Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 971540, 18 pages
doi:10.1155/2010/971540
Research Article
Existence and Uniqueness of Positive Solutions for
Discrete Fourth-Order Lidstone Problem with
a Parameter
Yanbin Sang,1, 2 Zhongli Wei,2, 3 and Wei Dong4
1
Department of Mathematics, North University of China, Taiyuan, Shanxi 030051, China
School of Mathematics, Shandong University, Jinan, Shandong 250100, China
3
Department of Mathematics, Shandong Jianzhu University, Jinan, Shandong 250101, China
4
Department of Mathematics, Hebei University of Engineering, Handan, Hebei 056021, China
2
Correspondence should be addressed to Yanbin Sang, sangyanbin@126.com
Received 9 January 2010; Revised 23 March 2010; Accepted 26 March 2010
Academic Editor: A. Pankov
Copyright q 2010 Yanbin Sang 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.
This work presents sufficient conditions for the existence and uniqueness of positive solutions for
a discrete fourth-order beam equation under Lidstone boundary conditions with a parameter; the
iterative sequences yielding approximate solutions are also given. The main tool used is monotone
iterative technique.
1. Introduction
In this paper, we are interested in the existence, uniqueness, and iteration of positive solutions
for the following nonlinear discrete fourth-order beam equation under Lidstone boundary
conditions with explicit parameter β given by
Δ4 yt − 2 − βΔ2 yt − 1 ht f1 yt f2 yt ,
ya 0 Δ2 ya − 1,
t ∈ a 1, b − 1
Z ,
yb 0 Δ2 yb − 1,
1.1
1.2
where Δ is the usual forward difference operator given by Δyt yt 1 − yt, Δn yt Δn−1 Δyt, c, d
Z : {c, c 1, . . . , d − 1, d}, and β > 0 is a real parameter.
In recent years, the theory of nonlinear difference equations has been widely applied
to many fields such as economics, neural network, ecology, and cybernetics, for details, see
2
Advances in Difference Equations
1–7
and references therein. Especially, there was much attention focused on the existence
and multiplicity of positive solutions of fourth-order problem, for example, 8–10
, and in
particular the discrete problem with Lidstone boundary conditions 11–17
. However, very
little work has been done on the uniqueness and iteration of positive solutions of discrete
fourth-order equation under Lidstone boundary conditions. We would like to mention some
results of Anderson and Minhós 11
and He and Su 12
, which motivated us to consider the
BVP 1.1 and 1.2.
In 11
, Anderson and Minhós studied the following nonlinear discrete fourth-order
equation with explicit parameters β and λ given by
Δ4 yt − 2 − βΔ2 yt − 1 λf t, yt ,
t ∈ a 1, b − 1
Z ,
1.3
with Lidstone boundary conditions 1.2, where β > 0 and λ > 0 are real parameters. The
authors obtained the following result.
Theorem 1.1 see 11
. Assume that the following condition is satisfied
A1 ft, y gtwy, where g : a 1, b − 1
Z → 0, ∞ with b−1
za1 gz > 0, w :
0, ∞ → 0, ∞ is continuous and nondecreasing, and there exists θ ∈ 0, 1 such that
wκy ≥ κθ wy for κ ∈ 0, 1 and y ∈ 0, ∞,
then, for any λ ∈ 0, ∞, the BVP 1.3 and 1.2 has a unique positive solution yλ . Furthermore,
such a solution yλ satisfies the following properties:
i limλ → 0 yλ 0 and limλ → ∞ yλ ∞;
ii yλ is nondecreasing in λ;
iii yλ is continuous in λ, that is, if λ → λ0 , then yλ − yλ0 → 0.
Very recently, in 12
, He and Su investigated the existence, multiplicity, and
nonexistence of nontrivial solutions to the following discrete nonlinear fourth-order
boundary value problem
Δ4 ut − 2 ηΔ2 ut − 1 − ξut λft, ut,
ua 0 Δ2 ua − 1,
t ∈ Za 1, b 1
,
ub 2 0 Δ2 ub 1,
1.4
where Δ denotes the forward difference operator defined by Δut ut 1 − ut, Δn ut ΔΔn−1 ut, Za 1, b 1
is the discrete interval given by {a 1, a 2, . . . , b 1} with a and
b a < b integers, η, ξ, λ are real parameters and satisfy
η < 8 sin2
π
,
2b − a 2
η2 4ξ ≥ 0,
ξ4η sin2
π
π
< 16 sin4
,
2b − a 2
2b − a 2
For the function f, the authors imposed the following assumption:
λ > 0.
1.5
B1 ft, x gthx, where g : Za 1, b 1
→ 0, ∞ with b1
ta1 gt > 0, h :
R → 0, ∞ is continuous and nondecreasing, and there exists θ ∈ 0, 1 such that
hμx ≥ μθ hx for μ ∈ 0, 1 and x ∈ 0, ∞.
Advances in Difference Equations
3
Their main result is the following theorem.
Theorem 1.2 see 12
. Assume that B1 holds. Then for any λ ∈ 0, ∞, the BVP 1.4 has a
unique positive solution uλ . Furthermore, such a solution uλ satisfies the properties (i)–(iii) stated in
Theorem 1.1.
The aim of this work is to relax the assumptions A1 and B1 on the nonlinear term,
without demanding the existence of upper and lower solutions, we present conditions for the
BVP 1.1 and 1.2 to have a unique solution and then study the convergence of the iterative
sequence. The ideas come from Zhai et al. 18, 19
and Liang 20
.
Let B denote the Banach space of real-valued functions on a − 1, b 1
Z , with the
supremum norm
y sup
yt.
1.6
t∈a−1,b1
Z
Throughout this paper, we need the following hypotheses:
H1 fi : 0, ∞ → 0, ∞ are continuous and fi y > 0 for y > 0 i 1, 2;
H2 h : a 1, b − 1
Z → 0, ∞ with b−1
za1 hz > 0;
H3 f1 : 0, ∞ → 0, ∞ is nondecreasing, f2 : 0, ∞ → 0, ∞ is nonincreasing,
and there exist ϕτ, ψτ on interval a1, b−1
Z with ϕ : a1, b−1
Z → 0, 1, for
all e0 ∈ 0, 1, there exists τ0 ∈ a1, b−1
Z such that ϕτ0 e0 , and ψτ > ϕτ, for
all τ ∈ a 1, b − 1
Z which satisfy
f1 ϕτy ≥ ψτf1 y ,
f2
1
y
ϕτ
≥ ψτf2 y ,
∀τ ∈ a 1, b − 1
Z , y ≥ 0.
1.7
2. Two Lemmas
To prove the main results in this paper, we will employ two lemmas. These lemmas are based
on the linear discrete fourth-order equation
Δ4 yt − 2 − βΔ2 yt − 1 ut,
t ∈ a 1, b − 1
Z ,
2.1
with Lidstone boundary conditions 1.2.
Lemma 2.1 see 11
. Let u : a 1, b − 1
Z → R be a function. Then the nonhomogeneous discrete
fourth-order Lidstone boundary value problem 2.1, 1.2 has solution
yt b−1
b sa za1
G2 t, sG1 s, zuz,
t ∈ a − 1, b 1
Z ,
2.2
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Advances in Difference Equations
where G2 t, s given by
⎧
⎨t, ab, s:
1
G2 t, s 1, 0b, a ⎩s, ab, t:
with t, s μt−s − μs−t for μ β 2 discrete boundary value problem
t ≤ s,
t, s ∈ a − 1, b 1
Z × a, b
Z
s ≤ t,
2.3
ββ 4/2, is the Green’s function for the second-order
− Δ2 yt − 1 − βyt 0,
t ∈ a, b
Z ,
2.4
ya 0 yb,
and G1 s, z given by
⎧
1 ⎨s − ab − z:
G1 s, z b − a ⎩z − ab − s:
s ≤ z,
z ≤ s,
s, z ∈ a, b
Z × a 1, b − 1
Z
2.5
is the Green’s function for the second-order discrete boundary value problem
−Δ2 xs − 1 0,
s ∈ a 1, b − 1
Z ,
xa 0 xb.
2.6
Lemma 2.2 see 11
. Let
m :
1, 0b, a 1
,
b − a 2 b, a
M :
b − a 2 b/2, a/2
.
41, 0b, a
2.7
Then, for t, s, z ∈ a 1, b − 1
3Z , one has
m ≤ G2 t, sG1 s, z ≤ M.
2.8
3. Main Results
Theorem 3.1. Assume that H1 –H3 hold. Then, the BVP 1.1 and 1.2 has a unique solution
y∗ t in D, where
D y ∈ B | ya 0 yb, yt > 0, t ∈ a 1, b − 1
Z .
3.1
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5
Moreover, for any x0 , y0 ∈ D, constructing successively the sequences
xn1 t b−1
b G2 t, sG1 s, zhz f1 xn z f2 yn z ,
sa za1
t ∈ a − 1, b 1
Z , n 0, 1, 2, . . . ,
yn1 t b b−1
G2 t, sG1 s, zhz f1 yn z f2 xn z ,
3.2
sa za1
t ∈ a − 1, b 1
Z , n 0, 1, 2, . . . ,
One has xn t, yn t converge uniformly to y∗ t in a − 1, b 1
Z .
Proof. First, we show that the BVP 1.1 and 1.2 has a solution.
It is easy to see that the BVP 1.1 and 1.2 has a solution y yt if and only if y is a
fixed point of the operator equation
b−1
b A y1 , y2 t G2 t, sG1 s, zhz f1 y1 z f2 y2 z ,
sa za1
t ∈ a − 1, b 1
Z .
3.3
In view of H3 and 3.3, Ay1 , y2 is nondecreasing in y1 and nonincreasing in y2 . Moreover,
for any τ ∈ a 1, b − 1
Z , we have
A ϕτy1 ,
b−1
b−1 1
1
y2 t y2 z
G2 t, sG1 s, zhz f1 ϕτy1 z f2
ϕτ
ϕτ
sa1 za1
≥ ψτ
b−1
b−1 G2 t, sG1 s, zhz f1 y1 z f2 y2 z
sa1 za1
ψτA y1 , y2 t
3.4
for t ∈ a, b
Z and y1 , y2 ∈ D.
Let
L b − a − 1
b−1
hz,
za1
3.5
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Advances in Difference Equations
condition H2 implies L > 0. Since fi y > 0 for y > 0 i 1, 2, by Lemma 2.2, we have
AL, L b−1
b−1 G2 t, sG1 s, zhz f1 L f2 L
sa1 za1
b−1
b−1 ≥ m f1 L f2 L
hz
3.6
sa1 za1
m f1 L f2 L L
for m in 2.1 and L in 3.5.
Moreover, we obtain
AL, L ≤ M f1 L f2 L L
3.7
m f1 L f2 L L ≤ AL, L ≤ M f1 L f2 L L.
3.8
for M in 2.1.
Thus
Therefore, we can choose a sufficiently small number e1 ∈ 0, 1 such that
e1 L ≤ AL, L ≤
L
,
e1
3.9
which together with H3 implies that there exists τ1 ∈ a 1, b − 1
Z such that ϕτ1 e1 , so
ϕτ1 L ≤ AL, L ≤
L
.
ϕτ1 3.10
Since ψτ1 /ϕτ1 > 1, we can take a sufficiently large positive integer k such that
ψτ1 ϕτ1 k
1
.
ϕτ1 3.11
≤ ϕτ1 .
3.12
≥
It is clear that
ϕτ1 ψτ1 k
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We define
⎧ k
⎪
− ϕτ1 L:
⎪
⎪
⎨
u0 t 0:
⎪
⎪
⎪
k
⎩
ϕτ1 L:
⎧
L
⎪
⎪
⎪
k :
⎪− ⎪
⎪
ϕτ
1
⎪
⎨
v0 t 0:
⎪
⎪
⎪
L
⎪
⎪
⎪
⎪
k :
⎩
ϕτ1 t a − 1, b 1,
t a, b,
t ∈ a 1, b − 1
Z ,
t a − 1, b 1,
3.13
t a, b,
t ∈ a 1, b − 1
Z .
Evidently, for t ∈ a, b
Z , u0 ≤ v0 . Take any λ ∈ 0, ϕτ1 2k , then λ ∈ 0, 1 and u0 ≥ λv0 .
By the mixed monotonicity of A, we have Au0 , v0 ≤ Av0 , u0 . In addition, combining
H3 with 3.10 and 3.11, we get
k
1
Au0 , v0 A ϕτ1 L, k L
ϕτ1 k−1
1
L,
A ϕτ1 ϕτ1 k−1 L
ϕτ1 ϕτ1 k−1
1
≥ ψτ1 A ϕτ1 L, k−1 L ≥ · · ·
ϕτ1 3.14
k
k
≥ ψτ1 AL, L ≥ ψτ1 ϕτ1 L
k
≥ ϕτ1 L u0 .
From H3 , we have
y1
1
,
ϕsy2
A y1 , y2 A ϕs
ϕs ϕs
y1
, ϕsy2 , ∀s ∈ a 1, b − 1
Z , y1 , y2 ≥ 0,
≥ ψsA
ϕs
3.15
y1
1
, ϕsy2 ≤
A y1 , y2 ,
A
ϕs
ψs
3.16
and hence
∀s ∈ a 1, b − 1
Z , y1 , y2 ≥ 0.
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Advances in Difference Equations
Thus, we have
Av0 , u0 A
L
ϕτ1 k
k , ϕτ1 L
L
A
k−1
L
k−1 , ϕτ1 ϕτ1 ϕτ1 ϕτ1 k−1
L
1
A L ≤ ···
≤
k−1 , ϕτ1 ψτ1 ϕτ1 3.17
L
1
1
≤
k AL, L ≤ k ϕτ .
1
ψτ1 ψτ1 In accordance with 3.12, we can see that
L
Av0 , u0 ≤ k v0 .
ϕτ1 3.18
Construct successively the sequences
un Aun−1 , vn−1 ,
vn Avn−1 , un−1 ,
n 1, 2, . . . .
3.19
By the mixed monotonicity of A, we have u1 Au0 , v0 ≤ Av0 , u0 v1 . By induction, we
obtain un ≤ vn , n 1, 2, . . .. It follows from 3.14, 3.18, and the mixed monotonicity of A
that
u0 ≤ u1 ≤ · · · ≤ un ≤ · · · ≤ vn ≤ · · · ≤ v1 ≤ v0 .
3.20
Note that u0 ≥ λv0 , so we can get un t ≥ u0 t ≥ λv0 t ≥ λvn t, t ∈ a, b
Z , n 1, 2, . . .. Let
λn sup{λ > 0 | un t ≥ λvn t, t ∈ a, b
Z },
n 1, 2, . . . .
3.21
Thus, we have
un t ≥ λn vn t,
t ∈ a, b
Z , n 1, 2, . . . ,
3.22
and then
un1 t ≥ un t ≥ λn vn t ≥ λn vn1 t,
t ∈ a, b
Z , n 1, 2, . . . .
3.23
Therefore, λn1 ≥ λn , that is, {λn } is increasing with {λn } ⊂ 0, 1
. Set λ limn → ∞ λn . We can
show that λ 1. In fact, if 0 < λ < 1, by H3 , there exists τ2 ∈ a1, b −1
Z such that ϕτ2 λ.
Consider the following two cases.
Advances in Difference Equations
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In this case, we have λn λ for all
i There exists an integer N such that λN λ.
n ≥ N holds. Hence, for n ≥ N, it follows from 3.4 and the mixed monotonicity of A that
n , 1 un A ϕτ2 vn , 1 un ≥ ψτ2 Avn , un ψτ2 vn1 .
un1 Aun , vn ≥ A λv
ϕτ2 λ
3.24
By the definition of λn , we have
λn1 λ ≥ ψτ2 > ϕτ2 λ.
3.25
This is a contradiction.
In this case, we have 0 < λn /λ < 1. In accordance with
ii For all integer n, λn < λ.
Hence, combining 3.4 with the
H3 , there exists θn ∈ a 1, b − 1
Z such that ϕθn λn /λ.
mixed monotonicity of A, we have
1
un1 Aun , vn ≥ A λn vn , un
λn
⎛
⎞
un
un
⎜ λn ⎟
A⎝ λv
⎠ A ϕθn ϕτ2 vn ,
n, ϕθn ϕτ2 λ
λn /λ λ
3.26
un
≥ ψθn A ϕτ2 vn ,
≥ ψθn ψτ2 Avn , un ϕτ2 ψθn ψτ2 vn1 .
By the definition of λn , we have
λn1 ≥ ψθn ψτ2 > ϕθn ψτ2 λn
ψτ2 .
λ
3.27
λψτ
Let n → ∞, we have λ ≥ λ/
2 > λ/λϕτ2 ϕτ2 λ, and this is also a contradiction.
Hence, limn → ∞ λn 1.
Thus, combining 3.20 with 3.22, we have
0 ≤ unl t − un t ≤ vn t − un t ≤ vn t − λn vn t 1 − λn vn t ≤ 1 − λn v0 t
3.28
for t ∈ a, b
Z , where l is a nonnegative integer. Thus,
unl − un ≤ vn − un ≤ 1 − λn v0 .
3.29
Therefore, there exists a function y∗ ∈ D such that
lim un t lim vn t y∗ t for t ∈ a − 1, b 1
Z .
n→∞
n→∞
3.30
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By the mixed monotonicity of A and 3.20, we have
un1 t Aun t, vn t ≤ A y∗ t, y∗ t ≤ Avn t, un t vn1 t.
3.31
Let n → ∞ and we get Ay∗ t, y∗ t y∗ t, t ∈ a − 1, b 1
Z . That is, y∗ is a nontrivial
solution of the BVP 1.1 and 1.2.
Next, we show the uniqueness of solutions of the BVP 1.1 and 1.2. Assume, to the
contrary, that there exist two nontrivial solutions y1 and y2 of the BVP 1.1 and 1.2 such
that Ay1 t, y1 t y1 t and Ay2 t, y2 t y2 t for t ∈ a − 1, b 1
Z . According to 3.9,
we can know that there exists 0 < η ≤ 1 such that ηy2 t ≤ y1 t ≤ 1/ηy2 t for t ∈ a, b
Z .
Let
!
1
η0 sup 0 < η ≤ 1 | ηy2 ≤ y1 ≤ y2 .
3.32
η
Then 0 < η0 ≤ 1 and η0 y2 t ≤ y1 t ≤ 1/η0 y2 t for t ∈ a, b
Z .
We now show that η0 1. In fact, if 0 < η0 < 1, then, in view of H3 , there exists
τ ∈ a 1, b − 1
Z such that ϕτ η0 . Furthermore, we have
1
1
y1 A y1 , y1 ≥ A η0 y2 , y2 A ϕτy2 ,
y2 ≥ ψτA y2 , y2 ψτy2 , 3.33
η0
ϕτ
y2
y2
1
1
y1 A y1 , y1 ≤ A
, η0 y2 A
, ϕτy2 ≤
A y2 , y2 y2 .
3.34
η0
ϕτ
ψτ
ψτ
In 3.34, we used the relation formula 3.16. Since ψτ > ϕτ η0 , this contradicts the
definition of η0 . Hence η0 1. Therefore, the BVP 1.1 and 1.2 has a unique solution.
Finally, we show that “moreover” part of the theorem. For any initial x0 , y0 ∈ D, in
accordance with 3.9, we can choose a sufficiently small number e2 ∈ 0, 1 such that
e2 L ≤ x0 ≤
1
L,
e2
e2 L ≤ y0 ≤
1
L.
e2
3.35
It follows from H3 that there exists τ3 ∈ a 1, b − 1
Z such that ϕτ3 e2 , and hence
ϕτ3 L ≤ x0 ≤
L
,
ϕτ3 ϕτ3 L ≤ y0 ≤
L
.
ϕτ3 3.36
Thus, we can choose a sufficiently large positive integer k such that
ψτ3 ϕτ3 k
≥
1
.
ϕτ3 3.37
L
v"0 k .
ϕτ3 3.38
Define
k
u
" 0 ϕτ3 L,
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Obviously, u
" 0 < x0 , y0 < v"0 . Let
u
" n A"
un−1 , v"n−1 ,
vn−1 , u
" n−1 ,
v"n A"
n 1, 2, . . . ,
b b−1
G2 t, sG1 s, zhz f1 xn−1 z f2 yn−1 z ,
xn t A xn−1 , yn−1 t sa za1
3.39
b−1
b yn t A yn−1 , xn−1 t G2 t, sG1 s, zhz f1 yn−1 z f2 xn−1 z
sa za1
for t ∈ a − 1, b 1
Z , n 1, 2, . . .. By induction, we get u
" n ≤ xn ≤ v"n , u
" n ≤ yn ≤ v"n , n 1, 2, . . ..
Similarly to the above proof, it follows that there exists y" ∈ D such that
" n lim v"n y,
"
lim u
n→∞
n→∞
A y,
" y" y.
"
3.40
By the uniqueness of fixed points A in D, we get y" y∗ . Therefore, we have
lim xn lim yn y∗ .
n→∞
n→∞
3.41
This completes the proof of the theorem.
Remark 3.2. From the proof of Theorem 3.1, we easily know that assume y Ay, x, x Ax, y, thus, let y0 y, x0 x, we have
yn y,
xn x,
n 1, 2, . . . .
3.42
Therefore y x y∗ .
Theorem 3.3. Assume that H2 holds, and the following conditions are satisfied:
C1 f : 0, ∞ → 0, ∞ is continuous and fy > 0 for y > 0;
C2 f : 0, ∞ → 0, ∞ is nondecreasing;
b−1
b b−1
b G2 t, sG1 s, zhzf ϕτyz ≥ ψ τ, y
G2 t, sG1 s, zhzf yz ,
sa za1
sa za1
3.43
for all τ ∈ a1, b−1
Z , y ∈ 0, ∞, where ϕ : a1, b−1
Z → 0, 1, for all e0 ∈ 0, 1,
there exists τ0 ∈ a 1, b − 1
Z such that ϕτ0 e0 , and ψ : a 1, b − 1
Z × 0, ∞ →
0, ∞, with ψτ, y > ϕτ, for all τ ∈ a 1, b − 1
Z , y ∈ 0, ∞;
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C3 for fixed τ ∈ a 1, b − 1
Z , one has
i ψτ, y is nonincreasing with respect to y, and there exists τ4 ∈ a 1, b − 1
Z such
that
mfL ≥ ϕτ4 ,
ψ τ4 , L/ϕτ4 ≥ MfL
ϕτ4 3.44
or
ii ψτ, y is nondecreasing with respect to y, and there exists τ5 ∈ a 1, b − 1
Z such
that
mfL ≥
ϕτ5 ,
ψτ5 , L
1
≥ MfL,
ϕτ5 3.45
where m, M are defined in 2.1, L is defined in 3.5. Then, the BVP
Δ4 yt − 2 − βΔ2 yt − 1 htf yt ,
ya 0 Δ2 ya − 1,
t ∈ a 1, b − 1
Z ,
yb 0 Δ2 yb − 1
3.46
has a unique solution y∗ .
Proof. For convenience, we still define the operator equation A by
Ayt b−1
b G2 t, sG1 s, zhzf yz ,
sa za1
t ∈ a − 1, b 1
Z .
3.47
In the following, we consider the following two cases.
i For fixed τ ∈ a 1, b − 1
Z , ψτ, y is nonincreasing with respect to y.
According to condition C3 and Lemma 2.2, we can know that there exists τ4 ∈ a 1, b − 1
Z such that
ϕτ4 L ≤ AL ≤
ψ τ4 , L/ϕτ4 L.
ϕτ4 3.48
Since ψτ4 , L/ϕτ4 > 1, we can find a sufficiently large positive integer k such that
ψτ4 , L
ϕτ4 k
≥
1
.
ϕτ4 3.49
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For t ∈ a 1, b − 1
Z , we still define
k
u0 t ϕτ4 L,
un t Aun−1 t,
L
v0 t k ,
ϕτ4 vn t Avn−1 t,
3.50
n 1, 2, . . . .
By the proof of Theorem 3.1, it is sufficient to show that
u0 ≤ u1 ≤ v1 ≤ v0 .
3.51
Obviously, u0 ≤ v0 and u1 ≤ v1 .
In this case, it follows from conditions C2 , C3 , and 3.49 that
u1 Au0 A
k ϕτ4 L
k−1 A ϕτ4 ϕτ4 L
k−1 k−1 L A ϕτ4 L
≥ ψ τ4 , ϕτ4 k−1 k−2 ψ τ4 , ϕτ4 L A ϕτ4 ϕτ4 L
k−1 k−2 k−2 ≥ ψ τ4 , ϕτ4 L ψ τ4 , ϕτ4 L A ϕτ4 L
3.52
≥ ···
k−1 k−2 ≥ ψ τ4 , ϕτ4 L ψ τ4 , ϕτ4 L · · · ψτ4 , LAL
k
≥ ψτ4 , L ϕτ4 L
k
≥ ϕτ4 L u0 .
In accordance with 3.16, we have
A
y
ϕs
≤
1
Ay,
ψ s, y/ϕs
3.53
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which together with condition C2 and 3.48 implies that
v1 Av0 A
A
L
ϕτ4 L
k
k−1
ϕτ4 ϕτ4 1
≤ k A
ψ τ4 , L/ ϕτ4 L
k−1
ϕτ4 1
k A
ψ τ4 , L/ ϕτ4 L
k−2
ϕτ4 ϕτ4 ≤
1
1
k k−1 A
ψ τ4 , L/ ϕτ4 ψ τ4 , L/ ϕτ4 ≤
1
1
1
k k−1 · · · ψ τ , L/ϕτ AL
4
4
ψ τ4 , L/ ϕτ4 ψ τ4 , L/ ϕτ4 ≤
≤
L
k−2
ϕτ4 3.54
1
AL
k−1 ψ τ4 , L/ϕτ4 ϕτ4 1
L
ϕτ4 k v0 .
ii For fixed τ ∈ a 1, b − 1
Z , ψτ, y is nondecreasing with respect to y.
In this case, by condition C3 and Lemma 2.2, we can know that there exists τ5 ∈
a 1, b − 1
Z such that
ϕτ5 L
L
≤ AL ≤
.
ψτ5 , L
ϕτ5 3.55
Since 0 < ϕτ5 /ψτ5 , L/ϕτ5 < 1, we can take a sufficiently large positive integer k such
that
#
ϕτ5 ψ τ5 , L/ϕτ5 $k
≤ ϕτ5 .
3.56
Advances in Difference Equations
15
For t ∈ a 1, b − 1
Z , we still define
k
u0 t ϕτ5 L,
un t Aun−1 t,
L
v0 t k ,
ϕτ5 vn t Avn−1 t,
3.57
n 1, 2, . . . .
We continue to prove that
u1 ≥ u0 ,
v1 ≤ v0 .
3.58
By 3.52, combining 3.55 with the monotonicity of ψ, we have
u1 Au0 A
k ϕτ5 L
k−1 k−2 L ψ τ5 , ϕτ5 L · · · ψτ5 , LAL
≥ ψ τ5 , ϕτ5 k−1
≥ ϕτ5 ψτ5 , LAL
3.59
k
≥ ϕτ5 L u0 .
In accordance with 3.54, combining the monotonicity of ψ and 3.55, we get
v1 Av0 A
≤
L
ϕτ5 k
1
1
1
k k−1 · · · ψ τ , L/ϕτ AL
5
5
ψ τ5 , L/ ϕτ5 ψ τ5 , L/ ϕτ5 3.60
1
L
≤ k ϕτ .
5
ψ τ5 , L/ϕτ5 An application of 3.56 yields
1
v1 ≤ k L v0 .
ϕτ5 3.61
u0 ≤ u1 ≤ v1 ≤ v0 .
3.62
Therefore, we obtain
For t a − 1, b 1, the proof is similar and hence omitted. This completes the proof of the
theorem.
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Advances in Difference Equations
Remark 3.4. In Theorem 3.1, the more general conditions are imposed on the nonlinear term
than Theorem 1.1. In particular, in Theorem 3.3, ψτ, y contains the variable y; therefore, the
more comprehensive functions can be incorporated.
4. An Example
Example 4.1. Consider the following discrete fourth-order Lidstone problem:
#
Δ yt − 2 − Δ yt − 1 t 1 y
4
2
1/4
t 2 y2 0 Δ y1,
2
$
1
,
y1/4 t
t ∈ 2 1, 7 − 1
Z ,
4.1
y7 0 Δ y6.
2
We claim that the BVP 4.1 and 1.2 has a unique solution y∗ t in D, where
D y ∈ B | y2 0 y7, yt > 0, t ∈ 3, 6
Z .
4.2
Moreover, for any x0 , y0 ∈ D, constructing successively the sequences
xn1 t 7 6
#
G2 t, sG1 s, zz 1 xn1/4 z
2
1
$
,
yn1/4 z
$
#
7 6
1
1/4
,
G2 t, sG1 s, zz 1 yn z 2 1/4
yn1 t xn z
s2 z3
t ∈ 1, 8
Z , n 0, 1, 2, . . . ,
s2 z3
t ∈ 2, 8
Z , n 0, 1, 2, . . . ,
4.3
we have xn t, yn t converge uniformly to y∗ t in 2, 8
Z .
In fact, we choose f1 y 1 y1/4 , f2 y 2 1/y1/4 , hz z, thus fi y > 0 for
y > 0 i 1, 2, 6z3 hz 6z3 z 18 > 0. It is easy to check that f1 is nondecreasing on
0, ∞, f2 is nonincreasing on 0, ∞. In addition, we set
τ
⎧
⎪
⎪
3,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨4,
⎪
⎪
⎪
5,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩6,
1
0, ,
4
1 1
, ,
ϕτ ∈
4 2
1 3
ϕτ ∈
, ,
2 4
3
ϕτ ∈
,1 ,
4
ϕτ ∈
4.4
Advances in Difference Equations
17
ψτ ϕτ
1/2 . It is easy to see that
1/4
f1 ϕτy 1 ϕτy
≥ ψτ 1 y1/4 ψτf1 y ,
f2
y
ϕτ
2 1
y/ϕτ
1/4 ≥ ψτ 2 1
y1/4
∀τ ∈ 3, 6
Z , y ≥ 0,
,
∀τ ∈ 3, 6
Z , y ≥ 0.
4.5
The conclusion then follows from Theorem 3.1.
Acknowledgments
The authors were supported financially by the National Natural Science Foundation of China
10971046, the Natural Science Foundation of Shandong Province ZR2009AM004, and the
Youth Science Foundation of Shanxi Province 2009021001-2.
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