# Asymptotic stability in totally nonlinear neutral difference equations

## DOI:

https://doi.org/10.4067/S0716-09172015000300005## Keywords:

Fixed point, Stability, Neutral difference equations, Variable delay.## Abstract

In this paper we use fixed point method to prove asymptotic stability results of the zero solution of the totally nonlinear neutral difference equation with variable delay

∆ x (n) = —a (n) f (x (n — τ (n))) + ∆g (n, x (n — τ (n))).

An asymptotic stability theorem with a sufficient condition is proved, which improves and generalizes some results due to Raffoul (2006) , Yankson (2009), Jin and Luo (2009) and Chen (2013).

## References

[1] A. Ardjouni and A. Djoudi, Fixed points and stability in linear neutral differential equations with variable delays, Nonlinear Analysis 74, pp. 2062-2070, (2011).

[2] A. Ardjouni and A. Djoudi, Stability in nonlinear neutral integrodifferential equations with variable delay using fixed point theory, J. Appl. Math. Comput. 44 : pp. 317-336, (2014).

[3] A. Ardjouni and A. Djoudi, Stability in nonlinear neutral difference equations, Afr. Mat. 26 : pp. 559—574, (2015).

[4] L. Berezansky and E. Braverman, On exponential dichotomy, BohlPerron type theorems and stability of difference equations, J. Math. Anal. Appl. 304, pp. 511-530, (2005).

[5] L. Berezansky, E. Braverman and E. Liz, Sufficient conditions for the global stability of nonautonomous higher order difference equations, J. Difference Equ. Appl. 11, No. 9, pp. 785-798, (2005).

[6] T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover, New York, (2006).

[7] T. A. Burton and T. Furumochi, Fixed points and problems in stability theory, Dynam. Systems Appl. 10, pp. 89-116, (2001).

[8] G. E. Chatzarakis, G.N. Miliaras, Asymptotic behavior in neutral difference equations with variable coefficients and more than one delay arguments. J. Math. Comput. Sci. 1, No. 1, pp. 32-52, (2011).

[9] G. Chen, A fixed point approach towards stability of delay differential equations with applications to neural networks, Ph. D. thesis, Leiden University, (2013).

[10] S. Elaydi, An Introduction to Difference Equations, Springer, New York, (1999).

[11] S. Elaydi, Periodicity and stability of linear Volterra difference systems, J. Math. Anal. Appl. 181, pp. 483-492, (1994).

[12] S. Elaydi and S. Murakami, Uniform asymptotic stability in linear Volterra difference equations, J. Difference Equ. Appl. 3, pp. 203-218, (1998).

[13] P. Eloe, M. Islam and Y. N. Raffoul, Uniform asymptotic stability in nonlinear Volterra discrete systems, Special Issue on Advances in Difference Equations IV, Computers Math. Appl. 45, pp. 1033-1039, (2003).

[14] I. Gyori and F. Hartung, Stability in delay perturbed differential and difference equations, Fields Inst. Commun. 29, pp. 181-194, (2001).

[15] M. Islam and Y. N. Raffoul, Exponential stability in nonlinear difference equations, J. Difference Equ. Appl. 9, pp. 819-825, (2003).

[16] M. Islam and E. Yankson, Boundedness and stability in nonlinear delay difference equations employing fixed point theory, Electronic Journal of Qualitative Theory of Differential Equations, No. 26, pp. 1-18, (2005).

[17] C. Jin and J. Luo, Stability by fixed point theory for nonlinear delay difference equations, Georgian Mathematical Journal, Volume 16, Number 4, pp. 683-691, (2009).

[18] W. G. Kelly and A. C. Peterson, Difference Equations : An Introduction with Applications, Academic Press, (2001).

[19] E. Liz, Stability of non-autonomous difference equations: simple ideas leading to useful results. J. Difference Equ. Appl. 17, No. 2, pp. 203- 220, (2011).

[20] E. Liz, On explicit conditions for the asymptotic stability of linear higher order difference equations, J. Math. Anal. Appl. 303, No. 2, pp. 492-498, (2005).

[21] V. V. Malygina and A. Y. Kulikov, On precision of constants in some theorems on stability of difference equations, Func. Differ. Equ. 15, No. 3-4, pp. 239-249, (2008).

[22] M. Pituk, A criterion for the exponential stability of linear difference equations, Appl. Math. Lett. 17, pp. 779-783, (2004).

[23] Y. N. Raffoul, Stability and periodicity in discrete delay equations. J. Math. Anal. Appl. 324, No. 2, pp. 1356-1362, (2006).

[24] Y. N. Raffoul, Periodicity in general delay nonlinear difference equations using fixed point theory. J. Difference Equ. Appl. 10, No. 13-15, pp. 1229-1242, (2004).

[25] Y. N. Raffoul, General theorems for stability and boundedness for nonlinear functional discrete systems, J. math. Anal. Appl. 279, pp. 639-650, (2003).

[26] D. R. Smart, Fixed point theorems, Cambridge Tracts in Mathematics, No. 66. Cambridge University Press, London-New York, (1974).

[27] E. Yankson, Stability in discrete equations with variable delays, Electronic Journal of Qualitative Theory of Differential Equations, No. 8, pp. 1-7, (2009).

[28] E. Yankson, Stability of Volterra difference delay equations, Electronic Journal of Qualitative Theory of Differential Equations, No. 20, pp. 1-14, (2006).

[29] B. Zhang, Fixed points and stability in differential equations with variable delays, Nonlinear Analysis 63, e233-e242, (2005).

[30] B. G. Zhang, C. J. Tian and P. J. Y. Wong, Global attractivity of difference equations with variable delay, Dynam. Contin. Discrete Impuls. Systems 6, No. 3, pp. 307-317, (1999).

[2] A. Ardjouni and A. Djoudi, Stability in nonlinear neutral integrodifferential equations with variable delay using fixed point theory, J. Appl. Math. Comput. 44 : pp. 317-336, (2014).

[3] A. Ardjouni and A. Djoudi, Stability in nonlinear neutral difference equations, Afr. Mat. 26 : pp. 559—574, (2015).

[4] L. Berezansky and E. Braverman, On exponential dichotomy, BohlPerron type theorems and stability of difference equations, J. Math. Anal. Appl. 304, pp. 511-530, (2005).

[5] L. Berezansky, E. Braverman and E. Liz, Sufficient conditions for the global stability of nonautonomous higher order difference equations, J. Difference Equ. Appl. 11, No. 9, pp. 785-798, (2005).

[6] T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover, New York, (2006).

[7] T. A. Burton and T. Furumochi, Fixed points and problems in stability theory, Dynam. Systems Appl. 10, pp. 89-116, (2001).

[8] G. E. Chatzarakis, G.N. Miliaras, Asymptotic behavior in neutral difference equations with variable coefficients and more than one delay arguments. J. Math. Comput. Sci. 1, No. 1, pp. 32-52, (2011).

[9] G. Chen, A fixed point approach towards stability of delay differential equations with applications to neural networks, Ph. D. thesis, Leiden University, (2013).

[10] S. Elaydi, An Introduction to Difference Equations, Springer, New York, (1999).

[11] S. Elaydi, Periodicity and stability of linear Volterra difference systems, J. Math. Anal. Appl. 181, pp. 483-492, (1994).

[12] S. Elaydi and S. Murakami, Uniform asymptotic stability in linear Volterra difference equations, J. Difference Equ. Appl. 3, pp. 203-218, (1998).

[13] P. Eloe, M. Islam and Y. N. Raffoul, Uniform asymptotic stability in nonlinear Volterra discrete systems, Special Issue on Advances in Difference Equations IV, Computers Math. Appl. 45, pp. 1033-1039, (2003).

[14] I. Gyori and F. Hartung, Stability in delay perturbed differential and difference equations, Fields Inst. Commun. 29, pp. 181-194, (2001).

[15] M. Islam and Y. N. Raffoul, Exponential stability in nonlinear difference equations, J. Difference Equ. Appl. 9, pp. 819-825, (2003).

[16] M. Islam and E. Yankson, Boundedness and stability in nonlinear delay difference equations employing fixed point theory, Electronic Journal of Qualitative Theory of Differential Equations, No. 26, pp. 1-18, (2005).

[17] C. Jin and J. Luo, Stability by fixed point theory for nonlinear delay difference equations, Georgian Mathematical Journal, Volume 16, Number 4, pp. 683-691, (2009).

[18] W. G. Kelly and A. C. Peterson, Difference Equations : An Introduction with Applications, Academic Press, (2001).

[19] E. Liz, Stability of non-autonomous difference equations: simple ideas leading to useful results. J. Difference Equ. Appl. 17, No. 2, pp. 203- 220, (2011).

[20] E. Liz, On explicit conditions for the asymptotic stability of linear higher order difference equations, J. Math. Anal. Appl. 303, No. 2, pp. 492-498, (2005).

[21] V. V. Malygina and A. Y. Kulikov, On precision of constants in some theorems on stability of difference equations, Func. Differ. Equ. 15, No. 3-4, pp. 239-249, (2008).

[22] M. Pituk, A criterion for the exponential stability of linear difference equations, Appl. Math. Lett. 17, pp. 779-783, (2004).

[23] Y. N. Raffoul, Stability and periodicity in discrete delay equations. J. Math. Anal. Appl. 324, No. 2, pp. 1356-1362, (2006).

[24] Y. N. Raffoul, Periodicity in general delay nonlinear difference equations using fixed point theory. J. Difference Equ. Appl. 10, No. 13-15, pp. 1229-1242, (2004).

[25] Y. N. Raffoul, General theorems for stability and boundedness for nonlinear functional discrete systems, J. math. Anal. Appl. 279, pp. 639-650, (2003).

[26] D. R. Smart, Fixed point theorems, Cambridge Tracts in Mathematics, No. 66. Cambridge University Press, London-New York, (1974).

[27] E. Yankson, Stability in discrete equations with variable delays, Electronic Journal of Qualitative Theory of Differential Equations, No. 8, pp. 1-7, (2009).

[28] E. Yankson, Stability of Volterra difference delay equations, Electronic Journal of Qualitative Theory of Differential Equations, No. 20, pp. 1-14, (2006).

[29] B. Zhang, Fixed points and stability in differential equations with variable delays, Nonlinear Analysis 63, e233-e242, (2005).

[30] B. G. Zhang, C. J. Tian and P. J. Y. Wong, Global attractivity of difference equations with variable delay, Dynam. Contin. Discrete Impuls. Systems 6, No. 3, pp. 307-317, (1999).

## How to Cite

[1]

A. Ardjouni and A. Djoudi, “Asymptotic stability in totally nonlinear neutral difference equations”,

*Proyecciones (Antofagasta, On line)*, vol. 34, no. 3, pp. 255-276, 1.## Issue

## Section

Artículos