### ON PERIODIC SOLUTIONS OF DIFFERENTIAL EQUATIONS WITH CONTINUOUS RIGHT-HAND SIDES ON LIE GROUPS

#### Abstract

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.

#### Keywords

#### Full Text:

PDF#### References

Gliklikh Yu., Kornev S., Obukhovskii V. Guiding Potentials and Periodic Solutions of Differential Equations on Manifolds. Global and Stochastic Analysis, 2019, vol. 6, no. 1, pp. 1-6.

Gliklikh Yu.E. Global and Stochastic Analysis with Applications to Mathematical Physics. London, Springer-Verlag, 2011.

Hirsch M.W. Differential Topology. New York, Springer-Verlag, 1994.

Krasnosel'skii M.A. Topological Methods in the Theory of Nonlinear Integral Equations. New York, Macmillan, 1964.

Krasnosel'skii M.A., Zabreiko P.P. Geometrical Methods of Nonlinear Analysis. Berlin, Springer-Verlag, 1984.

Borisovich Ju.G., Gliklih Ju.E. Fixed Points of Mappings of Banach Manifolds and Some Applications. Nonlinear Analysis: Theory, Methods and Applications, 1980, vol. 4, no. 1, pp. 165-192. DOI: 10.1016/0362-546X(80)90046-2

### Refbacks

- There are currently no refbacks.