Bounds for periodic solutions of vector second-order nonlinear differential equations
Consider the system of coupled second-order nonlinear differential equations \[ \ddot x+ f(x, \dot x,t)= 0,\tag{1} \] where \(x= (x_1,\dots, x_n)^T\in \mathbb{R}^n\), \(f\in C(\mathbb{R}^n\times \mathbb{R}^n\times \mathbb{R}, \mathbb{R}^n)\) and \(T\)-periodic in \(t\). Assume that (1) has a \(T\)-periodic solution \(x(t)\not\equiv\) constant. Let for \(1\leq i\leq n\) \[ A_i:= {1\over 2} \Biggl(\max_{t\in [0,T]} x_i(t)- \min_{t\in [0,T]} x_i(t)\Biggr),\quad V_i:= \max_{t\in [0,T]} |\dot x_i(t)|. \] The authors derive upper bounds for \(A_i\) and \(V_i\), \(1\leq i\leq n\), and a lower bound for the period \(T\) in the cases when (1) is autonomous or parametrically excited, that is \(f(0,0,t)\equiv 0\). To this end, the authors consider the auxiliary boundary value problem \[ \ddot x+ g(x,\dot x_0, t)= 0,\quad x(0)= \dot x(T/4)= 0,\tag{2} \] where the function \(g\) is associated with \(f\). They prove that the positive solutions of (2) provide the required bound.
scientific article; zbMATH DE number 5156432