Dynamic equation on time scales

Remarks

For $\mathbb{T} = \R$, the delta derivative is the ordinary derivative and the equation is the ordinary differential equation $x' = f(t, x)$. For $\mathbb{T} = \Z$, the equation reads $x(t + 1) = x(t) + f(t, x(t))$, a first-order recurrence relation. Example: on $\mathbb{T} = h\Z$ with $h > 0$, the function $x(t) = (1 + h p)^{t/h}$ solves $x^\Delta = p x$, since $\frac{x(t + h) - x(t)}{h} = \frac{(1 + hp) - 1}{h} x(t) = p x(t)$. Non-example: on $\mathbb{T} = \Z$, the function $x(t) = e^t$ does not solve $x^\Delta = x$, because $x(t + 1) - x(t) = (e - 1) e^t \ne e^t$. At a point $t$ with $\sigma(t) > t$, the equation fixes $x(\sigma(t))$ from $x(t)$, so solutions of initial value problems advance across gaps by an explicit step.