## Topological Methods in Nonlinear Analysis

### On existence of periodic solutions for Kepler type problems

#### Abstract

We prove existence and multiplicity of periodic motions for the forced $2$-body problem under conditions of topological character. In different cases, the lower bounds obtained for the number of solutions are related to the winding number of a curve in the plane and the homology of a space in $\mathbb R^3$.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 48, Number 2 (2016), 465-476.

Dates
First available in Project Euclid: 21 December 2016

https://projecteuclid.org/euclid.tmna/1482289224

Digital Object Identifier
doi:10.12775/TMNA.2016.053

Mathematical Reviews number (MathSciNet)
MR3642768

Zentralblatt MATH identifier
1375.34065

#### Citation

Amster, Pablo; Haddad, Julián. On existence of periodic solutions for Kepler type problems. Topol. Methods Nonlinear Anal. 48 (2016), no. 2, 465--476. doi:10.12775/TMNA.2016.053. https://projecteuclid.org/euclid.tmna/1482289224

#### References

• R. Abraham, J.E. Marsden and T.S. Ratiu, Manifolds, Tensor Analysis and Applications, Applied Mathematical Sciences 75, Springer, 1989.
• P. Amster, J. Haddad, R. Ortega and A. J. Ureña, Periodic motions in forced problems of Kepler type, Nonlinear Differential Equations Appl. 18 (2011), 649–657.
• P. Amster and M. Maurette, Periodic solutions of systems with singularities of repulsive type, Adv. Nonlinear Stud. 11 (2011), 201–220.
• A. Banyaga and D. Hurtubise, Lectures on Morse Homology, Kluwer Texts in the Mathematical Sciences. Vol. 29, Springer, 2004.
• J. Cronin, Fixed Points and Topological Degree in Nonlinear Analysis, Math. Surveys No. 11, Amer. Math. Soc. Providence R.I., 1964.
• J. Haddad and P. Amster, Critical point theory in knot complements, Differential Geom. Appl. 36 (2014), 56–65.
• J. Hale, Ordinary Differential Equations, Krieger Publishing Company, 1969.
• H. Hopf, Vektorfelder in $n$-dimensionalen Mannigfaltigkeiten, Math. Ann. 96 (1926/1927), 225–250.
• P. Martinez-Amores, J. Mawhin, R. Ortega and M. Willem, Generic results for the existence of nondegenerate periodic solutions of some differential systems with periodic nonlinearities, J. Differential Equations 91 (1991), 138–148.
• J. Mawhin, Topological degree methods in nonlinear boundary value problems, Regional Conf. Ser. Math. 40, Amer. Math. Soc., Providence, RI, 1979.
• J. Mawhin, Periodic solutions in the goldensSixties: the birth of a Continuation Theorem, Ten Mathematical Essays on Approximation in Analysis and Topology (J. Ferrera, J. López-Gómez, F.R. Ruiz del Portal, eds.), Elsevier 2005, 199–214.
• E.H. Spanier, Algebraic Topology, Springer, 1994.