R E S E A R C H
Open Access
Variationally stable dynamic systems on time
scales
Sung Kyu Choi
1, Yinhua Cui
2and Namjip Koo
1**Correspondence: [email protected] 1Department of Mathematics, Chungnam National University, Daejeon, 305-764, Korea Full list of author information is available at the end of the article
Abstract
In this paper we give a Lyapunov functional characterization ofh-stability for nonlinear dynamic systems on time scales under the condition ofu∞-similarity between their variational systems. Furthermore, we give some examples related to the notions ofu∞-similarity andh-stability.
MSC: 34N05; 39A30; 34D23; 34K20; 26E70
Keywords: dynamic system; variational system;u∞-similarity; Lyapunov function; h-stability;h-stability in variation; time scales
1 Introduction
The theory on time scales has been developed as a generalization of both continuous and discrete time theory and applied to many different fields of mathematics [, , , , , ]. It is widely known that the various types of stability of nonlinear differential equations or difference equations can be characterized by using Lyapunov’s second method, the method of variation of parameters, and inequalities,etc.[, , , , , ].
Pinto [] introduced the notion ofh-stability for differential equations with the inten-tion of obtaining results about stability for weakly stable differential systems under some perturbations. Also, Medina and Pinto [] applied theh-stability to obtain a uniform treatment for the various stability notions in difference systems and extended the study of exponential stability to a variety of reasonable systems calledh-systems. Pinto and Med-ina obtained the important properties abouth-stability for the various differential systems and difference systems [–, , –].
Choiet al.[] investigatedh-stability for the nonlinear differential systems by means of the notions of Lyapunov functions andt∞-similarity introduced by Conti []. Trench [] introduced summable similarity as a discrete analog of Conti’s definition oft∞-similarity
and investigated the various stabilities of linear difference systems by using summable similarity. Choi and Koo [] studied the variational stability for nonlinear difference sys-tems by means ofn∞-similarity. Also, Choiet al.studied the asymptotic property and the
h-stability of difference systems via discrete similarities and comparison principle [–]. For detailed results about the various stabilities including the notions ofh-stability and strong stability of dynamic systems on time scales, see [–, , , ].
In this paper we introduce the notion of u∞-similarity which extends the continuous
t∞-similarity [] and the discreten∞-summable similarity []. Then we give a Lyapunov functional characterization ofh-stability for nonlinear dynamic systems on time scales by
assuming the condition ofu∞-similarity between its variational systems. Furthermore, we
give some examples related to the notions ofu∞-similarity andh-stability on time scales.
2 Main results
We refer the reader to Ref. [] for all the basic definitions and results on time scales nec-essary for this work (e.g., delta differentiability, rd-continuity, exponential function and its properties).
It is assumed throughout that a time scaleTwill be unbounded above. IfThas a left-scattered maximumm, thenTκ=T–{m}. Otherwise,Tκ=T. LetRnbe then-dimensional
real Euclidean space.Crd(T×Rn,Rn) denotes the set of all rd-continuous functions from T×RntoRnandR
+= [,∞).
We consider the dynamic system
x=f(t,x), x(t) =x,t∈T, (.)
wheref∈Crd(T×Rn,Rn) withf(t, ) = , andxis the delta derivative ofx:T→Rnwith
respect tot∈T. We assume thatfx=∂∂fx exists and is rd-continuous onT×R
n. Letx(t) =
x(t,t,x) be the unique solution of (.) satisfying the initial conditionx(t,t,x) =x.
For the existence and uniqueness of solutions of nonlinear dynamic system (.), see []. Also, we consider its associated variational systems
v=fx(t, )v (.)
and
z=fx
t,x(t,t,x)
z, (.)
whereI+μ(t)fx(t,x(t)) is invertible for allt∈TandIdenotes then×nidentity matrix.
To establish our main results we will use the following lemmas.
Lemma .[, Theorem ..] Assume that x(t,t,x)and x(t,t,y)are the solutions
of system (.) through(t,x)and(t,y)respectively, which exist for each t∈Tand are
such that xand ybelong to a convex subset D ofRn. Then
x(t,t,y) –x(t,t,x) =
t,t,x+τ(y–x)
dτ·(y–x), t∈T,
whereis a fundamental matrix of (.) andT=T∩[t,∞).
This lemma can be proved in the same manner as that of Theorem .. in [], so we omit the detail.
Lemma .[, Lemma ..] Let f ∈Crd(Tκ ×D,Rn), where D is an open convex set
inRn. Suppose that f
xexists and is rd-continuous. Then
f(t,x) –f(t,x) =
fx
t,sx+ ( –s)x
In order to prove the variation of parameters formula on time scales, we need the fol-lowing result on differentiability of solutions with respect to initial values.
Lemma . Assume that f :T×Rn→Rnpossesses partial derivatives onT×Rnand
fx(t,x(t,t,x))is rd-continuous onT. Let x(t) =x(t,t,x)be the solution of (.), which
exists for t≥tand let
H(t,t,x) =
∂f(t,x(t,t,x))
∂x . (.)
Then
(t,t,x) =
∂x(t,t,x)
∂x
(.)
exists and is the solution of
(t,t,x) =H(t,t,x)(t,t,x), t∈T, (.)
(t,t,x) =I. (.)
The proof of Lemma . follows simply by differentiating the solution identity
x(t,t,x) =f
t,x(t,t,x)
, t≥t,
with respect tox. It is a special case of [, Satz ..].
Remark .[, Theorem ..] H(t,t,x) in Lemma . is given by
H(t,t,x) =lim ξ→
fx
t,sx(t,t,x) + ( –s)x(t,t,x+ξ)
ds.
It follows from Lemma . that the fundamental matrix solution(t,t, ) of (.) is
given by
(t,t, ) =
∂x(t,t, )
∂x
and the fundamental matrix solution(t,t,x) of (.) is given by
(t,t,x) =
∂x(t,t,x)
∂x
or equivalently
x(t,t,x) =
(t,t,sx)ds
x. (.)
LetMn(R) be the set of alln×nmatrices overR. The class of all rd-continuous functions
A:T→Mn(R) is denoted by
Crd
T,Mn(R)
Consider the quasilinear dynamic system
y=A(t)y+g(t,y), y(t) =y,t∈T, (.)
whereA∈Crd(T,Mn(R)) andg:T×Rn→Rnis rd-continuous in the first argument with
g(t, ) = .
We need the following result which is a slight modification of the variation of constants formula in [, Theorem ..].
Lemma .[] The solution y(t,t,y)of (.) satisfies the equation
y(t) =(t,t)y+
t t
t,σ(s)gs,y(s)s, t≥t, (.)
whereis a transition matrix of the linear system
y=A(t)y, y(t) =y, (.)
where A∈Crd(Tκ,Mn(R)).
For the Lyapunov-like functionV∈Crd(T×Rn,R+), we recall the following definition. Definition .[, Definition ..] We define thegeneralized derivative D+V
(.)(t,x(t))
ofV(t,x) relative to (.) as follows: givenε> , there exists a neighborhoodN(ε) oft∈T such that
σ(t) –s
Vσ(t),xσ(t)–Vs,xσ(t)–σ(t) –sft,x(t)
<D+V(.) t,x(t)+ε, s∈N(ε),s>t,
wherex(t) is any solution of (.) and theupper right Dini derivative V
* (t) ofV*is given
by
V*(t) = ⎧ ⎨ ⎩
limη→+,η+t∈TV*(t+η)–V*(t)
η , ift=σ(t),
V*(σ(t))–V*(t)
μ(t) , ift<σ(t),
(.)
whereV*(t) =V(t,x(t)).
Then it is well known that
D+V(.)
t,x(t)=V*(t)
ifV(t,x) is Lipschitzian inxfor eacht∈T[, ]. In caset∈Tis right dense, we have
D+V(.)
t,x(t)=DV(.)
t,x(t)= lim η→+
η
Vt+η,x(t) +ηft,x(t)–Vt,x(t)
= lim
η→+,η+t∈T
V(t+η,x(t+η)) –V(t,x(t))
η =D
In caset∈Tis right scattered andV(t,x(t)) is continuous att, we have
by the chain rule of a differentiable functionV(t,x(t)) [, Theorem ].
We note that the total difference of the functionValong the solutionsxof (.) is given by
V(.)(n,x) =V
n+ ,x(n+ ,n,x)–Vn,x(n,n,x).
Choiet al.[] investigatedh-stability for nonlinear differential systems using the notions oft∞-similarity and Lyapunov functions. Also, Choiet al.[] introduced the notion of
n∞-summable similarity which is the correspondingt∞-similarity for the discrete case
and then characterizedh-stability in variation and asymptotic equilibrium in variation for nonlinear difference system vian∞-summable similarity and comparison principle.
Now, we defineu∞-similarity on time scales in order to unify (continuous )t∞-similarity
and (discrete)n∞-similarity for matrix-valued functions.
LetMn(R) be the set of alln×ninvertible matrices overR, andCrd (Tκ,Mn(R)) be the
set of all rd-continuous differentiable functionsSfromTκ to
Mn(R) such thatSandS–
are bounded onT.
Definition . A functionA:Tκ→M
n(R) is calledregressiveif for eacht∈Tκthen×n
matrixI+μ(t)A(t) is invertible.
The class of all rd-continuous and regressive functions fromTκtoM
n(R) is denoted by
LetN(n) ={n,n+ , . . . ,n+k, . . .}, wherenis a nonnegative integer andMdenote
the set of alls×sinvertible matrix-valued functions defined onN(n).
Remark .[, Definition .] A matrix functionA∈Misn∞-summably similarto a
matrix functionB∈Mif there exists an absolutely summables×smatrixF(n) overN(n),
that is,
∞
l=n
F(l)<∞
such that
S(n) +S(n+ )B(n) –A(n)S(n) =F(n) (.)
for someS∈S.
For the example ofn∞-summable similarity, see [].
Remark . We can easily show that then∞-summable similarity is an equivalence
rela-tion in the similar manner of Trench in []. Also, ifAandBaren∞-summably similar withF(n) = , then we say that they arekinematically similar.
Pinto [] introduced the notion ofh-stability which is an extension of the notions of exponential stability and uniform stability of the solutions of differential equations. The symbol| · |will be used to denote any convenient vector norm inRn. We recall the notions
ofh-stability for dynamic systems on time scales in [].
Definition . System (.) is called anh-systemif there exist a positive rd-continuous functionh:T→R, a constantc≥ andδ> such that
x(t,t,x)≤c|x|h(t)h(t)–, t≥t
for|x|<δ(hereh(t)–=h(t)).
Moreover, system (.) is said to be
(hS) h-stableifhis a bounded function in the definition ofh-system,
(GhS) globallyh-stableif system (.) ishS for everyx∈D, whereD⊂Rnis a region
which includes the origin,
(hSV) h-stable in variationif system (.) ishS,
(GhSV) globallyh-stable in variationif system (.) is GhS.
For the various definitions of stability, we refer to [] and we obtain the following pos-sible implications for system (.) among the various types of stability:
h-stability⇒uniform exponential stability
⇒uniform Lipschitz stability
⇒uniform stability
We consider two linear dynamic systems
x=A(t)x (.)
and
y=B(t)y, (.)
whereA,B∈CrdR(Tκ,Mn(R)).
We say that ifAandBareu∞-similar, then systems (.) and (.) areu∞-similar.
Lemma .[, Lemma .] System (.) is an h-system if and only if there exist a positive rd-continuous function h defined onTand a constant c≥such that
A(t,t)≤ch(t)h(t)–, t∈T,
whereAis a fundamental matrix solution of (.).
We obtain the following result from Lemma . in [].
Lemma . Assume that A and B are u∞-similar. Then
B(t,t) =S–(t)
A(t,t)S(t) +
t t
A
t,σ(s)F(s)B(s,t)s
, t,t∈T,
whereAandBare the matrix exponential functions of (.) and (.) respectively.
Medina and Pinto [, Theorem ] showed thathSV implieshS. Also, they proved the converse when the condition
∞
l=n
h(l)
h(l+ )fx(l,n,x) –fx(l, )<∞, n≥ (.) for|x| ≤δholds [, Theorem ].
In order to establish our main results, we will introduce the following condition. (H):fx(t, ) andfx(t,x(t,t,x)) areu∞-similar fort≥tand|x| ≤δfor some constant
δ> and t∞
h(t)
h(σ(t))|F(t)|t<∞with the positive rd-continuous functionh(t) defined
onT.
Lemma .[, Theorem .] Assume that condition (H) is satisfied. Then variational system (.) is also an h-system if and only if variational system (.) is an h-system.
We can obtain the same result about Lemma . by assuming that fx(t, ) and
fx(t,x(t,t,x)) areu∞-quasisimilar fort≥tinstead of the condition (H) in Lemma .
[, Theorem .].
For nonlinear dynamic system (.), we can show that
GhSV⇔GhS, hS⇔hSV
We study the relation ofh-stability between two systems (.) and (.) by assuming the condition (H) is satisfied.
Theorem .[, Theorem ] Suppose that condition (H) is satisfied. If x= of (.) is h-stable, then v= of (.) is h-stable.
We obtain the following result from (.).
Theorem . If z= of (.) is h-stable, then x= of (.) is h-stable.
We can obtain the following result by using Lemma . and Theorem ..
Theorem . Assume that condition (H) is satisfied. If x= of (.) is h-stable, then z= of (.) is h-stable in variation.
Remark . For nonlinear dynamic system (.), we show that two concepts ofh-stability andh-stability in variation are equivalent under the condition that two variational systems (.) and (.) areu∞-similar.
Choiet al.investigated Massera type converse theorems for the nonlinear difference systemx(n+ ) =f(n,x(n)) vian∞-similarity in [, Theorem ] and [, Theorem .]. Fur-thermore, they characterizedh-stability in variation for the nonlinear difference system by using the notion ofn∞-summable similarity in [].
We need the following lemma to prove our main theorem.
Lemma .[, Korollar ..] If the delta differentiable function h:T→Ris positive, thenhh(t(t)) is positively regressive, and ep(t,t)satisfies
ep(t,t) =
h(t) h(t)
,
where p(t) = hh(t(t)).
We can obtain the following result that characterizesh-stability for nonlinear dynamic system (.) via the notions of Lyapunov functions andu∞-similarity. It is adapted from
Theorem .. in [] and Theorem . in [].
Theorem . Assume that condition (H) is satisfied. Suppose further that h(t)exists
and is rd-continuous onT. Then system (.) is GhS if and only if there exists a function V(t,x)defined onT×Rnsuch that the following properties hold:
(i) |x| ≤V(t,x)| ≤c|x|for(t,x)∈T×Rnand a constantc≥; (ii) |V(t,x) –V(t,x)| ≤c|x–x|fort∈Tandx,x∈Rn;
(iii) V(t,x)≤h(t)
h(t) V(t,x)for(t,x)∈T×R
n;
(iv) V(t,x)is continuous onT×Rn;
lim (ˆt,xˆ)→(t,x),ˆt≥t
Proof Necessity: Suppose that system (.) is GhS. Then system (.) is GhSV by Theo-rem .,i.e., there exist a constantc≥ and a positive rd-continuous bounded functionh defined onTsuch that for eachx∈Rn
(t,t,x)≤ch(t)h(t)–, t≥t, (.)
whereis a fundamental matrix solution of (.).
Fixt∈T. LetAt:={τ ∈R+:t+τ∈T}. Then we note thatAtis nonempty from ∈At.
Define the functionV:T×Rn→R
+by
V(t,x) = sup τ∈At
x(t+τ,t,x)h(t+τ)–h(t),
wherex(t+τ,t,x) is a unique solution of system (.) for (t,x)∈T×Rnwith the initial
valuex(t,t,x) =x. From GhS of (.) we have
x(t,t,x)≤c|x|h(t)h(t)–, t∈T,|x|<∞.
Furthermore, we obtain
|x|=x(t,t,x)≤sup τ∈At
x(t+τ,t,x)h(t+τ)–h(t)
≤c|x|h(t+τ)h(t)–h(t+τ)–h(t) =c|x|.
ThusVsatisfies property (i).
Let (t,x), (t,x)∈T×Rn. Then we have
V(t,x) –V(t,x)=sup τ∈At
x(t+τ,t,x)h(t+τ)–h(t)
–sup τ∈At
x(t+τ,t,x)h(t+τ)–h(t)
≤sup τ∈At
x(t+τ,t,x) –x(t+τ,t,x)h(t+τ)–h(t). (.)
It follows from Lemma . that for eachxandxin a convex subsetDofRn
x(t,t,x) –x(t,t,x)≤ |x–x|sup η∈D
(t,t,η). (.)
In view of (.), (.) and (.), we have
V(t,x) –V(t,x)≤ |x–x| sup τ∈At,η∈D
(t+τ,t,η)h(t+τ)–h(t)
≤ |x–x|c|h(t+τ)h(t)–h(t+τ)–h(t)
≤c|x–x|, t∈T,x,x∈Rn.
Next, we will prove property (iii). Letx(t,t,x) be a unique solution of system (.) for
each initial point (t,x)∈T×Rn. We will consider two cases,σ(t) =tandσ(t) >t, in the
proof.
Suppose thatσ(t) =t and letδ∈At. By the uniqueness of solutions of (.) and the
definition ofhS, we have
D+Vt,x(t)
Suppose thatσ(t) >t. Then it follows from the definition ofV(t) that
V * (t) =
V(σ(t)) –V(t) μ(t) .
Since the solution of (.) is unique, we have the following derivative:
≤
μ(t)
ht+μ(t)h(t)–– Vt,x(t)
= h
(t)
h(t) V
t,x(t).
Thus property (iii) was satisfied for two cases.
The continuity ofV(t,x) can be proved in a similar manner of Theorem .. in [] and Theorem . in []. It remains to show thatVis continuous in the sense of (iv): lett∈T, x∈Rnbe fixed and chooseε> arbitrary. Thenδ> andδ> must be found such that
V(t,ˆx) –ˆ V(t,x)<ε
holds for all
ˆ
t=t+ν, ν∈At, ≤ν<δ (.)
and allxˆ∈Bδ(x), whereBδ(x) is an open ball centered onxof radiusδ.
Ift∈Tis right scattered, then we can always choose a suitableδsuch thatˆt=tis the
only point satisfying condition (.) (see [, Theorem ..]). ThusV(t,x) is continuous in (t,x)∈T×RnsinceVis globally Lipschitz continuous inxfor fixedt∈T.
Suppose thatt∈Tis right dense and letˆt=t+νforν∈Atwithν≥. Then we have
V(t,ˆx) –ˆ V(t,x)=V(t+ν,x) –ˆ V(t,x)
≤V(t+ν,x) –ˆ V(t+ν,x) (.)
+V(t+ν,x) –Vt+ν,x(t+ν,t,x) (.)
+Vt+ν,x(t+ν,t,x)–V(t,x). (.)
SinceV(t,x) is Lipschitzian inxandx(t+ν,t,x) is continuous inν, the first two terms (.) and (.) on the right-hand side of the preceding inequality are small when|ˆx–x| andνare small. That is, we have
V(t+ν,x) –ˆ V(t+ν,x)<ε
for allxˆ∈Bδ(x) whenδ< ε c and
V(t+ν,x) –Vt+ν,x(t+ν,t,x)
≤cx–x(t+ν,t,x)
<ε
, ν∈At, ≤ν<δˆ,
since it follows fromlimν→,ν∈Atx(t+ν,t,x) =xthat there exists aδˆ> such that
x–x(t+ν,t,x)< ε c
Let us consider the third term in (.). We note that
xt+ν+τ,t+ν,x(t+ν,t,x)=x(t+ν+τ,t,x).
Thus we have
V
t+ν,x(t+ν,t,x)–V(t,x)
= sup τ∈At+ν
xt+ν+τ,t+ν,x(t+ν,t,x)h(t+ν+τ)–h(t+ν)
–sup τ∈At
xt+τ,t,x(t)h(t+τ)–h(t)
= sup
τ∈{τ∈[ν,∞):t+τ∈T}
xt+τ,t,x(t)h(t+τ)–h(t+ν)
–sup τ∈At
xt+τ,t,x(t)h(t+τ)–h(t)
=α(ν)h(t+ν) h(t) –α()
,
whereα(ν) =supτ∈{τ∈[ν,∞):t+τ∈T}|x(t+τ,t,x(t))|h(t+τ)–h(t) forν∈A
t.
We have α(ν)≤α() for allν∈At withν≥. Furthermore,α(ν) is a nonincreasing
function inνwith
lim ν→,ν∈At
α(ν) =α().
Hence, there exists aδˆ> such that
α(ν)h(t+ν) h(t) –α()
<ε
, ν∈At, ≤ν≤ ˆδ.
Now, chooseδ=min{ˆδ,δˆ}. Forˆt=t+νwithν∈Atwhere ≤ν<δandxˆ∈Bδ(x),
combining all of the above estimates of the terms in (.)-(.) gives
V(t,ˆx) –ˆ V(t,x)<ε +
ε +
ε =ε,
which proves the continuity ofV(t,x).
Sufficiency: Assume thatV(t,x) satisfies the properties (i)-(iv). Letx(t,t,x) be any
so-lution of system (.). Then it follows from condition (iii) ofV(t,x) that
Vt,x(t)≤V(t,x) +
t t
h(s)
h(s) V
s,x(s)s, t≥t.
From Gronwall’s inequality on time scale [] and Lemma . [], we obtain
Vt,x(t)≤V(t,x)ep(t)(t,t)
wherep(t) =hh(t(t)). From (.) and property (i) ofV(t,x), we have x(t,t,x)≤c|x|h(t)h(t)–,
for eacht≥tandx∈Rn. Hence the zero solutionx= of (.) is GhS. This completes
the proof of the theorem.
Remark . Assume that condition (H) is satisfied forT=R. Furthermore, suppose that
h(t) exists and is continuous onR+. Then we can obtain Theorems . and . in [] as a
continuous version of Theorem ..
Also, we can obtain the following result as a discrete version of Theorem ..
Corollary . [, Theorem .] Assume that fx(n, ) is n∞-summably similar to
fx(n,x(n,n,x))for n≥n≥and every x∈Rmwith hh(n(n+)) |F(n)| ∈l(N(n)). Then
sys-tem (.) is GhS if and only if there exists a function V(n,z)defined onN(n)×Rm such
that the following properties hold:
(i) V(n,z)is continuous onN(n)×Rm;
(ii) |x–y| ≤V(n,x–y)| ≤c|x–y|for(n,x,y)∈N(n)×Rm×Rm;
(iii) |V(n,z) –V(n,z)| ≤c|z–z|forn∈N(n),z,z∈Rm;
(iv) VV(n(n,x,x––yy))≤hh(n(n)) for(n,x,y)∈N(n)×Rm×Rmwithx=y.
Remark . Choiet al.[, ] introduced the notion ofn∞-similarity which is slightly different from n∞-summable similarity and studied a general variational stability for a
nonlinear difference system vian∞-similarity and Lyapunov functions. We can obtain the
discrete analogues [, Theorem , Corollary ] and [, Theorem .] as a discrete version of Theorem ..
Remark . Choiet al.[, Theorem .] studiedh-stability for linear dynamic
equa-tions on time scales by using the unified time scale quadratic Lyapunov funcequa-tions. Also, Mukdasai and Niamsup [, Theorem .] derived a sufficient condition forh-stability for a linear time-varying system with nonlinear perturbation on time scales by constructing appropriate Lyapunov functions.
We can obtain the following Massera type converse theorem for the uniform exponen-tial asymptotic stability of linear dynamic equations on time scales as a special case of Theorem ..
Corollary . [, Theorem .] Assume that f(t,x) = A(t)x is linear, where A ∈ CrdR(T,Rn×Rn). If system (.) is hS with h(t) =e–λton time scalesTfor a nonnegative
constantλ, then there exists a function V:T×Rn→Rnsuch that
(i) |x| ≤V(t,x)≤K|x|for allt∈T,x∈Rn.
(ii) |V(t,x) –V(t,x)| ≤K|x–x|for any fixedt∈Tand allx,x∈Rn.
(iii) The upper right Dini derivative ofV*exists and the estimates
V
functionξλ:T→Ris given by
In this section we give some examples which illustrate some results from the previous section.
To illustrate the notion ofu∞-similarity, we will give an example for scalar functions defined on time scales.
For the examples of nonscalar functions aboutt∞-similarity onT=Randn∞-summable
similarity onT=Z, see [, Example] and [, Example .] respectively.
Example . To illustrate Lemma ., we consider the linear dynamic system
where p(t) = +–ee–t–t andep(t,t) =exp
To illustrate that the converse of Theorem . does not hold in general, we give the following example.
Example .[, Example .] LetTbe the unbounded above time scales withμ(t) < for eacht∈T. We consider the nonlinear dynamic equation
x=f(t,x) = – x+x
, x(t
) =x= (.)
and its variational dynamic equation
v(t) =fx(t, )v(t) = –
v(t), v(t) =v= , (.)
wherefx(t,x) = –+ x. Thenv= of (.) ish-stable, butx= of (.) is noth-stable.
Proof Since the fundamental solution isφ(t) =e–
(t,t)vfor eacht≥t, Eq. (.) is
h-stable with a positive bounded functionh(t) =e–
(t,ˆt) for a fixed pointˆt∈T. But (.)
is noth-stable because there exists an unbounded solutionx(t,ˆt, ) of (.) such that
x(t,ˆt, ) =x(t) >t, t∈T.
Competing interests
Authors’ contributions
All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Chungnam National University, Daejeon, 305-764, Korea.2Department of Mathematics, Yanbian University, Yanji, 133002, China.
Acknowledgement
This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2010-0008835). The authors are thankful to the anonymous referees for their valuable comments and corrections to improve this paper.
Received: 21 December 2011 Accepted: 18 July 2012 Published: 31 July 2012 References
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doi:10.1186/1687-1847-2012-129
