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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 |
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). Google Scholar |
[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). Google Scholar |
[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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