Continuous-dependence estimate for ODEs

Statement

Suppose $y'=F(t,y)$ and $z'=F(t,z)+e(t)$ remain in a region where $|F(t,y)-F(t,z)|\le L(t)|y-z|$ , with $L\ge0$ integrable. Then for $t\ge s$ ,

|y(t)-z(t)|\le e^{\int_s^tL}|y(s)-z(s)| +\int_s^t e^{\int_\tau^tL}|e(\tau)|\,d\tau.

Derivation and scope

Subtract the integral equations and apply the source form of Gronwall. For perturbed vector fields, put their difference along the comparison trajectory into $e$ . The estimate measures stability on the specified common region; it does not itself keep solutions inside that region.