## Weak singularities may help periodic solutions to exist.(English)Zbl 1116.34036

There are given conditions for the existence of a positive T-periodic solution of the equation
$x''+ a(t)x= f(t,x)+ c(t),$
under the assumption, that the equation
$x''+ a(t)x= 0$
is nonresonant, that the corresponding Green’s function is nonnegative and
$0\leq f(x,t)\leq b(t)/x^\lambda,\qquad 0<\lambda< 1.$

### MSC:

 34C25 Periodic solutions to ordinary differential equations
Full Text:

### References:

 [1] Berkovich, L.M.; Rozov, N.Kh., Some remarks on differential equations of the form $$y'' + a_0(x) y = \varphi(x) y^\alpha$$, Differ. equ., 8, 1609-1612, (1972) · Zbl 0297.34001 [2] Bonheure, D.; De Coster, C., Forced singular oscillators and the method of upper and lower solutions, Topol. methods nonlinear anal., 22, 297-317, (2003) · Zbl 1108.34033 [3] Bonheure, D.; Fabry, C.; Smets, D., Periodic solutions of forced isochronous oscillators at resonance, Discrete contin. dyn. syst., 8, 4, 907-930, (2002) · Zbl 1021.34032 [4] Ding, T., A boundary value problem for the periodic Brillouin focusing system, Acta sci. natur. univ. pekinensis, 11, 31-38, (1965), (in Chinese) [5] Fonda, A., Periodic solutions of scalar second order differential equations with a singularity, Mém. cl. sciences acad. R. belguique (4), 8, 68-98, (1993) [6] Fonda, A.; Manásevich, R.; Zanolin, F., Subharmonics solutions for some second order differential equations with singularities, SIAM J. math. anal., 24, 1294-1311, (1993) · Zbl 0787.34035 [7] Franco, D.; Webb, J.K.L., Collisionless orbits of singular and nonsingular dynamical systems, Discrete contin. dyn. syst., 15, 747-757, (2006) · Zbl 1120.34029 [8] Gordon, W.B., Conservative dynamical systems involving strong forces, Trans. amer. math. soc., 204, 113-135, (1975) · Zbl 0276.58005 [9] Gordon, W.B., A minimizing property of Keplerian orbits, Amer. J. math., 99, 961-971, (1977) · Zbl 0378.58006 [10] Habets, P.; Sanchez, L., Periodic solutions of some Liénard equations with singularities, Proc. amer. math. soc., 109, 1135-1144, (1990) [11] Jiang, D.; Chu, J.; Zhang, M., Multiplicity of positive periodic solutions to superlinear repulsive singular equations, J. differential equations, 211, 2, 282-302, (2005) · Zbl 1074.34048 [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] del Pino, M.; Manásevich, R., Infinitely many T-periodic solutions for a problem arising in nonlinear elasticity, J. differential equations, 103, 260-277, (1993) · Zbl 0781.34032 [14] del Pino, M.; Manásevich, R.; Montero, A., T-periodic solutions for some second order differential equations with singularities, Proc. roy. soc. Edinburgh sect. A, 120, 3-4, 231-243, (1992) · Zbl 0761.34031 [15] del Pino, M.; Manásevich, R.; Murua, A., On the number of 2π-periodic solutions for u″+g(u)=s(1+h(t)) using the poincaré-birkhoff theorem, J. differential equations, 95, 240-258, (1992) · Zbl 0745.34035 [16] Rachunková, I.; Tvrdý, M.; Vrkoč, I., Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems, J. differential equations, 176, 445-469, (2001) · Zbl 1004.34008 [17] Rachunková, I.; Stanek, S.; Tvrdý, M., Singularities and Laplacians in boundary value problems for nonlinear ordinary differential equations, Handbook of differential equations (ordinary differential equations), vol. 3, (2006), Elsevier [18] Torres, P.J., Bounded solutions in singular equations of repulsive type, Nonlinear anal., 32, 117-125, (1998) · Zbl 1126.34326 [19] 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 [20] Torres, P.J., Twist solutions of a Hill’s equations with singular term, Adv. nonlinear stud., 2, 279-287, (2002) · Zbl 1016.34044 [21] 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 [22] P.J. Torres, Existence and stability of periodic solutions for second order semilinear differential equations with a singular nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A, in press · Zbl 1190.34050 [23] Torres, P.J.; Zhang, M., A monotone iterative scheme for a nonlinear second order equation based on a generalized anti-maximum principle, Math. nachr., 251, 101-118, (2003) · Zbl 1024.34030 [24] Torres, P.J.; Zhang, M., Twist periodic solutions of repulsive singular equations, Nonlinear anal., 56, 591-599, (2004) · Zbl 1058.34052 [25] Ye, Y.; Wang, X., Nonlinear differential equations in electron beam focusing theory, Acta math. appl. sin., 1, 13-41, (1978), (in Chinese) [26] Zhang, M., Periodic solutions of Liénard equations with singular forces of repulsive type, J. math. anal. appl., 203, 254-269, (1996) · Zbl 0863.34039 [27] 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 [28] Zhang, M., The best bound on the rotations in the stability of periodic solutions of a Newtonian equation, J. London math. soc. (2), 67, 137-148, (2003) · Zbl 1050.34075
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.