# Integralφ0-Stability in terms of Two Measures for Impulsive Differential Equations with “Supremum”

Authors: Peiguang Wang; Xiaojing Liu; Qing Xu
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

This paper establishes a criterion on integralφ0-stability in terms of two measures for impulsive differential equations with “supremum” by using the cone-valued piecewise continuous Lyapunov functions, Razumikhin method, and comparative method. Meantime, an example is given to illustrate our result.


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## Body

## 1. Introduction

In this paper, we discuss the integralφ0-stability in terms of two measures for impulsive differential equations with “supremum”:
(1)x′=F(t,x(t),sup⁡s∈[t-r,t]x(s))fort≥0,t≠τk,x(τk+0)=Ik(x(τk-0))fork=1,2,…,x(t)=ϕ(t),t∈[t0-r,t0],
and its perturbed impulsive differential equations with “supremum”
(2)x′=F(t,x(t),sup⁡s∈[t-r,t]x(s))+G(t,x(t),sup⁡s∈[t-r,t]x(s))nnnnnnnnnnnnnnnnnnnnnnnnnnnnnfort≥0,t≠τk,x(τk+0)=Ik(x(τk-0))+Jk(x(τk-0))nnnnnnnnnnnnnnnnnnnfork=1,2,…,x(t)=ϕ(t),t∈[t0-r,t0],
where x∈Rn, F,G:R+×Rn×Rn→Rn, F(t,0,0)=G(t,0,0)≡0, Ik,Jk:Rn→Rn, Ik(0)=Jk(0)≡0, k=1,2,…, r>0, t0∈R+, and ϕ∈(PC[t0-r,t0],Rn). Let Rn be n-dimensional Euclidean space with norm ∥x∥, R+=[0,∞), and {τk}1∞ a sequence of fixed points in R+ such that τk+1>τk and lim⁡k→∞τk=∞. We denote by x(t;t0,ϕ) the solution of (1). In our further investigation we will assume that solution x(t;t0,ϕ) is defined on [t0-r,∞) for any initial function ϕ∈PC([t0-r,t0],Rn).The research on impulsive differential equations with “supremum” problem, Bainov et al. [1] justified the partial averaging for impulsive differential equations, He et al. [2] discussed the periodic boundary value problem for first order impulsive differential equations, Agarwal and Hristova [3] studied the strict stability in terms of two measures for impulsive differential equations, Stamova and Stamov [4] investigated the global stability of models based on impulsive differential equations and variable impulsive perturbations, and Hristova [5, 6] obtained the φ0-stability in terms of two measures for impulsive differential equations.In recent years, the integral stability theory has been rapid development (see [7–12]). For example, Soliman and Abdalla [10] introduced integral φ0-stability of perturbed system of ordinary differential equations. Hristova [12] studied the integral stability in terms of two measures for impulsive differential equations with “supremum.” However, the corresponding theory of impulsive differential equations with “supremum” is still at an initial stage of its development, especially for integral φ0-stability in terms of two measures. Motivated by the idea of [5, 6, 10, 12], in this work, by employing the cone-valued piecewise continuous Lyapunov functions, Razumikhin method, and comparative method, we extend the notions of φ0-stability in terms of two measures to integral φ0-stability in terms of two measures for impulsive differential equations with “supremum.”


## 2. Preliminaries

Denote byPC(X,Y)(X⊂R,Y⊂Rn) the set of all functions u:X→Y which are piecewise continuous in X with points of discontinuity of the first kind at the points τk∈X and which are continuous from the left at the points τk∈X,u(τk)=u(τk-0).We denote byPC1(X,Y) the set of all function u∈PC(X,Y) which are continuously differentiable for t∈X,t≠τk.Letx,y∈Rn. Denote by (x·y) the dot product of both vectors x and y.LetK⊂Rn be a cone, and K*={φ∈Rn:(φ·x)≥0foranyx∈K} is adjoint cone.We give the following notations for convenience:(3)K={a∈C(R+,R+):nnna(s)is strictly  increasing,a(0)=0};CK={b∈C[R+×R+,R+]:nnnb(t,·)∈Kfor  any  fixedt∈[0,∞)};Γ={inf⁡x∈Rnh∈C[[-r,∞)×Rn,K]:nnninf⁡x∈Rnh(t,x)=0for  eacht∈[-r,∞)}.Leth0,h∈Γ,φ0∈K*,t∈R+, and ϕ∈PC([t0-r,t0],Rn). Define
(4)H0(t,ϕ,φ0)=sup⁡{(φ0·h0(t+s,ϕ(t+s))):s∈[-r,0]},(5)H(t,ϕ,φ0)=sup⁡{(φ0·h0(t+s,ϕ(t+s))):s∈[-r,0]}.Letρ,t and T>0 be constants, φ0∈K*, h∈Γ. Define sets:
(6)S(h,ρ,φ0)={(t,x)∈R+×Rn:(φ0·h(t,x))<ρ};Sc(h,ρ,φ0)={(t,x)∈R+×Rn:(φ0·h(t,x))≥ρ};Ω(t,T,ρ)={(φ0·h(t,x))(x,y)∈Rn×Rn:nnnn(φ0·h(t,x))<ρfors∈[t,t+T],nnnn(φ0·h(t,y))<ρfors∈[t-r,t+T]}.In our further investigations we use the following comparison scalar impulsive ordinary differential equation:(7)u′=g1(t,u),t≠τk,u(τk+0)=ξk(u(τk)),u(t0)=u0,nnnnnnnnnnk=1,2,…,
the scalar impulsive ordinary differential equation:
(8)w′=g2(t,w),t≠τk,w(τk+0)=ηk(w(τk)),w(t0)=w0,nnnnnnnnnnk=1,2,…,
and its perturbed scalar impulsive ordinary differential equation:
(9)w′=g2(t,w)+q(t),t≠τk,w(τk+0)=ηk(w(τk))+γk(w(τk)),w(t0)=w0,nnnnnnnnnnnnnnnnnnnk=1,2,…,
where u,w∈R, g1(t,0)=g2(t,0)≡0,ξk(0)=0,ηk(0)=0,k=1,2,….Assume that solutions of the scalar impulsive equations (7), (8), and (9) exist on [t0,∞) for any initial values. Meanwhile, we give some definitions and lemmas. The details can be found in [5].Definition 1 (see [5]).
We say that functionV(t,x):[-r,∞)×Rn→K,V=(V1,V2,…,Vn), belongs to the class Λ if(A1)
V(t,x)∈PC1([-r,∞)×Rn,K);(A2)
for eachk=1,2,… and x∈Rn there exist the finite limits
(10)V(τk-0,x)=lim⁡t↑τkV(t,x),V(τk+0,x)=lim⁡t↓τkV(t,x);(A3)
there exist constantsMi>0,i=1,2,…,n, such that |Vi(t,x)-Vi(t,y)|≤Mi∥x-y∥ for any t∈R+,x,y∈Rn.Definition 2 (see [5]).
Letφ0∈K*, h∈Γ be given. The function V(t,x)∈Λ is said to be φ0-strongly h-decrescent if there exist a constant δ>0 and a function a∈K such that (t,x)∈[-r,∞)×Rn:(φ0·h(t,x))<δ implies that (φ0·V(t,x))≤a(φ0·h(t,x)).
LetV(t,x)∈Λ,t∈Ω,t≠τk,x∈Rn, and ϕ∈PC([t-r,t],Rn). We define a derivative of the function V(t,x) along the trajectory of solution of (1) as follows:
(11)D(1)V(t,ϕ(t))=lim⁡ϵ→0sup⁡1ϵ{V((t,ϕ(t),sup⁡s∈[-r,0]ϕ(t+s))t+ϵ,ϕ(t)nnnnnnnnnnnnnnnnn+ϵF(t,ϕ(t),sup⁡s∈[-r,0]ϕ(t+s)))nnnnnnnnnnnnnnn-V(t,ϕ(t))(t+ϵ,ϕ(t)+ϵF(t,ϕ(t),sup⁡s∈[-r,0]ϕ(t+s)))}.
Similarly we define a derivative of the functionV(t,x)∈Λ along the trajectory of solution of the perturbed system (2) for t∈Ω,t≠τk,x∈Rn, and ϕ∈PC([t-r,t],Rn) as follows:
(12)D(2)V(t,ϕ(t))=lim⁡ϵ→0sup⁡1ϵ{V(+G(t,ϕ(t),sup⁡s∈[-r,0]ϕ(t+s)))t+ϵ,ϕ(t)nnnnnnnnnnnnnnnnn+ϵ(+G(t,ϕ(t),sup⁡s∈[-r,0]ϕ(t+s)))F(t,ϕ(t),sup⁡s∈[-r,0]ϕ(t+s))nnnnnnnnnnnnnnnnnnnn+G(t,ϕ(t),sup⁡s∈[-r,0]ϕ(t+s))))nnnnnnnnnnnnnnnnn-V(t,ϕ(t))+G(t,ϕ(t),sup⁡s∈[-r,0]ϕ(t+s)))}.Definition 3 (see [5]).
Letφ0∈K*, h,h0∈Γ be given. The function h0 is φ0-uniformly finer than h if there exist a constant δ>0 and a function a∈K, such that for any point (t,x)∈[0,∞)×Rn:(φ0·h0(t,x))<δ the inequality (φ0·h(t,x))≤a(φ0·h0(t,x)) holds.Lemma 4 (see [5]).
Leth,h0∈Γ, φ0∈K* be given, and h0(t,x) is φ0-uniformly finer than h(t,x) with a constant δ and a function a∈K. Then for any t∈R+ and ϕ∈PC([t-r,t],Rn) inequality H0(t,ϕ,φ0)<δ implies H(t,ϕ,φ0)≤a(H0(t,ϕ,φ0)), where functions H and H0 are defined by (4), (5).In our further investigations we use the following comparison result.Lemma 5 (see [5]).
Let the following conditions be fulfilled.(B1)
The vectorφ0∈K* and function V∈Λ are such that
(i)
for any numbert≥0:t≠τk and any function ψ∈PC([t-r,t],Rn) such that (φ0·V(t,ψ(t)))≥(φ0·V(t+s,ψ(t+s))) for s∈[-r,0) the inequality
(13)(φ0·D(1)V(t,ψ(t)))≤g1(t,(φ0·V(t,ψ(t))))holds, whereg1∈PC(R+×R+,R+).(ii)
(φ0·V(τk+0,Ik(x)))≤ξk(φ0·V(τk,x)),k=1,2,…, x∈Rn, and τk∈[t0,T], where functions ξk∈K.(B2)
Functionx(t;t0,ϕ) is a solution of (1) that is defined for t∈[t0-r,T], where ϕ∈PC([t0-r,t0],Rn).(B3)
Functionu*(t)=u*(t;t0,u0) is the maximal solution of (7) with initial condition u*(t0)=u0 that is defined for t∈[t0,T].
Then the inequalitysup⁡s∈[-r,0](φ0·V(t0+s,ϕ(t0+s)))≤u0 implies the validity of the inequality (φ0·V(t,x(t)))≤u*(t) for t∈[t0,T].Definition 6.
Leth0,h∈Γ. System of impulsive differential equations with “supremum” (1) is said to be(S1)
(H0,h)-equi-integral φ0-stable if for every α≥0 and for any t0≥0 there exists a positive function β=β(t0,α)∈CK which is continuous in t0 for each α and such that for maximal solution y*(t;t0,ϕ) of the perturbed system of impulsive differential equations with “supremum” (2) the inequality
(14)(φ0·h(t,y*(t;t0,ϕ)))<β,t≥t0holds, provided that(15)H0(t0,ϕ,φ0)≤α,and for everyT>0,
(16)∫t0t0+T‍sup⁡(x,y)∈Ω(t0,T,β)∥G(s,x,y*)∥ds+∑t0≤τk≤t0+T‍sup⁡x:h(τk,x)<β∥Jk(x)∥≤α,whereH0(t0,ϕ,φ0) is defined by (4) and ϕ∈PC([t0-r,t0],Rn);(S2)
(H0,h)-uniform-integrally φ0-stable if (S1) is satisfied, where δ is independent on t0.Remark 7.
We note that in the case whenh0(t,x)≡∥x∥ and h(t,x)≡∥x∥ the (H0,h)-equi-integral (uniform-integral)φ0-stability reduces to equi-integral (uniform-integral)φ0-stability.


## 3. Main Result

Theorem 8.
Let the following conditions be fulfilled.(H1)
Functionsh0,h∈Γ; h0 is φ0-uniformly finer than h.(H2)
There exists a functionV1∈Λ that is φ0-strongly h0-decrescent and
(i)
for any numbert≥0, t≠τk, and any function ψ∈PC([t-r,t],Rn), such that (φ0·V1(t,ψ(t)))>(φ0·V1(t+s,ψ(t+s))) for s∈[-r,0) and (t,ψ(t))∈S(h,ρ,φ0) the inequality
(17)(φ0·D(1)V1(t,ψ(t)))≤g1(t,(φ0·V1(t,ψ(t))))holds, whereρ>0 is a constant.(ii)
(φ0·V1(τk+0,Ik(x)))≤ξk(φ0·V1(τk,x)), for (τk,x)∈S(h,ρ,φ0), k=1,2,….(H3)
For any numberμ>0 there exists a function V2(μ)∈Λ such that
(iii)
b(φ0·h(t,x))≤(φ0·V2(μ)(t,x))≤a(φ0·h0(t,x)) for (t,x)∈[-r,∞)×Rn, where a,b∈K and lim⁡u→∞b(u)=∞.(iv)
For any numbert≥0,t≠τk, and any function ψ∈PC([t-r,t],Rn), such that (t,ψ(t))∈S(h,ρ,φ0)∩Sc(h0,μ,φ0) and (φ0·(V1(t,ψ(t))+V2(μ)(t,ψ(t))))>(φ0·(V1(t+s,ψ(t+s))+V2(μ)(t+s,ψ(t+s)))) for s∈[-r,0) the inequality
(18)(φ0·(D(1)V1(t,ψ(t))+D(2)V2(μ)(t,ψ(t))))≤g2(t,φ0·(V1(t,ψ(t))+V2(μ)(t,ψ(t))))holds.(v)
(φ0·(V1(τk+0,Ik(x))+V2(μ)(τk+0,Ik(x))))≤ηk(φ0·(V1(τk,x)+V2(μ)(τk,x))) for (τk,x)∈S(h,ρ,φ0)∩Sc(h0,μ,φ0), k=1,2,….(H4)
Zero solution of the scalar impulsive differential equation (7) is equi-stable.(H5)
Zero solution of the scalar impulsive differential equation (8) is uniform-integrally stable.
Then system of impulsive differential equations with “supremum” (1) is (H0,h)-uniform-integrally φ0-stable.Proof.
Since functionV1(t,x) is φ0-strongly h0-decrescent, there exist a constant ρ1∈(0,ρ) and a function ψ1∈K such that (φ0·h0(t,x))<ρ1 implies that
(19)(φ0·V1(t,x))≤ψ1((φ0·h0(t,x))).
Since h0(t,x) is φ0-uniformly finer than h(t,x), there exist a constant ρ0∈(0,ρ1) and a function ψ2∈K such that (φ0·h0(t,x))<ρ0 implies that
(20)(φ0·h(t,x))≤ψ2(φ0·h0(t,x)),
where ψ2(ρ0)<ρ1.
According to Lemma4, the inequality H0(t,ϕ,φ0)<ρ0 implies
(21)H(t,ϕ,φ0)≤ψ2(H0(t,ϕ,φ0)),ϕ∈PC([t-r,t],Rn).
Lett0≥0 be a fixed point. Choose a number α>0 such that α<ρ0.
According to condition(H3) of Theorem 8, there exists a function V2(α)(t,x) that is Lipshitz with a constant M2. Let M1 be the Lipshitz constant of function V(t,x).
Denote(M1+M2)α=α1. Without loss of generality we assume α1<b(ρ).
Since the zero solution of the scalar impulsive differential equation (7) is equi-stable, there exists a function δ1=δ1(t0,α1)>0 such that the inequality |u0|<δ1 implies
(22)|u(t;t0,u0)|<α12,t≥t0,
where u(t;t0,u0) is a solution of (7).
Since the functionψ1∈K there exists a δ2=δ2(δ1)>0,δ2<ρ1, such that for |u|<δ2 the inequality
(23)ψ1(u)<δ1
holds.
Since the zero solution of the scalar impulsive differential equation (8) is uniform-integrally stable, there exists a function β1=β1(α1)∈CK, b(ρ)>β1≥α1, such that for every solution of the perturbed impulsive equation (9) the inequality
(24)|w(t;t0,w0)|<β1,t≥t0,
holds, provided that
(25)|w0|<α1
and for every T>0,
(26)∫t0t0+T‍|q(s)|ds+∑t0≤τk≤t0+T‍|γk|≤α1.
Since the functionb∈K,lim⁡s→∞b(s)=∞, and ψ2(α)<ψ2(ρ0)<ρ1<ρ, we could choose a constant β=β(β1)>0,ρ>β>α,β>ψ2(α), such that
(27)b(β)≥β1.
Since the functiona,ψ2∈K, and β>ψ2(α), we can find a δ3=δ3(α1,β)>0,α<δ3<min⁡(δ2,ρ0), such that the inequalities
(28)a(δ3)<α12,ψ2(δ3)<β
hold.
From (21) and (28) it follows that H0(t0,ϕ,φ0)<α implies
(29)H(t0,ϕ,φ0)≤ψ2(H0(t0,ϕ,φ0))<ψ2(α)<ψ2(δ3)<β;
that is, h(t,ϕ,φ0)<β for t∈[t0-r,t0].
Now let the initial functionsϕ∈PC([t0-r,t0],Rn) be such that
(30)H0(t0,ϕ,φ0)<α
and let the perturbed functions in impulsive equation with “supremum” (2) be such that
(31)∫t0t0+T‍sup⁡x,y∈Ω(t0,T,β)∥G(s,x,y*)∥ds+∑t0≤τk≤t0+T‍∥Jk(x)∥≤α
for every T>0.
Lety*(t)=y*(t;t0,ϕ) be a solution of (2), where the initial function and the perturbed functions satisfy (30) and (31); then
(32)(φ0·h(t,y*(t;t0,ϕ)))<β,t≥t0.
Suppose it is not true. There exists a pointt*>t0 such that
(33)(φ0·h(t*,y*(t*;t0,ϕ)))=β,(φ0·h(t,y*(t;t0,ϕ)))<β,nnnnnnnnnnnnnnt∈[t0,t*).Case1. Let t*≠τk,k=1,2,…. Then from the continuity of the maximal solution y*(t;t0,ϕ) at point t* follows that (φ0·h(t*,y*(t*;t0,ϕ)))=β.
If we assume that(φ0·h0(t*,y*(t*)))≤δ3 then from the choice of δ3 and inequality (28) it follows (φ0·h(t*,y*(t*;t0,ϕ)))≤(φ0·h0(t*,y*(t*;t0,ϕ)))≤ψ2(δ3)<β that contradicts (33).
Therefore(34)(φ0·h0(t*,y*(t*)))≤δ3,H0(t0,ϕ,φ0)<α<δ1.Case1.1. Let there exist a point t0*∈(t0,t*), t0*≠τk,k=1,2,…, such that δ3=(φ0·h0(t*,y*(t*))) and (t,y*(t))∈S(h,β,φ0)∩Sc(h0,δ3,φ0). Since β<ρ and δ3>α it follows that
(35)(t,y*(t))∈S(h,ρ,φ0)∩Sc(h0,α,φ0),t∈[t0*,t*).
Define a functionϕ*(t)=y*(t) for t∈[t0*-r,t0*] and let r1(t;t0*,u0) be the maximal solution of impulsive scalar differential equation (7) where u0=sups∈[-r,0](φ0·V1(t0*,ϕ*(t0*))). Let x*(t)≡x*(t;t0*,ϕ*) be the solution of the impulsive equations (1), t∈[t0*-r,t0*]. From conditions (i), (ii) of Theorem 8, according to Lemma 5, it follows that
(36)(φ0·V1(t,x*(t)))≤r1(t;t0*,u0),t∈[t0*,t*].
From the choice of the pointt0* it follows that (φ0·h0(t0*,ϕ*(t0*)))=(φ0·h0(t0*,y*(t0*)))=δ3<δ2.
According to inequalities (19) and (23) we obtain
(37)u0=(φ0·V1(t0*,ϕ*(t0*)))≤ψ1(φ0·h0(t0*,ϕ*(t0*)))<δ1.
From inequalities (22) and (36) it follows that (φ0·V1(t,x*(t)))≤r1(t;t0*,u0)<α1/2 for t∈[t0*,t*], or
(38)(φ0·V1(t0*,y*(t0*)))<(φ0·V1(t0*,x(t0*)))<α12.
From inequality (28) and condition (iii) of Theorem 8, it follows that
(39)(φ0·V2(α)(t0*,y*(t0*)))<a(φ0·h0(t0*+s,y*(t0*+s)))=a(δ3)<α12.
Consider functionV2(α)(t,x) that is defined in condition (H7) of Theorem 8, and define the function
(40)V(t,x)=V1(t,x)+V2(α)(t,x),
the function V(t,x) satisfies the conditions of Lemma 5. Let point t∈[t0*,t*],t≠tk, and function ψ∈PC([t-r,t],Rn) be such that (t,ψ(t))∈S(h,β,φ0)∩Sc(h0,α,φ0), (ψ(t),sup⁡s∈[-r,0]ψ(t+s))∈Ω(t0*,T*,β), and V(t,ψ(t))>V(t+s,ψ(t+s)) for s∈[-r,0). Then using the Lipshitz conditions for functions V1(t,x) and V2(α)(t,x), and condition (iv) of Theorem 8, we obtain
(41)(φ0·D(2)V(t,ψ(t)))=(φ0·(D(2)V1(t,ψ(t))+D(2)V2(α)(t,ψ(t))))≤(φ0·D(1)V1(t,ψ(t))+D(1)V2(α)(t,ψ(t)))+(M1+M2)∥G(t,ψ(t),sup⁡s∈[-r,0]ψ(t+s))∥≤g2(t,(φ0·V(t,ψ(t))))+(M1+M2)×sup⁡(x,y)∈Ω(t0*,T*,β)∥G(t,x,y*)∥,
where T*=t*-t0*.
Letτk∈(t0*,t*),x∈Rn be such that (τk,x)∈S(h,β,φ0)∩Sc(h0,α,φ0). According to condition (v) of Theorem 8, we have
(42)(φ0·V(tk+0,Ik(x)+Jk(x)))=(φ0·V(tk+0,Ik(x)))+(φ0·(V(tk+0,Ik(x)+Jk(x))-V(tk+0,Ik(x))))≤ηk(φ0·V(tk,x))+(M1+M2)∥Jk(x)∥≤ηk(φ0·V(tk,x))+(M1+M2)×sup⁡x:h(τk,x)<β∥Jk(x)∥.
According to inequalities (41), (42) and Lemma 5, the inequality
(43)(φ0·V(t,y*(t)))≤r*(t;t0*,w0*),t∈[t0*,t*]
holds.
Consider the scalar impulsive differential equation (9), where
(44)q(t)=(M1+M2)sup⁡(x,y)∈Ω(t0*,T*,β)∥G(t,x,y*)∥,γk=(M1+M2)sup⁡x:h(τk,x)<β∥Jk(x)∥.
According to above notations and inequality (31) for T*=t*-t0*, we obtain
(45)∫t0*t*‍q(s)ds+∑t0*≤τk≤t*‍γk≤(M1+M2)α=α1.
Letr*(t;t0*,w0*) be the maximal solution of (9) through the point (t0*,w0*), where w0*=V(t0*+s,y*(t0*+s)), and perturbations q(t) and γk are defined above and satisfy inequality (45).
Choose a pointT*>t* such that
(46)∫t0*t*‍q(s)ds+12(T*-t*)q(t*)<α1.
Now define the continuous functionq*(t):[t0*,∞)→R:
(47)q*(t)={q(t)fort∈[t0*,t*]q(t*)t*-T*(t-T*)fort∈[t*,T*]0fort≥T*,
and the sequence of numbers {γk*}1∞:
(48)γk*={γkfork:τk∈(t0*,t*]0fork:τk>t*.
From (45), it follows that for every T>0(49)∫t0*t0*+T‍q*(s)ds+∑t0*≤τk≤t0*+T‍γk*≤∫t0*t0*+T‍q(s)ds+∑t0*≤τk≤t0*+T‍γk≤α1.
LetR(t;t0*,w0*) be the maximal solution of the scalar impulsive differential equation (9) through the point (t0*,w0*), where perturbations of the right parts are defined above function q*(t) and numbers γk*. We note that
(50)R(t;t0*,w0*)=r*(t;t0*,w0*),t∈[t0*,t*).
From inequalities (38) and (39), the definition of point w0*, and inequality (49) follows the validity of (24) for the solution R(t;t0*,w0*); that is,
(51)R(t;t0*,w0*)<β1,t≥t0*.
From inequalities (43) and (51), equality (50), the choice of point t*, and condition (iii) of Theorem 8, we obtain
(52)b(β)≥β1>R(t*;t0*,w0*)≥(φ0·V(t*,y*(t*;t0,ϕ)))≥(φ0·V2(α)(t*,y*(t*;t0,ϕ)))≥b((φ0·h(t*,y*(t*;t0,ϕ))))=b(β).
The obtained contradiction proves the validity of the inequality (32) for t≥t0.
Case
1.2. Let there exist a point τk∈(t0,t*) such that δ3<(φ0·h0(τk+0,y*(τk+0;t0,x0))), δ3>(φ0·h0(τk,y*(τk;t0,x0))), and (35) is true.
We choose a numberδ3~:δ3<δ3~<β such that δ3~=(φ0·h0(t0*,y*(t0*;t0,x0))) and t0*∈(t0,t*),t0*≠τk,k=1,2,…. We repeat the proof of Case 1.1, where instead of δ3 we use δ3~ and obtain a contradiction.
Case
2. Let there exist a natural number k such that (φ0·h(t,y*(t)))<β for t∈[t0,τk] and (φ0·h(τk,y*(τk+0)))=(φ0·h(τk,Ik(y*(τk))+Jk(y*(τk))))>β.
We repeat the proof of Case 1 as in this case we choose the constantβ=β(β1)>0, such that b(β)≥sup⁡k{ηk(β1)}.
As in the proof of Case 1.1, we obtain the validity of inequalities (51) and (43). We apply conditions (iii) and (v) of Theorem 8 and obtain
(53)b(β)≥ηk(r*(τk;t0*,w0*))≥ηk(φ0·V(τk,y*(τk)))=ηk((φ0·(V1(τk,y*(τk))+V2(α)(τk,y*(τk)))))≥(φ0·V1(τk,Ik(y*(τk))+Jk(y*(τk)))nnn+V2(α)(τk,Ik(y*(τk))+Jk(y*(τk))))≥(φ0·V2(α)(τk,Ik(y*(τk))+Jk(y*(τk))))≥b(φ0·h((τk,Ik(y*(τk))+Jk(y*(τk)))))>b(β),
and the obtained contradiction proves the validity of inequality (32) in this case. Inequality (32) proves (H0,h)-uniform-integral φ0-stabilities of the considered system of the impulsive differential equations with “supremum.”Next, we will provide an example which satisfies all the hypotheses of Theorem8.Example 9.
Consider the system of impulsive differential equations with “supremum”(54)x′=-e-tx(t)+2y(t)+e-tmax⁡s∈[t-r,t]x(s),t≠k,y′=-x(t)-e-ty(t)+12e-tmax⁡s∈[t-r,t]y(s),t≠k,x(k+0)=12k/2x(k),y(k+0)=12k/2y(k),nnnnnnnnk=1,2,…,x(t)=ϕ1(t-t0),y(t)=ϕ2(t-t0)nnnt∈[t0-r,t0],
and its perturbed impulsive differential equations with “supremum”
(55)x′=-e-tx(t)+2y(t)+e-tmax⁡s∈[t-r,t]x(s)+e-tmax⁡s∈[t-r,t]x2(s),nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnt≠k,y′=-x(t)-e-ty(t)+12e-tmax⁡s∈[t-r,t]y(s)+e-tmax⁡s∈[t-r,t]y2(s),nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnt≠k,x(k+0)=12k/2x(k),y(k+0)=12k/2y(k),nnnnnnnnk=1,2,…,x(t)=ϕ1(t-t0),y(t)=ϕ2(t-t0)t∈[t0-r,t0],
where x,y∈R,r>0 is enough small constant, t≥t0≥0. Without loss of generality we will assume further that 1≥t0≥0.
Leth0(t,x,y)=(∥x∥,∥y∥),h(t,x,y)=(x2,y2).
Consider functionV:R2→K,V=(V1,V2),V1(x,y)=(1/2)x2,V2(x,y)=(1/2)y2, where K={(x,y):x≥0,y≥0}⊂R2 is a cone.
Now, let us consider the vectorφ0=(1,2). It is easy to check that the function V1(t,x,y)=V(x,y) is φ0-strongly h0-decrescent with a function ψ2=x∈K and the condition (iii) is satisfied for the function V2(μ)=V(x,y), where b(u)=(1/2)u and a(u)=u2.
Lett≥0,t≠k,k=1,2…ψ∈PC([t-r,t],R2),ψ=(ψ1,ψ2) be such that the inequality
(56)(φ0·V(ψ1(t),ψ2(t)))≥(φ0·V(ψ1(t+s),ψ2(t+s))),s∈[-r,0]
or
(57)12ψ12(t)+ψ22(t)≥12ψ12(t+s)+ψ22(t+s),s∈[-r,0];
then
(58)ψ1(t)max⁡s∈[t-r,t]ψ1(s)≤2(φ0·V(ψ1(t),ψ2(t))),ψ2(t)max⁡s∈[t-r,t]ψ2(s)≤(φ0·V(ψ1(t),ψ2(t))).
Therefore if inequality (57) is satisfied then
(59)(φ0·D(54)V(ψ1(t),ψ2(t)))=e-t(max⁡s∈+ψ1(t)max⁡s∈[t-r,t]ψ1(s)+ψ2(t)max⁡s∈[t-r,t]ψ2(s))-(ψ1(t))2-2(ψ2(t))2+ψ1(t)max⁡s∈[t-r,t]ψ1(s)+ψ2(t)max⁡s∈[t-r,t]ψ2(s))≤e-t(-(ψ1(t))2-2(ψ2(t))2+2(φ0·V(ψ1(t),ψ2(t)))nnnnnnn+(φ0·V(ψ1(t),ψ2(t)))2(ψ2(t))2)=e-t(φ0·V(ψ1(t),ψ2(t)))
or
(60)(φ0·D(54)V(ψ1(t),ψ2(t)))≤e-t(φ0·V(ψ1(t),ψ2(t))).
Inequality (60) proves the validity of condition (i) of Theorem 8 for the function V1(t,x,y)=V(x,y), where g1(t,u)=ue-t. Meanwhile, inequality (60) proves the validity of condition (iv) of Theorem 8 for the function V2μ(t,x,y)=V(x,y), where g2(t,u)=2ue-t.
From jump conditions (54) and the choice of vector φ0 and function V we obtain the validity of conditions (ii) and (v) of Theorem 8 for the functions V1(t,x,y)=V(x,y) and V2μ(t,x,y)=V(x,y), where ξk(u)=(1/2k)u and ηk(u)=(1/2k)u.
Consider following comparison scalar impulsive differential equation:(61)u′=ue-t,t≠k,u(k+0)=12ku(k),(62)w′=2we-t,t≠k,w(k+0)=12kw(k).
The solutions of the impulsive differential equation (61) and (62), correspondingly, are equi-stable and uniform-integrally stable. Thus, according to Theorem 8 the system of impulsive differential equations with “supremum” (54) is (H0,h)-uniform-integrally φ0-stable.


## 4. Conclusion

This paper extends the notions ofφ0-stability in terms of two measures to integral φ0-stability in terms of two measures for impulsive differential equations with “supremum” and establishes a criterion on integral φ0-stability in terms of two measures for such system by using the cone-valued piecewise continuous Lyapunov functions, Razumikhin method and comparative method. Finally, an example is given to illustrate our result.



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*Source: 901853-2014-10-16.xml*