Ordinary differential equations Definition

Functional differential equation

Builds on Ordinary differential equation Initial value problem

Definition

Let $r \ge 0$ and $n \ge 1$ , and let $C = C([-r, 0], \R^n)$ be the space of continuous functions with the supremum norm. For a continuous function $x$ defined on an interval containing $[t - r, t]$ , let $x_t \in C$ be the segment $x_t(\theta) = x(t + \theta)$ for $\theta \in [-r, 0]$ . Let $\Omega \subseteq \R \times C$ be open and let $F \colon \Omega \to \R^n$ . A (retarded) functional differential equation is the equation $x'(t) = F(t, x_t).$ A solution through $(\sigma, \phi) \in \Omega$ is a continuous $x \colon [\sigma - r, \sigma + T) \to \R^n$ , for some $T > 0$ , such that $x_\sigma = \phi$ , $(t, x_t) \in \Omega$ and $x'(t) = F(t, x_t)$ for $t \in (\sigma, \sigma + T)$ .

Remarks

Example: $x'(t) = -\int_{-1}^{0} x(t + \theta) \, d\theta$ is of this form with $r = 1$ and $F(t, \psi) = -\int_{-1}^{0} \psi(\theta) \, d\theta$ .

Non-example: $x'(t) = x(t + 1)$ is not of this form, since $x'(t)$ depends on the value at the later time $t + 1$ , which is not part of $x_t$ .

For $r = 0$ , $C$ is $\R^n$ and the equation is an ordinary differential equation. If $F$ is continuous and locally Lipschitz in its second argument, the initial value problem has a unique local solution through each $(\sigma, \phi) \in \Omega$ . Equations whose derivative term also involves the past, such as $\frac{d}{dt}\bigl(x(t) - c\, x(t - r)\bigr) = F(t, x_t)$ , are called neutral.

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Initial value problem concept
Ordinary differential equation concept
Functional differential equation definition

Tag	`YRBPQS8GHM`
URI	`m2://YRBPQS8GHM`
Kind	definition
Writing	W0 stub
MSC	34K05
Defines	Functional differential equation
Uses	Ordinary differential equation, Initial value problem
Changed	2 November 2026

Functional differential equation

Definition

Remarks

Reading path

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