Periodic solutions of second order non-autonomous singular dynamical systems

Abstract In this paper, we establish two different existence results of positive periodic solutions for second order non-autonomous singular dynamical systems. The first one is based on a nonlinear alternative principle of Leray–Schauder and the result is applicable to the case of a strong singularity as well as the case of a weak singularity. The second one is based on Schauder's fixed point theorem and the result sheds some new light on problems with weak singularities and proves that in some situations weak singularities may help create periodic solutions. Recent results in the literature are generalized and significantly improved.

[1]  Meirong Zhang,et al.  Multiplicity of positive periodic solutions to superlinear repulsive singular equations , 2005 .

[2]  Martin Schechter,et al.  Periodic non-autonomous second-order dynamical systems , 2006 .

[3]  Patrick Habets,et al.  Periodic-solutions of Some Lienard Equations With Singularities , 1990 .

[4]  S. Terracini,et al.  On the existence of collisionless equivariant minimizers for the classical n-body problem , 2003, math-ph/0302022.

[5]  Manuel del Pino,et al.  T-periodic solutions for some second order differential equations with singularities , 1992, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[6]  Sergio Solimini,et al.  On periodic solutions of nonlinear differential equations with singularities , 1987 .

[7]  Denis Bonheure,et al.  Forced singular oscillators and the method of lower and upper solutions , 2003 .

[8]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[9]  Xiong Li,et al.  Periodic solutions for some second order differential equations with singularity , 2008 .

[10]  Alessandro Fonda,et al.  Subharmonic solutions for some second-order differential equations with singularities , 1993 .

[11]  Daniel Franco,et al.  COLLISIONLESS ORBITS OF SINGULAR AND NONSINGULAR DYNAMICAL SYSTEMS , 2006 .

[12]  D. O’Regan.  Existence Theory for Nonlinear Ordinary Differential Equations , 1997 .

[13]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[14]  S. Terracini.  Remarks on Periodic Orbits of Dynamical Systems with Repulsive Singularities , 1993 .

[15]  Meirong Zhang.  PERIODIC SOLUTIONS OF DAMPED DIFFERENTIAL SYSTEMS WITH REPULSIVE SINGULAR FORCES , 1999 .

[16]  Pedro J. Torres,et al.  Periodic solutions of singular systems without the strong force condition , 2007 .

[17]  Pedro J. Torres,et al.  Non-collision periodic solutions of forced dynamical systems with weak singularities , 2004 .

[18]  A. Ambrosetti,et al.  Periodic solutions of singular Lagrangian systems , 1993 .

[19]  M. A. Delpino,et al.  Infinitely Many T-Periodic Solutions for a Problem Arising in Nonlinear Elasticity , 1993 .

[20]  Kazunaga Tanaka.  A note on generalized solutions of singular Hamiltonian systems , 1994 .

[21]  Pedro J. Torres,et al.  Applications of Schauder's fixed point theorem to singular differential equations , 2007 .

[22]  Ravi P. Agarwal,et al.  Twin positive periodic solutions of second order singular differential systems , 2005 .

[23]  Sergio Solimini,et al.  On forced dynamical systems with a singularity of repulsive type , 1990 .

[24]  Weigu Li,et al.  A Lyapunov-type stability criterion using ^{} norms , 2002 .

[25]  Meirong Zhang,et al.  Periodic Solutions of Equations of Emarkov-Pinney Type , 2006 .

[26]  Shinji Adachi.  Non-collision periodic solutions of prescribed energy problem for a class of singular Hamiltonian systems , 2005 .

[27]  Pedro J. Torres,et al.  Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem , 2003 .

[28]  Donal O'Regan,et al.  Multiplicity of positive periodic solutions to second order differential equations , 2006 .

[29]  W. Gordon.  Conservative dynamical systems involving strong forces , 1975 .

[30]  Qing-Mei Zhou,et al.  Nonplanar and noncollision periodic solutions for $N$-body problems , 2004 .

[31]  Ravi P. Agarwal,et al.  Multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces , 2003 .

[32]  M. Ramos,et al.  NONCOLLISION PERIODIC SOLUTIONS TO SOME SINGULAR DYNAMICAL SYSTEMS WITH VERY WEAK FORCES , 1995 .

[33]  Irena Rachůnková,et al.  Existence of Nonnegative and Nonpositive Solutions for Second Order Periodic Boundary Value Problems , 2001 .

[34]  Pedro J. Torres,et al.  Weak singularities may help periodic solutions to exist , 2007 .