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 $\mathbb{X},\ \mathbb{Y},\ \mathbb{Z} \neq \varnothing$ be non-empty sets, then $\mathbb{Y}^ \mathbb{X},\ \mathbb{Z}^\mathbb{X}$  — sets of mappings from $\mathbb{X}$  with values in $\mathbb{Y}$  and $\mathbb{Z}$  respectively. The Nemytskii superposition operator $H\ \colon \mathbb{Y}^\mathbb{X} \to \mathbb{Z}^\mathbb{X}$ is the mapping induced by the function $h\ \colon \mathbb{X} \times \mathbb{Y} \to \mathbb{Z}$, and such that for any function $\varphi \in \mathbb{Y}^\mathbb{X}$  its image is given by the rule $$(H\varphi)(x) = h(x, \varphi(x)) \in \mathbb{Z}, \quad \mbox{for all}\ x\in \mathbb{X}.$$ The function $h$  is called the generator of the Nemytskii operator $H$.

Definition of Nemytskii operator
Let &Omega; be a domain (an open and connected set) in n-dimensional Euclidean space. A function f : &Omega; &times; Rm &rarr; R is said to satisfy the Carathéodory conditions if
 * f(x, u) is a continuous function of u for almost all x &isin; &Omega;;
 * f(x, u) is a measurable function of x for all u &isin; Rm.

Given a function f satisfying the Carathéodory conditions and a function u : &Omega; &rarr; Rm, define a new function F(u) : &Omega; &rarr; R by


 * $$F(u)(x) = f \big( x, u(x) \big).$$

The function F is called a Nemytskii operator.

Theorem on Lipschitzian Operators
Suppose that $h: [a, b] \times \mathbb{R} \to \mathbb{R}$, $X = \text{Lip} [a, b]$ and

$$H: \text{Lip} [a, b] \to \text{Lip} [a, b]$$

where operator $H$ is defined as $\left( Hf \right) \left(x\right)$  $= h(x, f(x))$  for any function $f : [a,b] \to \mathbb{R}$  and any $x \in [a,b]$. Under these conditions the operator $H$ is Lipschitz continuous if and only if there exist functions $G, H \in \text{Lip} [a, b]$  such that

$$h(x, y) = G(x)y + H(x), \quad x \in [a, b], \quad y \in \mathbb{R}.$$

Boundedness theorem
Let &Omega; be a domain, let 1 &lt; p &lt; +&infin; and let g &isin; Lq(&Omega;; R), with


 * $$\frac1{p} + \frac1{q} = 1.$$

Suppose that f satisfies the Carathéodory conditions and that, for some constant C and all x and u,


 * $$\big| f(x, u) \big| \leq C | u |^{p - 1} + g(x).$$

Then the Nemytskii operator F as defined above is a bounded and continuous map from Lp(&Omega;; Rm) into Lq(&Omega;; R).