## Topological Methods in Nonlinear Analysis

### Existence of non-collision periodic solutions for second order singular dynamical systems

Shuqing Liang

#### Abstract

In this paper, we study the existence of non-collision periodic solutions for the second order singular dynamical systems. We consider the systems where the potential has a repulsive or attractive type behavior near the singularity. The proof is based on Schauder's fixed point theorem involving a new type of cone. The so-called strong force condition is not needed and the nonlinearity could have sign changing behavior. We allow that the Green function is non-negative, so the critical case for the repulsive case is covered. Recent results in the literature are generalized and improved.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 35, Number 1 (2010), 127-137.

Dates
First available in Project Euclid: 21 April 2016

https://projecteuclid.org/euclid.tmna/1461249005

Mathematical Reviews number (MathSciNet)
MR2677434

Zentralblatt MATH identifier
1209.34048

#### Citation

Liang, Shuqing. Existence of non-collision periodic solutions for second order singular dynamical systems. Topol. Methods Nonlinear Anal. 35 (2010), no. 1, 127--137. https://projecteuclid.org/euclid.tmna/1461249005

#### References

• S. Adachi, Non-collision periodic solutions of prescribed energy problem for a class of singular Hamiltonian systems , Topol. Methods Nonlinear Anal., 25(2005), 275–296 \ref\key 2
• A. Ambrosetti and V. Coti Zelati, Periodic Solutions of Singular Lagrangian Systems, Birkhäuser Boston, Boston, MA (1993) \ref\key 3
• D. Bonheure and C. De Coster, Forced singular oscillators and the method of lower and upper solutions , Topol. Methods Nonlinear Anal., 22(2003), 297–317 \ref\key 4
• J. Chu and D. Franco, Non-collision periodic solutions of second order singular dynamical systems , J. Math. Anal. Appl., 344(2008), 898–905 \ref\key 5
• J. Chu and P. J. Torres, Applications of Schauder's fixed point theorem to singular differential equations , Bull. London Math. Soc., 39(2007), 653–660 \ref\key 6
• J. Chu, P. J. Torres and M. Zhang, Periodic solutions of second order non-autonomous singular dynamical systems , J. Differential Equations, 239(2007), 196–212 \ref\key 7
• M. Del Pino and R. Manásevich, Infinitely many $T$-periodic solutions for a problem arising in nonlinear elasticity , J. Differential Equations, 103(1993), 260–277 \ref\key 8
• M. Del Pino, R. Manásevich and A. Montero, $T$-periodic solutions for some second order differential equations with singularities , Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 231–243 \ref\key 9
• D. L. Ferrario and S. Terracini, On the existence of collisionless equivariant minimizers for the classical $n$-body problem , Invent. Math., 155(2004), 305–362 \ref\key 10
• A. Fonda, R. Manásevich and F. Zanolin, Subharmonic solutions for some second order differential equations with singularities , SIAM J. Math. Anal., 24(1993), 1294-1311 \ref\key 11
• D. Franco and P. J. Torres, Periodic solutions of singular systems without the strong force condition , Proc. Amer. Math. Soc., 136(2008), 1229–1236 \ref\key 12
• D. Franco and J. R. L. Webb, Collisionless orbits of singular and nonsingular dynamical systems , Discrete Contin. Dynam. Systems, 15(2006), 747–757 \ref\key 13
• W. B. Gordon, Conservative dynamical systems involving strong forces , Trans. Amer. Math. Soc., 204 (1975), 113–135 \ref\key 14
• D. Jiang, J. Chu and M. Zhang, Multiplicity of positive periodic solutions to superlinear repulsive singular equations , J. Differential Equations, 211 (2005), 282–302 \ref\key 15
• A. C. Lazer and S. Solimini, On periodic solutions of nonlinear differential equations with singularities , Proc. Amer. Math. Soc., 99 (1987), 109–114 \ref\key 16
• I. Rachunková, M. Tvrdý and I. Vrkoč, Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems , J. Differential Equations, 176 (2001), 445–469 \ref\key 17
• S. Solimini, On forced dynamical systems with a singularity of repulsive type , Nonlinear Anal., 14(1990), 489–500 \ref\key 18
• S. Terracini, Remarks on periodic orbits of dynamical systems with repulsive singularities , J. Funct. Anal., 111 (1993), 213–238 \ref\key 19
• P. J. Torres, Existence of one-signed periodic solutions of some second-order differential equations via a Krasnosel'skiĭ fixed point theorem , J. Differential Equations, 190(2003), 643–662 \ref\key 20 ––––, Non-collision periodic solutions of forced dynamical systems with weak singularities , Discrete Contin. Dynam. Systems, 11 (2004), 693–698 \ref\key 21
• M. Zhang, Periodic solutions of damped differential systems with repulsive singular forces , Proc. Amer. Math. Soc., 127 (1999), 401–407 \ref\key 22 ––––, Periodic solutions of equations of Emarkov–Pinney type , Adv. Nonlinear Stud., 6(2006), 57–67