New results on the existence of periodic solutions of periodically perturbed conservative systems
The system of ordinary differential equations \[ \ddot{x}(t)+ \nabla G(u(t))=f(t),\tag{1} \] is considered, where \(f:\mathbb{R}^1 \rightarrow \mathbb{R}^n\) is a continuous function with period \(2\pi\) and \(G:\mathbb{R}^n \rightarrow \mathbb{R}^1\) has a continuous second partial derivative. Equation (1) is interpreted as the Newtonian equation of motion of a mechanical system subject to conservative internal forces and periodic external forces. The investigation which began in [\textit{Z. Shen} and \textit{M. A. Wolfe}, J. Math. Anal. Appl. 153, No. 1, 78-83 (1990; Zbl 0726.34033)] is continued. Weak sufficient conditions for the existence of a unique \(2\pi\)-periodic solution of (1) with boundary conditions are given. It is shown that some conditions in the previous works are unnecessary.
system of ordinary differential equations
scientific article; zbMATH DE number 1005588