On conditions for existence of periodic solutions to a system of differential equations given integral characteristics (Q1921855)

Consider a smooth function \(f(x,y):D\to \mathbb{R}^r\), and a smooth system of equations (1) \(x=A(x,y)u+P(x,y)\), \(\dot y=Q(x,y,u)\), where \(x,u\in \mathbb{R}^m\), \(y\in \mathbb{R}^1\), \(P:D\to \mathbb{R}^m\), \(Q:D\times U\to \mathbb{R}^1\), \(A\) is an \(m\times m\)-matrix. Let \(A(x_0,y_0)u_0+P(x_0,y_0)=0\), \(Q(x_0,y_0,u_0)=0\). A periodic function \(u(t)\) such that system (1) admits a periodic solution \(x(t)\), \(y(t)\) of the same period satisfying the conditions \(T^{-1} \int^T_0 f(x(t),y(t))dt> f(x_0,y_0)\), \(T^{-1} \int^T_0 u(t)dt>u_0\) componentwise, is constructed.