Bounds for solutions to perturbed first order differential equations (Q1902756)

This author obtains bounds for the solutions of a perturbed nonlinear vector differential equation (1) \(dx/dt = \varphi (t,x) + f(t,x)\), \(x \in R^n\), by comparing the Cauchy matrix solution \(Y(t,s)\) of the linear variational equation along a solution \(y(t)\) of the corresponding unperturbed equation \(dy/dt = \varphi (t,y)\) with the solutions of a scalar equation \[ {dz \over dt} = K \exp \left( - \int^t_0 \psi (\tau) d \tau \right) F \left( t,z \exp \left( \int^t_0 \psi (\tau) d \tau \right) \right) \] with \(|Y(t,s) |\leq K \exp (\int^t_s \psi (\tau) d \tau)\) and \(|f(t,x) |\leq F(t,|x |)\). In addition he obtains certain results on stability and existence of periodic solutions for equation (1).