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 Zheng 1 and Qinghua Feng 1, 2 1 School of Science, Shandong University of Technology, Shandong, Zibo 255049, China 2 School of Mathematical Sciences, Qufu Normal University, Shandong, Qufu 273165, China Correspondence should be addressed to Bin Zheng, [email protected]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 estab- lished, 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 Gronwall- Bellman inequality 1, 2and its various generalizations which provide explicit bounds play a fundamental role in the research of this domain. Many such generalized inequalities e.g., see 3–30and the references thereinhave 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 Volterra- Fredholm-type discrete inequalities have been established. Recent results in this direction include the work of Ma 9to 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.
Transcript
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2011, Article ID 584951, 24 pagesdoi:10.1155/2011/584951
Research ArticleSome New Volterra-Fredholm-Type DiscreteInequalities and Their Applications in the Theoryof Difference Equations
Bin Zheng1 and Qinghua Feng1, 2
1 School of Science, Shandong University of Technology, Shandong, Zibo 255049, China2 School of Mathematical Sciences, Qufu Normal University, Shandong, Qufu 273165, China
Copyright q 2011 B. Zheng and Q. Feng. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.
Some new Volterra-Fredholm-type discrete inequalities in two independent variables are estab-lished, which provide a handy tool in the study of qualitative and quantitative properties ofsolutions of certain difference equations. The established results extend some known results inthe literature.
1. Introduction
In the research of solutions of certain differential and difference equations, if the solutionsare unknown, then it is necessary to study their qualitative and quantitative properties suchas boundedness, uniqueness, and continuous dependence on initial data. The Gronwall-Bellman inequality [1, 2] and its various generalizations which provide explicit bounds playa 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 theknown Ou-Liang’s inequality [3]. In [8], Ma generalized the discrete version of Ou-Liang’sinequality in two variables to Volterra-Fredholm form for the first time, which has provedto be very useful in the study of qualitative as well as quantitative properties of solutions ofcertain Volterra-Fredholm-type difference equations. But since then, few results on Volterra-Fredholm-type discrete inequalities have been established. Recent results in this directioninclude the work of Ma [9] to our knowledge. We notice, in the analysis of some certainVolterra-Fredholm-type difference equations with more complicated forms, that the boundsprovided by the earlier inequalities are inadequate and it is necessary to seek some newVolterra-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-typediscrete inequalities, which extend Ma’s work in [9], and provide new bounds for unknownfunctions lying in these inequalities. We will illustrate the usefulness of the establishedresults by applying them to study the boundedness, uniqueness, and continuous dependenceon initial data of solutions of certain more complicated Volterra-Fredholm-type differenceequations.
Throughout this paper, R denotes the set of real numbers R+ = [0,∞), and Z denotesthe 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 twoconstants, andKi > 0, i = 1, 2, 3, 4, are all constants. IfU is a lattice, then we denote the set ofall R-valued functions on U by ℘(U) and denote the set of all R+-valued functions on U by℘+(U). Finally, for a function f ∈ ℘+(U), we have
∑m1s=m0
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
pK(q−p)/pa +
p − q
pKq/p. (2.1)
Lemma 2.2. Suppose that u(m,n) ∈ ℘+(Ω), b(s, t,m, n) ∈ ℘+(Ω2), α ≥ 0 is a constant. If b isnondecreasing in the third variable, then, for (m,n) ∈ Ω,
u(m,n) ≤ α +m−1∑
s=m0
n−1∑
t=n0
b(s, t,m, n)u(s, t) (2.2)
implies that
u(m,n) ≤ α exp
{m−1∑
s=m0
n−1∑
t=n0
b(s, t,m, n)
}
. (2.3)
Lemma 2.3. Suppose that u(m,n), a(m,n), b(m,n) ∈ ℘+(Ω). If a(m,n) is nondecreasing in thefirst variable, then, for (m,n) ∈ Ω,
u(m,n) ≤ a(m,n) +m−1∑
s=m0
b(s, n)u(s, n) (2.4)
implies that
u(m,n) ≤ a(m,n)m−1∏
s=m0
[1 + b(s, n)]. (2.5)
Abstract and Applied Analysis 3
Remark 2.4. Lemma 2.3 is a direct variation of [19, Lemma 2.5(β1)], and we note a(m,n) ≥ 0here.
Theorem 2.5. Suppose that u(m,n), a(m,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 inthe last two variables. p, qi, ri are nonnegative constants with p ≥ qi, p ≥ ri, i = 1, 2, . . . , l1, p /= 0,while hi, ji are nonnegative constants with p ≥ hi, p ≥ ji, i = 1, 2, . . . , l2. If, for (m,n) ∈ Ω, u(m,n)satisfies
up(m,n) ≤ a(m,n) +l1∑
i=1
m−1∑
s=m0
n−1∑
t=n0
⎡
⎣bi(s, t,m, n)uqi(s, t) +s∑
ξ=m0
t∑
η=n0
ci(ξ, η,m, n
)uri(ξ, η
)⎤
⎦
+l2∑
i=1
M−1∑
s=m0
N−1∑
t=n0
⎡
⎣di(s, t,m, n)uhi(s, t) +s∑
ξ=m0
t∑
η=n0
ei(ξ, η,m, n
)uji(ξ, η
)⎤
⎦,
(2.6)
then
u(m,n) ≤{
a(m,n) +J(M,N)
1 − μ(M,N)C(m,n)
}1/p
, (2.7)
provided that μ(M,N) < 1, where
J(m,n) =l1∑
i=1
m−1∑
s=m0
n−1∑
t=n0
⎧⎨
⎩bi(s, t,m, n)
[qipK
(qi−p)/p1 a(s, t) +
p − qip
Kqi/p
1
]
+s∑
ξ=m0
t∑
η=n0
ci(ξ, η,m, n
)[ripK
(ri−p)/p2 a
(ξ, η
)+p − rip
Kri/p
2
]⎫⎬
⎭
+l2∑
i=1
M−1∑
s=m0
N−1∑
t=n0
⎧⎨
⎩di(s, t,m, n)
[hi
pK
(hi−p)/p3 a(s, t) +
p − hi
pK
hi/p
3
]
+s∑
ξ=m0
t∑
η=n0
ei(ξ, η,m, n
)[jipK
(ji−p)/p4 a
(ξ, η
)+p − jip
Kji/p
4
]⎫⎬
⎭,
(2.8)
μ(m,n) =l2∑
i=1
M−1∑
s=m0
N−1∑
t=n0
⎧⎨
⎩di(s, t,m, n)
hi
pK
(hi−p)/p3 C(s, t)+
s∑
ξ=m0
t∑
η=n0
ei(ξ, η,m, n
) jipK
(ji−p)/p4 C
(ξ, η
)
⎫⎬
⎭,
(2.9)
C(m,n) = exp
{m−1∑
s=m0
n−1∑
t=n0
B(s, t,m, n)
}
, (2.10)
B(s, t,m, n) =l1∑
i=1
⎡
⎣bi(s, t,m, n)qipK
(qi−p)/p1 +
s∑
ξ=m0
t∑
η=n0
ci(ξ, η,m, n
)ripK
(ri−p)/p2
⎤
⎦. (2.11)
4 Abstract and Applied Analysis
Proof. Denote
v(m,n) =l1∑
i=1
m−1∑
s=m0
n−1∑
t=n0
⎡
⎣bi(s, t,m, n)uqi(s, t) +s∑
ξ=m0
t∑
η=n0
ci(ξ, η,m, n
)uri(ξ, η
)⎤
⎦
+l2∑
i=1
M−1∑
s=m0
N−1∑
t=n0
⎡
⎣di(s, t,m, n)uhi(s, t) +s∑
ξ=m0
t∑
η=n0
ei(ξ, η,m, n
)uji(ξ, η
)⎤
⎦.
(2.12)
Then, we have
u(m,n) ≤ [a(m,n) + v(m,n)]1/p, (2.13)
and, furthermore, from Lemma 2.1 we have
uqi(m,n) ≤ [a(m,n) + v(m,n)]qi/p ≤ qipK
(qi−p)/p1 [a(m,n) + v(m,n)]
+p − qip
Kqi/p
1 , i = 1, 2, . . . , l1,
uri(m,n) ≤ [a(m,n) + v(m,n)]ri/p ≤ ripK
(ri−p)/p2 [a(m,n) + v(m,n)]
+p − rip
Kri/p
2 , i = 1, 2, . . . , l1,
uhi(m,n) ≤ [a(m,n) + v(m,n)]hi/p ≤ hi
pK
(hi−p)/p3 [a(m,n) + v(m,n)]
+p − hi
pK
hi/p
3 , i = 1, 2, . . . , l2,
uji(m,n) ≤ [a(m,n) + v(m,n)]ji/p ≤ jipK
(ji−p)/p4 [a(m,n) + v(m,n)]
+p − jip
Kji/p
4 , i = 1, 2, . . . , l2.
(2.14)
Abstract and Applied Analysis 5
So
v(m,n) ≤l1∑
i=1
m−1∑
s=m0
n−1∑
t=n0
⎧⎨
⎩bi(s, t,m, n)
[qipK
(qi−p)/p1 (a(s, t)+v(s, t))+
p − qip
Kqi/p
1
]
+s∑
ξ=m0
t∑
η=n0
ci(ξ, η,m, n
)[ripK
(ri−p)/p2
(a(ξ, η
)+ v
(ξ, η
))+p − rip
Kri/p
2
]⎫⎬
⎭
+l2∑
i=1
M−1∑
s=m0
N−1∑
t=n0
⎧⎨
⎩di(s, t,m, n)
[hi
pK
(hi−p)/p3 (a(s, t) + v(s, t)) +
p − hi
pK
hi/p
3
]
+s∑
ξ=m0
t∑
η=n0
ei(ξ, η,m, n
)[jipK
(ji−p)/p4
(a(ξ, η
)+ v
(ξ, η
))+p − jip
Kji/p
4
]⎫⎬
⎭
= H(m,n) +l1∑
i=1
m−1∑
s=m0
n−1∑
t=n0
⎡
⎣bi(s, t,m, n)qipK
(qi−p)/p1 v(s, t)
+s∑
ξ=m0
t∑
η=n0
ci(ξ, η,m, n
)ripK
(ri−p)/p2 v
(ξ, η
)⎤
⎦,
(2.15)
whereH(m,n) =J(m,n)+∑l2
i=1
∑M−1s=m0
∑N−1t=n0
{di(s, t,m, n)(hi/p)K(hi−p)/p3 v(s, t)+
∑sξ=m0
∑tη=n0
ei
(ξ, η,m, n)(ji/p)K(ji−p)/p4 v(ξ, η)} and J(m,n) is defined in (2.8). Then, using that H(m,n) is
nondecreasing in every variable, we obtain
v(m,n) ≤ H(M,N) +l1∑
i=1
m−1∑
s=m0
n−1∑
t=n0
⎡
⎣bi(s, t,m, n)qipK
(qi−p)/p1 v(s, t)
+s∑
ξ=m0
t∑
η=n0
ci(ξ, η,m, n
)ripK
(ri−p)/p2 v
(ξ, η
)⎤
⎦
≤ H(M,N) +l1∑
i=1
m−1∑
s=m0
n−1∑
t=n0
⎡
⎣bi(s, t,m, n)qipK
(qi−p)/p1
+s∑
ξ=m0
t∑
η=n0
ci(ξ, η,m, n
)ripK
(ri−p)/p2
⎤
⎦v(s, t)
= H(M,N) +m−1∑
s=m0
n−1∑
t=n0
B(s, t,m, n)v(s, t),
(2.16)
where B(s, t,m, n) is defined in (2.11).
6 Abstract and Applied Analysis
Since bi(s, t,m, n), ci(s, t,m, n) are nondecreasing in the last two variables, thenB(s, t,m, n) is also nondecreasing in the last two variables, and by a suitable application ofLemma 2.2 we obtain
v(m,n) ≤ H(M,N) exp
{m−1∑
s=m0
n−1∑
t=n0
B(s, t,m, n)
}
= H(M,N)C(m,n), (2.17)
where C(m,n) is defined in (2.10). Furthermore, considering the definition of H(m,n) and(2.17)we have
H(M,N) = J(M,N) +l2∑
i=1
M−1∑
s=m0
N−1∑
t=n0
⎧⎨
⎩di(s, t,M,N)
hi
pK
(hi−p)/p3 v(s, t)
+s∑
ξ=m0
t∑
η=n0
ei(ξ, η,M,N
) jipK
(ji−p)/p4 v
(ξ, η
)
⎫⎬
⎭
≤ J(M,N) +l2∑
i=1
M−1∑
s=m0
N−1∑
t=n0
⎧⎨
⎩di(s, t,M,N)
hi
pK
(hi−p)/p3 H(M,N)C(s, t)
+s∑
ξ=m0
t∑
η=n0
ei(ξ, η,m1, n1
) jipK
(ji−p)/p4 H(M,N)C
(ξ, η
)
⎫⎬
⎭
= J(M,N) +H(M,N)l2∑
i=1
M−1∑
s=m0
N−1∑
t=n0
⎧⎨
⎩di(s, t,M,N)
hi
pK
(hi−p)/p3 C(s, t)
+s∑
ξ=m0
t∑
η=n0
ei(ξ, η,M,N
) jipK
(ji−p)/p4 C
(ξ, η
)
⎫⎬
⎭
= J(M,N) +H(M,N)μ(M,N),(2.18)
where μ(m,n) is defined in (2.9). Then,
H(M,N) ≤ J(M,N)1 − μ(M,N)
. (2.19)
Combining (2.17) and (2.19)we deduce
v(m,n) ≤ J(M,N)1 − μ(M,N)
C(m,n). (2.20)
Then, combining (2.13) and (2.20), we obtain the desired result.
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, . . . , l1with g1i, g2i nondecreasing in every variable. d1i(m,n), e1i(m,n) ∈ ℘+(Ω), i = 1, 2, . . . , l2.u(m,n), a(m,n), p, qi, ri, hi, ji are defined as in Theorem 2.5. If, for (m,n) ∈ Ω, u(m,n) satisfies
up(m,n) ≤ a(m,n) +l1∑
i=1
g1i(m,n)m−1∑
s=m0
n−1∑
t=n0
⎡
⎣b1i(s, t)uqi(s, t) +s∑
ξ=m0
t∑
η=n0
c1i(ξ, η
)uri(ξ, η
)⎤
⎦
+l2∑
i=1
g2i(m,n)M−1∑
s=m0
N−1∑
t=n0
⎡
⎣d1i(s, t)uhi(s, t) +s∑
ξ=m0
t∑
η=n0
e1i(ξ, η
)uji(ξ, η
)⎤
⎦,
(2.21)
then
u(m,n) ≤{
a(m,n) +J(M,N)
1 − μ(M,N)C(M,N)
}1/p
, (2.22)
provided that μ(M,N) < 1, where
J(m,n) =l1∑
i=1
g1i(m,n)m−1∑
s=m0
n−1∑
t=n0
⎧⎨
⎩b1i(s, t)
[qipK
(qi−p)/p1 a(s, t) +
p − qip
Kqi/p
1
]
+s∑
ξ=m0
t∑
η=n0
c1i(ξ, η
)[ripK
(ri−p)/p2 a
(ξ, η
)+p − rip
Kri/p
2
]⎫⎬
⎭
+l2∑
i=1
g2i(m,n)M−1∑
s=m0
N−1∑
t=n0
⎧⎨
⎩d1i(s, t)
[hi
pK
(hi−p)/p3 a(s, t) +
p − hi
pK
hi/p
3
]
+s∑
ξ=m0
t∑
η=n0
e1i(ξ, η
)[jipK
(ji−p)/p4 a
(ξ, η
)+p − jip
Kji/p
4
]⎫⎬
⎭,
μ(m,n) =l2∑
i=1
⎧⎨
⎩g2i(m,n)
M−1∑
s=m0
N−1∑
t=n0
⎡
⎣d1i(s, t)hi
pK
(hi−p)/p3 C(s, t)
+s∑
ξ=m0
t∑
η=n0
e1i(ξ, η
) jipK
(ji−p)/p4 C
(ξ, η
)⎤
⎦
⎫⎬
⎭,
C(m,n) = exp
{m−1∑
s=m0
n−1∑
t=n0
B(s, t,m, n)
}
,
B(s, t,m, n) =l1∑
i=1
g1i(m,n)
⎡
⎣b1i(s, t)qipK
(qi−p)/p1 +
s∑
ξ=m0
t∑
η=n0
c1i(ξ, η
)ripK
(ri−p)/p2
⎤
⎦.
(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,n)b1i(s,t),ci(s,t,m,n) = g1i(m,n)c1i(s,t), di(s,t,m,n) = g2i(m,n)d1i(s,t), ei(s,t,m,n) = g2i(m,n)e1i(s,t) inTheorem 2.5.
Corollary 2.7. Suppose that u(m,n), a(m,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, u(m,n) satisfies
u(m,n) ≤ a(m,n) +l1∑
i=1
m−1∑
s=m0
n−1∑
t=n0
⎡
⎣bi(s, t,m, n)u(s, t) +s∑
ξ=m0
t∑
η=n0
ci(ξ, η,m, n
)u(ξ, η
)⎤
⎦
+l2∑
i=1
M−1∑
s=m0
N−1∑
t=n0
⎡
⎣di(s, t,m, n)u(s, t) +s∑
ξ=m0
t∑
η=n0
ei(ξ, η,m, n
)u(ξ, η
)⎤
⎦,
(2.24)
then
u(m,n) ≤ a(m,n) +J(M,N)
1 − μ(M,N)C(m,n), (2.25)
provided that μ(M,N) < 1, where
J(m,n) =l1∑
i=1
m−1∑
s=m0
n−1∑
t=n0
⎧⎨
⎩bi(s, t,m, n)a(s, t) +
s∑
ξ=m0
t∑
η=n0
ci(ξ, η,m, n
)a(ξ, η
)
⎫⎬
⎭
+l2∑
i=1
M−1∑
s=m0
N−1∑
t=n0
⎧⎨
⎩di(s, t,m, n)a(s, t) +
s∑
ξ=m0
t∑
η=n0
ei(ξ, η,m, n
)a(ξ, η
)
⎫⎬
⎭,
μ(m,n) =l2∑
i=1
M−1∑
s=m0
N−1∑
t=n0
⎧⎨
⎩di(s, t,m, n)C(s, t) +
s∑
ξ=m0
t∑
η=n0
ei(ξ, η,m, n
)C(ξ, η
)
⎫⎬
⎭,
C(m,n) = exp
{m−1∑
s=m0
n−1∑
t=n0
B(s, t,m, n)
}
,
B(s, t,m, n) =l1∑
i=1
⎡
⎣bi(s, t,m, n) +s∑
ξ=m0
t∑
η=n0
ci(ξ, η,m, n
)⎤
⎦.
(2.26)
Theorem 2.8. Suppose that w(m,n) ∈ ℘+(Ω), u, a, bi, ci, di, ei, p, qi, ri, hi, ji are defined as inTheorem 2.5. Furthermore, assume that a(m,n) is nondecreasing in the first variable. If, for (m,n) ∈Ω, u(m,n) satisfies
Obviously z(m,n) is nondecreasing in the first variable. So by Lemma 2.3 we obtain
up(m,n) ≤ z(m,n)m−1∏
s=m0
[1 +w(s, n)] = z(m,n)w(m,n), (2.37)
where w(m,n) =∏m−1
s=m0[1 +w(s, n)]. Define
v(m,n) =l1∑
i=1
m−1∑
s=m0
n−1∑
t=n0
⎡
⎣bi(s, t,m, n)uqi(s, t) +s∑
ξ=m0
t∑
η=n0
ci(ξ, η,m, n
)uri(ξ, η
)⎤
⎦
+l2∑
i=1
M−1∑
s=m0
N−1∑
t=n0
⎡
⎣di(s, t,m, n)uhi(s, t) +s∑
ξ=m0
t∑
η=n0
ei(ξ, η,m, n
)uji(ξ, η
)⎤
⎦.
(2.38)
Abstract and Applied Analysis 11
Then,
u(m,n) ≤ [(a(m,n) + v(m,n))w(m,n)]1/p, (2.39)
and, furthermore, by (2.39) and Lemma 2.1 we have
v(m,n) ≤l1∑
i=1
m−1∑
s=m0
n−1∑
t=n0
⎧⎨
⎩bi(s, t,m, n)[(a(s, t) + v(s, t))w(s, t)]qi/p
+s∑
ξ=m0
t∑
η=n0
ci(ξ, η,m, n
)[(a(ξ, η
)+v
(ξ, η
))w(ξ, η
)]ri/p
⎫⎬
⎭
+l2∑
i=1
M−1∑
s=m0
N−1∑
t=n0
⎧⎨
⎩di(s, t,m, n)[(a(s, t) + v(s, t))w(s, t)]hi/p
+s∑
ξ=m0
t∑
η=n0
ei(ξ, η,m, n
)[(a(ξ, η
)+v
(ξ, η
))w(ξ, η
)]ji/p
⎫⎬
⎭
≤l1∑
i=1
m−1∑
s=m0
n−1∑
t=n0
⎧⎨
⎩bi(s, t,m, n)(w(s, t))qi/p
[qipK
(qi−p)/p1 (a(s, t) + v(s, t)) +
p − qip
Kqi/p
1
]
+s∑
ξ=m0
t∑
η=n0
ci(ξ, η,m, n
)(w(ξ, η
))ri/p
×[ripK
(ri−p)/p2
(a(ξ, η
)+ v
(ξ, η
))+p − rip
Kri/p
2
]⎫⎬
⎭
+l2∑
i=1
M−1∑
s=m0
N−1∑
t=n0
⎧⎨
⎩di(s, t,m, n)(w(s, t))hi/p
[hi
pK
(qi−p)/p3 (a(s, t) + v(s, t)) +
p − hi
pK
hi/p
3
]
+s∑
ξ=m0
t∑
η=n0
ei(ξ, η,m, n
)(w(ξ, η
))ji/p
×[jipK
ji−p/p4
(a(ξ, η
)+v
(ξ, η
))+p − jip
Kji/p
4
]⎫⎬
⎭
=l1∑
i=1
m−1∑
s=m0
n−1∑
t=n0
⎧⎨
⎩bi(s, t,m, n)
[qipK
(qi−p)/p1 (a(s, t) + v(s, t)) +
p − qip
Kqi/p
1
]
+s∑
ξ=m0
t∑
η=n0
ci(ξ, η,m, n
)[ripK
(ri−p)/p2
(a(ξ, η
)+ v
(ξ, η
))+p − rip
Kri/p
2
]⎫⎬
⎭
12 Abstract and Applied Analysis
+l2∑
i=1
M−1∑
s=m0
N−1∑
t=n0
⎧⎨
⎩di(s, t,m, n)
[hi
pK
(qi−p)/p3 (a(s, t) + v(s, t)) +
p − hi
pK
hi/p
3
]
+s∑
ξ=m0
t∑
η=n0
ei(ξ, η,m, n
)[jipK
(ji−p)/p4
(a(ξ, η
)+ v
(ξ, η
))+p − jip
Kji/p
4
]⎫⎬
⎭
= H(m,n) +l1∑
i=1
m−1∑
s=m0
n−1∑
t=n0
⎡
⎣bi(s, t,m, n)qipK
(qi−p)/p1 v(s, t)
+s∑
ξ=m0
t∑
η=n0
ci(ξ, η,m, n
)ripK
(ri−p)/p2 v
(ξ, η
)⎤
⎦,
(2.40)
whereH(m,n) = J(m,n)+∑l2
i=1∑M−1
s=m0
∑N−1t=n0
{di(s, t,m, n)(hi/p)K(hi−p)/p3 v(s, t)+
∑sξ=m0
∑tη=n0
ei
(ξ, η,m, n)ji/pK(ji−p)/p4 v(ξ, η)}, and J(m,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
v(m,n) ≤ J(M,N)1 − μ(M,N)
C(m,n), (2.41)
where μ(m,n), C(m,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,n)b1i(s,t), ci(s,t,m,n) = g1i(m,n)c1i(s,t),di(s,t,m,n)=g2i(m,n)d1i(s,t), ei(s,t,m,n) = g2i(m,n)e1i(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 weomit 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, v)(u−v) foru ≥ v ≥ 0. If, for (m,n) ∈ Ω, u(m,n) satisfies
up(m,n) ≤ a(m,n) +l1∑
i=1
m−1∑
s=m0
n−1∑
t=n0
⎡
⎣bi(s, t,m, n)uqi(s, t) +s∑
ξ=m0
t∑
η=n0
ci(ξ, η,m, n
)uri(ξ, η
)⎤
⎦
+l2∑
i=1
M−1∑
s=m0
N−1∑
t=n0
⎡
⎣di(s, t,m, n)Li
(s, t, uhi(s, t)
)+
s∑
ξ=m0
t∑
η=n0
ei(ξ, η,m, n
)uji(ξ, η
)⎤
⎦,
(2.42)
Abstract and Applied Analysis 13
then
u(m,n) ≤{
a(m,n) +J(M,N)
1 − μ(M,N)C(m,n)
}1/p
, (2.43)
provided that μ(M,N) < 1, where
J(m,n) =l1∑
i=1
m−1∑
s=m0
n−1∑
t=n0
⎧⎨
⎩bi(s, t,m, n)
[qipK
(qi−p)/p1 a(s, t) +
p − qip
Kqi/p
1
]
+s∑
ξ=m0
t∑
η=n0
ci(ξ, η,m, n
)[ripK
(ri−p)/p2 a
(ξ, η
)+p − rip
Kri/p
2
]⎫⎬
⎭
+l2∑
i=1
M−1∑
s=m0
N−1∑
t=n0
⎧⎨
⎩di(s, t,m, n)Li
[
s, t,hi
pK
(hi−p)/p3 a(s, t) +
p − hi
pK
hi/p
3
]
+s∑
ξ=m0
t∑
η=n0
ei(ξ, η,m, n
)[jipK
(ji−p)/p4 a
(ξ, η
)+p − jip
Kji/p
4
]⎫⎬
⎭,
(2.44)
μ(m,n) =l2∑
i=1
M−1∑
s=m0
N−1∑
t=n0
⎧⎨
⎩di(s, t,m, n)
hi
pK
(hi−p)/p3 C(s, t)
+s∑
ξ=m0
t∑
η=n0
ei(ξ, η,m, n
) jipK
(ji−p)/p4 C
(ξ, η
)
⎫⎬
⎭,
(2.45)
di(s, t,m, n) = di(s, t,m, n)Ti[
s, t,hi
pK
(hi−p)/p3 a(s, t) +
p − hi
pK
hi/p
3
]
, i = 1, 2, . . . , l2, (2.46)
C(m,n) = exp
{m−1∑
s=m0
n−1∑
t=n0
B(s, t,m, n)
}
, (2.47)
B(s, t,m, n) =l1∑
i=1
⎡
⎣bi(s, t,m, n)qipK
(qi−p)/p1 +
s∑
ξ=m0
t∑
η=n0
ci(ξ, η,m, n
)ripK
(ri−p)/p2
⎤
⎦. (2.48)
14 Abstract and Applied Analysis
Proof. Denote
v(m,n) =l1∑
i=1
m−1∑
s=m0
n−1∑
t=n0
⎡
⎣bi(s, t,m, n)uqi(s, t) +s∑
ξ=m0
t∑
η=n0
ci(ξ, η,m, n
)uri(ξ, η
)⎤
⎦
+l2∑
i=1
M−1∑
s=m0
N−1∑
t=n0
⎡
⎣di(s, t,m, n)Li
(s, t, uhi(s, t)
)+
s∑
ξ=m0
t∑
η=n0
ei(ξ, η,m, n
)uji(ξ, η
)⎤
⎦.
(2.49)
Then
u(m,n) ≤ [a(m,n) + v(m,n)]1/p, (2.50)
and, furthermore, from Lemma 2.1 we have
v(m,n) ≤l1∑
i=1
m−1∑
s=m0
n−1∑
t=n0
⎧⎨
⎩bi(s, t,m, n)
[qipK
(qi−p)/p1 (a(s, t) + v(s, t)) +
p − qip
Kqi/p
1
]
+s∑
ξ=m0
t∑
η=n0
ci(ξ, η,m, n
)[ripK
(ri−p)/p2
(a(ξ, η
)+ v
(ξ, η
))+p − rip
Kri/p
2
]⎫⎬
⎭
+l2∑
i=1
M−1∑
s=m0
N−1∑
t=n0
⎧⎨
⎩di(s, t,m, n)Li
[
s, t,hi
pK
(hi−p)/p3 (a(s, t) + v(s, t)) +
p − hi
pK
hi/p
3
]
+s∑
ξ=m0
t∑
η=n0
ei(ξ, η,m, n
)[jipK
(ji−p)/p4
(a(ξ, η
)+ v
(ξ, η
))+p − jip
Kji/p
4
]⎫⎬
⎭
≤l1∑
i=1
m−1∑
s=m0
n−1∑
t=n0
⎧⎨
⎩bi(s, t,m, n)
[qipK
(qi−p)/p1 (a(s, t) + v(s, t)) +
p − qip
Kqi/p
1
]
+s∑
ξ=m0
t∑
η=n0
ci(ξ, η,m, n
)[ripK
(ri−p)/p2
(a(ξ, η
)+ v
(ξ, η
))+p − rip
Kri/p
2
]⎫⎬
⎭
+l2∑
i=1
M−1∑
s=m0
N−1∑
t=n0
⎧⎨
⎩di(s, t,m, n)
{
Li
[
s, t,hi
pK
(hi−p)/p3 (a(s, t) + v(s, t)) +
p − hi
pK
hi/p
3
]
Abstract and Applied Analysis 15
− Li
[
s, t,hi
pK
(hi−p)/p3 a(s, t) +
p − hi
pK
hi/p
3
]
+Li
[
s, t,hi
pK
(hi−p)/p3 a(s, t) +
p − hi
pK
hi/p
3
]}
+s∑
ξ=m0
t∑
η=n0
ei(ξ, η,m, n
)[jipK
(ji−p)/p4
(a(ξ, η
)+ v
(ξ, η
))+p − jip
Kji/p
4
]⎫⎬
⎭
≤l1∑
i=1
m−1∑
s=m0
n−1∑
t=n0
⎧⎨
⎩bi(s, t,m, n)
[qipK
(qi−p)/p1 (a(s, t) + v(s, t)) +
p − qip
Kqi/p
1
]
+s∑
ξ=m0
t∑
η=n0
ci(ξ, η,m, n
)[ripK
(ri−p)/p2
(a(ξ, η
)+ v
(ξ, η
))+p − rip
Kri/p
2
]⎫⎬
⎭
+l2∑
i=1
M−1∑
s=m0
N−1∑
t=n0
⎧⎨
⎩di(s, t,m, n)
{
Ti
[
s, t,hi
pK
(hi−p)/p3 a(s, t) +
p − hi
pK
hi/p
3
]hi
pK
(hi−p)/p3
×v(s, t) + Li
[
s, t,hi
pK
(hi−p)/p3 a(s, t) +
p − hi
pK
hi/p
3
]}
+s∑
ξ=m0
t∑
η=n0
ei(ξ, η,m, n
)[jipK
(ji−p)/p4
(a(ξ, η
)+ v
(ξ, η
))+p − jip
Kji/p
4
]⎫⎬
⎭
= H(m,n) +l1∑
i=1
m−1∑
s=m0
n−1∑
t=n0
⎡
⎣bi(s, t,m, n)qipK
(qi−p)/p1 v(s, t)
+s∑
ξ=m0
t∑
η=n0
ci(ξ, η,m, n
)ripK
(ri−p)/p2 v
(ξ, η
)⎤
⎦,
(2.51)
where H(m,n) = J(m,n)+∑l2
i=1
∑M−1s=m0
∑N−1t=n0
{di(s, t,m, n)(hi/p)K(hi−p)/p3 v(s, t)+
∑sξ=m0
∑tη=n0
ei
(ξ, η,m, n)(ji/p)K(ji−p)/p4 v(ξ, η)} and J(m,n), di(s, t,m, n) are defined in (2.44) and (2.46)
respectively.Similar to the process of (2.15)–(2.20) we deduce
v(m,n) ≤ J(M,N)1 − μ(M,N)
C(M,N), (2.52)
where μ(m,n), C(m,n) are defined in (2.45) and (2.47) respectively.Combining (2.50) and (2.52), we get the desired result.
16 Abstract and Applied Analysis
Theorem 2.11. Suppose that w(m,n) ∈ ℘+(Ω), u, a, bi, ci, di, ei, p, qi, ri, hi, ji are definedas in Theorem 2.5. Furthermore, assume a(m,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, u(m,n) satisfies
The proof for Theorem 2.11 is similar to the combination of Theorems 2.8 and 2.10, andwe 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, Theorem2.5]. If furthermore l1 = l2 = 1, then Corollary 2.6 reduces to [9, Theorem 2.1]. If wetake 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 showthat they are useful in the study of boundedness, uniqueness, and continuous dependence ofsolutions of certain difference equations.
Example 3.1. Consider the following Volterra-Fredholm sum-difference equation:
up(m,n) = a(m,n) +m−1∑
s=m0
n−1∑
t=n0
⎡
⎣F1(s, t,m, n, u(s, t)) +s∑
ξ=m0
t∑
η=n0
F2(ξ, η,m, n, u
(ξ, η
))⎤
⎦
+M−1∑
s=m0
M−1∑
t=n0
⎡
⎣G1(s, t,m, n, u(s, t)) +s∑
ξ=m0
t∑
η=n0
G2(ξ, η,m, n, u
(ξ, η
))⎤
⎦,
(3.1)
18 Abstract and Applied Analysis
where u(m,n), a(m,n) ∈ ℘(Ω), p ≥ 1 is an odd number, Fi, Gi : Ω2 × R → R, i = 1, 2, M, Nare two integers defined the same as in Theorem 2.5.
Theorem 3.2. Suppose that u(m,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
|u(m,n)| ≤{
|a(m,n)| + J(M,N)1 − μ(M,N)
C(m,n)}1/p
, (3.2)
provided that μ(M,N) < 1, where
J(m,n) =m−1∑
s=m0
n−1∑
t=n0
⎧⎨
⎩f1(s, t,m, n)
[q
pK
(q−p)/p1 |a(s, t)| + p − q
pK
q/p
1
]
+s∑
ξ=m0
t∑
η=n0
f2(ξ, η,m, n
)[r
pK
(r−p)/p2
∣∣a(ξ, η
)∣∣ +
p − r
pK
r/p
2
]⎫⎬
⎭
+M−1∑
s=m0
N−1∑
t=n0
⎧⎨
⎩g1(s, t,m, n)
[h
pK
(h−p)/p3 |a(s, t)| + p − h
pK
h/p
3
]
+s∑
ξ=m0
t∑
η=n0
g2(ξ, η,m, n
)[j
pK
(j−p)/p4
∣∣a(ξ, η
)∣∣ +
p − j
pK
j/p
4
]⎫⎬
⎭,
μ(m,n) =M−1∑
s=m0
N−1∑
t=n0
⎧⎨
⎩g1(s, t,m, n)
h
pK
(h−p)/p3 C(s, t)
+s∑
ξ=m0
t∑
η=n0
g2(ξ, η,m, n
) j
pK
(j−p)/p4 C
(ξ, η
)
⎫⎬
⎭,
C(m,n) = exp
{m−1∑
s=m0
n−1∑
t=n0
B(s, t,m, n)
}
,
B(s, t,m, n) = f1(s, t,m, n)q
pK
(q−p)/p1 +
s∑
ξ=m0
t∑
η=n0
f2(ξ, η,m, n
) r
pK
(r−p)/p2 .
(3.3)
Abstract and Applied Analysis 19
Proof. From (3.1)we have
|u(m,n)|p ≤ |a(m,n)| +m−1∑
s=m0
n−1∑
t=n0
⎡
⎣∣∣F1(s, t,m, n, u(s, t))
∣∣ +
s∑
ξ=m0
t∑
η=n0
∣∣F2
(ξ, η,m, n, u
(ξ, η
))∣∣
⎤
⎦
+M−1∑
s=m0
M−1∑
t=n0
⎡
⎣∣∣G1(s, t,m, n, u(s, t))
∣∣ +
s∑
ξ=m0
t∑
η=n0
∣∣G2
(ξ, η,m, n, u
(ξ, η
))∣∣
⎤
⎦
≤ |a(m,n)| +m−1∑
s=m0
n−1∑
t=n0
⎡
⎣f1(s, t,m, n
)∣∣u(s, t)
∣∣q +
s∑
ξ=m0
t∑
η=n0
f2(ξ, η,m, n
)∣∣u(ξ, η
)∣∣r
⎤
⎦
+M−1∑
s=m0
M−1∑
t=n0
⎡
⎣g1(s, t,m, n
)∣∣u(s, t)
∣∣h +
s∑
ξ=m0
t∑
η=n0
g2(ξ, η,m, n
)∣∣u(ξ, η
)∣∣j
⎤
⎦.
(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, wherefi, gi ∈ ℘+(Ω2), i = 1, 2 with fi, gi nondecreasing in the last two variables, and μ(M,N) =∑M−1
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,
∣∣∣u
p
1(m,n) − up
2(m,n)∣∣∣ ≤
m−1∑
s=m0
n−1∑
t=n0
⎡
⎣∣∣F1(s, t,m, n, u1(s, t)) − F1(s, t,m, n, u2(s, t))
∣∣
+s∑
ξ=m0
t∑
η=n0
∣∣F2
(ξ, η,m, n, u1
(ξ, η
))−F2(ξ, η,m, n, u2
(ξ, η
))∣∣
⎤
⎦
+M−1∑
s=m0
M−1∑
t=n0
⎡
⎣∣∣G1(s, t,m, n, u1(s, t)) −G1(s, t,m, n, u2(s, t))
∣∣
+s∑
ξ=m0
t∑
η=n0
∣∣G2
(ξ, η,m, n, u1
(ξ, η
))−G2(ξ, η,m, n, u2
(ξ, η
))∣∣
⎤
⎦
20 Abstract and Applied Analysis
≤m−1∑
s=m0
n−1∑
t=n0
⎡
⎣f1(s, t,m, n)∣∣∣u
p
1(s, t) − up
2(s, t)∣∣∣
+s∑
ξ=m0
t∑
η=n0
f2(ξ, η,m, n
)∣∣∣u
p
1
(ξ, η
) − up
2
(ξ, η
)∣∣∣
⎤
⎦
+M−1∑
s=m0
M−1∑
t=n0
⎡
⎣g1(s, t,m, n)∣∣∣u
p
1(s, t) − up
2(s, t)∣∣∣
+s∑
ξ=m0
t∑
η=n0
g2(ξ, η,m, n
)∣∣∣u
p
1
(ξ, η
) − up
2
(ξ, η
)∣∣∣
⎤
⎦.
(3.5)
Treat |up
1(m,n) − up
2(m,n)| as one variable, and a suitable application of Corollary 2.7yields |up
1(m,n) − up
2(m,n)| ≤ 0, which implies that up
1(m,n) ≡ up
2(m,n). Since p is an oddnumber, 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 functionsa, F1, F2, G1, G2.
Theorem 3.4. Suppose that u(m,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 holdfor ∀u, v ∈ R, where fi, gi ∈ ℘+(Ω2), i = 1, 2 with fi, gi nondecreasing in the last two variables, and,furthermore,
|a(m,n) − a(m,n)| +m−1∑
s=m0
n−1∑
t=n0
⎧⎨
⎩
∣∣∣F1(s, t, u(s, t)) − F1(s, t, u(s, t))
∣∣∣
+s∑
ξ=m0
t∑
η=n0
∣∣∣F2
(ξ, η, u
(ξ, η
)) − F2(ξ, η, u
(ξ, η
))∣∣∣
⎫⎬
⎭
+M−1∑
s=m0
N−1∑
t=n0
⎧⎨
⎩
∣∣∣G1(s, t, u(s, t)) −G1(s, t, u(s, t))
∣∣∣
+s∑
ξ=m0
t∑
η=n0
∣∣∣G2
(ξ, η, u
(ξ, η
)) −G2(ξ, η, u
(ξ, η
))∣∣∣
⎫⎬
⎭≤ ε,
(3.6)
where ε > 0 is a constant, and u(m,n) ∈ ℘(Ω) is the solution of the following difference equation:
Abstract and Applied Analysis 21
up(m,n) = a(m,n) +m−1∑
s=m0
n−1∑
t=n0
⎡
⎣F1(s, t,m, n, u(s, t)) +s∑
ξ=m0
t∑
η=n0
F2(ξ, η,m, n, u
(ξ, η
))⎤
⎦
+M−1∑
s=m0
M−1∑
t=n0
⎡
⎣G1(s, t,m, n, u(s, t)) +s∑
ξ=m0
t∑
η=n0
G2(ξ, η,m, n, u
(ξ, η
))⎤
⎦,
(3.7)
where Fi, Gi : Ω2 × R → R, i = 1, 2; then one has
∣∣up(m,n) − up(m,n)
∣∣ ≤ ε
[
1 +J(M,N)
1 − μ(M,N)C(m,n)
]
, (3.8)
provided that μ(M,N) < 1, where
J(m,n) =m−1∑
s=m0
n−1∑
t=n0
⎧⎨
⎩f1(s, t,m, n)+
s∑
ξ=m0
t∑
η=n0
f2(ξ, η,m, n
)
⎫⎬
⎭
+M−1∑
s=m0
N−1∑
t=n0
⎧⎨
⎩g1(s, t,m, n) +
s∑
ξ=m0
t∑
η=n0
g2(ξ, η,m, n
)
⎫⎬
⎭,
μ(m,n) =M−1∑
s=m0
N−1∑
t=n0
⎧⎨
⎩g1(s, t,m, n)C(s, t) +
s∑
ξ=m0
t∑
η=n0
g2(ξ, η,m, n
)C(ξ, η
)
⎫⎬
⎭,
C(m,n) = exp
{m−1∑
s=m0
n−1∑
t=n0
B(s, t,m, n)
}
,
B(s, t,m, n) = f1(s, t,m, n) +s∑
ξ=m0
t∑
η=n0
f2(ξ, η,m, n
).
(3.9)
Proof. From (3.1) and (3.7)we deduce
∣∣up(m,n)−up(m,n)
∣∣ ≤|a(m,n)−a(m,n)|
+m−1∑
s=m0
n−1∑
t=n0
⎧⎨
⎩
∣∣∣F1(s, t,m, n, u(s, t))−F1(s, t,m, n, u(s, t))
∣∣∣
+s∑
ξ=m0
t∑
η=n0
∣∣∣F2
(ξ, η,m, n, u
(ξ, η
))−F2(ξ, η,m, n, u
(ξ, η
))∣∣∣
⎫⎬
⎭
22 Abstract and Applied Analysis
+M−1∑
s=m0
N−1∑
t=n0
⎧⎨
⎩
∣∣∣G1(s, t,m, n, u(s, t)) −G1(s, t,m, n, u(s, t))
∣∣∣
+s∑
ξ=m0
t∑
η=n0
∣∣∣G2
(ξ, η,m, n, u
(ξ, η
)) −G2(ξ, η,m, n, u
(ξ, η
))∣∣∣
⎫⎬
⎭
≤ |a(m,n) − a(m,n)|
+m−1∑
s=m0
n−1∑
t=n0
⎧⎨
⎩
∣∣∣F1(s, t,m, n, u(s, t))−F1(s, t,m, n, u(s, t))
∣∣∣
+∣∣∣F1(s, t,m, n, u(s, t)) − F1(s, t,m, n, u(s, t))
∣∣∣
+s∑
ξ=m0
t∑
η=n0
∣∣F2
(ξ, η,m, n, u
(ξ, η
))−F2(ξ, η,m, n, u
(ξ, η
))∣∣
+∣∣∣F2
(ξ, η,m, n, u
(ξ, η
))−F2(ξ, η,m, n, u
(ξ, η
))∣∣∣
⎫⎬
⎭
+M−1∑
s=m0
N−1∑
t=n0
⎧⎨
⎩
∣∣∣G1(s, t,m, n, u(s, t)) −G1(s, t,m, n, u(s, t))
∣∣∣
+∣∣∣G1(s, t,m, n, u(s, t)) −G1(s, t,m, n, u(s, t))
∣∣∣
+s∑
ξ=m0
t∑
η=n0
∣∣G2
(ξ, η,m, n, u
(ξ, η
))−G2(ξ, η,m, n, u
(ξ, η
))∣∣
+∣∣∣G2
(ξ, η,m, n, u
(ξ, η
)) −G2(ξ, η,m, n, u
(ξ, η
))∣∣∣
⎫⎬
⎭
≤ ε +m−1∑
s=m0
n−1∑
t=n0
{∣∣F1(s, t,m, n, u(s, t)) − F1(s, t,m, n, u(s, t))
∣∣
+∣∣F2
(ξ, η,m, n, u
(ξ, η
)) − F2(ξ, η,m, n, u
(ξ, η
))∣∣}
+M−1∑
s=m0
N−1∑
t=n0
⎧⎨
⎩
∣∣G1(s, t,m, n, u(s, t)) −G1(s, t,m, n, u(s, t))
∣∣
+s∑
ξ=m0
t∑
η=n0
∣∣G2
(ξ, η,m, n, u
(ξ, η
))−G2(ξ, η,m, n, u
(ξ, η
))∣∣
⎫⎬
⎭
≤ ε +m−1∑
s=m0
n−1∑
t=n0
⎡
⎣f1(s, t,m, n)∣∣up(s, t) − up(s, t)
∣∣
Abstract and Applied Analysis 23
+s∑
ξ=m0
t∑
η=n0
f2(ξ, η,m, n
)∣∣up(ξ, η
)−up(ξ, η)∣∣
⎤
⎦
+M−1∑
s=m0
M−1∑
t=n0
⎡
⎣g1(s, t,m, n)∣∣up(s, t) − up(s, t)
∣∣
+s∑
ξ=m0
t∑
η=n0
g2(ξ, η,m, n
)∣∣up(ξ, η
) − up(ξ, η)∣∣
⎤
⎦.
(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 analysisabove.
Acknowledgment
The authors thank the referees very much for their careful comments and valuablesuggestions on this paper.
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