Existence of periodic solutions for a class of second order ordinary differential equations
This paper studies the existence of a \(2\pi-\)periodic solution to the equation \[ \ddot{x}+g(x)=\mu^{2n+1}p(t)+\mu^{4n+1}q(t,x,\dot{x},\mu), \] where \(n\) is a positive integer, \(\mu\) is a small parameter, the functions \(p\) and \(q(t,x,y,\mu)\) are smooth with period \(2\pi\) in the variable \(t\), and \[ g(x)=x+x^{2n+1}(b+xh(x)), \] where \(b\neq 0\) and \(h(x)\) is smooth too. After doing a re-scaling in the variables \((x,\dot{x})\), an approximation procedure for periodic solutions via averaging method is applied to solve the problem.
scientific article; zbMATH DE number 7341768
scientific article