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Yang, Liu

Existence and multiplicity of periodic solutions generated by impulses. (English) Zbl 1347.35125

Adv. Difference Equ. 2013, Paper No. 347, 12 p. (2013).
Summary: In this paper, we investigate the existence and multiplicity of periodic solutions for a class of \(p\)-Laplacian systems with impulses. By using variational methods and critical point theory, we obtain that such a system possesses at least one, two periodic solutions generated by impulses under different conditions, respectively. Recent results in the literature are generalized and significantly improved.

MSC:

35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35B10 Periodic solutions to PDEs
34A37 Ordinary differential equations with impulses

Keywords:

periodic solutions; solution generated by impulses; critical point
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[1] George PK, Nandakumaran AK, Arapostathis A: A note on controllability of impulsive systems. J. Math. Anal. Appl. 2000, 241: 276-283. 10.1006/jmaa.1999.6632 · Zbl 0965.93015 · doi:10.1006/jmaa.1999.6632
[2] Carter TE: Necessary and sufficient conditions for optimal impulsive rendezvous with linear equations of motion. Dyn. Control 2000, 10: 219-227. 10.1023/A:1008376427023 · Zbl 0980.93058 · doi:10.1023/A:1008376427023
[3] Pei Y, Li C, Chen L, Wang C: Complex dynamics of one-prey multi-predator system with defensive ability of prey and impulsive biological control on predators. Adv. Complex Syst. 2005, 8: 483-495. 10.1142/S0219525905000579 · Zbl 1082.92046 · doi:10.1142/S0219525905000579
[4] Liu X, Willms AR: Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft. Math. Probl. Eng. 1996, 2: 277-299. 10.1155/S1024123X9600035X · Zbl 0876.93014 · doi:10.1155/S1024123X9600035X
[5] Gao S, Chen L, Nieto JJ, Torres A: Analysis of a delayed epidemic model with pulse vaccination and saturation incidence. Vaccine 2006, 24: 6037-6045. 10.1016/j.vaccine.2006.05.018 · doi:10.1016/j.vaccine.2006.05.018
[6] Chu J, Nieto JJ: Impulsive periodic solutions of first-order singular differential equations. Bull. Lond. Math. Soc. 2008, 40: 143-150. 10.1112/blms/bdm110 · Zbl 1144.34016 · doi:10.1112/blms/bdm110
[7] Zavalishchin ST, Sesekin AN: Dynamic Impulse System. Theory and Applications. Kluwer Academic, Dordrecht; 1997. · Zbl 0880.46031 · doi:10.1007/978-94-015-8893-5
[8] Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations. World Scientific, Singapore; 1989. · Zbl 0719.34002 · doi:10.1142/0906
[9] Haddad WM, Chellaboina C, Nersesov SG, Sergey G: Impulsive and Hybrid Dynamical Systems. Stability, Dissipativity and Control. Princeton University Press, Princeton; 2006. · Zbl 1114.34001
[10] Samoilenko AM, Perestyuk NA: Impulsive Differential Equations. World Scientific, Singapore; 1995. · Zbl 0837.34003
[11] Agarwal RP, O’regan D: A multiplicity result for second order impulsive differential equations via the Leffett Williams fixed point theorem. Appl. Math. Comput. 2005, 161: 433-439. 10.1016/j.amc.2003.12.096 · Zbl 1070.34042 · doi:10.1016/j.amc.2003.12.096
[12] Wang H, Chen H: Boundary value problem for second-order impulsive functional differential equations. Appl. Math. Comput. 2007, 191: 582-591. 10.1016/j.amc.2007.02.140 · Zbl 1193.34134 · doi:10.1016/j.amc.2007.02.140
[13] Chen J, Tisdel CC, Yuan R: On the solvability of periodic boundary value problems with impulse. J. Math. Anal. Appl. 2007, 331: 902-912. 10.1016/j.jmaa.2006.09.021 · Zbl 1123.34022 · doi:10.1016/j.jmaa.2006.09.021
[14] Li J, Nieto JJ, Shen J: Impulsive periodic boundary value problems of first-order differential equations. J. Math. Anal. Appl. 2005, 303: 288-303. 10.1016/j.jmaa.2004.08.034 · Zbl 1071.34005 · doi:10.1016/j.jmaa.2004.08.034
[15] Hernandez E, Henriquez HR, Mckibben MA: Existence results for abstract impulsive second-order neutral functional differential equations. Nonlinear Anal. TMA 2009, 70: 2736-2751. 10.1016/j.na.2008.03.062 · Zbl 1173.34049 · doi:10.1016/j.na.2008.03.062
[16] Nieto JJ, O’Regan D: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl. 2009, 10: 680-690. 10.1016/j.nonrwa.2007.10.022 · Zbl 1167.34318 · doi:10.1016/j.nonrwa.2007.10.022
[17] Tian Y, Ge W: Applications of variational methods to boundary value problem for impulsive differential equations. Proc. Edinb. Math. Soc. 2008, 51: 509-527. · Zbl 1163.34015 · doi:10.1017/S0013091506001532
[18] Zhang Z, Yuan R: An application of variational methods to Dirichlet boundary value problem with impulses. Nonlinear Anal., Real World Appl. 2010, 11: 155-162. 10.1016/j.nonrwa.2008.10.044 · Zbl 1191.34039 · doi:10.1016/j.nonrwa.2008.10.044
[19] Zhou J, Li Y: Existence and multiplicity of solutions for some Dirichlet problems with impulsive effects. Nonlinear Anal. TMA 2009, 71: 2856-2865. 10.1016/j.na.2009.01.140 · Zbl 1175.34035 · doi:10.1016/j.na.2009.01.140
[20] Sun, J.; Chen, H., Variational method to the impulsive equation with Neumann boundary conditions, No. 2009 (2009) · Zbl 1184.34039
[21] Sun J, Chen H, Nieto JJ, Otero-Novoa M: The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects. Nonlinear Anal. TMA 2010, 72: 4575-4586. 10.1016/j.na.2010.02.034 · Zbl 1198.34036 · doi:10.1016/j.na.2010.02.034
[22] Zhang H, Li Z: Periodic and homoclinic solutions generated by impulses. Nonlinear Anal. 2011, 12(1):39-51. 10.1016/j.nonrwa.2010.05.034 · Zbl 1225.34019 · doi:10.1016/j.nonrwa.2010.05.034
[23] Han X, Zhang H: Periodic and homoclinic solutions generated by impulses for asymptotically linear and sublinear Hamiltonian system. J. Comput. Appl. Math. 2011, 235: 1531-1541. 10.1016/j.cam.2010.08.040 · Zbl 1211.34008 · doi:10.1016/j.cam.2010.08.040
[24] Han X, Zhang H, Li Z: Periodic solutions for p -Laplacian systems with impulsive conditions. Math. Appl. 2010, 23(1):213-218. · Zbl 1224.34120
[25] Teng K, Zhang C: Existence of solution to boundary value problem for impulsive differential equations. Nonlinear Anal., Real World Appl. 2010, 11(5):4431-4441. 10.1016/j.nonrwa.2010.05.026 · Zbl 1207.34034 · doi:10.1016/j.nonrwa.2010.05.026
[26] Zhang L, Ge W: Solvability of a kind of Sturm-Liouville boundary value problem with impulses via variational methods. Acta Appl. Math. 2010, 110: 1237-1248. 10.1007/s10440-009-9504-7 · Zbl 1194.34040 · doi:10.1007/s10440-009-9504-7
[27] Mawhin J, Willem M: Critical Point Theory and Hamiltonian Systems. Springer, Berlin; 1989. · Zbl 0676.58017 · doi:10.1007/978-1-4757-2061-7
[28] Rabinowitz, PH, CBMS Reg. Conf. Ser. in Math. 65 (1986), Providence · doi:10.1090/cbms/065
[29] Ricceri B: A further three critical points theorem. Nonlinear Anal. TMA 2009, 71: 4151-4157. 10.1016/j.na.2009.02.074 · Zbl 1187.47057 · doi:10.1016/j.na.2009.02.074
[30] Teng K: On variational methods to a generalized Emden-Fowler equation. Positivity 2010, 14: 335-351. 10.1007/s11117-009-0021-8 · Zbl 1206.34039 · doi:10.1007/s11117-009-0021-8
[31] Zeidler, E., 2 (1990), Berlin
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