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Positive periodic solutions of Hill’s equations with singular nonlinear perturbations. (English) Zbl 1148.34025

The paper proves existence and multiplicity of positive periodic solutions of the perturbation of Hill’s equation \[ x''+a(t)x=f(t,x)+e(t), (1) \] where \(a(t),e(t)\) are continuous, \(T\)-periodic functions. The nonlinearity \(f(t,x)\) is continuous in \((t,x)\) and \(T\)-periodic in \(t\) and has a singularity at \(x=0\). The case of a strong singularity as well as that of a weak singularity is considered, and \(e\) does not need to be positive. The proofs are based on Krasnoselskii’s fixed point theorem in cones and on the Leray-Schauder alternative together with a truncation technique. Some recent results in the literature are generalized and improved.

MSC:

34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
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[1] Bonheure, D.; De Coster, C., Forced singular oscillators and the method of lower and upper solutions, Topol. methods nonlinear anal., 22, 297-317, (2003) · Zbl 1108.34033
[2] Chu, J.; Lin, X.; Jiang, D.; O’Regan, D.; Agarwal, R.P., Multiplicity of positive solutions to second order differential equations, Bull. austral. math. soc., 73, 175-182, (2006) · Zbl 1096.34518
[3] De Coster, C.; Habets, P., Upper and lower solutions in the theory of ODE boundary value problems: classical and recent results, (), 1-78 · Zbl 0889.34018
[4] del Pino, M.A.; Manásevich, R.F.; Montero, A., \(T\)-periodic solutions for some second order differential equations with singularities, Proc. roy. soc. Edinburgh sect. A, 120, 231-243, (1992) · Zbl 0761.34031
[5] del Pino, M.A.; Manásevich, R.F., Infinitely many \(T\)-periodic solutions for a problem arising in nonlinear elasticity, J. differential equations, 103, 260-277, (1993) · Zbl 0781.34032
[6] Franco, D.; Webb, J.R.L., Collisionless orbits of singular and nonsingular dynamical systems, Discrete contin. dyn. syst., 15, 747-757, (2006) · Zbl 1120.34029
[7] D. Franco, P.J. Torres, Periodic solutions of singular systems without the strong force condition, Proc. Amer. Math. Soc. (in press) · Zbl 1129.37033
[8] Gordon, W.B., Conservative dynamical systems involving strong forces, Trans. amer. math. soc., 204, 113-135, (1975) · Zbl 0276.58005
[9] Habets, P.; Sanchez, L., Periodic solution of some Liénard equations with singularities, Proc. amer. math. soc., 109, 1135-1144, (1990)
[10] Jiang, D.; Chu, J.; Zhang, M., Multiplicity of positive periodic solutions to superlinear repulsive singular equations, J. differential equations, 211, 282-302, (2005) · Zbl 1074.34048
[11] Krasnosel’skii, M.A., Positive solutions of operator equations, (1964), Noordhoff Groningen
[12] Lazer, A.C.; Solimini, S., On periodic solutions of nonlinear differential equations with singularities, Proc. amer. math. soc., 99, 109-114, (1987) · Zbl 0616.34033
[13] Mawhin, J., Topological degree and boundary value problems for nonlinear differential equations, (), 74-142 · Zbl 0798.34025
[14] Rachunková, I.; Tvrdý, M.; Vrkoc˘, I., Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems, J. differential equations, 176, 445-469, (2001) · Zbl 1004.34008
[15] Torres, P.J., Existence and uniqueness of elliptic periodic solutions of the Brillouin electron beam focusing system, Math. methods appl. sci., 23, 1139-1143, (2000) · Zbl 0966.34038
[16] Torres, P.J., Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem, J. differential equations, 190, 643-662, (2003) · Zbl 1032.34040
[17] Torres, P.J., Weak singularities may help periodic solutions to exist, J. differential equations, 232, 277-284, (2007) · Zbl 1116.34036
[18] Yan, P.; Zhang, M., Higher order nonresonance for differential equations with singularities, Math. methods appl. sci., 26, 1067-1074, (2003) · Zbl 1031.34040
[19] Zhang, M., A relationship between the periodic and the Dirichlet BVPs of singular differential equations, Proc. roy. soc. Edinburgh sect. A., 128, 1099-1114, (1998) · Zbl 0918.34025
[20] Zhang, M.; Li, W., A Lyapunov-type stability criterion using \(L^\alpha\) norms, Proc. amer. math. soc., 130, 3325-3333, (2002) · Zbl 1007.34053
[21] Zhang, M., Periodic solutions of equations of ermakov – pinney type, Adv. nonlinear stud., 6, 57-67, (2006) · Zbl 1107.34037
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