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Periodic solutions for second order singular damped differential equations. (English) Zbl 1232.34065
Summary: We study the existence of positive periodic solutions for second order singular damped differential equations by combining the analysis of the sign of Green’s functions for the linear damped equation, together with a nonlinear alternative principle of Leray-Schauder. Recent results in the literature are generalized and significantly improved.
MSC:
34C25Periodic solutions of ODE
References:
[1]Barteneva, I. V.; Cabada, A.; Ignatyev, A. O.: Maximum and anti-maximum principles for the general operator of second order with variable coefficients, Appl. math. Comput. 134, 173-184 (2003) · Zbl 1037.34014 · doi:10.1016/S0096-3003(01)00280-6
[2]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
[3]Bravo, J. L.; Torres, P. J.: Periodic solutions of a singular equation with indefinite weight, Adv. nonlinear stud. 10, 927-938 (2010) · Zbl 1232.34064
[4]Cabada, A.; Cid, J. A.: On the sign of the Green’s function associated to Hill’s equation with an indefinite potential, Appl. math. Comput. 205, 303-308 (2008) · Zbl 1161.34014 · doi:10.1016/j.amc.2008.08.008
[5]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 · doi:10.1017/S0004972700038764
[6]Chu, J.; Torres, P. J.: Applications of Schauder’s fixed point theorem to singular differential equations, Bull. lond. Math. soc. 39, 653-660 (2007) · Zbl 1128.34027 · doi:10.1112/blms/bdm040
[7]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 · doi:10.1016/j.jde.2007.05.007
[8]Chu, J.; Li, M.: Positive periodic solutions of Hill’s equations with singular nonlinear perturbations, Nonlinear anal. 69, 276-286 (2008) · Zbl 1148.34025 · doi:10.1016/j.na.2007.05.016
[9]Chu, J.; Zhang, Z.: Periodic solutions of singular differential equations with sign-changing potential, Bull. austral. Math. soc. 82, 437-445 (2010)
[10]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 · doi:10.1006/jdeq.1993.1050
[11]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 · doi:10.3934/dcds.2006.15.747
[12]Franco, D.; Torres, P. J.: Periodic solutions of singular systems without the strong force condition, Proc. amer. Math. soc. 136, 1229-1236 (2008) · Zbl 1129.37033 · doi:10.1090/S0002-9939-07-09226-X
[13]Gaudenzi, M.; Habets, P.; Zanolin, F.: Positive solutions of singular boundary value problems with indefinite weight, Bull. belg. Math. soc. Simon stevin 9, 607-619 (2002) · Zbl 1048.34045
[14]Gordon, W. B.: Conservative dynamical systems involving strong forces, Trans. amer. Math. soc. 204, 113-135 (1975) · Zbl 0276.58005 · doi:10.2307/1997352
[15]Granas, A.; Guenther, R. B.; Lee, J. W.: Some general existence principles in the Carathéodory theory of nonlinear differential systems, J. math. Pures appl. 70, 153-196 (1991) · Zbl 0687.34009
[16]Granas, A.; Dugundji, J.: Fixed point theory, Springer monogr. Math. (2003)
[17]Habets, P.; Sanchez, L.: Periodic solution of some Liénard equations with singularities, Proc. amer. Math. soc. 109, 1135-1144 (1990) · Zbl 0695.34036 · doi:10.2307/2048134
[18]Halk, R.; Torres, P. J.: On periodic solutions of second-order differential equations with attractive-repulsive singularities, J. differential equations 248, 111-126 (2010) · Zbl 1187.34049 · doi:10.1016/j.jde.2009.07.008
[19]Halk, R.; Torres, P. J.: Maximum and antimaximum principles for a second order differential operator with variable coefficients of indefinite sign, Appl. math. Comput. 217, 7599-7611 (2011)
[20]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 · doi:10.1016/j.jde.2004.10.031
[21]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 · doi:10.2307/2046279
[22]Li, X.; Zhang, Z.: Periodic solutions for damped differential equations with a weak repulsive singularity, Nonlinear anal. 70, 2395-2399 (2009) · Zbl 1165.34349 · doi:10.1016/j.na.2008.03.023
[23]Meehan, M.; O’regan, D.: Existence theory for nonlinear Volterra integrodifferential and integral equations, Nonlinear anal. 31, 317-341 (1998) · Zbl 0891.45004 · doi:10.1016/S0362-546X(96)00313-6
[24]Meehan, M.; O’regan, D.: Multiple nonnegative solutions of nonlinear integral equations on compact and semi-infinite intervals, Appl. anal. 74, 413-427 (2000) · Zbl 1021.45007 · doi:10.1080/00036810008840824
[25]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 · doi:10.1006/jdeq.2000.3995
[26]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 · doi:10.1016/S0022-0396(02)00152-3
[27]Torres, P. J.: Non-collision periodic solutions of forced dynamical systems with weak singularities, Discrete contin. Dyn. syst. 11, 693-698 (2004) · Zbl 1063.34035 · doi:10.3934/dcds.2004.11.693
[28]Torres, P. J.: Weak singularities May help periodic solutions to exist, J. differential equations 232, 277-284 (2007) · Zbl 1116.34036 · doi:10.1016/j.jde.2006.08.006
[29]Torres, P. J.: Existence and stability of periodic solutions for second order semilinear differential equations with a singular nonlinearity, Proc. roy. Soc. Edinburgh sect. A 137, 195-201 (2007) · Zbl 1190.34050 · doi:10.1017/S0308210505000739
[30]Yan, P.; Zhang, M.: Higher order nonresonance for differential equations with singularities, Math. methods appl. Sci. 26, 1067-1074 (2003) · Zbl 1031.34040 · doi:10.1002/mma.413
[31]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 · doi:10.1017/S0308210500030080
[32]Zhang, M.: Periodic solutions of damped differential systems with repulsive singular forces, Proc. amer. Math. soc. 127, 401-407 (1999) · Zbl 0908.34024 · doi:10.1090/S0002-9939-99-05120-5
[33]Zhang, M.: Periodic solutions of equations of Ermakov-pinney type, Adv. nonlinear stud. 6, 57-67 (2006) · Zbl 1107.34037
[34]Zhang, M.: Sobolev inequalities and ellipticity of planar linear Hamiltonian systems, Adv. nonlinear stud. 8, 633-654 (2008) · Zbl 1165.34053
[35]Zhang, M.: Optimal conditions for maximum and antimaximum principles of the periodic solution problem, Bound. value probl. 2010 (2010) · Zbl 1200.34001 · doi:10.1155/2010/410986