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On general properties of retarded functional differential equations on manifolds
Multiple critical points for a class of periodic lower semicontinuous functionals
1. | Institute of Mathematics "Simion Stoilow", Romanian Academy, 21, Calea Griviţei, RO-010702-Bucharest, Sector 1, Romania |
2. | Department of Mathematics, West University of Timişoara, 4, Blvd. V. Pârvan RO-300223-Timişoara, Romania |
References:
[1] |
C. Bereanu and J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular $\phi$-Laplacian, J. Differential Equations, 243 (2007), 536-557.
doi: 10.1016/j.jde.2007.05.014. |
[2] |
C. Bereanu, P. Jebelean and J. Mawhin, Variational methods for nonlinear perturbation of singular $\phi$-Laplacians, Rend. Lincei Mat. Appl., 22 (2011), 89-111. |
[3] |
C. Bereanu and P. J. Torres, Existence of at least two periodic solutions of the forced relativistic pendulum, Proc. Amer. Math. Soc., 140 (2012), 2713-2719. |
[4] |
H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum, Differential Integral Equations, 23 (2010), 801-810. |
[5] |
H. Brezis and J. Mawhin, Periodic solutions of Lagrangian systems of relativistic oscillators, Comm. Appl. Anal., 15 (2011), 235-250. |
[6] |
K. C. Chang, On the periodic nonlinearity and the multiplicity of solutions, Nonlinear Anal., 13 (1989), 527-537.
doi: 10.1016/0362-546X(89)90062-X. |
[7] |
E. N. Dancer, On the use of asymptotics in nonlinear boundary value problems, Ann. Mat. Pura Appl., 131 (1982), 167-185.
doi: 10.1007/BF01765151. |
[8] |
W.-Y. Ding, A generalization of the Poincaré-Birkhoff theorem, Proc. Amer. Math. Soc., 88 (1983), 341-346.
doi: 10.1090/S0002-9939-1983-0695272-2. |
[9] |
P. Felmer, Periodic solutions of spatially periodic Hamiltonian systems, J. Differential Equations, 98 (1992), 143-168.
doi: 10.1016/0022-0396(92)90109-Z. |
[10] |
A. Fonda and R. Toader, Periodic solutions of pendulum-like Hamiltonian systems in the plane, Advanced Nonlinear Studies, 12 (2012), 395-408. |
[11] |
G. Fournier and M. Willem, Multiple solutions of the forced double pendulum equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 259-281. |
[12] |
J. Franks, Generalizations of the Poincaré-Birkhoff theorem, Ann. Math., 128 (1988), 139-151.
doi: 10.2307/1971464. |
[13] |
G. Hamel, Ueber erzwungene Schingungen bei endlischen Amplituden, Math. Ann., 86 (1922), 1-13.
doi: 10.1007/BF01458566. |
[14] |
J. Q. Liu, A generalized saddle point theorem, J. Differential Equations, 82 (1989), 372-385.
doi: 10.1016/0022-0396(89)90139-3. |
[15] |
R. Manásevich and J. R. Ward, On a result of Brezis and Mawhin, Proc. Amer. Math. Soc., 140 (2012), 531-539.
doi: 10.1090/S0002-9939-2011-11311-X. |
[16] |
S. Maró, Periodic solutions of a forced relativistic pendulum via twist dynamics, Topol. Meth. Nonlin. Anal., to appear. arXiv:1110.0851 |
[17] |
J. Mawhin, Forced second order conservative systems with periodic nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 415-434. |
[18] |
J. Mawhin, The forced pendulum: a paradigm for nonlinear analysis and dynamical systems, Exposition Math., 6 (1988), 271-287. |
[19] |
J. Mawhin, Multiplicity of solutions of variational systems involving $\phi$-Laplacians with singular $\phi$ and periodic nonlinearities, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 4015-4026. |
[20] |
J. Mawhin, Resonance problems for some non-autonomous ordinary differential equations, preprint. |
[21] |
J. Mawhin and M. Willem, Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations, J. Differential Equations, 52 (1984), 264-287.
doi: 10.1016/0022-0396(84)90180-3. |
[22] |
J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Springer, New York, 1989. |
[23] |
P. H. Rabinowitz, On a class of functionals invariant under a $Z_n$ action, Trans. Amer. Math. Soc., 310 (1988), 303-311.
doi: 10.1090/S0002-9947-1988-0965755-5. |
[24] |
A. Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 77-109. |
[25] |
A. Szulkin, A relative category and applications to critical point theory for strongly indefinite functionals, Nonlinear Anal., 15 (1990), 725-739.
doi: 10.1016/0362-546X(90)90089-Y. |
[26] |
M. Willem, Oscillations forcées de l'équation du pendule, Pub. IRMA Lille, 3 (1981), V-1-V-3. |
show all references
References:
[1] |
C. Bereanu and J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular $\phi$-Laplacian, J. Differential Equations, 243 (2007), 536-557.
doi: 10.1016/j.jde.2007.05.014. |
[2] |
C. Bereanu, P. Jebelean and J. Mawhin, Variational methods for nonlinear perturbation of singular $\phi$-Laplacians, Rend. Lincei Mat. Appl., 22 (2011), 89-111. |
[3] |
C. Bereanu and P. J. Torres, Existence of at least two periodic solutions of the forced relativistic pendulum, Proc. Amer. Math. Soc., 140 (2012), 2713-2719. |
[4] |
H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum, Differential Integral Equations, 23 (2010), 801-810. |
[5] |
H. Brezis and J. Mawhin, Periodic solutions of Lagrangian systems of relativistic oscillators, Comm. Appl. Anal., 15 (2011), 235-250. |
[6] |
K. C. Chang, On the periodic nonlinearity and the multiplicity of solutions, Nonlinear Anal., 13 (1989), 527-537.
doi: 10.1016/0362-546X(89)90062-X. |
[7] |
E. N. Dancer, On the use of asymptotics in nonlinear boundary value problems, Ann. Mat. Pura Appl., 131 (1982), 167-185.
doi: 10.1007/BF01765151. |
[8] |
W.-Y. Ding, A generalization of the Poincaré-Birkhoff theorem, Proc. Amer. Math. Soc., 88 (1983), 341-346.
doi: 10.1090/S0002-9939-1983-0695272-2. |
[9] |
P. Felmer, Periodic solutions of spatially periodic Hamiltonian systems, J. Differential Equations, 98 (1992), 143-168.
doi: 10.1016/0022-0396(92)90109-Z. |
[10] |
A. Fonda and R. Toader, Periodic solutions of pendulum-like Hamiltonian systems in the plane, Advanced Nonlinear Studies, 12 (2012), 395-408. |
[11] |
G. Fournier and M. Willem, Multiple solutions of the forced double pendulum equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 259-281. |
[12] |
J. Franks, Generalizations of the Poincaré-Birkhoff theorem, Ann. Math., 128 (1988), 139-151.
doi: 10.2307/1971464. |
[13] |
G. Hamel, Ueber erzwungene Schingungen bei endlischen Amplituden, Math. Ann., 86 (1922), 1-13.
doi: 10.1007/BF01458566. |
[14] |
J. Q. Liu, A generalized saddle point theorem, J. Differential Equations, 82 (1989), 372-385.
doi: 10.1016/0022-0396(89)90139-3. |
[15] |
R. Manásevich and J. R. Ward, On a result of Brezis and Mawhin, Proc. Amer. Math. Soc., 140 (2012), 531-539.
doi: 10.1090/S0002-9939-2011-11311-X. |
[16] |
S. Maró, Periodic solutions of a forced relativistic pendulum via twist dynamics, Topol. Meth. Nonlin. Anal., to appear. arXiv:1110.0851 |
[17] |
J. Mawhin, Forced second order conservative systems with periodic nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 415-434. |
[18] |
J. Mawhin, The forced pendulum: a paradigm for nonlinear analysis and dynamical systems, Exposition Math., 6 (1988), 271-287. |
[19] |
J. Mawhin, Multiplicity of solutions of variational systems involving $\phi$-Laplacians with singular $\phi$ and periodic nonlinearities, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 4015-4026. |
[20] |
J. Mawhin, Resonance problems for some non-autonomous ordinary differential equations, preprint. |
[21] |
J. Mawhin and M. Willem, Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations, J. Differential Equations, 52 (1984), 264-287.
doi: 10.1016/0022-0396(84)90180-3. |
[22] |
J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Springer, New York, 1989. |
[23] |
P. H. Rabinowitz, On a class of functionals invariant under a $Z_n$ action, Trans. Amer. Math. Soc., 310 (1988), 303-311.
doi: 10.1090/S0002-9947-1988-0965755-5. |
[24] |
A. Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 77-109. |
[25] |
A. Szulkin, A relative category and applications to critical point theory for strongly indefinite functionals, Nonlinear Anal., 15 (1990), 725-739.
doi: 10.1016/0362-546X(90)90089-Y. |
[26] |
M. Willem, Oscillations forcées de l'équation du pendule, Pub. IRMA Lille, 3 (1981), V-1-V-3. |
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