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September  2013, 18(7): 1995-1997. doi: 10.3934/dcdsb.2013.18.1995

An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system

Zaihong Wang 1, , Jin Li 2, and  Tiantian Ma 3,

1. 

School of Mathematical Sciences, Capital Normal University, Beijing 100048
2. 

School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China
3. 

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

Received  September 2012 Revised  April 2013 Published  May 2013

In this paper, we point out an error in the paper: Positive periodic solution for Brillouin electron beam focusing system, Discrete Contin. Dyn. Syst. Ser. B, 16(2011), 385-392. Meanwhile, it is pointed out that, for $0 < a < 1$, the conjecture that the Brillouin electron beam focusing system $x''+a(1+\cos 2t)x=1/x$ admits positive periodic solutions is still an open problem.
Keywords: Liénard equation, periodic solution., Brillouin electron beam focusing system.
Mathematics Subject Classification: Primary: 34C25; Secondary: 34B1.
Citation: Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete and Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995
References:
[1]

J. Ren, Z. Cheng and S. Siegmund, Positive periodic solution for Brillouin electron beam focusing system, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 385-392. doi: 10.3934/dcdsb.2011.16.385.

[2]

V. Bevc, J. L. Palmer and C. Süsskind, On the design of the transition region of axisymmetric magnetically focused beam values, J. British Inst. Radio Engineers, 18 (1958), 696-708.

[3]

R. Ortega, Periodic perturbations of an isochronous center, Qual. Theory Dyn. Syst., 3 (2002), 83-91. doi: 10.1007/BF02969334.

[4]

D. Bonheure, C. Fabry and D. Smets, Periodic solutions of forced isochronous oscillators at resonance, Discrete Contin. Dyn. Syst., 8 (2002), 907-930. doi: 10.3934/dcds.2002.8.907.

[5]

T. Ding, A boundary value problem for the periodic Brillouin focusing system, Acta Sci. Natur. Univ. Pekinensis, 11 (1965), 31-38.

[6]

Y. Ye and X. Wang, Nonlinear differential equations in electron beam focusing theory, Acta Math. Appl. Sinica, 1 (1978), 13-41.

[7]

M. R. Zhang, A relationship between the periodic and the Dirichlet BVPs of singular differential equation, Proc. Roy. Soc. Edinburgh, Sect. A, 128 (1998), 1099-1114. doi: 10.1017/S0308210500030080.

show all references

References:
[1]

J. Ren, Z. Cheng and S. Siegmund, Positive periodic solution for Brillouin electron beam focusing system, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 385-392. doi: 10.3934/dcdsb.2011.16.385.

[2]

V. Bevc, J. L. Palmer and C. Süsskind, On the design of the transition region of axisymmetric magnetically focused beam values, J. British Inst. Radio Engineers, 18 (1958), 696-708.

[3]

R. Ortega, Periodic perturbations of an isochronous center, Qual. Theory Dyn. Syst., 3 (2002), 83-91. doi: 10.1007/BF02969334.

[4]

D. Bonheure, C. Fabry and D. Smets, Periodic solutions of forced isochronous oscillators at resonance, Discrete Contin. Dyn. Syst., 8 (2002), 907-930. doi: 10.3934/dcds.2002.8.907.

[5]

T. Ding, A boundary value problem for the periodic Brillouin focusing system, Acta Sci. Natur. Univ. Pekinensis, 11 (1965), 31-38.

[6]

Y. Ye and X. Wang, Nonlinear differential equations in electron beam focusing theory, Acta Math. Appl. Sinica, 1 (1978), 13-41.

[7]

M. R. Zhang, A relationship between the periodic and the Dirichlet BVPs of singular differential equation, Proc. Roy. Soc. Edinburgh, Sect. A, 128 (1998), 1099-1114. doi: 10.1017/S0308210500030080.

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