Periodic solutions of the Brillouin electron beam focusing equation

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March 2014, 13(2): 961-975. doi: 10.3934/cpaa.2014.13.961

Periodic solutions of the Brillouin electron beam focusing equation

Maurizio Garrione ^1, and Manuel Zamora ^2,

1.	Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, 1, 60131, Ancona, Italy
2.	Departamento de Matemática Aplicada, Universidad de Granada, 18071, Granada, Spain

Received February 2013 Revised September 2013 Published October 2013

Abstract
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Quite unexpectedly with respect to the numerical and analytical results found in literature, we establish a new range for the real parameter $b$ for which the existence of $2\pi-$periodic solutions of the Brillouin focusing beam equation \begin{eqnarray} \ddot{x}+b(1+\cos t)x=\frac{1}{x} \end{eqnarray} is guaranteed. This is possible thanks to suitable nonresonance conditions acting on the rotation number of the solutions in the phase plane.

Keywords: Brillouin focusing system, Poincaré-Bohl theorem, singular nonlinearities, periodic solutions.

Mathematics Subject Classification: Primary: 34B15, 34C25; Secondary: 34B1.

Citation: Maurizio Garrione, Manuel Zamora. Periodic solutions of the Brillouin electron beam focusing equation. Communications on Pure and Applied Analysis, 2014, 13 (2) : 961-975. doi: 10.3934/cpaa.2014.13.961

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), 696-708.
[2]	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.
[3]	A. Boscaggin and M. Garrione, Resonance and rotation numbers for planar Hamiltonian systems: multiplicity results via the Poincaré-Birkhoff theorem, Nonlinear Anal., 74 (2011), 4166-4185. doi: 10.1016/j.na.2011.03.051.
[4]	H. Broer and M. Levi, Geometrical aspects of stability theory for Hill's equations, Arch. Rational Mech. Anal., 131 (1995), 225-240. doi: 10.1007/BF00382887.
[5]	A. Cabada and J. A. Cid, On comparison principles for the periodic Hill's equation, J. Lond. Math. Soc., 86 (2012), 272-290. doi: 10.1112/jlms/jds001.
[6]	M. del Pino, R. Manásevich and A. Montero, $T$-periodic solutions for some second order differential equations with singularities, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 231-243. doi: 10.1017/S030821050003211X.
[7]	T. Ding, A boundary value problem for the periodic Brillouin focusing system, Acta Sci. Natur. Univ. Pekinensis, 11 (1965), 31-38 (Chinese).
[8]	C. Fabry, Periodic solutions of the equation $x'' + f(t, x)=0$, Séminaire de Mathématique, 117 (1987), Louvain-la-Neuve.
[9]	C. Fabry and P. Habets, Periodic solutions of second order differential equations with superlinear asymmetric nonlinearities, Arch. Math. (Basel), 60 (1993), 266-276. doi: 10.1007/BF01198811.
[10]	A. Fonda and A. Sfecci, A general method for the existence of periodic solutions of differential systems in the plane, J. Differential Equations, 252 (2012), 1369-1391. doi: 10.1016/j.jde.2011.08.005.
[11]	A. Fonda and R. Toader, Radially symmetric systems with a singularity and asymptotically linear growth, Nonlinear Anal., 74 (2011), 2485-2496. doi: 10.1016/j.na.2010.12.004.
[12]	W. Magnus and S. Winkler, "Hill's Equation," corrected reprint of 1966 edition, Dover, New York, 1979.
[13]	J. Ren, Z. Cheng and S. Siegmund, Positive periodic solution for Brillouin electron beam focusing systems, Discrete Cont. Dyn. Syst. Ser. B, 16 (2011), 385-392. doi: 10.3934/dcdsb.2011.16.385.
[14]	P.J. Torres, Existence and uniqueness of elliptic periodic solutions of the Brillouin electron beam focusing system, Math. Methods Appl. Sci., 23 (2000), 1139-1143. doi: 10.1002/1099-1476(20000910)23:13<1139::AID-MMA155>3.0.CO;2-J.
[15]	P.J. Torres, Twist solutions of a Hill's equation with singular term, Adv. Nonlinear Stud., 2 (2002), 279-287.
[16]	P.J. Torres, Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem, J. Differential Equations, 190 (2003), 643-662. doi: 10.1016/S0022-0396(02)00152-3.
[17]	Q. Yao, Periodic positive solution to a class of singular second-order ordinary differential equations, Acta Appl. Math., 110 (2010), 871-883. doi: 10.1007/s10440-009-9482-9.
[18]	Y. Ye and X. Wang, Nonlinear differential equations arising in the theory of electron beam focusing, Acta Math. Appl. Sinica, 1 (1978), 13-41.
[19]	M. 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.
[20]	M. Zhang, A relationship between the periodic and the Dirichlet BVPs of singular differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 1099-1114. doi: 10.1017/S0308210500030080.
[21]	M. Zhang, Periodic solutions of equations of Emarkov-Pinney type, Adv. Nonlinear Stud., 6 (2006), 57-67.

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), 696-708.
[2]	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.
[3]	A. Boscaggin and M. Garrione, Resonance and rotation numbers for planar Hamiltonian systems: multiplicity results via the Poincaré-Birkhoff theorem, Nonlinear Anal., 74 (2011), 4166-4185. doi: 10.1016/j.na.2011.03.051.
[4]	H. Broer and M. Levi, Geometrical aspects of stability theory for Hill's equations, Arch. Rational Mech. Anal., 131 (1995), 225-240. doi: 10.1007/BF00382887.
[5]	A. Cabada and J. A. Cid, On comparison principles for the periodic Hill's equation, J. Lond. Math. Soc., 86 (2012), 272-290. doi: 10.1112/jlms/jds001.
[6]	M. del Pino, R. Manásevich and A. Montero, $T$-periodic solutions for some second order differential equations with singularities, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 231-243. doi: 10.1017/S030821050003211X.
[7]	T. Ding, A boundary value problem for the periodic Brillouin focusing system, Acta Sci. Natur. Univ. Pekinensis, 11 (1965), 31-38 (Chinese).
[8]	C. Fabry, Periodic solutions of the equation $x'' + f(t, x)=0$, Séminaire de Mathématique, 117 (1987), Louvain-la-Neuve.
[9]	C. Fabry and P. Habets, Periodic solutions of second order differential equations with superlinear asymmetric nonlinearities, Arch. Math. (Basel), 60 (1993), 266-276. doi: 10.1007/BF01198811.
[10]	A. Fonda and A. Sfecci, A general method for the existence of periodic solutions of differential systems in the plane, J. Differential Equations, 252 (2012), 1369-1391. doi: 10.1016/j.jde.2011.08.005.
[11]	A. Fonda and R. Toader, Radially symmetric systems with a singularity and asymptotically linear growth, Nonlinear Anal., 74 (2011), 2485-2496. doi: 10.1016/j.na.2010.12.004.
[12]	W. Magnus and S. Winkler, "Hill's Equation," corrected reprint of 1966 edition, Dover, New York, 1979.
[13]	J. Ren, Z. Cheng and S. Siegmund, Positive periodic solution for Brillouin electron beam focusing systems, Discrete Cont. Dyn. Syst. Ser. B, 16 (2011), 385-392. doi: 10.3934/dcdsb.2011.16.385.
[14]	P.J. Torres, Existence and uniqueness of elliptic periodic solutions of the Brillouin electron beam focusing system, Math. Methods Appl. Sci., 23 (2000), 1139-1143. doi: 10.1002/1099-1476(20000910)23:13<1139::AID-MMA155>3.0.CO;2-J.
[15]	P.J. Torres, Twist solutions of a Hill's equation with singular term, Adv. Nonlinear Stud., 2 (2002), 279-287.
[16]	P.J. Torres, Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem, J. Differential Equations, 190 (2003), 643-662. doi: 10.1016/S0022-0396(02)00152-3.
[17]	Q. Yao, Periodic positive solution to a class of singular second-order ordinary differential equations, Acta Appl. Math., 110 (2010), 871-883. doi: 10.1007/s10440-009-9482-9.
[18]	Y. Ye and X. Wang, Nonlinear differential equations arising in the theory of electron beam focusing, Acta Math. Appl. Sinica, 1 (1978), 13-41.
[19]	M. 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.
[20]	M. Zhang, A relationship between the periodic and the Dirichlet BVPs of singular differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 1099-1114. doi: 10.1017/S0308210500030080.
[21]	M. Zhang, Periodic solutions of equations of Emarkov-Pinney type, Adv. Nonlinear Stud., 6 (2006), 57-67.

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