linear difference equations with discrete transforms method abdul j jerri
Báo cáo hóa học: "PERTURBATION METHOD FOR LINEAR DIFFERENCE EQUATIONS WITH SMALL PARAMETERS" pot
Ngày tải lên :
22/06/2014, 22:20
... =
∞
j= 0
ε
j
y
( j)
t,k
+ ε
k
∞
j= 0
ε
j
w
( j)
0,k
+ ···+ ε
k−m+1
∞
j= 0
ε
j
w
( j)
m
−1,k
,0≤ k ≤ N, (3.8)
where
∞
j= 0
ε
j
y
( j)
t,k
is the outer series and
∞
j= 0
ε
j
w
( j)
s,k
, s = 0, ... (3.8 )wesee
that
y
k
(ε) =
+∞
j= 0
ε
j
y
( j)
k
,0≤ k ≤ N, (3.19)
where
y
( j)
k
= y
( j)
t,k
+ w
( j k)
0,k
+ w
( j k+1)
1,k
+ ···+ w
( j k+m−1)
m
−1,k
,0≤ k ≤ N, j ≥ 0. (3.20)
Proposition 3.4. ... j)
s,k+m
+ a
m−1
w
( j)
s,k+m
−1
+ ···+ a
0
w
( j)
s,k
=−a
m+1
w
( j 1)
s,k+m+1
− a
m+2
w
( j 2)
s,k+m+2
−···
−
a
m+n
w
( j n)
s,k+m+n
, k = 0,1, ,
(3.16)
w
( j)
s,s
=−y
( j)
t,s
, w
( j)
s,k
= 0ifk...
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Báo cáo hóa học: "ON LINEAR VOLTERRA DIFFERENCE EQUATIONS WITH INFINITE DELAY" pdf
Ngày tải lên :
22/06/2014, 22:20
... 1+
∞
j= 1
λ
− j
1 −
1 −
1
λ
j
G
j
+
1
λ
∞
j= 1
λ
− j
jK
j
≥ 1 −
∞
j= 1
λ
− j
1+
1+
1
λ
j
G
j
−
1
λ
∞
j= 1
λ
− j
j
K
j
> 1 −
∞
j= 1
γ
− j
1+
1+
1
γ
j
G
j
−
1
γ
∞
j= 1
γ
− ... λ>1
F
(λ) = 1 −
∞
j= 1
λ
− j
G
j
+
1 −
1
λ
∞
j= 1
λ
− j
j
G
j
−
1
λ
∞
j= 1
λ
− j
j
K
j
> 1 −
∞
j= 1
G
j
−
∞
j= 1
j
K
j
.
(4.75)
Hence, by ... e
γ
λ
0
=
∞
j= 1
λ
− j
0
1 −
1 −
1
λ
0
j
G
j
+
1
λ
0
∞
j= 1
λ
− j
0
jK
j
,
μ
λ
0
=
∞
j= 1
λ
− j
0
1+
1 −
1
λ
0
j
G
j
+
1
λ
0
∞
j= 1
λ
− j
0
j
K
j
.
(2.12)
Property...
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Báo cáo " Stability Radii for Difference Equations with Time-varying Coefficients " ppt
Ngày tải lên :
22/03/2014, 11:20
... M
2
x
0
K
d
,
(9)
for all x
0
∈ K
d
.
VNU Journal of Science, Mathematics - Physics 26 (2010) 175-184
Stability Radii for Difference Equations with Time-varying
Coefficients
Le Hong Lan*
Department ... Vietnam
Received 10 August 2010
Abstract. This paper deals with a formula of stability radii for an linear difference equation
(LDEs for short) with the coefficients varying in time under structured ... operator. The result is considered as an discrete version of
a previous result for time-varying ordinary differential equations [1].
Keywords: Robust stability, Linear difference equation, Input-output...
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Báo cáo hóa học: " Difference inequality for stability of impulsive difference equations with distributed delays" pdf
Ngày tải lên :
21/06/2014, 03:20
... with
distributed delays
⎧
⎪
⎪
⎨
⎪
⎪
⎩
x
i
(m +1)=a
i
x
i
(m)+
n
j= 1
b
ij
f
j
(x
j
(m)) +
n
j= 1
c
ij
∞
k=1
μ
ij
(k)g
j
(x
j
(m − k)), m ∈ Z
+
∞
, m = m
k
,
x
i
(m +1)=H
im
(x
1
(m), , x
m
(m)), ... doi:10.1016 /j. camwa.2007.10.020
10. Song, QK, Cao, JD: Dynamical behaviors of discrete- time fuzzy cellular neural networks with variable delays and
impulses. J Franklin Inst. 345,39–59 (2008). doi:10.1016 /j. jfranklin.2007.06.001
11. ... et al.: Difference inequality for stability of impulsive difference equations with distributed
delays. Journal of Inequalities and Applications 2011 2011:8.
Submit your manuscript to a
journal...
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báo cáo hóa học:" Research Article Error Bounds for Asymptotic Solutions of Second-Order Linear Difference Equations II: The First Case" pot
Ngày tải lên :
21/06/2014, 11:20
... approximation of j
q
b j y
1
j /y
1
j 2 is
j
q
b
j
y
1
j
y
1
j 2
b
0
j
ρ
2
1
1
α
2
− α
1
j
σ
j
,
3.39
where σ j is the reminder and σ
0
sup
j
{j
2
|σ j |} is a ... equation”
h
n
∞
j n
K
n, j
R
j
− j
p
φ
j
h
j 1
− j
q
ψ
j
h
j
.
2.1
Lemma 2.1. Let Kn, j , φ j , ψ j , R j be real or complex functions of integer variables n, j; p and
q ... h
s
n
|
∞
j n
K
n, j
j
p
φ
j
h
s
j 1
h
s−1
j 1
j
q
ψ
j
h
s
j
−h
s−1
j
∞
j n
2C
R
C
2
K
P
n
ςN θ
p
j
j
−θ
j
p−s
C
φ
λ
s−1
P
j...
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- 0
báo cáo hóa học:" Research Article Regularly Varying Solutions of Second-Order Difference Equations with Arbitrary Sign Coefficient" ppt
Ngày tải lên :
21/06/2014, 11:20
... get
w
k
∞
j k
p
j
∞
j k
w
2
j
1 w
j
;
3.5
8 Advances in Difference Equations
k ≥ a. Thanks to 3.22 and 3.24,weget
K
2
k
≤
1
h
k
∞
j k
h
j
j
j
Gw
j
≤
1
h
k
∞
j k
h
j
j
·
1 ... where H
1
k
|1/h
k
∞
j k
h
j
/j w
2
j
−v
2
j
|,
H
2
k
|1/h
k
∞
j k
h
j
/j Gw
j
− Gv
j
|,andH
3
k
|1/h
k
∞
j k
Δh
j
Δw
j
− v
j
|.Inview
of 3.22, we have
H
1
k
1
h
k
∞
j k
h
j
j
w
j
− ... that
lim
k
∞
j k
Δ
2
h
j
h
k
0.
3.19
Indeed, first note that
∞
j k
|Δh
j
|≤1 −2
1
2sup
j k
|ψ
j
|
∞
j k
h
j
/j < ∞,andso
∞
j k
|Δ
2
h
j
|≤
∞
j k
|Δh
j
| |Δh
j 1
| <...
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- 0
Báo cáo hóa học: " Research Article Inequalities among Eigenvalues of Second-Order Difference Equations with General Coupled Boundary Conditions" pptx
Ngày tải lên :
21/06/2014, 20:20
... k
12
N−1
j 0
w
j
ψ
j
Δϕ
N−1
ψ
j
− ϕ
j
Δψ
N−1
−
k
21
− k
22
N−2
j 0
w
j
ψ
j
ϕ
N−1
ψ
j
− ϕ
j
ψ
N−1
− k
12
N−1
j 0
w
j
ϕ
j
Δϕ
N−1
ψ
j
− ϕ
j
Δψ
N−1
N−1
j 0
w
j
δ
j
,
3.31
where
δ
j
:
k
11
− ... that
Δϕ
n−1
λ
n−1
j 0
w
j
ϕ
j
λ
Δϕ
n−1
λ
ψ
j
λ
− ϕ
j
λ
Δψ
n−1
λ
,
Δψ
n−1
λ
n−1
j 0
w
j
ψ
j
λ
Δϕ
n−1
λ
ψ
j
λ
− ϕ
j
λ
Δψ
n−1
λ
.
3.30
6 ... 3.27–3.28, we have
ϕ
n
λ
n−1
j 0
w
j
ϕ
j
λ
ϕ
n
λ
ψ
j
λ
− ϕ
j
λ
ψ
n
λ
,
ψ
n
λ
n−1
j 0
w
j
ψ
j
λ
ϕ
n
λ
ψ
j
λ
− ϕ
j
λ
ψ
n
λ
.
3.29
It...
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Báo cáo hoa học: " Research Article Asymptotic Behavior of Impulsive Infinite Delay Difference Equations with Continuous Variable" docx
Ngày tải lên :
21/06/2014, 20:20
... condition”
φ
i
0
a
i
φ
i
−τ
1
n
j 1
a
ij
f
j
φ
j
−τ
1
n
j 1
b
ij
g
j
φ
j
−τ
2
0
−∞
p
ij
−s
h
j
φ
j
s
ds I
i
.
2.2
By the method of steps, one can easily ... N
j
n
j 1
∞
0
q
rj
s
1 ε
z
j
e
−λ
t
∗
−s−t
0
N
j
ds I
r
n
j 1
p
rj
e
λτ
1
w
rj
e
λτ
2
∞
0
q
rj
s
e
λs
ds
z
j
1 ε
e
−λ
t
∗
−t
0
n
j 1
p
rj
... in Difference Equations
By using 3.1, 3.7–3.10,andq
ij
t ≥ 0, we have
u
r
t
∗
≤
n
j 1
p
rj
1 ε
z
j
e
−λ
t
∗
−τ
1
−t
0
N
j
n
j 1
w
rj
1 ε
z
j
e
−λ
t
∗
−τ
2
−t
0
...
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Báo cáo hoa học: " Research Article Summation Characterization of the Recessive Solution for Half-Linear Difference Equations" doc
Ngày tải lên :
21/06/2014, 20:20
... Project MSM0022162409 of the Czech Ministry of Education.
References
1 P.
ˇ
Reh
´
ak, “Hartman-Wintner type lemma, oscillation, and conjugacy criteria for half -linear difference
equations, ” Journal ... half -linear difference equations, ” Czechoslovak
Mathematical Journal, vol. 51, no. 2, pp. 303–321, 2001.
3 P.
ˇ
Reh
´
ak, “Generalized discrete Riccati equation and oscillation of half -linear ... Discrete Oscillation Theory, Hindawi, New
York, NY, USA, 2005.
7 O. Do
ˇ
sl
´
yandP.
ˇ
Reh
´
ak, “Recessive solution of half -linear second order difference equations, ” Journal
of Difference Equations...
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Báo cáo hóa học: "Research Article Stability of Equilibrium Points of Fractional Difference Equations with Stochastic Perturbations" pdf
Ngày tải lên :
22/06/2014, 11:20
... case,
A
j
p q if j 0, ,r, A
j
q if j r 1, ,m,
B
j
1ifj 0, ,r, B
j
0ifj r 1, ,m.
4.4
In the second case,
A
j
p q if j 0, ,m, A
j
p if j m 1, ,r, B
j
1ifj 0, ... fractional difference equation
x
n1
μ
k
j 0
a
j
x
n j
λ
k
j 0
b
j
x
n j
,n∈ Z {0, 1, }, 1.1
with the initial condition
x
j
φ
j
,j Z
0
{−k,−k 1, ,0}, 1.2
B. Paternoster ... the
equilibrium point of 2.1.
Together with nonlinear equation 2.2 we will consider and its linear part
z
n1
k
j 0
γ
j
z
n j
σz
n
ξ
n1
,γ
j
a
j
− b
j
x
λ B x
. 2.3
Two following...
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Báo cáo hóa học: "Research Article Exponential Stability of Difference Equations with Several Delays: Recursive Approach" docx
Ngày tải lên :
22/06/2014, 11:20
... equations with several delays. For comparison, we will cite the
following two results.
Theorem A 9–14. Suppose that
a
j
n
≥ 0, 0 ≤ j ≤ m,
m
j 0
a
j
n
> 0,
∞
n0
m
j 0
a
j
n
... 0,
∞
n0
m
j 0
a
j
n
∞,
sup
n≥m
n
kn−m
m
j 0
a
j
k
<
3
2
1
2
m 1
.
3.14
Then the equation with several delays
x
n 1
− x
n
−
m
j 0
a
j
n
x
n − j
,n 0, 1, 2, 3.15
is ... I
a
l
n
x
h
l
n
k∈I
a
k
n
n−1
j h
k
n
x
j 1
− x
j
x
n
−
k∈I
a
k
n
x
n
−
l
/
∈ I
a
l
n
x
h
l
n
−
k∈I
a
k
n
n−1
j h
k
n
m
l1
a
l
j
x
h
l
j
x
n
1...
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Báo cáo hóa học: "DIFFERENCE EQUATIONS ON DISCRETE POLYNOMIAL HYPERGROUPS" doc
Ngày tải lên :
22/06/2014, 22:20
... numbers
a
j, i
, not all equal to zero for j = 1, ,k and i = 0, ,l
j
such that
k
j= 1
l
j
i=0
a
j, i
P
(i)
n
λ
j
=
0 (3.10)
holds with a
j, l
j
= 0. Translating (3.10)by1wehave
k
j= 1
l
j
i=1
a
j, i
λ
j
P
(i)
n
λ
j
+ ... P
(l
1
)
n
(λ
1
):
l
1
−1
i=0
c
1,i
P
(i)
n
λ
1
+
k
j= 2
a
j, l
j
λ
1
− λ
j
P
(l
j
)
n
λ
j
+
k
j= 2
l
j
−1
i=0
c
j, i
P
(i)
n
λ
j
=
0 (3.12)
with some constants c
j, i
, and this means that either k ... a
j, l
j
= 0. Translating (3.10)by1wehave
k
j= 1
l
j
i=1
a
j, i
λ
j
P
(i)
n
λ
j
+ P
(i−1)
n
λ
j
+
k
j= 1
a
j, 0
λ
j
P
n
λ
j
=
0, (3.11)
and if we subtract (3.10)timesλ
1
from this...
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Báo cáo hóa học: "A REMARK ON kTH-ORDER LINEAR FUNCTIONAL EQUATIONS WITH CONSTANT COEFFICIENTS" pdf
Ngày tải lên :
22/06/2014, 22:20
... ON kTH-ORDER LINEAR FUNCTIONAL
EQUATIONS WITH CONSTANT COEFFICIENTS
JITKA LAITOCHOV
´
A
Received 30 January 2006; Accepted 18 May 2006
Abel functional equations are associated to a linear homogeneous ... <
λ
j
λ
1
< 1, j = 2,3, ,k, (3.13)
imply
lim
x→∞
c
j
(x)
λ
j
λ
1
X
∗
α(x)
= 0. (3.14)
Formally, at least,
lim
x→∞
f (x) = lim
x→∞
c
1
(x) λ
X
∗
α(x)
1
. (3.15)
8 A remark on linear ... element Φ in S according to multiplication ◦ in
group Ᏻ.
2. Linear functional equations of the kth order with constant coefficients
Let a
j
∈ R, j = 0,1,2, ,k. Then the equation
a
k
f ◦ Φ
k
(x)+a
k−1
f...
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