Abstract
We survey some recent results on the use of variational methods in proving the existence and multiplicity of periodic solutions of systems of differential equations of the type
or systems of difference equations of the type
when ϕ belongs to a class of suitable homeomorphisms between an open ball and the whole space and F is periodic in the components of q.
Keywords
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Bereanu, C., Jebelean, P., Mawhin, J.: Variational methods for nonlinear perturbations of singular ϕ-Laplacians. Rend. Lincei Mat. Appl. 22, 89–111 (2011)
Bereanu, C., Torres, P.: Existence of at least two periodic solutions of the forced relativistic pendulum. Proc. Am. Math. Soc. 140, 2713–2719 (2012)
Brezis, H., Mawhin, J.: Periodic solutions of the forced relativistic pendulum. Differ. Int. Equ. 23, 801–810 (2010)
Brezis, H., Mawhin, J.: Periodic solutions of Lagrangian systems of relativistic oscillators. Commun. Appl. Anal. 15, 235–250 (2011)
Chang, K.C.: On the periodic nonlinearity and the multiplicity of solutions. Nonlinear Anal. 13, 527–537 (1989)
Guo, Z.M., Yu, J.S.: The existence of periodic and subharmonic solutions of subquadratic second order difference equations. J. London Math. Soc. 68(2), 419–430 (2003)
Manásevich, R., Ward Jr, J.R.: On a result of Brezis and Mawhin. Proc. Amer. Math. Soc. 140, 531–539 (2012)
Mawhin, J.: Forced second order conservative systems with periodic nonlinearity. Ann. Inst. Henri-Poincaré Anal. Non Linéaire 5, 415–434 (1989)
Mawhin, J.: Global results for the forced pendulum equations. In: Cañada, A., Drábek, P., Fonda, A. (eds.) Handbook on Differential Equations. Ordinary Differential Equations, vol. 1, pp. 533–589. Elsevier, Amsterdam (2004)
Mawhin, J.: Periodic solutions of the forced pendulum: classical vs relativistic. Le Matematiche 65, 97–107 (2010)
Mawhin, J.: Multiplicity of solutions of variational systems involving ϕ-Laplacians with singular ϕ and periodic nonlinearities. Discrete Contin. Dyn. Syst. 32, 4015–4026 (2012)
Mawhin, J.: Periodic solutions of second order nonlinear difference systems with ϕ-Laplacian: a variational approach. Nonlinear Anal. 75, 4672–4687 (2012)
Mawhin, J.: Periodic solutions of second order Lagrangian difference systems with bounded or singular ϕ-Laplacian and periodic potential Discrete Contin. Dyn. Syst. S 6, 1065–1076 (2013)
Mawhin, J.: Stability and bifurcation theory for non-Autonomous differential equations. CIME (Cetraro, 2011), Jonhson R., Pera M.P. (eds.) Lect. Notes in Math, vol. 2065, pp. 103–184. Springer, Berlin (2013)
Mawhin, J., Willem, M.: Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations. J. Differ. Equ. 52, 264–287 (1984)
Mawhin, J., Willem, M.: Variational methods and boundary value problems for vector second order differential equations and applications to the pendulum equation. In: Vinti, C. (ed.) Nonlinear Analysis and Optimisation (Bologna, 1982), pp. 181–192, LNM 1107. Springer, Berlin (1984)
Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems. Springer, New York (1989)
Palais, R.S.: Ljusternik-Schnirelmann theory on Banach manifolds. Topology 5, 115–132 (1966)
Rabinowitz, P.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Regional Conference No. 65. American Mathematical Society, Providence (1986)
Rabinowitz, P.: On a class of functionals invariant under a Z n action. Trans. Am. Math. Soc. 310, 303–311 (1988)
Schwartz, J.T.: Nonlinear Functional Analysis. Gordon and Breach, New York (1969)
Szulkin, A.: Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 3, 77–109 (1986)
Szulkin, A.: A relative category and applications to critical point theory for strongly indefinite functionals. Nonlinear Anal. 15, 725–739 (1990)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this paper
Cite this paper
Mawhin, J. (2013). Periodic Solutions of Differential and Difference Systems with Pendulum-Type Nonlinearities: Variational Approaches. In: Pinelas, S., Chipot, M., Dosla, Z. (eds) Differential and Difference Equations with Applications. Springer Proceedings in Mathematics & Statistics, vol 47. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7333-6_7
Download citation
DOI: https://doi.org/10.1007/978-1-4614-7333-6_7
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-7332-9
Online ISBN: 978-1-4614-7333-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)