Nonlinear integral equation

Builds on Integral equation Linear integral equation

Definition

Let $a < b$ . A nonlinear integral equation on $[a, b]$ is an integral equation for an unknown function $\varphi \colon [a, b] \to \C$ that is not a linear integral equation. The most common form is the Urysohn equation $\varphi(x) = f(x) + \int_{I_x} H\left(x, s, \varphi(s)\right) ds \quad \text{for all } x \in [a, b].$ Here $I_x$ is $[a, b]$ or $[a, x]$ , $f$ is given, and $H \colon [a, b] \times [a, b] \times \C \to \C$ is a given function that is not affine in its third variable.

Remarks

Example: $\varphi(x) = x + \int_0^1 \varphi(s)^2 \, ds$ on $[0, 1]$ is nonlinear, with $H(x, s, u) = u^2$ . A solution has the form $\varphi(x) = x + c$ , where $c = \int_0^1 (s + c)^2 \, ds = 1/3 + c + c^2$ . This gives $c^2 = -1/3$ , so the equation has the two complex solutions $\varphi(x) = x \pm i/\sqrt{3}$ and no real one.

Non-example: $\varphi(x) = x + \int_0^1 (x s \, \varphi(s) + s^2) \, ds$ is linear, since $H(x, s, u) = x s u + s^2$ is affine in $u$ .

When $H(x, s, u) = K(x, s) \, g(s, u)$ the equation is called a Hammerstein equation.

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Tag	`N4NTJ6W38N`
URI	`m2://N4NTJ6W38N`
Kind	definition
Writing	W0 stub
MSC	45G10
Defines	Nonlinear integral equation
Uses	Integral equation, Linear integral equation

Nonlinear integral equation

Definition

Remarks

Reading path

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