Previous |  Up |  Next

Article

Keywords:
second-order $p$-Laplacian Hamiltonian systems; impulsive effect; critical point theory
Summary:
The purpose of this paper is to study the existence and multiplicity of a periodic solution for the non-autonomous second-order system \begin {gather} \frac {{\rm d}}{{\rm d}t}(|\dot {u}(t)|^{p-2}\dot {u}(t)) =\nabla F(t, u(t)),\quad \text {\rm a.e.}\ t\in [0,T],\nonumber \\ u(0)-u(T)=\dot {u}(0)-\dot {u}(T)=0,\nonumber \\ \Delta \dot {u}^i(t_{j})=\dot {u}^i(t_j^+)-\dot {u}^i(t_j^-)=I_{ij}(u^i(t_j)),\ i = 1, 2,\dots , N;\ j = 1, 2,\dots ,m.\nonumber \end {gather} By using the least action principle and the saddle point theorem, some new existence theorems are obtained for second-order $p$-Laplacian systems with or without impulse under weak sublinear growth conditions, improving some existing results in the literature.
References:
[1] Agarwal, R. P., O'Regan, D.: Multiple nonnegative solutions for second order impulsive differential equations. Appl. Math. Comput. 114 (2000), 51-59. DOI 10.1016/S0096-3003(99)00074-0 | MR 1775121 | Zbl 1047.34008
[2] Berger, M. S., Schechter, M.: On the solvability of semilinear gradient operator equations. Adv. Math. 25 (1977), 97-132. DOI 10.1016/0001-8708(77)90001-9 | MR 0500336 | Zbl 0354.47025
[3] Chen, P., Tang, X. H.: Existence of solutions for a class of $p$-Laplacian systems with impulsive effects. Taiwanese J. Math. 16 (2012), 803-828. MR 2917240 | Zbl 1251.34044
[4] Lakshmikantham, V., Baĭnov, D. D., Simeonov, P. S.: Theory of Impulsive Differential Equations. Series in Modern Applied Mathematics 6 World Scientific, Singapore (1989). MR 1082551
[5] Lee, E. K., Lee, Y. H.: Multiple positive solutions of singular two point boundary value problems for second order impulsive differential equations. Appl. Math. Comput. 158 (2004), 745-759. DOI 10.1016/j.amc.2003.10.013 | MR 2095700 | Zbl 1069.34035
[6] Lin, X. N., Jiang, D. Q.: Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations. J. Math. Anal. Appl. 321 (2006), 501-514. DOI 10.1016/j.jmaa.2005.07.076 | MR 2241134 | Zbl 1103.34015
[7] Mawhin, J.: Semi-coercive monotone variational problems. Bull. Cl. Sci., V. Sér., Acad. R. Belg. 73 (1987), 118-130. MR 0938142 | Zbl 0647.49007
[8] Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems. Applied Mathematical Sciences 74 Springer, New York (1989). DOI 10.1007/978-1-4757-2061-7 | MR 0982267 | Zbl 0676.58017
[9] Mawhin, J., Willem, M.: Critical points of convex perturbations of some indefinite quadratic forms and semilinear boundary value problems at resonance. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3 (1986), 431-453. MR 0870864 | Zbl 0678.35091
[10] Nieto, J. J., O'Regan, D.: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl. 10 (2009), 680-690. MR 2474254 | Zbl 1167.34318
[11] Rabinowitz, P. H.: On subharmonic solutions of hamiltonian systems. Commun. Pure Appl. Math. 33 (1980), 609-633. DOI 10.1002/cpa.3160330504 | MR 0586414 | Zbl 0425.34024
[12] Rabinowitz, P. H.: Minimax methods in critical point theory with applications to differential equations. Reg. Conf. Ser. Math. 65 American Mathematical Society, Providence (1986). MR 0845785 | Zbl 0609.58002
[13] Samoilenko, A. M., Perestyuk, N. A.: Impulsive Differential Equations. Transl. from the Russian. World Scientific Series on Nonlinear Science, Series A. 14. Singapore (1995). MR 1355787 | Zbl 0837.34003
[14] Sun, J. T., Chen, H. B., Yang, L.: The existence and multiplicity of solutions for an impulsive differential equation with two parameters via a variational method. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73 (2010), 440-449. DOI 10.1016/j.na.2010.03.035 | MR 2650827 | Zbl 1198.34037
[15] Tang, C. L.: Periodic solutions of non-autonomous second order systems with $\gamma$-quasisubadditive potential. J. Math. Anal. Appl. 189 (1995), 671-675. DOI 10.1006/jmaa.1995.1044 | MR 1312546 | Zbl 0824.34043
[16] Tang, C. L.: Periodic solutions of non-autonomous second order systems. J. Math. Anal. Appl. 202 (1996), 465-469. DOI 10.1006/jmaa.1996.0327 | MR 1406241 | Zbl 0857.34044
[17] Tang, C. L.: Periodic solutions of nonautonomous second order systems with sublinear nonlinearity. Proc. Am. Math. Soc. 126 (1998), 3263-3270. DOI 10.1090/S0002-9939-98-04706-6 | MR 1476396
[18] Tang, C. L., Wu, X. P.: Periodic solutions for second order systems with not uniformly coercive potential. J. Math. Anal. Appl. 259 (2001), 386-397. DOI 10.1006/jmaa.2000.7401 | MR 1842066 | Zbl 0999.34039
[19] Tang, X. H., Meng, Q.: Solutions of a second-order Hamiltonian system with periodic boundary conditions. Nonlinear Anal., Real World Appl. 11 (2010), 3722-3733. MR 2683825 | Zbl 1223.34024
[20] Willem, M.: Forced oscillations of Hamiltonian systems. Publ. Math. Fac. Sci. Besançon, Anal. Non Lineaire Annee 1980-1981, Expose No. 4 French (1981).
[21] Wu, X.: Saddle point characterization and multiplicity of periodic solutions of non-autonomous second order systems. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 58 (2004), 899-907. DOI 10.1016/j.na.2004.05.020 | MR 2086063 | Zbl 1058.34053
[22] Wu, X. P., Tang, C. L.: Periodic solutions of a class of non-autonomous second-order systems. J. Math. Anal. Appl. 236 (1999), 227-235. DOI 10.1006/jmaa.1999.6408 | MR 1704579 | Zbl 0971.34027
[23] Yang, X. X., Shen, J. H.: 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
[24] Zavalishchin, S. T., Sesekin, A. N.: Dynamics Impulse System: Theory and Applications. Mathematics and its Applications 394 Kluwer, Dordrecht (1997). MR 1441079
[25] Zhao, F., Wu, X.: Periodic solutions for a class of non-autonomous second order systems. J. Math. Anal. Appl. 296 (2004), 422-434. DOI 10.1016/j.jmaa.2004.01.041 | MR 2075174
[26] Zhao, F., Wu, X.: Existence and multiplicity of periodic solution for non-autonomous second-order systems with linear nonlinearity. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 60 (2005), 325-335. MR 2101882 | Zbl 1087.34022
[27] Zhou, J. W., Li, Y. K.: Existence and multiplicity of solutions for some Dirichlet problems with impulsive effects. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71 (2009), 2856-2865. DOI 10.1016/j.na.2009.01.140 | MR 2532812 | Zbl 1175.34035
[28] Zhou, J. W., Li, Y. K.: Existence of solutions for a class of second-order Hamiltonian systems with impulsive effects. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72 (2010), 1594-1603. DOI 10.1016/j.na.2009.08.041 | MR 2577560 | Zbl 1193.34057
Partner of
EuDML logo