Previous |  Up |  Next

Article

Keywords:
periodic boundary value problem; impulsive differential equation; fixed-point theorem; growth condition
Summary:
This paper deals with the periodic boundary value problem for nonlinear impulsive functional differential equation $$ \begin{cases} x'(t)=f(t,x(t),x(\alpha _1(t)),\cdots ,x(\alpha _n(t))) \text {for a.e.} \ t\in [0,T], \Delta x(t_k)=I_k(x(t_k)), \ k=1,\cdots ,m, x(0)=x(T). \end{cases} $$ We first present a survey and then obtain new sufficient conditions for the existence of at least one solution by using Mawhin's continuation theorem. Examples are presented to illustrate the main results.
References:
[1] Chen, J., Tisdell, C. C., Yuan, R.: On the solvability of periodic boundary value problems with impulse. J. Math. Anal. Appl. 331 (2007), 902-912. DOI 10.1016/j.jmaa.2006.09.021 | MR 2313690 | Zbl 1123.34022
[2] Franco, D., Nieto, J. J.: A new maximum principle for impulsive first-order problems. Int. J. Theor. Phys. 37 (1998), 1607-1616. DOI 10.1023/A:1026676105073 | MR 1627762 | Zbl 0946.34024
[3] Franco, D., Nieto, J. J.: First-order impulsive ordinary differential equations with anti-periodic and nonlinear boundary conditions. Nonlinear Anal., Theory Methods Appl. 42 (2000), 163-173. DOI 10.1016/S0362-546X(98)00337-X | MR 1773975 | Zbl 0966.34025
[4] Franco, D., Nieto, J. J.: Maximum principles for periodic impulsive first order problems. J. Comput. Appl. Math. 88 (1998), 144-159. DOI 10.1016/S0377-0427(97)00212-4 | MR 1609074 | Zbl 0898.34010
[5] Gaines, R. E., Mawhin, J. L.: Coincidence Degree, and Nonlinear Differential Equations. Lecture Notes in Math., Vol. 568. Springer Berlin (1977). MR 0637067
[6] He, Z., Yu, J.: Periodic boundary value problems for first order impulsive ordinary differential equations. J. Math. Anal. Appl. 272 (2002), 67-78. DOI 10.1016/S0022-247X(02)00133-6
[7] Hernández, E. M., Rabello, M., Henriquez, H. R.: Existence of solutions for impulsive partial neutral functional differential equations. J. Math. Anal. Appl. 331 (2007), 1135-1158. DOI 10.1016/j.jmaa.2006.09.043 | MR 2313705 | Zbl 1123.34062
[8] Jiang, D., Nieto, J. J., Zuo, W.: On monotone method for first and second order periodic boundary value problems and periodic solutions of functional differential equations. J. Math. Anal. Appl. 289 (2004), 691-699. DOI 10.1016/j.jmaa.2003.09.020 | MR 2026934 | Zbl 1134.34322
[9] Li, X., Lin, X., Jiang, D., Zhang, X.: Existence and multiplicity of positive periodic solutions to functional differential equations with impulse effects. Nonlinear Anal., Theory Methods Appl. 62 (2005), 683-701. DOI 10.1016/j.na.2005.04.005 | MR 2149910 | Zbl 1084.34071
[10] Li, J., Shen, J.: Periodic boundary value problems for delay differential equations with impulses. J. Comput. Appl. Math. 193 (2006), 563-573. DOI 10.1016/j.cam.2005.05.037 | MR 2229561 | Zbl 1101.34050
[11] Liang, R., Shen, J.: Periodic boundary value problem for the first order impulsive functional differential equations. J. Comput. Appl. Math. 202 (2007), 498-510. DOI 10.1016/j.cam.2006.03.017 | MR 2319972 | Zbl 1123.34050
[12] Liu, Y.: Further results on periodic boundary value problems for nonlinear first order impulsive functional differential equations. J. Math. Anal. Appl. 327 (2007), 435-452. DOI 10.1016/j.jmaa.2006.01.027 | MR 2277424 | Zbl 1119.34062
[13] Liu, X.: Nonlinear boundary value problems for first order impulsive integro-differential equations. Appl. Anal. 36 (1990), 119-130. DOI 10.1080/00036819008839925 | MR 1040882 | Zbl 0671.34018
[14] Liu, Y., Bai, Z., Gui, Z., Ge, W.: Positive periodic solutions of impulsive delay differential equations with sign-changing coefficients. Port. Math. (N.S.) 61 (2004), 177-191. MR 2066673
[15] Liu, Y., Ge, W.: Stability theorems and existence results for periodic solutions of nonlinear impulsive delay differential equations with variable coefficients. Nonlinear Anal., Theory Methods Appl. 57A (2004), 363-399. MR 2064097 | Zbl 1064.34051
[16] Nieto, J. J.: Basic theory for nonresonance impulsive periodic problems of first order. J. Math. Anal. Appl. 205 (1997), 423-433. DOI 10.1006/jmaa.1997.5207 | MR 1428357 | Zbl 0870.34009
[17] Nieto, J. J.: Impulsive resonance periodic problems of first order. Appl. Math. Lett. 15 (2002), 489-493. DOI 10.1016/S0893-9659(01)00163-X | MR 1902284 | Zbl 1022.34025
[18] Nieto, J. J.: Periodic boundary value problems for first-order impulsive ordinary differential equations. Nonlinear Anal., Theory Methods Appl. 51 (2002), 1223-1232. DOI 10.1016/S0362-546X(01)00889-6 | MR 1926625 | Zbl 1015.34010
[19] Nieto, J. J., Rodríguez-López, R.: Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations. J. Math. Anal. Appl. 318 (2006), 593-610. DOI 10.1016/j.jmaa.2005.06.014 | MR 2215172 | Zbl 1101.34051
[20] Pierson-Gorez, C.: Impulsive differential equations of first order with periodic boundary conditions. Differ. Equ. Dyn. Syst. 1 (1993), 185-196. MR 1258896 | Zbl 0868.34007
[21] Tang, S., Chen, L.: Global attractivity in a ``food-limited'' population model with impulsive effects. J. Math. Anal. Appl. 292 (2004), 211-221. DOI 10.1016/j.jmaa.2003.11.061 | MR 2050225 | Zbl 1062.34055
[22] Vatsala, A. S., Sun, Y.: Periodic boundary value problems of impulsive differential equations. Appl. Anal. 44 (1992), 145-158. DOI 10.1080/00036819208840074 | MR 1284994 | Zbl 0753.34008
[23] Yang, X., Shen, J.: Nonlinear boundary value problems for first order impulsive functional differential equations. Appl. Math. Comput. 189 (2007), 1943-1952. DOI 10.1016/j.amc.2006.12.085 | MR 2332147 | Zbl 1125.65074
Partner of
EuDML logo