Hostname: page-component-745bb68f8f-s22k5 Total loading time: 0 Render date: 2025-02-09T17:24:49.676Z Has data issue: false hasContentIssue false
  • English
  • Français

Positive Solutions of Impulsive Dynamic System on Time Scales

Published online by Cambridge University Press:  20 November 2018

Da-Bin Wang
Da-Bin Wang*
Affiliation:
Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, P. R. China e-mail: wangdb@lut.cn
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, some criteria for the existence of positive solutions of a class of systems of impulsive dynamic equations on time scales are obtained by using a fixed point theorem in cones.

Keywords

39A1034B15time scalepositive solutionfixed pointimpulsive dynamic equation
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 1 , 01 March 2012 , pp. 214 - 224
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Agarwal, R. P. and Bohner, M., Basic calculus on time scales and some of its applications. Results Math. 35(1999), no. 1–2, 3–22.Google Scholar
[2] Agarwal, R. P. and O’Regan, D., Multiple nonnegative solutions for second order impulsive differential equations. Appl. Math. Comput. 114(2000), no. 1, 51–59. doi:10.1016/S0096-3003(99)00074-0Google Scholar
[3] Agarwal, R. P., Otero-Espinar, V., Perera, K., and Vivero, D. R., Multiple positive solutions in the sense of distributions of singular BVPs on time scales and an application to Emden-Fowler equations. Adv. Difference Equ. 2008, Art. ID 796851, 1–13.Google Scholar
[4] Agarwal, R. P., Otero-Espinar, V., Perera, K., and Vivero, D. R., Wirtinger's inequalities on time scales. Canad. Math. Bull. 51(2008), no. 2, 161–171. doi:10.4153/CMB-2008-018-6Google Scholar
[5] Akhmet, M. U. and Turan, M., The differential equations on time scales through impulsive differential equations. Nonlinear Anal. 65(2006), no. 11, 2043–2060. doi:10.1016/j.na.2005.12.042Google Scholar
[6] Akhmet, M. U. and Turan, M., Differential equations on variable time scales. Nonlinear Anal. 70(2009), no. 3, 1175–1192. doi:10.1016/j.na.2008.02.020Google Scholar
[7] Bohner, M. and Peterson, A., Dynamic equations on time scales. An introduction with applications. Birkhäuser Boston Inc., Boston, MA, 2001.Google Scholar
[8] Bohner, M. and Peterson, A., Advances in dynamic equations on time scales. Birkhäuser Boston Inc., Boston, MA, 2003.Google Scholar
[9] Benchohra, M., Henderson, J., Ntouyas, S. K., and Ouahab, A., On first order impulsive dynamic equations on time scales. J. Difference Equ. Appl. 10(2004), no. 6, 541–548. doi:10.1080/10236190410001667986Google Scholar
[10] Benchohra, M., Ntouyas, S. K., and Ouahab, A., Existence results for second order boundary value problem of impulsive dynamic equations on time scales. J. Math. Anal. Appl. 296(2004), no. 1, 65–73. doi:10.1016/j.jmaa.2004.02.057Google Scholar
[11] Benchohra, M., Henderson, J., and Ntouyas, S. K., Impulsive differential equations and inclusions. Contemporary Mathematics and Its Applications, 2, Hindawi Publishing Corporation, New York, 2006.Google Scholar
[12] Chen, H. and Wang, H., Triple positive solutions of boundary value problems for p-Laplacian impulsive dynamic equations on time scales. Math. Comput. Modelling 47(2008), no. 9–10, 917–924. doi:10.1016/j.mcm.2007.06.012Google Scholar
[13] Guo, D. J. and Lakshmikantham, V., Nonlinear problems in abstract cones. Notes and Reports in Mathematics in Science and Engineering, 5, Academic Press, Inc., Boston, MA, 1988.Google Scholar
[14] Geng, F., Zhu, D., and Lu, Q., A new existence result for impulsive dynamic equations on timescales. Appl. Math. Lett. 20(2007), no. 2, 206–212. doi:10.1016/j.aml.2006.03.013Google Scholar
[15] Geng, F., Xu, Y., and Zhu, D., Periodic boundary value problems for first-order impulsive dynamic equations on time scales. Nonlinear Anal. 69(2008), no. 11, 4074–4087. doi:10.1016/j.na.2007.10.038Google Scholar
[16] Graef, J. R. and Ouahab, A., Extremal solutions for nonresonance impulsive functional dynamic equations on time scales. Appl. Math. Comput. 196(2008), no. 1, 333–339. doi:10.1016/j.amc.2007.05.056Google Scholar
[17] Hilger, S., Analysis on measure chains—a unified approach to continuous and discrete calculus. Results Math. 18(1990), no. 1–2, 18–56.Google Scholar
[18] He, Z. and Yu, J., Periodic boundary value problem for first-order impulsive functional differential equations. J. Comput. Appl. Math. 138(2002), no. 2, 205–217. doi:10.1016/S0377-0427(01)00381-8Google Scholar
[19] He, Z. and Zhang, X., Monotone iterative technique for first order impulsive differential equations with periodic boundary conditions. Appl. Math. Comput. 156(2004), no. 3, 605–620. doi:10.1016/j.amc.2003.08.013Google Scholar
[20] Jiao, J. J., Chen, L.-S., Nieto, J. J., and Torres, A., Permanence and global attractivity of stage-structured predator-prey model with continuous harvesting on predator and impulsive stocking on prey. Appl. Math. Mech. (English Ed.) 29(2008), no. 5, 653–663. doi:10.1007/s10483-008-0509-xGoogle Scholar
[21] Kaymakcalan, B., Lakshmikantham, V., and Sivasundaram, S., Dynamical systems on measure chains. Mathematics and its Applications, 370, Kluwer Academic Publishers Group, Dordrecht, 1996.Google Scholar
[22] Khan, R. A., Nieto, J. J., and Otero-Espinar, V., Existence and approximation of solution of three-point boundary value problems on time scales. J. Difference Equ. Appl. 14(2008), no. 7, 723–736. doi:10.1080/10236190701840906Google Scholar
[23] Krasnosel’skii, M. A., Positive solutions of operator equations. P. Noordhoff Ltd., Groningen, 1964.Google Scholar
[24] Lakshmikantham, V., Bainov, D. D., and Simeonov, P. S., Theory of impulsive differential equations. Series in Modern Applied Mathematics, 6, World Scientific, Teaneck, NJ, 1989.Google Scholar
[25] Li, J., Nieto, J. J., and Shen, J., Impulsive periodic boundary value problems of first-order differential equations. J. Math. Anal. Appl. 325(2007), no. 1, 226–236. doi:10.1016/j.jmaa.2005.04.005Google Scholar
[26] Li, J.-L. and Shen, J.-H., Existence of positive periodic solutions to a class of functional differential equations with impulses. Math. Appl. 17(2004), no. 3, 456–463.Google Scholar
[27] Li, J.-L. and Shen, J.-H., Existence results for second-order impulsive boundary value problems on time scales. Nonlinear Anal. 70(2009), no. 4, 1648–1655. doi:10.1016/j.na.2008.02.047Google Scholar
[28] Liu, H. and Xiang, X., A class of the first order impulsive dynamic equations on time scales. Nonlinear Anal. 69(2008), no. 9, 2803–2811. doi:10.1016/j.na.2007.08.052Google Scholar
[29] Nieto, J. J., Impulsive resonance periodic problems of first order. Appl. Math. Lett. 15(2002), no. 4, 489–493. doi:10.1016/S0893-9659(01)00163-XGoogle Scholar
[30] Nieto, J. J., Periodic boundary value problems for first-order impulsive ordinary differential equations. Nonlinear Anal. 51(2002), no. 7, 1223–1232. doi:10.1016/S0362-546X(01)00889-6Google Scholar
[31] Nieto, J. J. and O’Regan, D., Variational approach to impulsive differential equations. Nonlinear Anal. Real World Appl. 10(2009), no. 2, 680–690. doi:10.1016/j.nonrwa.2007.10.022Google Scholar
[32] Di Piazza, L. and Satco, B., A new result on impulsive differential equations involving non-absolutely convergent integrals. J. Math. Anal. Appl. 352(2009), no. 2, 954–963. doi:10.1016/j.jmaa.2008.11.048Google Scholar
[33] Zavalishchin, S. T. and Sesekin, A. N., Dynamic impulse systems. Theory and applications. Mathematics and Its Applications, 394, Kluwer Academic Publishers Group, Dordrecht, 1997.Google Scholar
[34] Sun, J.-P. and Li, W.-T., Existence of solutions to nonlinear first-order PBVPs on time scales. Nonlinear Anal. 67(2007), no. 3, 883–888. doi:10.1016/j.na.2006.06.046Google Scholar
[35] Sun, J.-P. and Li, W.-T., Existence and multiplicity of positive solutions to nonlinear first-order PBVPs on time scales. Comput. Math. Appl. 54(2007), no. 6, 861–871. doi:10.1016/j.camwa.2007.03.009Google Scholar
[36] Wang, D.-B., Positive solutions for nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales. Comput. Math. Appl. 56(2008), no. 6, 1496–1504. doi:10.1016/j.camwa.2008.02.038Google Scholar
[37] Zeng, G., Wang, F., and Nieto, J. J., Complexity of a delayed predator-prey model with impulsive harvest and Holling type II functional response. Adv. Complex Syst. 11(2008), no. 1, 77–97. doi:10.1142/S0219525908001519Google Scholar
[38] Zhang, H., Chen, L., and Nieto, J. J., A delayed epidemic model with stage-structure and pulses for pest management strategy. Nonlinear Anal. RealWorld Appl. 9(2008), no. 4, 1714–1726. doi:10.1016/j.nonrwa.2007.05.004Google Scholar
[39] Zhang, N., Dai, B., and Qian, X., Periodic solutions for a class of higher-dimension functional differential equations with impulses. Nonlinear Anal. 68(2008), no. 3, 629–638. doi:10.1016/j.na.2006.11.024Google Scholar