×

Existence for singular periodic problems: a survey of recent results. (English) Zbl 1277.34002

Summary: We present a survey on the existence of periodic solutions of singular differential equations. In particular, we pay attention to singular scalar differential equations, singular damped differential equations, singular impulsive differential equations, and singular differential systems.

MSC:

34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Agarwal, R. P.; Franco, D.; O’Regan, D., Singular boundary value problems for first and second order impulsive differential equations, Aequationes Mathematicae, 69, 1-2, 83-96 (2005) · Zbl 1073.34025
[2] Agarwal, R. P.; O’Regan, D.; Yan, B., Multiple positive solutions of singular Dirichlet second-order boundary-value problems with derivative dependence, Journal of Dynamical and Control Systems, 15, 1, 1-26 (2009) · Zbl 1203.34036
[3] Ambrosetti, A.; Coti Zelati, V., Periodic Solutions of Singular Lagrangian Systems (1993), Boston, Mass, USA: Birkhäuser Boston Inc., Boston, Mass, USA · Zbl 0785.34032
[4] Bahri, A.; Rabinowitz, P. H., A minimax method for a class of Hamiltonian systems with singular potentials, Journal of Functional Analysis, 82, 2, 412-428 (1989) · Zbl 0681.70018
[5] Barteneva, I. V.; Cabada, A.; Ignatyev, A. O., Maximum and anti-maximum principles for the general operator of second order with variable coefficients, Applied Mathematics and Computation, 134, 1, 173-184 (2003) · Zbl 1037.34014
[6] Boucherif, A.; Daoudi-Merzagui, N., Periodic solutions of singular nonautonomous second order differential equations, NoDEA. Nonlinear Differential Equations and Applications, 15, 1-2, 147-158 (2008) · Zbl 1152.34027
[7] Chu, J.; Nieto, J. J., Recent existence results for second-order singular periodic differential equations, Boundary Value Problems, 2009 (2009) · Zbl 1182.34057
[8] Habets, P.; Sanchez, L., Periodic solutions of some Liénard equations with singularities, Proceedings of the American Mathematical Society, 109, 4, 1035-1044 (1990) · Zbl 0695.34036
[9] Li, X.; Zhang, Z., Periodic solutions for damped differential equations with a weak repulsive singularity, Nonlinear Analysis. Theory, Methods & Applications, 70, 6, 2395-2399 (2009) · Zbl 1165.34349
[10] Rachunkova, I.; Tvrdý, M., Existence results for impulsive second-order periodic problems, Nonlinear Analysis. Theory, Methods & Applications, 59, 1-2, 133-146 (2004) · Zbl 1084.34031
[11] Yan, P.; Zhang, M., Higher order non-resonance for differential equations with singularities, Mathematical Methods in the Applied Sciences, 26, 12, 1067-1074 (2003) · Zbl 1031.34040
[12] Bevc, V.; Palmer, J. L.; Susskind, C., On the design of the transition region of axisymmetric, magnetically focused beam valves, Journal of the British Institution of Radio Engineers, 18, 12, 696-708 (1958)
[13] Ren, J.; Cheng, Z.; Siegmund, S., Positive periodic solution for Brillouin electron beam focusing system, Discrete and Continuous Dynamical Systems B, 16, 1, 385-392 (2011) · Zbl 1242.34072
[14] Torres, P. J., Existence and uniqueness of elliptic periodic solutions of the Brillouin electron beam focusing system, Mathematical Methods in the Applied Sciences, 23, 13, 1139-1143 (2000) · Zbl 0966.34038
[15] del Pino, M. A.; Manásevich, R. F., Infinitely many \(T\)-periodic solutions for a problem arising in nonlinear elasticity, Journal of Differential Equations, 103, 2, 260-277 (1993) · Zbl 0781.34032
[16] Chu, J.; Zhang, M., Rotation numbers and Lyapunov stability of elliptic periodic solutions, Discrete and Continuous Dynamical Systems A, 21, 4, 1071-1094 (2008) · Zbl 1161.37041
[17] Chu, J.; Li, M., Twist periodic solutions of second order singular differential equations, Journal of Mathematical Analysis and Applications, 355, 2, 830-838 (2009) · Zbl 1172.34029
[18] Zhang, M., Periodic solutions of equations of Emarkov-Pinney type, Advanced Nonlinear Studies, 6, 1, 57-67 (2006) · Zbl 1107.34037
[19] Sun, J.; O’Regan, D., Impulsive periodic solutions for singular problems via variational methods, Bulletin of the Australian Mathematical Society, 86, 2, 193-204 (2012) · Zbl 1280.34049
[20] Sun, J.; Chu, J.; Chen, H., Periodic solution generated by impulses for singular differential equations · Zbl 1304.34076
[21] Bonheure, D.; De Coster, C., Forced singular oscillators and the method of lower and upper solutions, Topological Methods in Nonlinear Analysis, 22, 2, 297-317 (2003) · Zbl 1108.34033
[22] Bravo, J. L.; Torres, P. J., Periodic solutions of a singular equation with indefinite weight, Advanced Nonlinear Studies, 10, 4, 927-938 (2010) · Zbl 1232.34064
[23] Chu, J.; Nieto, J. J., Impulsive periodic solutions of first-order singular differential equations, Bulletin of the London Mathematical Society, 40, 1, 143-150 (2008) · Zbl 1144.34016
[24] Fonda, A.; Toader, R., Periodic orbits of radially symmetric Keplerian-like systems: a topological degree approach, Journal of Differential Equations, 244, 12, 3235-3264 (2008) · Zbl 1168.34031
[25] Fonda, A.; Ureña, A. J., Periodic, subharmonic, and quasi-periodic oscillations under the action of a central force, Discrete and Continuous Dynamical Systems A, 29, 1, 169-192 (2011) · Zbl 1231.34073
[26] Torres, P. J., Non-collision periodic solutions of forced dynamical systems with weak singularities, Discrete and Continuous Dynamical Systems. Series A, 11, 2-3, 693-698 (2004) · Zbl 1063.34035
[27] Zhang, M., Periodic solutions of damped differential systems with repulsive singular forces, Proceedings of the American Mathematical Society, 127, 2, 401-407 (1999) · Zbl 0908.34024
[28] Zhang, M., A relationship between the periodic and the Dirichlet BVPs of singular differential equations, Proceedings of the Royal Society of Edinburgh A, 128, 5, 1099-1114 (1998) · Zbl 0918.34025
[29] Lazer, A. C.; Solimini, S., On periodic solutions of nonlinear differential equations with singularities, Proceedings of the American Mathematical Society, 99, 1, 109-114 (1987) · Zbl 0616.34033
[30] Gordon, W. B., Conservative dynamical systems involving strong forces, Transactions of the American Mathematical Society, 204, 113-135 (1975) · Zbl 0276.58005
[31] Chu, J.; Torres, P. J., Applications of Schauder’s fixed point theorem to singular differential equations, Bulletin of the London Mathematical Society, 39, 4, 653-660 (2007) · Zbl 1128.34027
[32] Chu, J.; Torres, P. J.; Zhang, M., Periodic solutions of second order non-autonomous singular dynamical systems, Journal of Differential Equations, 239, 1, 196-212 (2007) · Zbl 1127.34023
[33] Chu, J.; Li, M., Positive periodic solutions of Hill’s equations with singular nonlinear perturbations, Nonlinear Analysis. Theory, Methods & Applications, 69, 1, 276-286 (2008) · Zbl 1148.34025
[34] Chu, J.; Zhang, Z., Periodic solutions of second order superlinear singular dynamical systems, Acta Applicandae Mathematicae, 111, 2, 179-187 (2010) · Zbl 1204.34049
[35] Chu, J.; Fan, N.; Torres, P. J., Periodic solutions for second order singular damped differential equations, Journal of Mathematical Analysis and Applications, 388, 2, 665-675 (2012) · Zbl 1232.34065
[36] Franco, D.; Torres, P. J., Periodic solutions of singular systems without the strong force condition, Proceedings of the American Mathematical Society, 136, 4, 1229-1236 (2008) · Zbl 1129.37033
[37] Jiang, D.; Chu, J.; Zhang, M., Multiplicity of positive periodic solutions to superlinear repulsive singular equations, Journal of Differential Equations, 211, 2, 282-302 (2005) · Zbl 1074.34048
[38] Rachunkova, I.; Tvrdý, M.; Vrkoč, I., Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems, Journal of Differential Equations, 176, 2, 445-469 (2001) · Zbl 1004.34008
[39] Torres, P. J., Weak singularities may help periodic solutions to exist, Journal of Differential Equations, 232, 1, 277-284 (2007) · Zbl 1116.34036
[40] Cabada, A.; Cid, J., On the sign of the Green’s function associated to Hill’s equation with an indefinite potential, Applied Mathematics and Computation, 205, 1, 303-308 (2008) · Zbl 1161.34014
[41] Cheng, Z.; Ren, J., Periodic and subharmonic solutions for Duffing equation with a singularity, Discrete and Continuous Dynamical Systems. Series A, 32, 5, 1557-1574 (2012) · Zbl 1263.34054
[42] Chu, J.; Zhang, Z., Periodic solutions of singular differential equations with sign-changing potential, Bulletin of the Australian Mathematical Society, 82, 3, 437-445 (2010) · Zbl 1364.34056
[43] Franco, D.; Webb, J. R. L., Collisionless orbits of singular and non singular dynamical systems, Discrete and Continuous Dynamical Systems A, 15, 3, 747-757 (2006) · Zbl 1120.34029
[44] Hakl, R.; Torres, P. J., On periodic solutions of second-order differential equations with attractive-repulsive singularities, Journal of Differential Equations, 248, 1, 111-126 (2010) · Zbl 1187.34049
[45] Lin, X.; Jiang, D.; O’Regan, D.; Agarwal, R. P., Twin positive periodic solutions of second order singular differential systems, Topological Methods in Nonlinear Analysis, 25, 2, 263-273 (2005) · Zbl 1098.34032
[46] Torres, P. J., Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem, Journal of Differential Equations, 190, 2, 643-662 (2003) · Zbl 1032.34040
[47] Zhang, M.; Li, W., A Lyapunov-type stability criterion using \(L^\infty\) norms, Proceedings of the American Mathematical Society, 130, 11, 3325-3333 (2002) · Zbl 1007.34053
[48] Zhang, M., Optimal conditions for maximum and antimaximum principles of the periodic solution problem, Boundary Value Problems, 2010 (2010) · Zbl 1200.34001
[49] Granas, A.; Guenther, R. B.; Lee, J. W., Some general existence principles in the Carathéodory theory of nonlinear differential systems, Journal de Mathématiques Pures et Appliquées, 70, 2, 153-196 (1991) · Zbl 0687.34009
[50] Granas, A.; Dugundji, J., Fixed Point Theory. Fixed Point Theory, Springer Monographs in Mathematics (2003), New York, NY, USA: Springer-Verlag, New York, NY, USA · Zbl 1025.47002
[51] Krasnosel’skii, M. A., Positive Solutions of Operator Equations (1964), Groningen, TheNetherlands: P. Noordhoff Ltd., Groningen, TheNetherlands · Zbl 0121.10604
[52] Hakl, R.; Torres, P. J., Maximum and antimaximum principles for a second order differential operator with variable coefficients of indefinite sign, Applied Mathematics and Computation, 217, 19, 7599-7611 (2011) · Zbl 1235.34064
[53] Nieto, J. J.; O’Regan, D., Variational approach to impulsive differential equations, Nonlinear Analysis. Real World Applications, 10, 2, 680-690 (2009) · Zbl 1167.34318
[54] Sun, J.; Chen, H.; Nieto, J. J.; Otero-Novoa, M., The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects, Nonlinear Analysis. Theory, Methods & Applications, 72, 12, 4575-4586 (2010) · Zbl 1198.34036
[55] Mawhin, J.; Willem, M., Critical Point Theory and Hamiltonian Systems, 74 (1989), New York, NY, USA: Springer-Verlag, New York, NY, USA · Zbl 0676.58017
[56] Agarwal, R. P.; O’Regan, D.; Wong, P. J. Y., Constant-sign solutions of a system of integral equations: the semipositone and singular case, Asymptotic Analysis, 43, 1-2, 47-74 (2005) · Zbl 1081.45002
[57] Agarwal, R. P.; O’Regan, D.; Wong, P. J. Y., Constant-sign solutions of a system of integral equations with integrable singularities, Journal of Integral Equations and Applications, 19, 2, 117-142 (2007) · Zbl 1136.45007
[58] Agarwal, R. P.; O’Regan, D.; Wong, P. J. Y., Constant sign solutions for systems of Fredholm and Volterra integral equations: the singular case, Acta Applicandae Mathematicae, 103, 3, 253-276 (2008) · Zbl 1160.45002
[59] Agarwal, R. P.; O’Regan, D.; Wong, P. J. Y., Constant-sign solutions for systems of singular integral equations of Hammerstein type, Mathematical and Computer Modelling, 50, 7-8, 999-1025 (2009) · Zbl 1193.45023
[60] Agarwal, R. P.; O’Regan, D.; Wong, P. J. Y., Constant-sign solutions for singular systems of Fredholm integral equations, Mathematical Methods in the Applied Sciences, 33, 15, 1783-1793 (2010) · Zbl 1205.45006
[61] Agarwal, R. P.; O’Regan, D.; Wong, P. J. Y., Periodic constant-sign solutions for systems of Hill’s equations, Asymptotic Analysis, 67, 3-4, 191-216 (2010) · Zbl 1207.34048
[62] O’Regan, D.; Meehan, M., Existence Theory for Nonlinear Integral and Integrodifferential Equations (1998), Dordrecht, The Netherlands: Kluwer, Dordrecht, The Netherlands · Zbl 0932.45010
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.