| Peer-Reviewed

Oscillation Conditions for a Type of Second Order Neutral Differential Equations with Impulses

Received: 20 October 2016     Accepted: 14 November 2016     Published: 24 August 2017
Views:       Downloads:
Abstract

In this paper, we study a certain type of second order linear neutral differential equation with constant impulsive jumps. This type of equation is known always to possess an unbounded non-oscillatory solution. The method and technique of impulse imposition used here is due to studies by Bainov and Simeonov [1]. By assuming, amongst other conditions, that the constant coefficient of the equation in question lies between zero and one and the delay function is non-decreasing, it is shown that all bounded solutions of the said neutral impulsive equation are oscillatory.

Published in American Journal of Applied Mathematics (Volume 5, Issue 4)
DOI 10.11648/j.ajam.20170504.14
Page(s) 119-123
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), 2017. Published by Science Publishing Group

Keywords

Bounded, Oscillations, Second-order, Neutral, Delay, Impulsive, Differential Equation

References
[1] D. D. Bainov and P. S. Simeonov, Oscillation Theory of Impulsive Differential Equations, International Publications Orlando, Florida, 1998.
[2] R. D. Driver, A mixed neutral system, Nonlinear Anal. 8, 1984, pp. 155–158.
[3] G. S. Ladde, V. Lakshmikantham and B. G. Zhang, Oscillation Theory of Differential Equations with Deviating Arguments, Marcel Dekker, New York, 1987.
[4] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
[5] W. T. Li, Interval oscillation criteria of second-order half-linear functional differential equations, Appl. Math. Comput. 155, 2004, pp. 451–468.
[6] X. Lin, Oscillation of second-order nonlinear neutral differential equations, J. Math. Anal. Appl. 309, 2005, pp. 442–452.
[7] F. Meng and J. Wang, Oscillation criteria for second order quasilinear neutral delay differential equations, J. Indones. Math. Soc. (MIHMI) 10, 2004, pp. 61–75.
[8] I. O. Isaac and Z. Lipcsey, Oscillations of Scalar Neutral Impulsive Differential Equations of the First Order with variable Coefficients, Dynamic Systems and Applications, 19, 2010, pp. 45-62.
[9] I. O. Isaac, Z. Lipcsey and U. J. Ibok, Nonoscillatory and Oscillatory Criteria for First Order Nonlinear Neutral Impulsive Differential Equations, Journal of Mathematics Research; Vol. 3 Issue 2, 2011, pp. 52-65.
[10] L. H. Erbe, Q. Kong and B. G. Zhang, Oscillation Theory for Functional Differential Equations, Dekker, New York, 1995.
[11] I. O. Isaac and Z. Lipcsey, Oscillations in Linear Neutral Delay Impulsive Differential Equations with Constant Coefficients, Communications in Applied Analysis, 14, 2010, pp. 123-136.
[12] I. O. Isaac and Z. Lipcsey, Linearized Oscillations in Nonlinear Neutral Delay Impulsive Differential Equations, Journal of Modern Mathematics and Statistics, 3, 2009, pp. 17-21.
[13] I. O. Isaac and Z. Lipcsey, Oscillatory Conditions on Both Directions for a Nonlinear Impulsive Differential Equation with Deviating Arguments, Journal of Mathematics Research, 3, 2011, pp. 48-51.
[14] D. D. Bainov, M. B. Dimitrova and A. B. Dishliev, Oscillationof the Bounded Solutions of Impulsive Differential-Difference Equations of Second Order, Applied Mathematics and Computation, 114, 2000, pp. 61-68.
[15] D. D. Bainov and P. S. Simeonov, The Second Method of Lyapunov for Systems with an Impulse Effect, Tamkang J. Math. 16, 1985, pp. 19-40.
[16] D. D. Bainov and P. S. Simeonov, Stability with respect to Part of the Variables in Systema with Impulsive Effect, Journal of Mathematical Analysis and Applications, 1986, pp. 247-263.
[17] C. Yong-Shao and F. Wei-Zhen, Oscillation of Second Order Nonlinear Ordinary Differential Equations with Impulses, Journal of Mathematical Analysis and Applications, 210, 1997, pp. 150-169.
[18] A. B. Dishliev and D. D. Bainov, Dependence upon Initial Conditions and Parameter of Impulsive Differential Equations with Variable Structure, International Journal of Theoretical Physics, 29 No. 6, 1990, pp. 655-675.
[19] S. I. Gurgula, Investigation of the Stability of Solutions of Impulse Systems by Lyapunov's Second Method, Ukrainian Math., 1, 1982, pp. 400-103.
[20] S. V. Krishna, J. Vasundlara Devi and K. Satyavani, Boundedness and Dichotomies for Impulse Equations, Journal of Mathematical Analysis and Applications, 158, 1991, pp. 352-375.
[21] G. K. Kulev and D. D. Bainov, Lipschitz Stability of Impulsive Systems of Differential Equations, International Journal of Theoretical Physics 30, 1991, pp. 737-756.
[22] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific Publishing Company Ltd, Singapore, 1989.
[23] V. Lakshmikantham and X. Liu, On Quasistability for Impulsive Differential Systems, Nonlinear Analysis, 13, No. 7, 1989, pp. 819-828.
[24] M. Peng and Ge, Oscillation Criteria for Second Order Nonlinear Differential Equations with Impulses, Computers and Mathematics with Applications, 30, 2000, pp. 217-225.
[25] A. M. Samoilenko and N. A. Perestyuk, Stability of the Solutions of Differential Equations with Impulse Effect, Differential Equations, 11, 1977, pp. 1981-1992.
[26] P. P. Zabreiko, D. D. Bainov and S. I. Kostadinov, Characteristic Exponents of Impulsive Differential Equations in a Banach Space, International Journal of Theoretical Physics, 27, 1988, pp. 721-743.
[27] Y. Zhang, A. Zhao and J. Yan, Oscillation Criteria for Impulsive Differential Equations, Journal of Mathematical Analysis and Applications, 205, 1997, pp. 461-470.
[28] I. O. Isaac, Z. Lipcsey and U. J. Ibok, Linearized Oscillations in Autonomous Delay Impulsive Differential Equations, British Journal of Mathematics & Computer Science, 4(21): 2014, pp. 3068-3076.
[29] A. M. Samoilenko and N. A. Perestyuk, Impulsive differential Equations, World Scientific Publishing Company Ltd, Singapore, 1995.
[30] R. P. Agarwal, M. Benchohra, D. O'Regan and A. Ouahab, Second order Impulsive dynamic equations on time scales, Functional differential equations II, No. 3-4, 2004, pp. 223–234.
[31] S. G. Deo and S. G. Pandit, Differential systems involving impulses, Lecture Notes 954, Springer-Verlag, Berlin. differential equations Arch. Mat. 41, 1982, pp. 352-362.
[32] J. Cheng and Y. Chu, Oscillations of Second-Order Neutral Impulsive Differential Equations, Journal of Inequalities and Applications; doi: 10.1155/2010/493927: 1-29.
[33] E. M. Bonotto, L. P. Gimenes and M. Federson, Oscillation for a second-order neutral differential equation with impulses, Cadernos De Matematica; 09: 2008, pp. 169–190.
[34] Y. Sun, Oscillation of Certain Second-Order Sub-Half-Linear Neutral Impulsive Differential Equations, Discrete Dynamics in Nature and Society; doi: 10.1155/2011/195619: 2011, pp. 1-10.
[35] A. Tripathy and S. Santra, Oscillation Properties of a Class of Second Order Impulsive Differential Systems of Neutral Type, Functional Differential Equations; 23(1-2): 2016, pp. 57–71.
[36] D. D. Bainov and M. B. Dimitrova, Oscillation of sub and super linear impulsive differential equations with constant delay, Applicable Analysis; 64, 1977, pp. 57–67.
[37] U. A. Abasiekwere and I. U. Moffat, Oscillation Theorems for Linear Neutral Impulsive Differential Equations of the Second Order with Variable Coefficients and Constant Retarded Arguments, Applied Mathematics, Vol. 7 No. 3, 2017, pp. 39-43. doi: 10.5923/j.am.20170703.01.
[38] U. A. Abasiekwere, E. F. Nsien and I. U. Moffat, On the Existence of Bounded Oscillatory Solutions of Impulsive Delay Differential Equations of the Second Order, International Journal of Mathematics Trends and Technology, 2017, 48(2), pp. 6-10, ISSN: 2231-5373.
Cite This Article
  • APA Style

    Ubon Akpan Abasiekwere, Edwin Frank Nsien, Imoh Udo Moffat. (2017). Oscillation Conditions for a Type of Second Order Neutral Differential Equations with Impulses. American Journal of Applied Mathematics, 5(4), 119-123. https://doi.org/10.11648/j.ajam.20170504.14

    Copy | Download

    ACS Style

    Ubon Akpan Abasiekwere; Edwin Frank Nsien; Imoh Udo Moffat. Oscillation Conditions for a Type of Second Order Neutral Differential Equations with Impulses. Am. J. Appl. Math. 2017, 5(4), 119-123. doi: 10.11648/j.ajam.20170504.14

    Copy | Download

    AMA Style

    Ubon Akpan Abasiekwere, Edwin Frank Nsien, Imoh Udo Moffat. Oscillation Conditions for a Type of Second Order Neutral Differential Equations with Impulses. Am J Appl Math. 2017;5(4):119-123. doi: 10.11648/j.ajam.20170504.14

    Copy | Download

  • @article{10.11648/j.ajam.20170504.14,
      author = {Ubon Akpan Abasiekwere and Edwin Frank Nsien and Imoh Udo Moffat},
      title = {Oscillation Conditions for a Type of Second Order Neutral Differential Equations with Impulses},
      journal = {American Journal of Applied Mathematics},
      volume = {5},
      number = {4},
      pages = {119-123},
      doi = {10.11648/j.ajam.20170504.14},
      url = {https://doi.org/10.11648/j.ajam.20170504.14},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20170504.14},
      abstract = {In this paper, we study a certain type of second order linear neutral differential equation with constant impulsive jumps. This type of equation is known always to possess an unbounded non-oscillatory solution. The method and technique of impulse imposition used here is due to studies by Bainov and Simeonov [1]. By assuming, amongst other conditions, that the constant coefficient of the equation in question lies between zero and one and the delay function is non-decreasing, it is shown that all bounded solutions of the said neutral impulsive equation are oscillatory.},
     year = {2017}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - Oscillation Conditions for a Type of Second Order Neutral Differential Equations with Impulses
    AU  - Ubon Akpan Abasiekwere
    AU  - Edwin Frank Nsien
    AU  - Imoh Udo Moffat
    Y1  - 2017/08/24
    PY  - 2017
    N1  - https://doi.org/10.11648/j.ajam.20170504.14
    DO  - 10.11648/j.ajam.20170504.14
    T2  - American Journal of Applied Mathematics
    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
    SP  - 119
    EP  - 123
    PB  - Science Publishing Group
    SN  - 2330-006X
    UR  - https://doi.org/10.11648/j.ajam.20170504.14
    AB  - In this paper, we study a certain type of second order linear neutral differential equation with constant impulsive jumps. This type of equation is known always to possess an unbounded non-oscillatory solution. The method and technique of impulse imposition used here is due to studies by Bainov and Simeonov [1]. By assuming, amongst other conditions, that the constant coefficient of the equation in question lies between zero and one and the delay function is non-decreasing, it is shown that all bounded solutions of the said neutral impulsive equation are oscillatory.
    VL  - 5
    IS  - 4
    ER  - 

    Copy | Download

Author Information
  • Department of Mathematics and Statistics, Faculty of Science, University of Uyo, Uyo, Nigeria

  • Department of Mathematics and Statistics, Faculty of Science, University of Uyo, Uyo, Nigeria

  • Department of Mathematics and Statistics, Faculty of Science, University of Uyo, Uyo, Nigeria

  • Sections