Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem. (English) Zbl 1032.34040

The author provides new existence results for the periodic boundary value problem \[ x''=f(t,x), \quad x(0)=x(T),\;x'(0)=x'(T), (1) \] where \(f\) is a Carathéodory function. The proofs are based on the Krasnoselskii fixed-point theorem for completely continuous operators in a Banach space that exhibits a cone compression and expansion, and on the sign behaviour of Green’s function of the linearized equation.
The main results are contained in two theorems which give conditions guaranteeing the existence of a positive solution to (1). Modified assertions for negative solutions are shown.
As applications of these general results, the author obtains new existence results for equations with jumping nonlinearities and for equations with a repulsive or attractive singularity in the origin. Weak singularities are considered here, too.


34C25 Periodic solutions to ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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[1] Bevc, V.; Palmer, J. L.; Süsskind, C., On the design of the transition region of axi-symmetric magnetically focusing beam valves, J. Br Inst. Radio Eng., 18, 696-708 (1958)
[2] Ding, T., A boundary value problem for the periodic Brillouin focusing system, Acta Sci. Natur. Univ. Pekinensis, 11, 31-38 (1965), (in Chinese)
[3] De Coster, C.; Habets, P., Upper and lower solutions in the theory of ODE boundary value problems: classical and recent results, (Zanolin, F., Nonlinear Analysis and Boundary Value Problems for Ordinary Differential Equations. Nonlinear Analysis and Boundary Value Problems for Ordinary Differential Equations, CISM-ICMS, Vol. 371 (1996), Springer: Springer New York), 1-78 · Zbl 0889.34018
[4] del Pino, I.; Manásevich, R.; Montero, A., \(T\)-periodic solutions for some second order differential equations with singularities, Proc. Roy. Soc. Edinbourgh Sect. A, 120, 231-243 (1992) · Zbl 0761.34031
[5] Drábek, P., Landesman-Lazer condition for nonlinear problems with jumping nonlinearities, J. Differential Equations, 85, 186-199 (1990) · Zbl 0699.34019
[6] Esmailzadeh, E.; Nakhaie-Jazar, G., Periodic solutions of a Mathieu-Duffing type equation, J. Non. Linear Mech., 32, 905-912 (1997) · Zbl 0881.34064
[7] Erbe, L. H.; Mathsen, R. M., Positive solutions for singular nonlinear boundary value problems, Nonlinear Anal., 46, 979-986 (2001) · Zbl 1007.34020
[8] Erbe, L. H.; Wang, H., On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc., 120, 743-748 (1994) · Zbl 0802.34018
[9] Fonda, A., Periodic solutions of scalar second order differential equations with a singularity, Mém. Classe Sci. Acad. Roy. Belgique, 8-IV, 68-98 (1993)
[10] Fonda, A.; Habets, P., Periodic solutions of asymptotically positively homogeneous differential equations, J. Differential Equations, 81, 68-98 (1989) · Zbl 0692.34041
[11] Friedman, B., Principles and techniques of Applied Mathematics (1990), Dover Publications Inc: Dover Publications Inc New York · Zbl 1227.00033
[12] Gaete, S.; Manasevich, R. F., Existence of a pair of periodic solutions of an O.D.E. generalizing a problem in nonlinear Elasticity, via variational methods, J. Math. Anal. Appl., 134, 257-271 (1988) · Zbl 0672.34030
[13] Guo, D.; Laksmikantham, V., Multiple solutions of two-point boundary value problems of ordinary differential equations in Banach spaces, J. Math. Anal. Appl., 129, 211-222 (1988) · Zbl 0645.34014
[14] Habets, P.; Sanchez, L., Periodic solution of some Liénard equations with singularities, Proc. Amer. Math. Soc., 109, 1135-1144 (1990)
[15] Henderson, J.; Wang, H., Positive solutions for nonlinear eigenvalue problems, J. Math. Anal. Appl., 208, 252-259 (1997) · Zbl 0876.34023
[16] Ianacci, R.; Nkashama, M. N.; Omari, P.; Zanolin, F., Periodic solutions of forced Liénard equations with jumping nonlinearities under nonuniform conditions, Proc. Roy. Soc. Edinburgh Sect. A, 110, 183-198 (1988) · Zbl 0693.34046
[17] Krasnosel’skii, M. A., Positive Solutions of Operator Equations (1964), Noordhoff: Noordhoff Groningen · Zbl 0121.10604
[18] Lazer, A. C.; McKenna, P. J., Large-amplitude periodic oscillations in suspension bridgessome new connections with nonlinear analysis, SIAM Rev., 32, 537-578 (1990) · Zbl 0725.73057
[19] 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
[21] Mawhin, J., Topological degree and boundary value problems for nonlinear differential equations, (Furi, M.; Zecca, P., Topological methods for ordinary differential equations. Topological methods for ordinary differential equations, Lecture Notes in Mathematics, Vol. 1537 (1993), Springer: Springer New York/Berlin), 74-142 · Zbl 0798.34025
[22] Merivenci Atici, F.; Guseinov, G. Sh., On the existence of positive solutions for nonlinear differential equations with periodic boundary conditions, J. Comput. Appl. Math., 132, 341-356 (2001) · Zbl 0993.34022
[23] Omari, P.; Ye, W.-Y., Necessary and sufficient conditions for the existence of periodic solutions of second-order ordinary differential equations with singular nonlinearities, Differential Integral Equations, 8, 1843-1858 (1995) · Zbl 0831.34048
[24] Rachunková, I., Existence of two positive solutions of a singular nonlinear periodic boundary value problems, J. Differential Equations, 176, 445-469 (2001) · Zbl 1004.34008
[25] Rachunková, I.; Tvrdý, M.; Vrkoc̆, I., Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems, J. Differential Equations, 176, 445-469 (2001) · Zbl 1004.34008
[26] Torres, P., 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
[28] Ye, Y.; Wang, X., Nonlinear differential equations in electron beam focusing theory, Acta Math. Appl. Sinica, 1, 13-41 (1978), (in Chinese)
[29] Yujun, D., Invariance of homotopy and an extention of a theorem by Habets-Metzen on periodic solutions of Duffing equations, Nonlinear Anal., 46, 1123-1132 (2001) · Zbl 1005.34011
[30] Zhang, M., Periodic solutions of Liénard equations with singular forces of repulsive type, J. Math. Anal. Appl., 203, 254-269 (1996) · Zbl 0863.34039
[31] Zhang, M., A relationship between the periodic and the Dirichlet BVPs of singular differential equations, Proc. Roy. Soc. Edinbourgh Sect. A, 128, 1099-1114 (1998) · Zbl 0918.34025
[32] Zhang, M.; Li, W. G., A Lyapunov-type stability criterion using \(L^α\) norms, Proc. Amer. Math. Soc., 130, 11, 3325-3333 (2002) · Zbl 1007.34053
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