On the Asymptotic Stability of the Nonlinear Difference Equation System

Year 2021, Volume: 4 Issue: 2, 65 - 71, 31.08.2021

Serbun Ufuk Değer , Yaşar Bolat

https://doi.org/10.33187/jmsm.887537

Abstract

In this paper, we obtain some new results on the equi-boundedness of solutions and asymptotic stability for a class of nonlinear difference systems with variable delay of the form $$x(n+1)=ax\left(n\right)+B\left(n\right)F\left(x(n-m\left(n\right))\right),n=0,1,2,...$$ where $F$ is the real valued vector function, $m:Z\to Z^{+},$ which is bounded function and maximum value of $m$ is $k$ and is a $k\times k$ variable coefficient matrix. We carry out the proof of our results by using the Banach fixed point theorem and we use these results to determine the asymptotic stability conditions of an example.

Keywords

Asymptotic stability, Difference equation, Liapunov stable

References

• [1] RP. Agarwal, Difference equations and inequalities: theory, methods, and applications CRC Press, (2000).
• [2] MP. Chen, B. Liu, Asymptotic behavior of solutions of first order nonlinear delay difference equations Computers & Mathematics with Applications, (1996), 32.4: 9-13.
• [3] K. Conrad, The Contraction Mapping Theorem II, (2014).
• [4] J.T. Edwards, J.F. Neville, Boundedness and stability of solutions to difference equations Journal of Computational and Applied Mathematics, (2002), 140.1-2: 275-289.
• [5] D.V. Giang, D.C. Huong, Extinction, Persistence and global stability in models of population growth Journal of mathematical analysis and applications, (2005), 308.1: 195-207.
• [6] D.V. Giang, D.C. Huong, Nontrivial periodicity in discrete delay models of population growth Journal of mathematical analysis and applications, (2005), 305.1: 291-295.
• [7] J.R. Graef, C. Qian, Global stability in a nonlinear difference equation Journal of Difference Equations and Applications, (1999), 5.3: 251-270.
• [8] I. Gyori, G. Ladas, P.N. Vlahos, Global attractivity in a delay difference equation Nonlinear Analysis: Theory, Methods & Applications, (1991), 17.5: 473-479.
• [9] D.C. Huong, N.V. Mau, On a nonlinear difference equation with variable delay Demonstratio Mathematica, (2013), 46.1: 123-135.
• [10] D.C. Huong, On the asymptotic behaviour of solutions of a nonlinear difference equation with bounded multiple delay Vietnam Journal of Mathematics, (2006), 34.2: 163-170.
• [11] D.C. Huong, Oscillation for a Nonlinear Difference Equation Vietnam Journal of Mathematics, (2009), 37.4: 537-549.
• [12] D.C. Huong, P.T. Nam, On Oscillation, Convergence and Boundedness of Solutions of Some Nonlinear Difference Equations with Multiple DelayVietnam Journal of Mathematics, (2008), 36.2: 151-160.
• [13] D.C. Huong, Persistence and global attractivity for a discretized version of a general model of glucose-insulin interaction Demonstratio Mathematica, (2016), 49.3: 302-318.
• [14] D.C. Huong, Asymptotic Stability and Strict Boundedness for Non-autonomous Nonlinear Difference Equations with Time-varying Delay Vietnam Journal of Mathematics, (2016), 44.4: 789-800.
• [15] M. Liao, Global asymptotic stability of a family of nonlinear difference equations. Discrete Dynamics in Nature and Society, (2013).
• [16] Y. Muroya, E. Ishiwata, N. Guglielmi, Global stability for nonlinear difference equations with variable coefficients Journal of mathematical analysis and applications 334.1 (2007): 232-247.
• [17] S.U. Deger, Y.Bolat, Stability conditions a class of linear delay difference systems Cogent Mathematics, (2017), 4.1:129445.
• [18] V. Sree Hari Rao, G.Jaya Sudha, Global Behaviour of Solutions of Nonlinear Delay Difference Equations. The Journal of Difference Equations and Applications, (2002), 8.2: 101-124.
• [19] A. Asiri, E.M. Elsayed, Dynamics and Solutions of Some Recursive Sequences of Higher Order Journal of Computational Analysis& Applications, (2019), 27.1.
• [20] X. Zhang, J. Yan, Global asymptotic behavior of nonlinear difference equations Computers & Mathematics with Applications, (2005), 49.9-10: 1335-1345.
• [21] X. Zhang, Global attractivity for nonlinear difference equations with delay. Journal of Mathematical Analysis and Applications, (2007), 336 : 975-986.
• [22] A. Akgul, M.Inc, E. Karatas Reproducing kernel functions for difference equations. Discrete and Continuous Dynamical Systems, (2015), 8(6).
• [23] A. Akgul, On the solution of higher-order difference equationsy. Mathematical Methods in the Applied Sciences, (2015).