Periodic solutions of second-order systems with one-sided restrictions to the growth of the right-hand side with respect to the first derivative

Abstract

For the system a periodic solution exists if for each i one of the following inequalities holds:

$$\begin{array}{*{20}c} {f_i (t,u_1 , \ldots ,u_n ,p_1 , \ldots ,p_n ) \leqslant A(p_1 , \ldots ,p_{i - 1} )p_i^2 + B(p_1 , \ldots ,p_{i - 1} ),} \\ {f_i (t,u_1 , \ldots ,u_n ,p_1 , \ldots ,p_n ) \geqslant - A(p_1 , \ldots ,p_{i - 1} )p_i^2 - B(p_1 , \ldots ,p_{i - 1} ),} \\ {f_i (t,u_1 , \ldots ,u_n ,p_1 , \ldots ,p_n )signp_i \leqslant A(p_1 , \ldots ,p_{i - 1} )p_i^2 + B(p_1 , \ldots ,p_{i - 1} ),} \\ {f_i (t,u_1 , \ldots ,u_n ,p_1 , \ldots ,p_n )signp_i \geqslant - A(p_1 , \ldots ,p_{i - 1} )p_i^2 - B(p_1 , \ldots ,p_{i - 1} ),} \\ {f_i (t,u_1 , \ldots ,u_n ,p_1 , \ldots ,p_n )signu_i \geqslant - A(p_1 , \ldots ,p_{i - 1} )p_i^2 - B(p_1 , \ldots ,p_{i - 1} )} \\ \end{array} $$

for α(t)≤u≤β(t). Here α(t) and β(t) are the lower and upper vector functions for system (1) and the periodic conditions; A≥0, B≥0. Bibliography: 1 titles.