×

zbMATH — the first resource for mathematics

Existence and multiplicity of solutions for some Dirichlet problems with impulsive effects. (English) Zbl 1175.34035
The authors consider the following Dirichlet boundary value problem with impulses
\[ \left\{\begin{aligned} &-u''(t)+g(t)u(t)=f(t,u(t)) \quad \text{a.e.}\,\, t\in [0,T]\\ &u(0)=u(T)=0,\\ & \Delta u'(t_j)=u'(t_j^+)-u'(t_j^-)=I_j(u(t_j)), \,\,\, j=1,2,\dots, p, \end{aligned} \right. \] where \(t_0=0<t_1<t_2<\dots<t_p<t_{p+1}=T,\) \(g\in L^{\infty}[0,T],\) \(f: [0,T]\times {\mathbb R}\to {\mathbb R}\) is continuous and \(I_j: {\mathbb R}\to {\mathbb R},\) \(j=1,2,\dots,p\) are continuous. Existence and multiplicity results are obtained via Lax-Milgram theorem and critical points theorems. The main results are illustrated by examples.

MSC:
34B37 Boundary value problems with impulses for ordinary differential equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Benchohra, M.; Henderson, J.; Ntouyas, S., Impulsive differential equations and inclusions, (2006), Hindawi Publishing Corporation New York · Zbl 1130.34003
[2] Samoilenko, A.M.; Perestyuk, N.A., Impulsive differential equations, (1995), World Scientific Publishing Co., Inc. River Edge, NJ · Zbl 0837.34003
[3] Haddad, W.M.; Chellaboina, V.; Nersesov, S.G.; Sergey, G., Impulsive and hybrid dynamical systems. stability, dissipativity, and control, (2006), Princeton University Press Princeton, NJ · Zbl 1114.34001
[4] Nieto, J.J., Impulsive resonance periodic problems of first order, Appl. math. lett., 15, 489-493, (2002) · Zbl 1022.34025
[5] Nieto, J.J.; Rodriguez-Lopez, R., Boundary value problems for a class of impulsive functional equations, Comput. math. appl., 55, 2715-2731, (2008) · Zbl 1142.34362
[6] Chu, J.; Nieto, J.J., Impulsive periodic solutions of first-order singular differential equations, Bull. London math. soc., 40, 143-150, (2008) · Zbl 1144.34016
[7] Z. Zhang, R. Yuan, An application of variational methods to Dirichlet boundary value problem with impulse, Nonlinear Anal. RWA, in press (doi:10.1016/j.nonrwa.2008.10.044)
[8] Z. Liu, Impulsive perturbations in a periodic delay differential equation model of plankton allelopathy, Nonlinear Anal. RWA, in press (doi:10.1016/j.nonrwa.2008.11.017)
[9] Y.K. Chang, J.J. Nieto, W.S. Li, On impulsive hyperbolic differential inclusions with nonlocal initial conditions, J. Optim. Theory Appl., in press, (doi:10.1007/s10957-008-9468-1) · Zbl 1159.49042
[10] H. Zhang, Z. Li, Variational approach to impulsive differential equations with periodic boundary conditions, Nonlinear Anal. RWA, in press (doi:10.1016/j.nonrwa.2008.10.016)
[11] Carter, T.E., Necessary and sufficient conditions for optimal impulsive rendezvous with linear equations of motion, Dynam. control, 10, 219-227, (2000) · Zbl 0980.93058
[12] Zhang, H.; Xu, W.; Chen, L., A impulsive infective transmission SI model for pest control, Math. methods appl. sci., 30, 1169-1184, (2007) · Zbl 1155.34328
[13] Wang, W.B.; Shen, J.H.; Nieto, J.J., Permanence periodic solution of predator prey system with Holling type functional response and impulses, Discrete dyn. nat. soc., (2007), (Article ID 81756), 15 pages. doi:10.1155/2007/81756
[14] Zeng, G.Z.; Wang, F.Y.; Nieto, J.J., Complexity of delayed predator-prey model with impulsive harvest and Holling type-II functional response, Adv. complex syst., 11, 77-97, (2008) · Zbl 1168.34052
[15] Nieto, J.J.; Rodriguez-Lopez, R., Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations, J. math. anal. appl., 318, 593-610, (2006) · Zbl 1101.34051
[16] Luo, Z.; Nieto, J.J., New results of periodic boundary value problem for impulsive integro-differential equations, Nonlinear anal., 70, 2248-2260, (2009) · Zbl 1166.45002
[17] Nieto, J.J.; Rodriguez-Lopez, R., New comparison results for impulsive integro-differential equations and applications, J. math. anal. appl., 328, 1343-1368, (2007) · Zbl 1113.45007
[18] Wang, W.; Zhang, L.; Liang, Z., Initial value problems for nonlinear impulsive integro-differential equations in Banach space, J. math. anal. appl., 320, 510-527, (2006) · Zbl 1097.45011
[19] Ahmad, B.; Nieto, J.J., Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions, Nonlinear anal., 60, 3291-3298, (2008) · Zbl 1158.34049
[20] Liu, K.; Yang, G., Cone-valued Lyapunov functions and stability for impulsive functional differential equations, Nonlinear anal., 69, 2184-2191, (2008) · Zbl 1151.34063
[21] E. Hernandez, H.R. Henriquez, M.A. McKibben, Existence results for abstract impulsive second-order neutral functional differential equations, Nonlinear Anal., in press (doi:10.1016/j.na.2008.03.062) · Zbl 1173.34049
[22] Li, Y.K., Positive periodic solutions of nonlinear differential systems with impulses, Nonlinear anal., 68, 2389-2405, (2008) · Zbl 1162.34064
[23] Liu, X.; Willms, A.R., Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft, Math. probl. eng., 2, 277-299, (1996) · Zbl 0876.93014
[24] Prado, A.F.B.A., Bi-impulsive control to build a satellite constellation, Nonlinear dyn. syst. theory, 5, 169-175, (2005) · Zbl 1128.70015
[25] Zhang, W.; Fan, M., Periodicity in a generalized ecological competition system governed by impulsive differential equations with delays, Math. comput. modelling, 39, 479-493, (2004) · Zbl 1065.92066
[26] Yan, J.; Zhao, A.; Nieto, J.J., Existence and global attractivity of positive periodic solution of periodic single-species impulsive lotka – volterra systems, Math. comput. modelling, 40, 509-518, (2004) · Zbl 1112.34052
[27] Li, W.; Huo, H., Global attractivity of positive periodic solutions for an impulsive delay periodic model of respiratory dynamics, J. comput. appl. math., 174, 227-238, (2005) · Zbl 1070.34089
[28] Shen, J.H., Existence and global attractivity of positive periodic solutions for impulsive predator-prey model with dispersion and time delays, Nonlinear anal. RWA, 10, 227-243, (2009) · Zbl 1154.34372
[29] Zhang, X.; Shuai, Z.; Wang, K., Optimal impulsive harvesting policy for single population, Nonlinear anal. RWA, 4, 639-651, (2003) · Zbl 1011.92052
[30] d’Onofrio, A., On pulse vaccination strategy in the SIR epidemic model with vertical transmission, Appl. math. lett., 18, 729-732, (2005) · Zbl 1064.92041
[31] Zhang, H.; Chen, L.S.; Nieto, J.J., A delayed epidemic model with stage-structure and pulses for management strategy, Nonlinear anal. RWA, 9, 1714-1726, (2008) · Zbl 1154.34394
[32] Nieto, J.J.; O’Regan, D., Variational approach to impulsive differential equations, Nonlinear anal. RWA, 10, 680-690, (2009) · Zbl 1167.34318
[33] Pasquero, S., On the simultaneous presence of unilateral and kinetic constraints in time-dependent impulsive mechanics, J. math. phys., 47, (2006), p. 082903, 19 pages · Zbl 1112.70014
[34] Tian, Y.; Ge, W.G., Applications of variational methods to boundary value problem for impulsive differential equations, Proc. edinb. math. soc., 51, 509-527, (2008) · Zbl 1163.34015
[35] Lee, E.K.; Lee, Y.H., Multiple positive solutions of singular two point boundary value problems for second order impulsive differential equation, Appl. math. comput., 158, 745-759, (2004) · Zbl 1069.34035
[36] 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, 501-514, (2006) · Zbl 1103.34015
[37] Mawhin, J.; Willem, M., Critical point theory and Hamiltonian systems, (1989), Springer-Verlag Berlin · Zbl 0676.58017
[38] Chipot, M., Elements of nonlinear analysis, (2000), Birkhauser Verlag Basel · Zbl 0977.35050
[39] Rabinowitz, P.H., ()
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.