# h-Stability of Dynamic Equations on Time Scales with Nonregressivity

Authors: Sung Kyu Choi; Yoon Hoe Goo; Namjip Koo
Journal: Abstract and Applied Analysis
 (2008)

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## Abstract

We study theh-stability of dynamic equations on time scales,
without the regressivity condition on the right-hand side of dynamic equations.
This means that we can include noninvertible difference equations into our
results.


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## Body

## 1. Introduction

The theory of calculus on time
scales, which has recently received a lot of attention, was created by
Hilger [1] in order to
unify the theories of differential equations and of difference equations and in
order to extend those theories to other kinds of the so-called “dynamic
equations.” The two main features of the calculus on time scales are
unification and extension of continuous and discrete analysis.Time scale
calculus is especially useful when studying systems with discrete and
continuous elements in their domains, such as an insect population that is
continuous for parts of the year and discrete for other parts of the year.The calculus on
time scales and dynamic equations on time scales have applications in any field
that requires simultaneous modeling of continuous and discrete processes,
because they bridge the divide between continuous and discrete aspects of
processes. The applications include insect population models, epidemic models,
neural networks, and heat transfer. Foundational definitions and results from
the time scale calculus appear in an excellent
introductory text by Bohner and Peterson [2].In [3], we extended the concept of h-stability introduced by Pinto [4] to dynamic equations on
time scales. The notion of h-stability is quite flexible because it
includes the classical notions of uniform or exponential stability within one
common framework. For the detailed results of h-stability for differential and difference
systems, see [4–7].Regressivity
plays a crucial role in developing the fundamental theory of linear dynamic
equations. Consider the scalar dynamic equation on time scale𝕋:(1.1)xΔ(t)=a(t)x(t).Equation (1.1) is said to be
regressive if a(t) satisfies(1.2)1+a(t)μ(t)≠0for all t∈𝕋κ.
A system is nonregressive if it
is not regressive. When we consider the generalized time scale exponential
function, we need the concept of regressiveness since the exponential function ea(t,t0) is defined only for a(t) satisfying condition (1.2). The continuous
dynamical systems (e.g., ordinary differential equations) are always regressive
since 𝕋=ℝ has the graininess function μ(t)≡0.
However, nonregressivity is always a possibility in discrete dynamical systems
(e.g., difference equations), where the underlying domain consists of a mixture
of discrete and continuous parts. In fact, if there is even one point in 𝕋 with nonzero graininess, then nonregressivity
is possible [8].In this paper,
we investigate theh-stability for dynamic equations on time
scales, with the nonregressivity condition. Thus, we improve some results in
[3].


## 2. Calculus on Time Scales

We mention without
proof several foundational definitions and theorems, as well as a general
introduction to the theory of time scales in an
excellent introductory text by Bohner and Peterson [2].Definition 2.1.
Atime scale𝕋 is any nonempty closed subset of the real
numbers ℝ.One assumes
throughout that𝕋 has the topology that it inherits from the
standard topology on ℝ.It is also
assumed throughout that in𝕋 the interval [a,b] means the set {t∈𝕋:a≤t≤b} for the points a<b in 𝕋.
Since a time scale may or may not be connected, one needs the following concept
of jump operators.Definition 2.2.
The
functionsσ,ρ:𝕋→𝕋 defined by (2.1)σ(t)=inf{s∈𝕋:s>t},ρ(t)=sup{s∈𝕋:s<t}are called the
jump operators.The jump
operatorsσ and ρ allow the classification of points in 𝕋 in the following way.Definition 2.3.
A nonmaximal
elementt∈𝕋 is said to be
right-dense if σ(t)=t,
and  right-scattered if σ(t)>t.
Also, a nonminimal element t∈𝕋 is called
left-dense if ρ(t)=t,
and  left-scattered if ρ(t)<t.Definition 2.4.
The functionμ:𝕋→ℝ+ defined by μ(t)=σ(t)−t is called the
graininess function.If𝕋 has a left-scattered maximum m,
then 𝕋κ=𝕋−{m}.
Otherwise 𝕋κ=𝕋.Definition 2.5.
A functionf:𝕋→ℝ is called
differentiable at t∈𝕋κ with (delta) derivativefΔ(t)∈ℝ if given ε>0 there exists a neighborhood U of t such that, for all s∈U, (2.2)|fσ(t)−f(s)−fΔ(t)[σ(t)−s]|≤ε|σ(t)−s|,where fσ=f∘σ.
If f is delta differentiable for every t∈𝕋κ,
then f:𝕋→ℝ is called delta differentiable on 𝕋κ.Some basic
properties of delta derivatives are as follows [9, 10].(i)
Iff is differentiable at t∈𝕋κ,
then(2.3)fσ(t)=f(t)+μ(t)fΔ(t).(ii)
If bothf and g are differentiable at t∈𝕋κ,
then the product fg is also differentiable at t∈𝕋κ with(2.4)(fg)Δ(t)=fΔ(t)g(t)+fσ(t)gΔ(t)=f(t)gΔ(t)+fΔ(t)gσ(t).Definition 2.6.
The functionf:𝕋→ℝ is said to be
rd-continuous (denoted by f∈Crd(𝕋,ℝ)) if
(i)
f is continuous at every right-dense point t∈𝕋,(ii)
lims→t−f(s) exists and is finite at every left-dense point t∈𝕋.Definition 2.7.
Letf∈Crd(𝕋,ℝ).
The 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 t∈𝕋κ.
In this case, one
defines (2.5)∫atf(s)Δs=g(t)−g(a),t∈𝕋.
Some basic
properties of delta integral are as follows [2, 9, 10].
Letf,g:𝕋→ℝ be rd-continuous.
(i)
Ifr,s∈𝕋,α,β∈ℝ,
then(2.6)∫rs[αf(t)+βg(t)]Δt=α∫rsf(t)Δt+β∫rsg(t)Δt.(ii)
Ift∈𝕋κ,
then(2.7)∫tσ(t)f(τ)Δτ=μ(t)f(t).(iii)
If𝕋=ℝ and a,b∈𝕋,
then(2.8)∫abf(t)Δt=∫abf(t)dt,where the integral on the right
is the usual Riemann integral from calculus.(iv)
If[a,b] consists of only isolated points,
then(2.9)∫abf(t)Δt={∑t∈[a,b)μ(t)f(t)ifa<b,0ifa=b,−∑t∈[b,a)μ(t)f(t)ifa>b.(v)
If𝕋=δℤ,
where δ>0,
then(2.10)∫abf(t)Δt={∑k=a/δb/δ−1f(kδ)δifa<b,0ifa=b,−∑k=b/δa/δ−1f(kδ)δifa>b.


## 3.h-Stability

LetMn(ℝ) be the set of all n×n matrices over ℝ and the class of all rd-continuous operators A:𝕋→Mn(ℝ) denoted by(3.1)Crd(𝕋,Mn(ℝ)).We consider the linear
homogeneous dynamic system(3.2)xΔ=A(t)x,x(t0)=x0,where A∈Crd(𝕋,Mn(ℝ)).Definition 3.1 (See [8]).
System (3.2) is said to
be  regressive if (3.3)det[I+μ(t)A(t)]≠0for all t∈𝕋κ,
where I denotes the n×n identity matrix.It turns out
that condition (3.3) is equivalent to having all of the eigenvalues of A(t) regressive in the sense of (1.2) [8].The norm of ann×n matrix A is defined to be(3.4)|A|=max1≤j≤n|Aj|,where Aj is the jth column of A.We recall the
notion of the transition matrix of the linear dynamic systems without
regressivity.Definition 3.2 (See [11, Definition 1.3.5]).
Letτ∈𝕋 and A∈Crd(𝕋,Mn(ℝ)),
and assume that ΦA:{(t,τ)∈𝕋×𝕋:τ≤t}→Mn(ℝ) is an operator with ΦA(τ,τ)=I for all τ∈𝕋.
Then, the unique solution ΦA of the IVP (3.5)XΔ(t)=A(t)X(t),X(τ)=Iis called the
transition matrix of (3.2).Definition 3.3 (See [2, Definition 5.18]).
Letτ∈𝕋 and assume that A∈Crd(𝕋,Mn(ℝ)) is regressive. The unique solution ΦA(t,τ) of IVP (3.5) is called the  matrix exponential function of
(3.2).Note that the solution of (3.2) through (t0,x0) can be represented as x(t)=x(t,t0,x0)=ΦA(t,t0)x(t0).
The transition operator has the following properties.Lemma 3.4 (See [11, Theorem 1.3.9]).
IfA∈Crd(𝕋κ,Mn(ℝ)),
then there exists the transition matrix ΦA(t,τ) of (3.2) which satisfies the following
properties:
(i)
ΦA:{(t,τ)∈𝕋×𝕋:τ≤t}→Mn(ℝ) with ΦA(τ,τ)=I for all τ∈𝕋;(ii)
ΦA(t,τ)=ΦA(t,s)ΦA(s,τ) for all τ≤s≤t;(iii)
(3.6)ΦAΔt(t,τ)=A(t)ΦA(t,τ)∀τ≤t,ΦAΔτ(t,τ)=−ΦA(t,σ(τ))A(τ)∀σ(τ)≤t;(iv)
(3.7)ΦA(σ(τ),τ)=I+μ(τ)A(τ)∀τ∈𝕋κ;if A∈Crd(𝕋κ,Mn(ℝ)) is regressive, that is, (I+μ(t)A(t)) is invertible for all t∈𝕋κ,
then ΦA:𝕋×𝕋→Mn(ℝ) satisfies (ii) and (iii) for all t,τ∈𝕋;(v)
ΦA(t,τ) is invertible in Mn(ℝ) with ΦA(t,τ)−1=ΦA(τ,t) for all t,τ∈𝕋.Assume throughout
that𝕋 is unbounded above.We recall the
definitions about the various types of stability for the solutions of
(3.2).Definition 3.5.
The solutionx=0 of (3.2) is said to be  stable if to any pair of
numbers t0, ε>0,
there exists a δ=δ(t0,ε)>0 such that, for any solution x(t,t0,x0) of (3.2), the inequality |x0|≤δ implies |x(t)|<ε for all t≥t0∈𝕋.
A system is said to be  stable if all of
its solutions are stable.Definition 3.6.
The solutionx=0 of (3.2) is said to be  uniformly stable if it is stable and δ does not depend on t0.Definition 3.7.
The solutionx=0 of (3.2) is said to be  asymptotically stable if it is stable and if
there exists a δ0>0 such that |x0|≤δ0 implies |x(t)|→0 as t→∞.Pinto [4] introduced the notion of h-stability which is an extension of the
notions of exponential stability and uniform Lipschitz stability.Definition 3.8.
System (3.2) is said to be
(i)
anh-system if there exist a positive
function h:𝕋→ℝ and a constant c≥1 such that(3.8)|x(t,t0,x0)|≤c|x0|h(t)h(t0)−1,t≥t0,for |x0| small enough (here h(t)−1=1/h(t));(ii)
h-stable if system (3.2) is an h-system and the function h is bounded.Remark 3.9.
Ifh(t)=e−t,
then h-stability coincides with exponential
stability, and if h(t) is constant, then we have uniform Lipschitz
stability.Example 3.10.
A linear dynamic system on time
scale𝕋 with μ(t)≤1/2,(3.9)xΔ(t)=[−21−1−sint−2]x(t),is h-stable [3].For the various
definitions of stability, we refer to [12] and we obtain the following possible implications for
system (3.2) among the various types of stability:(3.10)h-stability⇒uniformexponentialstability⇒uniformLipschitzstability⇒uniformstabilityas in [6]. The above implications can
be proved by the characterization due to Pinto [7, Lemma 1] for the case 𝕋=ℝ,
in terms of the transition matrix for system (3.2).Now, we
consider the linear dynamic system (3.2) without the regressivity condition.Firstly, we
show that stability for solutions of (3.2) is equivalent to boundedness of
solutions.Theorem 3.11.
All solutions of (3.2) are stable if and only if they
are bounded for all t≥t0∈𝕋.Proof.
Suppose that the solutionx=0 of (3.2) is stable. Then, given any ε>0,
there exists a δ>0 such that |x0|<δ implies |x(t,t0,x0)|<ε.
However, |x(t,t0,x0)|=|ΦA(t,t0)x0|<ε for all t≥t0∈𝕋.
Now, let x0 be a vector (δ/2)ej for j=1,2,…,n.
Then,(3.11)|ΦA(t,t0)x0|=|xj(t,t0)|δ2<ε,t≥t0,j=1,…,n,where xj(t,t0) is the jth column of ΦA(t,t0).
Thus,(3.12)|ΦA(t,t0)|=max1≤j≤n|xj(t,t0)|<2εδ,t≥t0.Consequently, for any solution x(t,t0,x0) of (3.2),(3.13)|x(t,t0,x0)|=|ΦA(t,t0)x0|<2εδ|x0|=M|x0|,t≥t0,where M=2ε/δ.
That is, all solutions of (3.2) are bounded.
Similarly, we
can prove the converse.In [12, Theorem 2.1], DaCunha
obtained the characterization of uniform stability for the regressive system
(3.2). Also, we have the same characterization for the nonregressive system
(3.2) in what follows.Theorem 3.12.
Equation (3.2) is uniformly stable if and
only if there exists a constant γ>0 such that(3.14)|ΦA(t,t0)|≤γ,t≥t0∈𝕋,where ΦA(t,t0) is a transition matrix of (3.2).Proof.
Suppose that (3.2) is uniformly stable. Then, for any given ε>0,
there exists a δ=δ(ε)>0 such that t0≤t1∈𝕋 and |x(t1)|<δ imply |x(t)|<ε for all t≥t1∈𝕋.
Thus,(3.15)|ΦA(t,t1)x(t1)|<ε∀t≥t1≥t0∈𝕋.Since x(t1)=ΦA(t1,t0)x0 can be selected for any t0 and t1≥t0,
let x(t0) be a vector (δ/2)ej for each j=1,2,…,n.
Then,(3.16)|ΦA(t,t1)ΦA(t1,t0)x0|=|ΦA(t,t0)x0|=|xj(t,t0)|δ2<ε,t≥t1≥t0,where xj(t,t0) is the jth column of ΦA(t,t0).
Thus,(3.17)|ΦA(t,t0)|=max1≤j≤n|xj(t,t0)|<γ,t≥t0,where γ=2ε/δ.
We see that |ΦA(t,t0)|≤γ for all t,t0∈𝕋 with t≥t0.
Conversely,
suppose that there exists aγ>0 such that |ΦA(t,t0)|≤γ for all t,t0∈𝕋 with t≥t0.
For any t0 and x(t0)=x0, the solution of (3.2) satisfies(3.18)|x(t)|=|ΦA(t,t0)x0|≤|ΦA(t,t0)||x0|≤γ|x0|,t≥t0.Thus, uniform stability of
(3.2) is established.Pinto [4] gave the characterization of h-system. We obtain the time scale version in
what follows.Theorem 3.13.
Equation (3.2) is
an h-system if and only if there exist a positive
function h defined on 𝕋 and a constant c≥1 such that(3.19)|ΦA(t,t0)|≤ch(t)h(t0)−1,t≥t0∈𝕋,where ΦA(t,t0) is a transition matrix of (3.2).Proof.
Suppose
that (3.2) is an h-system. Let x0 be a vector (δ/2)ej for j=1,…,n.
Then, we have(3.20)|x(t,t0,x0)|=|ΦA(t,t0)x0|=|xj(t,t0)|δ2≤|x0|ch(t)h(t0)−1,t≥t0,j=1,2,…,n,where xj(t,t0) is the jth column of ΦA(t,t0).
Thus, we obtain(3.21)|ΦA(t,t0)|≤2δ|x0|ch(t)h(t0)−1=ch(t)h(t0)−1,t≥t0.
Conversely, we have(3.22)|x(t,t0,x0)|=|ΦA(t,t0)x0|≤|ΦA(t,t0)||x0|≤ch(t)h(t0)−1|x0|,t≥t0.Hence, (3.2) is an h-system.Remark 3.14.
Theorems3.11,
3.12, and
3.13 also hold for the linear dynamic systems with
regressivity.Now, we study theh-stability of the nonlinear perturbed dynamic
system without the regressivity condition on the right-hand side of the
equation via Gronwall's inequality and Bihari's inequality on time scales.We consider the
perturbed systems of (3.2):(3.23)xΔ=A(t)x+F(t)x,x(t0)=x0,where A,F∈Crd(𝕋,Mn(ℝ)).We can obtain the following variation-of-constants
formula [11].Lemma 3.15 (See [11]).
The solutionx(t,t0,x0) of system (3.24)
(3.24)xΔ(t)=A(t)x(t)+g(t,x),t∈𝕋,where A∈Crd(𝕋,Mn(ℝ))
and g:𝕋×ℝn→ℝn is rd-continuous in the first argument with g(t,0)=0,
with the initial value x(t0)=x0 is  given by(3.25)x(t,t0,x0)=ΦA(t,t0)x0+∫t0tΦA(t,σ(s))g(s,x(s))Δs,t≥t0,where ΦA(t,t0) is a transition matrix of (3.2).Theorem 3.16.
Suppose that (3.2) is h-stable. Then, (3.23) is
h-stable if there exists a positive constant β such that for all t0∈𝕋,(3.26)∫t0∞h(s)h(σ(s))|F(s)|Δs≤β.Proof.
Since (3.2) is h-stable, there exist a constant c≥1 and a positive bounded function h(t) such that(3.27)|ΦA(t,t0)|≤ch(t)h(t0)−1for all t≥t0∈𝕋.
By Lemma 3.15, the solution x(t) of (3.23) satisfies(3.28)x(t)=ΦA(t,t0)x0+∫t0tΦA(t,σ(s))F(s)x(s)Δs,t≥t0,where ΦA(t,t0) is a transition matrix of (3.2).
By taking the
norms of both sides of (3.28), we have(3.29)|x(t)|≤ch(t)h(t0)−1|x0|+c∫t0th(t)h(σ(s))−1|F(s)||x(s)|Δs,t≥t0.Dividing by h(t)>0 on both sides,(3.30)|x(t)|h(t)≤c|x0|h(t0)+c∫t0th(s)h(σ(s))|F(s)||x(s)|h(s)Δs,t≥t0.In view of Gronwall's inequality
on time scales in [9],
we obtain(3.31)|x(t)|h(t)≤c|x0|h(t0)ec(h(s)/h(σ(s)))|F(s)|(t,t0)={c|x0|h(t0)exp(∫t0tlog(1+μ(s)c(h(s)/h(σ(s)))|F(s)|)μ(s)Δs)ifμ≠0,c|x0|h(t0)exp(∫t0tch(s)h(σ(s))|F(s)|Δs)ifμ=0≤c|x0|h(t0)exp(∫t0tch(s)h(σ(s))|F(s)|Δs)≤c|x0|h(t0)exp(∫t0∞ch(s)h(σ(s))|F(s)|Δs)≤c|x0|h(t0)ecβfor all t≥t0∈𝕋.
Thus,(3.32)|x(t)|≤d|x0|h(t)h(t0)−1,t≥t0,where d=cecβ≥1.
Hence, (3.23) is h-stable.Corollary 3.17.
Suppose that (3.2) is h-stable with bounded h(t)/h(σ(t)) for each t∈𝕋.
Then, (3.23) is h-stable if there exists a positive constant β such that for all t0∈𝕋,(3.33)∫t0∞|F(s)|Δs≤β.Corollary 3.18.
When𝕋=ℝ,
(3.23) is h-stable if there exists a positive constant β such that for all t0∈ℝ,(3.34)∫t0∞|F(s)|ds≤β.Corollary 3.19.
When𝕋=δℤ with a positive constant δ,
(3.23) is h-stable if there exists a positive constant β such that for all t0∈δℤ,(3.35)∑s=t0∞h(s)h(s+δ)|F(s)|≤β.We consider the nonlinear perturbed dynamic
system(3.36)zΔ(t)=A(t)z(t)+G(t,z(t)),where A∈Crd(𝕋,Mn(ℝ)) and G is rd-continuous on 𝕋×ℝn.We recall the
notion of the classH^.Definition 3.20.
A functionw:ℝ+→ℝ+ belongs to the
class H^ if
(H1)
w(u) is nondecreasing and continuous for u≥0 and positive for u>0,(H2)
there exists a continuous functionϕ on ℝ+ with w(αu)≤ϕ(α)w(u) for α>0,u≥0,(H3)
limu→0+(w(u)/u) exists.Theorem 3.21.
Suppose that (3.2) is h-stable and(3.37)|G(t,x)|≤F(t)w(|x|),t≥t0,where F is positive and rd-continuous, and w∈H^ with corresponding multiplier function ϕ and(3.38)λ(t)=h(t0)F(t)h(σ(t))|z0|ϕ(|z0|h(t)h(t0)),d=W−1[W(c)+c∫t0∞λ(s)Δs]in [13, Theorem 5.8]. Then, (3.36)
is h-stable.Proof.
It follows from Lemma3.15 that the
solution z(t) of (3.36) is given by(3.39)z(t)=ΦA(t,t0)z0+∫t0tΦA(t,σ(s))G(s,z(s))Δs,t≥t0,where ΦA(t,t0) is a transition matrix of (3.2).
Byh-stability of (3.2), there exists a positive bounded function h(t) such that(3.40)|ΦA(t,t0)|≤ch(t)h(t0)−1for all t≥t0∈𝕋.
The rest of the
proof is the same as that of [3, Theorem 2.10].Now, we obtain some results about theh-stability of system (3.36) in the following
corollaries.Corollary 3.22.
Suppose that (3.2) is h-stable with a nondecreasing function h(t) and(3.41)|G(t,x)|≤F(t)w(|x|),t≥t0,where w∈H^ with corresponding multiplier function ϕ(α)=α for all α>0 and(3.42)d=W−1[W(c)+c∫t0∞F(s)Δs].Then, (3.36) is h-stable.Corollary 3.23.
Suppose that (3.2) is h-stable and(3.43)|G(t,x)|≤F(t)w(|x|),t≥t0∈𝕋=ℝ,where w∈H^ with corresponding multiplier function ϕ and(3.44)λ(t)=h(t0)F(t)h(t)|z0|ϕ(|z0|h(t)h(t0)),d=W−1[W(c)+c∫t0∞λ(s)ds].Then, (3.36) is h-stable.Corollary 3.24.
Suppose that (3.2) is h-stable and(3.45)|G(t,x)|≤F(t)w(|x|),t≥t0∈𝕋=δℤ,where w∈H^ with corresponding multiplier function ϕ and(3.46)λ(t)=h(t0)F(t)h(t+δ)|z0|ϕ(|z0|h(t)h(t0)),d=W−1[W(c)+cδ∑s∈δℤt0λ(s)].Then, (3.36) is h-stable.We examine the property ofh-stability for the perturbed dynamic system
(3.24) on time scale 𝕋.Lemma 3.25.
Suppose thatk∈Crd(𝕋×ℝ+,ℝ) is nondecreasing in the second argument x for each fixed t≥t0 with the property
(3.47)x(t)−∫t0tk(s,x(s))Δs≤y(t)−∫t0tk(s,y(s))Δs,t≥t0∈𝕋,for x,y∈Crd(𝕋,ℝ+).
If x(t0)<y(t0),
then x(t)<y(t) for all t≥t0∈𝕋.Proof.
Suppose that there exists aτ∈𝕋 with t0<τ such that x(s)<y(s) for t0≤s<τ and x(τ)≥y(τ).
Also, we obtain(3.48)x(τ)−∫t0τk(s,x(s))Δs>y(τ)−∫t0τk(s,y(s))Δs,τ≥t0∈𝕋.This is a contradiction.If𝕋=ℝ and 𝕋=δℤ with a positive constant δ,
then we can obtain the following results as corollaries of Lemma
3.25.Corollary 3.26 (See [14, Lemma 2.1]).
Suppose thatk∈C(ℝ×ℝ+,ℝ) is nondecreasing in the second argument x for t≥t0 with the property(3.49)x(t)−∫t0tk(s,x(s))ds≤y(t)−∫t0tk(s,y(s))ds,t≥t0∈ℝ,for x,y∈C(𝕋,ℝ+).
If x(t0)<y(t0),
then x(t)<y(t) for all t≥t0∈ℝ.Corollary 3.27 (See [5, Lemma 9]).
Suppose thatk:δℤ×ℝ+→ℝ is nondecreasing in the second argument x for t≥t0 with the property(3.50)x(t)−∑s=t0t−δk(s,x(s))δ≤y(t)−∑s=t0t−δk(s,y(s))δ,t≥t0∈δℤ,for x,y:δℤ→ℝ+.
If x(t0)<y(t0),
then x(t)<y(t) for all t≥t0∈δℤ.Theorem 3.28.
Assume thatx=0 of (3.2) is uniformly stable. Suppose
that(3.51)|g(t,x)|≤ι(t,|x|),where ι∈Crd(𝕋×ℝ+,ℝ+) is strictly increasing in u for each fixed t∈𝕋t0 with ι(t,0)=0.
Consider the scalar dynamic equation(3.52)uΔ=cι(t,u),u(t0)=u0,t∈𝕋t0,where c≥1 is a constant. If u=0 of (3.52) is h-stable, then (3.24) is h-stable
whenever u0=c|x0|.Proof.
It
follows from Lemma3.15 that the solution x(t) of (3.24) is given by(3.53)x(t)=ΦA(t,t0)x0+∫t0tΦA(t,σ(s))g(s,x(s))Δs,t∈𝕋t0.Since x=0 of (3.2) is uniformly stable, we
obtain(3.54)|x(t)|≤|ΦA(t,t0)||x0|+∫t0t|ΦA(t,σ(s))||g(s,x(s))|Δs≤c|x0|+c∫t0tι(s,|x(s)|)Δs.Thus, we have(3.55)|x(t)|−c∫t0tι(s,|x(s)|)Δs≤c|x0|≤u0=u(t)−c∫t0tι(s,u(s))Δs,t∈𝕋t0.By Lemma 3.25, we have |x(t)|<u(t) for all t∈𝕋t0.
Since u=0 of (3.52) is h-stable, we obtain(3.56)|x(t)|≤c1u0h(t)h(t0)−1≤c1c|x0|h(t)h(t0)−1=M|x0|h(t)h(t0)−1,t∈𝕋t0,where M=c1c≥1 is a constant. This completes the proof.



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*Source: 632473-2008-07-01.xml*