## Periodic solutions of impulse differential systems in critical cases.(Russian)Zbl 0739.34016

The following linear impulse differential system is considered: $\dot z=A(t)z+f(t), \;t\neq\tau_ i,\;\Delta z\mid_{t=\tau_ i}=S_ iz+a_ i.$ Here $$z\in \mathbb R^ n$$, $$A(t)$$ and $$f(t)$$ are respectively $$n\times n$$ and $$n\times 1$$ piecewise continuous and $$T$$-periodic matrix functions, the constant matrices $$S_ i$$ and the moments $$\tau_ i$$ satisfy $$S_{i+p}=S_ i$$, $$a_{i+p}=a_ i$$, $$\tau_{i+p}=\tau_ i+T(\tau_ 0<0<\tau_ 1<\cdots<\tau_ p<T)$$, the matrices $$I_ n+S_ i$$ are non-degenerate ($$I_ n$$ is the $$n\times n$$ unit matrix). Each solution $$z(t,c)$$, $$z(0,c)=c\in\mathbb R^ n$$, is of the type $z(t,c)=X(t)c+\int^ T_ 0 K(t,\tau)f(\tau)\,d\tau+\sum^ p_{i=1}K(t,\tau_ i)a_ i,$ where $$X(t)$$ $$(X(0)=I_ n)$$ is the matrix fundamental solution of the homogeneous system $\dot z=A(t)z,\;t\neq\tau_ i,\;\Delta z\mid_{t=\tau_ i}=S_ iz$ and $K(t,\tau)=\begin{cases} X(t)X^{-1}(\tau), & 0\leq\tau\leq t\leq T, \\ 0, & 0\leq t\leq\tau\leq T,\end{cases}$ $$K(t,\tau_ i)=K(t,\tau_ i+0)$$, is the Green function for the considered impulse system. The solution $$z(t,c)$$ is $$T$$-periodic if and only if $$z(0,c)=z(T,c)$$, which leads to the algebraic system $Qc=\int^ T_ 0K(t,\tau)f(\tau)\,d\tau+\sum^ p_{i=1}K(t,\tau_ i)a_ i$ with $$Q=I_ n-X(t)$$. The paper deals with the critical case $$\det Q=0$$. The main result is that if $$\text{rank}\, Q=n_ 1$$, then the homogeneous impulse system admits exactly $$r=n-n_ 1$$ linearly independent periodic solutions. A condition the non- homogeneous system to be solvable is given, in which case it is shown that it admits an $$r$$-parametric family of periodic solutions. Similar problems for nonlinear differential systems with small parameters of the type $\dot z=A(t)z+f(t)+\varepsilon Z(z,t,\varepsilon),\;t\neq\tau_ i,\;\Delta z\mid_{t=t_ i}=S_ iz+a_ i+\varepsilon J_ i(z,\varepsilon)$ are considered.

### MSC:

 34A37 Ordinary differential equations with impulses 34C25 Periodic solutions to ordinary differential equations