## Periodic solutions to systems of differential equations with a random impulse action.(English. Ukrainian original)Zbl 0999.34057

Theory Probab. Math. Stat. 63, 131-136 (2001); translation from Teor. Jmovirn. Mat. Stat. 63, 119-124 (2000).
The authors consider $$T$$-periodic systems of differential equations ${dx\over dt}=f(t,x,\xi(t)),\;t\neq t_{i},\quad \Delta x|_{t=t_{i}}=x(t_{i}+0)-x(t_{i}-0)=I_{i}(x,\eta_{i}),$ with $$t\in \mathbb{R}$$, $$x\in \mathbb{R}^{n}$$, $$i\in \mathbb{Z}$$. The function $$f(t,x,y)$$, $$y\in \mathbb{R}^{k},$$ is periodic in $$t$$ with the period $$T$$; the functions $$I_{i}(x,z)$$, $$z\in \mathbb{R}^{l}$$, and time moments $$t_{i}$$ are such that for some positive integer $$p$$: $$I_{i+p}(x,z)=I_{i}(x,z),\;t_{i+p}=t_{i}+T$$. Let $$\xi(t)$$ be a stochastically continuous random process and let $$\eta_{i}$$ be a sequence of random variables which are periodically connected. The authors obtain necessary and sufficient conditions for the existence of periodic solutions to the considered system. Also, for the system $dx/dt=F(x)+\varepsilon g(t,x,\xi(t)), \;t\neq t_{i}, \quad \Delta x|_{t=t_{i}}=\varepsilon I_{i}(x,\eta_{i}),$ where $$\varepsilon>0$$ is a small parameter, sufficient conditions for the existence of periodic solutions are derived.

### MSC:

 34F05 Ordinary differential equations and systems with randomness 34C25 Periodic solutions to ordinary differential equations 34K45 Functional-differential equations with impulses 34K50 Stochastic functional-differential equations