A Priori Estimate and Existence of Periodic Solutions for a Certain Class of Systems of Nonlinear Second-Order Ordinary Differential Equations on the Plane

摘要

In the present paper, we consider ω-periodic solutions of the system z``=(z`-B_1(t, z))(z`-B_2(t,z))+f(t,z,z`), where ω＞0, z=x1+ix2 ∈C, C is the space of complex numbers, the upper bar stands for complex conjugation, (Bl,B2) 6 Mw, and / 6 i?w,2- Here Mw is the set of all pairs (A1,A2) of continuous mappings A1(t, z) and A2(t, z) that map RxC into C, are w-periodic with respect to t, and are positively homogeneous of order 1 with respect to z;i.e., A_j(t, λz)=λA_j(t,z) for all λ＞0, j = 1,2. Next, Rw,2 is the set of all continuous mappings g (t, Zj,z2) of R x C x C into C that are w-periodic with respect to t and satisfy the condition g (t, z1,z2)/(|z1|+|z2|~2 = 0 (uniformly with respect to t) as |z1|+|z2|→∞.