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# A new nonlinear impulsive delay differential inequality and its applications

- Huali Wang
^{1}and - Changming Ding
^{1}Email author

**2011**:11

https://doi.org/10.1186/1029-242X-2011-11

© Wang and Ding; licensee Springer. 2011

**Received:**19 February 2011**Accepted:**20 June 2011**Published:**20 June 2011

## Abstract

In this article, a new nonlinear impulsive delay differential inequality is established, which can be applied in the dynamical analysis of nonlinear systems to improve many extant results. Using the inequality, we obtain some sufficient conditions to guarantee the exponential stability of nonlinear impulsive functional differential equations. Two examples are given to illustrate the effectiveness and advantages of our results.

## Keywords

- Impulsive delay differential inequality
- exponential stability
- nonlinear functional differential systems

## 1. Introduction

It is well known that the theory of differential inequalities plays an important role in the qualitative and quantitative studies of differential equations [1–3]. In recent years, various inequalities have been established such as the Halanay inequalities in [4–6], the delay inequalities in [7–10], and the impulsive differential inequalities in [11–13]. Using the linear inequality techniques, many results have been done on the stability and dynamical behavior for differential systems, see [4–13] and the references cited therein. For example, [11] presents an extended impulsive delay Halanay inequality and deals with the global exponential stability of impulsive Hopfield neural networks with time delays. In [13], the authors establish a delay differential inequality with impulsive initial conditions and derive some sufficient conditions ensuring the exponential stability of solutions for the impulsive differential equations. However, linear differential inequalities do not work in the studies for nonlinear differential equations. With the development of the theory on nonlinear differential equations (e.g., see [14, 15]), it is necessary to study the corresponding nonlinear differential inequalities. In [16], the authors develop a new nonlinear delay differential inequality that works well in studying a class of nonlinear delay differential systems. Indeed, nonlinear delay differential inequalities with impulses are seldom discussed in the literature. Therefore, in further researches of nonlinear systems, it is beneficial to obtain some new nonlinear impulsive delay differential inequalities. Our goal in this article is to do some investigations on such problems.

Indeed, impulsive effects and delay effects widely exist in the real world. Impulsive delay differential equations provide mathematical models for many phenomena and processes in the field of natural science and technology. In the last few decades, the stability theory of impulsive functional differential equations has obtained a rapid development, and many interesting results have been reported, see [1–8, 16–23]. Recently, exponential stability has attracted increasing interest in both theoretical research and applications [20–22, 24]. However, the existing works mainly focus on linear impulsive functional differential equations [17–19, 25]. There exist very little works devoted to the investigations of exponential stabilities for nonlinear impulsive functional differential systems.

Motivated by the above discussions, in this article, we shall establish a new nonlinear impulsive delay differential inequality, which improves some recent works in the literature [9–12, 16, 18] and can be applied to the dynamical analysis of nonlinear systems. Based on the inequality, some sufficient conditions guaranteeing the local exponential stability of nonlinear impulsive functional differential equations are derived. Finally, two examples are given to show the effectiveness and advantages of our proposed results.

## 2. Preliminaries

_{+}the set of nonnegative real numbers, ℤ

_{+}the set of positive integers, and ℝ

^{ n }the

*n*-dimensional real space equipped with the Euclidean norm | · |. Consider the following impulsive functional differential systems

where *x* ∈ ℝ ^{
n
} , the impulse times {*t*_{
k
} } satisfy 0 ≤ *t*_{0} < *t*_{1} < ⋯ < *t*_{
k
} < ⋯ and lim_{k→+∞}*t*_{
k
} = +∞, and *x*' denotes the right-hand derivative of *x*. Also, assume *f* ∈ *C*([*t*_{k-1}, *t*_{
k
} ) × Ψ, ℝ ^{
n
} ), meanwhile *φ* ∈ Ψ, Ψ is an open set in *PC*([-*τ*, 0], ℝ ^{
n
} ), where *PC*([-*τ*, 0], ℝ ^{
n
} ) = {*ψ*: [-*τ*, 0] → ℝ ^{
n
} |*ψ* is continuous except at a finite number of points *t*_{
k
} , at which
and
exist and
}. For *ψ* ∈ Ψ , the norm of *ψ* is defined by ||*ψ*|| = sup_{-τ≤θ≤0}|*ψ*(*θ*)|. For each *t* ≥ *t*_{0}, *x*_{
t
} ∈ Ψ is defined by *x*_{
t
} (*s*) = *x*(*t* + *s*), *s* ∈ [-*τ*, 0]. For each *k* ∈ ℤ_{+}, *I*_{
k
} (*t*,*x*) ∈ *C*(*t*_{0}, ∞) × ℝ ^{
n
} , ℝ ^{
n
} ).

In this article, we suppose that there exists a unique solution of system (2.1) through each (*t*_{0}, *φ*), see [23] for the details. Furthermore, we assume that *f*(*t*, 0) = 0, and *I*_{
k
} (*t*, 0) = 0, *k* ∈ ℤ_{+}, so that *x*(*t*) = 0 is a solution of system (2.1), which is called the trivial solution.

We now introduce some definitions that will be used in the sequel.

**Definition 2.1**.

*([5]) A function V*: [-

*τ*, ∞) × Ψ → ℝ

_{+}belongs to class

*v*

_{0}if

- (i)
*V*is continuous on each set [*t*_{k - 1},*t*_{ k }) × Ψ and exists, - (ii)
*V*(*t*,*x*) is locally Lipschitzian in*x*and*V*(*t*, 0) ≡ 0.

**Definition 2.2**.

*([5])*Let

*V*∈

*v*

_{0}, for any (

*t*,

*ψ*) ∈ [

*t*

_{k - 1},

*t*

_{ k }) × Ψ, the upper right-hand Dini derivative of

*V*(

*t*,

*x*) along a solution of system (2.1) is defined by

**Definition 2.3**.

*([6])*The trivial solution of system (2.1) is said to be exponentially stable, if for any initial data , there exists a

*λ*> 0, and for every

*ε*> 0, there exists

*δ*=

*δ*(

*ε*) > 0 such that

whenever || *φ* || < *δ* and *t*_{0} ∈ ℝ_{+}.

## 3. Main results

*First, we present a nonlinear impulsive delay differential inequality*.

**Lemma 3.1**.

*Assume that there exist constants p*> 0,

*q*> 0,

*θ*> 1,

*and function m*(

*t*) ∈

*PC*([

*t*

_{0}-

*τ*, ∞), ℝ

_{+})

*satisfying the scalar impulsive differential inequality*

*a*

_{ k },

*b*

_{ k }∈ ℝ

_{+}, .

*Moreover, there exists a constant M*≥ 1

*such that*.

*Then*

*implies*

*Proof*. We first note that
implies that there exists a scalar *λ* > 0 such that the inequality (3.2) holds.

where *a*_{0} = 1, *b*_{0} = 0.

It is clear that
for *t* ∈ [*t*_{0} - *τ*, *t*_{0}] by the definition of
.

Take *k* = 1, we can prove that
for *t* ∈ [*t*_{0}, *t*_{1}). Suppose on the contrary, then there exists some *t* ∈ [*t*_{0}, *t*_{1}) such that
.

*t*∈ [

*t*

_{0}-

*τ*,

*t**], and

*D*

^{+}

*L*(

*t**) ≥ 0. Calculating the upper right-hand Dini derivative of

*L*(

*t*) along the solution of (2.1), it can be deduced that

which is a contradiction. So we have proven
for all *t* ∈ [*t*_{0}, *t*_{1}).

*t*∈ [

*t*

_{1},

*t*

_{2}) such that . Let , then , and ,

*t*∈ [

*t*

_{0}-

*τ*,

*t*** ],

*D*

^{+}

*L*(

*t***) ≥ 0. Calculating the upper right-hand Dini derivative of

*L*(

*t*) along the solution of (2.1), it can be deduced that

which is a contradiction. So we have proven
for all *t* ∈ [*t*_{1}, *t*_{2}).

So (3.1) holds. The proof of Lemma 3.1 is complete.

*Remark 3.1*. In [9–18], the authors got some results for linear differential inequalities under the assumption that *p* > *q*. Note in our result, the restriction *p* > *q* is completely removed if the initial value satisfies some certain conditions.

*Remark 3.2*. It should be noted that if *p* > *q*, then
, which implies that (3.1) holds for any initial value
. In this sense, Lemma 3.1 becomes the well known case, see [16]. Hence, our development result has wider adaptive range than those in [9–12, 16, 18].

Next, based on Lemma 3.1, we shall construct a suitable Lyapunov function to derive some conditions guaranteeing the exponential stability of the trivial solution of system (2.1).

**Theorem 3.1**.

*Assume that there exist function V*(

*t*,

*x*) ∈

*v*

_{0},

*and constants*0 <

*c*

_{1}≤

*c*

_{2},

*m*> 0,

*p*> 0,

*q*> 0,

*θ*> 1

*such that the following conditions hold:*

- (i)
*c*_{1}||*x*||^{ m }≤*V*(*t*,*x*) ≤*c*_{2}||*x*||^{ m }, (*t*,*x*) ∈ (ℝ_{+},ℝ^{ n }); - (ii)

*where*
.

*Then the trivial solution of system* (2.1) *is exponentially stable*.

*Proof*. Let *x*(*t*) = *x*(*t*, *t*_{0}, *φ*) be any solution of system (2.1) with initial value (*t*_{0}, *φ*). By condition (iii), we know that there exists *M* ≥ 1 such that
.

*δ*> 0 such that . When ||

*φ*|| <

*δ*<

*ε*, we have . Using Lemma 3.1 and the condition (

*i*), we derive

By the definition 2.3, the trivial solution of system (2:1) is exponentially stable. This completes the proof.

*Remark 3.3*. From the proof of the Theorem 3.1, it follows that if *p* > *q*, *θ* → 1^{+}, the trivial solution of system (2.1) is globally exponentially stable.

## 4. Examples

In this section, we shall give two examples to illustrate the effectiveness of our results.

where *θ* > 1, *a*(*t*) ≥ *a* > 0, 0 < |*b*(*t*)| ≤ b, 0 ≤ *τ* (*t*) ≤ *τ*, for all *t* ≥ *t*_{0}.

**Property 4.1**. *The trivial solution of system* (4.1) *is exponentially stable*.

*Proof*. Choose

*V*(

*t*) = |

*x*(

*t*)|. When

*t*≠

*t*

_{ k }, calculating the derivative of

*D*

^{+}

*V*(

*t*) along the solution of (4.1), we get

Hence, by Theorem 3.1, the trivial solution of system (4.1) is exponentially stable.

**Example 2**. Consider the following impulsive functional differential equation with distributed delays:

where *θ* > 1, *a* > 0, *b* ∈ ℝ.

**Property 4.2**. *The trivial solution of system* (4.2) *is exponentially stable*.

*Proof*. Choose

*V*(

*t*) = |

*x*(

*t*)|. When

*t*≠

*t*

_{ k }, calculating the derivative of

*D*

^{+}

*V*(

*t*) along the solution of (4:2), we get

Hence, by Theorm 3.1, the trivial solution of system (4.2) is exponentially stable.

*Remark 3.4* It should be noted that the sufficient conditions ensuring the exponential stabilities of (4.1) and (4.2) are easily to check, which show the advantages of our results.

## Declarations

### Acknowledgements

The authors sincerely thanks the referees for their valuable suggestions.

## Authors’ Affiliations

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