Periodic Solutions for a Class of 2n-Order Ordinary Differential Equations

Wenjin Li, Jiaxuan Sun () and Yanni Pang
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Wenjin Li: School of Statistics and Data Science, Jilin University of Finance and Economics, Changchun 130117, China
Jiaxuan Sun: School of Statistics and Data Science, Jilin University of Finance and Economics, Changchun 130117, China
Yanni Pang: School of Mathematics, Jilin University, Changchun 130021, China

Mathematics, 2025, vol. 13, issue 17, 1-16

Abstract: Periodic solutions of high-order nonlinear differential equations are fundamental in dynamical systems, yet they remain challenging to establish with traditional methods. This paper addresses the existence of periodic solutions in general 2 n -order autonomous and nonautonomous ordinary differential equations. By extending Carathéodory’s variational technique from the calculus of variations, we reformulate the original periodic solution problem as an equivalent higher-order variational problem. The approach constructs a convex function and introduces an auxiliary transformation to enforce convexity in the highest-order term, enabling a tractable operator-theoretic analysis. Within this framework, we prove two main theorems that provide sufficient conditions for periodic solutions in both autonomous and nonautonomous cases. These results generalize the known theory for second-order equations to arbitrary higher-order systems and highlight a connection to the Hamilton–Jacobi theory, offering new insights into the underlying variational structure. Finally, numerical examples validate our theoretical results by confirming the periodic solutions predicted by the theory and demonstrating the approach’s practical applicability.

Keywords: periodic solution; differential equations; calculus of variations; convex functions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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