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Existence and uniqueness of singular solutions for elliptic equation on the hyperbolic space
Periodic solutions of the Brillouin electron beam focusing equation
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 |
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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