Skip Navigation Links
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email:

On the Asymptotic Stability Behaviours of Solutions of Non-linear Differential Equations with Multiple Variable Advanced Arguments

Journal of Environmental Accounting and Management 8(2) (2019) 239--249 | DOI:10.5890/JAND.2019.06.007

Emel Biçer$^{1}$, Cemil Tunç$^{2}$

$^{1}$ Department of Mathematics, Faculty of Arts and Sciences, Bingol University, 12000 Bingol, Turkey

$^{2}$ Department of Mathematics, Faculty of Sciences, Van Yuzuncu Yil University, 65080, Van, Turkey

Download Full Text PDF



We pay our attention to a non-linear differential equation of first order with multiple two variable advanced arguments. We find sufficient conditions satisfying the convergence (C) and exponential convergence (EC) of solutions of the considered non-linear advanced differential equation (NADE) by contraction mapping principle (CMP). The obtained results improve and extend the results can be found in the relevant literature from a case of linear advanced differential equation (LADE) of first order to a case of (NADE) of first order with multiple two variable advanced arguments. We give examples for illustrations by applying MATLAB-Simulink. It is also clearly shown the behaviors of the orbits for the special cases of the considered (NADE).


  1. [1]  Asif, N.A., Talib, I., and Tunç, C. (2015), Existence of solution for first order coupled system with nonlinear coupled boundary conditions, Bound. Value Probl., 2015(134), 1-9.
  2. [2]  Berezansky, L. and Braverman, E. (2009), On exponential stability of a linear delay differential equation with an oscillating coefficient, Appl. Math. Lett., 22, 1833-1837.
  3. [3]  Berezansky, L. and Braverman E. (2011), On non-oscillation of advanced differential equations with several terms, Abstr. Appl. Anal., 14 pp.
  4. [4]  Burton, T.A. and Furumochi, T. (2001), Fixed points and problems in stability theory for ordinary and functional differential equations, Dynam. Systems Appl., 10, 89-116.
  5. [5]  Dung, N.T. (2015), Asymptotic behavior of linear advanced differential equations, Acta Mathematica Scientia, 35B(3), 610-618.
  6. [6]  Jankowski, T. (2005), Advanced differential equations with nonlinear boundary conditions, J. Math. Anal. Appl., 304, 490-503.
  7. [7]  Kitamura, Y. and Kusano, T. (1980), Oscillation of first order nonlinear differential equations with deviating arguments, Proc. Amer. Math. Soc., 78, 64-68.
  8. [8]  Korkmaz, E. and Tunç, C. (2017), Inequalities and exponential decay of certain differential equations of first order in time varying delay, Dynam. Systems Appl., 26, 157-166.
  9. [9]  Li, X. and Zhu, D. (2002), Oscillation and non-oscillation of advanced differential equations with variable coefficients, J. Math. Anal. Appl., 269, 462-488.
  10. [10]  Liu, B. and Tunç, C. (2015), Pseudo almost periodic solutions for a class of first order differential iterative equations, Appl. Math. Lett., 40, 29-34.
  11. [11]  Pravica, D.W., Randriampiry, N., and Spurr, M.J. (2009), Applications of an advanced differential equation in the study of wavelets. , Appl. Comput. Harmon. Anal., 27, 2-11.
  12. [12]  Shah, S.M. and Wiener, J. (1983), Advanced differential equations with piecewise constant argument deviations, Internat J. Math. Math. Sci., 6, 671-703.
  13. [13]  Tunç, C. (2014), On the uniform asymptotic stability to certain first order neutral differential equations, Cubo, 16, 111-119.
  14. [14]  Tunç, C., (2014), Asymptotic stability of solutions of a class of neutral differential equations with multiple deviating arguments, Bull. Math. Soc. Sci. Math. Roumanie. Tome, 57(105), no. 1, 121-130.
  15. [15]  Tunç, C. (2015), Convergence of solutions of nonlinear neutral differential equations with multiple delays, Bol. Soc. Mat. Mex., 21, 219-231.
  16. [16]  Tunc, C. (2017), Stability and boundedness in Volterra-integro differential equations with delays, Dynam. Systems Appl., 26, 121-130.
  17. [17]  Tunc, C. (2017), Qualitative properties in nonlinear Volterra integro-differential equations with delay. Journal of Taibah University for Science, 11(2), 309-314.
  18. [18]  Tunç, C. and Bicer, E. (2015), Hyers-Ulam-Rassias stability for a first order functional differential equation, J. Math. Fundam. Sci., 47(2), 143-153.
  19. [19]  Tunç, C. and Mohammed, S.A. (2017), On the stability and instability of functional Volterra integrodifferential equations of first order, Bull. Math. Anal. Appl., 9, 151-160.
  20. [20]  Wiener, J., Debnath, L., and Shah, S.M. (1986), Analytic solutions of nonlinear neutral and advanced differential equations, Internat. J. Math. Sci. 9, 365-372.