On periodic solutions of differential equations with continuous right-hand sides on Lie groups

Yu. E. Gliklikh

Journal of computational and engineering mathematics, Tome 11 (2024) no. 1, pp. 3-10.

Résumé

We apply the method of guiding potentials to obtain an existence of periodic solution theorem to a differential equation with continuous periodic right-hand side on a Lie group, i.e., the solution of the Cauchy problem for this equation is not unique. To avoid this difficulty we elaborate the machinery of integral operators with parallel translation such that for a $T$-periodic ordinary differential equation (i.e., a vector field) on a Lie group with continuous right-hand side the fixed points of those operators are $T$-periodic solutions. It is shown that under some natural conditions the second iteration of such operator is completely continuous. The method of guiding potentials with those operators allows us to obtain the existence theorem we are looking for. The paper contains a short survey of the theory of integral operators with parallel translation and a modification of the construction of topological index applicable on the manifolds.

Détail

Keywords: Lie groups, ordinary differential equations, integral operators with parallel translation, topological index on manifold, periodic solutions.