Volume 6, Issue 4, August 2018, Page: 135-141
Formulation of Impulsive Differential Equations with Time-Dependent Continuous Delay
Ita Micah Esuabana, Department of Mathematics, Faculty of Science, University of Calabar, Calabar, Nigeria
Ubon Akpan Abasiekwere, Department of Mathematics and Statistics, Faculty of Science, University of Uyo, Uyo, Nigeria
Received: Sep. 10, 2018;       Accepted: Oct. 9, 2018;       Published: Oct. 31, 2018
DOI: 10.11648/j.ajam.20180604.11      View  169      Downloads  15
Research in impulsive delay differential equations has been undergoing some exciting growth in recent times. This to a large extent can be attributed to the quest by mathematicians in particular and the science community as a whole to unveil nature the way it truly is. The realization that differential equations, in general, and indeed impulsive delay differential equations are very important models for describing the true state of several real-life processes/phenomena may have been the tunic. One can attest that in most human processes or natural phenomena, the present state is most often affected significantly by their past state and those that were thought of as continuous may indeed undergo abrupt change at several points or even be stochastic. In this study, a special strictly ascending continuous delay is constructed for a class of system of impulsive differential equations. It is demonstrated that even though the dynamics of the system and the delay have ideal continuity properties, the right side may not even have limits at some points due to the impact of past impulses in the present. The integral equivalence of the formulated system of equations is also obtained via a scheme similar to that of Perron by making use of certain assumptions.
Impulsive, Differential Equation, Continuous Delay, Integral Equivalence
To cite this article
Ita Micah Esuabana, Ubon Akpan Abasiekwere, Formulation of Impulsive Differential Equations with Time-Dependent Continuous Delay, American Journal of Applied Mathematics. Vol. 6, No. 4, 2018, pp. 135-141. doi: 10.11648/j.ajam.20180604.11
Copyright © 2018 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Bereketoglu, H. and Karakoc, F. (2008). Asymptotic, constancy for impulsive delay differential equations. Dynamic Systems and Applications, 17, pp. 71–84.
Dubeau, F. and Karrakchou, J. (2002). State–dependent impulsive delay differential equations, Applied Mathematics Letters, 15, pp. 333–338.
Ballinger, H. G. (1999). Qualitative Theory of impulsive Delay Differential equations, Unpublished PhD Thesis, University of Waterloo, Canada.
Liu, X. and Ballinger, G. (2002). Existence and continuability of solutions for differential equations with delays and state–dependent impulses, Nonlinear Analysis TMA, 51, pp. 633–647.
Oyelami, B., Ale, S. O., Ogidi, J. A., and Onumanyi, P. (2003), Impulsive hiv – 1 model in the presence of antiretroval drugs using b transform method, Proceedings of African Mathematical Union, 1, pp. 62–76.
Yan, J. (2004), Oscillation properties of a second order impulsive delay differential equation, Comp. and Math. with Applications, 47, pp. 253–258.
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.
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.
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.
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.
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.
J. A. Ugboh, I. M. Esuabana, Existence and Uniqueness Result for a Class of Impulsive Delay Differential Equations, International Journal of Chemistry, Mathematics and Physics, Vol. 2 (4), 27-32, 2018, doi:10.22161/ijcmp.2.4.1.
Abasiekwere, U. A., Esuabana, I. M., Asymptotic behaviour of nonoscillating solutions of neutral delay differential equations of the second order with impulses, Journal of Mathematical and Computational Science, Vol 8, No 5 (2018), 620-629, doi:10.28919/jmcs/3765.
Esuabana, I. M., Abasiekwere, U. A., Ugboh, J. A., Lipcsey, Z. Equivalent Construction of Ordinary Differential Equations from Impulsive System, Academic Journal of Applied Mathematical Sciences, Vol. 4, No. 8, 2018, 77-89.
Bainov, D. D. and Simeonov, P. S. (1995), Impulsive Differential Equations – Asymptotic Properties of the Solutions, World Scientific Pub. Coy. Pte. Ltd, Singapore.
Lakshmikantham, V., Bainov, D. D. and Simeonov, P. S. (1989), Theory of Impulsive Differential Equations, World Scientific Publishing Company Limited, Singapore.
Oyelami, B. O. (1999), On military model for impulsive reinforcement functions using exclusion and marginalization techniques, Nonlinear Analysis, Vol 35, pp. 947–958.
Samaoilenko, A. M. and Perestyuk, J. J. J. (1995), Impulsive Differential Equations, World Scientific, New York.
Bainov, D. D. and Simeonov, P. S. (1989), Systems with Impulse Effects, Stability, Theory and Applications, Ellis Horwood Series in Mathematics and its Applications.
Bainov, D. D. and Simeonov, P. S. (1993), Impulsive Differential Equations – Periodic Solutions and Applications, Longman Scientific and Technical.
Simeonov, P. S. and Bainov, D. D. (1988). Differentiability of solutions of systems with impulse effects with respect to initial data and parameters, Proceedings of Edinburgh Maths Society, 31, pp. 353–368.
Arino, O. and Pituk, M. (2001), More on linear differential systems with small delays, Journal of Differential Equations, Vol 17, pp. 381–407.
Bastinec, J., Diblik, J. and Smarda, Z. (2000), Convergence tests for one scalar differential equation with vanishing delay, Arch. Math. (Brno), 36, pp. 405–414. CDDE Issue.
Oyelami, O. B. and Ale, S. O. (2002). B – transform and its application to fish – hyacinth model, International Journal of Mathematics, Education, Science and Technology, 33 (4), pp. 565–573.
Bainov, D. D. and Stamova, I. M. (1999), Existence, uniqueness and continuability of solutions of differential-difference equations, Journal of Applied Mathematics and Stochastic Analysis, 12: pp. 293–300.
Gopalsamy, K. (1992), Stability and Oscillation in Delay Differential Equations of Population Dynamics, Kluwer Academics, Dordrecht.
Coddington, A. E. and Levinson, N. (1955), Theory of Ordinary Differential Equations, McGraw–Hill Book Company, New York.
Browse journals by subject