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

Previous Article
Slow motion for equal depth multiple-well gradient systems: The degenerate case
DCDS Home
This Issue
Next Article
Lyapunov inequalities for partial differential equations at radial higher eigenvalues

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

Alberto Boscaggin ^1, and Fabio Zanolin ^2,

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

Abstract
Related Papers

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.

Keywords: parameter dependent equations., lower and upper solutions, subharmonic solutions, Periodic solutions, Poincaré-Birkhoff twist theorem.

Mathematics Subject Classification: Primary: 34C25, 54H25; Secondary: 37J10, 37J4.

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[4]	A. Boscaggin, Subharmonic solutions of planar Hamiltonian systems: a rotation number approach,, Adv. Nonlinear Stud., 11 (2011), 77. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	C. De Coster and P. Habets, "Two-point Boundary Value Problems: Lower and Upper Solutions,", Elsevier B. V., (2006). Google Scholar
[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. Google Scholar
[15]	W.-Y. Ding, Fixed points of twist mappings and periodic solutions of ordinary differential equations, (Chinese), Acta Math. Sinica, 25 (1982), 227. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[22]	J. Franks, Generalizations of the Poincaré-Birkhoff theorem,, Ann. of Math. (2), 128 (1988), 139. doi: 10.2307/1971464. Google Scholar
[23]	M. Furi, M. P. Pera and M. Spadini, Multiplicity of forced oscillations for scalar differential equations,, Electron. J. Differential Equations, 36 (2001). Google Scholar
[24]	E. Gaines and J. Mawhin, "Coincidence Degree, and Nonlinear Differential Equations,", Lecture Notes in Mathematics, 568 (1977). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[29]	J. Mawhin, "Topological Degree Methods in Nonlinear Boundary Value Problems,", CBMS Regional Conference Series in Mathematics, 40 (1979). Google Scholar
[30]	J. Mawhin, Recent results on periodic solutions of the forced pendulum equation,, Rend. Istit. Mat. Univ. Trieste, 19 (1987), 119. Google Scholar
[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. Google Scholar
[32]	J. Mawhin, Global results for the forced pendulum equation,, in, (2004), 533. Google Scholar
[33]	J. Mawhin, Periodic solutions of the forced pendulum: classical vs relativistic,, Matematiche (Catania), 65 (2010), 97. Google Scholar
[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. Google Scholar
[35]	J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,", Applied Mathematical Sciences, 74 (1989). Google Scholar
[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. Google Scholar
[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. Google Scholar
[38]	M. Pliss, "Nonlocal Problems of the Theory of Oscillations,", Academic Press, (1966). Google Scholar
[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. Google Scholar
[40]	P. H. Rabinowitz, On subharmonic solutions of Hamiltonian systems,, Comm. Pure Appl. Math., 33 (1980), 609. doi: 10.1002/cpa.3160330504. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[46]	J. R. Ward, Periodic solutions of ordinary differential equations with bounded nonlinearities,, Topol. Methods Nonlinear Anal., 19 (2002), 275. Google Scholar
[47]	J. Yu, The minimal period problem for the classical forced pendulum equation,, J. Differential Equations, 247 (2009), 672. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[4]	A. Boscaggin, Subharmonic solutions of planar Hamiltonian systems: a rotation number approach,, Adv. Nonlinear Stud., 11 (2011), 77. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	C. De Coster and P. Habets, "Two-point Boundary Value Problems: Lower and Upper Solutions,", Elsevier B. V., (2006). Google Scholar
[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. Google Scholar
[15]	W.-Y. Ding, Fixed points of twist mappings and periodic solutions of ordinary differential equations, (Chinese), Acta Math. Sinica, 25 (1982), 227. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[22]	J. Franks, Generalizations of the Poincaré-Birkhoff theorem,, Ann. of Math. (2), 128 (1988), 139. doi: 10.2307/1971464. Google Scholar
[23]	M. Furi, M. P. Pera and M. Spadini, Multiplicity of forced oscillations for scalar differential equations,, Electron. J. Differential Equations, 36 (2001). Google Scholar
[24]	E. Gaines and J. Mawhin, "Coincidence Degree, and Nonlinear Differential Equations,", Lecture Notes in Mathematics, 568 (1977). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[29]	J. Mawhin, "Topological Degree Methods in Nonlinear Boundary Value Problems,", CBMS Regional Conference Series in Mathematics, 40 (1979). Google Scholar
[30]	J. Mawhin, Recent results on periodic solutions of the forced pendulum equation,, Rend. Istit. Mat. Univ. Trieste, 19 (1987), 119. Google Scholar
[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. Google Scholar
[32]	J. Mawhin, Global results for the forced pendulum equation,, in, (2004), 533. Google Scholar
[33]	J. Mawhin, Periodic solutions of the forced pendulum: classical vs relativistic,, Matematiche (Catania), 65 (2010), 97. Google Scholar
[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. Google Scholar
[35]	J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,", Applied Mathematical Sciences, 74 (1989). Google Scholar
[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. Google Scholar
[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. Google Scholar
[38]	M. Pliss, "Nonlocal Problems of the Theory of Oscillations,", Academic Press, (1966). Google Scholar
[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. Google Scholar
[40]	P. H. Rabinowitz, On subharmonic solutions of Hamiltonian systems,, Comm. Pure Appl. Math., 33 (1980), 609. doi: 10.1002/cpa.3160330504. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[46]	J. R. Ward, Periodic solutions of ordinary differential equations with bounded nonlinearities,, Topol. Methods Nonlinear Anal., 19 (2002), 275. Google Scholar
[47]	J. Yu, The minimal period problem for the classical forced pendulum equation,, J. Differential Equations, 247 (2009), 672. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

[1]	Massimo Tarallo, Zhe Zhou. Limit periodic upper and lower solutions in a generic sense. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 293-309. doi: 10.3934/dcds.2018014
[2]	João Fialho, Feliz Minhós. The role of lower and upper solutions in the generalization of Lidstone problems. Conference Publications, 2013, 2013 (special) : 217-226. doi: 10.3934/proc.2013.2013.217
[3]	Luisa Malaguti, Cristina Marcelli. Existence of bounded trajectories via upper and lower solutions. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 575-590. doi: 10.3934/dcds.2000.6.575
[4]	Daniel Núñez, Pedro J. Torres. Periodic solutions of twist type of an earth satellite equation. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 303-306. doi: 10.3934/dcds.2001.7.303
[5]	Zhibo Cheng, Jingli Ren. Periodic and subharmonic solutions for duffing equation with a singularity. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1557-1574. doi: 10.3934/dcds.2012.32.1557
[6]	Zhengxin Zhou. On the Poincaré mapping and periodic solutions of nonautonomous differential systems. Communications on Pure & Applied Analysis, 2007, 6 (2) : 541-547. doi: 10.3934/cpaa.2007.6.541
[7]	Armengol Gasull, Hector Giacomini, Joan Torregrosa. Explicit upper and lower bounds for the traveling wave solutions of Fisher-Kolmogorov type equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3567-3582. doi: 10.3934/dcds.2013.33.3567
[8]	Nakao Hayashi, Chunhua Li, Pavel I. Naumkin. Upper and lower time decay bounds for solutions of dissipative nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2089-2104. doi: 10.3934/cpaa.2017103
[9]	Luca Biasco, Laura Di Gregorio. Periodic solutions of Birkhoff-Lewis type for the nonlinear wave equation. Conference Publications, 2007, 2007 (Special) : 102-109. doi: 10.3934/proc.2007.2007.102
[10]	Ana Maria Bertone, J.V. Goncalves. Discontinuous elliptic problems in $R^N$: Lower and upper solutions and variational principles. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 315-328. doi: 10.3934/dcds.2000.6.315
[11]	Yuxiang Zhang, Shiwang Ma. Some existence results on periodic and subharmonic solutions of ordinary $P$-Laplacian systems. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 251-260. doi: 10.3934/dcdsb.2009.12.251
[12]	Benjamin B. Kennedy. Multiple periodic solutions of state-dependent threshold delay equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1801-1833. doi: 10.3934/dcds.2012.32.1801
[13]	V. Mastropietro, Michela Procesi. Lindstedt series for periodic solutions of beam equations with quadratic and velocity dependent nonlinearities. Communications on Pure & Applied Analysis, 2006, 5 (1) : 1-28. doi: 10.3934/cpaa.2006.5.1
[14]	Anne Mund, Christina Kuttler, Judith Pérez-Velázquez. Existence and uniqueness of solutions to a family of semi-linear parabolic systems using coupled upper-lower solutions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5695-5707. doi: 10.3934/dcdsb.2019102
[15]	Sergey V. Bolotin, Piero Negrini. Variational approach to second species periodic solutions of Poincaré of the 3 body problem. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1009-1032. doi: 10.3934/dcds.2013.33.1009
[16]	Anouar Bahrouni, Marek Izydorek, Joanna Janczewska. Subharmonic solutions for a class of Lagrangian systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1841-1850. doi: 10.3934/dcdss.2019121
[17]	Peter Poláčik, Eiji Yanagida. Stable subharmonic solutions of reaction-diffusion equations on an arbitrary domain. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 209-218. doi: 10.3934/dcds.2002.8.209
[18]	Xiying Sun, Qihuai Liu, Dingbian Qian, Na Zhao. Infinitely many subharmonic solutions for nonlinear equations with singular $ \phi $-Laplacian. Communications on Pure & Applied Analysis, 2020, 19 (1) : 279-292. doi: 10.3934/cpaa.20200015
[19]	Xiao-Fei Zhang, Fei Guo. Multiplicity of subharmonic solutions and periodic solutions of a particular type of super-quadratic Hamiltonian systems. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1625-1642. doi: 10.3934/cpaa.2016005
[20]	Chungen Liu, Xiaofei Zhang. Subharmonic solutions and minimal periodic solutions of first-order Hamiltonian systems with anisotropic growth. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1559-1574. doi: 10.3934/dcds.2017064

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences