Positive periodic solution for Brillouin electron beam focusing system

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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

Abstract
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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.
[2]	T. R. Ding, "Applications of Qualitative Methods of Ordinary Differential Equations,", Higher Education Press, (2004).
[3]	T. R. Ding, A boundary value problem for the periodic Brillouin focusing system,, Acta Sci. Natru. Univ. Pekinensis, 11 (1965), 31.
[4]	Weigao Ge, "Boundary Value Problems for Nonlinear Ordinary Differential Equations,", Science Press, (2007).
[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.
[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.
[7]	Y. Ye and X. Wang, Nonlinear differential equations in electron beam focusing theory,, Acta Math. Appl. Sinica, 1 (1978), 13.
[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.
[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.

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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.
[2]	T. R. Ding, "Applications of Qualitative Methods of Ordinary Differential Equations,", Higher Education Press, (2004).
[3]	T. R. Ding, A boundary value problem for the periodic Brillouin focusing system,, Acta Sci. Natru. Univ. Pekinensis, 11 (1965), 31.
[4]	Weigao Ge, "Boundary Value Problems for Nonlinear Ordinary Differential Equations,", Science Press, (2007).
[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.
[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.
[7]	Y. Ye and X. Wang, Nonlinear differential equations in electron beam focusing theory,, Acta Math. Appl. Sinica, 1 (1978), 13.
[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.
[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.

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