ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Email: miguel.sanjuan@urjc.es

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: aluo@siue.edu

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

Abstract

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).

References

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.