American Institute of Mathematical Sciences

Advanced
July  2011, 16(1): 385-392. doi: 10.3934/dcdsb.2011.16.385

Positive periodic solution for Brillouin electron beam focusing system

Jingli Ren 1, , Zhibo Cheng 1, and  Stefan Siegmund 2,

1. 

Dept. of Math., Zhengzhou University, Zhengzhou 450001, China, China
2. 

Dept. of Math., Dresden University of Technology, Dresden 01062, Germany

Received  March 2010 Revised  July 2010 Published  April 2011

An experimental conjecture on the existence of positive periodic solutions for the Brillouin electron beam focusing system $x''+a(1+\cos2t)x=\frac{1}{x}$ for $0 < a < 1$ is proved, using a topological degree theorem by Mawhin.
Keywords: Brillouin electron beam focusing system, singularity., Liénard equation.
Mathematics Subject Classification: Primary: 34B27; Secondary: 37C5.
Citation: Jingli Ren, Zhibo Cheng, Stefan Siegmund. Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 385-392. doi: 10.3934/dcdsb.2011.16.385
References:
[1]

V. Bevc, J. L. Palmer and C. Süsskind, On the design of the transition region of axisymmetric, magnetically focused beam valves,, J. British Inst. Radio Engineer, 18 (1958), 696.   Google Scholar

[2]

T. R. Ding, "Applications of Qualitative Methods of Ordinary Differential Equations,", Higher Education Press, (2004).   Google Scholar

[3]

T. R. Ding, A boundary value problem for the periodic Brillouin focusing system,, Acta Sci. Natru. Univ. Pekinensis, 11 (1965), 31.   Google Scholar

[4]

Weigao Ge, "Boundary Value Problems for Nonlinear Ordinary Differential Equations,", Science Press, (2007).   Google Scholar

[5]

J. Mawhin, Topological degree and boundary value problems for nonlinear differental equations,, Topological Methods for Ordinary Differential Equations, 1537 (1993), 74.  doi: 10.1007/BFb0085076.  Google Scholar

[6]

P. J. Torres, Existence and uniquenness of elliptic periodic solutions of the Brillouin electron beam focusing system,, Math. Meth. Appl. Sci., 23 (2000), 1139.  doi: 10.1002/1099-1476(20000910)23:13<1139::AID-MMA155>3.0.CO;2-J.  Google Scholar

[7]

Y. Ye and X. Wang, Nonlinear differential equations in electron beam focusing theory,, Acta Math. Appl. Sinica, 1 (1978), 13.   Google Scholar

[8]

M. R. Zhang, Periodic solutions of Liénard equations with singular forces of repulsive type,, J. Math. Anal. Appl., 203 (1996), 254.  doi: 10.1006/jmaa.1996.0378.  Google Scholar

[9]

M. R. Zhang, Nonuniform nonresonance at the first eigenvalue of the $p$-Laplacian,, Nonlinear Analysis TMA, 29 (1997), 41.  doi: 10.1016/S0362-546X(96)00037-5.  Google Scholar

show all references

References:
[1]

V. Bevc, J. L. Palmer and C. Süsskind, On the design of the transition region of axisymmetric, magnetically focused beam valves,, J. British Inst. Radio Engineer, 18 (1958), 696.   Google Scholar

[2]

T. R. Ding, "Applications of Qualitative Methods of Ordinary Differential Equations,", Higher Education Press, (2004).   Google Scholar

[3]

T. R. Ding, A boundary value problem for the periodic Brillouin focusing system,, Acta Sci. Natru. Univ. Pekinensis, 11 (1965), 31.   Google Scholar

[4]

Weigao Ge, "Boundary Value Problems for Nonlinear Ordinary Differential Equations,", Science Press, (2007).   Google Scholar

[5]

J. Mawhin, Topological degree and boundary value problems for nonlinear differental equations,, Topological Methods for Ordinary Differential Equations, 1537 (1993), 74.  doi: 10.1007/BFb0085076.  Google Scholar

[6]

P. J. Torres, Existence and uniquenness of elliptic periodic solutions of the Brillouin electron beam focusing system,, Math. Meth. Appl. Sci., 23 (2000), 1139.  doi: 10.1002/1099-1476(20000910)23:13<1139::AID-MMA155>3.0.CO;2-J.  Google Scholar

[7]

Y. Ye and X. Wang, Nonlinear differential equations in electron beam focusing theory,, Acta Math. Appl. Sinica, 1 (1978), 13.   Google Scholar

[8]

M. R. Zhang, Periodic solutions of Liénard equations with singular forces of repulsive type,, J. Math. Anal. Appl., 203 (1996), 254.  doi: 10.1006/jmaa.1996.0378.  Google Scholar

[9]

M. R. Zhang, Nonuniform nonresonance at the first eigenvalue of the $p$-Laplacian,, Nonlinear Analysis TMA, 29 (1997), 41.  doi: 10.1016/S0362-546X(96)00037-5.  Google Scholar

[1]

Teresa D'Aprile. Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains. Communications on Pure & Applied Analysis, 2021, 20 (1) : 159-191. doi: 10.3934/cpaa.2020262
[2]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298
[3]

Michiel Bertsch, Danielle Hilhorst, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3117-3142. doi: 10.3934/dcds.2019226
[4]

Jintai Ding, Zheng Zhang, Joshua Deaton. The singularity attack to the multivariate signature scheme HIMQ-3. Advances in Mathematics of Communications, 2021, 15 (1) : 65-72. doi: 10.3934/amc.2020043
[5]

Xin Zhong. Singularity formation to the nonhomogeneous magneto-micropolar fluid equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021021
[6]

Kai Yang. Scattering of the focusing energy-critical NLS with inverse square potential in the radial case. Communications on Pure & Applied Analysis, 2021, 20 (1) : 77-99. doi: 10.3934/cpaa.2020258
[7]

Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $ \beta $-transformation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267
[8]

Zongyuan Li, Weinan Wang. Norm inflation for the Boussinesq system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020353
[9]

Marc Homs-Dones. A generalization of the Babbage functional equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 899-919. doi: 10.3934/dcds.2020303
[10]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021002
[11]

Bilel Elbetch, Tounsia Benzekri, Daniel Massart, Tewfik Sari. The multi-patch logistic equation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021025
[12]

Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301
[13]

Craig Cowan, Abdolrahman Razani. Singular solutions of a Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 621-656. doi: 10.3934/dcds.2020291
[14]

Michael Winkler, Christian Stinner. Refined regularity and stabilization properties in a degenerate haptotaxis system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 4039-4058. doi: 10.3934/dcds.2020030
[15]

Xing-Bin Pan. Variational and operator methods for Maxwell-Stokes system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3909-3955. doi: 10.3934/dcds.2020036
[16]

Peter Giesl, Sigurdur Hafstein. System specific triangulations for the construction of CPA Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020378
[17]

Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021003
[18]

Chao Xing, Zhigang Pan, Quan Wang. Stabilities and dynamic transitions of the Fitzhugh-Nagumo system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 775-794. doi: 10.3934/dcdsb.2020134
[19]

Marcos C. Mota, Regilene D. S. Oliveira. Dynamic aspects of Sprott BC chaotic system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1653-1673. doi: 10.3934/dcdsb.2020177
[20]

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

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (37)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences