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Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2008, Article ID 712913, 12 pages
doi:10.1155/2008/712913
Research Article
WKB Estimates for 2 ×2 Linear Dynamic
Systems on Time Scales
Gro Hovhannisyan
Kent State University, Stark Campus, 6000 Frank Avenue NW, Canton, OH 44720-7599, USA
Correspondence should be addressed to Gro Hovhannisyan,
Received 3 May 2008; Accepted 26 August 2008
Recommended by Ond
ˇ
rej Do
ˇ
sl
´
y
We establish WKB estimates for 2 × 2 linear dynamic systems with a s mall parameter ε on a time
scale unifying continuous and discrete WKB method. We introduce an adiabatic invariant for 2 ×
2 dynamic system on a time scale, which is a generalization of adiabatic invariant of Lorentz’s
pendulum. As an application we prove that the change of adiabatic invariant is vanishing as ε
approaches zero. This result was known before only for a continuous time scale. We show that it is
true for the discrete scale only for the appropriate choice of graininess depending on a parameter
ε. The proof is based on the truncation of WKB s eries and WKB estimates.
Copyright q 2008 Gro Hovhannisyan. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Adiabatic invariant of dynamic systems on time scales
Consider the following system with a small parameter ε>0 on a time scale:
v
Δ
tAtεvt, 1.1
where v
Δ
is the delta derivative, vt is a 2-vector function, and
AtεAτ
a
11
τ ε
k
a
12
τ
ε
−k
a
21
τ a
22
τ
,τ tε, k is an integer. 1.2
WKB method 1, 2 is a powerful method of the description of behavior of solutions
of 1.1 by using asymptotic expansions. It was developed by Carlini 1817, Liouville, Green
1837 and became very useful in the development of quantum mechanics in 1920 1, 3.The
discrete WKB approximation was introduced and developed in 4–8.
The calculus of times scales was initiated by Aulbach and Hilger 9–11 to unify the
discrete and continuous analysis.
In this paper, we are developing WKB approximations for the linear dynamic systems
on a time scale to unify the discrete and continuous WKB theory. Our formulas for WKB series
2 Advances in Difference Equations
are based on the representation of fundamental solutions of dynamic system 1.1 given in
12. Note that the WKB estimate see 2.21 below has double asymptotical character and it
shows that the error could be made small by either ε→0, or t→∞.
It is well known 13, 14 that the change of adiabatic invariant of harmonic oscillator
is vanishing with the exponential speed as ε approaches zero, if the frequency is an analytic
function.
In this paper, we prove that for the discrete harmonic oscillator even for a harmonic
oscillator on a time scale the change of adiabatic invariant approaches zero with the power
speed when the graininess depends on a parameter ε in a special way.
A time scale T is an arbitrary nonempty closed subset of the real numbers. If T has a
left-scattered minimum m, then T
k
T − m, otherwise T
k
T. Here we consider the time
scales with t ≥ t
0
, and sup T ∞.
For t ∈ T, we define forward jump operator
σtinf{s ∈ T,s>t}. 1.3
The forward graininess function μ : T→0, ∞ is defined by
μtσt − t. 1.4
If σt >t, we say that t is right scattered. If t<∞ and σtt, then t is called right dense.
For f : T→R and t ∈ T
k
define the delta see 10, 11 derivative f
Δ
t to be the
number provided it exists with the property that f or given any >0, there exist a δ>0and
a neighborhood U t − δ, t δ ∩ T of t such that
|fσt − fs − f
Δ
tσt − s|≤|σt − s| 1.5
for all s ∈ U.
For any positive ε define auxilliary “slow” time scales
T
ε
{εt τ, t ∈ T} 1.6
with forward jump operator and graininess function
σ
1
τinf{sε ∈ T
ε
,sε>τ},μ
1
τεμt,τ tε. 1.7
Further frequently we are suppressing dependence on τ tε or t. To distinguish the
differentiation by t or τ we show the argument of differentiation in parenthesizes: f
Δ
t
f
Δ
t
t or f
Δ
τf
Δ
τ
τ.
Assuming A, θ
j
∈ C
1
rd
see 10 for the definition of rd-differentiable function, denote
TrAτa
11
τa
22
τ, det Aτa
11
τa
22
τ − a
12
τa
21
τ,
λτ
TrAτ
2
− 4|Aτ|
2a
12
τ
,
1.8
Hov
j
tθ
2
j
t − θ
j
tTrAτdet Aτ − εa
12
τ1 μθ
j
a
11
− θ
j
a
12
Δ
τ, 1.9
Q
0
τ
Hov
1
− Hov
2
θ
1
− θ
2
,Q
1
τ
θ
1
− a
11
Hov
2
− θ
2
− a
11
Hov
1
a
12
θ
1
− θ
2
, 1.10
Kτ2μtmax
j1,2
1
e
j
e
3−j
2Hov
j
θ
1
− θ
2
εa
12
1 μθ
j
θ
1
− θ
2
a
11
− θ
j
a
12
Δ
τ
|θ
j
|
, 1.11
Gro Hovhannisyan 3
where j 1, 2,θ
1,2
t are unknown phase functions, · is the Euclidean matrix norm, and
{e
j
t}
j1,2
are the exponential functions on a time scale 10, 11:
e
j
t ≡ e
θ
j
t, t
0
exp
t
t
0
lim
pμs
log1 pθ
j
sΔs
p
< ∞,j 1, 2. 1.12
Using the ratio of Wronskians formula proposed in 15 we introduce a new definition
of adiabatic invariant of system 1.1
Jt, θ, v, ε−
v
1
tθ
1
t − a
11
τ − v
2
ta
12
τv
1
tθ
2
t − a
11
τ − v
2
a
12
τ
θ
1
− θ
2
2
te
θ
1
te
θ
2
t
, 1.13
Theorem 1.1. Assume a
12
τ
/
0,A,θ∈ C
1
rd
T
ε
, and for some positive number β and any natural
number m conditions
|1 μTrA Q
0
μ
2
det A θ
1
Q
0
− Hov
1
|τ ≥ β, ∀τ ∈ T
ε
, 1.14
Kτ ≤ const, ∀τ ∈ T
ε
, 1.15
∞
tε
1
e
j
e
3−j
Hov
j
θ
1
− θ
2
τΔτ ≤ C
0
ε
m1
,j 1, 2, 1.16
are satisfied, where the positive parameter ε is so small that
0 ≤
2C
0
1 Kτ
β
ε
m
≤ 1. 1.17
Then for any solution vt of 1.1 and for all t
1
,t
2
∈ T, the estimate
Jv, ε ≡|Jt
1
,v,ε − Jt
2
,v,ε|≤C
3
ε
m
1.18
is true for some positive constant C
3
.
Checking condition 1.16 of Theorem 1.1 is based on the construction of asymptotic
solutions in the form of WKB series
vtC
1
e
θ
1
t, t
0
C
2
e
θ
2
t, t
0
, 1.19
where τ tε, and
θ
1,2
t
∞
j0
ε
j
ζ
j±
τ,θ
Δ
1,2
t
∞
k0
ε
k1
ζ
Δ
k±
τ. 1.20
Here the functions ζ
0
τ,ζ
0−
τ are defined as
ζ
0±
τ
TrA
2
± a
12
λ, ζ
1±
τ−
1 μζ
0±
2λ
λ ∓
a
11
− a
22
2a
12
Δ
τ, 1.21
4 Advances in Difference Equations
where λτ is defined in 1.8,andζ
k
τ,ζ
k−
τ,k 2, 3, are defined by recurrence
relations
ζ
k±
τ∓
1 μζ
0±
2λ
ζ
k−1±
a
12
Δ
τ∓
k−1
j1
ζ
j±
2λ
ζ
k−j±
a
12
μ
ζ
k−1−j±
− a
11
δ
j,k−1
a
12
Δ
τ
,
1.22
δ
jk
is the Kroneker symbol δ
jk
1, if k j,andδ
kj
0 otherwise.
Denote
Z
1
τZζ
0
τ,Z
2
τZζ
0−
τ, 1.23
Zζ
0
a
12
1 μζ
0
ζ
m
a
12
Δ
m
j1
ζ
j
ζ
m1−j
εζ
m2−j
a
12
μ
ζ
m−j
− a
11
δ
j,m
εζ
m1−j
a
12
Δ
τ
.
1.24
In the next Theorem 1.2 by truncating series 1.20:
θ
1
t
m
k0
ε
k
ζ
k
,θ
2
t
m
k0
ε
k
ζ
k−
, 1.25
where ζ
k±
t,k 1, 2, ,m are given in 1.21 and 1.22, we deduce estimate 1.16 from
condition 1.26 below given directly in the t erms of matrix Aτ.
Theorem 1.2. Assume that a
12
τ
/
0,A,θ∈ C
1
rd
T
ε
, and conditions 1.14, 1.15 , 1.17, and
∞
tε
1
e
j
e
3−j
Z
j
τ
θ
1
− θ
2
Δτ ≤ C
0
,j 1, 2, 1.26
are satisfied. Then, estimate 1.18 is true.
Note that if a
11
a
22
, then formulas 1.21 and 1.22 are simplified:
ζ
0±
τa
11
τ ± a
12
λτ,ζ
1±
−
1 μζ
0±
τλ
Δ
τ
2λτ
, 1.27
where from 1.8
λτ
a
12
τa
21
τ
a
12
τ
. 1.28
Taking m 1in1.25 and ζ
0±
t,ζ
1±
t as in 1.21, we have
θ
1
tζ
0
tεζ
1
t,θ
2
tζ
0−
tεζ
1−
t, 1.29
which means that in 1.20 ζ
2±
ζ
3±
··· 0, and from 1.24
Zζ
0
ζ
2
1
a
12
1 μζ
0
ζ
1
a
12
Δ
μa
12
ζ
1
ζ
0
− a
11
εζ
1
a
12
Δ
. 1.30
Gro Hovhannisyan 5
Example 1.3. Consider system 1.1 with a
11
a
22
. Then for continuous time scale T R we
have μ 0, and by picking m 1in1.25 we get by direct calculations ζ
1
ζ
1−
and
Hovθ
1
Hovθ
2
Zζ
0
Zζ
0−
. 1.31
In view of
Z
1
Z
2
ζ
2
1
a
12
ζ
1
a
12
Δ
λ
τ
λ
2
− 2a
12
λ
τ
a
12
λ
τ
λ
1/2
τ
a
−1
12
τλ
−1/2
τ
τ
τ
, 1.32
condition 1.26 under the assumption Rλ0 turns to
∞
0
a
−1
12
τλ
−1/2
τ
a
−1
12
τλ
−1/2
τ
τ
τ
Δτ<C
0
, 1.33
and from Theorem 1.2 we have the following corollary.
Corollary 1.4. Assume that a
−1
12
∈ C
1
0, ∞,λ∈ C
2
0, ∞, Rλτ ≡ 0,a
11
τ ≡ a
22
τ, and
1.33 is satisfied. Then for ε ≤ 1/C
0
estimate 1.18 with m 1 is true for all solutions vt of
system 1.1 on continuous time scale T R.
If a
12
1,then1.33 turns to
∞
t
0
ε
|λ
−1/2
τλ
−1/2
τ
ττ
|Δτ<C
0
, 1.34
and for λτ
√
a
21
iτ
−2γ
it is satisfied for any real γ.
If λτ is an analytic function, then it is known (see [13]) that the change of adiabatic invariant
approaches zero with exponential speed as ε approaches zero.
Example 1.5. Consider harmonic oscillator on a discrete time scale T εZ,
u
ΔΔ
tw
2
tεut0,t∈ εZ, 1.35
which could be written in form 1.1, where
A
01
−w
2
tε 0
,v
u
u
Δ
. 1.36
Choosing m 1 from formulas 1.27 and 1.29 we have λτiwτ, and
θ
1
tζ
0
εζ
1
iwτ −
εw
Δ
τ
2wτ
−
iεμw
Δ
τ
2
,τ tε,
θ
2
tζ
0−
εζ
1−
−iwτ −
εw
Δ
τ
2wτ
iεμw
Δ
τ
2
.
1.37
From 1.13 we get
Jt, v, ε
v
2
tiwτv
1
tv
2
t − iwτv
1
t
2wτ − εμtw
Δ
τ
2
e
θ
1
te
θ
2
t
, 1.38
6 Advances in Difference Equations
or
Jt, u, ε
u
Δ
t
2
w
2
τu
2
t
2wτ − εμtw
Δ
τ
2
e
η
, 1.39
η θ
1
θ
2
μθ
1
θ
2
−
εw
Δ
τ
w
μεw
Δ
2
4w
2
μ
w −
εμw
Δ
2
2
. 1.40
If we choose
wτ
aε
2
τ
2
bε
3
τ
3
a
t
2
b
t
3
,λτ
a
21
τiwτ, 1.41
then all conditions of Theorem 1.2 are satisfied see proof of Example 1.5 in the next section
for any real numbers b, a
/
0, and estimate 1.18 with m 1istrue.
Note that for continuous time scale we have μ 0, and 1.39 turns to the formula of
adiabatic invariant for Lorentz’s pendulum 13:
Jt, v, ε
u
2
t
tw
2
tεu
2
t
4wtε
. 1.42
2. WKB series and WKB estimates
Fundamental system of solutions of 1.1 could be represented in form
vtΨtC δt, 2.1
where Ψt is an approximate fundamental matrix function and δt is an error vector
function.
Introduce the matrix function
Ht1 μtΨ
−1
tΨ
Δ
t
−1
Ψ
−1
tAtΨt − Ψ
Δ
t. 2.2
In 16, the following theory was proved.
Theorem 2.1. Assume there exists a matrix function Ψt ∈ C
1
rd
T
∞
such that H∈R
rd
, the
matrix function ΨμΨ
∇
is invertible, and the following exponential function on a time scale is
bounded:
e
Ht
∞,texp
∞
t
lim
pμs
log1 pHsΔs
p
< ∞. 2.3
Then every solution of 1.1 can be represented in form 2.1 and the error vector function δt can be
estimated as
δt≤Ce
H
∞,t − 1, 2.4
where · is the Euclidean vector (or matrix) norm.
Gro Hovhannisyan 7
Remark 2.2. If μt ≥ 0, then from 2.4 we get
δt≤C
e
∞
t
HsΔs
− 1
. 2.5
Proof of Remark 2.2. Indeed if x ≥ 0, the function fxx − log1 x is increasing, so fx ≥
f0, log1 x ≤ x, and from p ≥ 0, Ht≥0weget
log1 pHs
p
≤Hs, 2.6
and by integration
∞
t
lim
pμs
log1 pHs
p
Δs ≤
∞
t
HsΔs, 2.7
or
e
H
t, ∞ − 1 ≤−1 exp
∞
t
HsΔs. 2.8
Note that from the definition
σ
1
τεσt,μ
1
τεμt,q
Δ
tεq
Δ
τ
τ. 2.9
Indeed
εσtε inf
s∈T
{s, s > t} inf
εs∈T
ε
{εs, s > t} inf
εs∈T
ε
{εs, εs > εt} σ
1
εtσ
1
τ,
σ
1
τεσt,μ
1
τσ
1
tε − εt εσt − tεμt,
qεσt qtεεμtq
Δ
τ
τqtεμtq
Δ
t.
2.10
If a
12
τ
/
0, then the fundamental matrix Ψt in 2.1 is given by see 12
Ψt
e
θ
1
t e
θ
2
t
U
1
te
θ
1
t U
2
te
θ
2
t
,U
j
t
θ
j
t − a
11
t
a
12
t
. 2.11
Lemma 2.3. If conditions 1.14, 1.15 are satisfied, then
Ht≤
21 Kτ
β
max
j1,2
1
e
j
t
e
3−j
t
Hov
j
t
θ
1
t − θ
2
t
,t∈ T, 2.12
where the functions Hov
j
t,Kτ are defined in 1.9, 1.11.
Proof. Denote
Ω1 μΨ
−1
Ψ
Δ
,MΨ
−1
AΨ − Ψ
Δ
. 2.13
8 Advances in Difference Equations
By direct calculations see 12,wegetfrom2.11
M
1
θ
1
− θ
2
⎛
⎜
⎝
−Hov
1
−
e
2
Hov
2
e
1
e
1
Hov
1
e
2
Hov
2
⎞
⎟
⎠
, Ψ
Δ
Ψ
−1
a
11
a
12
a
21
Q
1
a
22
Q
0
. 2.14
Using 2.14,weget
det ΩdetΨΩΨ
−1
det1 μΨ
Δ
Ψ
−1
1 μQ
0
TrAμ
2
det A a
11
Q
0
− a
12
Q
1
,
2.15
and from 1.14
|detΩ| |1 μQ
0
TrAμ
2
det A a
11
Q
0
− a
12
Q
1
|≥β>0,
Ω
−1
Ω
co
|det Ω|
≤
Ω
|det Ω|
≤
Ω
β
,HΩ
−1
M,
Ψ
−1
AΨ
1
θ
1
− θ
2
⎛
⎜
⎜
⎝
−θ
2
1
θ
1
TrA − det A −
e
2
θ
2
2
− θ
2
TrA det A
e
1
e
1
θ
2
1
− θ
1
TrA det A
e
2
θ
2
2
− θ
2
TrA det A
⎞
⎟
⎟
⎠
θ
1
0
0 θ
2
,
M≤2 max
j1,2
1
e
j
e
3−j
Hov
j
θ
1
− θ
2
.
2.16
So by using 1.9, we have
Ψ
−1
AΨ≤2 max
j1,2
1
e
j
e
3−j
Hov
j
θ
1
− θ
2
εa
12
1 μθ
j
a
11
− θ
j
/a
12
Δ
τ
θ
1
− θ
2
|θ
j
|
Ω 1 μΨ
−1
AΨ − M≤1 μΨ
−1
AΨ M.
2.17
From 2.2, 2.13, 2.17,weget2.12 in view of
H≤Ω
−1
·M≤
Ω
β
M≤
1 K
β
M. 2.18
Proof of Theorem 1.1. From 1.16 changing variable of integration τ εs, we get
∞
t
MsΔs ≤
∞
t
2 max
j1,2
1
e
j
s
e
3−j
s
Hov
j
s
θ
1
s − θ
2
s
Δs ≤ 2C
0
ε
m
,j 1, 2. 2.19
So using 2.12,weget
∞
t
HsΔs ≤
∞
t
1 Kεs
β
MsΔs ≤ cC
0
ε
m
. 2.20
Gro Hovhannisyan 9
From this estimate and 2.5, we have
δt≤C
e
∞
t
HsΔs
− 1
≤C
e
C
0
cε
m
− 1
≤ eCC
0
cε
m
, 2.21
where ε is so small that 1.17 is satisfied. The last estimate follows from the inequality e
x
−1 ≤
ex, x ∈ 0, 1. Indeed because gxex 1 − e
x
is increasing for 0 ≤ x ≤ 1, we have
gx ≥ g0.
Further from 2.1, 2.11, we have
v
1
C
1
δ
1
e
θ
1
C
2
δ
2
e
θ
2
,v
2
C
1
δ
1
U
1
e
θ
1
C
2
δ
2
U
2
e
θ
2
. 2.22
Solving these equation for C
j
δ
j
,weget
C
1
δ
1
v
1
U
2
− v
2
U
2
− U
1
e
θ
1
,C
2
δ
2
v
2
− v
1
U
1
U
2
− U
1
e
θ
2
. 2.23
By multiplication see 1.12,weget
JtC
1
δ
1
tC
2
δ
2
t C
1
C
2
C
2
δ
1
tC
1
δ
2
tδ
1
tδ
2
t,
Jt
1
− Jt
2
C
2
δ
1
t
1
− δ
1
t
2
C
1
δ
2
t
1
− δ
2
t
2
δ
1
t
1
δ
2
t
1
− δ
1
t
2
δ
2
t
2
,
2.24
and using estimate 2.21, we have
|Jt
1
,θ,v,ε − Jt
2
,θ,v,ε|≤C
3
ε
m
. 2.25
Proof of Theorem 1.2. Let us look for solutions of 1.1 in the form
vtΨtC, 2.26
where Ψ is given by 2.11, and functions θ
j
are given via WKB series 1.20.
Substituting series 1.20 in 1.9 ,weget
Hovθ
1
∞
r,j0
ζ
r
ε
r
ζ
j
ε
j
− TrA
∞
r0
ζ
r
ε
r
det A
a
12
ε
1 μ
∞
r0
ζ
r
ε
r
∞
j0
ζ
j
ε
j
− a
11
a
12
Δ
τ,
2.27
or
Hovθ
1
≡
∞
k0
b
k
τε
k
. 2.28
To make Hovθ
1
asymptotically equal zero or Hovθ
1
≡ 0 we must solve for ζ
k
the equations
b
k
τ0,k 0, 1, 2 2.29
10 Advances in Difference Equations
By direct calculations from the first quadratic equation
b
0
ζ
2
0
− ζ
0
TrA detA0, 2.30
and the second one
b
1
τ2ζ
1
ζ
0
− ζ
1
TrA a
12
1 μζ
0
ζ
0
− a
11
a
12
Δ
0, 2.31
we get two solutions ζ
j±
given by 1.21 and 1.22.Notethat
ζ
0
− a
11
a
12
a
22
− a
11
2a
12
λ,
ζ
0−
− a
11
a
12
a
22
− a
11
2a
12
− λ,
ζ
1
− ζ
1−
a
12
μλ
Δ
2 μTrA
2λ
a
11
− a
22
2a
12
Δ
.
2.32
Furthermore from k 1th equation
b
k
2ζ
0
− TrAζ
k
a
12
1 μζ
0
ζ
k−1
a
12
Δ
k−1
j1
ζ
j
ζ
k−j
a
12
μ
ζ
k−1−j
− a
11
δ
j,k−1
a
12
Δ
τ
0,
2.33
we get recurrence relations 1.22.
In view of Theorem 1.1, to prove Theorem 1.2 it is enough to deduce condition 1.16
from 1.26. By truncation of series 1.20 or by taking
ζ
k
ζ
k−
0,k m 1,m 2, , 2.34
we get 1.25. Defining ζ
j±
,j 1, 2, ,mas in 1.21 and 1.22, we have
b
0
b
1
··· b
m−1
b
m
b
m3
b
m4
··· 0,
b
m1
a
12
1 μζ
0
ζ
m
a
12
Δ
m
j1
ζ
j
ζ
m1−j
a
12
μ
ζ
m−j
− a
11
δ
j,m
a
12
Δ
τ
,
b
m2
m
j1
ζ
j
ζ
m2−j
a
12
μ
ζ
m1−j
a
12
Δ
τ
.
2.35
Now 1.16 follows from 1.26 in view of
Hovθ
k
ε
m1
b
m1
b
m2
εε
m1
Z
k
,k 1, 2. 2.36
Gro Hovhannisyan 11
Note that from 1.13 and the estimates
log |1 pθ|≤log
1 2pRθp
2
|θ|
2
≤
1
2
|2pRθp
2
|θ|
2
|,
log |1 pθ|≤log
1 2pRθp
2
|θ|
2
≤
|2pRθp
2
|θ|
2
|,
2.37
it follows
|e
θ
t, t
0
|≤exp
t
t
0
Rθs
μs|θs|
2
2
Δs, 2.38
|e
θ
t, t
0
|≤exp
t
t
0
|θs|
2
2Rθs
μs
Δs, μs > 0. 2.39
Proof of Example 1.5. From 1.37, 1.41, we have
θ
1
− θ
2
i2wτ − εμw
Δ
τ,θ
1
θ
2
−
εw
Δ
τ
w
,θ
1
θ
2
εw
Δ
2
4w
2
w −
εμw
Δ
2
2
,
η
1
t
θ
1
− θ
2
1 μθ
2
2ia
t
2
Ot
−3
,η
2
t
θ
2
− θ
1
1 μθ
1
−2ia
t
2
Ot
−3
,τ−→ ∞,
2.40
and using 2.39,weget
e
θ
1
e
θ
2
≤|e
η
1
|≤const,
e
θ
2
e
θ
1
≤|e
η
2
|≤const. 2.41
Further for τ→∞
ζ
1±
−
λ
Δ
2λ
∓
λ
Δ
2
1
τ
bε − 3aμ
2aτ
2
1
τ
3
2μ
2
−
3bεμ
2a
−
b
2
ε
2
2a
2
± iaε
2
Oτ
−4
,
Z
1
ζ
2
1
ζ
Δ
1
εζ
Δ
1
ζ
1
Oτ
−4
μ − ε
τ
3
Oτ
−4
Oτ
−4
,Z
2
Z
1
Oτ
−4
.
2.42
So if μ ε, then 1.26 and all other conditions of Theorem 1.2 are satisfied, and 1.18 is true
with m 1.
Acknowledgment
The author wants to thank Professor Ondrej Dosly for his comments that helped improving
the original manuscript.
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12 Advances in Difference Equations
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