Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 795145, 14 pages
doi:10.1155/2010/795145
Research Article
Nonlinear Delay Discrete Inequalities and Their
Applications to Volterra Type Difference Equations
Yu Wu,1 Xiaopei Li,2 and Shengfu Deng3
1
Yibin University, Yibin, Sichuan 644007, China
Department of Mathematics, Zhanjiang Normal University, Zhanjiang, Guangdong 524048, China
3
Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, China
2
Correspondence should be addressed to Shengfu Deng, sf deng@sohu.com
Received 7 September 2009; Accepted 14 January 2010
Academic Editor: Abdelkader Boucherif
Copyright q 2010 Yu Wu et al. 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.
Delay discrete inequalities with more than one nonlinear term are discussed, which generalize
some known results and can be used in the analysis of various problems in the theory of certain
classes of discrete equations. Application examples to show boundedness and uniqueness of
solutions of a Volterra type difference equation are also given.
1. Introduction
Gronwall-Bellman inequalities and their various linear and nonlinear generalizations play
very important roles in the discussion of existence, uniqueness, continuation, boundedness,
and stability properties of solutions of differential equations and difference equations. The
literature on such inequalities and their applications is vast. For example, see 1–12 for
continuous cases, and 13–20 for discrete cases. In particular, the book 21 written by
Pachpatte considered three types of discrete inequalities:
un ≤ an n−1
fswus,
s0
n−1
u2 n ≤ an 2 fsus,
s0
u2 n ≤ an n−1
s0
fswus.
1.1
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In this paper, we consider a delay discrete inequality
un ≤ an n−1
m bi fi n, swi us,
n ∈ N0
1.2
i1 sbi 0
which has m nonlinear terms where N0 {0, 1, 2, . . .}. We will show that many discrete
inequalities like 1.1 can be reduced to this form. Our main result can be applied to analyze
properties of solutions of discrete equations. We also give examples to show boundedness
and uniqueness of solutions of a Volterra type difference equation.
2. Main Results
Assume that
C1 an is nonnegative for n ∈ N0 and a0 > 0;
C2 bi n i 1, . . . , m are nondecreasing for n ∈ N0 , the range of each bi belongs to N0 ,
and bi n ≤ n;
C3 all fi n, j i 1, . . . , m are nonnegative for n, j ∈ N0 ;
C4 all wi i 1, . . . , m are continuous and nondecreasing functions on 0, ∞ and are
positive on 0, ∞. They satisfy the relationship w1 ∝ w2 ∝ · · · ∝ wm where wi ∝ wi1
means that wi1 /wi is nondecreasing on 0, ∞ see 10.
u
Let Wi u ui dz/wi z for u ≥ ui where ui > 0 is a given constant. Then, Wi is strictly
increasing so its inverse Wi−1 is well defined, continuous, and increasing in its corresponding
domain. Define bi −1 −1, Δun un 1 − un and Δ2 rn, j rn, j 1 − rn, j.
Theorem 2.1. Suppose that (C1 )–(C4 ) hold and un is a nonnegative function for n ∈ N0 satisfying
1.2. Then
⎡
un ≤
−1 ⎣
a0
Wm
Wm bm
n−1
fm n, s sbm 0
n−1
s0 φm
Δ2 rm n, s
⎤
⎦,
−1
Wm−1
rm 0, s
n ≤ N1 ,
2.1
where an max0≤τ≤n,τ∈N0 aτ, fi n, j max0≤τ≤n,τ∈N0 fi τ, j, rm n, j is determined recursively
by
r1 n, j a j ,
bi
j−1
j−1
ri1 n, j Wi r1 n, 0 fi n, s sbi 0
s0 φi
Δ2 ri n, s
−1
Wi−1
ri 0, s
,
i 1, . . . , m − 1,
2.2
φi u wi u/wi−1 u, φ1 u w1 u, W0 I (Identity), and N1 is the largest positive integer
such that
Wi a0 bi N
1 −1
sbi 0
fi N1 , s N
1 −1
s0
Δ2 ri N1 , s
≤
−1
φi Wi−1 ri 0, s
∞
dz
,
w
i z
ui
i 1, . . . , m.
2.3
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Remark
2.2. 1 N1 is defined by 2.3 and N1 ∞ when all wi i 1, . . . , m satisfy
∞
dz/w
i z ∞. Different choices of ui in Wi do not affect our results see 2.
ui
2 If bi n n for i 1, . . . , m, then 2.1 gives the estimate of the following inequality:
un ≤ an m n−1
fi n, swi us,
n ∈ N0
2.4
i1 s0
by replacing bm n − 1, bm 0, bi j − 1, bi 0, and bi N1 − 1 with n − 1, 0, j − 1, 0 and N1 − 1,
respectively. Especially, if b1 n n and f1 n, s fs, then 1.2 for m 1 becomes the first
inequality of 1.1. Equation 2.1 shows the same estimate given by b1 of Theorem 4.2.3 in
the book 21.
Lemma 2.3. Δ2 ri n, j is nonnegative and nondecreasing in n, and ri n, j is nonnegative and
nondecreasing in n and j for i 1, . . . , m.
Proof. By the definitions of an and fi n, j, it is easy to check that they are nonnegative and
nondecreasing in n, and an ≥ an and fi n, j ≥ fi n, j for each fixed j where i 1, . . . , m.
a0 > 0 in C1 implies that an > 0 for all n ≤ N1 . Clearly,
Δ2 r1 n 1, j − Δ2 r1 n, j 0,
Δ2 r1 n 1, j − Δ2 r1 n, j
≥ 0,
Δ2 r2 n 1, j − Δ2 r2 n, j f1 n 1, b1 j − f1 n, b1 j w1 r1 0, j
2.5
where r1 0, j aj > 0 is used, which yields that Δ2 r1 n, j and Δ2 r2 n, j are nondecreasing
in n. Assume that Δ2 rl n, j is nondecreasing in n. Then
Δ2 rl1 n 1, j − Δ2 rl1
Δ2 rl n 1, j − Δ2 rl n, j
≥ 0,
n, j fl n 1, bl j − fl n, bl j −1 φl Wl−1
rl 0, j
2.6
which implies that Δ2 rl1 n, j is nondecreasing in n. By induction, Δ2 ri n, j i 1, . . . , m
are nondecreasing in n. Similarly, we can prove that they are nonnegative by induction again.
Then ri n, j i 1, . . . , m are nonnegative and nondecreasing in n and j.
Proof of Theorem 2.1. Take any arbitrary positive integer n
≤ N1 and consider the auxiliary
inequality
un ≤ r1 n, n n−1
m bi i1 sbi 0
n, swi us,
fi n≤n
.
2.7
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Claim that un in 2.7 satisfies
⎡
−1 ⎣
Wm r1 un ≤ Wm
n, 0 bm
n−1
n−1
sbm 0
s0 φm
n, s fm Δ2 rm n, s
−1
Wm−1
rm 0, s
⎤
⎦
2.8
for n ≤ {
n, N2 } where N2 is the largest positive integer such that
n, 0 Wi r1 bi N
2 −1
n, s fi sbi 0
N
2 −1
Δ2 ri n, s
−1
s0 φi Wi−1 ri 0, s
≤
∞
dz
,
w
i z
ui
2.9
i 1, . . . , m.
n, n, Δ2 ri n, n, and fi n, n are
Before we prove 2.8, notice that N1 ≤ N2 . In fact, ri is chosen larger.
nondecreasing in n
by Lemma 2.3. Thus, N2 satisfying 2.9 gets smaller as n
N1 if r1 n, 0 a0 is applied.
In particular, N2 satisfies the same 2.3 as N1 for n
We divide the proof of 2.8 into two steps by using induction.
b1 n−1 n, sw1 us for n ≤ n
and z0 0. It is clear that zn
f1 Step 1 m 1. Let zn sb
1 0
n, n zn
is nonnegative and nondecreasing. Observe that 2.7 is equivalent to un ≤ r1 for n ≤ n
and by assumptions C2 and C4 and Lemma 2.3,
Δzn f1 n, b1 nw1 ub1 n ≤ f1 n, b1 nw1 r1 n, b1 n zb1 n
≤ f1 n, b1 nw1 r1 n, n zn.
2.10
n, n an > 0, we have
Since w1 is nondecreasing and r1 Δzn Δ2 r1 n, n
n, n
Δ2 r1 ≤ f1 n, b1 n w1 r1 w1 r1 n, n zn
n, n zn
n, n
Δ2 r1 .
≤ f1 n, b1 n w1 r1 0, n
2.11
Then
zn1r1 n,n1
znr1 n,n
dτ
≤
w1 τ
≤
zn1r1 n,n1
znr1 n,n
dτ
w1 zn r1 n, n
Δzn Δ2 r1 n, n
w1 zn r1 n, n
n, n
Δ2 r1 ,
≤ f1 n, b1 n w1 r1 0, n
2.12
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and so
znr1 n,n
n−1
dτ
z0r1 n,0 w1 τ
s0
≤
zs1r1 n,s1
zsr1 n,s
dτ
w1 τ
n−1
n−1
Δ2 r1 n, s
n, b1 s f1 w
s
r
0,
s0
s0 1 1
b1
n−1
n, s f1 sb1 0
2.13
n−1
Δ2 r1 n, s
.
w
s
r
0,
1
1
s0
The definition of W1 in Theorem 2.1 and z0 0 show
W1 zn r1 n, n ≤ W1 r1 n, 0 b1
n−1
n, s f1 sb1 0
n−1
Δ2 r1 n, s
,
w r 0, s
s0 1 1
n≤n
.
2.14
Equation 2.9 shows that the right side of 2.14 is in the domain of W1−1 for all n ≤ n
. Thus
the monotonicity of W1−1 implies
⎡
⎤
n−1
Δ
r
n
,
s
2 1
⎦
un ≤ zn r1 n, n ≤ W1−1 ⎣W1 r1 n, 0 n, s f1 w
r1 0, s
1
s0
sb 0
b1
n−1
2.15
1
for n ≤ n
; that is, 2.8 is true for m 1.
Step 2 m k 1. Assume that 2.8 is true for m k. Consider
un ≤ r1 n, n n−1
k1 bi n, swi us,
fi n≤n
.
2.16
i1 sbi 0
k1 bi n−1 Let zn n, swi us and z0 0. Then zn is nonnegative and
f i1
sbi 0 i
n, n zn for n ≤ n
. Moreover, we have
nondecreasing and satisfies un ≤ r1 Δzn k1
k1
n, bi nwi ubi n ≤
n, bi nwi r1 n, bi n zbi n.
fi fi i1
i1
2.17
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Advances in Difference Equations
n, n > 0, we have by the
Since wi and r1 are nondecreasing in their arguments and r1 assumption bi n ≤ n
n, n
Δzn Δ2 r1 ≤
w1 zn r1 n, n
k1 n, bi nwi zbi n r1 n, bi n
n, n
Δ2 r1 i1 fi w1 zn r1 w1 r1 n, n
n, n
≤ f1 n, b1 n k1
wi zbi n r1 n, bi n
n, n
Δ2 r1 n, bi n
fi w1 zbi n r1 n, bi n w1 r1 0, n
i2
≤ f1 n, b1 n k
n, bi1 nφi1 zbi1 n r1 n, bi1 n
fi1 i1
n, n
Δ2 r1 w1 r1 0, n
2.18
for n ≤ n
where φi1 u wi1 u/w1 u for i 1, . . . , k, which gives
zn1r1 n,n1
znr1 n,n
dτ
≤
w1 τ
zn1r1 n,n1
znr1 n,n
dτ
w1 zn r1 n, n
Δzn Δ2 r1 n, n
w1 r1 n, n zn
≤
2.19
n, n
Δ2 r1 ≤ f1 n, b1 n w1 r1 0, n
k
n, bi1 nφi1 zbi1 n r1 n, bi1 n.
fi1 i1
Therefore,
znr1 n,n
b1
n−1
n−1
dτ
Δ2 r1 n, s
≤
n, s f1 w r 0, s
z0r1 n,0 w1 τ
s0 1 1
sb 0
1
n−1
k n, bi1 sφi1 zbi1 s r1 n, bi1 s,
fi1 2.20
i1 s0
that is,
W1 zn r1 n, n ≤ W1 r1 n, 0 b1
n−1
n, s f1 sb1 0
n−1
Δ2 r1 n, s
w
s
r
0,
s0 1 1
n−1
k bi1
n, sφi1 zs r1 n, s,
fi1 i1 sbi1 0
2.21
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or equivalently
ξn ≤ c1 n, n n−1
k bi1
i1 sbi1 0
n, sφi1 W1−1 ξs ,
fi1 n≤n
,
2.22
the same as 2.7 for m k where ξn W1 zn r1 n, n and
c1 n, n W1 r1 n, 0 b1
n−1
n, s f1 sb1 0
n−1
Δ2 r1 n, s
.
w r 0, s
s0 1 1
2.23
From the assumption C4 , each φi1 W1−1 , i 1, . . . , k, is continuous and nondecreasing
on 0, ∞ and is positive on 0, ∞ since W1−1 is continuous and nondecreasing on 0, ∞.
Moreover, φ2 W1−1 ∝ φ3 W1−1 ∝ · · · ∝ φk1 W1−1 . By the inductive assumption, we have
⎡
⎣
ξn ≤ Φ−1
n, 0 k1 Φk1 c1 bk1
n−1
n, s fk1 sbk1 0
⎤
n−1
Δ2 ck n, s
⎦
−1
ψ
Φ
c
k 0, s
s0 k1
k
2.24
u
dz/φi1 W1−1 z, u > 0, Φ1 I Identity, u
i1 −1
−1
−1
−1
W1 ui1 , Φi1 is the inverse of Φi1 , ψi1 u φi1 W1 u/φi W1 u wi1 W1 u/
wi W1−1 u, i 1, . . . , k,
for n ≤ min{
n, N3 } where Φi1 u u
i1
ci1 n, n Φi1 c1 n, 0 bi1
n−1
n, s fi1 sbi1 0
n−1
Δ2 ci n, s
,
−1
s0 ψi1 Φi ci 0, s
2.25
i 1, . . . , k − 1, and N3 is the largest positive integer such that
Φi1 c1 n, 0 bi1
N3 −1
n, s fi1 sbi1 0
≤
W1 ∞
u
i1
dz
,
φi1 W1−1 z
N
3 −1
s0
Δ2 ci n, s
−1
ψi1 Φi ci 0, s
i 1, . . . , k.
2.26
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Note that
w1 W1−1 z dz
Φi u −1 i W −1 z
u
i φ
W1 ui wi W1 z
1
u
u
dz
W −1 u
dz
Wi ◦ W1−1 u, i 2, . . . , k 1,
w
i z
ui
wi1 W −1 Φ−1 u wi1 W −1 W1 W −1 u
i
i
1
1
−1
ψi1 Φi u −1 −1 wi W1−1 Φ−1
w
W
W
W
u
i
1
i
i u
1
−1 wi1 Wi u
−1 φi1 Wi−1 u , i 1, . . . , k 1.
wi Wi u
1
2.27
Thus, we have from 2.24 that
un ≤ r1 n, n zn W1−1 ξn
⎡
n−1
n−1
bk1
−1 ⎣
Wk1 W1−1 c1 ≤ Wk1
n, 0 n, s fk1 ⎤
Δ2 ck n, s
⎦
−1
s0 φk1 Wk ck 0, s
sbk1 0
⎡
−1 ⎣
≤ Wk1
n, 0 Wk1 r1 bk1
n−1
n, s fk1 sbk1 0
2.28
⎤
n−1
Δ2 ck n, s
⎦
−1
s0 φk1 Wk ck 0, s
for n ≤ min{
n, N3 } since c1 n, 0 W1 r1 n, 0.
In the following, we prove that ci n, n ri1 n, n by induction again.
n, n r2 n, n for i 1. Suppose that cl n, n rl1 n, n for i l. We
It is clear that c1 have
n, n Φl1 c1 n, 0 cl1 bl1
n−1
n, s fl1 sbl1 0
Wl1 r1 n, 0 bl1
n−1
sbl1 0
n, s fl1 n−1
Δ2 cl n, s
−1
s0 ψl1 Φl cl 0, s
n−1
Δ2 rl1 n, s
−1
s0 φl1 Wl rl1 0, s
2.29
rl2 n, n,
n, 0 W1 r1 n, 0 is applied. It implies that it is true for i l 1. Thus, ci n, n where c1 n, n for i 1, . . . , k.
ri1 Advances in Difference Equations
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Equation 2.26 becomes
Wi1 r1 n, 0 bi1
N3 −1
n, s fi1 sbi1 0
≤
W1 ∞
u
i1
N
3 −1
s0
dz
φi1 W1−1 z
W1 ∞
u
i1
Δ2 ri1 n, s
−1
φi1 Wi ri1 0, s
2.30
∞
w1 W1−1 z
dz
−1 dz w
wi1 W1 z
ui1 i1 z
for i 1, . . . , k. It implies that N2 N3 . Thus, 2.28 becomes
⎡
−1 ⎣
Wk1 r1 un ≤ Wk1
n, 0 bk1
n−1
n, s fk1 sbk1 0
⎤
n−1
Δ2 rk1 n, s
⎦
−1
φ
W
r
k1 0, s
s0 k1
k
2.31
for n ≤ n
. It shows that 2.8 is true for m k 1. Thus, the claim is proved.
Now we prove 2.1. Replacing n by n
in 2.8, we have
⎡
−1 ⎣
u
n ≤ Wm
Wm r1 n, 0 bm
n−1
n, s fm sbm 0
n
−1
s0 φm
Δ2 rm n, s
−1
Wm−1
rm 0, s
⎤
⎦.
2.32
Since 2.8 is true for any n
≤ N1 , we replace n
by n and get
⎡
−1 ⎣
Wm r1 n, 0 un ≤ Wm
bm
n−1
fm n, s sbm 0
n−1
s0 φm
Δ2 rm n, s
−1
Wm−1
rm 0, s
⎤
⎦.
2.33
This is exactly 2.1 since r1 n, 0 a0. This proves Theorem 2.1.
Remark 2.4. If an 0 for all n ∈ N0 , then a0 0. Let r1,u1 n, j : r1 n, j u1 where u1 > 0
u
is given in W1 u u1 dz/w1 z. Using the same arguments as in 2.11 where r1 n, j is
n, s 0 and 2.14 becomes
replaced with the positive r1,u1 n, j, we have Δ2 r1,u1 W1 zn r1,u1 n, n ≤ W1 r1,u1 n, 0 b1
n−1
n, s
f1 sb1 0
W1 u1 b1
n−1
sb1 0
n, s f1 b1
n−1
2.34
n, s,
f1 sb1 0
that is,
un ≤ zn r1,u1 n, n zn u1
⎤
⎡
b1
n−1
≤ W1−1 ⎣
n, s⎦, n ≤ n
,
f1 sb1 0
2.35
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which is the same as 2.15 with a complementary definition that W1 0 0. From 1 of
Remark 2.2, the estimate of 2.35 is independent of u1 . Then we similarly obtain 2.1 and all
ri are defined by the same formula 2.2 where we define Wi 0 0 for i 1, . . . , m.
3. Some Corollaries
In this section, we apply Theorem 2.1 and obtain some corollaries.
Assume that ϕ ∈ CR , R is a strictly increasing function with ϕ∞ ∞ where
R 0, ∞. Consider the inequality
ϕun ≤ an n−1
m bi fi n, swi us,
n ∈ N0 .
3.1
i1 sbi 0
Corollary 3.1. Suppose that C1 )–(C4 ) hold. If un in3.1is nonnegative for n ∈ N0 , then
⎡
⎛
⎢ −1 ⎜ un ≤ ϕ−1 ⎣W
a0 m ⎝Wm bm
n−1
fm n, s sbm 0
⎞⎤
n−1
Δ2 rm n, s
⎟⎥
⎠⎦
m W
−1 rm 0, s
s0 φ
m−1
3.2
i u u dz/wi ϕ−1 z, W
−1 is the inverse of W
i , W
0 IIdentity,
for n ≤ N1 where W
i
ui
φi u wi ϕ−1 u/wi−1 ϕ−1 u, φ1 u w1 ϕ−1 u, and other related functions are defined
as in Theorem 2.1 by replacing wi u with wi ϕ−1 u.
Proof. Let ξn ϕun. Then 3.1 becomes
ξn ≤ an n−1
m bi fi n, swi ϕ−1 ξs ,
n ∈ N0 .
3.3
i1 sbi 0
Note that wi ϕ−1 u satisfy C4 for i 1, . . . , m. Using Theorem 2.1, we obtain the estimate
about ξn by replacing wi u with wi ϕ−1 u. Then use the fact that un ϕ−1 ξn and
we get Corollary 3.1.
If ϕu up where p > 0, then 3.1 reads
up n ≤ an n−1
m bi fi n, swi us,
n ∈ N0 .
3.4
i1 sbi 0
Directly using Corollary 3.1, we have the following result.
Corollary 3.2. Suppose that C1 )–(C4 ) hold. If un in 3.4 is nonnegative for n ∈ N0 , then
⎡
⎛
⎢ −1 ⎜ un ≤ ⎣W
a0 m ⎝Wm bm
n−1
sbm 0
fm n, s n−1
⎞⎤1/p
Δ2 rm n, s
m W
−1 rm 0, s
s0 φ
m−1
⎟⎥
⎠⎦
3.5
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i u u dz/wi z1/p , W
i is the inverse of W
i , W
0 IIdentity, φi u for n ≤ N1 where W
ui
wi u1/p /wi−1 u1/p , φ1 u w1 u1/p , and other related functions are defined as in Theorem 2.1
by replacing wi u with wi u1/p .
If m 1, p 2, b1 n n, 3.4 becomes the second inequality of 1.1 with f1 n, s 2fs and w1 u u, and the third inequality of 1.1 with f1 n, s fs and w1 u wu,
which are discussed in the book 21. Equation 3.5 yields the same estimates of Theorem
4.2.4 in the book 21.
4. Applications to Volterra Type Difference Equations
In this section, we apply Theorem 2.1 to study boundedness and uniqueness of solutions of a
nonlinear delay difference equation of the form
yn βn b1
n−1
n−1
b2
F n, s, ys H n, s, ys ,
sb1 0
n ∈ N0 ,
4.1
sb2 0
where y : N0 → R is an unknown function, β maps from N0 to R, F and H map from
N0 × N0 × R to R, and bi satisfies the assumption C2 for i 1, 2.
Theorem 4.1. Suppose that β0 /
0 and the functions F and H in 4.1 satisfy the conditions
F n, s, y ≤ f1 n, s y,
4.2
H n, s, y ≤ f2 n, sy,
where f1 , f2 : N0 × N0 → 0, ∞. If yn is a solution of 4.1 on N0 , then
⎡
⎤
b2
n−1
n−1 f1 n, b1 s Δ
as/ as
⎥
yn ≤ a0 exp⎢
f2 n, s ⎣
⎦,
hs
s0
sb 0
4.3
2
where
as max βτ,
0≤τ≤s,τ∈N0
f2 n, s hn a0 f1 n, s max
0≤τ≤n,τ∈N0
f1 τ, s,
max f2 τ, s,
0≤τ≤n,τ∈N0
b n−1
n−1
Δ
as
1 1 1
.
f1 0, s 2 sb 0
2 s0 as
1
4.4
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Proof. Using 4.1 and 4.2, the solution yn satisfies
un ≤ an b1
n−1
f1 n, sw1 us sb1 0
b2
n−1
f2 n, sw2 us,
4.5
n ∈ N0 ,
sb2 0
where
un yn,
an βn,
w1 u √
u,
w2 u u.
4.6
0. For positive constants u1 , u2 , we have
Clearly, an > 0 for all n ∈ N0 since β0 /
u
u √ 2
√
√ dz
u1 ,
2 u − u1 , W1−1 u 2
u1 w1 z
u
dz
u
W2 u ln , W2−1 u u2 expu,
w
u
z
2
2
u2
r1 n, j a j > 0, r1 n, 0 a0,
W1 u 4.7
b1
j−1
j−1
√
Δ
as
,
r2 n, j 2
a0 − u1 f1 n, s as
s0
sb1 0
a j
Δ
w2 u √
Δ2 r2 n, j f1 n, b1 j , φ2 u u.
w1 u
a j
It is obvious that w1 and w2 satisfy C4 . Applying Theorem 2.1 gives
⎡
b2
n−1
⎢
un ≤ a0 exp⎣
sb2 0
f2 n, s n−1 f1 n, b1 s
s0
⎤
Δ
as/ as
⎥
⎦
hs
4.8
which implies 4.3.
Theorem 4.2. Suppose that β0 /
0 and the functions F and H in 4.1 satisfy the conditions
F n, s, y1 − F n, s, y2 ≤ f1 n, s y1 − y2 ,
4.9
H n, s, y1 − H n, s, y2 ≤ f2 n, sy1 − y2 ,
where f1 , f2 : N0 × N0 → 0, ∞. Then 4.1 has at most one solution on N0 .
Proof. Let y1 n and y2 n be two solutions of 4.1 on N0 . From 4.9, we have
|un| ≤
b1
n−1
b2
n−1
sb1 0
sb2 0
f1 n, sw1 us f2 n, sw2 us,
n ∈ N0 ,
4.10
Advances in Difference Equations
13
√
where un |y1 n − y2 n|, an 0, w1 u u and w2 u u. Appling Theorem 2.1,
Remark 2.4, and the notation Wi 0 0 for i 1, 2, we obtain that un 0 which implies that
the solution is unique.
Acknowledgments
This paper was supported by Guangdong Provincial natural science Foundation 07301595.
The authors would like to thank Professor Boling Guo for his great help.
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