Oscillation of a class of nonlinear partial difference equations with continuous variables
This paper is concerned with a class of nonlinear partial difference equations with continuous variables. Some oscillation criteria are obtained using an integral transformation and inequalities. Розглянуто клас нелiнiйних частково рiзницевих рiвнянь з неперервними змiнними. Отримано деякi критерi...
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Guo, Y. Liu, A. Liu, T. Ma, Q. 2021-01-28T19:31:37Z 2021-01-28T19:31:37Z 2010 Oscillation of a class of nonlinear partial difference equations with continuous variables / Y. Guo, A. Liu, T. Liu, Q. Ma // Нелінійні коливання. — 2010. — Т. 13, № 3. — С. 305-313. — Бібліогр.: 11 назв. — англ. 1562-3076 https://nasplib.isofts.kiev.ua/handle/123456789/174955 517.9 This paper is concerned with a class of nonlinear partial difference equations with continuous variables. Some oscillation criteria are obtained using an integral transformation and inequalities. Розглянуто клас нелiнiйних частково рiзницевих рiвнянь з неперервними змiнними. Отримано деякi критерiї осциляцiї з використанням iнтегральних перетворень та нерiвностей. This work was supported by Natural Science Foundation of China (Grant Nos.10661002), Guangxi Natural Science Foundation Grant No.0832065 and Research Foundation for Outstanding Young Teachers, China University of Geosciences (Wuhan) (No.CUGQNL0841). en Інститут математики НАН України Нелінійні коливання Oscillation of a class of nonlinear partial difference equations with continuous variables Осциляція класу нелінійних частково різницевих рівнянь з неперервними змінними Осцилляция класса нелинейных частично разностных уравнений с непрерывными переменными Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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Oscillation of a class of nonlinear partial difference equations with continuous variables |
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Oscillation of a class of nonlinear partial difference equations with continuous variables Guo, Y. Liu, A. Liu, T. Ma, Q. |
| title_short |
Oscillation of a class of nonlinear partial difference equations with continuous variables |
| title_full |
Oscillation of a class of nonlinear partial difference equations with continuous variables |
| title_fullStr |
Oscillation of a class of nonlinear partial difference equations with continuous variables |
| title_full_unstemmed |
Oscillation of a class of nonlinear partial difference equations with continuous variables |
| title_sort |
oscillation of a class of nonlinear partial difference equations with continuous variables |
| author |
Guo, Y. Liu, A. Liu, T. Ma, Q. |
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Guo, Y. Liu, A. Liu, T. Ma, Q. |
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2010 |
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English |
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Нелінійні коливання |
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Інститут математики НАН України |
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Article |
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Осциляція класу нелінійних частково різницевих рівнянь з неперервними змінними Осцилляция класса нелинейных частично разностных уравнений с непрерывными переменными |
| description |
This paper is concerned with a class of nonlinear partial difference equations with continuous variables.
Some oscillation criteria are obtained using an integral transformation and inequalities.
Розглянуто клас нелiнiйних частково рiзницевих рiвнянь з неперервними змiнними. Отримано
деякi критерiї осциляцiї з використанням iнтегральних перетворень та нерiвностей.
|
| issn |
1562-3076 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/174955 |
| citation_txt |
Oscillation of a class of nonlinear partial difference equations with continuous variables / Y. Guo, A. Liu, T. Liu, Q. Ma // Нелінійні коливання. — 2010. — Т. 13, № 3. — С. 305-313. — Бібліогр.: 11 назв. — англ. |
| work_keys_str_mv |
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| first_indexed |
2025-11-27T02:36:02Z |
| last_indexed |
2025-11-27T02:36:02Z |
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1850794727742373888 |
| fulltext |
UDC 517 . 9
OSCILLATION OF A CLASS OF NONLINEAR
PARTIAL DIFFERENCE EQUATIONS
WITH CONTINUOUS VARIABLES*
ОСЦИЛЯЦIЯ КЛАСУ НЕЛIНIЙНИХ
ЧАСТКОВО РIЗНИЦЕВИХ РIВНЯНЬ
З НЕПЕРЕРВНИМИ ЗМIННИМИ
Y. Guo
Graduate School China Acad. Eng. Phys.
P. O. Box 2101, Beijing, 100088, P. R. China
Guangxi Univ. Technology
Liuzhou, 545006, P. R. China
A. Liu, T. Liu, Q. Ma
School Math. and Phys. China Univ. Geosci.
Wuhan, 430074, P. R. China
e-mail:wh_apliu@sina.com
This paper is concerned with a class of nonlinear partial difference equations with continuous variables.
Some oscillation criteria are obtained using an integral transformation and inequalities.
Розглянуто клас нелiнiйних частково рiзницевих рiвнянь з неперервними змiнними. Отримано
деякi критерiї осциляцiї з використанням iнтегральних перетворень та нерiвностей.
1. Introduction. Partial difference equations are difference equations which involve functions
with two or more independent variables. Such equations arise in investigation of random walk
problems, molecular structure problems [1], and numerical difference approximation problems
[2], etc. Recently, oscillation problems for partial difference equations with invariable coeffici-
ents and discrete variables have been investigated in [3 – 8]. We can further investigate oscillati-
on properties of nonlinear equations with variable coefficients and continuous variables and
obtain some oscillation criteria.
In this paper, we consider a class of nonlinear partial difference equations with continuous
variables,
p1(x, y)A(x + a, y + b) + p2(x, y)A(x + a, y) + p3(x, y)A(x, y + b)− p4(x, y)A(x, y)+
+
m∑
i=1
hi(x, y, A(x− σi, y − τi)) = 0, (1)
∗ This work was supported by Natural Science Foundation of China (Grant Nos.10661002), Guangxi Natural
Science Foundation Grant No.0832065 and Research Foundation for Outstanding Young Teachers, China University
of Geosciences (Wuhan) (No.CUGQNL0841).
c© Y. Guo, A. Liu, T. Liu, Q. Ma, 2010
ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 3 305
306 Y. GUO, A. LIU, T. LIU, Q. MA
where p1(x, y) ∈ C(R+×R+, [0,∞)); p2(x, y), p3(x, y), p4(x, y) ∈ C(R+×R+, (0,∞)); a, b, σi,
τi are negative and hi(x, y, u) ∈ C(R+ ×R+ ×R,R), i = 1, . . . ,m.
Let σ = maxi=1,...,m{σi}, τ = maxi=1,...,m{τi}. A solution of (1) is defined to be a conti-
nuous function A(x, y), for all x ≥ −σ, y ≥ −τ, which satisfies (1) on R+ × R+. A solution
A(x, y) of (1) is said to be oscillatory if it is neither eventually positive nor eventually negative.
Some oscillation criteria for a solution of (1) are obtained using integral transformation and
inequalities. Our results extend some oscillation properties of nonlinear equations with invari-
able coefficients and discrete variables to nonlinear equations with variable coefficients and
continuous variables.
2. Main lemmas. We assume that the following conditions are satisfied throughout this
paper:
(I) p1(x, y) ≥ p1 ≥ 0, p2(x, y) ≥ p2 > 0, p3(x, y) ≥ p3 > 0, 0 < p4(x, y) ≤ p4, and pi,
i = 1, 2, 3, 4, are constants and also satisfy p2, p3 ≥ p4;
(II) τi = kia + θi, σ = lib + ξi, i = 1, . . . ,m, where ki, li are nonnegative integers and
θi ∈ (a, 0], ξi ∈ (b, 0].
Lemma 1. Assume that
(i) hi ∈ C(R+ × R+ × R,R), uhi(x, y, u) > 0 for u 6= 0, and hi(x, y, u), i = 1, . . . ,m, is a
nondecreasing function in u;
(ii) hi(x, y, u), i = 1, . . . ,m, is convex in u for u ≥ 0.
Let A(x, y) be an eventually positive solution of (1), then there exists a positive function
Z(x, y) =
1
ab
∫ x
x+a
∫ y
y+b
A(u, v) du dv eventually satisfying the following results:
(1) if mini=1,...,m{ki} 6= 0 and mini=1,...,m{li} 6= 0, then
p1Z(x + a, y + b) + p2Z(x + a, y) + p3Z(x, y + b)− p4Z(x, y)+
+
m∑
i=1
hi(x, y, Z(x− σi, y − τi)) ≤ 0; (2)
(2) if mini=1,...,m{ki} = 0 and mini=1,...,m{li} = 0, then
p1Z(x + a, y + b) + p2Z(x + a, y) + p3Z(x, y + b)− p4Z(x, y)+
+
m∑
i=1
hi(x, y, Z(x− kia, y − lib)) ≤ 0; (3)
(3) if mini=1,...,m{ki} = 0 and mini=1,...,m{li} 6= 0, then
p1Z(x + a, y + b) + p2Z(x + a, y) + p3Z(x, y + b)− p4Z(x, y)+
+
m∑
i=1
hi(x, y, Z(x− σi, y − lib)) ≤ 0; (4)
ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 3
OSCILLATION OF A CLASS OF NONLINEAR PARTIAL DIFFERENCE EQUATIONS . . . 307
(4) if mini=1,...,m{ki} 6= 0 and mini=1,...,m{li} = 0, then
p1Z(x + a, y + b) + p2Z(x + a, y) + p3Z(x, y + b)− p4Z(x, y)+
+
m∑
i=1
hi(x, y, Z(x− kia, y − τi)) ≤ 0. (5)
Proof. From (I), we have the following inequality:
p4(A(x + a, y) + A(x, y + b)−A(x, y)) ≤
≤ p1A(x + a, y + b) + p2A(x + a, y) + p3A(x, y + b)− p4A(x, y) ≤
≤ p1(x, y)A(x + a, y + b) + p2(x, y)A(x + a, y)+
+ p3(x, y)A(x, y + b)− p4(x, y)A(x, y) < 0
eventually.
Since
Z(x, y) =
1
ab
x∫
x+a
y∫
y+b
A(u, v) du dv, (6)
we have
∂Z(x, y)
∂x
=
1
ab
y∫
y+b
(A(x, v)−A(x + a, v))dv > 0 (7)
and
∂Z(x, y)
∂y
=
1
ab
x∫
x+a
(A(u, y)−A(u, y + b))du > 0. (8)
From the above, we have Z(x, y) is nondecreasing in x and y eventually.
Integrating (1), from (I) we have
p1Z(x + a, y + b) + p2Z(x + a, y) + p3Z(x, y + b)− p4Z(x, y)+
+
1
ab
m∑
i=1
x+a∫
x
y+b∫
y
hi(u, v,A(u− σi, v − τi)) dv du ≤ 0.
By (i), (ii), and Jensen’s inequality, we obtain the following inequality:
p1Z(x+a, y + b)+p2Z(x+a, y)+p3Z(x, y + b)−p4Z(x, y)+
m∑
i=1
hi(x, y, Z(x−σi, y− τi)) ≤ 0.
ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 3
308 Y. GUO, A. LIU, T. LIU, Q. MA
Thus (2) holds.
Since a, b, τi, σi are negative real numbers, there exist nonnegative integers ki and li sati-
sfying σi = kia + θi, τi = lib + ξi, where a < θi ≤ 0, b < ξi ≤ 0, i = 1, 2, . . . ,m. From (7) and
(8), we obtain Z(x, y) is nondecreasing eventually. So if
min
i=1,...,m
{ki} 6= 0, min
i=1,...,m
{li} 6= 0,
we have Z(x−σi, y−τi) ≥ Z(x−kia, y− lib), i = 1, 2, . . . ,m. Since hi(x, y, u), i = 1, 2, . . . ,m,
is nondecreasing in u, we have
p1Z(x+a, y+b)+p2Z(x+a, y)+p3Z(x, y+b)−p4Z(x, y)+
m∑
i=1
hi(x, y, Z(x−kia, y− lib)) ≤ 0.
Hence, (3) holds.
Similarly if mini=1,...,m{ki} = 0, mini=1,...,m{li} 6= 0, Z(x, y) is nondecreasing in x and y
eventually, we have Z(x − σi, y − τi) ≥ Z(x − σi, y − lib). Since hi(x, y, u), i = 1, 2, . . . ,m, is
nondecreasing in u eventually, we have the following inequality:
p1Z(x+a, y+b)+p2Z(x+a, y)+p3Z(x, y+b)−p4Z(x, y)+
m∑
i=1
hi(x, y, Z(x−σi, y− lib)) ≤ 0,
implying (4).
Similarly if mini=1,...,m{ki} 6= 0, mini=1,...,m{li} = 0, i = 1, 2, . . . ,m, Z(x, y) is nondecrea-
sing in x and y eventually, we have Z(x − σi, y − τi) ≥ Z(x − kia, y − τi). Since hi(x, y, u) is
nondecresing in u, we have the following inequality:
p1Z(x+a, y+b)+p2Z(x+a, y)+p3Z(x, y+b)−p4Z(x, y)+
m∑
i=1
hi(x, y, Z(x−kia, y−τi)) ≤ 0.
Hence, (5) holds.
The proof is completed.
By a similar method, we can obtain properties of an eventually negative solution of (1).
Lemma 2. Assume that
(i) hi ∈ C(R+ × R+ × R,R), uhi(x, y, u) > 0 for u 6= 0 and hi(x, y, u), i = 1, . . . ,m, is a
nondecreasing function in u;
(ii) hi(x, y, u), i = 1, . . . ,m, is concave in u for u ≤ 0.
Let A(x, y) be an eventually negative solution of (1), then there exists a negative function
Z(x, y) =
1
ab
∫ x
x+a
∫ y
y+b
A(u, v)dudv eventually satisfying the following results:
(1) if mini=1,...,m{ki} 6= 0 and mini=1,...,m{li} 6= 0, then
p1Z(x + a, y + b) + p2Z(x + a, y) + p3Z(x, y + b)− p4Z(x, y)+
+
m∑
i=1
hi(x, y, Z(x− σi, y − τi)) ≥ 0; (9)
ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 3
OSCILLATION OF A CLASS OF NONLINEAR PARTIAL DIFFERENCE EQUATIONS . . . 309
(2) if mini=1,...,m{ki} = 0 and mini=1,...,m{li} = 0, then
p1Z(x + a, y + b) + p2Z(x + a, y) + p3Z(x, y + b)− p4Z(x, y)+
+
m∑
i=1
hi(x, y, Z(x− kia, y − lib)) ≥ 0; (10)
(3) if mini=1,...,m{ki} = 0 and mini=1,...,m{li} 6= 0, then
p1Z(x + a, y + b) + p2Z(x + a, y) + p3Z(x, y + b)− p4Z(x, y)+
+
m∑
i=1
hi(x, y, Z(x− σi, y − lib)) ≥ 0; (11)
(4) if mini=1,...,m{ki} 6= 0 and mini=1,...,m{li} = 0, then
p1Z(x + a, y + b) + p2Z(x + a, y) + p3Z(x, y + b)− p4Z(x, y)+
+
m∑
i=1
hi(x, y, Z(x− kia, y − τi)) ≥ 0. (12)
3. Main results. In the following, we investigate oscillatory properties of a solution of (1)
and obtain the main results of this paper.
Theorem 1. Assume that
(i) hi(x, y, u) ∈ C(R+ × R+ × R,R) is nondecreasing in u and uhi(x, y, u) > 0, i =
= 1, 2, . . . ,m, for all u 6= 0,
(ii) lim inf
x,y→+∞,u→0
hi(x, y, u)/u = Si ≥ 0,
m∑
i=1
Si > 0, i = 1, 2, . . . ,m,
(iii) hi(x, y, u) is convex in u for u > 0, hi(x, y, u), i = 1, . . . ,m, is concave in u for u < 0,
(iv) one of the following conditions holds:
m∑
i=1
Si
(ηi + 1)ηi+1
ηηi
i
(p1 + p2 + p3)ηi
p
(ηi+1)
4
> 1, ηi = min{ki, li} > 0, i = 1, . . . ,m, (13)
m∑
i=1
Si
kki
i
(ki − 1)ki−1
pki−1
2
pki
4
> 1, min
i=1,...,m
{ki} > 0, min
i=1,...,m
{li} = 0, (14)
m∑
i=1
Si
llii
(li − 1)li−1
pli−1
3
pli
4
> 1, min
i=1,...,m
{ki} = 0, min
i=1,...,m
{li} > 0, (15)
1
p4
m∑
i=1
Si > 1, min
i=1,...,m
{ki} = min
i=1,...,m
{li} = 0. (16)
Then every solution of (1) is oscillatory.
ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 3
310 Y. GUO, A. LIU, T. LIU, Q. MA
Proof. Assume the contrary. Let A(x, y) be an eventually positive solution of (1), and
Z(x, y) be defined by (6). Then by Lemma 1, we obtain limx,y→+∞ Z(x, y) = ζ ≥ 0. In the
following, we claim that ζ = 0. Otherwise, let ζ > 0. By Lemma 1, we know that (2) holds.
From (2) and condition (I), we have
p1Z(x + a, y + b) + p4(Z(x + a, y) + Z(x, y + b)− Z(x, y)) ≤
≤ p1Z(x + a, y + b) + p2Z(x + a, y) + p3Z(x, y + b)− p4Z(x, y) ≤ 0.
So
Z(x + a, y) + Z(x, y + b)− Z(x, y) ≤ 0. (17)
Taking the limit on both side of (17), we have ζ ≤ 0. Consider ζ ≥ 0. Then we have ζ = 0.
If mini=1,...,m{ki} > 0, mini=1,...,m{li} > 0, in the view of (2), we have
(p1 + p2 + p3)Z(x + a, y + b)
Z(x, y)
− p4 ≤
p1Z(x + a, y + b) + p2Z(x + a, y) + p3Z(x, y + b)
Z(x, y)
− p4 ≤
≤ −
m∑
i=1
hi(x, y, Z(x− σi, y − τi))
Z(x, y)
. (18)
Since Z(x, y) is nondecreasing eventually, from (18) for all large x and y we have
(p1 + p2 + p3)Z(x + a, y + b)
Z(x, y)
− p4 ≤ −
m∑
i=1
hi(x, y, Z(x− ηia, y − ηib))
Z(x, y)
=
= −
m∑
i=1
hi(x, y, Z(x− ηia, y − ηib))
Z(x− ηia, y − ηib)
ηi∏
j=1
Z(x− ja, y − jb)
Z(x− (j − 1)a, y − (j − 1)b)
, (19)
where ηi = min{ki, li}, i = 1, . . . ,m.
Let α(x, y) = Z(x, y)/Z(x + a, y + b). Then α(x, y) > 1 for all large x and y. From (19), we
have
p1 + p2 + p3
α(x, y)
+
m∑
i=1
hi(x, y, Z(x− ηia, y − ηib))
Z(x− ηia, y − ηib)
ηi∏
j=1
α(x− ja, y − jb) ≤ p4,
i.e.,
(p1 + p2 + p3) +
m∑
i=1
hi(x, y, Z(x− ηia, y − ηib))
Z(x− ηia, y − ηib)
ηi∏
j=1
α(x− ja, y − jb)α(x, y) ≤ p4α(x, y).
(20)
By (ii), (20) implies that α(x, y) is bounded.
Let lim inf
x,y→+∞
α(x, y) = β. Taking the limit inferior on both sides of (20), we obtain
(p1 + p2 + p3) +
m∑
i=1
Siβ
ηi+1 ≤ p4β,
ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 3
OSCILLATION OF A CLASS OF NONLINEAR PARTIAL DIFFERENCE EQUATIONS . . . 311
i.e.,
p1 + p2 + p3
β
≤ p4 −
m∑
i=1
Siβ
ηi < p4. (21)
Hence, β >
p1 + p2 + p3
p4
. Since
∑m
i=1 Siβ
ηi+1/(p4β − (p1 + p2 + p3)) ≤ 1, computing the
minimum of the function f(x) = xηi+1/(p4x− (p1 + p2 + p3)) as x >
p1 + p2 + p3
p4
we obtain
min
β>
p1+p2+p3
p4
βηi+1
p4β − (p1 + p2 + p3)
=
(ηi + 1)ηi+1
ηηi
i
(p1 + p2 + p3)ηi
pηi+1
4
.
So we have
∑m
i=1 Si
(ηi + 1)ηi+1
ηηi
i
(p1 + p2 + p3)ηi
pηi+1
4
≤ 1 which is contrary to (13). Therefore if
(13) holds we can obtain that every solution of (1) is oscillatory.
If mini=1,...,m{ki} > 0, mini=1,...,m{li} = 0, by Lemma 1, we obtain
p1Z(x+a, y+b)+p2Z(x+a, y)+p3Z(x, y+b)−p4Z(x, y)+
m∑
i=1
hi(x, y, Z(x−kia, y−τi)) ≤ 0.
Then we have
p2Z(x + a, y)
Z(x, y)
− p4 ≤ −
m∑
i=1
hi(x, y, Z(x− kia, y − τi)
Z(x, y)
=
= −
m∑
i=1
hi(x, y, Z(x− kia, y − τi))
Z(x− kia, y − τi)
Z(x− a, y − τi)
Z(x, y)
ki∏
j=2
Z(x− ja, y − τi)
Z(x− (j − 1)a, y − τi)
. (22)
Since Z(x, y) is nondecreasing in x, y eventually, we have Z(x − a, y − τi)/Z(x, y) > 1 for
all large x and y. From (22), we have
p2Z(x + a, y)
Z(x, y)
+
m∑
i=1
hi(x, y, Z(x− kia, y − τi))
Z(x− kia, y − τi)
ki∏
j=2
Z(x− ja, y − τi)
Z(x− (j − 1)a, y − τi)
≤ p4. (23)
Let α(x, y) = Z(x, y)/Z(x + a, y) > 1. From (23), we have
p2
α(x, y)
+
m∑
i=1
hi(x, y, Z(x− kia, y − τi))
Z(x− kia, y − τi)
ki∏
j=2
α(x− ja, y − τi) ≤ p4. (24)
By condition (ii) the above inequality implies that α(x, y) is bounded. Let lim inf
x,y→+∞
α(x, y) =
= β. From (22), we can obtain
p2 +
m∑
i=1
hi(x, y, Z(x− kia, y − τi))
Z(x− kia, y − τi)
ki∏
j=1
α(x− ja, y − τi)α(x, y) ≤ p4α(x, y). (25)
ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 3
312 Y. GUO, A. LIU, T. LIU, Q. MA
Taking the limit inferior on both sides of (25), we have p2 +
∑m
i=1 Siβ
ki ≤ p4β.
Hence we have
p2
β
+
∑m
i=1 Siβ
ki−1 ≤ p4, i.e.,
p2
β
≤ p4 −
∑m
i=1 Siβ
ki−1 < p4.
Then we obtain β >
p2
p4
and
∑m
i=1 Siβ
ki−1/(p4β − p2) ≤ 1.
Since minβ>
p2
p4
βki
p4β − p2
=
pki−1
2
pki
4
kki
i
(ki − 1)ki−1
, we have
∑m
i=1 Si
pki−1
2
pki
4
kki
i
(ki − 1)ki−1
≤ 1,
which contradicfs (14). So if (14) holds we can obtain that every solution of (1) is oscillatory.
Similarly, we can prove that if (15) holds then we can also obtain every solution of (1) is
oscillatory.
If mini=1,...,m{ki} = mini=1,...,m{li} = 0, from Lemma 1, we know that (3) holds.
Hence we have
p1Z(x + a, y + b) + p2Z(x + a, y) + p3Z(x, y + b)− p4Z(x, y)+
+
m∑
i=1
hi(x, y, Z(x, y)) ≤ p1Z(x + a, y + b) + p2Z(x + a, y)+
+ p3Z(x, y + b)− p4Z(x, y) +
m∑
i=1
hi(x, y, Z(x− σi, y − τi)) ≤ 0.
Then
m∑
i=1
hi(x, y, Z(x, y))
Z(x, y)
− p4 ≤ 0. (26)
Taking the limit inferior on both sides of (26), we have
∑m
i=1 Si ≤ p4, which is contrary to
(16). So if (16) holds we can obtain that every solution of (1) is oscillatory.
If A(x, y) is the eventually negative solution of (1), we can obtain a contradiction by assu-
ming that A(x, y) is an eventually negative solution of equation (1). Therefore we know the
result is correct.
The proof is over.
The results indicate that there are some criteria of oscillatory properties of solutions of some
partial difference equations with forward front difference. In some sense, the results play some
roles in investigating properties of solutions of advanced partial differential equations.
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Received 07.02.06,
after revision — 11.04.09
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