Urysohn integral equation

Builds on Nonlinear integral equation Continuous function

Definition

Let $\Omega$ be a measurable subset of $\R^m$ , for example a closed interval $[a, b]$ . Let $K \colon \Omega \times \Omega \times \R \to \R$ and $f \colon \Omega \to \R$ be given functions. A Urysohn integral equation for an unknown function $\varphi \colon \Omega \to \R$ is $\varphi(x) = f(x) + \int_\Omega K\bigl(x, t, \varphi(t)\bigr)\, dt, \quad x \in \Omega.$ The function $K$ is the Urysohn kernel, and the map $\varphi \mapsto \int_\Omega K(x, t, \varphi(t))\, dt$ is the Urysohn operator.

Remarks

In the classical setting $\Omega = [a, b]$ , $K$ and $f$ are continuous, and $\varphi$ is sought among continuous functions. If $K(x, t, u) = k(x, t)\, u$ , the equation is a linear Fredholm equation of the second kind. If $K(x, t, u) = k(x, t)\, g(t, u)$ , it is a Hammerstein equation. Some authors also call $\int_\Omega K(x, t, \varphi(t))\, dt = f(x)$ a Urysohn equation of the first kind.

Example: on $[0, 1]$ , take $f(x) = x$ and $K(x, t, u) = \sin\bigl(x(u - t)\bigr)$ . Then $\varphi(x) = x$ is a solution: the integrand is $\sin 0 = 0$ , so the right side is $x$ .

Non-example: $\varphi(x)^3 = f(x) + \int_\Omega k(x, t)\, \varphi(t)\, dt$ is a nonlinear integral equation, but not a Urysohn equation, since the unknown appears outside the integral through $\varphi(x)^3$ and not alone.

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Tag	`24RRYYGC7S`
URI	`m2://24RRYYGC7S`
Kind	definition
Writing	W0 stub
MSC	45G10
Defines	Urysohn integral equation
Uses	Nonlinear integral equation, Continuous function

Urysohn integral equation

Definition

Remarks

Reading path

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