This article was downloaded by: [Universidad Autonoma de Barcelona]On: 27 October 2014, At: 02:14Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Mathematical Modelling andAnalysisPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/tmma20

Eventual Stability and EventualBoundedness for ImpulsiveDifferential Equations with“Supremum”Ivanka Stamova aa Burgas Free University , San Stefano St., 8000, Burgas,BulgariaPublished online: 24 Jun 2011.

To cite this article: Ivanka Stamova (2011) Eventual Stability and Eventual Boundednessfor Impulsive Differential Equations with “Supremum”, Mathematical Modelling andAnalysis, 16:2, 304-314, DOI: 10.3846/13926292.2011.580470

To link to this article: http://dx.doi.org/10.3846/13926292.2011.580470

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information(the “Content”) contained in the publications on our platform. However, Taylor& Francis, our agents, and our licensors make no representations or warrantieswhatsoever as to the accuracy, completeness, or suitability for any purposeof the Content. Any opinions and views expressed in this publication are theopinions and views of the authors, and are not the views of or endorsed byTaylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor andFrancis shall not be liable for any losses, actions, claims, proceedings, demands,costs, expenses, damages, and other liabilities whatsoever or howsoever causedarising directly or indirectly in connection with, in relation to or arising out of theuse of the Content.

This article may be used for research, teaching, and private study purposes.Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly

forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Dow

nloa

ded

by [

Uni

vers

idad

Aut

onom

a de

Bar

celo

na]

at 0

2:14

27

Oct

ober

201

4

Mathematical Modelling and Analysis www.tandf.co.uk/journals/TMMA

Volume 16 Number 2, June 2011, 304–314 Publisher: Taylor&Francis and VGTU

Doi:10.3846/13926292.2011.580470 Online ISSN: 1648-3510

c©Vilnius Gediminas Technical University, 2011 Print ISSN: 1392-6292

Eventual Stability and Eventual Boundedness

for Impulsive Differential Equations with

“Supremum”

Ivanka Stamova

Burgas Free University

San Stefano St., 8000 Burgas, Bulgaria

E-mail(corresp.): stamova@bfu.bg

Received August 24, 2010; revised March 9, 2011; published online May 1, 2011

Abstract. Eventual stability and eventual boundedness for nonlinear impulsive dif-ferential equations with supremums are studied. The impulses take place at fixed mo-ments of time. Piecewise continuous Lyapunov functions have been applied. Methodof Razumikhin as well as comparison method for scalar impulsive ordinary differentialequations have been employed.

Keywords: Eventual stability, eventual boundedness, impulsive differential equations,

supremum, Lyapunov functions.

AMS Subject Classification: 34K45.

1 Introduction

The stability of solutions of differential equations via Lyapunov method hasbeen intensively investigated in the past. In many real cases, it is obligatoryto study the stability of such sets, which are not invariant with respect to agiven system of differential equations. This immediately excludes the stabilityin the sense of Lyapunov. Examples for that can be found when self-controlledsystems of management are being studied [2]. For the problem, arisen in thissituation, to be solved, a new notion is introduced – eventual stability [7, 17].In this case, the set under consideration, despite not being invariant in theusual sense, is invariant in the asymptotic sense.

Impulsive differential equations are found in almost every domain of appliedsciences. Numerous examples were given in Bainov’s and his collaborators’book [6]. Some impulsive differential equations have been recently introducedin population dynamics [15], neural networks [12], the chemostat [16], etc. Inthe mathematical simulation in various important branches of control theory,pharmacokinetics, economics, etc. one has to analyse the influence of both themaximum of the function investigated and its impulsive changes. An adequate

Dow

nloa

ded

by [

Uni

vers

idad

Aut

onom

a de

Bar

celo

na]

at 0

2:14

27

Oct

ober

201

4

Eventual Stability and Eventual Boundedness 305

mathematical apparatus for simulation of such processes are the impulsive dif-ferential equations with supremums [1, 4, 8]. To the best of our knowledge,there are no results considering the stability of nonlinear impulsive differentialequations with supremums, which is very important in theories and applicationsand also is a very challenging problem.

In the present paper eventual stability of x = 0 and eventual boundednessof the solutions with respect to the system of impulsive differential equationswith “supremum” is defined. By employing a class of piecewise continuousfunctions which are generalization of the classical Lyapunov’s functions [6, 11]coupled with the Razumikhin technique [3, 5, 9, 10, 13, 14, 15] some sufficientconditions are found.

2 Preliminary Notes and Definitions

Let Rn be the n-dimensional Euclidean space with norm |.|; Ω be a domain inRn containing the origin; R+ = [0,∞); R = (−∞,∞); t0 ∈ R+, τ > 0.

Let J ⊆ R. Define the following class of functions: PC [J,Ω] = σ : J →Ω: σ(t) is a piecewise continuous function with points of discontinuity t ∈ J atwhich σ(t− 0) and σ(t+ 0) exist and σ(t− 0) = σ(t).

Consider the following system of impulsive differential equations with “supre-mum”

x(t) = f(t, x(t), sups∈[t−τ,t]

x(s)), t 6= tk,

∆x(tk) = Ik(x(tk − 0)), tk > t0, k = 1, 2, . . . ,(2.1)

where f : [t0,∞) × Ω × Ω → Rn; Ik : Ω → Rn, k = 1, 2, . . .; ∆x(t) =x(t+0)−x(t−0); t0 < t1 < t2 < · · · ; limk→∞ tk = ∞. Let ϕ0 ∈ PC [[−τ, 0], Ω].Denote by x(t) = x(t; t0, ϕ0), x ∈ Ω the solution of system (2.1), satisfying theinitial conditions:

x(t; t0, ϕ0) = ϕ0(t− t0), t0 − τ ≤ t ≤ t0,

x(t0 + 0; t0, ϕ0) = ϕ0(0)(2.2)

and by J+(t0, ϕ0) – the maximal interval of type [t0, β) in which the solutionx(t; t0, ϕ0) is defined.

The solution x(t; t0, ϕ0) of problem (2.1), (2.2) is a piecewise continuousfunction in interval J+(t0, ϕ0) with points of discontinuity of the first kindt = tk, k = 1, 2, . . . at which it is continuous from the left, i.e. the followingrelations are satisfied:

x(tk − 0) = x(tk), x(tk + 0) = x(tk) + Ik(x(tk)), tk ∈ J+(t0, ϕ0).

Introduce the following notations:

Gk = (t, x) ∈ [t0,∞)×Ω: tk−1 < t < tk, k = 1, 2, . . . ; G =⋃∞

k=1Gk;

‖φ‖ = supt∈[t0−τ,t0]

|φ(t− t0)| is the norm of the function φ ∈ PC [[−τ, 0], Ω];

Sρ = x ∈ Rn: |x| < ρ, ρ = const > 0;

Math. Model. Anal., 16(2):304–314, 2011.

Dow

nloa

ded

by [

Uni

vers

idad

Aut

onom

a de

Bar

celo

na]

at 0

2:14

27

Oct

ober

201

4

306 I. Stamova

K = a ∈ C[R+, R+]: a(u) is strictly increasing and such that a(0) = 0.

Introduce the following conditions:

H2.1. f ∈ C[[t0,∞)×Ω ×Ω,Rn].

H2.2. f(t, 0, 0) = 0, t ∈ [t0,∞).

H2.3. The function f is Lipschitz continuous with respect to its second andthird arguments in [t0,∞)×Ω ×Ω, uniformly on t ∈ [t0,∞).

H2.4. Ik ∈ C[Ω,Rn], k = 1, 2, . . . .

H2.5. Ik(0) = 0, k = 1, 2, . . . .

H2.6. The functions (I+ Ik) : Ω → Ω, k = 1, 2, . . . where I is the identity in Ω.

H2.7. t0 < t1 < t2 < · · · .

H2.8. limk→∞ tk = ∞.

In the further considerations we shall use the class V0 of piecewise continuousauxiliary functions V : [t0,∞) × Ω → R+ which are analogues of Lyapunov’sfunctions [11].

Definition 1. A function V : [t0,∞)×Ω → R+ belongs to class V0, if:

1. V is continuous in G, locally Lipschitz continuous with respect to itssecond argument on each of the sets Gk, k = 1, 2, . . . and V (t, 0) = 0,t ∈ [t0,∞).

2. For each k = 1, 2, . . . and x ∈ Ω

V (tk − 0, x) = V (tk, x) and V (tk + 0, x) = limt→t

k

t>tk

V (t, x) exists.

Definition 2. Given a function V ∈ V0. For any t ≥ t0, t 6= tk, k = 1, 2, . . .and any function φ ∈ PC [[t − τ, t], Ω] the upper right-hand derivative of Vwith respect to system (2.1) is defined by

D+V (t, φ) = limh→0+

sup1

h[V (t+ h, φ(t) + hf(t, φ(t),

sups∈[−τ,0]

φ(t + s)))− V (t, φ(t))]. (2.3)

Note that in Definition 2, D+V (t, φ) is a functional whereas V is a function.

Introduce the set PC 1[J,Rn] = σ ∈ PC [J,Rn]: σ(t) is continuously dif-ferentiable everywhere except some points tk at which σ(tk − 0) and σ(tk + 0)exist and σ(tk − 0) = σ(tk), k = 1, 2, . . .. We remark that if V ∈ V0 andV ∈ PC 1[[t0,∞)×Ω,R+], then (2.3) reduces to

D+V (t, φ) =∂V (t, φ(t))

∂t+∆xV (t, φ(t)).f(t, φ(t), sup

s∈[−τ,0]

φ(t+ s)).

Dow

nloa

ded

by [

Uni

vers

idad

Aut

onom

a de

Bar

celo

na]

at 0

2:14

27

Oct

ober

201

4

Eventual Stability and Eventual Boundedness 307

Together with system (2.1), we shall consider the comparison equation

u(t) = g(t, u), t ≥ t0, t 6= tk,

∆u(tk) = Bk(u(tk)), tk > t0, k = 1, 2, . . . ,(2.4)

where g : [t0,∞)×R+ → R+, Bk : R+ → R, k = 1, 2, . . ..In the proof of the main results we shall use the following lemmas:

Lemma 1. Let the conditions H2.1, H2.3, H2.4, H2.6, H2.7 and H2.8 hold.Then J+(t0, ϕ0) = [t0,∞).

Proof. Since the conditions H2.1, H2.3, H2.4 and H2.6 hold then from theexistence theorem for the equation x(t) = f(t, x(t),maxs∈[t−τ,t] x(s)) withoutimpulses [3] it follows that the solution x(t) = x(t; t0, ϕ0) of the problem (2.1),(2.2) is defined on each of the intervals (tk−1, tk], k = 1, 2, . . .. From conditionsH2.7 and H2.8 we conclude that it can be prolonged continuously for t ≥ t0.ut

Let us note that the problems of existence, uniqueness, and continuabilityof the solutions of functional differential equations without impulses has beeninvestigated in the monograph [3].

Lemma 2. Assume that :

1. Conditions H2.1, H2.3, H2.4, H2.6, H2.7 and H2.8 hold.

2. The function g : [t0,∞) × R+ → R+ is continuous in each of the sets(tk−1, tk]×R+, k = 1, 2, . . . .

3. Bk ∈ C[R+, R], and ψk(u) = u + Bk(u) ≥ 0, k = 1, 2, . . . are non-decreasing with respect to u.

4. The maximal solution u+(t; t0, u0) of the scalar equation (2.4) withu+(t0 + 0; t0, u0) = u0, u0 ∈ R+ is defined in the interval [t0,∞).

5. The function V ∈ V0 is such that V (t0 + 0, ϕ0(0)) ≤ u0,

V (t+ 0, x+ Ik(x)) ≤ ψk(V (t, x)), x ∈ Ω, t = tk, k = 1, 2, . . . ,

and the inequality D+V (t, φ(t)) ≤ g(t, V (t, φ(t))), t 6= tk, k = 1, 2, . . . isvalid for any t ∈ [t0,∞) and any φ ∈ PC [[t − τ, t], Ω] such that V (t +s, φ(t+ s)) < V (t, φ(t)), s ∈ [−τ, 0).

Then

V (t, x(t; t0, ϕ0)) ≤ u+(t; t0, u0), t ∈ [t0,∞). (2.5)

Proof. From Lemma 1 it follows that J+(t0, ϕ0) = [t0,∞) and the solutionx = x(t; t0, ϕ0) of the problem (2.1), (2.2) is such that

x ∈ PC [(t0 − τ,∞), Ω] ∩ PC 1[[t0,∞), Ω].

Math. Model. Anal., 16(2):304–314, 2011.

Dow

nloa

ded

by [

Uni

vers

idad

Aut

onom

a de

Bar

celo

na]

at 0

2:14

27

Oct

ober

201

4

308 I. Stamova

The maximal solution u+(t; t0, u0) of the system (2.4) is defined by the equality

u+(t; t0, u0) =

r0(t; t0, u+0 ), t0 < t ≤ t1,

r1(t; t1, u+1 ), t1 < t ≤ t2,

. . .rk(t; tk, u

+k ), tk < t ≤ tk+1,

. . . ,

where rk(t; tk, u+k ) is the maximal solution of the system without impulses

u = F (t, u) = f(t, u,maxs∈[t−τ,t] u(s)) in the interval (tk, tk+1], k = 0, 1, 2, . . . ,

for which u+k = ψk(rk−1(tk; tk−1, u+k−1)), k = 1, 2, . . . and u+0 = u0.

Let t ∈ (t0, t1]. Then, from the corresponding comparison theorem for thecontinuous case [7], it follows that V (t, x(t; t0, ϕ0)) ≤ u+(t; t0, u0), i.e. theinequality (2.5) is valid for t ∈ (t0, t1]. Suppose that (2.5) is satisfied fort ∈ (tk−1, tk], k > 1. Then, using condition 5 of Lemma 2 and the fact that thefunction ψk is non-decreasing, we obtain

V (tk + 0, x(tk + 0; t0, ϕ0)) ≤ ψk(V (tk, x(tk; t0, ϕ0)))

≤ ψk(u+(tk; t0, ϕ0)) = ψk(rk−1(tk; tk−1, u

+k−1)) = u+k .

We apply again the comparison theorem for the continuous case in the interval(tk, tk+1] and obtain V (t, x(t; t0, ϕ0)) ≤ rk(t; tk, u

+k ) = u+(t; t0, u0), i.e. the

inequality (2.5) is valid for t ∈ (tk, tk+1]. ut

3 Main Results

3.1 Eventual stability

We shall also use the following notations:

Bα = (t, x) ∈ [t0,∞)×Rn: |x| < α;

Bα = (t, x) ∈ [t0,∞)×Rn: |x| ≤ α, α > 0.

We shall use the following definitions of eventual stability of x = 0 for thesystem (2.1).

Definition 3. The set x(t) ≡ 0 is said to be:

(a) eventually stable set of system (2.1), if

(∀ε> 0) (∃T = T (ε) > 0) (∀t0 ≥ T ) (∃δ = δ(t0, ε) > 0)

(∀ϕ0 ∈ PC [[−τ, 0], Ω]: ‖ϕ0‖ < δ) (∀t ≥ t0): |x(t; t0, ϕ0)| < ε;

(b) uniformly eventually stable set of system (2.1), if the number δ in (a) isindependent of t0 ∈ R.

Theorem 1. Assume that :

1. Conditions H2.1–H2.8 hold.

Dow

nloa

ded

by [

Uni

vers

idad

Aut

onom

a de

Bar

celo

na]

at 0

2:14

27

Oct

ober

201

4

Eventual Stability and Eventual Boundedness 309

2. g(t, 0) = 0, t ∈ [t0,∞).

3. Bk(0) = 0, k = 1, 2, . . ..

4. The conditions of Lemma 2 hold, and there exists a function a ∈ K suchthat

a(|x|) ≤ V (t, x), a ∈ K, (t, x) ∈ [t0,∞)×Ω, (3.1)

where V ∈ V0.

Then if the set u = 0 is an eventually stable set of system (2.4), then the setx = 0 is an eventually stable set of system (2.1).

Proof. Let ε > 0 be such that Sε ⊂ Ω and the set u = 0 be an eventuallystable of system (2.4). Then there exist T > 0 and δ1 > 0 such that

u0 < δ1 implies u+(t; t0, u0) < a(ε), t ≥ t0

for some given t0 ≥ T , where the maximal solution u+(t; t0, u0) of (2.4) isdefined in the interval [t0,∞).

From the properties of the function V , it follows that there exists a constantδ = δ(t0, ε) > 0 such that if (t0 + 0, x) ∈ Bδ, then V (t0 + 0, x) ∈ Sδ1 . Let ϕ0 ∈PC [[−τ, 0], Ω]: ‖ϕ0‖ < δ and x(t) = x(t; t0, ϕ0) be the solution of problem (2.1),(2.2). Then |ϕ0(0)| ≤ ‖ϕ0‖ < δ, (t0 + 0, ϕ0(0)) ∈ Bδ, hence V (t0 + 0, ϕ0(0)) ∈Sδ1 . Thus

u+(t; t0, V (t0 + 0, ϕ0(0))) < a(ε). (3.2)

Setting u0 = V (t0 + 0, ϕ0(0)), we get by Lemma 2,

V (t, x(t; t0, ϕ0)) ≤ u+(t; t0, V (t0 + 0, ϕ0(0))) for t ≥ t0. (3.3)

Consequently, from (3.1), (3.3) and (3.2), we obtain

a(|x(t; t0, ϕ0)|) ≤ V (t, x(t; t0, ϕ0)) ≤ u+(t; t0, V (t0 + 0, ϕ0(0)) < a(ε), t ≥ t0.

Hence, |x(t; t0, ϕ0)| < ε, t ≥ t0 for the given t0 > T , which proves the eventualstability of the set x = 0 of (2.1). ut

Theorem 2. Let the conditions of Theorem 1 hold, and a function b ∈ K existssuch that

V (t, x) ≤ b(|x|), (t, x) ∈ [t0,∞)×Ω. (3.4)

Then if the set u = 0 is an uniformly eventually stable set of system (2.4), thenthe set x = 0 is an uniformly eventually stable set of system (2.1).

Proof. Let ε > 0 be such that Sε ⊂ Ω. Suppose now, that the set u = 0 bean uniformly eventually stable of system (2.4). Therefore, we have that

u0 < δ1 implies u+(t; t0, u0) < a(ε), t ≥ t0 (3.5)

for every t0 > T and δ1 > 0 independent of t0 ∈ R. Let δ > 0 be such thatδ < b−1(δ1) and

‖ϕ0‖ < δ. (3.6)

Math. Model. Anal., 16(2):304–314, 2011.

Dow

nloa

ded

by [

Uni

vers

idad

Aut

onom

a de

Bar

celo

na]

at 0

2:14

27

Oct

ober

201

4

310 I. Stamova

Then, from (3.4) and (3.6), it follows

V (t0 + 0, ϕ0(0)) ≤ b(|ϕ0(0)|) ≤ b(‖ϕ0‖) < b(δ) < δ1

which due to (3.5) implies

u+(t; t0, V (t0 + 0, ϕ0(0))) < a(ε), t ≥ t0. (3.7)

We claim that ‖ϕ0‖ < δ implies |x(t; t0, ϕ0)| < ε, t ≥ t0 for every t0 > T . Ifthe claim is not true, there exists t0 > T , a corresponding solution x(t; t0, ϕ0)of (2.1) with ‖ϕ0‖ < δ, and t∗ > t0 such that,

|x(t∗; t0, ϕ0)| ≥ ε, |x(t; t0, ϕ0)| < ε, t0 ≤ t ≤ tk,

where t∗ ∈ (tk, tk+1] for some k. Then, due to H2.6 and condition 5 of Lemma 2,we can find t0 ∈ (tk, t

∗) such that

|x(t0; t0, ϕ0)| ≥ ε and x(t0; t0, ϕ0) ∈ Ω. (3.8)

Hence, setting u0 = V (t0, ϕ0(t0 − tk)), since all the conditions of Lemma 2 are

satisfied, we get

V (t, x(t; t0, ϕ0)) ≤ u+(t; t0, V (t0, ϕ0(t0 − tk))) for t0 ≤ t ≤ t0. (3.9)

From (3.8), (3.1), (3.9) and (3.7), it follows that

a(ε) ≤ a(|x(t0; t0, ϕ0)|) ≤ V (t0, x(t; t0, ϕ0))

≤ u+(t0; t0, V (t0, ϕ0(t0 − tk))) < a(ε).

The contradiction obtained proves that (2.1) is uniformly practically stable.ut

Remark 1. We have assumed in Theorems 1 and 2 stronger requirements onV only to unify all the stability criteria for the comparison equation and forthe system under consideration. This obviously puts burden on the comparisonequation (2.4). However, to obtain only non-uniform stability criteria, we couldweaken certain assumption, as it is stated in the next result.

Theorem 3. Assume that :

1. Conditions H2.1–H2.8 hold.

2. There exists a function V ∈ V0 such that (3.1) holds,

V (t+ 0, x+ Ik(x)) ≤ V (t, x), x ∈ Ω, t = tk, k = 1, 2, . . . , (3.10)

and the inequality

D+V (t, φ(t)) ≤ p(t)q(t, φ(t)), t 6= tk, k = 1, 2, . . . (3.11)

is valid for any t ∈ [t0,∞) and any φ ∈ PC [[t− τ, t], Ω] such that V (t+s, φ(t+s)) < V (t, φ(t)), s ∈ [−τ, 0), p : [t0,∞) → R, q : [t0,∞)×Ω → R.

Dow

nloa

ded

by [

Uni

vers

idad

Aut

onom

a de

Bar

celo

na]

at 0

2:14

27

Oct

ober

201

4

Eventual Stability and Eventual Boundedness 311

3. There exists a number Γ > 0 such that |q(t, x)| ≤ Γ , (t, x) ∈ [t0,∞)×Ω.

4.∫∞

t0|p(t)| dt <∞.

Then the set x = 0 is an eventually stable set of system (2.1).

Proof. Let ε > 0 be such that Sε ⊂ Ω and Γ > 0. Let the number T = T (ε) >0 be chosen so that for t ≥ T

∫ ∞

t

|p(s)| ds <a(ε)

2Γ. (3.12)

(This is possible in view of condition 4 of Theorem 3.)Let t0 ≥ T . From the properties of the function V , it follows that there

exists a constant δ = δ(t0, ε) > 0 such that if (t0+0, x) ∈ Bδ, then V (t0+0, x) <12a(ε). Let ϕ0 ∈ PC [[−τ, 0], Ω]: ‖ϕ0‖ < δ and x(t) = x(t; t0, ϕ0) be the solutionof problem (2.1), (2.2). Then |ϕ0(0)| ≤ ‖ϕ0‖ < δ, (t0 + 0, ϕ0(0)) ∈ Bδ, hence

V (t0 + 0, ϕ0(0)) <1

2a(ε). (3.13)

From condition 3 of Theorem 3, (3.11) and (3.12), we have∫ t

t0

D+V (s, x(s)) ds ≤ Γ

∫ t

t0

|p(s)| ds <1

2a(ε), t ≥ t0. (3.14)

Let tk+l < t < tk+l+1. Then, we have

∫ t

t0

D+V (s, x(s)) ds =

∫ t1

t0

D+V (s, x(s)) ds +

k+l∑

j=2

∫ tj

tj−1

D+V (s, x(s)) ds

+

∫ t

tk+l

D+V (s, x(s)) ds = V (t1, x(t1))− V (t0 + 0, ϕ0(0))

+

k+l∑

j=2

[V (tj , x(tj))− V (tj−1 + 0, x(tj−1 + 0)] + V (t, x(t))

− V (tk+l + 0, x(tk+l + 0)] ≥ V (t, x(t)) − V (t0 + 0, ϕ0(0)). (3.15)

From (3.1), (3.12)–(3.15), we obtain

a(|x(t; t0, ϕ0)|) ≤ V (t0 + 0, ϕ0(0)) +

∫ t

t0

D+V (s, x(s)) ds

< V (t0 + 0, ϕ0(0)) +1

2a(ε) < a(ε).

Therefore, |x(t; t0, ϕ0)| < ε for t ≥ t0. ut

Remark 2. It is well known that, in the stability theory of functional differentialequations, the condition D+V (t, x(t)) ≤ p(t)q(t, x(t)) allows the derivative ofthe Lyapunov function to be positive which may not even guarantee the stabil-ity of a functional differential system (see [3, 5]). However, as we can see fromTheorem 3, impulses have played an important role in stabilizing a functionaldifferential system [14].

Math. Model. Anal., 16(2):304–314, 2011.

Dow

nloa

ded

by [

Uni

vers

idad

Aut

onom

a de

Bar

celo

na]

at 0

2:14

27

Oct

ober

201

4

312 I. Stamova

Theorem 4. Let the conditions of Theorem 3 hold, and a function b ∈ K existssuch that

V (t, x) ≤ b(|x|), (t, x) ∈ [t0,∞)×Ω. (3.16)

Then the set x = 0 is an uniformly eventually stable set of system (2.1).

Proof. Let ε > 0 be given. Choose δ = δ(ε) < b−1(12a(ε)), 0 < δ < ε

and Γ = Γ (ε) > 0 so that |q(t, x)| ≤ Γ for (t, x) ∈ Bδ. Let the numberT = T (ε) > 0 be chosen so that

∫ ∞

t

|p(s)| ds <b(δ)

Γ, t ≥ T. (3.17)

Let t0 ≥ T , ϕ0 ∈ PC [[−τ, 0], Ω]: ‖ϕ0‖ < δ and let x(t) = x(t; t0, ϕ0) be thesolution of problem (2.1), (2.2). From (3.1), (3.10), (3.11), (3.16)) and (3.17),we have

a(|x(t; t0, ϕ0)|) ≤ V (t0 + 0, ϕ0(0)) +

∫ t

t0

D+V (s, x(s)) ds

≤ b(|ϕ0(0)|) + Γ

∫ t

t0

|p(s)| ds < 2b(δ) < a(ε)

for t ≥ t0. Therefore, |x(t; t0, ϕ0)| < ε for t ≥ t0. ut

Example 1. Consider the equation

x(t) = p(t) sups∈[t−τ,t]

x(s), t 6= tk,

∆x(tk) = ck, tk > 0, k = 1, 2, . . . ,(3.18)

where t ≥ 0; x ∈ R+; τ > 0; p ∈ C[R+, R]; ck < 0 and |ck + x| < |x| fork = 1, 2, . . . , 0 < t1 < t2 < · · · and limk→∞ tk = ∞.

The set x = 0 is not stable in the sense of Lyapunov, because it is not anequilibrium for the equation (3.18).

Let α > 0. Consider the function V (t, x) = |x|. For t ≥ 0, t 6= tk, and forany φ ∈ PC [[t − τ, t], R] such that V (t + s, φ(t + s)) < V (t, φ(t)), s ∈ [−r, 0),we have

D+V (t, φ(t)) = sign(φ(t))[p(t) sups∈[t−τ,t]

φ(s)] ≤ |p(t)|| sups∈[t−τ,t]

φ(s)| ≤ |p(t)||φ(t)|

for φ ∈ Sα. Also, for t = tk, k = 1, 2, . . ., we obtain

V (t+ 0, x(t) + ck) = |ck + x(t)| ≤ V (t, x(t)).

If∫∞

0|p(t)| dt <∞, then all conditions of Theorem 3 are satisfied, and the

set x = 0 is an eventually stable set with respect to (3.18).

Dow

nloa

ded

by [

Uni

vers

idad

Aut

onom

a de

Bar

celo

na]

at 0

2:14

27

Oct

ober

201

4

Eventual Stability and Eventual Boundedness 313

3.2 Eventual boundedness

In this part of Section 3, we shall apply the direct method of Lyapunov forinvestigation of eventual boundedness of the solutions of system of the type(2.1) for Ω ≡ Rn, i.e. we shall consider the system

x(t) = f(t, x(t), sups∈[t−τ,t]

x(s)), t 6= tk,

∆x(tk) = Ik(x(tk)), k = 1, 2, . . . ,(3.19)

where f : [t0,∞) × Rn × Rn → Rn; Ik : Rn → Rn, k = 1, 2, . . .; ∆x(t) =x(t+ 0)− x(t− 0); t0 < t1 < t2 < · · · ; limk→∞ tk = ∞.

Let ϕ0 ∈ PC [[−τ, 0], Rn]. Denote by x(t) = x(t; t0, ϕ0) the solution of(3.19), satisfying the initial conditions

x(t; t0, ϕ0) = ϕ0(t− t0), t0 − τ ≤ t ≤ t0,

x(t0 + 0; t0, ϕ0) = ϕ0(0).

Definition 4. The solutions of (3.19) are said to be:

(a) eventually equi-bounded, if

(∀α> 0) (∃T = T (α) > 0) (∀t0 ≥ T ) (∃β = β(t0, α) > 0)

(∀ϕ0 ∈ PC [[−τ, 0], Rn]: ‖ϕ0‖ < α) (∀t ≥ t0): |x(t; t0, ϕ0)| < β;

(b) uniformly eventually bounded, if the number β in (a) is independent oft0 ∈ R.

The proofs of the next theorems are similar to the proofs of Theorems 3 and4. Piecewise continuous Lyapunov functions V : [t0,∞) × Rn → R+, V ∈ V0are used.

Theorem 5. Let the conditions of Theorem 3 hold for Ω ≡ Rn, and a(u) → ∞as u→ ∞. Then the solutions of system (3.19) are eventually equi-bounded.

Theorem 6. Let the conditions of Theorem 5 hold, and a function b ∈ K existssuch that V (t, x) ≤ b(|x|), (t, x) ∈ [t0,∞)× Rn. Then the solutions of system(3.19) are uniformly eventually bounded.

References

[1] D. Bainov, Yu. Domshlak and S. Milusheva. Partial averaging for impulsivedifferential equations with supremum. Georgian Math. J., 3(1):11–26, 1996.Doi:10.1007/BF02256795.

[2] J.R. Haddock. Some new results on stability and convergence of solutions ofordinary and functional differential equations. Funkcial. Ekvac., 19:247–269,1976.

[3] J. Hale. Theory of Functional Differential Equations. Springer-Verlag, New York,Heidelberg, Berlin, 1977.

Math. Model. Anal., 16(2):304–314, 2011.

Dow

nloa

ded

by [

Uni

vers

idad

Aut

onom

a de

Bar

celo

na]

at 0

2:14

27

Oct

ober

201

4

314 I. Stamova

[4] Z. He, P. Wang and W. Ge. Periodic boundary value problem for first orderimpulsive differential equations with supremum. Indian J. Pure Appl. Math.,34(1):133–143, 2003.

[5] J. Kato. On Liapunov–Razumikhin type theorems for functional differentialequations. Funkcial. Ekvac., 16:225–239, 1973.

[6] V. Lakshmikantham, D.D. Bainov and P.S. Simeonov. Theory of Impulsive Dif-

ferential Equations. World Scientific, Singapore, New Jersey, London, 1989.

[7] V. Lakshmikantham, S. Leela and A. Martynyuk. Practical Stability Analysis of

Nonlinear Systems. World Scientific Publishers, Singapore, New Jersey, London,Hong Kong, 1990.

[8] S. Milusheva and D. Bainov. Averaging method for neutral type impulsive dif-ferential equations with supremums. Ann. Fac. Sci. Toulouse, 12(3):391–403,1991.

[9] B.S. Razumikhin. Stability of Hereditary Systems. Nauka, Moscow, 1988. (inRussian)

[10] J. Shen and J. Yan. Razumikhin type stability theorems for impulsivefunctional differential equations. Nonlinear Anal., 33(5):519–531, 1998.Doi:10.1016/S0362-546X(97)00565-8.

[11] P. Simeonov and D. Bainov. Stability with respect to part of the variablesin systems with impulse effect. J. Math. Anal. Appl., 117(1):247–263, 1986.Doi:10.1016/0022-247X(86)90259-3.

[12] G. Stamov. Almost periodic models of impulsive hopfield neural network. J.

Math. Kyoto Univ., 49(1):57–67, 2009.

[13] I. Stamova. Vector Lyapunov functions for practical stability of nonlinear im-pulsive functional differential equations. J. Math. Anal. Appl., 325(1):612–623,2007. Doi:10.1016/j.jmaa.2006.02.019.

[14] I. Stamova. Boundedness of impulsive functional differential equations with vari-able impulsive perturbations. Bull. Austral. Math. Soc., 77(2):331–345, 2008.Doi:10.1017/S0004972708000439.

[15] I. Stamova and G. Stamov. Lyapunov–Razumikhin method for im-pulsive functional differential equations and applications to the popu-lation dynamics. J. Comput. Appl. Math., 130(1–2):163–171, 2001.Doi:10.1016/S0377-0427(99)00385-4.

[16] S. Sun and L. Chen. Dynamic behaviors of Monod type chemostat model with im-pulsive perturbation on the nutrient concentration. J. Math. Chem., 42(4):837–848, 2007. Doi:10.1007/s10910-006-9144-3.

[17] T. Yoshizawa. Stability Theory by Lyapunov’s Second Method. The MathematicalSociety of Japan, 1966.

Dow

nloa

ded

by [

Uni

vers

idad

Aut

onom

a de

Bar

celo

na]

at 0

2:14

27

Oct

ober

201

4