Asymptotic stability in totally nonlinear neutral difference equations
Keywords:Fixed point, Stability, Neutral difference equations, Variable delay.
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).
 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).
 A. Ardjouni and A. Djoudi, Stability in nonlinear neutral difference equations, Afr. Mat. 26 : pp. 559—574, (2015).
 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).
 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).
 T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover, New York, (2006).
 T. A. Burton and T. Furumochi, Fixed points and problems in stability theory, Dynam. Systems Appl. 10, pp. 89-116, (2001).
 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).
 G. Chen, A fixed point approach towards stability of delay differential equations with applications to neural networks, Ph. D. thesis, Leiden University, (2013).
 S. Elaydi, An Introduction to Difference Equations, Springer, New York, (1999).
 S. Elaydi, Periodicity and stability of linear Volterra difference systems, J. Math. Anal. Appl. 181, pp. 483-492, (1994).
 S. Elaydi and S. Murakami, Uniform asymptotic stability in linear Volterra difference equations, J. Difference Equ. Appl. 3, pp. 203-218, (1998).
 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).
 I. Gyori and F. Hartung, Stability in delay perturbed differential and difference equations, Fields Inst. Commun. 29, pp. 181-194, (2001).
 M. Islam and Y. N. Raffoul, Exponential stability in nonlinear difference equations, J. Difference Equ. Appl. 9, pp. 819-825, (2003).
 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).
 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).
 W. G. Kelly and A. C. Peterson, Difference Equations : An Introduction with Applications, Academic Press, (2001).
 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).
 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).
 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).
 M. Pituk, A criterion for the exponential stability of linear difference equations, Appl. Math. Lett. 17, pp. 779-783, (2004).
 Y. N. Raffoul, Stability and periodicity in discrete delay equations. J. Math. Anal. Appl. 324, No. 2, pp. 1356-1362, (2006).
 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).
 Y. N. Raffoul, General theorems for stability and boundedness for nonlinear functional discrete systems, J. math. Anal. Appl. 279, pp. 639-650, (2003).
 D. R. Smart, Fixed point theorems, Cambridge Tracts in Mathematics, No. 66. Cambridge University Press, London-New York, (1974).
 E. Yankson, Stability in discrete equations with variable delays, Electronic Journal of Qualitative Theory of Differential Equations, No. 8, pp. 1-7, (2009).
 E. Yankson, Stability of Volterra difference delay equations, Electronic Journal of Qualitative Theory of Differential Equations, No. 20, pp. 1-14, (2006).
 B. Zhang, Fixed points and stability in differential equations with variable delays, Nonlinear Analysis 63, e233-e242, (2005).
 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).