×

Existence of solution to boundary value problem for impulsive differential equations. (English) Zbl 1207.34034

The authors deal with the homogeneous Dirichlet problem
\[ \begin{aligned} &-(|u'(t)|^{p-2}u'(t))' = f(t,u(t),u'(t)), \\ &\Delta u'(t_i) = I_i(u(t_i)), \quad i = 1,2,\dots,\ell,\\ &u(0) = u(T) = 0, \end{aligned} \]
where \(p \geq 2\), \(0 < t_1 < \dots < t_\ell < T\), \(\Delta u'(t_i) = (|u'|^{p-2}u')(t_i^+) - (|u'|^{p-2}u')(t_i^-)\). The existence of a nontrivial solution is obtained using variational and iterative methods.

MSC:

34B37 Boundary value problems with impulses for ordinary differential equations
58E30 Variational principles in infinite-dimensional spaces
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Benchohra, M.; Henderson, J.; Ntouyas, S.K., Impulsive differential equations and inclusions, vol. 2, (2006), Hindawi Publishing Corporation New York · Zbl 1130.34003
[2] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002
[3] Samoilenko, A.M.; Perestyuk, N.A., Impulsive differential equations, (1995), World Scientific Singapore · Zbl 0837.34003
[4] Zavalishchin, S.T.; Sesekin, A.N., Dynamic impulse systems. theory and applications, Mathematics and its applications, vol. 394, (1997), Kluwer Academic Publishers Group Dordrecht · Zbl 0880.46031
[5] Choisy, M.; Guegan, J.F.; Rohani, P., Dynamics of infectious diseases and pulse vaccination: teasing apart the embedded resonance effects, Physica D, 22, 26-35, (2006) · Zbl 1110.34031
[6] Gao, S.; Chen, L.; Nieto, J.J.; Torres, A., Analysis of a delayed epidemic model with pulse vaccination and saturation incidence, Vaccine, 24, 6037-6045, (2006)
[7] Jiao, J., Effect of delayed response in growth on the dynamics of a chemostat model with impulsive input, Chaos solitons fractals, 42, 2280-2287, (2009) · Zbl 1198.34134
[8] 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
[9] Meng, X., Dynamic analysis of michaelis – menten chemostat-type competition models with time delay and pulse in a polluted environment, J. math. chem., 47, 123-144, (2010) · Zbl 1194.92075
[10] 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
[11] Tang, S.; Chen, L., Density-dependent birth rate, birth pulses and their population dynamic consequences, J. math. biol., 44, 185-199, (2002) · Zbl 0990.92033
[12] Wang, L., The dynamics of an epidemic model for pest control with impulsive effect, Nonlinear anal. RWA, 11, 1374-1386, (2010) · Zbl 1188.93038
[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., 15, (2007), Article ID 81756 · Zbl 1146.37370
[14] 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
[15] Zeng, G.; Wang, F.; Nieto, J.J., Complexity of a delayed predator – prey model with impulsive harvest and Holling-type II functional response, Adv. complex syst., 11, 77-97, (2008) · Zbl 1168.34052
[16] 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
[17] Zhang, H., Impulsive perturbation and bifurcation of solutions for a model of chemostat with variable yield, Appl. math. mech. (English ed.), 30, 933-944, (2009) · Zbl 1178.34053
[18] Nieto, J.J.; O’ Regan, D., Variational approach to impulsive differential equations, Nonlinear anal. RWA, 10, 680-690, (2009) · Zbl 1167.34318
[19] 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
[20] Nieto, J.J., Variational formulation of a damped Dirichlet impulsive problem, Appl. math. lett., 23, 940-942, (2010) · Zbl 1197.34041
[21] Sun, J.; Chen, H., Variational method to the impulsive equation with Neumann boundary conditions, Bound. value prob., 2009, (2009), Article ID 316812 · Zbl 1184.34039
[22] Sun, J., The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects, Nonlinear anal. TMA, 72, 4575-4586, (2010) · Zbl 1198.34036
[23] Sun, J.; Chen, H., Multiplicity of solutions for a class of impulsive differential equations with Dirichlet boundary conditions via variant Fountain theorems, Nonlinear anal. RWA, 11, 5, 4062-4071, (2010) · Zbl 1208.34031
[24] 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
[25] Tian, Y.; Ge, W.G., Variational methods to sturm – liouville boundary value problem for impulsive differential equations, Nonlinear anal. TMA, 72, 277-287, (2010) · Zbl 1191.34038
[26] Zhang, H.; Li, Z., Variational approach to impulsive differential equations with periodic boundary condition, Nonlinear anal. RWA, 11, 67-78, (2010) · Zbl 1186.34089
[27] Agarwal, R.P.; Regan, D.O’, Multiple nonnegative solutions for second-order impulsive differential equations, Appl. math. comput., 114, 51-59, (2000) · Zbl 1047.34008
[28] Lin, X.; Jiang, D., 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
[29] Yao, M.; Zhao, A.; Yun, J., Periodic boundary value problems of second-order impulsive differential equations, Nonlinear anal. TMA, 70, 262-273, (2009) · Zbl 1176.34032
[30] Ahmad, B.; Sivasundaram, S., The monotone iterative technique for impulsive hybrid set valued integro-differential equations, Nonlinear anal. TMA, 65, 2260-2276, (2006) · Zbl 1111.45006
[31] Franco, D.; Nieto, J.J., First-order impulsive ordinary differential equations with antiperiodic and nonlinear boundary conditions, Nonlinear anal. TMA, 42, 163-173, (2000) · Zbl 0966.34025
[32] 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, 745-759, (2004) · Zbl 1069.34035
[33] De Figueiredo, D.; Girardi, M.; Matzeu, M., Semilinear elliptic equations with dependence on the gradient via mountain-pass techniques, Differential integral equations, 17, 119-126, (2004) · Zbl 1164.35341
[34] Del Pino, M.A.; Elgueta, M.; Mansevich, R.F., A homotopic deformation along p of a leray – schauder degree result and existence for \((| u^\prime |^{p - 2} u^\prime)^\prime + f(t, u) = 0\), \(u(0) = u(T) = 0\), \(p > 1\), J. differential equations, 80, 1, 1-13, (1989) · Zbl 0708.34019
[35] Mawhin, J.; Willem, M., Critical point theory and Hamiltonian systems, (1989), Springer-Verlag New York · Zbl 0676.58017
[36] Lindqvist, P., On the equation \(d i v(| \nabla u |^{p - 2} \nabla u) + \lambda | u |^{p - 2} u = 0\), Proc. amer. math. soc., 109, 157-164, (1990) · Zbl 0714.35029
[37] 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.