Method of the small parameter for a class of impulse differential equations. Critical case (Q2706031)

The authors of this interesting paper investigate the \(T\)-periodic impulsive differential equation NEWLINE\[NEWLINE\dot x=f(t,x)+\varepsilon g(t,x,\varepsilon), \quad \dot x \equiv d x/dt,NEWLINE\]NEWLINE \(x\overline\in \sigma (\varepsilon)\) with impulsive source \(\triangle x= I(t,x)+\varepsilon J(t,x,\varepsilon)\), where \(x \in \sigma (\varepsilon)\), \(\varepsilon \in (-\overline\varepsilon ,\overline \varepsilon)\) is a small real parameter, \(x\in \mathbb R^n\), \(n\geq 2\), \(\sigma (\varepsilon)\) is a set of a finite number of hypersurfaces. The moments of impulse effect arise always in case the integral curves of the equation reach a hypersurface whose equation does not depend on time \(t\). The critical case \(\varepsilon =0\) is of great interest due to the fact that the problem under consideration possesses a \(T\)-periodic solution \(p(t)\) and then the corresponding to this solution equation in variations has at least one \(T\)-periodic solution. The authors obtain necessary conditions for the existence of a \(T\)-periodic solution which is close to \(p(t)\) for small \(\varepsilon \). The number of the \(T\)-periodic solutions is discussed.