Abstract
We derive new nonlinear discrete analogue of the continuous Halanay-type inequality. These inequalities can be used as basic tools in the study of the global asymptotic stability of the equilibrium of certain generalized difference equations.
Highlights
The investigation of stability of nonlinear difference equations with delays has attracted a lot of attention from many researchers such as Agarwal et al 1–3, Baınov and Simeonov 4, Bay and Phat 5, Cooke and Ivanov 6, Gopalsamy 7, Liz et al 8–10, Niamsup et al 11, Mohamad and Gopalsamy, Pinto and Trofimchuk, and references sited therein
In 2, 6, 10, 12, 13, the authors considered discrete Halanay-type inequalities to study some discrete version of functional differential equations
In the following results of Liz et al 10, authors showed that some discrete versions of these maximum inequalities can be applied to study the global asymptotic stability of a family of difference equations
Summary
The investigation of stability of nonlinear difference equations with delays has attracted a lot of attention from many researchers such as Agarwal et al 1–3 , Baınov and Simeonov 4 , Bay and Phat 5 , Cooke and Ivanov 6 , Gopalsamy 7 , Liz et al 8–10 , Niamsup et al 11, , Mohamad and Gopalsamy , Pinto and Trofimchuk , and references sited therein. In , Halanay proved an asymptotic formula for the solutions of a differential inequality involving the “maximum” functional and applied it in the stability theory of linear systems with delay. In the following results of Liz et al 10 , authors showed that some discrete versions of these maximum inequalities can be applied to study the global asymptotic stability of a family of difference equations. Xn−r , a > 0, 1.6 one has xn ≤ max xi λn[0 ], n ≥ 0, 1.7 where λ0 can be calculated in the form established in Theorem A. The main aim of the present paper is to establish some new nonlinear retarded Halanay-type inequalities, which extend Theorem A, along with the derivation of new global stability conditions for nonlinear difference equations
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