Formulation of Impulsive Differential Equations with Time-Dependent Continuous Delay

Ita Micah Esuabana; Ubon Akpan Abasiekwere

doi:doi:10.11648/j.ajam.20180604.11

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Formulation of Impulsive Differential Equations with Time-Dependent Continuous Delay

Ita Micah Esuabana, Ubon Akpan Abasiekwere

Published in American Journal of Applied Mathematics (Volume 6, Issue 4)

Received: 10 September 2018 Accepted: 9 October 2018 Published: 31 October 2018

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Abstract

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.

Published in	American Journal of Applied Mathematics (Volume 6, Issue 4)
DOI	10.11648/j.ajam.20180604.11
Page(s)	135-141
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2018. Published by Science Publishing Group

Keywords

Impulsive, Differential Equation, Continuous Delay, Integral Equivalence

References

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[8]	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.
[9]	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.
[10]	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.
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[12]	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.
[13]	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.
[14]	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.
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APA Style

Ita Micah Esuabana, Ubon Akpan Abasiekwere. (2018). Formulation of Impulsive Differential Equations with Time-Dependent Continuous Delay. American Journal of Applied Mathematics, 6(4), 135-141. https://doi.org/10.11648/j.ajam.20180604.11

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ACS Style

Ita Micah Esuabana; Ubon Akpan Abasiekwere. Formulation of Impulsive Differential Equations with Time-Dependent Continuous Delay. Am. J. Appl. Math. 2018, 6(4), 135-141. doi: 10.11648/j.ajam.20180604.11

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AMA Style

Ita Micah Esuabana, Ubon Akpan Abasiekwere. Formulation of Impulsive Differential Equations with Time-Dependent Continuous Delay. Am J Appl Math. 2018;6(4):135-141. doi: 10.11648/j.ajam.20180604.11

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@article{10.11648/j.ajam.20180604.11,
  author = {Ita Micah Esuabana and Ubon Akpan Abasiekwere},
  title = {Formulation of Impulsive Differential Equations with Time-Dependent Continuous Delay},
  journal = {American Journal of Applied Mathematics},
  volume = {6},
  number = {4},
  pages = {135-141},
  doi = {10.11648/j.ajam.20180604.11},
  url = {https://doi.org/10.11648/j.ajam.20180604.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20180604.11},
  abstract = {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.},
 year = {2018}
}

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TY  - JOUR
T1  - Formulation of Impulsive Differential Equations with Time-Dependent Continuous Delay
AU  - Ita Micah Esuabana
AU  - Ubon Akpan Abasiekwere
Y1  - 2018/10/31
PY  - 2018
N1  - https://doi.org/10.11648/j.ajam.20180604.11
DO  - 10.11648/j.ajam.20180604.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 135
EP  - 141
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20180604.11
AB  - 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.
VL  - 6
IS  - 4
ER  -

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Author Information

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

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Plain Text BibTeX RIS

APA Style

Ita Micah Esuabana, Ubon Akpan Abasiekwere. (2018). Formulation of Impulsive Differential Equations with Time-Dependent Continuous Delay. American Journal of Applied Mathematics, 6(4), 135-141. https://doi.org/10.11648/j.ajam.20180604.11

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ACS Style

Ita Micah Esuabana; Ubon Akpan Abasiekwere. Formulation of Impulsive Differential Equations with Time-Dependent Continuous Delay. Am. J. Appl. Math. 2018, 6(4), 135-141. doi: 10.11648/j.ajam.20180604.11

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AMA Style

Ita Micah Esuabana, Ubon Akpan Abasiekwere. Formulation of Impulsive Differential Equations with Time-Dependent Continuous Delay. Am J Appl Math. 2018;6(4):135-141. doi: 10.11648/j.ajam.20180604.11

Copy | Download

@article{10.11648/j.ajam.20180604.11,
  author = {Ita Micah Esuabana and Ubon Akpan Abasiekwere},
  title = {Formulation of Impulsive Differential Equations with Time-Dependent Continuous Delay},
  journal = {American Journal of Applied Mathematics},
  volume = {6},
  number = {4},
  pages = {135-141},
  doi = {10.11648/j.ajam.20180604.11},
  url = {https://doi.org/10.11648/j.ajam.20180604.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20180604.11},
  abstract = {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.},
 year = {2018}
}

Copy | Download

TY  - JOUR
T1  - Formulation of Impulsive Differential Equations with Time-Dependent Continuous Delay
AU  - Ita Micah Esuabana
AU  - Ubon Akpan Abasiekwere
Y1  - 2018/10/31
PY  - 2018
N1  - https://doi.org/10.11648/j.ajam.20180604.11
DO  - 10.11648/j.ajam.20180604.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 135
EP  - 141
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20180604.11
AB  - 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.
VL  - 6
IS  - 4
ER  -

Copy | Download