# American Institute of Mathematical Sciences

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

## Positive periodic solution for Brillouin electron beam focusing system

 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.
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

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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. 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
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