Article Contents
Article Contents

# Positive periodic solution for Brillouin electron beam focusing system

• 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.
Mathematics Subject Classification: Primary: 34B27; Secondary: 37C55.

 Citation:

•  [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), 696-708. [2] T. R. Ding, "Applications of Qualitative Methods of Ordinary Differential Equations," Higher Education Press, BeiJing, 2004. [3] T. R. Ding, A boundary value problem for the periodic Brillouin focusing system, Acta Sci. Natru. Univ. Pekinensis, 11 (1965), 31-38. [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), 74-142.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), 1139-1143.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), 13-41. [8] M. R. Zhang, Periodic solutions of Liénard equations with singular forces of repulsive type, J. Math. Anal. Appl., 203 (1996), 254-269.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), 41-51.doi: 10.1016/S0362-546X(96)00037-5.