Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2009, Article ID 538431, 15 pages
doi:10.1155/2009/538431
Research Article
Existence of Three Positive Solutions for
m-Point Discrete Boundary Value Problems with
p-Laplacian
Yanping Guo, Wenying Wei, and Yuerong Chen
College of Sciences, Hebei University of Science and Technology, Shijiazhuang, Hebei 050018, China
Correspondence should be addressed to Wenying Wei, wwyhhb@163.com
Received 23 May 2009; Accepted 18 October 2009
Recommended by Leonid Shaikhet
We consider the multi-point discrete boundary value problem with one-dimensional p-Laplacian
operator Δφp Δut − 1 qtft, ut, Δut 0, t ∈ {1, . . . , n − 1} subject to the boundary
p−2
conditions: u0 0, un m−2
s, p > 1, ξi ∈ {2, . . . , n − 2} with
i1 ai uξi , where φp s |s|
m−2
1 < ξ1 < · · · < ξm−2 < n − 1 and ai ∈ 0, 1, 0 < i1 ai < 1. Using a new fixed point theorem
due to Avery and Peterson, we study the existence of at least three positive solutions to the above
boundary value problem.
Copyright q 2009 Yanping Guo 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.
1. Introduction
The second-order differential and difference boundary value problems arise in many
branches of both applied and basic mathematics and have been extensively studied in
literature. We refer the reader to some recent results for second-order nonlinear two-point 1–
6 and multipoint 7–9 boundary value problems. The main tools used in the above works
are fixed point theorems.
Recently, Feng and Ge in 9 considered the following multipoint BVPs:
φp ut − 1 qtf t, ut, u t 0,
u0 0,
un m−2
ai uξi .
t ∈ 0, 1,
1.1
i1
The authors obtained sufficient conditions that guarantee the existence of at least three
positive solutions by using fixed point theorems due to Avery-Peterson.
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In this work, we study the existence of multiple positive solutions to the discrete
boundary value problem for the one-dimensional p-Laplacian:
Δ φp Δut − 1 qtft, ut, Δut 0,
u0 0,
un t ∈ {1, . . . , n − 1}
m−2
ai uξi ,
1.2
i1
where Δut ut 1 − ut for t ∈ {0, 1, . . . , n − 1}, Δ2 ut ut 2 − 2ut 1 ut, for
t ∈ {0, 1, . . . , n − 2}, and φp s |s|p−2 s, p > 1, ξi ∈ {2, . . . , n − 2} with 1 < ξ1 < · · · < ξm−2 < n − 1.
In order to study the existence of at least three positive solutions to the above boundary
value problem, we assume that ai , f, q satisfy the following.
H1 ai ∈ 0, 1 satisfy 0 < m−2
i1 ai < 1.
H2 f : {1, . . . , n − 1} × 0, ∞ × R → 0, ∞ is continuous.
H3 qt > 0 for t ∈ {1, 2, . . . , n − 1}.
We will depend on an application of a fixed point theorems due to Avery and Peterson,
which deals with fixed points of a cone-preserving operator defined on an ordered Banach
space to obtain our main results.
2. Preliminaries
For the convenience of readers, we provide some background material from the theory of
cones in Banach spaces. In this section, we also state Avery-Peterson’s fixed point theorem.
Definition 2.1. Let E be a real Banach space over R. A nonempty convex closed set P ⊂ E is
said to be a cone of E if it satisfies the following conditions:
i au bv ∈ P for all u, v ∈ p and all a ≥ 0, b ≥ 0;
ii u, −u ∈ P implies u 0.
Every cone P ⊂ E induces an ordering in E given by x ≤ y if and only if y − x ∈ P .
Definition 2.2. An operator is called completely continuous if it is continuous and maps
bounded sets into precompact sets.
Definition 2.3. The map α is said to be a nonnegative continuous concave functional on a cone
P of a real Banach space E provided that α : P → 0, ∞ is continuous and
α tx 1 − ty ≥ tαx 1 − tα y
2.1
for all x, y ∈ P and 0 ≤ t ≤ 1. Similarly, we say that the map β is a nonnegative continuous
convex functional on a cone P of a real Banach space E provided that β : P → 0, ∞ is
continuous and
β tx 1 − ty ≤ tβx 1 − tβ y
for all x, y ∈ P and 0 ≤ t ≤ 1.
2.2
Discrete Dynamics in Nature and Society
3
Let γ and θ be nonnegative continuous convex functionals on a cone P , let α be
a nonnegative continuous concave functional on a cone P , and let ψ be a nonnegative
continuous functional on a cone P . Then for positive real numbers a, b, c, and d, we define
the following convex sets:
P γ, d u ∈ P | γu < d ,
P γ, α, b, d u ∈ P | b ≤ αu, γu ≤ d ,
P γ, θ, α, b, c, d u ∈ P | b ≤ αu, θu ≤ c, γu ≤ d
2.3
R γ, ψ, a, d u ∈ P | a ≤ ψu, γu ≤ d .
2.4
and a closed set
To prove our results, we need the following fixed point theorem due to Avery and
Peterson in 1.
Theorem 2.4. Let P be a cone in a real Banach space E. Let γ and θ be nonnegative continuous
convex functionals on P , let α be a nonnegative continuous concave functional on P , and let ψ be a
nonnegative continuous functional on P satisfying ψλu ≤ λψu for 0 ≤ λ ≤ 1, such that for some
positive numbers M and d,
αu ≤ ψu,
u ≤ Mγu
2.5
for all u ∈ P γ , d. Suppose that
T : P γ, d −→ P γ, d
is completely continuous and there exist positive numbers a, b, and c with a < b such that
∅ and αT u > b for u ∈ P γ, θ, α, b, c, d;
S1 {u ∈ P γ, θ, α, b, c, d | αu > b} /
S2 αT u > b for u ∈ P γ, α, b, d with θT u > c;
S3 0 ∈ Rγ, ψ, a, d and ψT u < a for u ∈ Rγ, ψ, a, d with ψu a.
Then T has at least three fixed points u1 , u2 , u3 ∈ P γ, d, such that
γui ≤ d for i 1, 2, 3,
b < αu1 a < ψu2 , with αu2 < b,
ψu3 < a.
2.6
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3. Related Lemmas
Let the Banach space E {u : {0, 1, . . . , n} → R} be endowed with the ordering x ≤ y if
xt ≤ yt for all t ∈ {0, 1, . . . , n}, and the maximum norm
u max
max |ut| ,
t∈{0,1,...,n}
max
t∈{0,1,...,n−1}
|Δut| .
3.1
Then, we define the cone P in E by
P u ∈ E | ut ≥ 0, t ∈ {0, 1, . . . , n}; u0 0, Δ2 ut ≤ 0, t ∈ {0, 1, . . . , n − 2} .
3.2
Let k be a natural number, such that k < min{ξ1 , n − ξm−2 }.
Let the nonnegative continuous concave functional α, the nonnegative continuous
convex functionals θ, γ, and the nonnegative continuous functional ψ be defined on the cone
P by
γu ψu θu |Δut|,
max
t∈{0,1,...,n−1}
αu max |ut|,
t∈{0,1,...,n}
|ut|
min
3.3
t∈{k1,...,n−k−1}
for u ∈ P .
In order to prove our main results, we need the following lemma.
Lemma 3.1. If u ∈ P , then
max |ut| ≤ n
t∈{0,1,...,n}
max
that is, θu ≤ nγu.
|Δut|,
t∈{0,1,...,n−1}
3.4
Proof. Suppose that the maximum of u occurs at t0 ∈ {0, 1, . . . , n}; by the definition of the cone
P , we know Δut 1 ≤ Δut, and then,
ut0 ut0 − u0 Δu0 Δu1 · · · Δut0 − 1
≤ t0 Δu0 ≤ nΔu0 ≤ n
max
t∈{0,1,...,n−1}
|Δut|.
3.5
So, we have
max |ut| ≤ n
t∈{0,1,...,n}
The proof is complete.
max
|Δut|.
t∈{0,1,...,n−1}
3.6
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By Lemma 3.1 and the definitions, the functionals defined above satisfy
1
θu ≤ αu ≤ θu ψu,
n/k 1
u max θu, γu ≤ nγu
3.7
for all u ∈ P γ, d ⊂ P . Therefore, condition 2.5 is satisfied.
Now, we show that 1/n/k 1θu ≤ αu. Here, we also suppose θu ut0 ,
and by the definitions of α and the cone P , we can distinguish two cases.
i αu uk 1, then we certainly have t0 ≥ k 1, and
ut0 Δu0 · · · Δuk Δuk 1 · · · Δu2k
t0
Δu2k 1 · · · Δu3k · · · Δu
k
k
t0
Δu
k 1 · · · Δut0 − 1
k
n
t0
1 uk 1 ≤
1 uk 1,
≤
k
k
3.8
that is, 1/n/k 1ut0 ≤ uk 1.
ii αu un − k − 1, then t0 ≤ n − k − 1, un ≤ un − k − 1 and
ut0 un − Δun − 1 − · · · − Δun − k − 1
− Δun − k − 2 − · · · − Δun − 2k − 1
n − t0
− Δun − 2k − 2 − · · · − Δu n −
k−1
k
n − t0
−Δu n −
k − 2 − · · · − Δut0 k
n − t0
1 un − k − 1 − un
≤ un k
n − t0
n
≤
1 un − k − 1 ≤
1 un − k − 1,
k
k
3.9
that is, 1/n/k 1ut0 ≤ un − k − 1. So, we have 1/n/k 1θu ≤ αu.
Lemma 3.2. Assume that H1 –H3 hold. Then, for any x ∈ E, xt ≥ 0,
Δ φp Δut − 1 qtft, xt, Δxt 0,
u0 0,
un m−2
t ∈ {1, . . . , n − 1},
ai uξi i1
3.10
3.11
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Discrete Dynamics in Nature and Society
has a unique solution ut given by
j
t−1
−1
ut φp Au − qifi, xi, Δxi
j0
3.12
i1
or
ut m−2
ξ
i −1
i1
j0
ai
φp−1
Au −
j
j
n−1
−1
− φp Au − qifi, xi, Δxi ,
qifi, xi, Δxi
jt
i1
i1
3.13
where Au satisfies
m−2
ξ
i −1
i1
j0
ai
φp−1
j
j
n−1
−1
Au − qifi, xi, Δxi φp Au − qifi, xi, Δxi .
j0
i1
i1
3.14
Proof. For any x ∈ E suppose that u is a solution of the BVPs 3.7 and 3.11. According to
the property of the difference operator, it follows that
φp Δu1 − φp Δu0 −q1f1, x1, Δx1,
φp Δu2 − φp Δu1 −q2f2, x2, Δx2,
···
3.15
φp Δut − φp Δut − 1 −qtft, xt, Δxt,
then
φp Δut − φp Δu0 −
t
qifi, xi, Δxi,
i1
φp Δut φp Δu0 −
t
3.16
qifi, xi, Δxi.
i1
Let φp Δu0 Au ,
t
Au − qifi, xi, Δxi ,
Δut φp−1
i1
3.17
Discrete Dynamics in Nature and Society
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and by
Δu0 u1 − u0 φp−1 Au ,
1
−1
Δu1 u2 − u1 φp Au − qifi, xi, Δxi ,
i1
3.18
···
Δut − 1 ut − ut − 1 φp−1
t−1
Au − qifi, xi, Δxi ,
i1
then
ut − u0 t−1
j
Au − qifi, xi, Δxi .
φp−1
j0
3.19
i1
By
n−1
Au − qifi, xi, Δxi ,
Δun − 1 un − un − 1 φp−1
i1
n−2
Au − qifi, xi, Δxi ,
Δun − 2 un − 1 − un − 2 φp−1
i1
3.20
···
t
Au − qifi, xi, Δxi ,
Δut ut 1 − ut φp−1
i1
so that
un − ut n−1
j
Au − qifi, xi, Δxi ,
φp−1
jt
ut un −
n−1
i1
j
Au − qifi, xi, Δxi ,
φp−1
jt
3.21
i1
Using the boundary condition 3.11, we can easily obtain
j
t−1
−1
ut φp Au − qifi, xi, Δxi
j0
i1
3.22
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Discrete Dynamics in Nature and Society
or
ut m−2
ξ
i −1
i1
j0
ai
φp−1
Au −
j
qifi, xi, Δxi
j
n−1
−1
− φp Au − qifi, xi, Δxi ,
jt
i1
i1
3.23
where Au satisfies 3.14.
On the other hand, it is easy to verify that if u is the solution of 3.12 or 3.13, then u
is a solution of 3.10 and 3.11.
Lemma 3.3. For any u ∈ E, ut ≥ 0, there exists a unique Au ∈ −∞, ∞ satisfying 3.14.
Moreover, there is a unique n0 ∈ {1, . . . , n − 1}, such that
n
0 −1
n0
i1
i1
qifi, ui, Δui < Au ≤
qifi, ui, Δui.
3.24
Proof. Let
ϕt m−2
ξ
i −1
i1
j0
ai
−
n−1
φp−1
j
t − qifi, ui, Δui
i1
j
t − qifi, ui, Δui ,
3.25
φp−1
j0
t ∈ −∞, ∞,
i1
so that
ϕ0 m−2
ξ
i −1
i1
j0
ai
j
− qifi, ui, Δui
φp−1
j
n−1
−1
− φp − qifi, ui, Δui
j0
i1
i1
j
j
n−1
m−2
i −1
ξ
−1
−1
φp
qifi, ui, Δui −
ai φp
qifi, ui, Δui
j0
i1
i1
j0
i1
⎡
ξ −1 j
⎤
j
m−2
n−1
i
>
ai ⎣ φp−1
qifi, ui, Δui − φp−1
qifi, ui, Δui ⎦
i1
j0
j0
i1
i1
≥ 0,
n−1
ϕ
qifi, ui, Δui
i1
m−2
ξ
i −1
i1
j0
ai
⎛
φp−1 ⎝
n−1
⎞
⎛
n−1
n−1
qifi, ui, Δui⎠ − φp−1 ⎝
qifi, ui, Δui⎠
ij1
⎞
j0
ij1
Discrete Dynamics in Nature and Society
9
⎞
⎞⎤
⎛
⎛
⎡
ξ
m−2
n−1
n−1
n−1
i −1
−1
−1
<
ai ⎣ φp ⎝
qifi, ui, Δui⎠ − φp ⎝
qifi, ui, Δui⎠⎦
j0
i1
j0
ij1
ij1
≤ 0.
3.26
By the continuity of ϕt, we know that there exists at least one
n−1
0, qifi, ui, Δui ⊂ −∞, ∞
Au ∈
3.27
i1
satisfying 3.14.
On the other hand,
ϕ t m−2
ξ
i −1
i1
j0
ai
−
φp−1
n−1 j
t − qtfi, ui, Δui
i1
φp−1
j
t − qifi, ui, Δui
j0
<
i1
m−2
⎡
ξ
i −1
ai ⎣
i1
φp−1
j
t − qifi, ui, Δui
j0
−
i1
n−1 φp−1
j0
3.28
t−
j
⎤
qifi, ui, Δui ⎦ ≤ 0.
i1
Then ϕt is strictly increasing on t ∈ −∞, ∞.
So, there exists a unique Au ∈ −∞, ∞ satisfying 3.14.
Moreover, there exists a n0 ∈ {1, . . . , n − 1}, such that
n
0 −1
n0
i1
i1
qifi, ui, Δui < Au ≤
qifi, ui, Δui.
3.29
Lemma 3.4. Assume that H1 –H3 hold. If x ∈ E, xt ≥ 0, then the unique solution ut of the
BVPs 3.10 and 3.11 has the following properties:
i Δ2 ut ≤ 0, and ut ≥ 0;
ii there exists a unique n0 ∈ {0, 1, . . . , n}, such that un0 maxt∈{0,1,...,n} ut, which n0 is
given in Lemma 3.3.
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Discrete Dynamics in Nature and Society
Proof. Suppose that ut is the solution of 3.10 and 3.11. Then we have the following.
i By Lemma 3.2, it is easy to see that Δ2 ut ≤ 0. Without loss of generality,
we assume that un min{u0, un}. By Δ2 ut ≤ 0, we know that ut ≥
m−2
m−2
un, t ∈ {0, 1, . . . , n}. So we get un i1 ai uξi ≥
i1 ai un, that is,
m−2
1 − i1 ai un ≥ 0. Hence un ≥ 0. So, from the concavity of ut, we know
that ut ≥ 0, t ∈ {0, 1, . . . , n}.
ii From
j
n
0 −1
−1
un0 φp Au − qifi, ui, Δui ,
j0
un0 1 n
0 −1
φp−1
Au −
j0
j
i1
qifi, ui, Δui
n0
Au − qifi, ui, Δui ,
φp−1
i1
i1
3.30
n0
and by Lemma 3.3, we have Au −
Also
un0 − 1 i1
n
0 −2
φp−1
qifi, ui, Δui < 0, so that un0 > un0 1.
j
Au − qifi, ui, Δui
j0
i1
un0 − φp−1 Au −
n
0 −1
3.31
qifi, ui, Δui ,
i1
and one arrives at un0 ≥ un0 − 1. So, un0 maxt∈{0,1,...,n} ut.
If there exist n1 , n2 ∈ {0, 1, . . . , n}, n1 < n2 such that Δun1 Δun2 , then
0 φp Δun1 − φp Δun2 n2
qifi, xi, Δxi > 0,
n1 1
3.32
which is a contradiction.
Lemma 3.5. For any u ∈ P, define the operator
⎧
j
t−1
⎪
⎪
⎪
−1
⎪ φp Au − qifi, ui, Δui ,
⎪
⎪
⎪
⎪ j0
i1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
j
⎨m−2
i −1
ξ
−1
ai φp Au − qifi, ui, Δui
T ut ⎪
⎪
j0
i1
i1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
j
⎪
n−1
⎪
⎪
−1
⎪
⎪
− φp Au − qifi, ui, Δui ,
⎪
⎩
jt
i1
Then T : P → P is completely continuous.
0 ≤ t ≤ n0 ,
3.33
n0 1 ≤ t ≤ n.
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Proof. Using the continuity of f and the definition of T , it is easy to show that T : P → P is
continuous. Next, we prove that T is completely continuous.
Suppose that the sequence {ui } ⊆ P is bounded, then there exists M > 0, such that
ui j ≤ M, for any i 1, 2, . . . , j 0, 1, . . . , n. By the continuity of f, φp−1 and Au are bounded,
and we know that there exists M > 0, such that |T ui t| ≤ M , for t ∈ {0, 1, . . . , n} and
i 1, 2, . . . , n, . . .. In view of the bounded sequence {T ui 0}, there exists {ui0 } ⊆ {ui }, such
that limi → ∞ T ui0 0 a0 . For the bounded sequence {T ui0 1}, there exists {ui1 } ⊆ {ui0 }, such
that limi → ∞ T ui1 1 a1 . By repetition in this way, we have that there exists {uij } ⊆ {uij−1 } for
j 2, 3, . . . , n, such that limi → ∞ T uij j aj . Let y {a0 , a1 , . . . , an }; by the definition of the
norm on E, there exists {uin } ⊆ {ui }, such that limi → ∞ T uin j y.
Hence, T : P → P is completely continuous.
4. Existence of Triple Positive Solutions to 1.2
We are now ready to apply Avery-Peterson’s fixed point theorem to the operator T to give
sufficient conditions for the existence of at least three positive solutions to the BVPs 1.2.
Now for convenience we introduce the following notations. Let
L
φp−1
n−1
qi ,
i1
⎞⎫
⎬
M min
φp−1 ⎝
qi⎠,
φp−1 ⎝
qi⎠ ,
⎭
⎩jk1
ij1
jn/2
in/21
⎧
⎨ n/2
⎛
n/2
⎞
n−k−1
⎛
j
⎧
⎞
⎞
⎞⎫
⎛
⎛
⎛
j
n/21
m−2
n−1
n−1
⎨n/2
⎬
i −1
ξ
N max
φp−1 ⎝
qi⎠,
ai φp−1 ⎝
qi⎠ φp−1 ⎝
qi⎠ .
⎩ j0
⎭
j0
ij1
i1
ij1
jn/2
in/21
4.1
Theorem 4.1. Assume that conditions H1 –H3 hold. Let 0 < a < b < n/k 1b < d, and
suppose that f satisfies the following conditions:
A1 ft, u, v ≤ φp d/L, for t, u, v ∈ {0, 1, . . . , n} × 0, nd × −d, d;
A2 ft, u, v > φp n/k 1b/M, for t, u, v ∈ {k 1, . . . , n − k − 1} × b, n/k 1b × −d, d;
A3 ft, u, v < φp a/N, for t, u, v ∈ {0, 1, . . . , n} × 0, a × −d, d.
Then BVPs 1.2 have at least three positive solutions u1 , u2 , and u3 such that
max
|Δui t| ≤ d,
min
|u1 t|,
t∈{0,1,...,n−1}
b<
t∈{k1,...,n−k−1}
n
1 b,
a < max |u2 t| <
k
t∈{0,1,...,n}
i 1, 2, 3,
max |u1 t| ≤ b,
t∈{0,1,...,n}
4.2
with
max |u3 t| < a.
t∈{0,1,...,n}
min
|u2 t| < b,
t∈{k1,...,n−k−1}
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Proof. The BVPs 1.2 have a solution u ut if and only if u solves the operator equation
u T u. Thus we set out to verify that the operator T satisfies Avery-Peterson’s fixed point
theorem which will prove the existence of three fixed points of T which satisfy the conclusion
of the theorem. Now the proof is divided into some steps.
1 We will show that A1 implies that T : P r, d → P r, d.
In fact, for u ∈ P r, d, there is γu maxt∈{0,1,...,n−1} |Δut| ≤ d. With Lemma 3.1,
there is maxt∈{0,1,...,n} |ut| ≤ nd, and then condition A1 implies ft, u, v ≤ φp d/L. On
the other hand, for u ∈ P , there is T u ∈ P , then T u is concave on t ∈ {0, 1, . . . , n}, and
max |ΔT ut| max{|ΔT u0|, |ΔT un − 1|}, and so
γT u |ΔT ut| max{|ΔT u0|, |ΔT un − 1|}
max
t∈{0,1,...,n−1}
$ $%
n−1
$
$
$
$
max φp−1 Au , $φp−1 Au − qifi, ui, Δui $
$
$
i1
#
≤
φp−1
4.3
n−1
d
d
L d.
qiφp
L
L
i1
Thus, T : P r, d → P r, d holds.
2 We show that condition S1 in Theorem 2.4 holds.
We take ut n/k 1b, for t ∈ {1, 2, . . . , n}, and u0 0. It is easy to see that
ut ∈ P γ, θ, α, b, n/k1b, d and αu n/k1b > b. Hence {u ∈ P γ, θ, α, b, n/k
1b, d | αu > b} /
∅. Thus for u ∈ P γ, θ, α, b, n/k 1b, d, there is b ≤ ut ≤ n/k 1b, |Δut| ≤ d. Hence by condition A2 of this theorem, one has ft, u, v > φp n/k 1b/M for t ∈ k 1, n − k − 1. By Lemma 3.4 and combining the conditions on α and P ,
we have
αT u
min
|T ut|
k1≤t≤n−k−1
1
1
max |T ut| T un0 n/k 1 t∈{0,...,n}
n/k 1
⎧
j
⎨n
0 −1
1
−1
Au − qifi, ui, Δui ,
min
φ
⎩ j0 p
n/k 1
i1
≥
m−2
ξ
i −1
i1
j0
ai
−
n−1
jn0
φp−1
Au −
j
qifi, ui, Δui
i1
φp−1 Au −
j
i1
qifi, ui, Δui
⎫
⎬
⎭
Discrete Dynamics in Nature and Society
⎧
⎞
⎛
n
⎨ n
0 −1
0 −1
1
−1 ⎝
≥
min
φ
qifi, ui, Δui⎠,
⎩jk1 p
n/k 1
ij1
n−k−1
φp−1
jn0
j
qifi, ui, Δui
in0 1
13
⎫
⎬
⎭
⎧
⎞
⎛
n/2
⎨ n/2
1
≥
min
φ−1 ⎝
qifi, ui, Δui⎠,
⎩jk1 p
n/k 1
ij1
⎞⎫
⎬
φp−1 ⎝
qifi, ui, Δui⎠
⎭
jn/2
in/21
n−k−1
⎛
j
⎧
⎛
⎞
n/2
⎨ n/2
1
1b
n/k
⎠,
>
min
φ−1 ⎝
qiφp
⎩jk1 p
M
n/k 1
ij1
n−k−1
jn/2
⎛
φp−1 ⎝
j
qiφp
in/21
⎞⎫
n/k 1b ⎠⎬
⎭
M
⎧
⎞
⎞⎫
⎛
⎛
j
n/2
n−k−1
⎬
⎨ n/2
1
n/k 1b
−1 ⎝
−1
min
φp
qi⎠,
φp ⎝
qi⎠
⎭
⎩jk1
M
n/k 1
ij1
jn/2
in/21
b
M b.
M
4.4
Therefore we have
αT u > b
n
∀u ∈ P γ, θ, α, b,
1 b, d .
k
4.5
Consequently, condition S1 in Theorem 2.4 is satisfied.
3 We now prove that S2 in Theorem 2.4 holds.
With 3.7, we have
αT u ≥
1
1
θT u >
n/k 1
n/k 1
n
1 b b
k
4.6
for u ∈ P γ, α, b, d with θT u > n/k 1b. Hence, condition S2 in Theorem 2.4 is
satisfied.
4 Finally, we prove that S3 in Theorem 2.4 is also satisfied.
14
Discrete Dynamics in Nature and Society
Since ψ0 0 < a, so 0∈Rγ, ψ, a, d. Suppose that u ∈ Rγ, ψ, a, d with ψu a.
Then, by the condition S3 of this theorem,
ψT u
max |T ut| T un0 t∈{0,1,...,n}
⎧
j
⎨n
0 −1
−1
max
Au − qifi, ui, Δui ,
φ
⎩ j0 p
i1
m−2
ξ
i −1
i1
j0
ai
−
n−1
j
Au − qifi, ui, Δui
φp−1
i1
φp−1
Au −
j
jn0
≤ max
⎩
qifi, ui, Δui
⎫
⎬
i1
⎧
⎨n/2
⎭
⎞
⎛
n/21
φp−1 ⎝
qifi, ui, Δui⎠,
j0
ij1
m−2
ξ
i −1
i1
j0
ai
⎛
n−1
φp−1 ⎝
⎞
qifi, ui, Δui⎠
ij1
⎞⎫
⎬
φp−1 ⎝
qifi, ui, Δui⎠
⎭
jn/2
in/21
⎛
n−1
≤ max
⎧
⎨n/2
⎩
j
⎛
⎞
a
⎠,
qiφp
N
n/21
φp−1 ⎝
j0
ij1
⎛
⎞
⎞⎫
j
n−1
a
a ⎬
⎠
⎠
ai φp−1 ⎝
qiφp
φp−1 ⎝
qiφp
⎭
N
N
j0
i1
ij1
jn/2
in/21
m−2
ξ
i −1
⎛
n−1
⎧
⎞
⎞
⎞⎫
⎛
⎛
⎛
j
n/21
m−2
n−1
n−1
⎬
⎨n/2
i −1
ξ
a
max
φp−1 ⎝
qi⎠,
ai φp−1 ⎝
qi⎠ φp−1 ⎝
qi⎠
⎭
⎩ j0
N
j0
ij1
i1
ij1
jn/2
in/21
a
N a.
N
Thus condition S3 in Theorem 2.4 holds.
4.7
Discrete Dynamics in Nature and Society
15
Therefore an application of Theorem 2.4 implies that BVPs 1.2 have at least three
positive solutions u1 , u2 , and u3 such that
|Δui t| ≤ d,
max
t∈{0,1,...,n−1}
b<
min
|u1 t|,
t∈{k1,...,n−k−1}
n
1 b,
a < max |u2 t| <
k
t∈{0,1,...,n}
i 1, 2, 3,
max |u1 t| ≤ b,
t∈{0,1,...,n}
4.8
with
min
|u2 t| < b,
t∈{k1,...,n−k−1}
max |u3 t| < a.
t∈{0,1,...,n}
Acknowledgments
The project is supported by the Natural Science Foundation of China 10971045, and the
Natural Science Foundation of Hebei Province A2009000664.
References
1 R. I. Avery and A. C. Peterson, “Three positive fixed points of nonlinear operators on ordered Banach
spaces,” Computers and Mathematics with Applications, vol. 42, no. 3–5, pp. 313–322, 2001.
2 R. I. Avery and J. Henderson, “Three symmetric positive solutions for a second-order boundary value
problem,” Applied Mathematics Letters, vol. 13, no. 3, pp. 1–7, 2000.
3 R. W. Leggett and L. R. Williams, “Multiple positive fixed points of nonlinear operators on ordered
Banach spaces,” Indiana University Mathematics Journal, vol. 28, no. 4, pp. 673–688, 1979.
4 Z. Bai, Y. Wang, and W. Ge, “Triple positive solutions for a class of two-point boundary-value
problems,” Electronic Journal of Differential Equations, vol. 2004, no. 6, pp. 1–8, 2004.
5 Y. Guo, J. Zhang, and Y. Ji, “Existence of triple positive solutions for second-order discrete boundary
value problems,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 60534, 10 pages, 2007.
6 Y. Wang and W. Ge, “Triple positive solutions for two-point boundary-value problems with onedimensional p-Laplacian,” Applicable Analysis, vol. 84, no. 8, pp. 821–831, 2005.
7 Y. Wang and W. Ge, “Positive solutions for multipoint boundary value problems with a onedimensional p-Laplacian,” Nonlinear Analysis: Theory, Methods & Application, vol. 66, no. 6, pp. 1246–
1256, 2007.
8 Y. Wang and C. Hou, “Existence of multiple positive solutions for one-dimensional p-Laplacian,”
Journal of Mathematical Analysis and Applications, vol. 315, no. 1, pp. 144–153, 2006.
9 H. Feng and W. Ge, “Existence of three positive solutions for m-point boundary-value problems with
one-dimensional p-Laplacian,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 7, pp.
2017–2026, 2008.