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Existence of at least two periodic solutions of the forced relativistic pendulum. (English) Zbl 1275.34057
Summary: Using Szulkin’s critical point theory, we prove that the relativistic forced pendulum under periodic boundary value conditions \[ \left(\frac{u^\prime}{\sqrt{1-u\prime ^2}}\right)^\prime +\mu \sin u=h(t), \quad u(0)-u(T)=0=u'(0)-u'(T) \] has at least two solutions not differing by a multiple of \( 2\pi \) for any continuous function \( h:[0,T]\to \mathbb R\) with \( \int _0^Th(t)\,dt=0\) and any \( \mu \neq 0\). The existence of at least one solution has recently been proved by Brezis and Mawhin.

MSC:
34C25 Periodic solutions to ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
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[1] Cristian Bereanu, Petru Jebelean, and Jean Mawhin, Radial solutions for Neumann problems involving mean curvature operators in Euclidean and Minkowski spaces, Math. Nachr. 283 (2010), no. 3, 379 – 391. · Zbl 1185.35113
[2] C. Bereanu, P. Jebelean, and J. Mawhin, Variational methods for nonlinear perturbations of singular \( \phi \)-Laplacians, Rend. Lincei Mat. Appl. 22 (2011), 89-111. · Zbl 1219.35062
[3] C. Bereanu, P. Jebelean, and J. Mawhin, Radial solutions of Neumann problems involving mean extrinsic curvature and periodic nonlinearities, preprint. · Zbl 1262.35088
[4] C. Bereanu and J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular \?-Laplacian, J. Differential Equations 243 (2007), no. 2, 536 – 557. · Zbl 1148.34013
[5] Haim Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011. · Zbl 1220.46002
[6] Haïm Brezis and Jean Mawhin, Periodic solutions of the forced relativistic pendulum, Differential Integral Equations 23 (2010), no. 9-10, 801 – 810. · Zbl 1240.34207
[7] E. N. Dancer, On the use of asymptotics in nonlinear boundary value problems, Ann. Mat. Pura Appl. (4) 131 (1982), 167 – 185. · Zbl 0519.34011
[8] N. Ghoussoub and D. Preiss, A general mountain pass principle for locating and classifying critical points, Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989), no. 5, 321 – 330 (English, with French summary). · Zbl 0711.58008
[9] Georg Hamel, Über erzwungene Schwingungen bei endlichen Amplituden, Math. Ann. 86 (1922), no. 1-2, 1 – 13 (German). · JFM 48.0519.03
[10] Salvatore A. Marano and Dumitru Motreanu, A deformation theorem and some critical point results for non-differentiable functions, Topol. Methods Nonlinear Anal. 22 (2003), no. 1, 139 – 158. · Zbl 1213.58010
[11] J. Mawhin and M. Willem, Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations, J. Differential Equations 52 (1984), no. 2, 264 – 287. · Zbl 0557.34036
[12] F. Obersnel and P. Omari, Multiple bounded variation solutions of a periodicaly perturbed sine-curvature equation, preprint. · Zbl 1234.34020
[13] Patrizia Pucci and James Serrin, Extensions of the mountain pass theorem, J. Funct. Anal. 59 (1984), no. 2, 185 – 210. · Zbl 0564.58012
[14] Andrzej Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), no. 2, 77 – 109 (English, with French summary). · Zbl 0612.58011
[15] M. Willem, Oscillations forcées de l’équation du pendule, Pub. IRMA Lille, 3 (1981), V-1-V-3.
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