January  2013, 33(1): 89-110. doi: 10.3934/dcds.2013.33.89

Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions

1. 

SISSA - ISAS, International School for Advanced Studies, Via Bonomea 265, 34136 Trieste, Italy

2. 

University of Udine, Department of Mathematics and Computer Science, Via delle Scienze 206, 33100 Udine, Italy

Received  August 2011 Revised  October 2011 Published  September 2012

We study the problem of existence and multiplicity of subharmonic solutions for a second order nonlinear ODE in presence of lower and upper solutions. We show how such additional information can be used to obtain more precise multiplicity results. Applications are given to pendulum type equations and to Ambrosetti-Prodi results for parameter dependent equations.
Citation: Alberto Boscaggin, Fabio Zanolin. Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 89-110. doi: 10.3934/dcds.2013.33.89
References:
[1]

J. Belmonte-Beitia and P. J. Torres, Existence of dark soliton solutions of the cubic nonlinear Schródinger equation with periodic inhomogeneous nonlinearity,, J. Nonlinear Math. Phys., 15 (2008), 65.

[2]

C. Bereanu and J. Mawhin, Multiple periodic solutions of ordinary differential equations with bounded nonlinearities and $\varphi$-Laplacian,, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 159. doi: 10.1007/s00030-007-7004-x.

[3]

G. D. Birkhoff and D. C. Lewis, On the periodic motions near a given periodic motion of a dynamical system,, Ann. Mat. Pura Appl., 12 (1934), 117. doi: 10.1007/BF02413852.

[4]

A. Boscaggin, Subharmonic solutions of planar Hamiltonian systems: a rotation number approach,, Adv. Nonlinear Stud., 11 (2011), 77.

[5]

A. Boscaggin and M. Garrione, Resonance and rotation numbers for planar Hamiltonian systems: multiplicity results via the Poincaré-Birkhoff theorem,, Nonlinear Anal., 74 (2011), 4166. doi: 10.1016/j.na.2011.03.051.

[6]

N. P. Các and A. C. Lazer, On second order, periodic, symmetric, differential systems having subharmonics of all sufficiently large orders,, J. Differential Equations, 127 (1996), 426. doi: 10.1006/jdeq.1996.0076.

[7]

A. Capietto, J. Mawhin and F. Zanolin, Continuation theorems for periodic perturbations of autonomous systems,, Trans. Amer. Math. Soc., 329 (1992), 41. doi: 10.1090/S0002-9947-1992-1042285-7.

[8]

J. \'A. Cid and L. Sanchez, Periodic solutions for second order differential equations with discontinuous restoring forces,, J. Math. Anal. Appl., 288 (2003), 349. doi: 10.1016/j.jmaa.2003.08.005.

[9]

C. V. Coffman and D. F. Ullrich, On the continuation of solutions of a certain non-linear differential equation,, Monatsh. Math., 71 (1967), 385. doi: 10.1007/BF01295129.

[10]

E. N. Dancer, On the use of asymptotics in nonlinear boundary value problems,, Ann. Mat. Pura Appl. (4), 131 (1982), 167. doi: 10.1007/BF01765151.

[11]

E. N. Dancer and R. Ortega, The index of Lyapunov stable fixed points in two dimensions,, J. Dynam. Differential Equations, 6 (1994), 631. doi: 10.1007/BF02218851.

[12]

C. De Coster and P. Habets, "Two-point Boundary Value Problems: Lower and Upper Solutions,", Elsevier B. V., (2006).

[13]

J. P. Den Hartog, "Mechanical Vibrations,", Dover, (1985).

[14]

T. Ding and F. Zanolin, Subharmonic solutions of second order nonlinear equations: a time-map approach,, Nonlinear Anal., 20 (1993), 509. doi: 10.1016/0362-546X(93)90036-R.

[15]

W.-Y. Ding, Fixed points of twist mappings and periodic solutions of ordinary differential equations, (Chinese), Acta Math. Sinica, 25 (1982), 227.

[16]

W.-Y. Ding, A generalization of the Poincaré-Birkhoff theorem,, Proc. Amer. Math. Soc., 88 (1983), 341. doi: 10.1090/S0002-9939-1983-0695272-2.

[17]

C. Fabry and P. Habets, Periodic solutions of second order differential equations with superlinear asymmetric nonlinearities,, Arch. Math. (Basel), 60 (1993), 266. doi: 10.1007/BF01198811.

[18]

C. Fabry, J. Mawhin and M. N. Nkashama, A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations,, Bull. London Math. Soc., 18 (1986), 173. doi: 10.1112/blms/18.2.173.

[19]

A. Fonda and M. Ramos, Large-amplitude subharmonic oscillations for scalar second-order differential equations with asymmetric nonlinearities,, J. Differential Equations, 109 (1994), 354. doi: 10.1006/jdeq.1994.1055.

[20]

A. Fonda and M. Willem, Subharmonic oscillations of forced pendulum-type equations,, J. Differential Equations, 81 (1989), 215. doi: 10.1016/0022-0396(89)90120-4.

[21]

A. Fonda and F. Zanolin, On the use of time-maps for the solvability of nonlinear boundary value problems,, Arch. Math. (Basel), 59 (1992), 245. doi: 10.1007/BF01197322.

[22]

J. Franks, Generalizations of the Poincaré-Birkhoff theorem,, Ann. of Math. (2), 128 (1988), 139. doi: 10.2307/1971464.

[23]

M. Furi, M. P. Pera and M. Spadini, Multiplicity of forced oscillations for scalar differential equations,, Electron. J. Differential Equations, 36 (2001).

[24]

E. Gaines and J. Mawhin, "Coincidence Degree, and Nonlinear Differential Equations,", Lecture Notes in Mathematics, 568 (1977).

[25]

P. Hartman, On boundary value problems for superlinear second order differential equations,, J. Differential Equations, 26 (1977), 37. doi: 10.1016/0022-0396(77)90097-3.

[26]

P. Le Calvez and J. Wang, Some remarks on the Poincaré-Birkhoff theorem,, Proc. Amer. Math. Soc., 138 (2010), 703. doi: 10.1090/S0002-9939-09-10105-3.

[27]

A. Margheri, C. Rebelo and F. Zanolin, Maslov index, Poincaré-Birkhoff theorem and periodic solutions of asymptotically linear planar Hamiltonian systems,, J. Differential Equations, 183 (2002), 342. doi: 10.1006/jdeq.2001.4122.

[28]

R. Martins and A. J. Ureña, The star-shaped condition on Ding's version of the Poincaré- Birkhoff theorem,, Bull. London Math. Soc., 39 (2007), 803. doi: 10.1112/blms/bdm064.

[29]

J. Mawhin, "Topological Degree Methods in Nonlinear Boundary Value Problems,", CBMS Regional Conference Series in Mathematics, 40 (1979).

[30]

J. Mawhin, Recent results on periodic solutions of the forced pendulum equation,, Rend. Istit. Mat. Univ. Trieste, 19 (1987), 119.

[31]

J. Mawhin, Topological degree and boundary value problems for nonlinear differential equations, in "Topological Methods for Ordinary Differential Equations'' (Montecatini Terme, 1991),, Lecture Notes in Math., 1537 (1993), 74.

[32]

J. Mawhin, Global results for the forced pendulum equation,, in, (2004), 533.

[33]

J. Mawhin, Periodic solutions of the forced pendulum: classical vs relativistic,, Matematiche (Catania), 65 (2010), 97.

[34]

J. Mawhin and M. Willem, Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations,, J. Differential Equations, 52 (1984), 264. doi: 0.1016/0022-0396(84)90180-3.

[35]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,", Applied Mathematical Sciences, 74 (1989).

[36]

R. Ortega, The twist coefficient of periodic solutions of a time-dependent Newton's equation,, J. Dynam. Differential Equations, 4 (1992), 651. doi: 10.1007/BF01048263.

[37]

R. Ortega, Periodic solutions of a Newtonian equation: stability by the third approximation,, J. Differential Equations, 128 (1996), 491. doi: 10.1006/jdeq.1996.0103.

[38]

M. Pliss, "Nonlocal Problems of the Theory of Oscillations,", Academic Press, (1966).

[39]

D. Qian and P. J. Torres, Periodic motions of linear impact oscillators via the successor map,, SIAM J. Math. Anal., 36 (2005), 1707. doi: 10.1137/S003614100343771X.

[40]

P. H. Rabinowitz, On subharmonic solutions of Hamiltonian systems,, Comm. Pure Appl. Math., 33 (1980), 609. doi: 10.1002/cpa.3160330504.

[41]

C. Rebelo, A note on the Poincaré-Birkhoff fixed point theorem and periodic solutions of planar systems,, Nonlinear Anal., 29 (1997), 291. doi: 10.1016/S0362-546X(96)00065-X.

[42]

C. Rebelo and F. Zanolin, Multiplicity results for periodic solutions of second order ODEs with asymmetric nonlinearities,, Trans. Amer. Math. Soc., 348 (1996), 2349. doi: 10.1090/S0002-9947-96-01580-2.

[43]

K. Schmitt and Z. Q. Wang, On critical points for noncoercive functionals and subharmonic solutions of some Hamiltonian systems,, Proceedings of the Conference on Nonlinear Differential Equations (Coral Gables, 5 (2000), 237.

[44]

E. Serra, M. Tarallo and S. Terracini, Subharmonic solutions to second-order differential equations with periodic nonlinearities,, Nonlinear Anal., 41 (2000), 649. doi: 10.1016/S0362-546X(98)00302-2.

[45]

A. J. Ureña, Dynamics of periodic second-order equations between an ordered pair of lower and upper solutions,, Adv. Nonlinear Stud., 11 (2011), 675.

[46]

J. R. Ward, Periodic solutions of ordinary differential equations with bounded nonlinearities,, Topol. Methods Nonlinear Anal., 19 (2002), 275.

[47]

J. Yu, The minimal period problem for the classical forced pendulum equation,, J. Differential Equations, 247 (2009), 672.

[48]

C. Zanini, Rotation numbers, eigenvalues, and the Poincaré-Birkhoff theorem,, J. Math. Anal. Appl., 279 (2003), 290. doi: 10.1016/S0022-247X(03)00012-X.

[49]

C. Zanini and F. Zanolin, A multiplicity result of periodic solutions for parameter dependent asymmetric non-autonomous equations,, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 12 (2005), 343.

[50]

F. Zanolin, Remarks on multiple periodic solutions for nonlinear ordinary differential systems of Liénard type,, Boll. Un. Mat. Ital. B (6), 1 (1982), 683.

show all references

References:
[1]

J. Belmonte-Beitia and P. J. Torres, Existence of dark soliton solutions of the cubic nonlinear Schródinger equation with periodic inhomogeneous nonlinearity,, J. Nonlinear Math. Phys., 15 (2008), 65.

[2]

C. Bereanu and J. Mawhin, Multiple periodic solutions of ordinary differential equations with bounded nonlinearities and $\varphi$-Laplacian,, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 159. doi: 10.1007/s00030-007-7004-x.

[3]

G. D. Birkhoff and D. C. Lewis, On the periodic motions near a given periodic motion of a dynamical system,, Ann. Mat. Pura Appl., 12 (1934), 117. doi: 10.1007/BF02413852.

[4]

A. Boscaggin, Subharmonic solutions of planar Hamiltonian systems: a rotation number approach,, Adv. Nonlinear Stud., 11 (2011), 77.

[5]

A. Boscaggin and M. Garrione, Resonance and rotation numbers for planar Hamiltonian systems: multiplicity results via the Poincaré-Birkhoff theorem,, Nonlinear Anal., 74 (2011), 4166. doi: 10.1016/j.na.2011.03.051.

[6]

N. P. Các and A. C. Lazer, On second order, periodic, symmetric, differential systems having subharmonics of all sufficiently large orders,, J. Differential Equations, 127 (1996), 426. doi: 10.1006/jdeq.1996.0076.

[7]

A. Capietto, J. Mawhin and F. Zanolin, Continuation theorems for periodic perturbations of autonomous systems,, Trans. Amer. Math. Soc., 329 (1992), 41. doi: 10.1090/S0002-9947-1992-1042285-7.

[8]

J. \'A. Cid and L. Sanchez, Periodic solutions for second order differential equations with discontinuous restoring forces,, J. Math. Anal. Appl., 288 (2003), 349. doi: 10.1016/j.jmaa.2003.08.005.

[9]

C. V. Coffman and D. F. Ullrich, On the continuation of solutions of a certain non-linear differential equation,, Monatsh. Math., 71 (1967), 385. doi: 10.1007/BF01295129.

[10]

E. N. Dancer, On the use of asymptotics in nonlinear boundary value problems,, Ann. Mat. Pura Appl. (4), 131 (1982), 167. doi: 10.1007/BF01765151.

[11]

E. N. Dancer and R. Ortega, The index of Lyapunov stable fixed points in two dimensions,, J. Dynam. Differential Equations, 6 (1994), 631. doi: 10.1007/BF02218851.

[12]

C. De Coster and P. Habets, "Two-point Boundary Value Problems: Lower and Upper Solutions,", Elsevier B. V., (2006).

[13]

J. P. Den Hartog, "Mechanical Vibrations,", Dover, (1985).

[14]

T. Ding and F. Zanolin, Subharmonic solutions of second order nonlinear equations: a time-map approach,, Nonlinear Anal., 20 (1993), 509. doi: 10.1016/0362-546X(93)90036-R.

[15]

W.-Y. Ding, Fixed points of twist mappings and periodic solutions of ordinary differential equations, (Chinese), Acta Math. Sinica, 25 (1982), 227.

[16]

W.-Y. Ding, A generalization of the Poincaré-Birkhoff theorem,, Proc. Amer. Math. Soc., 88 (1983), 341. doi: 10.1090/S0002-9939-1983-0695272-2.

[17]

C. Fabry and P. Habets, Periodic solutions of second order differential equations with superlinear asymmetric nonlinearities,, Arch. Math. (Basel), 60 (1993), 266. doi: 10.1007/BF01198811.

[18]

C. Fabry, J. Mawhin and M. N. Nkashama, A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations,, Bull. London Math. Soc., 18 (1986), 173. doi: 10.1112/blms/18.2.173.

[19]

A. Fonda and M. Ramos, Large-amplitude subharmonic oscillations for scalar second-order differential equations with asymmetric nonlinearities,, J. Differential Equations, 109 (1994), 354. doi: 10.1006/jdeq.1994.1055.

[20]

A. Fonda and M. Willem, Subharmonic oscillations of forced pendulum-type equations,, J. Differential Equations, 81 (1989), 215. doi: 10.1016/0022-0396(89)90120-4.

[21]

A. Fonda and F. Zanolin, On the use of time-maps for the solvability of nonlinear boundary value problems,, Arch. Math. (Basel), 59 (1992), 245. doi: 10.1007/BF01197322.

[22]

J. Franks, Generalizations of the Poincaré-Birkhoff theorem,, Ann. of Math. (2), 128 (1988), 139. doi: 10.2307/1971464.

[23]

M. Furi, M. P. Pera and M. Spadini, Multiplicity of forced oscillations for scalar differential equations,, Electron. J. Differential Equations, 36 (2001).

[24]

E. Gaines and J. Mawhin, "Coincidence Degree, and Nonlinear Differential Equations,", Lecture Notes in Mathematics, 568 (1977).

[25]

P. Hartman, On boundary value problems for superlinear second order differential equations,, J. Differential Equations, 26 (1977), 37. doi: 10.1016/0022-0396(77)90097-3.

[26]

P. Le Calvez and J. Wang, Some remarks on the Poincaré-Birkhoff theorem,, Proc. Amer. Math. Soc., 138 (2010), 703. doi: 10.1090/S0002-9939-09-10105-3.

[27]

A. Margheri, C. Rebelo and F. Zanolin, Maslov index, Poincaré-Birkhoff theorem and periodic solutions of asymptotically linear planar Hamiltonian systems,, J. Differential Equations, 183 (2002), 342. doi: 10.1006/jdeq.2001.4122.

[28]

R. Martins and A. J. Ureña, The star-shaped condition on Ding's version of the Poincaré- Birkhoff theorem,, Bull. London Math. Soc., 39 (2007), 803. doi: 10.1112/blms/bdm064.

[29]

J. Mawhin, "Topological Degree Methods in Nonlinear Boundary Value Problems,", CBMS Regional Conference Series in Mathematics, 40 (1979).

[30]

J. Mawhin, Recent results on periodic solutions of the forced pendulum equation,, Rend. Istit. Mat. Univ. Trieste, 19 (1987), 119.

[31]

J. Mawhin, Topological degree and boundary value problems for nonlinear differential equations, in "Topological Methods for Ordinary Differential Equations'' (Montecatini Terme, 1991),, Lecture Notes in Math., 1537 (1993), 74.

[32]

J. Mawhin, Global results for the forced pendulum equation,, in, (2004), 533.

[33]

J. Mawhin, Periodic solutions of the forced pendulum: classical vs relativistic,, Matematiche (Catania), 65 (2010), 97.

[34]

J. Mawhin and M. Willem, Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations,, J. Differential Equations, 52 (1984), 264. doi: 0.1016/0022-0396(84)90180-3.

[35]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,", Applied Mathematical Sciences, 74 (1989).

[36]

R. Ortega, The twist coefficient of periodic solutions of a time-dependent Newton's equation,, J. Dynam. Differential Equations, 4 (1992), 651. doi: 10.1007/BF01048263.

[37]

R. Ortega, Periodic solutions of a Newtonian equation: stability by the third approximation,, J. Differential Equations, 128 (1996), 491. doi: 10.1006/jdeq.1996.0103.

[38]

M. Pliss, "Nonlocal Problems of the Theory of Oscillations,", Academic Press, (1966).

[39]

D. Qian and P. J. Torres, Periodic motions of linear impact oscillators via the successor map,, SIAM J. Math. Anal., 36 (2005), 1707. doi: 10.1137/S003614100343771X.

[40]

P. H. Rabinowitz, On subharmonic solutions of Hamiltonian systems,, Comm. Pure Appl. Math., 33 (1980), 609. doi: 10.1002/cpa.3160330504.

[41]

C. Rebelo, A note on the Poincaré-Birkhoff fixed point theorem and periodic solutions of planar systems,, Nonlinear Anal., 29 (1997), 291. doi: 10.1016/S0362-546X(96)00065-X.

[42]

C. Rebelo and F. Zanolin, Multiplicity results for periodic solutions of second order ODEs with asymmetric nonlinearities,, Trans. Amer. Math. Soc., 348 (1996), 2349. doi: 10.1090/S0002-9947-96-01580-2.

[43]

K. Schmitt and Z. Q. Wang, On critical points for noncoercive functionals and subharmonic solutions of some Hamiltonian systems,, Proceedings of the Conference on Nonlinear Differential Equations (Coral Gables, 5 (2000), 237.

[44]

E. Serra, M. Tarallo and S. Terracini, Subharmonic solutions to second-order differential equations with periodic nonlinearities,, Nonlinear Anal., 41 (2000), 649. doi: 10.1016/S0362-546X(98)00302-2.

[45]

A. J. Ureña, Dynamics of periodic second-order equations between an ordered pair of lower and upper solutions,, Adv. Nonlinear Stud., 11 (2011), 675.

[46]

J. R. Ward, Periodic solutions of ordinary differential equations with bounded nonlinearities,, Topol. Methods Nonlinear Anal., 19 (2002), 275.

[47]

J. Yu, The minimal period problem for the classical forced pendulum equation,, J. Differential Equations, 247 (2009), 672.

[48]

C. Zanini, Rotation numbers, eigenvalues, and the Poincaré-Birkhoff theorem,, J. Math. Anal. Appl., 279 (2003), 290. doi: 10.1016/S0022-247X(03)00012-X.

[49]

C. Zanini and F. Zanolin, A multiplicity result of periodic solutions for parameter dependent asymmetric non-autonomous equations,, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 12 (2005), 343.

[50]

F. Zanolin, Remarks on multiple periodic solutions for nonlinear ordinary differential systems of Liénard type,, Boll. Un. Mat. Ital. B (6), 1 (1982), 683.

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