Open Access

Advanced Discrete Halanay-Type Inequalities: Stability of Difference Equations

Journal of Inequalities and Applications20092009:535849

DOI: 10.1155/2009/535849

Received: 7 December 2008

Accepted: 21 January 2009

Published: 5 February 2009

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.

1. Introduction

The investigation of stability of nonlinear difference equations with delays has attracted a lot of attention from many researchers such as Agarwal et al. [13], Baĭnov and Simeonov [4], Bay and Phat [5], Cooke and Ivanov [6], Gopalsamy [7], Liz et al. [810], Niamsup et al. [11, 12], Mohamad and Gopalsamy [13], Pinto and Trofimchuk [14], and references sited therein. In [15], 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. Such an inequality was called Halanay inequality in several works. Some generalizations as well as new applications can be found, for instance, in Agarwal et al. [2], Gopalsamy [7], Liz et al. [810], Niamsup et al. [11, 12], Mohamad and Gopalsamy [13], and Pinto and Trofimchuk [14]. In particular, 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.

Theorem 1.

Assume that satisfies the system of inequalities
(1.1)
where and is a natural number. If and
(1.2)
then there exist constants and such that
(1.3)
Moreover, can be chosen as the smallest root in the interval of equation where
(1.4)

By a simple use of Theorem A, authors also demonstrated the validity of the following statement, namely, Theorem B.

Theorem 1.

Assume that satisfies the following inequalities:
(1.5)

If either

holds, then there exist and such that for every solution of
(1.6)
one has
(1.7)

where can be calculated in the form established in Theorem A. As a consequence, the trivial solution of (1.6) is globally asymptotically stable.

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.

2. Halanay-Type Discrete Inequalities

Let denote the set of all real numbers, the set of positive real numbers, the set of nonnegative real numbers, the set of integers, the set of positive integers, and . Consider the following nonlinear difference equation:
(2.1)
where , and . Equation (2.1) is a generalized difference equation (see [3, Section 21] and [11]). The initial value problem for this equation requires the knowledge of the initial data . This vector is called the initial string in [6]. For every initial string, there exists a unique solution of (2.1) that can be calculated using the explicit recurrence formula
(2.2)

In this section, we introduce new discrete inequalities which will be used to derive global stability conditions in the next section.

Theorem 2.1.

Let and
(2.3)
where Also, let be a sequence of nonnegative real numbers satisfying the system of inequalities
(2.4)
where is a constant. Then there exist constants and such that
(2.5)
where , and with Moreover, can be chosen as the smallest root in the interval of equation where
(2.6)

with

Proof.

Let be a sequence of nonnegative real numbers satisfying the system of inequalities
(2.7)

where Since it is easy to prove by induction that if and for then and for all

On the other hand, the system (2.7) is equivalent to
(2.8)
where Next we prove, under the assumptions of the theorem, that there exists a solution to system (2.8) in the form with Indeed, such is a solution of (2.8) if and only if
(2.9)

This is equivalent to the existence of a solution of equation where is the polynomial defined by (2.6).

Now, in view of On the other hand, in view of (2.3). As a consequence, there exists such that Hence, is a solution of (2.9) with

For this value of , the pair is a solution of (2.8) for every Thus, choosing we have that , and for all

Hence, using the first part of the proof, we can conclude that , and for all

By the similar argument used in Theorem 2.1, we obtain the following result.

Theorem 2.2.

Let and
(2.10)
with Also, let be a sequence of nonnegative real numbers satisfying the system of inequalities
(2.11)
Then there exist constants and such that
(2.12)
where , and with Moreover, can be chosen as the smallest root in the interval of equation where
(2.13)

with

Proof.

Let be a sequence of nonnegative real numbers satisfying the system of inequalities
(2.14)

Since it is easy to prove by induction that if and for then and for all

Next we prove that, under the assumptions of the theorem, there exists a solution to system (2.14) in the form with Indeed, such is a solution of (2.14) if and only if
(2.15)

This is equivalent to the existence of a solution of equation where is the polynomial defined in (2.13).

Now, in view of , we have in case in case and in case

On the other hand, in view of (2.10). As a consequence, there exists such that Hence, is a solution of (2.15) with

For this value of , the pair is a solution of (2.14) for every Thus, choosing we have , and for all These imply , and for all Hence, using the first part of the proof, we can conclude that , and for all

Remark 2.3.

In [10], a discrete Halanay-type inequality was given as in Theorem A, where the inequalities were replaced by
(2.16)
where and is a natural number. Note that if a sequence of positive real numbers satisfies (2.16), then it also satisfies (2.4). On the other hand, let and Then we might easily show that the sequence satisfies (2.4) but not (2.16). Indeed,
(2.17)
with On the other hand,
(2.18)

Therefore, in the case of positive sequences, the discrete inequality (2.4) is less conservative than the discrete Halanay-type inequality given by (2.16).

3. Global Stability of Difference Equations

In order to show the applicability of the previous result, in this section we consider the generalized difference equation
(3.1)

where

Although, for every initial string the solution of (3.1) can be explicitly calculated by a recurrence formula similar to (2.2), it is in general difficult to investigate the asymptotic behavior of the solutions using that formula. The next result gives an asymptotic estimate by a simple use of the discrete Halanay inequality.

Theorem 3.1.

For all assume that satisfies the following inequalities:
(3.2)
(3.3)

where and with If either

(a) and or

(b) and

hold, then there exists a constant for every solution of (3.1) such that
(3.4)
where and can be chosen as the smallest root in the interval of equation where
(3.5)

with

As a consequence, the trivial solution of (3.1) is globally asymptotically stable.

Proof.

Let be a solution of (3.1). Equation (3.1) can be written in the form
(3.6)
Hence, we know that
(3.7)
where Thus, using inequality (3.3), we obtain
(3.8)
Denote for and
(3.9)
for Then we have and, from inequality (3.9), we obtain
(3.10)
for On the other hand, using hypothesis (3.2) in (3.1), we have
(3.11)
Denote We can apply Theorem 2.1 to the system of inequalities (3.10) and (3.11) with and Consequently, Theorem 2.1 ensures the validity of the following inequality:
(3.12)

where and are chosen as in Theorem 3.1. This completes the proof of the theorem.

Next, we obtain new conditions for the asymptotic stability of (3.1) using inequality (3.13) instead of (3.3).

Corollary 3.2.

For all assume that satisfies inequality (3.2) and the following condition:
(3.13)
where and with If
(3.14)
holds, where then there exists a constant for every solution of (3.1) such that
(3.15)
where , and can be chosen as the smallest root in the interval of equation where
(3.16)

with

As a consequence, the trivial solution of (3.1) is globally asymptotically stable.

Similarly, using Theorem 2.2 instead of Theorem 2.1, we obtain the following result.

Theorem 3.3.

For all assume that satisfies the following inequalities:
(3.17)
where and with If and
(3.18)
then there exists a constant for every solution of (3.1) such that
(3.19)
where , and can be chosen as the smallest root in the interval of equation where
(3.20)

with

As a consequence, the trivial solution of (3.1) is globally asymptotically stable.

Remark 3.4.

Equation (3.1) covers a variety of difference equations. For instance, we can consider the following difference equation:
(3.21)

Next, we study the asymptotic behavior of the solutions of (3.21). We can apply Theorem 3.1, Corollary 3.2, or Theorem 3.3 to obtain some relations between coefficients and that ensure the global asymptotic stability of the zero solution. Moreover, from Theorem 3.1 we know that if there exists such that for all and if either

(a) and or

(b) and

hold, then all solutions of (3.21) converge to zero.

Declarations

Acknowledgment

The authors thank the referees of this paper for their careful and insightful critique.

Authors’ Affiliations

(1)
Department of Mathematical Sciences, Florida Institute of Technology
(2)
Department of Applied Mathematics, Changwon National University

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© Ravi P. Agarwal et al. 2009

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