Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2009, Article ID 486895, 17 pages
doi:10.1155/2009/486895
Research Article
An Extension to Nonlinear Sum-Difference
Inequality and Applications
Wu-Sheng Wang1 and Xiaoliang Zhou2
1
2
Department of Mathematics, Hechi University, Yizhou, Guangxi 546300, China
Department of Mathematics, Guangdong Ocean University, Zhanjiang 524088, China
Correspondence should be addressed to Xiaoliang Zhou, zjhdzxl@yahoo.com.cn
Received 31 March 2009; Revised 31 March 2009; Accepted 17 May 2009
Recommended by Martin J. Bohner
We establish a general form of sum-difference inequality in two variables, which includes both
more than two distinct nonlinear sums without an assumption of monotonicity and a nonconstant
term outside the sums. We employ a technique of monotonization and use a property of stronger
monotonicity to give an estimate for the unknown function. Our result enables us to solve those
discrete inequalities considered in the work of W.-S. Cheung 2006. Furthermore, we apply our
result to a boundary value problem of a partial difference equation for boundedness, uniqueness,
and continuous dependence.
Copyright q 2009 W.-S. Wang and X. Zhou. 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. Introduction
Gronwall-Bellman inequality 1, 2 is a fundamental tool in the study of existence,
uniqueness, boundedness, stability, invariant manifolds and other qualitative properties
of solutions of differential equations and integral equation. There are a lot of papers
investigating them such as 3–15. Along with the development of the theory of integral
inequalities and the theory of difference equations, more attentions are paid to some discrete
versions of Bellman-Gronwall type inequalities e.g., 16–18. Starting from the basic form
un ≤ an n−1
fsus,
1.1
s0
discussed in 19, an interesting direction is to consider the inequality
n−1 αu2 s Qgsus ,
u2 n ≤ P 2 u2 0 2
s0
1.2
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a discrete version of Dafermos’ inequality 20, where α, P, Q are nonnegative constants and
u, g are nonnegative functions defined on {1, 2, . . . , T } and {1, 2, . . . , T − 1}, respectively. Pang
and Agarwal 21 proved for 1.2 that un ≤ 1 αn P u0 n−1
s0 Qgs for all 0 ≤ n ≤ T .
Another form of sum-difference inequality
n−1
u2 n ≤ c2 2
f1 suswus f2 sus
1.3
s0
n−1
estimated by Pachpatte 22 as un ≤ Ω−1 Ωc n−1
s0 f2 s s0 f1 s, where Ωu :
was
u
ds/ws.
Recently,
Pachpatte
23,
24
discussed
the
inequalities
of
two variables
u0
um, n ≤ c m−1
n−1
us, t as, t log us, t bs, tg log us, t ,
s0 t0
um, n ≤ c m−1
n−1
f1 s, tgus, t s0 t0
s0 t0
κs, t, σ, τguσ, τ
σ0 τ0
1.4
⎞⎞
⎛
τ−1
s−1 σ−1 t−1
⎝
⎝
h s, t, σ, τ, ξ, η g u ξ, η ⎠⎠,
⎛
s−1 t−1
m−1
n−1
m−1
n−1
s0 t0
σ0 τ0
ξ0 η0
where g is nondecreasing. In 25 another form of inequality of two variables
u2 m, n ≤ c2 m−1
n−1
as, tus, t sm0 tn0
m−1
n−1
bs, tus, twus, t
1.5
sm0 tn0
was discussed. Later, this result was generalized in 26 to the inequality
up m, n ≤ c m−1
n−1
ds, tuq s, t sm0 tn0
m−1
n−1
es, tuq s, twus, t,
1.6
sm0 tn0
where c, p, and q are all constants, c ≥ 0, p > q > 0, and d, e are both nonnegative real-valued
functions defined on a lattice in Z2 , and w is a continuous nondecreasing function satisfying
wu > 0 for all u > 0.
In this paper we establish a more general form of sum-difference inequality with
positive integers m, n,
ψum, n ≤ am, n k m−1
n−1
fi m, n, s, tϕi us, t,
1.7
i1 sm0 tn0
where k ≥ 2. In 1.7 we replace the constant c, the functions up , ds, t, es, t, uq and uq wu
in 1.6 with a function am, n, more general functions ψu, f1 m, n, s, t, f2 m, n, s, t,
ϕ1 u and ϕ2 u, respectively. Moreover, we consider more than two nonlinear terms and
do not require the monotonicity of every ϕi i 1, 2, . . . , k. We employ a technique of
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3
monotonization to construct a sequence of functions which possesses stronger monotonicity
than the previous one. Unlike the work in 26 for two sum terms, the maximal regions of
validity for our estimate of the unknown function u are decided by boundaries of more than
two planar regions. Thus we have to consider the inclusion of those regions and find common
regions. We demonstrate that inequalities 1.6 and other inequalities considered in 26 can
also be solved with our result. Furthermore, we apply our result to boundary value problems
of a partial difference equation for boundedness, uniqueness, and continuous dependence.
2. Main Result
Throughout this paper, let R −∞, ∞, R 0, ∞, and N0 {0, 1, 2, . . .}, m0 , n0 ∈ N0 , X, Y ∈
N0 ∪ {∞} are given nonnegative integers. For any integers s < t, let diss, t {j : s ≤ j ≤
t, j ∈ N0 }, I dism0 , X, and J disn0 , Y . Define Λ I × J ⊂ N20 , and let Λs,t denote the
sublattice dism0 , s × disn0 , t in Λ for any s, t ∈ Λ.
For functions gm, n, m, n ∈ N0 , their first-order differences are defined by
Δ1 gm, n gm 1, n − gm, n and Δ2 gm, n gm, n 1 − gm, n. Obviously, the
linear difference equation Δxm bm with the initial condition xm0 0 has the solution
m0 −1
m−1
sm0 bs. In the sequel, for convenience, we complementarily define that
sm0 bs 0.
We give the following basic assumptions for the inequality 1.7.
H1 ψ is a strictly increasing continuous function on R satisfying that ψ∞ ∞ and
ψu > 0 for all u > 0.
H2 All ϕi i 1, 2, . . . , k are continuous and positive functions on R .
H3 am, n ≥ 0 on Λ.
H4 All fi i 1, 2, . . . , k are nonnegative functions on Λ × Λ.
With given functions ϕ1 , ϕ2 , and ψ, we technically consider a sequence of functions
wi s, which can be calculated recursively by
w1 s : max ϕ1 τ ,
τ∈0,s
wi1 s : max
τ∈0,s
2.1
ϕi1 τ
wi s,
wi τ
i 1, . . . , k − 1.
For given constants ui > 0 and variable u > 0, we define
Wi u, ui :
u
dx
−1 ,
w
ψ
x
ui i
i 1, 2, . . . , k.
2.2
Obviously, Wi is strictly increasing in u > 0 and therefore the inverses Wi−1 are well defined,
continuous, and increasing. Let
fi m, n, s, t :
max
τ,ξ∈m0 ,m×n0 ,n
fi τ, ξ, s, t,
2.3
which is nondecreasing in m and n for each fixed s and t and satisfies fi x, y, t, s ≥
fi x, y, t, s ≥ 0 for all i 1, . . . , k.
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Theorem 2.1. Suppose that H1 –H4 hold and um, n is a nonnegative function on Λ satisfying
1.7. Then, for m, n ∈ ΛM1 ,N1 , a sublattice in Λ,
um, n ≤ ψ
−1
Wk−1
m−1
n−1
Wk Υk m, n fk m, n, s, t
,
2.4
sm0 tn0
where Υk m, n is determined recursively by
Υ1 m, n : am0 , n0 m−1
|as 1, n0 − as, n0 | sm0
Υi1 m, n :
|am, t 1 − am, t|,
tn0
Wi−1
n−1
Wi Υi m, n m−1
n−1
2.5
fi m, n, s, t ,
i 1, . . . , k − 1,
sm0 tn0
and M1 , N1 ∈ Λ is arbitrarily given on the boundary of the lattice
dx
fi m, n, s, t ≤
−1 , i 1, 2, . . . , k .
m, n ∈ Λ : Wi Υi m, n ui wi ψ x
sm0 tn0
2.6
U :
∞
m−1
n−1
Remark 2.2. As explained in 3, Remark 2, since different choices of ui in Wi i 1, 2, . . . , k
do not affect our results, we simply let Wi u denote Wi u, ui when there is no confusion. For
i u u dx/wi ψ −1 x. Obviously, W
i u Wi u W
i ui ui , let W
positive constants vi /
vi
i ui . It follows that
−1 v W −1 v − W
and W
i
i
−1
W
i
i Υi m, n W
m−1
n−1
fi m, n, s, t sm0 tn0
Wi−1
Wi Υi m, n m−1
n−1
fi m, n, s, t ,
sm0 tn0
2.7
i , i 1, 2, . . . , k.
that is, we obtain the same expression in 2.4 if we replace Wi with W
Moreover, by replacing Wi with Wi , the condition in the definition of U in Theorem 2.1 reads
i Υi M1 , N1 W
M
1 −1 N
1 −1
fi m, n, s, t ≤
sm0 tn0
∞
dx
,
−1
vi wi ψ x
i 1, 2, . . . , k,
2.8
the left-hand side of which is equal to
i ui Wi Υi M1 , N1 W
M
1 −1 N
1 −1
sm0 tn0
fi m, n, s, t,
2.9
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and the right-hand side of which equals
ui
dx
−1 vi wi ψ x
∞
dx
i ui −1 W
ui wi ψ x
∞
dx
.
−1
ui wi ψ x
2.10
The comparison between the both sides implies that 2.8 is equivalent to the condition given
in the definition of U in Theorem 2.1 with m, n M1 , N1 .
Remark 2.3. If we choose k 2, ψu up , ϕ1 u uq , ϕ2 u uq wu with p > q > 0,
f1 m, n, s, t ds, t and f2 m, n, s, t es, t and restrict am, n to be a constant c in 1.7,
then we can apply Theorem 2.1 to inequality 1.6 discussed in 26.
3. Proof of Theorem
First of all, we monotonize some given functions ϕi , fi in the sums. Obviously, the sequence
wi s defined by ϕi i 1, . . . , k in 2.1 consists of nondecreasing nonnegative functions and
satisfies wi s ≥ ϕi s, for i 1, . . . , k. Moreover,
wi ∝ wi1 ,
i 1, 2, . . . , k − 1,
3.1
as defined in 27 for comparison of monotonicity of functions wi s i 1, . . . , k, because
every ratio wi1 s/wi s is nondecreasing. By the definitions of functions wi , fi , ψ, and Υ1 ,
from 1.7 we get
um, n ≤ ψ
−1
Υ1 m, n k m−1
n−1
fi m, n, s, twi us, t ,
∀m, n ∈ Λ.
3.2
i1 sm0 tn0
Then, we discuss the case that am, n > 0 for all m, n ∈ Λ. Because Υ1 satisfies
m−1
Υ1 m, n am0 , n0 |as 1, n0 − as, n0 | sm0
n−1
|am, t 1 − am, t|
tn0
3.3
≥ am, n,
it is positive and nondecreasing on Λ. We consider the auxiliary inequality to 3.2, for all
m, n ∈ ΛM,N ,
um, n ≤ ψ
−1
Υ1 M, N k m−1
n−1
i1 sm0 tn0
fi M, N, s, twi us, t ,
3.4
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where M ∈ dism0 , M1 and N ∈ disn0 , N1 are chosen arbitrarily, and claim that, for
m, n ∈ Λmin{M2 ,M},min{N2 ,N} , a sublattice in ΛM1 ,N1 ,
um, n ≤ ψ
−1
Wk−1
Wk
n−1
m−1
k M, N, m, n fk M, N, s, t
Υ
,
3.5
sm0 tn0
k M, N, m, n is determined recursively by
where Υ
1 M, N, m, n : Υ1 M, N,
Υ
n−1
m−1
−1
Υi1 M, N, m, n : Wi Wi Υi M, N, m, n fi M, N, s, t ,
3.6
sm0 tn0
i 1, 2, . . . , k − 1, and M2 , N2 ∈ ΛM1 ,N1 is arbitrarily chosen on the boundary of the lattice
U1 :
n−1
m−1
i M, N, m, n fi M, N, s, t
m, n ∈ Λ : Wi Υ
sm0 tn0
dx
≤
−1 , i 1, 2, . . . , k .
ui wi ψ x
∞
3.7
We note that M2 , N2 can be chosen appropriately such that
M2 M, N M1 ,
N2 M, N N1 ,
∀M, N ∈ ΛM1 ,N1 .
3.8
In fact, from the fact of M1 , N1 being on the boundary of the lattice U, we see that
M
1 −1 N
1 −1
i M1 , N1 , M1 , N1 Wi Υ
fi M1 , N1 , s, t
sm0 tn0
Wi Υi M1 , N1 ≤
∞
M
1 −1 N
1 −1
fi M1 , N1 , s, t
3.9
sm0 tn0
dx
,
−1
ui wi ψ x
i 1, 2, . . . , k.
Thus, it means that we can take M2 M1 , N2 N1 . Moreover, M min{M2 , M}, N min{N2 , N}.
In the following, we will use mathematical induction to prove 3.5.
n−1 For k 1, let zm, n m−1
sm0
tn0 f1 M, N, s, tw1 us, t. Then z is nonnegative
and nondecreasing in each variable on ΛM,N . From 3.4 we observe that
um, n ≤ ψ −1 Υ1 M, N zm, n,
∀m, n ∈ ΛM N .
3.10
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Moreover, we note that w1 is nondecreasing and satisfies w1 u > 0 for u > 0 and that
Υ1 M, N zm, n > 0. From 3.10 we have
n−1 Δ1 Υ1 M, N zm, n
tn0 f1 M, N, m, tw1 um, t
−1
w1 ψ Υ1 M, N zm, n
w1 ψ −1 Υ1 M, N zm, n
≤
n−1
3.11
f1 M, N, m, t.
tn0
On the other hand, by the Mean Value Theorem for integral and by the monotonicity of w1
and ψ, for arbitrarily given m, n, m 1, n ∈ ΛM N there exists ξ in the open interval
Υ1 M, N zm, n, Υ1 M, N zm 1, n such that
W1 Υ1 M, N zm 1, n − W1 Υ1 M, N zm, n
Υ1 M,Nzm1,n
du
−1 w
ψ
u
1
Υ1 M,Nzm,n
Δ1 Υ1 M, N zm, n
w1 ψ −1 ξ
≤
Δ1 Υ1 M, N zm, n
.
w1 ψ −1 Υ1 M, N zm, n
3.12
It follows from 3.11 and 3.12 that
W1 Υ1 M, N zm 1, n − W1 Υ1 M, N zm, n ≤
n−1
f1 M, N, m, t.
3.13
tn0
Substituting m with s and summing both sides of 3.13 from s m0 to m − 1, we get, for all
m, n ∈ ΛM N ,
W1 Υ1 M, N zm, n ≤ W1 Υ1 M, N m−1
n−1
f1 M, N, s, t.
3.14
sm0 tn0
We note from the definition of zm, n in 3.2 and the definition of
zm0 , n 0. By the monotonicity of W −1 and 3.10 we obtain
um, n ≤ ψ
−1
W1−1
W1 Υ1 M, N that is, 3.5 is true for k 1.
m−1
n−1
sm0 tn0
m0 −1
sm0
in Section 2 that
f1 M, N, s, t
,
∀m, n ∈ ΛM N , 3.15
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Next, we make the inductive assumption that 3.5 is true for k l. Consider
um, n ≤ ψ
−1
Υ1 M, N l1 m−1
n−1
fi M, N, s, twi us, t ,
3.16
i1 sm0 tn0
l1 m−1 n−1 for all m, n ∈ ΛM N . Let ym, n sm0
tn0 fi M, N, s, twi us, t, which is
i1
nonnegative and nondecreasing in each variable on ΛM,N . Then 3.16 is equivalent to
um, n ≤ ψ −1 Υ1 M, N ym, n ,
∀m, n ∈ ΛM N .
3.17
Since wi is nondecreasing and satisfies wi u > 0 for u > 0 i 1, 2, . . . , l 1 and Υ1 K, L ym, n > 0, from 3.17 we obtain, for all m, n ∈ ΛM N ,
n−1 Δ1 Υ1 M, N ym, n
tn0 f1 M, N, m, tw1 um, t
−1 w1 ψ Υ1 M, N ym, n
w1 ψ −1 Υ1 M, N ym, n
l1 n−1 tn fi M, N, m, twi um, t
i2
−10
w1 ψ Υ1 M, N ym, n
≤
n−1
f1 M, N, m, t tn0
n−1
l fi1 M, N, m, tφi1 um, t,
i1 tn0
3.18
where
φi u :
wi u
,
w1 u
i 2, 3, . . . , l 1.
3.19
On the other hand, by the Mean Value Theorem for integrals and by the monotonicity of
w1 and ψ, for arbitrarily given m, n, m 1, n ∈ ΛM,N there exists ξ in the open interval
Υ1 M, N ym, n, Υ1 M, N ym 1, n such that
W1 Υ1 M, N ym 1, n − W1 Υ1 M, N ym, n
Υ1 M,Nym1,n
du
−1 w
ψ
u
1
Υ1 M,Nym,n
Δ1 Υ1 M, N ym, n
w1 ψ −1 ξ
Δ1 Υ1 M, N ym, n
≤
.
w1 ψ −1 Υ1 M, N ym, n
3.20
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Therefore, it follows from 3.18 and 3.20 that
W1 Υ1 M, N ym 1, n − W1 Υ1 M, N ym, n
≤
n−1
f1 M, N, m, t tn0
n−1
l 3.21
fi1 M, N, m, tφi1 um, t.
i1 tn0
substituting m with s in 3.21 and summing both sides of 3.21 from s m0 to m − 1, we
get, for all m, n ∈ ΛM,N ,
W1 Υ1 M, N ym, n − W1 Υ1 M, N
≤
m−1
n−1
f1 M, N, s, t sm0 tn0
l m−1
n−1
fi1 M, N, s, tφi1 us, t,
3.22
i1 sm0 tn0
where we note that ym0 , n 0. For convenience, let
ψΞm, n : W1 Υ1 M, N ym, n ,
θM, N, m, n : W1 Υ1 M, N m−1
n−1
3.23
f1 M, N, s, t.
sm0 tn0
From 3.17 and 3.22 we can get
Ξm, n ≤ ψ
−1
θM, N, M, N l m−1
n−1
i1 sm0 tn0
−1
−1
,
fi1 M, N, s, tφi1 ψ W1 ψΞm, n
3.24
the same form as 3.4 for k l, for all m, n ∈ ΛM,N , where we note that θM, N, M, N ≥
θM, N, m, n for all m, n ∈ ΛM,N . We are ready to use the inductive assumption for 3.24.
In order to demonstrate the basic condition of monotonicity, let hs ψ −1 W1−1 ψs,
obviously which is a continuous and nondecreasing function on R . Thus each φi hs is
continuous and nondecreasing on R and satisfies φi hs > 0 for s > 0. Moreover,
φi1 hs wi1 hs
ϕi1 τ
max
,
φi hs
wi hs
wi τ
τ∈0,hs
3.25
which is also continuous nondecreasing on R and positive on R . This implies that φi hs ∝
φi1 hs, for i 2, . . . , l. Therefore, the inductive assumption for 3.5 can be used to 3.24
and we obtain, for all m, n ∈ Λmin{M,M3 },min{N,N3 } ,
Ξm, n ≤ ψ
−1
Φ−1
l1
Φl1 θl1 M, N, m, n m−1
n−1
sm0 tn0
fl1 M, N, s, t
,
3.26
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u
where Φi u : ui ds/φi hs, u > 0, u ψ −1 W1 u, Φ−1
i is the inverse of Φi for
i 2, 3, . . . , l 1, θl1 M, N, m, n is determined recursively by
θ1 M, N, m, n : θM, N, M, N,
θi1 M, N, m, n :
Φ−1
i
m−1
n−1
Φi θi M, N, m, n fi M, N, s, t ,
i 1, 2, . . . , l,
3.27
sm0 tn0
and M3 , N3 are functions of M, N such that M3 M, N, N3 M, N ∈ ΛM1 ,N1 lie on the
boundary of the lattice
U2 :
m, n ∈ Λ : Φi θi M, N, m, n m−1
n−1
fi M, N, s, t
sm0 tn0
ds
, i 2, 3, . . . , l 1 ,
≤
ui φi hs
∞
3.28
where ∞ denotes either limu → ∞ u if it converges or ∞. Note that
u
ds
W1−1 ψs
ui θ
u
w1 ψ −1 W1−1 ψs ds
−1 −1 W1 ψs
ui wi ψ
Φi u ψ −1
W −1 ψu
3.29
dx
−1 w
ψ
x
i
ui
Wi W1−1 ψu , i 2, 3, . . . , l 1.
1
Thus, from 3.17, 3.23, and 3.27, 3.26 can be equivalently written as
um, n ≤ ψ −1 W1−1 ψΞm, n
−1
≤ ψ −1 Wl1
Wl1 W1−1 ψθl1 M, N, m, n
m−1
n−1
fl1 M, N, s, t
,
3.30
∀m, n ∈ Λmin{M,M3 },min{N,N3 } .
sm0 tn0
i M, N, m, n, defined
We further claim that the term W1−1 ψθi M, N, m, n is the same as Υ
in 3.6, i 1, 2, . . . , l 1. For convenience, let θi M, N, m, n W1−1 ψθi M, N, m, n.
1 M, N, m, n.
Obviously, it is that θ1 M, N, m, n Υ
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The remainder case is that am, n 0 for some m, n ∈ Λ. Let
Υ1,ε m, n Υ1 m, n ε,
3.31
where ε > 0 is an arbitrary small number. Obviously, Υ1,ε m, n > 0 for all m, n ∈ Λ. Using
the same arguments as above and replacing Υ1 m, n with Υ1,ε m, n, we get
um, n ≤ ψ
−1
W2−1
W2
W1−1
W1 Υ1, m, n m−1
n−1
f1 s, t
sm0 tn0
m−1
n−1
sm0 tn0
f2 s, t
3.32
for all m, n ∈ Λm1 ,n1 .
Considering continuities of Wi and Wi−1 for i 1, 2 as well as of Υi,ε in ε and letting
ε → 0 , we obtain 2.4. This completes the proof.
We remark that m1 , n1 lie on the boundary of the lattice U. In particular, 2.4 is true
∞
for all m, n ∈ Λ when every wi i 1, 2 satisfies ui dx/wi ψ −1 x ∞. Therefore, we may
take m1 M, n1 N.
4. Applications to a Difference Equation
In this section we apply our result to the following boundary value problem simply called
BVP for the partial difference equation:
Δ1 Δ2 ψzm, n Fm, n, zm, n,
m, n ∈ Λ,
4.1
zm, n0 fm,
zm0 , n gn,
m, n ∈ Λ,
where Λ : I × J is defined as in the beginning of Section 2, ψ ∈ C0 R, R is strictly increasing
odd function satisfying ψu > 0 for u > 0, F : Λ × R → R satisfies
|Fm, n, u| ≤ h1 m, nϕ1 |u| h2 m, nϕ2 |u|,
4.2
for given functions h1 , h2 : Λ → R and ϕi ∈ C0 R , R i 1, 2 satisfying ϕi u > 0 for
u > 0, and functions f : I → R and g : J → R satisfy that fm0 gn0 0. Obviously,
4.1 is a generalization of the BVP problem considered by 26, Section 3, and the theorems
of 26 are not able to solve it. In the following we first apply our main result to the discussion
of boundedness of 4.1.
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Corollary 4.1. All solutions zm, n of BVP 4.1 have the following estimation for all m, n ∈
Λm1 ,n1 |zm, n| ≤ ψ
−1
W2−1
W2 Υ2 m, n m−1
n−1
h2 s, t
,
4.3
sm0 tn0
where m1 , n1 are given as in Theorem 2.1 and
W2 u u
dx
,
−1 −1
maxτ∈0,x ϕ1 ψ −1 τ
1 maxτ∈0,x ϕ2 ψ τ /maxτ1 ∈0,τ ϕ1 ψ τ1 u
dx
−1 ,
max
τ∈0,x ϕ1 ψ τ
1
m−1
n−1
−1
Υ2 m, n W1 W1 Υ1 m, n h1 t, s ,
W1 u sm0 tn0
Υ1 m, n ≤
m−1
n−1 ψ fs 1 − ψ fs ψ gt 1 − ψ gt .
sm0
tn0
4.4
Proof. Clearly, the difference equation of BVP 4.1 is equivalent to
n−1
m−1
ψzm, n ψ fm ψ gn Fs, t, zs, t.
4.5
sm0 tn0
It follows, by 4.2, that
n−1
m−1
ψzm, n ≤ ψ fm ψ gn h1 s, tϕ1 |zs, t|
sm0 tn0
m−1
n−1
4.6
h2 s, tϕ2 |zs, t|.
sm0 tn0
Let am, n |ψfm ψgn|. Since |ψzm, n| ψ|zm, n|, 4.6 is of the form 1.6.
Applying our Theorem 2.1 to inequality 4.6, we obtain the estimate of zm, n as given in
this corollary.
Corollary 4.1 gives a condition of boundedness for solutions. Concretely, if
Υ1 m, n < ∞,
m−1
n−1
sm0 tn0
h1 s, t < ∞,
m−1
n−1
h2 s, t < ∞
sm0 tn0
for all m, n ∈ Λm1 ,n1 , then every solution zm, n of BVP 4.1 is bounded on Λm1 ,n1 .
4.7
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13
Next, we discuss the uniqueness of solutions for BVP 4.1.
Corollary 4.2. Suppose additionally that
|Fm, n, u1 − Fm, n, u2 | ≤ h1 m, nϕ1 ψu1 − ψu2 h2 m, nϕ2 ψu1 − ψu2 4.8
for u1 , u2 ∈ R and m, n ∈ Λ : I × J, where I m0 , M ∩ N0 , J n0 , N ∩ N0 as assumed in the
beginning of Section 2 with natural numbers M and N, h1 , h2 are both nonnegative functions defined
on the lattice Λ, ϕ1 , ϕ2 ∈ C0 R , R are both nondecreasing with the nondecreasing ratio ϕ2 /ϕ1 such
1
that ϕi 0 0, ϕi u > 0 for all u > 0 and 0 ds/ϕi s ∞ for i 1, 2 and ψ ∈ C0 R, R is strictly
increasing odd function satisfying ψu > 0 for u > 0. Then BVP 4.1 has at most one solution on Λ.
Proof. Assume that both zm, n and zm, n are solutions of BVP 4.1. From the equivalent
form 4.5 of 4.1 we have
n−1
m−1
ψzm, n − ψ
zm, n ≤
zs, t
h1 s, tϕ1 ψzs, t − ψ
sm0 tn0
m−1
n−1
zs, t
h2 s, tϕ2 ψzs, t − ψ
4.9
sm0 tn0
for all m, n ∈ Λ, which is an inequality of the form 1.7, where am, n ≡ 0. Applying
our Theorem 2.1 with the choice that u1 u2 1, we obtain an estimate of the difference
|ψzm, n − ψ
zm, n| in the form 2.4, where Υ1 m, n ≡ 0 because am, n ≡ 0.
Furthermore, by the definition of Wi we see that
lim Wi u −∞,
u→0
lim Wi−1 u 0,
u → −∞
i 1, 2.
4.10
It follows that
W1 Υ1 m, n m−1
n−1
h1 s, t −∞,
4.11
sm0 tn0
since m < M, n < N. Thus, by 4.10,
Υ2 m, n W1−1
W1 Υ1 m, n m−1
n−1
sm0 tn0
h1 s, t 0.
4.12
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Advances in Difference Equations
Similarly, we get W2 Υ2 m, n m−1 n−1
sm0
tn0
h2 s, t −∞ and therefore
W2−1
W2 Υ2 m, n m−1
n−1
h2 s, t 0.
4.13
sm0 tn0
Thus we conclude from 2.4 that |ψzm, n − ψ
zm, n| ≤ 0, implying that zm, n zm, n for all m, n ∈ Λ since ψ is strictly increasing. It proves the uniqueness.
Remark 4.3. If h1 ≡ 0 or h2 ≡ 0 in 4.8, the conclusion of the Corollary 4.2 also can be obtained.
Finally, we discuss the continuous dependence of solutions of BVP 4.1 on the given
functions F, f, and g. Consider a variation of BVP 4.1
Δ1 Δ2 ψzm, n Fm,
n, zm, n,
m, n ∈ Λ,
zm, n0 fm,
m, n ∈ Λ,
zm0 , n g n,
4.14
where ψ ∈ C0 R, R is strictly increasing odd function satisfying ψu > 0 for u > 0, F ∈
0 g n0 0.
C0 Λ × R, R, and f : I → R, g : J → R are functions satisfying fm
Corollary 4.4. Let F be a function as assumed in the beginning of Section 4 and satisfy 4.2 and
4.8 on the same lattice Λ as assumed in Corollary 4.2. Suppose that the three differences
maxf − f ,
m∈I
max g − g ,
n∈J
max
s,t,u∈Λ×R
Fs, t, u − Fs, t, u
4.15
are all sufficiently small. Then solution zm, n of BVP 4.14 is sufficiently close to the solution
zm, n of BVP 4.1.
Proof. By Corollary 4.2, the solution zm, n is unique. By the continuity and the strict
monotonicity of ψ, we suppose that
maxψ fm
− ψ fm < ,
m∈I
max
maxψ g n − ψ gn < ,
n∈J
Fs, t, u − Fs, t, u < ,
s,t,u∈I×J×R
4.16
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15
where > 0 is a small number. By the equivalent difference equation 4.5 and the inequality
4.8 we get
ψ zm, n − ψzm, n ≤ ψ fm
− ψ fm ψ g n − ψ gn Fs, t, zs, t − Fs, t, zs, t
m−1
n−1 sm0 tn0
≤ 2 Fs, t, zs, t − Fs, t, zs, t
m−1
n−1 sm0 tn0
m−1
n−1
|Fs, t, zs, t − Fs, t, zs, t|
4.17
sm0 tn0
≤ {2 m1 − m0 n1 − n0 }
m−1
n−1
zs, t − ψzs, t
h1 s, tϕ1 ψ
sm0 tn0
m−1
n−1
zs, t − ψzs, t ,
h2 s, tϕ2 ψ
sm0 tn0
that is an inequality of the form 1.7. Applying Theorem 2.1 to 4.17, we obtain, for all
m, n ∈ Λm1 ,n1 , that
m−1
n−1
ψ zm, n − ψzm, n ≤ W −1 W2 Υ2 m, n h2 s, t ,
2
4.18
sm0 tn0
where m1 , n1 are given as in Theorem 2.1,
Υ2 m, n W1−1
W1 Υ1 m, n m−1
n−1
h1 t, s ,
sm0 tn0
4.19
Υ1 m, n {2 m1 − m0 n1 − n0 }.
By 4.10 we see that Υi m, n → 0 i 1, 2 as → 0. It follows from 4.18 that
zm, n − ψzm, n| 0 and hence zm, n depends continuously on F, f, and
lim → 0 |ψ
g.
Remark 4.5. Our requirement of the small difference F − F in Corollary 4.4 is stronger than the
condition iii in 26, Theorem 3.3, but it is easier to check than the condition of them.
16
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Acknowledgments
The authors thank Professor Weinian Zhang Sichuan University for his valuable discussion.
The authors also thank the referees for their helpful comments and suggestions. This work is
supported by the Natural Science Foundation of Guangxi Autonomous Region 200991265,
by Scientific Research Foundation of the Education Department of Guangxi Autonomous
Region of China 200707MS112 and by Foundation of Natural Science of Hechi University
2006A-N001 and Key Discipline of Applied Mathematics of Hechi University of China
200725.
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