Nemytskii operator

The Nemytskii- or superposition operator is a nonlinear operator in mathematics , which occurs when studying differential and integral equations . It has many favorable properties, for example it maintains continuity and maps limited sets back to limited sets. It is named after the Russian mathematician Viktor Wladimirowitsch Nemyzki .

Motivation and Definition

Considering an ordinary differential equation of the form

${\displaystyle {\frac {dx}{dt}}(t)=f(t,x(t))}$

with functions , and , as you can this by using the by -induced Nemytskii operator ${\displaystyle x:\mathbb {R} \to \mathbb {R} ^{n}}$ ${\displaystyle t\mapsto x(t)}$ ${\displaystyle f:\mathbb {R} \times \mathbb {R} ^{n}\to \mathbb {R} ^{n}}$ ${\displaystyle (t,x)\mapsto f(t,x)}$ ${\displaystyle f}$

${\displaystyle N_{f}:{\text{Abb}}(\mathbb {R} ,\mathbb {R} ^{n})\to {\text{Abb}}(\mathbb {R} ,\mathbb {R} ^{n})}$, ${\displaystyle x\mapsto f(\cdot ,x(\cdot ))}$

understand it as an operator equation:

${\displaystyle {\frac {dx}{dt}}=N_{f}x}$

This can then be examined with the tools of functional analysis and operator theory.

In general, if one has a mapping , where open subsets of Banach spaces and is a compact metric space , one defines the Nemytskii operator induced by ${\displaystyle f:T\times X\to Y}$ ${\displaystyle X,Y}$ ${\displaystyle E,F}$ ${\displaystyle T}$ ${\displaystyle f}$

${\displaystyle N_{f}:{\text{Abb}}(T,X)\to {\text{Abb}}(T,Y)}$, . ${\displaystyle x\mapsto f(\cdot ,x(\cdot ))}$

The conditions for the quantities and are chosen so that it has the properties claimed at the beginning. ${\displaystyle X,Y}$ ${\displaystyle T}$ ${\displaystyle N_{f}}$

Aside from differential equations, the Nemytskii operator can also be used to study integral operators and equations. For example, if one formulates parameter integrals with the Nemytskii operator, one sees that their differentiability is a consequence of the chain rule for the Fréchet derivative . Concatenations of (linear) integral and Nemytskii operators are also called Hammerstein operators.

literature

• Herbert Amann , Joachim Escher : Analysis 2 . 2nd corrected edition. Birkhäuser, Basel - Boston - Berlin 2005, ISBN 3-7643-7105-6 , VII.6 Nemytskii operators and calculus of variations, pp. 204-209 .
• Winfried Kaballo: Basic Course Functional Analysis . 2nd Edition. Springer, Berlin 2017, ISBN 978-3-662-54747-2 , Part I, 4.5 Nonlinear Integral Equations, p. 76-68 .