×

On the existence of solutions for impulsive Duffing dynamic equations on time scales with Dirichlet boundary conditions. (English) Zbl 1223.34126

The authors study the following impulsive Duffing dynamic equations on time scales \[ u^{\Delta \Delta}(t)+Cu^{\Delta}(\sigma(t))-r(t)u(\sigma(t))+f(\sigma(t),u(\sigma(t)))=h(t), \text{ a.e. } t \in [0,\sigma(T)]^{\kappa^2}_{\mathbb{T}}, \]
\[ \Delta u^{\Delta}(t_j)=u^{\Delta}(t_j^+)-u^{\Delta}(t_j^-)= I_{j}(u(t_j)),\quad j=1,\dots,p, \]
\[ u(0)=0=u(\sigma(T)) \]
with \(T>0\), \(C\) a regressive constant, \(\{t_j\}_{j=1}^{p}\) a finite increasing sequence in \((0,\sigma(T))\), \(t_0=0\), \(t_{p+1}=\sigma(T)\), \(r \in L^{\infty}[0,\sigma(T)]_{\mathbb{T}}\), \(h \in L^{2}[0,\sigma(T)]_{\mathbb{T}}\), \(f\) continuous and \(I_j\) continuous, for \(j=1,\dots,p\).
They establish the variational framework for the problem and, hence, sufficient conditions for the existence of weak solutions by using critical point theorems, providing some examples of application.

MSC:

34N05 Dynamic equations on time scales or measure chains
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
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] G. Duffing, Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz und Ihre Technische Beduetung, Sammlung Vieweg, Braunschweig, Germany, 1918. · JFM 46.1168.01
[2] E. C. Zeeman, “Duffing’s equation in brain modelling,” Journal of the Institute of Mathematics and Its Applications, vol. 12, no. 7, pp. 207-214, 1976.
[3] F.-G. Xie, W.-M. Zheng, and B.-L. Hao, “Symbolic dynamics of the two-well Duffing equation,” Communications in Theoretical Physics, vol. 24, no. 1, pp. 43-52, 1995.
[4] D. Hao and S. Ma, “Semilinear Duffing equations crossing resonance points,” Journal of Differential Equations, vol. 133, no. 1, pp. 98-116, 1997. · Zbl 0877.34036 · doi:10.1006/jdeq.1996.3193
[5] S. Ma, Z. Wang, and J. Yu, “Coincidence degree and periodic solutions of Duffing equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 34, no. 3, pp. 443-460, 1998. · Zbl 0931.34048 · doi:10.1016/S0362-546X(97)00664-0
[6] B. Liu, “Boundedness of solutions for semilinear Duffing equations,” Journal of Differential Equations, vol. 145, no. 1, pp. 119-144, 1998. · Zbl 0913.34032 · doi:10.1006/jdeq.1997.3406
[7] B. Liu, “Boundedness in nonlinear oscillations at resonance,” Journal of Differential Equations, vol. 153, no. 1, pp. 142-174, 1999. · Zbl 0926.34028 · doi:10.1006/jdeq.1998.3553
[8] H. Khammari, C. Mira, and J.-P. Carcassés, “Behavior of harmonics generated by a Duffing type equation with a nonlinear damping, part I,” International Journal of Bifurcation and Chaos, vol. 15, no. 10, pp. 3181-3221, 2005. · Zbl 1093.70506 · doi:10.1142/S0218127405014076
[9] A. Elías-Zúñiga, “A general solution of the Duffing equation,” Nonlinear Dynamics, vol. 45, no. 3-4, pp. 227-235, 2006. · Zbl 1121.70016 · doi:10.1007/s11071-006-1858-z
[10] S. W. Ma and J. H. Wu, “A small twist theorem and boundedness of solutions for semilinear Duffing equations at resonance,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 1, pp. 200-237, 2007. · Zbl 1187.34043 · doi:10.1016/j.na.2006.04.023
[11] H. Chen and Y. Li, “Rate of decay of stable periodic solutions of Duffing equations,” Journal of Differential Equations, vol. 236, no. 2, pp. 493-503, 2007. · Zbl 1169.34027 · doi:10.1016/j.jde.2007.01.023
[12] A. Caneco, C. Grácio, and J. L. Rocha, “Kneading theory analysis of the Duffing equation,” Chaos, Solitons and Fractals, vol. 42, no. 3, pp. 1529-1538, 2009. · Zbl 1198.37070 · doi:10.1016/j.chaos.2009.03.040
[13] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, Mass, USA, 2001. · Zbl 0978.39001
[14] V. Spedding, Taming Nature’s Numbers, New Scientist, 2003.
[15] M. A. Jones, B. Song, and D. M. Thomas, “Controlling wound healing through debridement,” Mathematical and Computer Modelling, vol. 40, no. 9-10, pp. 1057-1064, 2004. · Zbl 1061.92036 · doi:10.1016/j.mcm.2003.09.041
[16] R. P. Agarwal, M. Bohner, and W.-T. Li, Nonoscillation and Oscillation: Theory for Functional Differential Equations, vol. 267 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2004. · Zbl 1068.34002
[17] M. Benchohra, J. Henderson, and S. Ntouyas, Impulsive Differential Equations and Inclusions, vol. 2 of Contemporary Mathematics and Its Applications, Hindawi Publishing Corporation, New York, NY, USA, 2006. · Zbl 1130.34003 · doi:10.1155/9789775945501
[18] W. M. Haddad, V. Chellaboina, and S. G. Nersesov, Impulsive and Hybrid Dynamical Systems, Stability, Dissipativity, and Control, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, USA, 2006. · Zbl 1114.34001
[19] J. J. Nieto and R. Rodríguez-López, “Boundary value problems for a class of impulsive functional equations,” Computers & Mathematics with Applications, vol. 55, no. 12, pp. 2715-2731, 2008. · Zbl 1142.34362 · doi:10.1016/j.camwa.2007.10.019
[20] J. Chu and J. J. Nieto, “Impulsive periodic solutions of first-order singular differential equations,” Bulletin of the London Mathematical Society, vol. 40, no. 1, pp. 143-150, 2008. · Zbl 1144.34016 · doi:10.1112/blms/bdm110
[21] J. Zhou and Y. Li, “Existence of solutions for a class of second-order Hamiltonian systems with impulsive effects,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1594-1603, 2010. · Zbl 1193.34057 · doi:10.1016/j.na.2009.08.041
[22] Z. Zhang and R. Yuan, “An application of variational methods to Dirichlet boundary value problem with impulses,” Nonlinear Analysis: Real World Applications, vol. 11, no. 1, pp. 155-162, 2010. · Zbl 1191.34039 · doi:10.1016/j.nonrwa.2008.10.044
[23] C. Li, Y. Li, and Y. Ye, “Exponential stability of fuzzy Cohen-Grossberg neural networks with time delays and impulsive effects,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 11, pp. 3599-3606, 2010. · Zbl 1222.34090 · doi:10.1016/j.cnsns.2010.01.001
[24] Y. Li and T. Zhang, “Existence and uniqueness of anti-periodic solution for a kind of forced Rayleigh equation with state dependent delay and impulses,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 12, pp. 4076-4083, 2010. · Zbl 1222.34096 · doi:10.1016/j.cnsns.2010.01.023
[25] H. Zhang, W. Xu, and L. Chen, “A impulsive infective transmission SI model for pest control,” Mathematical Methods in the Applied Sciences, vol. 30, no. 10, pp. 1169-1184, 2007. · Zbl 1155.34328 · doi:10.1002/mma.834
[26] G. Zeng, F. Wang, and J. J. Nieto, “Complexity of a delayed predator-prey model with impulsive harvest and Holling type II functional response,” Advances in Complex Systems, vol. 11, no. 1, pp. 77-97, 2008. · Zbl 1168.34052 · doi:10.1142/S0219525908001519
[27] K. Liu and G. Yang, “Cone-valued-Lyapunov functions and stability for impulsive functional differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 7, pp. 2184-2191, 2008. · Zbl 1151.34063 · doi:10.1016/j.na.2007.07.057
[28] B. Ahmad and J. J. Nieto, “Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 10, pp. 3291-3298, 2008. · Zbl 1158.34049 · doi:10.1016/j.na.2007.09.018
[29] Y. Li, “Positive periodic solutions of nonlinear differential systems with impulses,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 8, pp. 2389-2405, 2008. · Zbl 1162.34064 · doi:10.1016/j.na.2007.01.066
[30] H. Zhang, L. Chen, and J. J. Nieto, “A delayed epidemic model with stage-structure and pulses for pest management strategy,” Nonlinear Analysis: Real World Applications, vol. 9, no. 4, pp. 1714-1726, 2008. · Zbl 1154.34394 · doi:10.1016/j.nonrwa.2007.05.004
[31] Z. Luo and J. J. Nieto, “New results for the periodic boundary value problem for impulsive integro-differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 6, pp. 2248-2260, 2009. · Zbl 1166.45002 · doi:10.1016/j.na.2008.03.004
[32] Y. Li, X. Chen, and L. Zhao, “Stability and existence of periodic solutions to delayed Cohen-Grossberg BAM neural networks with impulses on time scales,” Neurocomputing, vol. 72, no. 7-9, pp. 1621-1630, 2009. · Zbl 05718962 · doi:10.1016/j.neucom.2008.08.010
[33] L. Jiang and Z. Zhou, “Existence of weak solutions of two-point boundary value problems for second-order dynamic equations on time scales,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 4, pp. 1376-1388, 2008. · Zbl 1189.34170 · doi:10.1016/j.na.2007.06.034
[34] J. Zhou and Y. Li, “Existence and multiplicity of solutions for some Dirichlet problems with impulsive effects,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 2856-2865, 2009. · Zbl 1175.34035 · doi:10.1016/j.na.2009.01.140
[35] M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2003. · Zbl 1025.34001
[36] B. P. Rynne, “L2 spaces and boundary value problems on time-scales,” Journal of Mathematical Analysis and Applications, vol. 328, no. 2, pp. 1217-1236, 2007. · Zbl 1116.34021 · doi:10.1016/j.jmaa.2006.06.008
[37] R. Agarwal, M. Bohner, and A. Peterson, “Inequalities on time scales: a survey,” Mathematical Inequalities & Applications, vol. 4, no. 4, pp. 535-557, 2001. · Zbl 1021.34005
[38] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74 of Applied Mathematical Sciences, Springer, Boston, Mass, USA, 1989. · Zbl 0676.58017
[39] D. J. Guo, Nonlinear Functional Analysis, Shandong Science and Technology Press, Jinan, China, 2001.
[40] P. H. Rabinowitz, “Minimax methods in critical point theory with applications to differential equations,” in Proceedings of the CBMS Regional Conference in the Mathematical Sciences, vol. 65, American Mathematical Society, Providence, RI, USA, 1986. · Zbl 0609.58002
[41] R. P. Agarwal, M. Bohner, and P. J. Y. Wong, “Sturm-Liouville eigenvalue problems on time scales,” Applied Mathematics and Computation, vol. 99, no. 2-3, pp. 153-166, 1999. · Zbl 0938.34015 · doi:10.1016/S0096-3003(98)00004-6
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.