American Institute of Mathematical Sciences

Advanced
January  2013, 33(1): 141-152. doi: 10.3934/dcds.2013.33.141

On the existence and stability of periodic solutions for pendulum-like equations with friction and $\phi$-Laplacian

J. Ángel Cid 1, and  Pedro J. Torres 2,

1. 

Departamento de Matemáticas, Universidade de Vigo, Higher Technical School of Computer Engineering, 32004, Ourense, Spain
2. 

Departamento de Matemática Aplicada, Universidad de Granada, Facultad de Ciencias, Granada, Spain

Received  April 2011 Revised  January 2012 Published  September 2012

In this paper we study the existence, multiplicity and stability of T-periodic solutions for the equation $\left(\phi(x')\right)'+c\, x'+g(x)=e(t)+s.$
Keywords: pendulum equation, Periodic solution, stability., $\phi$-Laplacian, degree theory.
Mathematics Subject Classification: Primary: 34C25.
Citation: J. Ángel Cid, Pedro J. Torres. On the existence and stability of periodic solutions for pendulum-like equations with friction and $\phi$-Laplacian. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 141-152. doi: 10.3934/dcds.2013.33.141
References:
[1]

H. Amann, "Ordinary Differential Equations. An Introduction to Nonlinear Analysis,", Walter de Gruyter, (1990).   Google Scholar

[2]

C. Bereanu, P. Jebelean and J. Mawhin, Periodic solutions of pendulum-like perturbations of singular and bounded $\Phi$-Laplacians,, Journal of Dynamics and Differential Equations, 22 (2010), 463.  doi: 10.1007/s10884-010-9172-3.  Google Scholar

[3]

C. Bereanu and J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular $\phi$-Laplacian,, J. Differential Equations, 243 (2007), 536.  doi: 10.1016/j.jde.2007.05.014.  Google Scholar

[4]

C. Bereanu and J. Mawhin, Multiple periodic solutions of ordinary differential equations with bounded nonlinearities and $\phi$-Laplacian,, NoDEA, 15 (2008), 159.  doi: 10.1007/s00030-007-7004-x.  Google Scholar

[5]

C. Bereanu and J. Mawhin, Boundary value problems for some nonlinear systems with singular $\phi$-Laplacian,, J. Fixed Point Theory Appl., 4 (2008), 57.   Google Scholar

[6]

C. Bereanu and P. J. Torres, Existence of at least two periodic solutions of the forced relativistic pendulum,, Proc. Am. Math. Soc. 140 (2012), 140 (2012), 2713.   Google Scholar

[7]

H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum,, Differential Integral Equations, 23 (2010), 801.   Google Scholar

[8]

J. Čepička, P. Drábek and J. Jenšíková, On the stability of periodic solutions of the damped pendulum equation,, J. Math. Anal. Appl., 209 (1997), 712.  doi: 10.1006/jmaa.1997.5380.  Google Scholar

[9]

H. Chen and Y. Li, Rate of decay of stable periodic solutions of Duffing equations,, J. Differential Equations, 236 (2007), 493.  doi: 10.1016/j.jde.2007.01.023.  Google Scholar

[10]

J. Chu, J. Lei and M. Zhang, The stability of the equilibrium of a nonlinear planar system and application to the relativistic oscillator,, J. Differential Equations, 247 (2009), 530.   Google Scholar

[11]

J. A. Cid and P. J. Torres, Solvability for some boundary value problems with $\phi$-Laplacian operators,, Discrete Contin. Dyn. Syst., 23 (2009), 727.  doi: 10.3934/dcds.2009.23.727.  Google Scholar

[12]

G. Hamel, Ueber erzwungene Schingungen bei endlischen Amplituden,, Math. Ann., 86 (1922), 1.  doi: 10.1007/BF01458566.  Google Scholar

[13]

J. Mawhin, Global results for the forced pendulum equation,, in, 1 (2004), 533.   Google Scholar

[14]

J. Mawhin, Periodic solutions of the forced pendulum: classical vs relativistic,, Le Matematiche, LXV (2010), 97.   Google Scholar

[15]

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.  doi: 10.1016/0022-0396(84)90180-3.  Google Scholar

[16]

D. S. Mitrinović, J. E. Pečarić and A. M. Fink, "Inequalities Involving Functions and Their Integrals and Derivatives,", Kluwer Academic Publishers, (1991).   Google Scholar

[17]

R. Ortega, Stability and index of periodic solutions of an equation of Duffing type,, Boll. Un. Mat. Italiana, 3-B (1989), 533.   Google Scholar

[18]

R. Ortega, Some applications of the topological degree to stability theory,, in, 472 (1995), 377.   Google Scholar

[19]

R. Ortega, Stability of a periodic problem of Ambrosetti-Prodi type,, Diff. Int. Equ., 3 (1990), 275.   Google Scholar

[20]

P. J. Torres, Periodic oscillations of the relativistic pendulum with friction,, Physics Letters A, 372 (2008), 6386.  doi: 10.1016/j.physleta.2008.08.060.  Google Scholar

[21]

P. J. Torres, Nondegeneracy of the periodically forced Liénard differential equation with $\phi$-Laplacian,, Communications in Contemporary Mathematics, 13 (2011), 283.  doi: 10.1142/S0219199711004208.  Google Scholar

show all references

References:
[1]

H. Amann, "Ordinary Differential Equations. An Introduction to Nonlinear Analysis,", Walter de Gruyter, (1990).   Google Scholar

[2]

C. Bereanu, P. Jebelean and J. Mawhin, Periodic solutions of pendulum-like perturbations of singular and bounded $\Phi$-Laplacians,, Journal of Dynamics and Differential Equations, 22 (2010), 463.  doi: 10.1007/s10884-010-9172-3.  Google Scholar

[3]

C. Bereanu and J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular $\phi$-Laplacian,, J. Differential Equations, 243 (2007), 536.  doi: 10.1016/j.jde.2007.05.014.  Google Scholar

[4]

C. Bereanu and J. Mawhin, Multiple periodic solutions of ordinary differential equations with bounded nonlinearities and $\phi$-Laplacian,, NoDEA, 15 (2008), 159.  doi: 10.1007/s00030-007-7004-x.  Google Scholar

[5]

C. Bereanu and J. Mawhin, Boundary value problems for some nonlinear systems with singular $\phi$-Laplacian,, J. Fixed Point Theory Appl., 4 (2008), 57.   Google Scholar

[6]

C. Bereanu and P. J. Torres, Existence of at least two periodic solutions of the forced relativistic pendulum,, Proc. Am. Math. Soc. 140 (2012), 140 (2012), 2713.   Google Scholar

[7]

H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum,, Differential Integral Equations, 23 (2010), 801.   Google Scholar

[8]

J. Čepička, P. Drábek and J. Jenšíková, On the stability of periodic solutions of the damped pendulum equation,, J. Math. Anal. Appl., 209 (1997), 712.  doi: 10.1006/jmaa.1997.5380.  Google Scholar

[9]

H. Chen and Y. Li, Rate of decay of stable periodic solutions of Duffing equations,, J. Differential Equations, 236 (2007), 493.  doi: 10.1016/j.jde.2007.01.023.  Google Scholar

[10]

J. Chu, J. Lei and M. Zhang, The stability of the equilibrium of a nonlinear planar system and application to the relativistic oscillator,, J. Differential Equations, 247 (2009), 530.   Google Scholar

[11]

J. A. Cid and P. J. Torres, Solvability for some boundary value problems with $\phi$-Laplacian operators,, Discrete Contin. Dyn. Syst., 23 (2009), 727.  doi: 10.3934/dcds.2009.23.727.  Google Scholar

[12]

G. Hamel, Ueber erzwungene Schingungen bei endlischen Amplituden,, Math. Ann., 86 (1922), 1.  doi: 10.1007/BF01458566.  Google Scholar

[13]

J. Mawhin, Global results for the forced pendulum equation,, in, 1 (2004), 533.   Google Scholar

[14]

J. Mawhin, Periodic solutions of the forced pendulum: classical vs relativistic,, Le Matematiche, LXV (2010), 97.   Google Scholar

[15]

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.  doi: 10.1016/0022-0396(84)90180-3.  Google Scholar

[16]

D. S. Mitrinović, J. E. Pečarić and A. M. Fink, "Inequalities Involving Functions and Their Integrals and Derivatives,", Kluwer Academic Publishers, (1991).   Google Scholar

[17]

R. Ortega, Stability and index of periodic solutions of an equation of Duffing type,, Boll. Un. Mat. Italiana, 3-B (1989), 533.   Google Scholar

[18]

R. Ortega, Some applications of the topological degree to stability theory,, in, 472 (1995), 377.   Google Scholar

[19]

R. Ortega, Stability of a periodic problem of Ambrosetti-Prodi type,, Diff. Int. Equ., 3 (1990), 275.   Google Scholar

[20]

P. J. Torres, Periodic oscillations of the relativistic pendulum with friction,, Physics Letters A, 372 (2008), 6386.  doi: 10.1016/j.physleta.2008.08.060.  Google Scholar

[21]

P. J. Torres, Nondegeneracy of the periodically forced Liénard differential equation with $\phi$-Laplacian,, Communications in Contemporary Mathematics, 13 (2011), 283.  doi: 10.1142/S0219199711004208.  Google Scholar

[1]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432
[2]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276
[3]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445
[4]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074
[5]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451
[6]

Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240
[7]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458
[8]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051
[9]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450
[10]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348
[11]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440
[12]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118
[13]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075
[14]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442
[15]

Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020275
[16]

A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441
[17]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469
[18]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136
[19]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345
[20]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences