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Periodic solutions for damped differential equations with a weak repulsive singularity. (English) Zbl 1165.34349

Summary: This paper deals with the existence of positive \(T\)-periodic solutions for the damped differential equation
\[ \ddot x +p(t)\dot x+q(t)x = f(t,x)+c(t) \]
where \(p, q, c \in L^1(\mathbb R)\) are \(T\)-periodic functions and \(f \in Car(\mathbb R\times \mathbb R^+, \mathbb R)\) is \(T\)-periodic in the first variable. We will prove that a weak repulsive singularity enables the achievement of new existence criteria through a basic application of Schauder’s fixed point theorem.

MSC:

34C25 Periodic solutions to ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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[1] 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
[2] Chu, J.; Torres, P. J.; Zhang, M., Periodic solutions of second order non-autonomous singular dynamical systems, J. Differential Equations, 239, 196-212 (2007) · Zbl 1127.34023
[3] Chu, J.; Torres, P. J., Applications of Schauder’s fixed point theorem to singular differential equations, Bull. London Math. Soc., 39, 653-660 (2007) · Zbl 1128.34027
[4] 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
[5] Forbat, F.; Huaux, A., Détermination approché et stabilité locale de la solution périodique d’une équation différentielle non liné, Mém. Public. Soc. Sciences, Artts Lettres du Hainaut, 76, 3-13 (1962) · Zbl 0107.07203
[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] Habets, P.; Sanchez, L., Periodic solutions of some Liénard equations with singularities, Proc. Amer. Math. Soc., 109, 1135-1144 (1990)
[8] Habets, P.; Sanchez, L., Periodic solutions of dissipative dynamical systems with singular potentials, Differential Integral Equations, 3, 1139-1149 (1990) · Zbl 0724.34049
[9] Huaux, A., Sur L’existence d’une solution périodique de l’équation différentielle non linéaire \(\ddot{x} + 0, 2 \dot{x} + \frac{x}{1 - x} = (0, 5) \cos \omega t\), Bull. Cl. Sci. Acad. R. Belguique, 48, 494-504 (1962) · Zbl 0112.06105
[10] Jebelean, P.; Mawhin, J., Periodic solutions of forced dissipative \(p\)-Liénard equations with singularities, Vietnam J. Math., 32, 97-103 (2004) · Zbl 1098.34031
[11] 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
[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] X. Li, Z. Zhang, Periodic solutions for some second order differential equations with singularity, Z. Angew. Math. Phys. (in press); X. Li, Z. Zhang, Periodic solutions for some second order differential equations with singularity, Z. Angew. Math. Phys. (in press) · Zbl 1157.34034
[14] Liu, B., Periodic solutions of dissipative dynamical systems with singular potential and \(p\)-Laplacian, Ann. Polon. Math., 79, 109-120 (2002) · Zbl 1024.34037
[15] Manásevich, R.; Mawhin, J., Periodic solutions for nonlinear systems with \(p\)-Laplacian-like operators, J. Differential Equations, 145, 367-393 (1998) · Zbl 0910.34051
[16] 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
[17] 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
[18] Rachunková, I.; Stanek, S.; Tvrdý, M., Singularities and Laplacians in boundary value problems for nonlinear ordinary differential equations, (Ordinary Differential Equations. Ordinary Differential Equations, Handbook of Differential Equations, vol. 3 (2006), Ed. Elsevier)
[19] Torres, P. J., Weak singularities may help periodic solutions to exist, J. Differential Equations, 232, 277-284 (2007) · Zbl 1116.34036
[20] Wang, Y.; Lian, H.; Ge, W., Periodic solutions for a second order nonlinear functional differential equation, Appl. Math. Lett., 20, 110-115 (2007) · Zbl 1151.34056
[21] Zhang, M., Periodic solutions of Liénard equations with singular forces of repulsive type, J. Math. Anal. Appl., 203, 254-269 (1996) · Zbl 0863.34039
[22] Zhang, M., Periodic solutions of damped differential systems with repulsive singular forces, Proc. Amer. Math. Soc., 127, 401-407 (1999) · Zbl 0908.34024
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