Single-step autonomous delay differential equation

Definition

Form of the differential equation

This is a particular type of first-order first-degree autonomous delay differential equation, given explicitly as:

$\frac{d(x(t))}{dt}=f(x(t),x(t-\tau ))$

where $f$ is a known function and $\tau >0$ is also known.

Nature of initial value specification

The initial value specification for this type of delay differential equation is a description of $x$ as a function of $t$ on an interval of length $\tau$, typically the left-most such interval in our domain.

Solution method

The solution method is called the method of steps. The idea is that, if the function is known on an interval of the form $[a-\tau ,a]$, we can figure out what it is on $[a,a+\tau ]$, and then repeat the process to determine what the function is on $[a+\tau ,a+2\tau ]$, and continue to proceed in this way to determine the function everywhere on $[a,\mathrm{\infty })$.

Let us say that we know that $x(t)=\phi (t)$ on the interval $[a-\tau ,a]$. Then, $x(t)$ is the solution $\psi (t)$ to the following equation on $[a,a+\tau ]$ subject to the condition $\psi (a)=\phi (a)$:

$\frac{d}{dt}(\psi (t))=f(\psi (t),\phi (t-\tau ))$

This is an ordinary first-order first-degree differential equation in $\phi$ with an initial-value specification, so we expect it to have a unique solution.