# American Institute of Mathematical Sciences

October  2014, 19(8): 2593-2601. doi: 10.3934/dcdsb.2014.19.2593

## Periodic solutions to differential equations with a generalized p-Laplacian

 1 Centre of Mathematics and Physics, Technical University of Łódź, 90-924 Łódź, ul. Wólczańska 215, Poland 2 Institute of Mathematics, Technical University of Łódź, 90-924 Łódź, ul. Wólczańska 215, Poland, Poland

Received  October 2013 Revised  February 2014 Published  August 2014

The existence of a periodic solution to nonlinear ODEs with $\varphi$-Laplacian is proved under conditions on functions given in the equation (not on the unknown solutions). The results are applied to a relativistic pendulum equation in a general form.
Citation: Adam Lipowski, Bogdan Przeradzki, Katarzyna Szymańska-Dębowska. Periodic solutions to differential equations with a generalized p-Laplacian. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2593-2601. doi: 10.3934/dcdsb.2014.19.2593
##### References:
 [1] C. Bereanu, P. Jebelean and J. Mawhin, Periodic solutions of pendulum-like perturbations of singular and bounded $\phi$-Laplacians, J. Dynam. Differential Equations, 22 (2010), 463-471. doi: 10.1007/s10884-010-9172-3. [2] 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. [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. doi: 10.1090/S0002-9939-2011-11101-8. [4] H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum, Differential Integral Equations, 23 (2010), 801-810. [5] J. A. Cid and P. J. Torres, On the existence and stability of periodic solutions for pendulum-like equations with friction and $\phi$-Laplacian, Discrete Contin. Dyn. Syst., 33 (2013), 141-152. doi: 10.3934/dcds.2013.33.141. [6] W. Ge and J. Ren, An extension of Mawhin's continuation theorem and its application to boundary value problems with a p-Laplacian, Nonl. Anal. TMA, 58 (2004), 477-488. doi: 10.1016/j.na.2004.01.007. [7] S. Ma and Y. Zhang, Existence of infinitely many periodic solutions for ordinary p-Laplacian systems, J. Math. Anal. Appl., 351 (2009), 469-479. doi: 10.1016/j.jmaa.2008.10.027. [8] R. Manásevich and J. Mawhin, Periodic solutions for nonlinear systems with p-Laplacian-like operators, J. Differential Equations, 145 (1998), 367-393. doi: 10.1006/jdeq.1998.3425. [9] R. Manásevich and J. Mawhin, Boundary value problems for nonlinear perturbations of vector p-Laplacian-like operators, J. Korean Math. Soc., 37 (2000), 665-685. [10] J. Mawhin, Periodic solutions of the forced pendulum: Classical vs relativistic, Le Mathematiche, 65 (2010), 97-107. [11] Xiang Lv, Shiping Lu and Ping Yan, Periodic solutions of non-autonomous ordinary p-Laplacian systems, J. Appl. Math. Comput., 35 (2011), 11-18. doi: 10.1007/s12190-009-0336-4. [12] P. J. Torres, Nondegeneracy of the periodically forced Liénard differential equation with $\phi$-Laplacian, Commun. Contemp. Mathematics, 13 (2011), 283-292. doi: 10.1142/S0219199711004208.

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##### References:
 [1] C. Bereanu, P. Jebelean and J. Mawhin, Periodic solutions of pendulum-like perturbations of singular and bounded $\phi$-Laplacians, J. Dynam. Differential Equations, 22 (2010), 463-471. doi: 10.1007/s10884-010-9172-3. [2] 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. [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. doi: 10.1090/S0002-9939-2011-11101-8. [4] H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum, Differential Integral Equations, 23 (2010), 801-810. [5] J. A. Cid and P. J. Torres, On the existence and stability of periodic solutions for pendulum-like equations with friction and $\phi$-Laplacian, Discrete Contin. Dyn. Syst., 33 (2013), 141-152. doi: 10.3934/dcds.2013.33.141. [6] W. Ge and J. Ren, An extension of Mawhin's continuation theorem and its application to boundary value problems with a p-Laplacian, Nonl. Anal. TMA, 58 (2004), 477-488. doi: 10.1016/j.na.2004.01.007. [7] S. Ma and Y. Zhang, Existence of infinitely many periodic solutions for ordinary p-Laplacian systems, J. Math. Anal. Appl., 351 (2009), 469-479. doi: 10.1016/j.jmaa.2008.10.027. [8] R. Manásevich and J. Mawhin, Periodic solutions for nonlinear systems with p-Laplacian-like operators, J. Differential Equations, 145 (1998), 367-393. doi: 10.1006/jdeq.1998.3425. [9] R. Manásevich and J. Mawhin, Boundary value problems for nonlinear perturbations of vector p-Laplacian-like operators, J. Korean Math. Soc., 37 (2000), 665-685. [10] J. Mawhin, Periodic solutions of the forced pendulum: Classical vs relativistic, Le Mathematiche, 65 (2010), 97-107. [11] Xiang Lv, Shiping Lu and Ping Yan, Periodic solutions of non-autonomous ordinary p-Laplacian systems, J. Appl. Math. Comput., 35 (2011), 11-18. doi: 10.1007/s12190-009-0336-4. [12] P. J. Torres, Nondegeneracy of the periodically forced Liénard differential equation with $\phi$-Laplacian, Commun. Contemp. Mathematics, 13 (2011), 283-292. doi: 10.1142/S0219199711004208.
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