Survey of Impulsive Differential Equations with
Continuous Delay

I. M. Esuabana; J. A. Ugboh

doi:https://doi.org/10.14445/22315373/IJMTT-V60P504

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 60 | Number 1 | Year 2018 | Article Id. IJMTT-V60P504 | DOI : https://doi.org/10.14445/22315373/IJMTT-V60P504

Survey of Impulsive Differential Equations with Continuous Delay

I. M. Esuabana, J. A. Ugboh

Abstract

In this study, we investigate first order impulsive differential equations with continuous time dependent delays and revisit some of the fundamental concepts in literature. Consequently, we analyse the change in the formulation of impulsive differential equation problems from studying piecewise continuously differentiable trajectories to working with piecewise absolute continuous trajectories and determine a suitable solution space for them.

Keywords

Impulsive, differential equation, continuous delay, trajectories, impulse points

References

[1] Bereketoglu, H. and Karakoc, F., Asymptotic constancy for impulsive delay differential equations,Dynamic Systems and Applications, 17, pp. 71 – 84, 2008.
[2] Oyelami, B. and Ale, S., On existence of solution, oscillation and non-oscillation properties of delay equation containing maximum,Acta Applicande Mathematical Journal, Springer Publication,Vol. 109, No. 3, http://dx.doi.org/10.1007/s10440-008-9340-1, p 683-701, 2010.
[3] Ale, S. O. and Oyelami, B., B–stability and its applications to someconstantdelayimpulsivecontrolmodels. InNMC–COMSATSProceedings on International Confrence on Mathematical Modeling of some Global Challenges in the 21st Century, pages pp56–65, 2009,NMC-COMSATS.
[4] Ballinger,H.G., Qualitative Theory of impulsive Delay Differential equations, Unpublished PhD Thesis, 1999, University of Waterloo, Canada.
[5] Isaac, I. O., Oscilliation Theory of Neutral Impulsive Differential Equations. PhD thesis, Postgraduate School, 2008, University ofCalabar, Calabar, Nigeria.
[6] Liu, X. and Ballinger, G.,Existence and continuability of solutionsfor differentialequationswithdelaysandstate–dependentimpulses,Nonlinear Analysis TMA, 51: pp. 633 – 647, 2002.
[7] Oyelami, B., Ale, S. O., Ogidi, J. A., and Onumanyi, P.,Impulsive hiv – 1 model in the presence of antiretroval drugs usingb transform method. Proceedings of African Mathematical Union,1: pp. 62 – 76, 2003.
[8] Oyelami,B.O.,Onmilitarymodelforimpulsivereinforcement functions using exclusion and marginalization techniques. Nonlinear Analysis, Vol 35:pp. 947 – 958, 1999.
[9] Yan, J.,Oscillation properties of a second order impulsive delay differential equation. Comp. and Math. with Applications, 47:pp. 253 – 258, 2004.
[10] Bainov, D. D. and Simeonov, P. S.,Impulsive DifferentialEquations – Asymptotic Properties of the Solutions, World Scientific Pub. Coy. Pte. Ltd, Singapore, 1995.
[11] Ladde, G. S., Lakshmikantham, V., and Zhang, B. G.,Oscillation Theory of Differential Equations with Deviating Arguments.Maarcel-Dekker Inc., New York, 1989.
[12] Michiels, W. and Niculescu, S., Stability and Stabilization of time delay systems. An eigen value approach. Wikipedia, 2007.
[13] Du, B. and Zhang, X., Delay dependent stability analysis and synthesis for uncertain impulsive switched system with mixeddelays. Hindawi Publishing Corporation – Discrete Dynamic in Nature and Society, 2011:pp. 1 – 9, 2011.
[14] Isaac, I. O., Lipcsey, Z., and Ibok, U. J., Non–oscillatory and oscillatory creteria for a first order nonlinear impulsive differential equations. Journal of Mathematics Research, Canada, 3(2): pp. 52– 65, 2011a.
[15] Isaac, I. O., Lipcsey, Z., and Ibok, U. J., Oscillatory conditions on both directions for a nonlinear impulsive di?erential equation with deviating arguments. Journal of Mathematics Research, Canada, 3(3): pp. 49 –51, 2011b.
[16] Esuabana, I. M. and Abasiekwere, U. A., On stability of first order linear impulsive differential equations,International Journal of Statistics and Applied, Mathematics,Volume 3, Issue 3C, 231-236, 2018.
[17] Abasiekwere, U. A., Esuabana, I. M., Isaac, I. O., Lipcsey, Z, Classification of Non-Oscillatory Solutions of Nonlinear Neutral Delay Impulsive Differential Equations, Global Journal of Science Frontier Research: Mathematics and Decision Sciences (USA), Volume 18, Issue 1, 49-63, 2018,doi:10.17406/GJSFR.
[18] Abasiekwere, U. A., Esuabana, I. M., Oscillation Theorem for Second Order Neutral Delay Differential Equations with Impulses, International Journal of Mathematics Trends and Technology, Vol. 52(5), 330-333, 2017, doi:10.14445/22315373/IJMTT-V52P548.
[19] Abasiekwere, U. A., Esuabana, I. M., Isaac, I. O., Lipcsey, Z., Existence Theorem For Linear Neutral Impulsive Differential Equations of the Second Order, Communications and Applied Analysis, USA, Vol. 22, No. 2, 135-147, 2018,doi:10.12732/caa.v22i2.1.
[20] Abasiekwere, U. A., Esuabana, I. M., Isaac, I. O., Lipcsey, Z., Oscillations of Second order Impulsive Differential Equations with Advanced Arguments, Global Journal of Science Frontier Research: Mathematics and Decision Sciences (USA), Volume 18, Issue 1, 25-32, 2018, doi:10.17406/GJSFR.
[21] Bodine, S. and Lutz, D. A., Integration under weak dichotomies,Rocky Mountain Journal of Mathematics, 40(1):pp. 1 –51, 2010.
[22] Roghothama, A.and Narayanan, S., Non–linear dynamics of a two dimensional air–foil by incremental harmonic balance method,Journal of Sound and Vibrations, 226(3):pp. 493 – 517, 1999.
[23] Gopalsamy, K.,Stability and Oscillation in Delay Differential Equations of Population Dynamics, Kluwer Academics, Dordrecht, 1999.
[24] Rodney, D. D.,Introduction to Ordinary Differential Equations, Harper and Row Publishers, Inc., 1978,New York.
[25] Thompson, S., Delay differential equations. Scholarpedia, 2(3):2367, 2007
[26] Roussel, M. R., Delay – differential equations. Lecture Notes, 2005
[27] Brauer, F. and Wang, J. S., On the asymptotic relationships between solutions of two systems of ordinary differential equations. Journal of Differential Equations, 9: pp. 527 – 543, 1969.
[28] Yu, J. and Zhang, B. G., Stability theorem for delay differential equations with impulses,J. Math. Anal. Appl., 199: pp. 162– 175, 1996.
[29] Ballinger, G. and Liu, X., Existence and uniqueness results for impulsive delay differential equations, Dyn. Contin. Discrete Impuls. Syst., 5, pp. 579-591, 1999.
[30] Holmes, P. and Shea-Brown, E. T., Stability and asympotic stability. Scholarpedia, 1(10):1838, 2006.
[31] Pugh, C. and Peixoto, M. M., Structural stability. Scholarpedia, 3(9):4008, 2008.
[32] Quckenheimeir, J., Structural failure: Burification. Scholarpedia, 2(6):1517, 2007.
[33] Rajput, R. K., Fluid Mechanics. S.Chand and Company Ltd, New Delhi, 2009.
[34] Benchohra, M., Henderson, J. and Ntouyas, S. K., Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, New York, 2006

Citation :

I. M. Esuabana, J. A. Ugboh, "Survey of Impulsive Differential Equations with Continuous Delay," International Journal of Mathematics Trends and Technology (IJMTT), vol. 60, no. 1, pp. 22-28, 2018. Crossref, https://doi.org/10.14445/22315373/IJMTT-V60P504