# Stability in Terms of Two Measures for Nonlinear Impulsive Systems on Time Scales

Authors: Kexue Zhang; Xinzhi Liu
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We investigate some stability problems in terms of two measures for nonlinear dynamic systems on time scales with fixed moments of impulsive effects. Sufficient conditions for (uniform) stability, (uniform) asymptotic stability, and instability in terms of two measures are derived by using the method of Lyapunov functions. Our results include the existing results as special cases when the time scale reduces to the set of real numbers. Particularly, our results provide stability criteria for impulsive discrete systems in terms of two measures, which have not been investigated extensively. Two examples are presented to illustrate the efficiency of the proposed results.


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## Body

## 1. Introduction

It is well known that the theory of impulsive differential equations provides a general framework for mathematical modeling of many real world phenomena [1, 2]. In particular, it serves as an adequate mathematical tool for studying evolution processes that are subjected to abrupt changes in their states. At the present time, the qualitative theory of such equations has been extensively studied. Many results on the stability and boundedness of their solutions are obtained [1–4]. Due to the needs of applications, the concepts of Lyapunov stability have given rise to many new notions, for example, partial stability, conditional stability, eventual stability, practical stability, and so on. A notion which unifies and includes the above concepts of stability is the notion of stability in terms of two measures which was initialed by Movchan [5]. Since the publication of Salvadori's paper [6], this unified theory in terms of two measures became popular. For a systematic introduction to the theory of stability in terms of two measures, refer to [7].On the other hand, a theory of time scales or calculus on measure chains was introduced by Hilger in his Ph.D. thesis [8] in 1988, with the purpose of incorporating both the existing theory of dynamic systems on continuous and discrete time scales, namely, time scale as arbitrary closed subset of real numbers, and extending the existing theory to dynamic systems on generalized hybrid (continuous/discrete) time scales. The theory of time scales recently has gained much attention and is undergoing rapid development. Recently, various work has been done on the stability problem of dynamic systems on time scales [9–14]. For more details about the theory of time scales, refer to [15–17].Motivated by the above discussion, in this paper, we will consider the stability problems in terms of two measures for impulsive systems on time scales. Several new stability criteria and instability criteria are obtained by using the method of Lyapunov functions. As far as we know, there are very few studies on stability analysis of impulsive discrete systems in terms of two measures. Moreover, our results can be applied to impulsive systems on other time scales in addition to the set of integers and the set of real numbers.The rest of this paper is organized as follows. In Section2, we introduce some basic knowledge of dynamic systems on time scales. In Section 3, we formulate the problem and present several definitions of stability and instability in terms of two measures. In Section 4, several (h0,h)-stability and instability criteria are established by employing the Lyapunov function approach. For illustration of our results, two examples are shown in Section 5. Finally, some conclusions are drawn in Section 6.


## 2. Preliminaries

In this section, we briefly introduce some basic definitions and results concerning time scales for later use.Letℝ be the set of real numbers, ℝ+ be the set of nonnegative real numbers, ℤ be the set of integers, ℤ+ be the set of nonnegative integers, ℕ={1,2,…}, and 𝕋 be an arbitrary nonempty closed subset of ℝ. We assume that 𝕋 is a topological space with relative topology induced from ℝ. Then, 𝕋 is called a time scale.Definition 1.
The mappingsσ,θ:𝕋→𝕋 defined as
(1)σ(t)=inf{s∈𝕋:s>t},θ(t)=sup{s∈𝕋:s<t}
are called forward and backward jump operators, respectively.A nonmaximal elementt∈𝕋 is called right-scattered (rs) if σ(t)>t and right-dense (rd) if σ(t)=t. A nonminimal element t∈𝕋 is called left-scattered (ls) if θ(t)<t and left-dense (ld) if θ(t)=t. If 𝕋 has a ls maximum m, then we define 𝕋k=𝕋∖{m}, otherwise, 𝕋k=𝕋.Definition 2.
The graininess functionμ:𝕋→ℝ+ is defined by
(2)μ(t)=σ(t)-t.Definition 3.
Fory:𝕋→ℝ and t∈𝕋k, one defines the delta derivative yΔ(t) of y(t), to be the number (when it exists) with the property that for any ɛ>0, there is a neighborhood U of t (i.e., U=(t-δ,t+δ)⋂‍𝕋 for some δ>0) such that
(3)|y(σ(t))-y(s)-yΔ(t)(σ(t)-s)|≤ɛ|σ(t)-s|,∀s∈U.A functionf:𝕋→ℝ is rd-continuous if it is continuous at rd points in 𝕋 and its left-side limits exist at ld points in 𝕋. The set of rd-continuous functions f:𝕋→ℝ will be denoted by Crd=Crd(𝕋,ℝ). If f is continuous at each rd point and each ld point, f is said to be continuous function on 𝕋. If a,b∈𝕋, then one defines the interval [a,b] on 𝕋 by [a,b]:={t∈𝕋:a≤t≤b}. Open intervals and half-open intervals can be defined similarly.Definition 4.
Letf∈Crd. A function g:𝕋→ℝ is called the antiderivative of f on 𝕋 if it is differentiable on 𝕋 and satisfies gΔ(t)=f(t) for all t∈𝕋. In this case, one defines
(4)∫at‍f(s)Δs=g(t)-g(a),a,t∈𝕋.One  says that a functionp:𝕋→ℝ is regressive if 1+μ(t)p(t)≠0 for all t∈𝕋. The set of all regressive and rd-continuous functions f:𝕋→ℝ is denoted by Crdℛ=Crdℛ(𝕋,ℝ), and the set of all positively regressive elements of Crdℛ is denoted by Crdℛ+=Crdℛ+(𝕋,ℝ)={p∈Crdℛ:1+μ(t)p(t)>0 for all t∈𝕋}.Definition 5.
Ifp∈Crdℛ, then one defines the exponential function on time scale 𝕋 by
(5)ep(t,s)=exp(∫st‍ξμ(τ)(p(τ))Δτ),fort,s∈𝕋,
where the cylinder transformation
(6)ξh(z)={Log(1+hz)h,h≠0,z,h=0,
where Log is the principal logarithm function.It is known thatx(t)=ep(t,t0) is the unique solution of the initial value problem xΔ(t)=p(t)x(t), x(t0)=1.Remark 6.
Letα∈Crdℛ be a constant. If 𝕋=ℤ, then eα(t,t0)=(1+α)t-t0 for all t∈𝕋. If 𝕋=ℝ, then eα(t,t0)=eα(t-t0) for all t∈𝕋.Definition 7.
One says that a functionm:𝕋→ℝ is right-nondecreasing at a point t∈𝕋 provided (i)
ift is rs, then m(σ(t))≥m(t);(ii)
ift is rd, then there is a neighborhood U of t such that
(7)m(s)≥m(t),∀s∈Uwiths>t.
Similarly, one says thatm is right-nonincreasing if above in (i) m(σ(t))≤m(t) and in (ii) m(s)≤m(t). If m is right-nondecreasing (right-nonincreasing) at every t∈𝕋, one says that m is right-nondecreasing (right-nonincreasing) on 𝕋.Lemma 8.
Letm∈C(𝕋,ℝ). Then m(t) is right-nondecreasing (right-nonincreasing) on 𝕋 if and only if D+mΔ(t)≥0(D+mΔ(t)≤0) for every t∈𝕋, where
(8)D+mΔ(t)={m(σ(t))-m(t)μ(t),σ(t)>t,limsups→t+m(s)-m(t)s-t,σ(t)=t.Proof.
The condition is obviously necessary. Let us prove that it is sufficient. We only assumeD+mΔ(t)≥0 for t∈𝕋 as the second statement can be shown similarly.
Ift is rs, then
(9)D+mΔ(t)=m(σ(t))-m(t)μ(t)≥0,
and hence m(σ(t))≥m(t).
Let nowt to be rd, and N be a neighborhood of t. We need to show that m(s)≥m(t) for s>t with s∈N. This follows directly from Lemma 1.1.1 in [7].
Thus the proof of the lemma is complete.


## 3. Problem Formulation

Consider the following nonlinear impulsive system on time scale𝕋:
(10)xΔ(t)=f(t,x),t≠tk,t∈𝕋,Δx(t)=Ik(x(t)),t=tk,k∈ℕ,x(t0+)=x0
under the following assumptions.(a)
𝕋 is a time scale with t0≥0 as minimal element and no maximal element.(b)
{tk}∈𝕋,  t0<t1<t2<⋯<tk<⋯ and limk→∞tk=∞.(c)
x∈ℝn and Δx(tk)=x(tk+)-x(tk-). If tk is rd point, x(tk+) denotes the right limit of x at tk; if tk is rs point, x(tk+) denotes the state of x at tk with the impulse. If tk is ld point, x(tk-) denotes the left limit of x at tk with x(tk-)=x(tk) if tk is ls point. Here, we assume that x(tk-)=x(tk).(d)
f:𝕋×ℝn→ℝn is continuous in (tk-1,tk]×ℝn for k∈ℕ, f(t,0)=0, and for each x∈ℝn, k∈ℕ, lim(t,y)→(tk+,x)f(t,y)=f(tk+,x);(e)
Ik:ℝn→ℝn and Ik(0)=0.Throughout this paper, we denote byx(t)=x(t;t0,x0) the solution of system (10) satisfying initial condition x(t0+)=x0. Obviously, system (10) admits the trivial solution. Moreover, f is assumed to satisfy necessary assumptions so that the following initial value problems:
(11)xΔ=f(t,x),t∈[t0,t1],x(t0)=x0,xΔ=f(t,x),t∈(tk,tk+1],x(tk+)=x(tk)+Ik(x(tk)),
have unique solutions x0(t), t∈[t0,t1], and xk(t), t∈(tk,tk+1], k∈ℕ, respectively (e.g., see [17] for existence and uniqueness results for dynamical systems on time scales.). Thus, if we define
(12)x(t;t0,x0)={x0(t),t∈[t0,t1]x1(t),t∈(t1,t2]⋮⋮xk(t),t∈(tk,tk+1]⋮⋮
then it is easy to see that x(t;t0,x0) is the unique solution of system (10).Let us list the classes of functions and definitions for convenience.PC={δ:𝕋→ℝ+,
continuous on(tk-1,tk] and limt→tk+δ(t)=δ(tk+) exists for k∈ℕ};𝒦={δ∈C(ℝ+,ℝ+), strictly increasing and δ(0)=0};PC𝒦={δ:𝕋×ℝ+→ℝ+,δ(·,u)∈PC for each u∈ℝ+ and δ(t,·)∈𝒦 for each t∈𝕋};Γ={h:𝕋×ℝn→ℝ+,h(·,x)∈PC for each x∈ℝn,h(t,·)∈C(ℝn,ℝ+) for each t∈𝕋 and infh(t,x)=0};ν0={V:𝕋×ℝn→ℝ+, continuous on (tk-1,tk]×ℝn, k∈ℕ, and for all x∈ℝn and k∈ℕ, lim(t,y)→(tk+,x)V(t,y)=v(tk+,x) exists}.ForV∈ν0, (t,x)∈(tk-1,tk]×ℝn, k∈ℕ, we define the upper right-hand Dini delta derivative of V(t,x) relative to (10) as follows:
(13)D+VΔ(t,x)={V(σ(t),x(σ(t)))-V(t,x(t))μ(t),σ(t)>t,limsups→t+V(s,x(t)+(s-t)f(t,x(t)))-V(t,x(t))s-t,σ(t)=t.Definition 9.
Leth0,h∈Γ. Then one says that (i)
h0 is finer than h if there exists a constant ρ>0 and a function φ∈𝒦 such that h0(t,x)<ρ implies h(t,x)≤φ(h0(t,x));(ii)
h0 is weakly finer than h if there exists a constant ρ>0 and a function φ∈PC𝒦 such that h0(t,x)<ρ implies h(t,x)≤φ(t,h0(t,x)).Definition 10.
LetV∈ν0 and h0,h∈Γ. Then V(t,x) is said to be (i)
h-positive definite if there exist a ρ>0 and a function b∈𝒦 such that h(t,x)<ρ implies b(h(t,x))≤V(t,x);(ii)
h0-decrescent if there exist a ρ>0 and a function a∈𝒦 such that h0(t,x)<ρ implies V(t,x)≤a(h0(t,x));(iii)
h0-weakly decrescent if there exist a ρ>0 and a function a∈PC𝒦 such that h0(t,x)<ρ implies V(t,x)≤a(t,h0(t,x)).Definition 11.
The impulsive system (10) is said to be (S1)
(h0,h)-stable, if for each ɛ>0, t0∈𝕋, there exists a δ=δ(t0,ɛ)>0 such that h0(t0,x0)<δ implies h(t,x(t))<ɛ, t≥t0 for any solution x(t)=x(t;t0,x0) of (10);(S2)
(h0,h)-uniformly stable, if the δ in (S1) is independent of t0;(S3)
(h0,h)-attractive, if for each ɛ>0, t0∈𝕋, there exist two positive constants δ=δ(t0,ɛ) and T=T(t0,ɛ) such that h0(t0,x0)<δ implies h(t,x(t))<ɛ, t≥t0+T;(S4)
(h0,h)-uniformly attractive, if (S3) holds with δ and T being independent of t0;(S5)
(h0,h)-asymptotically stable, if (S1) and (S3) hold simultaneously;(S6)
(h0,h)-uniformly asymptotically stable, if (S2) and (S4) hold together;(S7)
(h0,h)-unstable, if (S1) fails to hold.


## 4. Main Results

Let us establish, in this section, sufficient conditions for(h0,h)-(uniform) stability, (h0,h)-(uniform) asymptotic stability, and (h0,h)-instability properties of impulsive systems (10) in the following subsections, respectively. Let
(14)S(h,ρ)={(t,x)∈𝕋×ℝn:h(t,x)<ρ}.


### 4.1.(h0,h)-(Uniform) Stability

Theorem 12.
Assume that(i)
h,h0∈Γ, and h0 is weakly finer than h;(ii)
V∈ν0, V(t,x) is h-positive definite on S(h,ρ), h0-weakly decrescent, locally Lipschtiz in x for t∈𝕋 which is rd, and
(15)D+VΔ(t,x)≤ckV(t,x),t∈(tk,tk+1),(t,x)∈S(h,ρ),whereck≥0, k∈ℤ+;(iii)
there existsς≥ςk>0, such that
(16)V(tk+,xk+Ik(xk))≤ςkV(tk,xk),(tk,xk)∈S(h,ρ),wherexk=x(tk), k∈ℕ;(iv)
supk∈ℤ+{∏i=0k‍ςieci(ti+1,ti)}=M<∞ where ς0=1;(v)
there exists a constantρ1, 0<ρ1<ρ, such that h(t,x)<ρ1 implies h(σ(t),x(σ(t)))<ρ;(vi)
there exists a constantρ0, 0<ρ0<ρ1, such that h(tk,xk)<ρ0 implies h(tk+,xk+Ik(xk))<ρ1.
Then system (10) is (h0,h)-stable.Proof.
SinceV(t,x) is h0-weakly decrescent, there exist a constant δ0>0 and a function a∈PC𝒦 such that
(17)V(t,x)≤a(t,h0(t,x)),ifh0(t,x)<δ0.
There exists, in view of (ii), a function b∈𝒦 such that
(18)b(h(t,x))≤V(t,x),ifh(t,x)<ρ.
By (i), there exist δ1>0 and φ∈PC𝒦 such that
(19)h(t,x)≤φ(t,h0(t,x)),ifh0(t,x)<δ1.
Letɛ∈(0,ρ0) and t0∈𝕋 be given. There exists δ2=δ2(t0,ɛ) such that
(20)φ(t0,δ2)<ρ0,max{1,M}a(t0,δ2)<b(ɛ).
Choose δ=min{δ0,δ1,δ2}. Let (t0,x0)∈𝕋×ℝn such that h0(t0,x0)<δ and x(t)=x(t;t0,x0) be any solution of (10). Then, from (17) to (20), we get
(21)b(h(t0,x0))≤V(t0,x0)≤a(t0,h0(t0,x0))<b(ɛ),
which implies h(t0,x0)<ɛ.
We now claim that, for every solutionx(t)=x(t;t0,x0) of (10), h0(t0,x0)<δ implies
(22)h(t,x(t))<ɛ,t≥t0.
If this is not true, then there exist a solution x(t) with h0(t0,x0)<δ and a t*>t0 such that tk<t*≤tk+1, for some k, satisfying
(23)h(t*,x(t*))≥ɛ,h(t,x(t))<ɛfort∈[t0,tk].
Since0<ɛ<ρ0, it follows form condition (vi) that
(24)h(tk+,xk+):=h(tk+,xk+Ik(xk))<ρ1,
where xk+=x(tk+) and h(tk,xk)<ɛ. Next, we will show that there exists a t0, tk<t0≤t*, such that
(25)ɛ≤h(t0,x(t0))<ρ,h(t,x(t))<ρfort∈[t0,t0].
To do this, we consider the following two cases:(1)
there exists at*, tk<t*≤t*, such that h(t*,x(t*))≥ρ;(2)
h(t,x(t))<ρ for all t∈(tk,t*].
Case  1. Let t¯=inf{t∈(tk,t*],h(t,x(t))≥ρ}. As h(tk+,xk+)<ρ1<ρ, we know that tk<t¯≤t*. If t¯ is left-dense, from the selection of t¯, we know that there exists a left-hand neighborhood Uϵ=(t¯-ϵ,t¯)⊂(tk,t*] for some ϵ>0, such that ρ1<h(t,x(t))<ρ for all t∈Uϵ. Then, we can choose t0∈Uϵ.
Ift¯ is left-scattered, from the selection of t¯ and h(tk+,xk+)<ρ1, we know that tk<θ(t¯)<t* and h(θ(t¯),x(θ(t¯)))<ρ. Here, we claim that h(θ(t¯),x(θ(t¯)))≥ρ1. If this is not true, that is, h(θ(t¯),x(θ(t¯)))<ρ1, form condition (v), we know that
(26)h(t¯,x(t¯))=h(σ(θ(t¯)),x(σ(θ(t¯))))<ρ,
which is a contradiction. Thus, ρ1≤h(θ(t¯),x(θ(t¯)))<ρ. Then, we can choose t0=θ(t¯).
Case  2. If h(t,x(t))<ρ for all t∈(tk,t*], then we can choose t0=t*.
Hence, we can find at0, tk<t0≤t*, such that (25) holds.
Fort0≤t≤t0 and by conditions (ii) and (iii), we obtain
(27)D+VΔ(t,x)≤ciV(t,x),t∈(ti,ti+1),t≤t0,(28)V(ti+,xi+Ii(xi))≤ςiV(ti,xi),t0<ti<t0.
By (27), we will show that
(29)V(t,x)≤ec0(t,t0)V(t0,x0),t∈[t0,t1].
To do this, we apply the induction principle ([17], Theorem 1.7) on [t0,t1] to the statement
(30)A(t):V(t,x)≤ec0(t,t0)V(t0,x0).
(1) The statementA(t0) is true since ec0(t0,t0)V(t0,x0)=V(t0,x0).
(2) Lett be rs and A(t) be true. We have to prove that A(σ(t)) is true.
By the definition of upper right-hand derivative, we see that(31)D+VΔ(t,x)=V(σ(t),x(σ(t)))-V(t,x(t))μ(t)≤c0V(t,x),
then
(32)V(σ(t),x(σ(t)))≤(1+c0μ(t))V(t,x)=ec0(σ(t),t)V(t,x)≤ec0(σ(t),t)ec0(t,t0)V(t0,x0)=ec0(σ(t),t0)V(t0,x0)
which implies that A(σ(t)) is true.
(3) Lett be rd, A(t) be true and N be a neighborhood of t. We need to show that A(s) is true for s>t, s∈N. By (27) and Remark 6, we get
(33)V(s,x(s))≤ec0(s-t)V(t,x)≤ec0(s-t)ec0(t,t0)V(t0,x0)=ec0(s,t)ec0(t,t0)V(t0,x0)=ec0(s,t0)V(t0,x0)
which implies that A(s) is true.
(4) Lett be ld and A(s) be true for all s<t. We need to show that A(t) is true. By the continuous property of function V and the exponential function, it follows that
(34)V(t,x)=lims→t-V(s,x(s))≤lims→t-ec0(s,t0)V(t0,x0)=ec0(t,t0)V(t0,x0)
which implies that A(t) is true.
Hence, we conclude that (29) is true.
Similarly, we can prove that(35)V(t,x)≤eci(t,ti)V(ti+,x(ti+)),fort∈(ti,ti+1],t≤t0.
Then, by (28), (35), and (20), we obtain
(36)V(t0,x(t0))≤eck(t0,tk)V(tk+,x(tk+))≤eck(t0,tk)ςkV(tk,x(tk))≤eck(t0,tk)ςkeck-1(tk,tk-1)V(tk-1+,x(tk-1+))≤⋯≤eck(t0,tk)ςkeck-1(tk,tk-1)ςk-1⋯ec1(t2,t1)×ς1ec0(t1,t0)V(t0,x0)≤∏i=0k‍ςieci(ti+1,ti)V(t0,x0)≤MV(t0,x0)≤Ma(t0,h0(t0,x0))≤Ma(t0,δ)<b(ɛ)
that is, V(t0,x(t0))<b(ɛ). Thus, by (18) and (25),
(37)b(ɛ)≤b(h(t0,x(t0)))≤V(t0,x(t0))<b(ɛ)
which is a contradiction. Therefore (22) is true and system (10) is (h0,h)-stable.Theorem 13.
Assume that all conditions of Theorem12 hold with the following changes: (i)*h0 is finer than h;(ii)*V(t,x) is h0-decrescent.
Then, system (10) is (h0,h)-uniformly stable.Proof.
From conditions (i)* and (ii)*, the numberδ in the proof of Theorem 12 can be chosen independent of t0. Then following the same reasoning of Theorem 12, we can get the (h0,h)-uniform stability of system (10). The details are omitted.Ifck≡0 in condition (ii) of the previous theorems, then the Lyapunov function V is monotone along the solutions of system (10) in each impulsive intervals. In this case, we have the following conservative result.Corollary 14.
Assume that(i)
h0,h∈Γ, V∈ν0, V(t,x) is h-positive definite on S(h,ρ), locally Lipschtiz in x for each t∈𝕋 which is rd, and D+VΔ(t,x)≤0 for t≠tk and (t,x)∈S(h,ρ);(ii)
V(tk+,xk+Ik(xk))-V(tk,xk)≤dkV(tk,xk) for (tk,xk)∈S(h,ρ), where dk≥0 and ∑i=1∞‍di<∞,
and conditions (v), (vi) of Theorem12 hold. Then (A)
if, in addition,h0 is weakly finer than h, and V(t,x) is h0-weakly decrescent, then system (10) is (h0,h)-stable;(B)
if, in addition,h0 is finer than h, and V(t,x) is h0-decrescent, then system (10) is (h0,h)-uniformly stable.Proof.
Notice(38)supk∈ℤ+{∏i=0k‍ςieci(ti+1,ti)}=supk∈ℤ+{∏i=0k‍(1+di)}≤∏i=1∞‍(1+di)<∞,
where d0=0. Then, by Theorems 12 and 13, the result holds.Remark 15.
The continuous version of Corollary14 with dk≡0, k∈ℕ, can be found in [7], while the discrete one for impulsive discrete systems is brand new, and the discrete version of Corollary 14 with V(tk+,xk+Ik(xk))=V(tk,x(tk)), k∈ℕ, reduces to Theorems 3.1 and 3.2 in [18] for discrete systems with no impulses.


### 4.2.(h0,h)-(Uniform) Asymptotic Stability

Theorem 16.
Assume that conditions (v), (vi) of Theorem12 and condition (i) of Corollary 14 hold and the following conditions are satisfied: (i)
h0 is weakly finer than h, and V(t,x) is h0-weakly decrescent;(ii)
V(tk+,xk+Ik(xk))-V(tk,xk)≤-λkψ(V(tk,xk)) for (tk,xk)∈S(h,ρ), where λk≥0, ∑k=1∞‍λk=∞, ψ:ℝ+→ℝ+, ψ(0)=0 and ψ(s)>0 if s>0.
Then system (10) is (h0,h)-asymptotically stable.Proof.
By Theorem12, system (10) is (h0,h)-stable. Thus, for ρ>0, there exists δ=δ(t0,ρ)>0 such that h0(t0,x0)<δ implies h(t,x)<ρ, t≥t0. To prove the theorem, it remains to show that limt→∞h(t,x(t))=0.
Letm(t)=V(t,x(t)). Then it follows from assumptions that m(t) is right-nonincreasing and bounded from below, and consequently limt→∞m(t)=ω≥0 exists. If ω>0 for some solution x(t)=x(t;t0,x0) of (10), we let γ=minω≤s≤m(t0)ψ(s). Then, by condition (ii), we have
(39)m(tk+)-m(tk)≤-λkψ(m(tk))≤-λkγ.
Thus we obtain from (39) that
(40)m(tk+)≤m(tk)-λkγ≤m(tk-1+)-λkγ≤m(tk-1)-λk-1γ-λkγ≤⋯≤m(t1)-γ∑j=1k‍λj,
which implies, in view of the assumption ∑j=1k‍λj=∞, that limk→∞m(tk+)=-∞. This is a contradiction. Thus we must have ω=0 and consequently limt→∞h(t,x(t))=0. Hence system (10) is (h0,h)-attractive and the proof is complete.In the following theorems, two auxiliary functions of classν0 are used to investigate the (h0,h)-asymptotic stability property of system (10).Theorem 17.
Let conditions(v), (vi) of Theorem 12 hold and assume that (i)
h0,h∈Γ and h0 is weakly finer than h;(ii)
there exists a functionV∈ν0 such that V(t,x) is locally Lipschtiz in x for each t∈𝕋 which is rd, h-positive definite on S(h,ρ), h0-weakly decrescent and
(41)D+VΔ(t,x)≤-c(W(t,x)),t≠tk,(t,x)∈S(h,ρ),wherec∈𝒦, W∈ν0;(iii)
V(tk+,xk+Ik(xk))-V(tk,xk)≤dkV(tk,xk), (tk,xk)∈S(h,ρ), where dk≥0 and ∑i=1∞‍di<∞;(iv)
W(t,x) is h-positive definite on S(h,ρ), locally Lipschtiz in x for every t∈𝕋 which is rd and
(42)D+WΔ(t,x)≤0,t≠tk,(t,x)∈S(h,ρ),W(tk+,xk+Ik(xk))-W(tk,xk)≤d¯kW(tk,xk),(tk,xk)∈S(h,ρ),whered¯k≥0 and ∑i=1∞‍d¯i<∞.
Then system (10) is (h0,h)-asymptotically stable.Proof.
From Corollary14, it follows that system (10) is (h0,h)-stable. Thus, for ρ>0, there exists δ=δ(t0,ρ)>0 such that h0(t0,x0)<δ implies h(t,x)<ρ, t≥t0. To prove the theorem, it remains to show that for every solution x(t) of (10) with h0(t0,x0)<δ, limt→∞h(t,x)=0.
Suppose that this is not true. Then there exists a sequence{ξi}i=1∞ diverging to ∞ as i→∞ and such that h(ξi,x(ξi))≥r(i∈ℕ) for some positive number r. From condition (iv), we know that there exists a function b∈𝒦 such that b(h(t,x))≤W(t,x), if h(t,x)<ρ. Then
(43)W(ξi,x(ξi))>b(r),i∈ℕ.
For any givent∈𝕋, there exists a i>0 such that t∈(ξi,ξi+1]. Then for ξi+1, there exists a k>0 such that ξi+1∈(tk,tk+1]. If tk<t≤tk+1, from (43) and condition (iv), we have
(44)W(t,x)≥W(ξi+1,x(ξi+1))>b(r)≥b(r)M0,
where M0=∏j=1∞‍(1+d¯j)<∞. If tk-1<t≤tk, we have
(45)W(t,x)≥W(tk,xk)≥11+d¯kW(tk+,x(tk+))>11+d¯kb(r)≥b(r)M0.
Following this procedure, we conclude that(46)W(t,x)>b(r)M0,t∈𝕋.
Let(47)L(t,x)=V(t,x)+∫tkt‍c(W(s,x(s)))Δs,t∈(tk,tk+1],k∈ℤ+.
Then, by condition (ii),
(48)D+LΔ(t,x)=D+VΔ(t,x)+c(W(t,x))≤0,t≠tk,
which implies
(49)V(t,x)≤V(tk+,x(tk+))-∫tkt‍c(W(s,x(s)))Δs,t∈(tk,tk+1],
for k∈ℤ+.
Hence, forξi+1∈(tk,tk+1], we obtain, form (46), (49), and condition (iii),
(50)V(ξi+1,x(ξi+1))≤V(tk+,x(tk+))-∫tkξi+1‍c(W(s,x(s)))Δs≤(1+dk)V(tk,x(tk))-∫tkξi+1‍c(W(s,x(s)))Δs≤(1+dk)(V(tk-1+,x(tk-1+))-∫tk-1tk‍c(W(s,x(s)))Δs)-∫tkξi+1‍c(W(s,x(s)))Δs≤(1+dk)(1+dk-1)V(tk-1,x(tk-1))-∫tk-1ξi+1‍c(W(s,x(s)))Δs≤⋯≤∏j=1k‍(1+dj)V(t0,x0)-∫t0ξi+1‍c(W(s,x(s)))Δs≤MV(t0,x0)-c(b(r)M0)(ξi+1-t0)→-∞,fori→∞,
where M=∏j=1∞‍(1+dj)<∞. This is a contradiction, hence limt→∞h(t,x)=0. Theorem 17 is proved.In Theorem17, the function may have a special form. In the case when W(t,x)=V(t,x) and dk=d¯k, k∈ℕ, we deduce the following corollary.Corollary 18.
Let conditions (v), (vi) of Theorem12 hold and assume that (i)
h0,h∈Γ and h0 is weakly finer than h;(ii)
there existc∈𝒦, and function V∈ν0 such that V(t,x) is locally Lipschtiz in x for each t∈𝕋 which is rd, h-positive definite on S(h,ρ), h0-weakly decrescent and
(51)D+VΔ(t,x)≤-c(V(t,x)),t≠tk,(t,x)∈S(h,ρ);(iii)
V(tk+,xk+Ik(xk))-V(tk,xk)≤dkV(tk,xk), (tk,xk)∈S(h,ρ), where dk≥0 and ∑i=1∞‍di<∞.
Then system (10) is (h0,h)-asymptotically stable.Theorem 19.
Assume that all conditions of Theorem13 hold. Suppose further, that there exists a function W∈ν0 such that W(t,x) is locally Lipschtiz in x for each t∈𝕋 which is rd, and the following conditions hold: (i)
D+WΔ(t,x)≤-p(t)c(h0(t,x))+q(t), t≠tk, (t,x)∈S(h,ρ), where c∈𝒦, p,q∈Crd(𝕋,ℝ+), ∫t0∞‍p(τ)Δτ=∞, and ∫t0∞‍q(τ)Δτ<∞;(ii)
W(tk+,xk+Ik(xk))-W(tk,xk)≤d¯kW(tk,xk), (tk,xk)∈S(h,ρ), where d¯k≥0 and ∑i=1∞‍d¯i<∞.
Then system (10) is (h0,h)-attractive.Proof.
By Theorem13, system (10) is (h0,h)-uniformly stable. Thus, for ρ>0, there exists a δ0=δ0(ρ)>0 such that h0(t0,x0)<δ0 implies that h(t,x)<ρ, t≥t0, where x(t)=x(t;t0,x0) is any solution of (10).
Letɛ∈(0,ρ) be given, δ=δ(ɛ)>0 be the same as defined in the definition of (h0,h)-uniform stability, and h0(t0,x0)<δ0. We claim that there exists a t*≥t0 such that
(52)h0(t*,x(t*))<δ.
If this is not true, then h0(t,x)≥δ for all t≥t0.
Let(53)L(t,x)=W(t,x)+∫tkt‍p(τ)c(h0(τ,x(τ)))Δτ-∫tkt‍q(τ)Δτ,t∈(tk,tk+1],
for k∈ℤ+. By condition (i), we obtain
(54)D+LΔ(t,x)=D+WΔ(t,x)+p(t)h0(t,x)-q(t)≤0,t≠tk,
which implies that, for t∈(tk,tk+1], k∈ℤ+,
(55)L(t,x)≤L(tk+,x(tk+)).
Then, it follows from (55) and condition (ii) that, for t∈(tk,tk+1],
(56)W(t,x)≤W(tk+,x(tk+))-∫tkt‍p(τ)c(h0(τ,x(τ)))Δτ+∫tkt‍q(τ)Δτ≤(1+d¯k)W(tk,x(tk))-∫tkt‍p(τ)c(h0(τ,x(τ)))Δτ+∫tkt‍q(τ)Δτ≤(1+d¯k)×(∫tk-1tk‍W(tk-1+,x(tk-1+))-∫tk-1tk‍p(τ)c(h0(τ,x(τ)))Δτ+∫tk-1tk‍q(τ)Δτ)-∫tkt‍p(τ)c(h0(τ,x(τ)))Δτ+∫tkt‍q(τ)Δτ≤(1+d¯k)(1+d¯k-1)W(tk-1,x(tk-1))-∫tk-1t‍p(τ)c(h0(τ,x(τ)))Δτ+(1+d¯k)∫tk-1t‍q(τ)Δτ≤⋯≤∏i=1k‍(1+d¯i)W(t0,x0)-∫t0t‍p(τ)c(h0(τ,x(τ)))Δτ+∏i=2k‍(1+d¯i)∫t0t‍q(τ)Δτ≤M0(W(t0,x0)+∫t0t‍q(τ)Δτ)-c(δ)∫t0t‍p(τ)Δτ,
where M0=∏i=1∞(1+d¯i)<∞. Then, (56) implies that W(t,x)→-∞, for t→∞. This contradiction shows that (52) is true, and hence,
(57)h(t,x)<ɛ,t≥t*.
Thus we conclude that system (10) is (h0,h)-attractive.Remark 20.
When𝕋=ℤ, and dk=d¯k=0, k∈ℕ, Theorem 19 contains Theorem  3.4 in [18] for discrete systems without impulse effects.
Next, we will give two results on uniform asymptotic stability in terms of two measures.Theorem 21.
Let all the conditions of Theorem19 and the following additional conditions hold: (i)
W(t,x) is h0-decrescent, and p(t)≡p  (p is a positive constant), for t∈𝕋;(ii)
there exists a constantτ>0 such that
(58){t+kτ:t∈𝕋,k∈ℕ}⊂𝕋.
Then system (10) is uniformly asymptotically stable.Proof.
SinceW(t,x) is h0-decrescent, there exist δ1>0 and a function b∈𝒦 such that
(59)W(t,x)≤b(h0(t,x)),ifh0(t,x)<δ1.
By Theorem 19, system (10) is (h0,h)-uniformly stable. Thus, there exists a δ0=δ0(ρ)∈(0,δ1) such that h0(t0,x0)<δ0 implies h(t,x)<ρ, t≥t0, for any solution x(t)=x(t;t0,x0) of (10).
Letɛ∈(0,ρ) be given and δ=δ(ɛ)>0 be the same as defined in the definition of (h0,h)-uniformly stability. Let m>0 be the smallest integer such that
(60)m>M0(b(δ0)+N)pτc(δ),
where M0=∏k=1∞‍(1+d¯k)<∞ and N=∫t0∞‍q(s)Δs<∞.
ChooseT=mτ and let x(t)=x(t;t0,x0) be any solution of (10) with h0(t0,x0)<δ0. We claim that there exists a t*∈[t0,t0+T] such that h0(t*,x(t*))<δ. If this is not true, then h0(t,x)≥δ for all t*∈[t0,t0+T]. By (56), (60), and condition (ii), we have
(61)W(t0+T,x(t0+T))≤M0W(t0,x0)-pc(δ)T+M0∫t0t0+T‍q(s)Δs≤M0b(δ0)-mc(δ)pτ+M0N<0
which is a contradiction. Thus, our claim is true and by the uniform stability we have
(62)h(t,x)<ɛ,t≥t0+T≥t*.
Hence, system (10) is (h0,h)-uniformly attractive. This completes the proof.Theorem 22.
Let conditions (v), (vi) of Theorem12 hold and assume that (i)
h0,h∈Γ and h0 is finer than h;(ii)
there exist ac∈𝒦, and function V∈ν0 such that V(t,x) is locally Lipschtiz in x for each t∈𝕋 which is rd, h-positive definite on S(h,ρ), h0-decrescent and
(63)D+VΔ(t,x)≤-c(h0(t,x)),t≠tk,(t,x)∈S(h,ρ);(iii)
V(tk+,xk+Ik(xk))-V(tk,xk)≤dkV(tk,xk), (tk,xk)∈S(h,ρ), where dk≥0 and ∑i=1∞‍di<∞.
Then system (10) is (h0,h)-uniformly asymptotically stable.Proof.
SinceV(t,x) is h0-decrescent, there exist a constant δ0 and a function a∈𝒦 such that
(64)V(t,x)≤a(h0(t,x)),ifh0(t,x)<δ0.
The fact that V(t,x) is h-positive definite on S(h,ρ) implies that there exists a function b∈𝒦 such that
(65)b(h(t,x))≤V(t,x),ifh(t,x)<ρ.
It follows from Corollary14 that system (10) is (h0,h)-uniformly stable. Thus for ρ>0, there exist a δ1=δ1(ρ)∈(0,δ0) such that h0(t0,x0)<δ1 implies
(66)h(t,x)<ρ,t≥t0.
By the choice of δ1, we get
(67)V(t0,x0)≤a(h0(t0,x0))≤a(δ1).
To prove the theorem, it is enough to show that system (10) is (h0,h)-uniformly attractive.
Given0<ɛ<ρ, let δ=δ(ɛ)>0 be the same as defined in the definition of (h0,h)-uniformly stability. Then for any solution x(t)=x(t;t0,x0) of system (10) with h0(t0,x0)<δ1, we claim that there exists a T=T(ɛ)>Ma(δ1)/c(δ) such that, for some t*∈[t0,t0+T],
(68)h0(t*,x(t*))<δ,
where M=∏k=1∞‍(1+dk)<∞. Suppose that this is false. Then for any T>Ma(δ1)/c(δ) there exists a solution x(t)=x(t;t0,x0) of (10) satisfying h0(t0,x0)<δ1, such that
(69)h0(t,x(t))≥δ,t∈[t0,t0+T].
By setting(70)L(t,x)=V(t,x)+∫tkt‍c(h0(s,x(s)))Δs,t∈(tk,tk+1],k∈ℤ+,
and condition (ii), we have
(71)D+LΔ(t,x)=D+VΔ(t,x)+c(h0(t,x))≤0,t≠tk,
which implies L(t,x)≤L(tk+,x(tk+)), for t∈(tk,tk+1]. Then, for t∈(tk,tk+1], we get
(72)V(t,x)≤V(tk+,x(tk+))-∫tkt‍c0(s)Δs≤(1+dk)V(tk,x(tk))-∫tkt‍c0(s)Δs≤(1+dk)(V(tk-1+,x(tk-1+))-∫tk-1tk‍c0(s)Δs)-∫tkt‍c0(s)Δs≤(1+dk)(1+dk-1)V(tk-1+,x(tk-1+))-∫tk-1t‍c0(s)Δs≤V(t0,x0)∏i=1k‍(1+di)-∫t0t‍c0(s)Δs≤MV(t0,x0)-∫t0t‍c0(s)Δs,
where c0(t):=c(h0(t,x)). Hence, for t=t0+T, we obtain
(73)V(t,x(t))|t=t0+T≤MV(t0,x0)-∫t0t0+T‍c(h0(s,x(s)))Δs≤Ma(δ1)-c(δ)T<0
which is a contradiction. Hence there exists a number t*∈(t0,t0+T] such that h0(t*,x(t*))<δ. Then for t≥t*, thus for t≥t0+T as well, we have
(74)h(t,x)<ɛ,
that is, h(t,x)<ɛ holds for t≥t0+T which means that system (10) is uniformly attractive. Theorem 22 is proved.


### 4.3.(h0,h)-Instability

Theorem 23.
Assume that(i)
h0,h∈Γ, V∈ν0, V(t,x) is locally Lipschtiz in x for each t∈𝕋 which is rd, h-positive definite on S(h,ρ), and
(75)D+VΔ(t,x)≥0,t≠tk,(t,x)∈S(h,ρ),and for anys≥t0 and α>0, there is β>0 such that V(t,x)≥α for t≥s implies h(t,x)≥β for t≥s;(ii)
for(tk,xk)∈S(h,ρ), k∈ℕ,
(76)V(tk+,x(tk+))-V(tk,xk)≥λkψ(V(tk,xk)),whereλk≥0 with ∑k=1∞‍λk=∞, ψ:ℝ+→ℝ+ is nondecreasing and ψ(0)=0, ψ(s)>0 for s>0.
Then system (10) is (h0,h)-unstable.Proof.
Let us assume on the contrary that system (10) is (h0,h)-stable. Then for 0<ɛ<ρ, there exists a δ=δ(t0,ɛ)>0 such that h0(t0,x0)<δ implies that h(t,x)<ɛ for t≥t0.
By settingL(t)=V(t,x), we know from condition (i) and (ii) that L(t) is right-nondecreasing for t≥t0. Then, L(t)≥L(t0), if t≥t0. Thus, it follows from condition (i) that there exists a β0=β0(L(t0))>0 such that h(t,x)≥β0 for t≥t0. Since V(t,x) is h-positive definite on S(h,ρ), there exists b∈𝒦 such that b(h(t,x))≤V(t,x), if h(t,x)<ρ. Then, we have
(77)L(tk)≥b(h(tk,xk))≥b(β0),k∈ℕ.
From condition (ii) and (77), we have
(78)L(tk+)-L(tk-1+)≥L(tk+)-L(tk)≥λkψ(L(tk))≥λkψ(b(β0)),
and so
(79)L(tk+)≥L(tk-1+)+λkψ(b(β0))≥L(tk-2+)+(λk-1+λk)ψ(b(β0))≥⋯≥L(t1)+ψ(b(β0))∑i=1k‍λi→∞,ask→∞,
which implies that, for a given number M≫0, there exists a s≥t0 such that L(t)≥M for t≥s. Thus by condition (i), there is a β=β(M)>0 such that h(t,x)≥β for t≥s which is a contradiction to the (h0,h)-stability. Therefore, system (10) is (h0,h)-unstable.Theorem 24.
Assume that(i)
h0,h∈Γ, V∈ν0, V(t,x) is locally Lipschtiz in x for each t∈𝕋 which is rd, h-decrescent on S(h,ρ), and
(80)-g(t)c(V(t,x))≤D+VΔ(t,x)≤0,t≠tk,(t,x)∈S(h,ρ),whereg∈Crd(𝕋,ℝ+) and c∈𝒦;(ii)
there existψk:ℝ+→ℝ+, k∈ℕ, such that ψk(s)>s, ψk(s)-s≥ψk(t)-t for t≥s≥0, and
(81)V(tk+,xk+Ik(xk))≥ψk(V(tk,xk)),(tk,xk)∈S(h,ρ),k∈ℕ;(iii)
for anyu>0,
(82)-∫tktk+1‍g(τ)Δτ+ψk(u)-uc(u)≥rk,k∈ℕ,whererk≥0 and ∑i=1∞‍rk=∞.
Then system (10) is (h0,h)-unstable.Proof.
For the sake of contradiction, we assume that system (10) is (h0,h)-stable. Then for 0<ɛ<ρ, there exists a δ=δ(t0,ɛ)>0 such that h0(t0,x0)<δ implies h(t,x)<ɛ for t≥t0. Since V(t,x) is h-decrescent on S(h,ρ), there exists a function a∈𝒦 such that
(83)V(t,x)≤a(h(t,x))<a(ρ),ifh(t,x)<ρ.
Letm(t)=V(t,x) and L(t)=V(t,x)+∫tkt‍g(τ)c(V(τ,x(τ)))Δτ, for t∈(tk,tk+1], k∈ℤ+. Then it follows from condition (i) that
(84)D+LΔ(t)=D+VΔ(t,x)+g(t)c(V(t,x))≥0,t≠tk,
which implies
(85)L(t)≥L(tk+),t∈(tk,tk+1].
Since D+VΔ(t,x)≤0, it follows from (85), that
(86)m(tk+1)-m(tk+)≥-∫tktk+1‍g(τ)c(m(τ))Δτ≥-c(m(tk+))∫tktk+1‍g(τ)Δτ.
From condition (ii), we have
(87)m(tk+)-m(tk)≥ψk(m(tk))-m(tk)≥ψk(m(tk+))-m(tk+)
which, together with (86) and condition (iii), yields
(88)m(tk+1)-m(tk)=m(tk+1)-m(tk+)+m(tk+)-m(tk)≥ψk(m(tk+))-m(tk+)-c(m(tk+))∫tktk+1‍g(τ)Δτ≥c(m(tk+))rk≥c(m(tk))rk≥c(m(t1))rk,k∈ℕ.
Thus,
(89)m(tk+1)≥m(t1)+c(m(t1))∑i=1k‍ri→∞,ask→∞,
which contradicts (83). Therefore, system (10) is (h0,h)-unstable and the proof is complete.


## 4.1.(h0,h)-(Uniform) Stability

Theorem 12.
Assume that(i)
h,h0∈Γ, and h0 is weakly finer than h;(ii)
V∈ν0, V(t,x) is h-positive definite on S(h,ρ), h0-weakly decrescent, locally Lipschtiz in x for t∈𝕋 which is rd, and
(15)D+VΔ(t,x)≤ckV(t,x),t∈(tk,tk+1),(t,x)∈S(h,ρ),whereck≥0, k∈ℤ+;(iii)
there existsς≥ςk>0, such that
(16)V(tk+,xk+Ik(xk))≤ςkV(tk,xk),(tk,xk)∈S(h,ρ),wherexk=x(tk), k∈ℕ;(iv)
supk∈ℤ+{∏i=0k‍ςieci(ti+1,ti)}=M<∞ where ς0=1;(v)
there exists a constantρ1, 0<ρ1<ρ, such that h(t,x)<ρ1 implies h(σ(t),x(σ(t)))<ρ;(vi)
there exists a constantρ0, 0<ρ0<ρ1, such that h(tk,xk)<ρ0 implies h(tk+,xk+Ik(xk))<ρ1.
Then system (10) is (h0,h)-stable.Proof.
SinceV(t,x) is h0-weakly decrescent, there exist a constant δ0>0 and a function a∈PC𝒦 such that
(17)V(t,x)≤a(t,h0(t,x)),ifh0(t,x)<δ0.
There exists, in view of (ii), a function b∈𝒦 such that
(18)b(h(t,x))≤V(t,x),ifh(t,x)<ρ.
By (i), there exist δ1>0 and φ∈PC𝒦 such that
(19)h(t,x)≤φ(t,h0(t,x)),ifh0(t,x)<δ1.
Letɛ∈(0,ρ0) and t0∈𝕋 be given. There exists δ2=δ2(t0,ɛ) such that
(20)φ(t0,δ2)<ρ0,max{1,M}a(t0,δ2)<b(ɛ).
Choose δ=min{δ0,δ1,δ2}. Let (t0,x0)∈𝕋×ℝn such that h0(t0,x0)<δ and x(t)=x(t;t0,x0) be any solution of (10). Then, from (17) to (20), we get
(21)b(h(t0,x0))≤V(t0,x0)≤a(t0,h0(t0,x0))<b(ɛ),
which implies h(t0,x0)<ɛ.
We now claim that, for every solutionx(t)=x(t;t0,x0) of (10), h0(t0,x0)<δ implies
(22)h(t,x(t))<ɛ,t≥t0.
If this is not true, then there exist a solution x(t) with h0(t0,x0)<δ and a t*>t0 such that tk<t*≤tk+1, for some k, satisfying
(23)h(t*,x(t*))≥ɛ,h(t,x(t))<ɛfort∈[t0,tk].
Since0<ɛ<ρ0, it follows form condition (vi) that
(24)h(tk+,xk+):=h(tk+,xk+Ik(xk))<ρ1,
where xk+=x(tk+) and h(tk,xk)<ɛ. Next, we will show that there exists a t0, tk<t0≤t*, such that
(25)ɛ≤h(t0,x(t0))<ρ,h(t,x(t))<ρfort∈[t0,t0].
To do this, we consider the following two cases:(1)
there exists at*, tk<t*≤t*, such that h(t*,x(t*))≥ρ;(2)
h(t,x(t))<ρ for all t∈(tk,t*].
Case  1. Let t¯=inf{t∈(tk,t*],h(t,x(t))≥ρ}. As h(tk+,xk+)<ρ1<ρ, we know that tk<t¯≤t*. If t¯ is left-dense, from the selection of t¯, we know that there exists a left-hand neighborhood Uϵ=(t¯-ϵ,t¯)⊂(tk,t*] for some ϵ>0, such that ρ1<h(t,x(t))<ρ for all t∈Uϵ. Then, we can choose t0∈Uϵ.
Ift¯ is left-scattered, from the selection of t¯ and h(tk+,xk+)<ρ1, we know that tk<θ(t¯)<t* and h(θ(t¯),x(θ(t¯)))<ρ. Here, we claim that h(θ(t¯),x(θ(t¯)))≥ρ1. If this is not true, that is, h(θ(t¯),x(θ(t¯)))<ρ1, form condition (v), we know that
(26)h(t¯,x(t¯))=h(σ(θ(t¯)),x(σ(θ(t¯))))<ρ,
which is a contradiction. Thus, ρ1≤h(θ(t¯),x(θ(t¯)))<ρ. Then, we can choose t0=θ(t¯).
Case  2. If h(t,x(t))<ρ for all t∈(tk,t*], then we can choose t0=t*.
Hence, we can find at0, tk<t0≤t*, such that (25) holds.
Fort0≤t≤t0 and by conditions (ii) and (iii), we obtain
(27)D+VΔ(t,x)≤ciV(t,x),t∈(ti,ti+1),t≤t0,(28)V(ti+,xi+Ii(xi))≤ςiV(ti,xi),t0<ti<t0.
By (27), we will show that
(29)V(t,x)≤ec0(t,t0)V(t0,x0),t∈[t0,t1].
To do this, we apply the induction principle ([17], Theorem 1.7) on [t0,t1] to the statement
(30)A(t):V(t,x)≤ec0(t,t0)V(t0,x0).
(1) The statementA(t0) is true since ec0(t0,t0)V(t0,x0)=V(t0,x0).
(2) Lett be rs and A(t) be true. We have to prove that A(σ(t)) is true.
By the definition of upper right-hand derivative, we see that(31)D+VΔ(t,x)=V(σ(t),x(σ(t)))-V(t,x(t))μ(t)≤c0V(t,x),
then
(32)V(σ(t),x(σ(t)))≤(1+c0μ(t))V(t,x)=ec0(σ(t),t)V(t,x)≤ec0(σ(t),t)ec0(t,t0)V(t0,x0)=ec0(σ(t),t0)V(t0,x0)
which implies that A(σ(t)) is true.
(3) Lett be rd, A(t) be true and N be a neighborhood of t. We need to show that A(s) is true for s>t, s∈N. By (27) and Remark 6, we get
(33)V(s,x(s))≤ec0(s-t)V(t,x)≤ec0(s-t)ec0(t,t0)V(t0,x0)=ec0(s,t)ec0(t,t0)V(t0,x0)=ec0(s,t0)V(t0,x0)
which implies that A(s) is true.
(4) Lett be ld and A(s) be true for all s<t. We need to show that A(t) is true. By the continuous property of function V and the exponential function, it follows that
(34)V(t,x)=lims→t-V(s,x(s))≤lims→t-ec0(s,t0)V(t0,x0)=ec0(t,t0)V(t0,x0)
which implies that A(t) is true.
Hence, we conclude that (29) is true.
Similarly, we can prove that(35)V(t,x)≤eci(t,ti)V(ti+,x(ti+)),fort∈(ti,ti+1],t≤t0.
Then, by (28), (35), and (20), we obtain
(36)V(t0,x(t0))≤eck(t0,tk)V(tk+,x(tk+))≤eck(t0,tk)ςkV(tk,x(tk))≤eck(t0,tk)ςkeck-1(tk,tk-1)V(tk-1+,x(tk-1+))≤⋯≤eck(t0,tk)ςkeck-1(tk,tk-1)ςk-1⋯ec1(t2,t1)×ς1ec0(t1,t0)V(t0,x0)≤∏i=0k‍ςieci(ti+1,ti)V(t0,x0)≤MV(t0,x0)≤Ma(t0,h0(t0,x0))≤Ma(t0,δ)<b(ɛ)
that is, V(t0,x(t0))<b(ɛ). Thus, by (18) and (25),
(37)b(ɛ)≤b(h(t0,x(t0)))≤V(t0,x(t0))<b(ɛ)
which is a contradiction. Therefore (22) is true and system (10) is (h0,h)-stable.Theorem 13.
Assume that all conditions of Theorem12 hold with the following changes: (i)*h0 is finer than h;(ii)*V(t,x) is h0-decrescent.
Then, system (10) is (h0,h)-uniformly stable.Proof.
From conditions (i)* and (ii)*, the numberδ in the proof of Theorem 12 can be chosen independent of t0. Then following the same reasoning of Theorem 12, we can get the (h0,h)-uniform stability of system (10). The details are omitted.Ifck≡0 in condition (ii) of the previous theorems, then the Lyapunov function V is monotone along the solutions of system (10) in each impulsive intervals. In this case, we have the following conservative result.Corollary 14.
Assume that(i)
h0,h∈Γ, V∈ν0, V(t,x) is h-positive definite on S(h,ρ), locally Lipschtiz in x for each t∈𝕋 which is rd, and D+VΔ(t,x)≤0 for t≠tk and (t,x)∈S(h,ρ);(ii)
V(tk+,xk+Ik(xk))-V(tk,xk)≤dkV(tk,xk) for (tk,xk)∈S(h,ρ), where dk≥0 and ∑i=1∞‍di<∞,
and conditions (v), (vi) of Theorem12 hold. Then (A)
if, in addition,h0 is weakly finer than h, and V(t,x) is h0-weakly decrescent, then system (10) is (h0,h)-stable;(B)
if, in addition,h0 is finer than h, and V(t,x) is h0-decrescent, then system (10) is (h0,h)-uniformly stable.Proof.
Notice(38)supk∈ℤ+{∏i=0k‍ςieci(ti+1,ti)}=supk∈ℤ+{∏i=0k‍(1+di)}≤∏i=1∞‍(1+di)<∞,
where d0=0. Then, by Theorems 12 and 13, the result holds.Remark 15.
The continuous version of Corollary14 with dk≡0, k∈ℕ, can be found in [7], while the discrete one for impulsive discrete systems is brand new, and the discrete version of Corollary 14 with V(tk+,xk+Ik(xk))=V(tk,x(tk)), k∈ℕ, reduces to Theorems 3.1 and 3.2 in [18] for discrete systems with no impulses.


## 4.2.(h0,h)-(Uniform) Asymptotic Stability

Theorem 16.
Assume that conditions (v), (vi) of Theorem12 and condition (i) of Corollary 14 hold and the following conditions are satisfied: (i)
h0 is weakly finer than h, and V(t,x) is h0-weakly decrescent;(ii)
V(tk+,xk+Ik(xk))-V(tk,xk)≤-λkψ(V(tk,xk)) for (tk,xk)∈S(h,ρ), where λk≥0, ∑k=1∞‍λk=∞, ψ:ℝ+→ℝ+, ψ(0)=0 and ψ(s)>0 if s>0.
Then system (10) is (h0,h)-asymptotically stable.Proof.
By Theorem12, system (10) is (h0,h)-stable. Thus, for ρ>0, there exists δ=δ(t0,ρ)>0 such that h0(t0,x0)<δ implies h(t,x)<ρ, t≥t0. To prove the theorem, it remains to show that limt→∞h(t,x(t))=0.
Letm(t)=V(t,x(t)). Then it follows from assumptions that m(t) is right-nonincreasing and bounded from below, and consequently limt→∞m(t)=ω≥0 exists. If ω>0 for some solution x(t)=x(t;t0,x0) of (10), we let γ=minω≤s≤m(t0)ψ(s). Then, by condition (ii), we have
(39)m(tk+)-m(tk)≤-λkψ(m(tk))≤-λkγ.
Thus we obtain from (39) that
(40)m(tk+)≤m(tk)-λkγ≤m(tk-1+)-λkγ≤m(tk-1)-λk-1γ-λkγ≤⋯≤m(t1)-γ∑j=1k‍λj,
which implies, in view of the assumption ∑j=1k‍λj=∞, that limk→∞m(tk+)=-∞. This is a contradiction. Thus we must have ω=0 and consequently limt→∞h(t,x(t))=0. Hence system (10) is (h0,h)-attractive and the proof is complete.In the following theorems, two auxiliary functions of classν0 are used to investigate the (h0,h)-asymptotic stability property of system (10).Theorem 17.
Let conditions(v), (vi) of Theorem 12 hold and assume that (i)
h0,h∈Γ and h0 is weakly finer than h;(ii)
there exists a functionV∈ν0 such that V(t,x) is locally Lipschtiz in x for each t∈𝕋 which is rd, h-positive definite on S(h,ρ), h0-weakly decrescent and
(41)D+VΔ(t,x)≤-c(W(t,x)),t≠tk,(t,x)∈S(h,ρ),wherec∈𝒦, W∈ν0;(iii)
V(tk+,xk+Ik(xk))-V(tk,xk)≤dkV(tk,xk), (tk,xk)∈S(h,ρ), where dk≥0 and ∑i=1∞‍di<∞;(iv)
W(t,x) is h-positive definite on S(h,ρ), locally Lipschtiz in x for every t∈𝕋 which is rd and
(42)D+WΔ(t,x)≤0,t≠tk,(t,x)∈S(h,ρ),W(tk+,xk+Ik(xk))-W(tk,xk)≤d¯kW(tk,xk),(tk,xk)∈S(h,ρ),whered¯k≥0 and ∑i=1∞‍d¯i<∞.
Then system (10) is (h0,h)-asymptotically stable.Proof.
From Corollary14, it follows that system (10) is (h0,h)-stable. Thus, for ρ>0, there exists δ=δ(t0,ρ)>0 such that h0(t0,x0)<δ implies h(t,x)<ρ, t≥t0. To prove the theorem, it remains to show that for every solution x(t) of (10) with h0(t0,x0)<δ, limt→∞h(t,x)=0.
Suppose that this is not true. Then there exists a sequence{ξi}i=1∞ diverging to ∞ as i→∞ and such that h(ξi,x(ξi))≥r(i∈ℕ) for some positive number r. From condition (iv), we know that there exists a function b∈𝒦 such that b(h(t,x))≤W(t,x), if h(t,x)<ρ. Then
(43)W(ξi,x(ξi))>b(r),i∈ℕ.
For any givent∈𝕋, there exists a i>0 such that t∈(ξi,ξi+1]. Then for ξi+1, there exists a k>0 such that ξi+1∈(tk,tk+1]. If tk<t≤tk+1, from (43) and condition (iv), we have
(44)W(t,x)≥W(ξi+1,x(ξi+1))>b(r)≥b(r)M0,
where M0=∏j=1∞‍(1+d¯j)<∞. If tk-1<t≤tk, we have
(45)W(t,x)≥W(tk,xk)≥11+d¯kW(tk+,x(tk+))>11+d¯kb(r)≥b(r)M0.
Following this procedure, we conclude that(46)W(t,x)>b(r)M0,t∈𝕋.
Let(47)L(t,x)=V(t,x)+∫tkt‍c(W(s,x(s)))Δs,t∈(tk,tk+1],k∈ℤ+.
Then, by condition (ii),
(48)D+LΔ(t,x)=D+VΔ(t,x)+c(W(t,x))≤0,t≠tk,
which implies
(49)V(t,x)≤V(tk+,x(tk+))-∫tkt‍c(W(s,x(s)))Δs,t∈(tk,tk+1],
for k∈ℤ+.
Hence, forξi+1∈(tk,tk+1], we obtain, form (46), (49), and condition (iii),
(50)V(ξi+1,x(ξi+1))≤V(tk+,x(tk+))-∫tkξi+1‍c(W(s,x(s)))Δs≤(1+dk)V(tk,x(tk))-∫tkξi+1‍c(W(s,x(s)))Δs≤(1+dk)(V(tk-1+,x(tk-1+))-∫tk-1tk‍c(W(s,x(s)))Δs)-∫tkξi+1‍c(W(s,x(s)))Δs≤(1+dk)(1+dk-1)V(tk-1,x(tk-1))-∫tk-1ξi+1‍c(W(s,x(s)))Δs≤⋯≤∏j=1k‍(1+dj)V(t0,x0)-∫t0ξi+1‍c(W(s,x(s)))Δs≤MV(t0,x0)-c(b(r)M0)(ξi+1-t0)→-∞,fori→∞,
where M=∏j=1∞‍(1+dj)<∞. This is a contradiction, hence limt→∞h(t,x)=0. Theorem 17 is proved.In Theorem17, the function may have a special form. In the case when W(t,x)=V(t,x) and dk=d¯k, k∈ℕ, we deduce the following corollary.Corollary 18.
Let conditions (v), (vi) of Theorem12 hold and assume that (i)
h0,h∈Γ and h0 is weakly finer than h;(ii)
there existc∈𝒦, and function V∈ν0 such that V(t,x) is locally Lipschtiz in x for each t∈𝕋 which is rd, h-positive definite on S(h,ρ), h0-weakly decrescent and
(51)D+VΔ(t,x)≤-c(V(t,x)),t≠tk,(t,x)∈S(h,ρ);(iii)
V(tk+,xk+Ik(xk))-V(tk,xk)≤dkV(tk,xk), (tk,xk)∈S(h,ρ), where dk≥0 and ∑i=1∞‍di<∞.
Then system (10) is (h0,h)-asymptotically stable.Theorem 19.
Assume that all conditions of Theorem13 hold. Suppose further, that there exists a function W∈ν0 such that W(t,x) is locally Lipschtiz in x for each t∈𝕋 which is rd, and the following conditions hold: (i)
D+WΔ(t,x)≤-p(t)c(h0(t,x))+q(t), t≠tk, (t,x)∈S(h,ρ), where c∈𝒦, p,q∈Crd(𝕋,ℝ+), ∫t0∞‍p(τ)Δτ=∞, and ∫t0∞‍q(τ)Δτ<∞;(ii)
W(tk+,xk+Ik(xk))-W(tk,xk)≤d¯kW(tk,xk), (tk,xk)∈S(h,ρ), where d¯k≥0 and ∑i=1∞‍d¯i<∞.
Then system (10) is (h0,h)-attractive.Proof.
By Theorem13, system (10) is (h0,h)-uniformly stable. Thus, for ρ>0, there exists a δ0=δ0(ρ)>0 such that h0(t0,x0)<δ0 implies that h(t,x)<ρ, t≥t0, where x(t)=x(t;t0,x0) is any solution of (10).
Letɛ∈(0,ρ) be given, δ=δ(ɛ)>0 be the same as defined in the definition of (h0,h)-uniform stability, and h0(t0,x0)<δ0. We claim that there exists a t*≥t0 such that
(52)h0(t*,x(t*))<δ.
If this is not true, then h0(t,x)≥δ for all t≥t0.
Let(53)L(t,x)=W(t,x)+∫tkt‍p(τ)c(h0(τ,x(τ)))Δτ-∫tkt‍q(τ)Δτ,t∈(tk,tk+1],
for k∈ℤ+. By condition (i), we obtain
(54)D+LΔ(t,x)=D+WΔ(t,x)+p(t)h0(t,x)-q(t)≤0,t≠tk,
which implies that, for t∈(tk,tk+1], k∈ℤ+,
(55)L(t,x)≤L(tk+,x(tk+)).
Then, it follows from (55) and condition (ii) that, for t∈(tk,tk+1],
(56)W(t,x)≤W(tk+,x(tk+))-∫tkt‍p(τ)c(h0(τ,x(τ)))Δτ+∫tkt‍q(τ)Δτ≤(1+d¯k)W(tk,x(tk))-∫tkt‍p(τ)c(h0(τ,x(τ)))Δτ+∫tkt‍q(τ)Δτ≤(1+d¯k)×(∫tk-1tk‍W(tk-1+,x(tk-1+))-∫tk-1tk‍p(τ)c(h0(τ,x(τ)))Δτ+∫tk-1tk‍q(τ)Δτ)-∫tkt‍p(τ)c(h0(τ,x(τ)))Δτ+∫tkt‍q(τ)Δτ≤(1+d¯k)(1+d¯k-1)W(tk-1,x(tk-1))-∫tk-1t‍p(τ)c(h0(τ,x(τ)))Δτ+(1+d¯k)∫tk-1t‍q(τ)Δτ≤⋯≤∏i=1k‍(1+d¯i)W(t0,x0)-∫t0t‍p(τ)c(h0(τ,x(τ)))Δτ+∏i=2k‍(1+d¯i)∫t0t‍q(τ)Δτ≤M0(W(t0,x0)+∫t0t‍q(τ)Δτ)-c(δ)∫t0t‍p(τ)Δτ,
where M0=∏i=1∞(1+d¯i)<∞. Then, (56) implies that W(t,x)→-∞, for t→∞. This contradiction shows that (52) is true, and hence,
(57)h(t,x)<ɛ,t≥t*.
Thus we conclude that system (10) is (h0,h)-attractive.Remark 20.
When𝕋=ℤ, and dk=d¯k=0, k∈ℕ, Theorem 19 contains Theorem  3.4 in [18] for discrete systems without impulse effects.
Next, we will give two results on uniform asymptotic stability in terms of two measures.Theorem 21.
Let all the conditions of Theorem19 and the following additional conditions hold: (i)
W(t,x) is h0-decrescent, and p(t)≡p  (p is a positive constant), for t∈𝕋;(ii)
there exists a constantτ>0 such that
(58){t+kτ:t∈𝕋,k∈ℕ}⊂𝕋.
Then system (10) is uniformly asymptotically stable.Proof.
SinceW(t,x) is h0-decrescent, there exist δ1>0 and a function b∈𝒦 such that
(59)W(t,x)≤b(h0(t,x)),ifh0(t,x)<δ1.
By Theorem 19, system (10) is (h0,h)-uniformly stable. Thus, there exists a δ0=δ0(ρ)∈(0,δ1) such that h0(t0,x0)<δ0 implies h(t,x)<ρ, t≥t0, for any solution x(t)=x(t;t0,x0) of (10).
Letɛ∈(0,ρ) be given and δ=δ(ɛ)>0 be the same as defined in the definition of (h0,h)-uniformly stability. Let m>0 be the smallest integer such that
(60)m>M0(b(δ0)+N)pτc(δ),
where M0=∏k=1∞‍(1+d¯k)<∞ and N=∫t0∞‍q(s)Δs<∞.
ChooseT=mτ and let x(t)=x(t;t0,x0) be any solution of (10) with h0(t0,x0)<δ0. We claim that there exists a t*∈[t0,t0+T] such that h0(t*,x(t*))<δ. If this is not true, then h0(t,x)≥δ for all t*∈[t0,t0+T]. By (56), (60), and condition (ii), we have
(61)W(t0+T,x(t0+T))≤M0W(t0,x0)-pc(δ)T+M0∫t0t0+T‍q(s)Δs≤M0b(δ0)-mc(δ)pτ+M0N<0
which is a contradiction. Thus, our claim is true and by the uniform stability we have
(62)h(t,x)<ɛ,t≥t0+T≥t*.
Hence, system (10) is (h0,h)-uniformly attractive. This completes the proof.Theorem 22.
Let conditions (v), (vi) of Theorem12 hold and assume that (i)
h0,h∈Γ and h0 is finer than h;(ii)
there exist ac∈𝒦, and function V∈ν0 such that V(t,x) is locally Lipschtiz in x for each t∈𝕋 which is rd, h-positive definite on S(h,ρ), h0-decrescent and
(63)D+VΔ(t,x)≤-c(h0(t,x)),t≠tk,(t,x)∈S(h,ρ);(iii)
V(tk+,xk+Ik(xk))-V(tk,xk)≤dkV(tk,xk), (tk,xk)∈S(h,ρ), where dk≥0 and ∑i=1∞‍di<∞.
Then system (10) is (h0,h)-uniformly asymptotically stable.Proof.
SinceV(t,x) is h0-decrescent, there exist a constant δ0 and a function a∈𝒦 such that
(64)V(t,x)≤a(h0(t,x)),ifh0(t,x)<δ0.
The fact that V(t,x) is h-positive definite on S(h,ρ) implies that there exists a function b∈𝒦 such that
(65)b(h(t,x))≤V(t,x),ifh(t,x)<ρ.
It follows from Corollary14 that system (10) is (h0,h)-uniformly stable. Thus for ρ>0, there exist a δ1=δ1(ρ)∈(0,δ0) such that h0(t0,x0)<δ1 implies
(66)h(t,x)<ρ,t≥t0.
By the choice of δ1, we get
(67)V(t0,x0)≤a(h0(t0,x0))≤a(δ1).
To prove the theorem, it is enough to show that system (10) is (h0,h)-uniformly attractive.
Given0<ɛ<ρ, let δ=δ(ɛ)>0 be the same as defined in the definition of (h0,h)-uniformly stability. Then for any solution x(t)=x(t;t0,x0) of system (10) with h0(t0,x0)<δ1, we claim that there exists a T=T(ɛ)>Ma(δ1)/c(δ) such that, for some t*∈[t0,t0+T],
(68)h0(t*,x(t*))<δ,
where M=∏k=1∞‍(1+dk)<∞. Suppose that this is false. Then for any T>Ma(δ1)/c(δ) there exists a solution x(t)=x(t;t0,x0) of (10) satisfying h0(t0,x0)<δ1, such that
(69)h0(t,x(t))≥δ,t∈[t0,t0+T].
By setting(70)L(t,x)=V(t,x)+∫tkt‍c(h0(s,x(s)))Δs,t∈(tk,tk+1],k∈ℤ+,
and condition (ii), we have
(71)D+LΔ(t,x)=D+VΔ(t,x)+c(h0(t,x))≤0,t≠tk,
which implies L(t,x)≤L(tk+,x(tk+)), for t∈(tk,tk+1]. Then, for t∈(tk,tk+1], we get
(72)V(t,x)≤V(tk+,x(tk+))-∫tkt‍c0(s)Δs≤(1+dk)V(tk,x(tk))-∫tkt‍c0(s)Δs≤(1+dk)(V(tk-1+,x(tk-1+))-∫tk-1tk‍c0(s)Δs)-∫tkt‍c0(s)Δs≤(1+dk)(1+dk-1)V(tk-1+,x(tk-1+))-∫tk-1t‍c0(s)Δs≤V(t0,x0)∏i=1k‍(1+di)-∫t0t‍c0(s)Δs≤MV(t0,x0)-∫t0t‍c0(s)Δs,
where c0(t):=c(h0(t,x)). Hence, for t=t0+T, we obtain
(73)V(t,x(t))|t=t0+T≤MV(t0,x0)-∫t0t0+T‍c(h0(s,x(s)))Δs≤Ma(δ1)-c(δ)T<0
which is a contradiction. Hence there exists a number t*∈(t0,t0+T] such that h0(t*,x(t*))<δ. Then for t≥t*, thus for t≥t0+T as well, we have
(74)h(t,x)<ɛ,
that is, h(t,x)<ɛ holds for t≥t0+T which means that system (10) is uniformly attractive. Theorem 22 is proved.


## 4.3.(h0,h)-Instability

Theorem 23.
Assume that(i)
h0,h∈Γ, V∈ν0, V(t,x) is locally Lipschtiz in x for each t∈𝕋 which is rd, h-positive definite on S(h,ρ), and
(75)D+VΔ(t,x)≥0,t≠tk,(t,x)∈S(h,ρ),and for anys≥t0 and α>0, there is β>0 such that V(t,x)≥α for t≥s implies h(t,x)≥β for t≥s;(ii)
for(tk,xk)∈S(h,ρ), k∈ℕ,
(76)V(tk+,x(tk+))-V(tk,xk)≥λkψ(V(tk,xk)),whereλk≥0 with ∑k=1∞‍λk=∞, ψ:ℝ+→ℝ+ is nondecreasing and ψ(0)=0, ψ(s)>0 for s>0.
Then system (10) is (h0,h)-unstable.Proof.
Let us assume on the contrary that system (10) is (h0,h)-stable. Then for 0<ɛ<ρ, there exists a δ=δ(t0,ɛ)>0 such that h0(t0,x0)<δ implies that h(t,x)<ɛ for t≥t0.
By settingL(t)=V(t,x), we know from condition (i) and (ii) that L(t) is right-nondecreasing for t≥t0. Then, L(t)≥L(t0), if t≥t0. Thus, it follows from condition (i) that there exists a β0=β0(L(t0))>0 such that h(t,x)≥β0 for t≥t0. Since V(t,x) is h-positive definite on S(h,ρ), there exists b∈𝒦 such that b(h(t,x))≤V(t,x), if h(t,x)<ρ. Then, we have
(77)L(tk)≥b(h(tk,xk))≥b(β0),k∈ℕ.
From condition (ii) and (77), we have
(78)L(tk+)-L(tk-1+)≥L(tk+)-L(tk)≥λkψ(L(tk))≥λkψ(b(β0)),
and so
(79)L(tk+)≥L(tk-1+)+λkψ(b(β0))≥L(tk-2+)+(λk-1+λk)ψ(b(β0))≥⋯≥L(t1)+ψ(b(β0))∑i=1k‍λi→∞,ask→∞,
which implies that, for a given number M≫0, there exists a s≥t0 such that L(t)≥M for t≥s. Thus by condition (i), there is a β=β(M)>0 such that h(t,x)≥β for t≥s which is a contradiction to the (h0,h)-stability. Therefore, system (10) is (h0,h)-unstable.Theorem 24.
Assume that(i)
h0,h∈Γ, V∈ν0, V(t,x) is locally Lipschtiz in x for each t∈𝕋 which is rd, h-decrescent on S(h,ρ), and
(80)-g(t)c(V(t,x))≤D+VΔ(t,x)≤0,t≠tk,(t,x)∈S(h,ρ),whereg∈Crd(𝕋,ℝ+) and c∈𝒦;(ii)
there existψk:ℝ+→ℝ+, k∈ℕ, such that ψk(s)>s, ψk(s)-s≥ψk(t)-t for t≥s≥0, and
(81)V(tk+,xk+Ik(xk))≥ψk(V(tk,xk)),(tk,xk)∈S(h,ρ),k∈ℕ;(iii)
for anyu>0,
(82)-∫tktk+1‍g(τ)Δτ+ψk(u)-uc(u)≥rk,k∈ℕ,whererk≥0 and ∑i=1∞‍rk=∞.
Then system (10) is (h0,h)-unstable.Proof.
For the sake of contradiction, we assume that system (10) is (h0,h)-stable. Then for 0<ɛ<ρ, there exists a δ=δ(t0,ɛ)>0 such that h0(t0,x0)<δ implies h(t,x)<ɛ for t≥t0. Since V(t,x) is h-decrescent on S(h,ρ), there exists a function a∈𝒦 such that
(83)V(t,x)≤a(h(t,x))<a(ρ),ifh(t,x)<ρ.
Letm(t)=V(t,x) and L(t)=V(t,x)+∫tkt‍g(τ)c(V(τ,x(τ)))Δτ, for t∈(tk,tk+1], k∈ℤ+. Then it follows from condition (i) that
(84)D+LΔ(t)=D+VΔ(t,x)+g(t)c(V(t,x))≥0,t≠tk,
which implies
(85)L(t)≥L(tk+),t∈(tk,tk+1].
Since D+VΔ(t,x)≤0, it follows from (85), that
(86)m(tk+1)-m(tk+)≥-∫tktk+1‍g(τ)c(m(τ))Δτ≥-c(m(tk+))∫tktk+1‍g(τ)Δτ.
From condition (ii), we have
(87)m(tk+)-m(tk)≥ψk(m(tk))-m(tk)≥ψk(m(tk+))-m(tk+)
which, together with (86) and condition (iii), yields
(88)m(tk+1)-m(tk)=m(tk+1)-m(tk+)+m(tk+)-m(tk)≥ψk(m(tk+))-m(tk+)-c(m(tk+))∫tktk+1‍g(τ)Δτ≥c(m(tk+))rk≥c(m(tk))rk≥c(m(t1))rk,k∈ℕ.
Thus,
(89)m(tk+1)≥m(t1)+c(m(t1))∑i=1k‍ri→∞,ask→∞,
which contradicts (83). Therefore, system (10) is (h0,h)-unstable and the proof is complete.


## 5. Examples

In this section, as applications of the above-derived theoretical criteria, two representative examples are given as follows.Example 25.
Consider the system(90)xΔ(t)=y(t)2(1+x2(t))-x(t),t≠tk,yΔ(t)=x(t)2(1+y2(t))-y(t),t≠tk,Δx(tk)=32kx(tk),k∈ℕ,Δy(tk)=52ky(tk),k∈ℕ,
on time scale 𝕋.LetV(x,y)=x2+y2. Then we have
(91)VΔ(x,y)=(x2+y2)Δ=xΔx+xσxΔ+yΔy+yσyΔ=xΔ(2x+μxΔ)+yΔ(2y+μyΔ)=2x(y2(1+x2)-x)+2y(x2(1+y2)-y)+μ[(y2(1+x2)-x)2+(x2(1+y2)-y)2]=(1-μ)xy(1+x2)+(μ/4)y2(1+x2)2+(μ-2)(x2+y2)+(1-μ)xy(1+y2)+(μ/4)x2(1+y2)2≤(1/2)(1-μ)(x2+y2)(1+x2)+(μ/4)(x2+y2)(1+x2)2+(μ-2)(x2+y2)+(1/2)(1-μ)(x2+y2)(1+y2)+(μ/4)(x2+y2)(1+y2)2≤12(μ-2)V(x,y),t≠tk,
where xσ(t)=x(σ(t)), and
(92)V(x(tk+),y(tk+))=x2(tk+)+y2(tk+)=(1+32k)2x2(tk)+(1+52k)2y2(tk)≤V(x(tk),y(tk))+12k-5V(x(tk),y(tk)),k∈ℕ,
that is,
(93)V(x(tk+),y(tk+))-V(x(tk),y(tk))≤dkV(x(tk),y(tk)),
where dk=1/2k-5 and ∏k=1∞‍dk<∞.Ifμ(t)≤2, then by (91) we have
(94)VΔ(x,y)≤0.
Then for h(x,y)=|x| and h0(x,y)=x2+y2, we have
(95)h2(x,y)≤V(t,x)≤h02(x,y).
From Corollary 14, we conclude that system (90) is (h0,h)-uniformly stable. It should be noted that (h0,h)-stability in this case implies, by the choice of h and h0, that the trivial solution of (90) is partially stable with respect to x.If, for a given positive constantm≤1, μ(t)≤2(1-m), then we have
(96)VΔ(x,y)≤-mV(x,y).
Then for h(x,y)=2|xy| and h0(x,y)=x2+y2, we have
(97)h(x,y)≤V(x,y)≤h0(x,y).
By Corollary 18, we conclude that system (90) is (h0,h)-asymptotically stable. Moreover, since h0=V, we conclude, by Theorem 22, that system (90) is (h0,h)-uniformly asymptotically stable.Example 26.
Consider the following system [14]
(98)x1Δ=-(e-t+1)x1-x1x221+x22,t≠tk,x2Δ=-(e-t+1)x2-x12x21+x12,t≠tk,Δx1=12x1,t=tk,k∈ℕ,Δx2=12x2,t=tk,k∈ℕ,
on time scale 𝕋 such that μ(t)≥2/(e-t+1), for all t∈𝕋.LetV(t,x)=x12+x22, then we get
(99)VΔ(t,x)=x1Δx1+x1σx1Δ+x2Δx2+x2σx2Δ=x1Δx1+(x1+μx1σ)x1Δ+x2Δx2+(x2+μx2σ)x2Δ=2x1Δx1+2x2Δx2+μ[(x1Δ)2+(x2Δ)2],
where xiσ=xi(σ(t)), i=1,2. Substituting (98) into (99) yields
(100)VΔ(t,x)=[μ(e-t+1)2-2(e-t+1)](x12+x22)+μx12x24(1+x22)2+μx14x22(1+x12)2+2[μ(e-t+1)-1](x12x221+x12+x12x221+x22)≥x12x24(1+x22)2+x14x22(1+x12)2≥0.Whent=tk, we have
(101)V(tk+,x(tk+))=x12(tk+)+x22(tk+)=94x12(tk)+94x22(tk)=V(tk,x(tk))+λkψ(V(tk,x(tk))),
where λk≡5/4 and ψ(s)=s.Forh(t,x)=|x1|+|x2| and h0∈Γ, we have
(102)h2(t,x)=(|x1|+|x2|)2=x12+x22+2x1x2≥V(t,x).
Then, for any s≥t0 and α>0, V(t,x)≥α implies that h(t,x)≥β=α. Hence, all the conditions of Theorem 23 are satisfied, and system (98) is (h0,h)-unstable.


## 6. Conclusions

We have generalized the concepts of stability in terms of two measures relative to ordinary impulsive systems to impulsive systems on time scales. By employing the approach of Lyapunov function, we have established some criteria ensuring stability and instability in terms of two measures for nonlinear impulsive systems on time scales. Two examples have been worked out to demonstrate the main results.



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*Source: 313029-2013-02-24.xml*