Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 584951, 24 pages
doi:10.1155/2011/584951
Research Article
Some New Volterra-Fredholm-Type Discrete
Inequalities and Their Applications in the Theory
of Difference Equations
Bin Zheng1 and Qinghua Feng1, 2
1
2
School of Science, Shandong University of Technology, Shandong, Zibo 255049, China
School of Mathematical Sciences, Qufu Normal University, Shandong, Qufu 273165, China
Correspondence should be addressed to Bin Zheng, zhengbin2601@126.com
Received 21 March 2011; Accepted 29 June 2011
Academic Editor: Martin D. Schechter
Copyright q 2011 B. Zheng and Q. Feng. 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.
Some new Volterra-Fredholm-type discrete inequalities in two independent variables are established, which provide a handy tool in the study of qualitative and quantitative properties of
solutions of certain difference equations. The established results extend some known results in
the literature.
1. Introduction
In the research of solutions of certain differential and difference equations, if the solutions
are unknown, then it is necessary to study their qualitative and quantitative properties such
as boundedness, uniqueness, and continuous dependence on initial data. The GronwallBellman inequality 1, 2 and its various generalizations which provide explicit bounds play
a fundamental role in the research of this domain. Many such generalized inequalities e.g.,
see 3–30 and the references therein have been established in the literature including the
known Ou-Liang’s inequality 3. In 8, Ma generalized the discrete version of Ou-Liang’s
inequality in two variables to Volterra-Fredholm form for the first time, which has proved
to be very useful in the study of qualitative as well as quantitative properties of solutions of
certain Volterra-Fredholm-type difference equations. But since then, few results on VolterraFredholm-type discrete inequalities have been established. Recent results in this direction
include the work of Ma 9 to our knowledge. We notice, in the analysis of some certain
Volterra-Fredholm-type difference equations with more complicated forms, that the bounds
provided by the earlier inequalities are inadequate and it is necessary to seek some new
Volterra-Fredholm-type discrete inequalities in order to obtain a diversity of desired results.
2
Abstract and Applied Analysis
Our aim in this paper is to establish some new generalized Volterra-Fredholm-type
discrete inequalities, which extend Ma’s work in 9, and provide new bounds for unknown
functions lying in these inequalities. We will illustrate the usefulness of the established
results by applying them to study the boundedness, uniqueness, and continuous dependence
on initial data of solutions of certain more complicated Volterra-Fredholm-type difference
equations.
Throughout this paper, R denotes the set of real numbers R 0, ∞, and Z denotes
the set of integers, while N0 denotes the set of nonnegative integers. Let Ω : m0 , M ×
n0 , N Z2 , where m0 , n0 ∈ Z and M, N ∈ Z {∞} are two constants. l1 , l2 ∈ Z are two
constants, and Ki > 0, i 1, 2, 3, 4, are all constants. If U is a lattice, then we denote the set of
all R-valued functions on U by ℘U and denote the set of all R -valued functions on U by
1
℘ U. Finally, for a function f ∈ ℘ U, we have m
sm0 f 0 provided m0 > m1 .
2. Main Results
Lemma 2.1 see 15. Assume that a ≥ 0, p ≥ q ≥ 0, and p /
0 then for any K > 0
aq/p ≤
q q−p/p
p − q q/p
K
K .
a
p
p
2.1
Lemma 2.2. Suppose that um, n ∈ ℘ Ω, bs, t, m, n ∈ ℘ Ω2 , α ≥ 0 is a constant. If b is
nondecreasing in the third variable, then, for m, n ∈ Ω,
um, n ≤ α m−1
n−1
bs, t, m, nus, t
2.2
sm0 tn0
implies that
um, n ≤ α exp
m−1
n−1
bs, t, m, n .
2.3
sm0 tn0
Lemma 2.3. Suppose that um, n, am, n, bm, n ∈ ℘ Ω. If am, n is nondecreasing in the
first variable, then, for m, n ∈ Ω,
um, n ≤ am, n m−1
bs, nus, n
2.4
sm0
implies that
um, n ≤ am, n
m−1
sm0
1 bs, n.
2.5
Abstract and Applied Analysis
3
Remark 2.4. Lemma 2.3 is a direct variation of 19, Lemma 2.5β1 , and we note am, n ≥ 0
here.
Theorem 2.5. Suppose that um, n, am, n ∈ ℘ Ω, bi s, t, m, n, ci s, t, m, n ∈ ℘ Ω2 , i 1, 2, . . . , l1 , di s, t, m, n, ei s, t, m, n ∈ ℘ Ω2 , i 1, 2, . . . , l2 with bi , ci , di , ei nondecreasing in
0,
the last two variables. p, qi , ri are nonnegative constants with p ≥ qi , p ≥ ri , i 1, 2, . . . , l1 , p /
while hi , ji are nonnegative constants with p ≥ hi , p ≥ ji , i 1, 2, . . . , l2 . If, for m, n ∈ Ω, um, n
satisfies
l1 m−1
n−1
up m, n ≤ am, n ⎤
s t
⎣bi s, t, m, nuqi s, t ci ξ, η, m, n uri ξ, η ⎦
⎡
i1 sm0 tn0
ξm0 ηn0
⎤
⎡
l2 M−1
s t
N−1
h
j
⎣di s, t, m, nu i s, t ei ξ, η, m, n u i ξ, η ⎦,
i1 sm0 tn0
2.6
ξm0 ηn0
then
um, n ≤
am, n JM, N
Cm, n
1 − μM, N
1/p
2.7
,
provided that μM, N < 1, where
⎧
l1 m−1
n−1 ⎨
p − qi qi /p
qi qi −p/p
K1
K1
bi s, t, m, n
as, t Jm, n ⎩
p
p
i1 sm0 tn0
⎫
s t
ri ri −p/p p − ri ri /p ⎬
K
K2
ci ξ, η, m, n
a ξ, η ⎭
p 2
p
ξm ηn0
0
⎧
l2 M−1
N−1
⎨
p − hi hi /p
hi hi −p/p
K3
K3
as, t di s, t, m, n
⎩
p
p
i1 sm0 tn0
2.8
⎫
s t
ji ji −p/p p − ji ji /p ⎬
,
K
K4
ei ξ, η, m, n
a ξ, η ⎭
p 4
p
ξm ηn0
0
⎫
⎧
l2 M−1
s t
N−1
⎨
ji ji −p/p ⎬
hi hi −p/p
μm, n d s, t, m, n K3
Cs, t
ei ξ, η, m, n K4
C ξ, η ,
⎭
⎩ i
p
p
i1 sm tn
ξm ηn
0
0
0
Cm, n exp
Bs, t, m, n ,
2.9
2.10
sm0 tn0
⎤
s t
ri ri −p/p
qi qi −p/p ⎣bi s, t, m, n K
⎦.
Bs, t, m, n ci ξ, η, m, n K2
p 1
p
i1
ξm ηn0
l1
⎡
m−1
n−1
0
0
2.11
4
Abstract and Applied Analysis
Proof. Denote
vm, n l1 m−1
n−1
⎤
s t
⎣bi s, t, m, nuqi s, t ci ξ, η, m, n uri ξ, η ⎦
⎡
i1 sm0 tn0
ξm0 ηn0
⎡
l2 M−1
N−1
s
i1 sm0 tn0
ξm0 ηn0
⎣di s, t, m, nuhi s, t t
⎤
2.12
ei ξ, η, m, n uji ξ, η ⎦.
Then, we have
2.13
um, n ≤ am, n vm, n1/p ,
and, furthermore, from Lemma 2.1 we have
uqi m, n ≤ am, n vm, nqi /p ≤
qi qi −p/p
K
am, n vm, n
p 1
uri m, n ≤ am, n vm, nri /p ≤
p − ri ri /p
K2 ,
p
i 1, 2, . . . , l1 ,
hi hi −p/p
K
am, n vm, n
p 3
uji m, n ≤ am, n vm, nji /p ≤
i 1, 2, . . . , l1 ,
ri ri −p/p
K
am, n vm, n
p 2
uhi m, n ≤ am, n vm, nhi /p ≤
p − qi qi /p
K1 ,
p
p − hi hi /p
K3 ,
p
i 1, 2, . . . , l2 ,
ji ji −p/p
K
am, n vm, n
p 4
p − ji ji /p
K4 ,
p
i 1, 2, . . . , l2 .
2.14
Abstract and Applied Analysis
5
So
vm, n ≤
⎧
l1 m−1
n−1 ⎨
bi s, t, m, n
i1 sm0 tn0
⎩
qi qi −p/p
p − qi qi /p
K1
K1
as, tvs, t
p
p
⎫
s t
ri ri −p/p p − ri ri /p ⎬
K
K2
ci ξ, η, m, n
a ξ, η v ξ, η ⎭
p 2
p
ξm ηn0
0
⎧
l2 M−1
N−1
⎨
p − hi hi /p
hi hi −p/p
di s, t, m, n
K3
K3
as, t vs, t ⎩
p
p
i1 sm0 tn0
⎫
s t
ji ji −p/p p − ji ji /p ⎬
K
K4
ei ξ, η, m, n
a ξ, η v ξ, η ⎭
p 4
p
ξm ηn0
0
Hm, n l1 m−1
n−1
⎡
⎣bi s, t, m, n
i1 sm0 tn0
qi qi −p/p
K
vs, t
p 1
⎤
s t
ri ri −p/p ci ξ, η, m, n K2
v ξ, η ⎦,
p
ξm ηn0
0
2.15
where Hm, n Jm,n l2 M−1 N−1
i1
j −p/p
ξ, η, m, nji /pK4 i
vξ, η}
sm0
hi −p/p
vs, t sξm0
tn0 {di s, t, m, nhi /pK3
t
ηn0
ei
and Jm, n is defined in 2.8. Then, using that Hm, n is
nondecreasing in every variable, we obtain
vm, n ≤ HM, N l1 m−1
n−1
⎡
⎣bi s, t, m, n
i1 sm0 tn0
qi qi −p/p
K
vs, t
p 1
⎤
s t
ri ri −p/p ci ξ, η, m, n K2
v ξ, η ⎦
p
ξm ηn0
0
≤ HM, N l1 m−1
n−1
⎡
⎣bi s, t, m, n
i1 sm0 tn0
qi qi −p/p
K
p 1
⎤
s t
ri ri −p/p
⎦vs, t
ci ξ, η, m, n K2
p
ξm ηn0
0
HM, N m−1
n−1
Bs, t, m, nvs, t,
sm0 tn0
where Bs, t, m, n is defined in 2.11.
2.16
6
Abstract and Applied Analysis
Since bi s, t, m, n, ci s, t, m, n are nondecreasing in the last two variables, then
Bs, t, m, n is also nondecreasing in the last two variables, and by a suitable application of
Lemma 2.2 we obtain
vm, n ≤ HM, N exp
m−1
n−1
Bs, t, m, n
HM, NCm, n,
2.17
sm0 tn0
where Cm, n is defined in 2.10. Furthermore, considering the definition of Hm, n and
2.17 we have
HM, N JM, N ⎧
l2 M−1
N−1
⎨
di s, t, M, N
i1 sm0 tn0
⎩
hi hi −p/p
K
vs, t
p 3
⎫
s t
ji ji −p/p ⎬
ei ξ, η, M, N K4
v ξ, η
⎭
p
ξm ηn0
0
≤ JM, N ⎧
l2 M−1
N−1
⎨
di s, t, M, N
i1 sm0 tn0
⎩
hi hi −p/p
K
HM, NCs, t
p 3
⎫
s t
⎬
ji ji −p/p
ei ξ, η, m1 , n1 K4
HM, NC ξ, η
⎭
p
ξm ηn0
0
JM, N HM, N
⎧
l2 M−1
N−1
⎨
di s, t, M, N
i1 sm0 tn0
⎩
hi hi −p/p
K
Cs, t
p 3
⎫
s t
ji ji −p/p ⎬
ei ξ, η, M, N K4
C ξ, η
⎭
p
ξm ηn0
0
JM, N HM, NμM, N,
2.18
where μm, n is defined in 2.9. Then,
HM, N ≤
JM, N
.
1 − μM, N
2.19
Combining 2.17 and 2.19 we deduce
vm, n ≤
JM, N
Cm, n.
1 − μM, N
Then, combining 2.13 and 2.20, we obtain the desired result.
2.20
Abstract and Applied Analysis
7
Corollary 2.6. Suppose that g1i m, n, g2i m, n, b1i m, n, c1i m, n ∈ ℘ Ω, i 1, 2, . . . , l1
with g1i , g2i nondecreasing in every variable. d1i m, n, e1i m, n ∈ ℘ Ω, i 1, 2, . . . , l2 .
um, n, am, n, p, qi , ri , hi , ji are defined as in Theorem 2.5. If, for m, n ∈ Ω, um, n satisfies
u m, n ≤ am, n p
l1
g1i m, n
⎡
⎣b1i s, tu s, t i1
c1i ξ, η u
ri
⎤
M−1
N−1
s t
sm0 tn0
ξm0 ηn0
⎣d1i s, tuhi s, t ji
ξ, η ⎦
ξm0 ηn0
⎡
g2i m, n
s t
qi
sm0 tn0
i1
l2
m−1
n−1
2.21
⎤
e1i ξ, η u ξ, η ⎦,
then
um, n ≤
1/p
JM, N
am, n CM, N
,
1 − μM, N
2.22
provided that μM, N < 1, where
Jm, n l1
⎧
g1i m, n
b1i s, t
sm0 tn0
i1
m−1
n−1 ⎨
⎩
p − qi qi /p
qi qi −p/p
K1
K1
as, t p
p
s t
c1i
ξm0 ηn0
l2
⎧
g2i m, n
d1i s, t
sm0 tn0
i1
M−1
N−1
⎨
⎩
⎫
ri ri −p/p p − ri ri /p ⎬
K
K2
a ξ, η ξ, η
⎭
p 2
p
p − hi hi /p
hi hi −p/p
K3
K3
as, t p
p
s t
e1i
μm, n ⎫
ji ji −p/p p − ji ji /p ⎬
,
K
K4
a ξ, η ξ, η
⎭
p 4
p
⎡
g2i m, n
⎩
i1
M−1
N−1
⎣d1i s, t hi K hi −p/p Cs, t
p 3
sm0 tn0
s t
ξm0 ηn0
Cm, n exp
ξm0 ηn0
⎧
l2 ⎨
m−1
n−1
e1i
⎤⎫
ji ji −p/p ⎬
C ξ, η ⎦ ,
ξ, η K4
⎭
p
Bs, t, m, n ,
sm0 tn0
⎡
⎤
s t
ri ri −p/p
qi qi −p/p ⎦.
Bs, t, m, n g1i m, n⎣b1i s, t K1
c1i ξ, η K2
p
p
ηn
i1
ξm
0
l1
0
2.23
8
Abstract and Applied Analysis
The proof of Corollary 2.6 can be completed by setting bi s, t, m, n g1i m,nb1i s,t,
ci s,t,m,n g1i m,nc1i s,t, di s,t,m,n g2i m,nd1i s,t, ei s,t,m,n g2i m,ne1i s,t in
Theorem 2.5.
Corollary 2.7. Suppose that um, n, am, n, bi s, t, m, n, ci s, t, m, n, di s, t, m, n, ei s, t, m, n
are defined as in Theorem 2.5. If, for m, n ∈ Ω, um, n satisfies
um, n ≤ am, n l1 m−1
n−1
⎤
s t
⎣bi s, t, m, nus, t ci ξ, η, m, n u ξ, η ⎦
⎡
i1 sm0 tn0
ξm0 ηn0
⎤
l2 M−1
s t
N−1
⎣di s, t, m, nus, t ei ξ, η, m, n u ξ, η ⎦,
⎡
i1 sm0 tn0
2.24
ξm0 ηn0
then
um, n ≤ am, n JM, N
Cm, n,
1 − μM, N
2.25
provided that μM, N < 1, where
Jm, n ⎧
l1 m−1
n−1 ⎨
i1 sm0
⎫
s t
⎬
b s, t, m, nas, t ci ξ, η, m, n a ξ, η
⎩ i
⎭
tn
ξm ηn
0
i1 sm0 tn0
μm, n ⎫
s t
⎬
ei ξ, η, m, n a ξ, η ,
d s, t, m, nas, t ⎩ i
⎭
ξm ηn
0
0
⎧
l2 M−1
N−1
⎨
i1 sm0 tn0
Cm, n exp
Bs, t, m, n 0
0
⎧
l2 M−1
N−1
⎨
l1
i1
⎫
s t
⎬
d s, t, m, nCs, t ei ξ, η, m, n C ξ, η ,
⎩ i
⎭
ξm ηn
m−1
n−1
0
2.26
0
Bs, t, m, n ,
sm0 tn0
⎤
s t
⎣bi s, t, m, n ci ξ, η, m, n ⎦.
⎡
ξm0 ηn0
Theorem 2.8. Suppose that wm, n ∈ ℘ Ω, u, a, bi , ci , di , ei , p, qi , ri , hi , ji are defined as in
Theorem 2.5. Furthermore, assume that am, n is nondecreasing in the first variable. If, for m, n ∈
Ω, um, n satisfies
Abstract and Applied Analysis
up m, n ≤ am, n m−1
9
ws, nup m, n
sm0
l1 m−1
n−1
⎤
s t
⎣bi s, t, m, nuqi s, t ci ξ, η, m, n uri ξ, η ⎦
⎡
i1 sm0 tn0
ξm0 ηn0
2.27
⎤
⎡
l2 M−1
s t
N−1
⎣di s, t, m, nuhi s, t ei ξ, η, m, n uji ξ, η ⎦,
i1 sm0 tn0
ξm0 ηn0
then
um, n ≤
1/p
JM, N
am, n Cm, n wm, n
,
1 − μM, N
2.28
provided that μM, N < 1, where
⎧
l1 m−1
n−1 ⎨
p − qi qi /p
qi qi −p/p
K1
K1
Jm, n bi s, t, m, n
as, t ⎩
p
p
i1 sm0 tn0
⎫
s t
ri ri −p/p p − ri ri /p ⎬
K
K2
ci ξ, η, m, n
a ξ, η ⎭
p 2
p
ξm ηn0
0
⎧
⎪
l2 M−1
⎨
N−1
⎪
p − hi hi /p
hi hi −p/p
K3
K3
di s, t, m, n
as, t ⎪
p
p
i1 sm0 tn0 ⎪
⎩
⎫
s t
ji ji −p/p p − ji ji /p ⎬
,
K
K4
ei ξ, η, m, n
a ξ, η ⎭
p 4
p
ξm ηn0
0
2.29
bi s, t, m, n bi s, t, m, nws, tqi /p ,
ci s, t, m, n ci s, t, m, nws, tri /p ,
i 1, 2, . . . , l1 ,
di s, t, m, n di s, t, m, nws, thi /p ,
ei s, t, m, n ei s, t, m, nws, tji /p ,
i 1, 2, . . . , l2 ,
2.30
2.31
10
Abstract and Applied Analysis
wm, n m−1
1 ws, n,
sm0
⎧
l2 M−1
N−1
⎨
hi h −p/p
μm, n di s, t, m, n K3 i
Cs, t
⎩
p
i1 sm tn
0
2.32
0
⎫
s t
ji ji −p/p ⎬
ei ξ, η, m, n K4
C ξ, η
⎭
p
ξm ηn0
0
Cm, n exp
m−1
n−1
2.33
Bs, t, m, n ,
sm0 tn0
⎡
⎤
l1
s t
ri ri −p/p
qi qi −p/p ⎣bi s, t, m, n K
⎦.
Bs, t, m, n ci ξ, η, m, n K2
p 1
p
i1
ξm ηn0
2.34
0
Proof. Denote
zm, n am, n l1 m−1
n−1
⎡
⎤
s t
ri ⎣bi s, t, m, nu s, t ci ξ, η, m, n u ξ, η ⎦
qi
i1 sm0 tn0
ξm0 ηn0
⎤
l2 M−1
s t
N−1
⎣di s, t, m, nuhi s, t ei ξ, η, m, n uji ξ, η ⎦.
⎡
i1 sm0 tn0
2.35
ξm0 ηn0
Then, we have
up m, n ≤ zm, n m−1
ws, nup m, n.
2.36
sm0
Obviously zm, n is nondecreasing in the first variable. So by Lemma 2.3 we obtain
up m, n ≤ zm, n
m−1
1 ws, n zm, nwm, n,
2.37
sm0
where wm, n vm, n m−1
sm0 1
l1 m−1
n−1
ws, n. Define
⎤
s t
ri ⎣bi s, t, m, nu s, t ci ξ, η, m, n u ξ, η ⎦
⎡
qi
i1 sm0 tn0
ξm0 ηn0
⎤
l2 M−1
s t
N−1
⎣di s, t, m, nuhi s, t ei ξ, η, m, n uji ξ, η ⎦.
⎡
i1 sm0 tn0
ξm0 ηn0
2.38
Abstract and Applied Analysis
11
Then,
um, n ≤ am, n vm, nwm, n1/p ,
2.39
and, furthermore, by 2.39 and Lemma 2.1 we have
vm, n ≤
⎧
l1 m−1
n−1 ⎨
bi s, t, m, nas, t vs, tws, tqi /p
i1 sm0
⎩
tn
0
⎫
s t
ri /p⎬
ci ξ, η, m, n a ξ, η v ξ, η w ξ, η
⎭
ξm ηn
0
0
⎧
l2 M−1
N−1
⎨
di s, t, m, nas, t vs, tws, thi /p
i1 sm0 tn0
⎩
⎫
s t
ji /p⎬
ei ξ, η, m, n a ξ, η v ξ, η w ξ, η
⎭
ξm ηn
0
≤
0
⎧
l1 m−1
n−1 ⎨
bi s, t, m, nws, tqi /p
i1 sm0 tn0
⎩
qi qi −p/p
p − qi qi /p
K
K1
as, t vs, t p 1
p
s t
r /p
ci ξ, η, m, n w ξ, η i
ξm0 ηn0
⎫
p − ri ri /p ⎬
ri ri −p/p K2
× K2
a ξ, η v ξ, η ⎭
p
p
⎧
l2 M−1
N−1
⎨
di s, t, m, nws, thi /p
i1 sm0 tn0
⎩
p − hi hi /p
hi qi −p/p
K3
K3
as, t vs, t p
p
s t
j /p
ei ξ, η, m, n w ξ, η i
ξm0 ηn0
⎫
ji ji −p/p p − ji ji /p ⎬
K4
× K4
a ξ, η v ξ, η ⎭
p
p
⎧
l1 m−1
n−1 ⎨
qi qi −p/p
p − qi qi /p
K1
K1
bi s, t, m, n
as, t vs, t ⎩
p
p
i1 sm0 tn0
⎫
s t
ri ri −p/p p − ri ri /p ⎬
K
K2
ci ξ, η, m, n
a ξ, η v ξ, η ⎭
p 2
p
ξm ηn0
0
12
Abstract and Applied Analysis
⎧
l2 M−1
N−1
⎨
p − hi hi /p
hi qi −p/p
K3
K3
di s, t, m, n
as, t vs, t ⎩
p
p
i1 sm0 tn0
⎫
s t
ji ji −p/p p − ji ji /p ⎬
a ξ, η v ξ, η K
K4
ei ξ, η, m, n
⎭
p 4
p
ξm ηn0
0
Hm, n l1 m−1
n−1
⎡
⎣bi s, t, m, n
i1 sm0 tn0
qi qi −p/p
K
vs, t
p 1
⎤
s
t
ri ri −p/p ci ξ, η, m, n K2
v ξ, η ⎦,
p
ηn
ξm
0
0
2.40
where Hm, n Jm, n
l2 M−1 N−1
i1
j −p/p
ξ, η, m, nji /pK4 i
vξ, η},
sm0
hi −p/p
vs, t sξm0
tn0 {di s, t, m, nhi /pK3
t
ηn0
ei
and Jm, n, bi , ci , di , ei are defined in 2.29–2.31 respec-
tively.
Similar to the process of 2.15–2.20 we deduce
vm, n ≤
JM, N
Cm, n,
1 − μM, N
2.41
where μm, n, Cm, n are defined in 2.32 and 2.33.
Combining 2.39 and 2.41, we get the desired result.
Remark 2.9. If we set bi s,t,m,n g1i m,nb1i s,t, ci s,t,m,n g1i m,nc1i s,t, di s,t,m,n
g2i m,nd1i s,t, ei s,t,m,n g2i m,ne1i s,t or set p qi ri hi ji 1 in Theorem 2.8,
then immediately we get two corollaries which are similar to Corollaries 2.6 and 2.7, and we
omit the details for them.
Theorem 2.10. Suppose that u, a, bi , ci , di , ei , p, qi , ri , hi , ji are defined as in Theorem 2.5.
Li , Ti : Ω × R → R , i 1, 2, . . . , l2 , satisfies 0 ≤ Li m, n, u − Li m, n, v ≤ Ti m, n, vu − v for
u ≥ v ≥ 0. If, for m, n ∈ Ω, um, n satisfies
up m, n ≤ am, n l1 m−1
n−1
i1 sm0 tn0
⎤
s t
⎣bi s, t, m, nuqi s, t ci ξ, η, m, n uri ξ, η ⎦
⎡
ξm0 ηn0
⎡
⎤
l2 M−1
s t
N−1
⎣di s, t, m, nLi s, t, uhi s, t ei ξ, η, m, n uji ξ, η ⎦,
i1 sm0 tn0
ξm0 ηn0
2.42
Abstract and Applied Analysis
13
then
um, n ≤
!
JM,
N !
am, n Cm, n
1−μ
! M, N
1/p
2.43
,
provided that μ
! M, N < 1, where
!
Jm,
n ⎧
l1 m−1
n−1 ⎨
bi s, t, m, n
i1 sm0 tn0
⎩
p − qi qi /p
qi qi −p/p
K1
K1
as, t p
p
⎫
s t
ri ri −p/p p − ri ri /p ⎬
K
K2
ci ξ, η, m, n
a ξ, η ⎭
p 2
p
ξm ηn0
0
⎧
l2 M−1
N−1
⎨
di s, t, m, nLi s, t,
i1 sm0 tn0
⎩
p − hi hi /p
hi hi −p/p
K
K3
as, t p 3
p
2.44
⎫
s t
ji ji −p/p p − ji ji /p ⎬
,
K
K4
ei ξ, η, m, n
a ξ, η ⎭
p 4
p
ξm ηn0
0
μ
! m, n ⎧
l2 M−1
N−1
⎨
hi h −p/p
d!i s, t, m, n K3 i
Cs, t
⎩
p
i1 sm0 tn0
s
ξm0
2.45
⎫
⎬
t
ji j −p/p ei ξ, η, m, n K4 i
C ξ, η ,
⎭
p
ηn0
p − hi hi /p
hi h −p/p
d!i s, t, m, n di s, t, m, nTi s, t, K3 i
K3
,
as, t p
p
!
Cm,
n exp
m−1
n−1
i 1, 2, . . . , l2 ,
2.46
! t, m, n ,
Bs,
2.47
sm0 tn0
⎤
s t
q
ri r −p/p ⎦
! t, m, n ⎣bi s, t, m, n i K qi −p/p .
Bs,
ci ξ, η, m, n K2 i
1
p
p
i1
ξm ηn0
l1
⎡
0
2.48
14
Abstract and Applied Analysis
Proof. Denote
vm, n l1 m−1
n−1
⎤
s t
ri ⎣bi s, t, m, nu s, t ci ξ, η, m, n u ξ, η ⎦
⎡
qi
i1 sm0 tn0
ξm0 ηn0
⎤ 2.49
⎡
l2 M−1
s t
N−1
⎣di s, t, m, nLi s, t, uhi s, t ei ξ, η, m, n uji ξ, η ⎦.
i1 sm0 tn0
ξm0 ηn0
Then
2.50
um, n ≤ am, n vm, n1/p ,
and, furthermore, from Lemma 2.1 we have
vm, n ≤
⎧
l1 m−1
n−1 ⎨
bi s, t, m, n
i1 sm0 tn0
⎩
qi qi −p/p
p − qi qi /p
K1
K1
as, t vs, t p
p
⎫
s t
ri ri −p/p p − ri ri /p ⎬
K
K2
ci ξ, η, m, n
a ξ, η v ξ, η ⎭
p 2
p
ξm ηn0
0
⎧
l2 M−1
N−1
⎨
p − hi hi /p
hi h −p/p
di s, t, m, nLi s, t, K3 i
K3
as, t vs, t ⎩
p
p
i1 sm0 tn0
⎫
s t
ji ji −p/p p − ji ji /p ⎬
K
K4
ei ξ, η, m, n
a ξ, η v ξ, η ⎭
p 4
p
ξm ηn0
0
⎧
l1 m−1
n−1 ⎨
qi qi −p/p
p − qi qi /p
K1
K1
bi s, t, m, n
≤
as, t vs, t ⎩
p
p
i1 sm0 tn0
⎫
s t
ri ri −p/p p − ri ri /p ⎬
K
K2
ci ξ, η, m, n
a ξ, η v ξ, η ⎭
p 2
p
ξm ηn0
0
⎧
l2 M−1
N−1
⎨
p − hi hi /p
hi h −p/p
K3
di s, t, m, n Li s, t, K3 i
as, t vs, t ⎩
p
p
i1 sm0 tn0
Abstract and Applied Analysis
15
p − hi hi /p
hi h −p/p
K3
as, t − Li s, t, K3 i
p
p
p − hi hi /p
hi h −p/p
K3
Li s, t, K3 i
as, t p
p
⎫
s t
ji ji −p/p p − ji ji /p ⎬
K
K4
ei ξ, η, m, n
a ξ, η v ξ, η ⎭
p 4
p
ξm ηn0
0
⎧
l1 m−1
n−1 ⎨
qi qi −p/p
p − qi qi /p
bi s, t, m, n
≤
K1
K1
as, t vs, t ⎩
p
p
i1 sm0 tn0
⎫
s t
ri ri −p/p p − ri ri /p ⎬
K
K2
ci ξ, η, m, n
a ξ, η v ξ, η ⎭
p 2
p
ξm ηn0
0
⎧
l2 M−1
N−1
⎨
p − hi hi /p hi hi −p/p
hi h −p/p
di s, t, m, n Ti s, t, K3 i
K3
K
as, t ⎩
p
p
p 3
i1 sm0 tn0
p − hi hi /p
hi h −p/p
K3
×vs, t Li s, t, K3 i
as, t p
p
⎫
s t
ji ji −p/p p − ji ji /p ⎬
K
K4
ei ξ, η, m, n
a ξ, η v ξ, η ⎭
p 4
p
ξm ηn0
0
"
Hm,
n l1 m−1
n−1
⎡
⎣bi s, t, m, n
i1 sm0 tn0
qi qi −p/p
K
vs, t
p 1
⎤
s t
ri ri −p/p ci ξ, η, m, n K2
v ξ, η ⎦,
p
ξm ηn0
0
2.51
2 M−1 N−1 !
hi −p/p
"
!
where Hm,
n Jm,
n li1
vs, t sξm0 tηn0 ei
sm0
tn0 {di s, t, m, nhi /pK3
j −p/p
!
ξ, η, m, nji /pK4 i
vξ, η} and Jm,
n, d!i s, t, m, n are defined in 2.44 and 2.46
respectively.
Similar to the process of 2.15–2.20 we deduce
vm, n ≤
!
JM,
N !
CM, N,
1−μ
! M, N
!
where μ
! m, n, Cm,
n are defined in 2.45 and 2.47 respectively.
Combining 2.50 and 2.52, we get the desired result.
2.52
16
Abstract and Applied Analysis
Theorem 2.11. Suppose that wm, n ∈ ℘ Ω, u, a, bi , ci , di , ei , p, qi , ri , hi , ji are defined
as in Theorem 2.5. Furthermore, assume am, n is nondecreasing in the first variable. Li , Ti , i 1, 2, . . . , l2 , are defined as in Theorem 2.10. If, for m, n ∈ Ω, um, n satisfies
up m, n ≤ am, n m−1
ws, nup m, n
sm0
l1 m−1
n−1
⎤
s t
⎣bi s, t, m, nuqi s, t ci ξ, η, m, n uri ξ, η ⎦
⎡
i1 sm0 tn0
ξm0 ηn0
⎡
⎤
l2 M−1
s t
N−1
⎣di s, t, m, nLi s, t, uhi s, t ei ξ, η, m, n uji ξ, η ⎦,
i1 sm0 tn0
ξm0 ηn0
2.53
then
1/p
#
JM,
N #
#
n
am, n Cm, n wm,
,
1−μ
# M, N
um, n ≤
2.54
provided that μ
# M, N < 1, where
#
Jm,
n ⎧
l1 m−1
n−1 ⎨
p − qi qi /p
qi qi −p/p
K1
K1
as, t b#i s, t, m, n
⎩
p
p
i1 sm0 tn0
⎫
s t
ri ri −p/p p − ri ri /p ⎬
K
K2
c#i ξ, η, m, n
a ξ, η ⎭
p 2
p
ξm ηn0
0
⎧
l2 M−1
N−1
⎨
$
%
p−hi hi /p
hi hi−p/p
di s, t, m, nLi s, t, ws,
K3
K3
as, t
# thi /p
⎩
p
p
i1 sm0 tn0
⎫
s t
ji ji −p/p p − ji ji /p ⎬
K
K4
e#i ξ, η, m, n
a ξ, η ,
⎭
p 4
p
ξm ηn0
0
# tqi /p ,
b#i s, t, m, n bi s, t, m, nws,
c#i s, t, m, n ci s, t, m, nws,
# tri /p ,
i 1, 2, . . . , l1 ,
$
%
p − hi hi /p
hi hi −p/p
K3
K3
d#i s, t, m, n di s, t, m, nws,
as, t # thi /p
,
# thi /p Ti s, t, ws,
p
p
e#i s, t, m, n ei s, t, m, nws,
# tji /p ,
i 1, 2, . . . , l2 ,
Abstract and Applied Analysis
wm,
#
n m−1
17
1 ws, n,
sm0
μ
# m, n ⎧
l2 M−1
N−1
⎨
hi h −p/p #
d#i s, t, m, n K3 i
Cs, t
⎩
p
i1 sm0 tn0
⎫
s t
ji ji −p/p ⎬
# ξ, η ,
e#i ξ, η, m, n K4
C
⎭
p
ηn
ξm
0
0
#
Cm,
n exp
m−1
n−1
#
Bs, t, m, n ,
sm0 tn0
⎡
⎤
l1
s t
q
ri r −p/p ⎦
# t, m, n ⎣bi s, t, m, n i K qi −p/p .
Bs,
ci ξ, η, m, n K2 i
1
p
p
i1
ξm ηn0
0
2.55
The proof for Theorem 2.11 is similar to the combination of Theorems 2.8 and 2.10, and
we omit the details here.
Remark 2.12. If we take g1i m, n ≡ 1, c1i m, n ≡ 0, i 1, 2, . . . , l1 , and g2i m, n ≡
1, e1i m, n ≡ 0, i 1, 2, . . . , l2 in Corollary 2.6, then Corollary 2.6 reduces to 9, Theorem
2.5. If furthermore l1 l2 1, then Corollary 2.6 reduces to 9, Theorem 2.1. If we
take bi s, t, m, n b1i s, t, ci s, t, m, n ≡ 0, i 1, 2, . . . , l1 and di s, t, m, n d1i s, t,
ei s, t, m, n ≡ 0, hi 1, i 1, 2, . . . , l2 in Theorem 2.10, then Theorem 2.10 reduces to 9,
Theorem 2.7. If furthermore l1 l2 1, then Theorem 2.10 reduces to 9, Theorem 2.6.
3. Applications
In this section, we will present some applications for the established results above and show
that they are useful in the study of boundedness, uniqueness, and continuous dependence of
solutions of certain difference equations.
Example 3.1. Consider the following Volterra-Fredholm sum-difference equation:
m−1
n−1
u m, n am, n p
⎡
⎣F1 s, t, m, n, us, t sm0 tn0
⎡
s t
ξm0 ηn0
M−1
M−1
s t
sm0 tn0
ξm0 ηn0
⎣G1 s, t, m, n, us, t ⎤
F2 ξ, η, m, n, u ξ, η ⎦
⎤
G2 ξ, η, m, n, u ξ, η ⎦,
3.1
18
Abstract and Applied Analysis
where um, n, am, n ∈ ℘Ω, p ≥ 1 is an odd number, Fi , Gi : Ω2 × R → R, i 1, 2, M, N
are two integers defined the same as in Theorem 2.5.
Theorem 3.2. Suppose that um, n is a solution of 3.1, and |F1 s, t, m, n, u| ≤ f1 s, t, m, n|u|q ,
|F2 s, t, m, n, u| ≤ f2 s, t, m, n|u|r , |G1 s, t, m, n, u| ≤ g1 s, t, m, n|u|h , |G2 s, t, m, n, u| ≤
g2 s, t, m, n|u|j , where q, r, h, j are nonnegative constants satisfying p ≥ q, p ≥ r, p ≥ h, p ≥ j,
fi , gi ∈ ℘ Ω2 , i 1, 2 and fi , gi are nondecreasing in the last two variables; then one has
|um, n| ≤
|am, n| JM, N
Cm, n
1 − μM, N
1/p
3.2
,
provided that μM, N < 1, where
⎧
Jm, n m−1
n−1 ⎨
f1 s, t, m, n
sm0 tn0
⎩
⎧
M−1
N−1
⎨
g1 s, t, m, n
⎩
sm0 tn0
⎫
r r−p/p & & p − r r/p ⎬
&a ξ, η & K2
K2
f2 ξ, η, m, n
⎭
p
p
ηn0
s t
ξm0
q q−p/p
p − q q/p
K1
K1
|as, t| p
p
s t
ξm0 ηn0
p − h h/p
h h−p/p
K3
K3
|as, t| p
p
⎫
j j−p/p & & p − j j/p ⎬
&a ξ, η & K
K4
,
g2 ξ, η, m, n
⎭
p 4
p
⎧
μm, n M−1
N−1
⎨
sm0 tn0
h h−p/p
Cs, t
g1 s, t, m, n K3
p
⎩
ξm0
Cm, n exp
m−1
n−1
⎫
j j−p/p ⎬
g2 ξ, η, m, n K4
C ξ, η ,
⎭
p
ηn0
s t
Bs, t, m, n ,
sm0 tn0
s t
r r−p/p
q q−p/p Bs, t, m, n f1 s, t, m, n K1
f2 ξ, η, m, n K2
.
p
p
ξm ηn0
0
3.3
Abstract and Applied Analysis
19
Proof. From 3.1 we have
|um, n|p ≤ |am, n| m−1
n−1
sm0 tn0
⎤
⎡
s t &
&
& &
&F2 ξ, η, m, n, u ξ, η &⎦
⎣&F s, t, m, n, us, t& 1
ξm0 ηn0
⎤
⎡
M−1
s t &
M−1
&
& &
&G2 ξ, η, m, n, u ξ, η &⎦
⎣&G1 s, t, m, n, us, t& sm0 tn0
ξm0 ηn0
≤ |am, n| m−1
n−1
⎡
⎤
s t
&q &
& &r
⎣f1 s, t, m, n &us, t& f2 ξ, η, m, n &u ξ, η & ⎦
sm0 tn0
⎡
ξm0 ηn0
⎤
s t
&h &
& &j
⎣g1 s, t, m, n &us, t& g2 ξ, η, m, n &u ξ, η & ⎦.
M−1
M−1
sm0 tn0
ξm0 ηn0
3.4
Then a suitable application of Theorem 2.5 with l1 l2 1 to 3.4 yields the desired result.
The following theorem deals with the uniqueness of the solutions of 3.1.
Theorem 3.3. Suppose that |Fi s, t, m, n, u − Fi s, t, m, n, v| ≤ fi s, t, m, n|up − vp |, |
Gi s, t, m, n, u − Gi s, t, m, n, v| ≤ gi s, t, m, n|up − vp |, i 1, 2 hold for ∀u, v ∈ R, where
f , gi ∈ ℘ Ω2 , i 1, 2 with fi , gi nondecreasing in the last two variables, and μM, N s
N−1
t
i M−1
sm0
tn0 {g1 s, t, M, NCs, t ηn0 g2 ξ, η, M, NCξ, η} < 1, where Cm, n ξm0
m−1 n−1
exp{ sm0 tn0 Bs, t, m, n}, and Bs, t, m, n f1 s, t, m, n sξm0 tηn0 f2 ξ, η, m, n, then
3.1 has at most one solution.
Proof. Suppose that u1 m, n, u2 m, n are two solutions of 3.1. Then,
⎡
n−1
& m−1
&
&
p
& ⎣&&
& p
F1 s, t, m, n, u1 s, t − F1 s, t, m, n, u2 s, t&
&u1 m, n − u2 m, n& ≤
sm0 tn0
⎤
s t &
&
&F2 ξ, η, m, n, u1 ξ, η −F2 ξ, η, m, n, u2 ξ, η &⎦
ξm0 ηn0
⎡
&
&
⎣&G1 s, t, m, n, u1 s, t − G1 s, t, m, n, u2 s, t&
M−1
M−1
sm0 tn0
⎤
s t &
&
&G2 ξ, η, m, n, u1 ξ, η −G2 ξ, η, m, n, u2 ξ, η &⎦
ξm0 ηn0
20
Abstract and Applied Analysis
≤
m−1
n−1
sm0 tn0
⎡
&
&
⎣f1 s, t, m, n&&up s, t − up s, t&&
2
1
s t
ξm0 ηn0
⎤
&
&& p p
&
f2 ξ, η, m, n &u1 ξ, η − u2 ξ, η &⎦
⎡
&
&
⎣g1 s, t, m, n&&up s, t − up s, t&&
2
1
M−1
M−1
sm0 tn0
s t
ξm0 ηn0
⎤
&& p &&
p
g2 ξ, η, m, n &u1 ξ, η − u2 ξ, η &⎦.
3.5
p
p
Treat |u1 m, n − u2 m, n| as one variable, and a suitable application of Corollary 2.7
p
p
p
p
yields |u1 m, n − u2 m, n| ≤ 0, which implies that u1 m, n ≡ u2 m, n. Since p is an odd
number, then we have u1 m, n ≡ u2 m, n, and the proof is complete.
Finally we study the continuous dependence of the solutions of 3.1 on the functions
a, F1 , F2 , G1 , G2 .
Theorem 3.4. Suppose that um, n is a solution of 3.1, |Fi s, t, m, n, u − Fi s, t, m, n, v| ≤
fi s, t, m, n|up − vp |, |Gi s, t, m, n, u − Gi s, t, m, n, v| ≤ gi s, t, m, n|up − vp |, i 1, 2 hold
for ∀u, v ∈ R, where fi , gi ∈ ℘ Ω2 , i 1, 2 with fi , gi nondecreasing in the last two variables, and,
furthermore,
⎧
|am, n − am, n| m−1
n−1 ⎨&
&
&
&
&F1 s, t, us, t − F 1 s, t, us, t&
sm0 tn0
⎩
⎫
s t &
&⎬
&
&
&F2 ξ, η, u ξ, η − F 2 ξ, η, u ξ, η &
⎭
ηn
ξm
0
0
⎧
M−1
&
N−1
⎨&&
&
&G1 s, t, us, t − G1 s, t, us, t&
⎩
sm tn
0
3.6
0
⎫
s t &
&&⎬
& &G2 ξ, η, u ξ, η − G2 ξ, η, u ξ, η & ≤ ε,
⎭
ξm ηn
0
0
where ε > 0 is a constant, and um, n ∈ ℘Ω is the solution of the following difference equation:
Abstract and Applied Analysis
21
m−1
n−1
p
u m, n am, n ⎡
⎣F 1 s, t, m, n, us, t sm0 tn0
⎤
F 2 ξ, η, m, n, u ξ, η ⎦
ξm0 ηn0
⎡
s t
M−1
M−1
s t
sm0 tn0
ξm0 ηn0
⎣G1 s, t, m, n, us, t ⎤
G2 ξ, η, m, n, u ξ, η ⎦,
3.7
where F i , Gi : Ω2 × R → R, i 1, 2; then one has
&
& p
&u m, n − up m, n& ≤ ε 1 JM, N
Cm, n ,
1 − μM, N
3.8
provided that μM, N < 1, where
⎧
Jm, n m−1
n−1 ⎨
f1 s, t, m, n
sm0 tn0
⎩
s t
ξm0 ηn0
⎧
M−1
N−1
⎨
μm, n ⎩
ξm0
⎧
M−1
N−1
⎨
g1 s, t, m, nCs, t sm0 tn0
Cm, n exp
⎫
⎬
g2 ξ, η, m, n ,
⎭
ηn
s t
g1 s, t, m, n sm0 tn0
⎫
⎬
f2 ξ, η, m, n
⎭
⎩
0
⎫
⎬
g2 ξ, η, m, n C ξ, η ,
⎭
ηn
s t
ξm0
m−1
n−1
3.9
0
Bs, t, m, n ,
sm0 tn0
Bs, t, m, n f1 s, t, m, n s t
f2 ξ, η, m, n .
ξm0 ηn0
Proof. From 3.1 and 3.7 we deduce
& p
&
&u m, n−up m, n& ≤ |am, n−am, n|
⎧
m−1
n−1 ⎨&
&
&
&
&F1 s, t, m, n, us, t−F 1 s, t, m, n, us, t&
⎩
sm tn
0
0
⎫
s t &
&⎬
&
&
&F2 ξ, η, m, n, u ξ, η −F 2 ξ, η, m, n, u ξ, η &
⎭
ξm ηn
0
0
22
Abstract and Applied Analysis
⎧
M−1
&
N−1
⎨&&
&
&G1 s, t, m, n, us, t − G1 s, t, m, n, us, t&
⎩
sm tn
0
0
⎫
s t &
&⎬
&
&
&G2 ξ, η, m, n, u ξ, η − G2 ξ, η, m, n, u ξ, η &
⎭
ξm ηn
0
0
≤ |am, n − am, n|
⎧
m−1
n−1 ⎨&
&
&
&
&F1 s, t, m, n, us, t−F1 s, t, m, n, us, t&
⎩
sm tn
0
0
&
&
&
&
&F1 s, t, m, n, us, t − F 1 s, t, m, n, us, t&
s t &
&
&F2 ξ, η, m, n, u ξ, η −F2 ξ, η, m, n, u ξ, η &
ξm0 ηn0
⎫
& &&⎬
&
&F2 ξ, η, m, n, u ξ, η −F 2 ξ, η, m, n, u ξ, η &
⎭
⎧
M−1
N−1
⎨&
sm0 tn0
⎩
&
&
&
&G1 s, t, m, n, us, t − G1 s, t, m, n, us, t&
&
&
&
&
&G1 s, t, m, n, us, t − G1 s, t, m, n, us, t&
s t &
&
&G2 ξ, η, m, n, u ξ, η −G2 ξ, η, m, n, u ξ, η &
ξm0 ηn0
⎫
& &&⎬
&
&G2 ξ, η, m, n, u ξ, η − G2 ξ, η, m, n, u ξ, η &
⎭
≤ε
m−1
n−1
&
'&
&F1 s, t, m, n, us, t − F1 s, t, m, n, us, t&
sm0 tn0
& &(
&F2 ξ, η, m, n, u ξ, η − F2 ξ, η, m, n, u ξ, η &
⎧
M−1
N−1
⎨&
&
&G1 s, t, m, n, us, t − G1 s, t, m, n, us, t&
⎩
sm tn
0
0
⎫
s t &
&⎬
&G2 ξ, η, m, n, u ξ, η −G2 ξ, η, m, n, u ξ, η &
⎭
ξm ηn
0
≤ε
m−1
n−1
sm0 tn0
⎡
0
&
&
⎣f1 s, t, m, n&up s, t − up s, t&
Abstract and Applied Analysis
23
s t
⎤
& p p &
f2 ξ, η, m, n &u ξ, η −u ξ, η &⎦
ξm0 ηn0
⎡
M−1
M−1
&
&
⎣g1 s, t, m, n&up s, t − up s, t&
sm0 tn0
s t
⎤
&
& p g2 ξ, η, m, n &u ξ, η − up ξ, η &⎦.
ξm0 ηn0
3.10
Then a suitable application of Corollary 2.7 yields the desired result.
Remark 3.5. We note that the results in 1–30 are not available here to establish the analysis
above.
Acknowledgment
The authors thank the referees very much for their careful comments and valuable
suggestions on this paper.
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