Applied and Computational Mathematics

| Peer-Reviewed |

Oscillations of Solutions of Neutral Nonlinear Differential Equations

Received: 18 July 2018    Accepted:     Published: 19 July 2018
Views:       Downloads:

Share This Article

Abstract

This paper aims to establish a new class of differential equations and study the oscillatory behavior of a kind of first-order neutral nonlinear differential equation with time delay arguments. The oscillatory properties of the solutions of the type of first order neutral functional differential equations applied in chemomedical problems are studied. Sufficient conditions for the oscillations of solutions of the above equations are obtained. Also, some results which demonstrate in literature [1-4] will be extended, and the paper focuses on expanding the main finding of literature [2, 3]. Moreover, a new kind of method to be used to discuss the properties of oscillation of the first-order neutral nonlinear differential equations and some theorems are obtained in the paper.

DOI 10.11648/j.acm.20180703.16
Published in Applied and Computational Mathematics (Volume 7, Issue 3, June 2018)
Page(s) 112-120
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), 2024. Published by Science Publishing Group

Keywords

Oscillation, Differential Equations, Neutral, Piecewise Constant Arguments

References
[1] G. Ladas and I. P. Stavroulakis, On Delay Differential Inequalities of First Order, Funkcialaj Ekvacioj, 25 (1982), 105-113.
[2] L. C. Lin and G. Q. Wang, A Class Of Nonlinear Differential Inequalities With Time Delay Arguments, Annals of Differential Equations, Vol. 7 No. 1 (1991), 58-67.
[3] Weng Peixuan, Oscillations Of Solutions Of First-order Neutral Differential Difference Equations, Annals of Differential Equations, Vol. 7 No. 1 (1991), 103-116.
[4] Ruan Jiang, Oscillations of Neutral Differential Difference Equations With Several Retarded Arguments, Scientia Sinica (Series A) 5 (1986), 467-477.
[5] Busenberg S, Cooke K L. Models of vertically transmitted diseases with sequential continuous dynamics [A]. In: Lakshmikantham V, ed. Nonlinear Phenomena in Mathematical Sciences [M]. New York: Academic Press, 1982. 179-187.
[6] Tomaras A. Oscillatory behavior of an equation arising from an industrial problem Bull. Austral. Math. Soc. 13 (1975), 255-260.
[7] G. Ladas, Sharp conditions for oscillation caused by delays, Applicable Analysis, 9 (1979), 93-98.
[8] K. L. Cooke and J. Wiener, Retarded differential equations with piecewise constant delays, J. Math. Anal. Appl. 99 (1984), 265-297.
[9] A. R. Aftabizadeh and J. Wiener, oscillatory properties of first order linear functional differential equation, Applicable Anal. 20 (1985), 165-187.
[10] G. S. Ladde and B. G. Zhang, oscillation and non-oscillation for systems of two first order linear differential equations with delay, J. Math. Anal. Appl. 115 (1986), 57-75.
[11] K. L. Cooke and J. Wiener, An equation alternately of retarded and advanced type, Proc. Amer. Math. Soc. 99 (1987), 726-732.
[12] J. Wiener and A. R. Aftabizadeh, Differential equations alternately of retarded and advanced type, J. Math. Anal. Appl. 129 (1988), 243-255.
[13] Joseph Wiener, oscillations in systems of differential equations with piecewise constant argument, J. Math. Anal. Appl. 137 (1989), 221-239.
[14] B. G. Zhang and N. Pathi. oscillatory and non-oscillatory properties of first order differential equations with piecewise constant deviating arguments, J. Math. Anal. Appl. 139 (1989), 23-25.
[15] Y. Kitamure and T. Kusano, Oscillation of first order nonlinear differential equations with deviating argument, Proc. Amer. Math. Soc. 78 (1990), 64-68.
[16] K. Gopalsamy, M. R. S. Kulenovic and G. Ladas, On a logistic equation with piecewise constant arguments, Differential and Integral equations, 4 (1991), 215-223.
[17] Y. G. Sficas and I. P. Starvoulakis, Necessary and sufficient conditions of neutral differential equations, J. Math. Anal. Appl. 123 (1987), 494-507.
[18] B. G. Zhang, oscillation of first order neutral functional differential equations, J. Math. Anal. Appl. 2 (1989), 311-318.
[19] R. Yuan, Almost period solutions of neutral differential equations with piecewise constant delays, Chinese Ann. Math, 1998, 19A (4):499-506.
[20] Muminov, I. M., On the method of finding periodic solutions of second-order neutral differential equations with piecewise constant arguments, Adv. Differ. Equ. (2017) 2017:336.
[21] Shyam S. S., Tripathy, A. K., On oscillatory first nonlinear neutral differential equations with nonlinear impulses, J. Appl. Math. Comput. 2018.
[22] Tripathy, A. K., Shyam S. S., Pulsatile constant and characterization of first order neutral impulsive differential equations, Commun. Appl. Anal. 20 (2016), 65-76.
[23] Candan, T.: Existence of non-oscillatory solutions to first-order neutral differential equations. Elect. J. Diff. Equ. 2016 (39), pp1-11.
[24] Candan, T.: Existence of positive solutions higher-order nonlinear neutral equations. J. Ineq. Appl. 2013, 573 (2013).
[25] G. Infante, R. Koplatadze, and I. P. Stavroulakis, Oscillation criteria for differential equations with several retarded arguments, Funkcial Ekvac., 58, No. 3, pp347-364 (2015).
[26] E. Braverman, G. E. Chatzarakis, and I. P. Stavroulakis, Iterative oscillation tests for differential equations with several non-monotone arguments, Adv. Differ. Equ. 2016, 87, pp1-18. (2016).
[27] K. Chudinov, Note on oscillation conditions for first-order delay differential equations, Electron. J. Qual. Theory Differ. Eqs. 2, pp1-10 (2016).
[28] Diblik, J., Positive solutions of nonlinear delayed differential equations with impulses, Appl. Math. Lett. 72, pp16-22 (2017).
[29] Tripathy, A. K., Santra S. S., Pinelas, S, Necessary and sufficient conditions of for asymptotic behavior of solutions of a class of first order impulsive systems, Adv. Dyn. Syst. Appl. 11 (2), pp135-145 (2016).
[30] Tripathy, A. K., Santra S. S., Necessary and sufficient conditions of for oscillations of solutions of a class of first order impulsive differential equations, Funct. Differ. Equ. 22 (3-4), pp149-167 (2015).
[31] K. M. Chudinov, Exact conditions of oscillation of solutions to differential equations with several Delays, J. Math. Sci. Vol. 230, pp790-793 (2018).
[32] A. Ozbekler, A. Zafer, Wong’s oscillation theorem for the second-order delay differential equations, J. Math. Sci. Vol. 222, pp304-311 (2017).
Author Information
  • School of Finance, Guangdong University of Foreign Studies, Guangzhou, China

  • School of Finance, Guangdong University of Foreign Studies, Guangzhou, China

  • School of Finance, Guangdong University of Foreign Studies, Guangzhou, China

Cite This Article
  • APA Style

    Jinyu Wang, Min Xi, Ailing Xiao. (2018). Oscillations of Solutions of Neutral Nonlinear Differential Equations. Applied and Computational Mathematics, 7(3), 112-120. https://doi.org/10.11648/j.acm.20180703.16

    Copy | Download

    ACS Style

    Jinyu Wang; Min Xi; Ailing Xiao. Oscillations of Solutions of Neutral Nonlinear Differential Equations. Appl. Comput. Math. 2018, 7(3), 112-120. doi: 10.11648/j.acm.20180703.16

    Copy | Download

    AMA Style

    Jinyu Wang, Min Xi, Ailing Xiao. Oscillations of Solutions of Neutral Nonlinear Differential Equations. Appl Comput Math. 2018;7(3):112-120. doi: 10.11648/j.acm.20180703.16

    Copy | Download

  • @article{10.11648/j.acm.20180703.16,
      author = {Jinyu Wang and Min Xi and Ailing Xiao},
      title = {Oscillations of Solutions of Neutral Nonlinear Differential Equations},
      journal = {Applied and Computational Mathematics},
      volume = {7},
      number = {3},
      pages = {112-120},
      doi = {10.11648/j.acm.20180703.16},
      url = {https://doi.org/10.11648/j.acm.20180703.16},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.acm.20180703.16},
      abstract = {This paper aims to establish a new class of differential equations and study the oscillatory behavior of a kind of first-order neutral nonlinear differential equation with time delay arguments. The oscillatory properties of the solutions of the type of first order neutral functional differential equations applied in chemomedical problems are studied. Sufficient conditions for the oscillations of solutions of the above equations are obtained. Also, some results which demonstrate in literature [1-4] will be extended, and the paper focuses on expanding the main finding of literature [2, 3]. Moreover, a new kind of method to be used to discuss the properties of oscillation of the first-order neutral nonlinear differential equations and some theorems are obtained in the paper.},
     year = {2018}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - Oscillations of Solutions of Neutral Nonlinear Differential Equations
    AU  - Jinyu Wang
    AU  - Min Xi
    AU  - Ailing Xiao
    Y1  - 2018/07/19
    PY  - 2018
    N1  - https://doi.org/10.11648/j.acm.20180703.16
    DO  - 10.11648/j.acm.20180703.16
    T2  - Applied and Computational Mathematics
    JF  - Applied and Computational Mathematics
    JO  - Applied and Computational Mathematics
    SP  - 112
    EP  - 120
    PB  - Science Publishing Group
    SN  - 2328-5613
    UR  - https://doi.org/10.11648/j.acm.20180703.16
    AB  - This paper aims to establish a new class of differential equations and study the oscillatory behavior of a kind of first-order neutral nonlinear differential equation with time delay arguments. The oscillatory properties of the solutions of the type of first order neutral functional differential equations applied in chemomedical problems are studied. Sufficient conditions for the oscillations of solutions of the above equations are obtained. Also, some results which demonstrate in literature [1-4] will be extended, and the paper focuses on expanding the main finding of literature [2, 3]. Moreover, a new kind of method to be used to discuss the properties of oscillation of the first-order neutral nonlinear differential equations and some theorems are obtained in the paper.
    VL  - 7
    IS  - 3
    ER  - 

    Copy | Download

  • Sections