On the existence of periodic solutions for scalar second order differential equations when only the asymptotic behaviour of the potential is known

Fonda, Alessandro

doi:10.1090/S0002-9939-1993-1154246-4

On the existence of periodic solutions for scalar second order differential equations when only the asymptotic behaviour of the potential is known

Author: Alessandro Fonda
Journal: Proc. Amer. Math. Soc. 119 (1993), 439-445
MSC: Primary 34C25; Secondary 34B15
DOI: https://doi.org/10.1090/S0002-9939-1993-1154246-4
MathSciNet review: 1154246
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Assuming only asymptotic conditions on the potential function, we prove the existence of periodic solutions for equations whose nonlinearity stays below the first curve of Fučik's spectrum.

[1] A. Adje, Sur et sous solutions dans les équations différentielles discontinues avec conditions aux limites non linéaires, Ph.D. Thesis, Louvain-la-Neuve, 1987.
[2] D. G. de Figueiredo and B. Ruf, On a superlinear Sturm-Liouville equation and a related bouncing problem, J. Reine Angew. Math. 421 (1991), 1–22. MR 1129572, https://doi.org/10.1515/crll.1991.421.1
[3] Tong Ren Ding, Rita Iannacci, and Fabio Zanolin, Existence and multiplicity results for periodic solutions of semilinear Duffing equations, J. Differential Equations 105 (1993), no. 2, 364–409. MR 1240400, https://doi.org/10.1006/jdeq.1993.1093
[4] Tong Ren Ding and Fabio Zanolin, Time-maps for the solvability of periodically perturbed nonlinear Duffing equations, Nonlinear Anal. 17 (1991), no. 7, 635–653. MR 1128965, https://doi.org/10.1016/0362-546X(91)90111-D
[5] Lourdes Fernandes and Fabio Zanolin, Periodic solutions of a second order differential equation with one-sided growth restrictions on the restoring term, Arch. Math. (Basel) 51 (1988), no. 2, 151–163. MR 959391, https://doi.org/10.1007/BF01206473
[6] Alessandro Fonda and Fabio Zanolin, On the use of time-maps for the solvability of nonlinear boundary value problems, Arch. Math. (Basel) 59 (1992), no. 3, 245–259. MR 1174402, https://doi.org/10.1007/BF01197322
[7] S. Fučik, Solvability of nonlinear equations and boundary value problems, Reidel, Dortrecht, 1980.
[8] Jean-Pierre Gossez and Pierpaolo Omari, Periodic solutions of a second order ordinary differential equation: a necessary and sufficient condition for nonresonance, J. Differential Equations 94 (1991), no. 1, 67–82. MR 1133541, https://doi.org/10.1016/0022-0396(91)90103-G
[9] Jean-Pierre Gossez and Pierpaolo Omari, A note on periodic solutions for a second order ordinary differential equation, Boll. Un. Mat. Ital. A (7) 5 (1991), no. 2, 223–231 (English, with Italian summary). MR 1120383
[10] Jean Mawhin, Points fixes, points critiques et problèmes aux limites, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 92, Presses de l’Université de Montréal, Montreal, QC, 1985 (French). MR 789982
[11] Z. Opial, Sur les périodes des solutions de l’équation différentielle 𝑥′′+𝑔(𝑥)=0, Ann. Polon. Math. 10 (1961), 49–72 (French). MR 121544, https://doi.org/10.4064/ap-10-1-49-72
[12] Paul H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, vol. 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. MR 845785