Journal of Inequalities in Pure and
Applied Mathematics
TRIPLE SOLUTIONS FOR A HIGHER-ORDER DIFFERENCE
EQUATION
volume 6, issue 1, article 10,
2005.
ZENGJI DU, CHUNYAN XUE AND WEIGAO GE
Department of Mathematics
Beijing Institute of Technology
Beijing 100081
People’s Republic of China
Received 09 April, 2004;
accepted 24 December, 2004.
Communicated by: S.S. Dragomir
EMail: duzengji@163.com
EMail: xuechunyanab@bit.edu.cn
EMail: gew@bit.edu.cn
Abstract
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c
2000
Victoria University
ISSN (electronic): 1443-5756
080-04
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Abstract
In this paper, we are concerned with the following nth difference equations
∆n y(k − 1) + f(k, y(k)) = 0, k ∈ {1, . . . , T },
∆i y(0) = 0, i = 0, 1, . . . , n − 2,
∆n−2 y(T + 1) = α∆n−2 y(ξ),
where f is continuous, n ≥ 2, T ≥ 3 and ξ ∈ {2, . . . , T − 1} are three fixed
positive integers, constant α > 0 such that αξ < T + 1. Under some suitable
conditions, we obtain the existence result of at least three positive solutions for
the problem by using the Leggett-Williams fixed point theorem.
2000 Mathematics Subject Classification: 39A10.
Key words: Discrete three-point boundary value problem; Multiple solutions; Green’s
function; Cone; Fixed point.
Sponsored by the National Natural Science Foundation of China (No. 10371006).
The authors express their sincere gratitude to the referee for very valuable suggestions.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Background Definitions and Green’s Function . . . . . . . . . . . . 5
3
Existence of Triple Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
References
Triple Solutions for a
Higher-order Difference
Equation
Zengji Du, Chunyan Xue and
Weigao Ge
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1.
Introduction
This paper deals with the following three-point discrete boundary value problem
(BVP, for short):
(1.1)
(1.2)
∆n y(k − 1) + f (k, y(k)) = 0,
∆i y(0) = 0, i = 0, 1, . . . , n − 2,
k ∈ {1, . . . , T },
∆n−2 y(T + 1) = α∆n−2 y(ξ),
where ∆y(k − 1) = y(k) − y(k − 1), ∆n y(k − 1) = ∆n−1 (∆y(k − 1)),
k ∈ {1, . . . , T }.
Throughout, we assume that the following conditions are satisfied:
(H1 ) T ≥ 3 and ξ ∈ {2, . . . , T − 1} are two fixed positive integers, α > 0 such
that αξ < T + 1.
(H2 ) f ∈ C({1, . . . , T } × [0, +∞), [0, +∞)) and f (k, ·) ≡ 0 does not hold on
{1, . . . , ξ − 1} and {ξ, . . . , T }.
In the few past years, there has been increasing interest in studying the existence of multiple positive solutions for differential and difference equations, for
example, we refer the reader to [1] – [8].
Recently, Ma [9] studied the following second-order three-point boundary
value problem
(1.3)
u00 + λa(t)f (u) = 0, t ∈ (0, 1),
u(0) = 0,
Triple Solutions for a
Higher-order Difference
Equation
Zengji Du, Chunyan Xue and
Weigao Ge
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αu(η) = u(1),
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by applying fixed-point index theorems and Leray-Schauder degree and upper
and lower solutions. In the case λ = 1, under the conditions that f is superlinear
or sublinear, Ma [10] considered the existence of at least one positive solution
of problem (1.3) by using Krasnosel’skii’s fixed-point theorem.
However, in [9] – [11], the author did not give the associate Green’s function and exceptional work was carried out for higher order multi-point difference equations. In the current work, we give the associate Green’s function
and obtain the existence of multiple positive solutions for BVP (1.1) – (1.2) by
employing the Leggett-Williams fixed point theorem. Our results are new and
different from those in [9] – [11]. Particularly, we do not require the assumption
that f is either superlinear or sublinear.
Triple Solutions for a
Higher-order Difference
Equation
Zengji Du, Chunyan Xue and
Weigao Ge
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2.
Background Definitions and Green’s Function
For the convenience of the reader, we present here the necessary definitions
from cone theory in Banach space, which can be found in [3].
Let N be the nonnegative integers, we let Ni,j = {k ∈ N : i ≤ k ≤ j} and
Np = N0,p .
We say that y is a positive solution of BVP (1.1) – (1.2), if y : NT +n−1 −→
R, y satisfies (1.1) on N1,T , y fulfills (1.2) and y is nonnegative on NT +n−1 and
positive on Nn−1,T .
Definition 2.1. Let E be a Banach space, a nonempty closed set K ⊂ E is said
to be a cone provided that
Triple Solutions for a
Higher-order Difference
Equation
Zengji Du, Chunyan Xue and
Weigao Ge
(i) if x ∈ K and λ ≥ 0 then λx ∈ K;
(ii) if x ∈ K and −x ∈ K then x = 0.
If K ⊂ E is a cone, we denote the order induced by K on E by ≤. For
x, y ∈ K, we write x ≤ y if and only if y − x ∈ K.
Definition 2.2. A map h is a nonnegative continuous concave functional on the
cone K which is convex, provided that
(i) h : K −→ [0, ∞) is continuous;
(ii) h(tx + (1 − t)y) ≥ th(x) + (1 − t)h(y) for all x, y ∈ K and 0 ≤ t ≤ 1.
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Now we shall denote
Kc = {y ∈ K : kyk < c}
and
K(h, a, b) = {y ∈ K : h(y) ≥ a, kyk ≤ b},
where k · k is the maximum norm.
Next we shall state the fixed point theorem due to Leggett-Williams [12] also
see [3].
Theorem 2.1. Let E be a Banach space, and let K ⊂ E be a cone in E.
Assume that h is a nonnegative continuous concave functional on K such that
h(y) ≤ kyk for all y ∈ Kc , and let S : Kc −→ Kc be a completely continuous
operator. Suppose that there exist 0 < a < b < d ≤ c such that
Triple Solutions for a
Higher-order Difference
Equation
Zengji Du, Chunyan Xue and
Weigao Ge
(A1 ) {y ∈ K(h, b, d) : h(y) > b} =
6 ∅ and h(Sy) > b for all y ∈ K(h, b, d);
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(A2 ) kSyk < a for kyk < a;
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(A3 ) h(Sy) > b for all y ∈ K(h, b, c) with kSyk > d.
Then S has at least three fixed points y1 , y2 and y3 in Kc such that ky1 k < a,
h(y2 ) > b and ky3 k > a with h(y3 ) < b.
In the following, we assume that the function G(k, l) is the Green’s function
of the problem −∆n y(k − 1) = 0 with the boundary condition (1.2).
It is clear that (see [3])
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g(k, l) = ∆n−2 G(k, l), (with respect to k)
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is the Green’s function of the problem −∆2 y(k − 1) = 0 with the boundary
condition
y(0) = 0,
(2.1)
y(T + 1) = αy(ξ).
We shall give the Green’s function of the problem −∆2 y(k − 1) = 0 with the
boundary condition (2.1).
Lemma 2.2. The problem
(2.2)
2
∆ y(k − 1) + u(k) = 0,
Triple Solutions for a
Higher-order Difference
Equation
k ∈ N1,T ,
with the boundary condition (2.1) has the unique solution
k−1
X
(2.3) y(k) = − (k − l)u(l) +
l=1
Zengji Du, Chunyan Xue and
Weigao Ge
T
X
k
(T + 1 − l)u(l)
T + 1 − αξ l=1
ξ−1
X
αk
(ξ − l)u(l),
−
T + 1 − αξ l=1
Proof. From (2.2), one has
∆y(k) − ∆y(k − 1) = −u(k),
∆y(k − 1) − ∆y(k − 2) = −u(k − 1),
..
.
∆y(1) − ∆y(0) = −u(1).
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k ∈ NT +1 .
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We sum the above equalities to obtain
∆y(k) = ∆y(0) −
k
X
u(l),
k ∈ NT ,
l=1
Pq
here and in the following, we denote l=p u(l) = 0, if p > q. Similarly, we
sum the equalities from 0 to k and change the order of summation to obtain
y(k + 1) = y(0) + (k + 1)∆y(0) −
k X
l
X
Triple Solutions for a
Higher-order Difference
Equation
u(j)
l=1 j=1
= y(0) + (k + 1)∆y(0) −
k
X
(k + 1 − l)u(l),
k ∈ NT ,
Zengji Du, Chunyan Xue and
Weigao Ge
l=1
Title Page
i.e.,
(2.4)
y(k) = y(0) + k∆y(0) −
k−1
X
Contents
(k − l)u(l),
k ∈ NT +1 .
l=1
By using the boundary condition (2.1), we have
ξ−1
T
X
X
1
α
(2.5) ∆y(0) =
(T + 1 − l)u(l) −
(ξ − l)u(l).
T + 1 − αξ l=1
T + 1 − αξ l=1
By (2.4) and (2.5), we have shown that (2.3) holds.
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Lemma 2.3. The function
 l[T +1−k−α(ξ−k)]
,

T +1−αξ





 l(T +1−k)+αξ(k−l)
,
T +1−aξ
(2.6)
g(k, l) =
k[T +1−l−α(ξ−l)]


,

T +1−αξ



 k(T +1−l)
,
T +1−αξ
l ∈ N1,k−1
T
N1,ξ−1 ;
l ∈ Nξ,k−1 ;
l ∈ Nk,ξ−1 ;
T
l ∈ Nk,T Nξ,T .
is the Green’s function of the following problem
−∆2 y(k − 1) = 0, k ∈ N1,T ,
(2.7)
Triple Solutions for a
Higher-order Difference
Equation
Zengji Du, Chunyan Xue and
Weigao Ge
y(0) = 0, y(T + 1) = αy(ξ).
(2.1)
Proof. We shall divide the proof into the following two steps.
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Step 1. We suppose k < ξ. Then the unique solution of problem (2.7), (2.1)
can be written as
Contents
k−1
X
y(k) = − (k − l)u(l) +
l=1
k−1
X
k
(T + 1 − l)u(l)
T + 1 − αξ l=1
" ξ−1
#
T
X
X
k
+
(T + 1 − l)u(l) +
(T + 1 − l)u(l)
T + 1 − αξ l=k
l=ξ
" k−1
#
ξ−1
X
X
αk
(ξ − l)u(l) +
(ξ − l)u(l)
−
T + 1 − αξ l=1
l=k
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k−1
X
l[T + 1 − k − α(ξ − k)]
=
u(l)
T
+
1
−
αξ
l=1
ξ−1
T
X
X
k[T + 1 − l − α(ξ − l)]
k(T + 1 − l)
+
u(l) +
u(l)
T
+
1
−
αξ
T
+
1
−
αξ
l=k
l=ξ
=
T
X
g(k, l)u(l).
l=1
Step 2. We suppose k ≥ ξ. Then the unique solution of problem (2.7), (2.1)
can be written as
" ξ−1
#
k−1
X
X
y(k) = −
(k − l)u(l) +
(k − l)u(l)
l=1
+
Zengji Du, Chunyan Xue and
Weigao Ge
Title Page
l=ξ
k
T + 1 − αξ
" ξ−1
X
(T + 1 − l)u(l) +
k−1
X
l=1
(T + 1 − l)u(l)
l=ξ
ξ−1
X
αk
(ξ − l)u(l)
+
(T + 1 − l)u(l) −
T + 1 − αξ l=1
l=k
T
X
#
ξ−1
=
X l[T + 1 − k − α(ξ − k)]
u(l)
T + 1 − αξ
l=1
+
k−1
X
l=ξ
Triple Solutions for a
Higher-order Difference
Equation
l(T + 1 − k) + αξ(k − l)
u(l) +
T + 1 − αξ
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T
X
l=k
k(T + 1 − l)
u(l)
T + 1 − αξ
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=
T
X
g(k, l)u(l).
l=1
Thus
PT the unique solution of problem (2.7), (2.1) can be written as y(k) =
l=1 g(k, l)u(l).
We observe that the condition αξ < T + 1 implies that g(k, l) is nonnegative
on NT +1 × N1,T , and positive on N1,T × N1,T . From (2.3), we have
y(k) =
T
X
g(k, l)u(l),
l=1
Triple Solutions for a
Higher-order Difference
Equation
Zengji Du, Chunyan Xue and
Weigao Ge
where
g(k, l) := (T + 1 − αξ)−1 k(T + 1 − l)
− (k − l)(T + 1 − αξ)χ[1,k−1] (l) − αk(ξ − l)χ[1,ξ−1] (l) .
This is a positive function, which means that the finite set
{g(k, l)/g(k, k) : k, l = 1, 2, . . . , T }
takes positive values. Let M1 , M2 be its minimum and maximum values, respectively.
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3.
Existence of Triple Solutions
In the following, we denote
m = min
k∈Nξ,T
T
X
g(k, l),
l=ξ
M = max
k∈NT +1
T
X
g(k, l)
l=1
and
m
e = min g(k, k),
k∈Nξ,T
f = max g(k, k).
M
k∈NT +1
f.
Then 0 < m < M , 0 < m
e <M
Let E be the Banach space defined by
Triple Solutions for a
Higher-order Difference
Equation
Zengji Du, Chunyan Xue and
Weigao Ge
E = {y : NT +n−1 −→ R, ∆i y(0) = 0, i = 0, 1, . . . , n − 2}.
Define
n−2
n−2
K = y ∈ E : ∆ y(k) ≥ 0 for k ∈ NT +1 and min ∆ y(k) ≥ σkyk
k∈Nξ,T
where σ =
M1 m
e
f
M2 M
∈ (0, 1), kyk = max |∆n−2 y(k)|. It is clear that K is a cone
k∈NT +1
in E.
Finally, let the nonnegative continuous concave functional h : K −→ [0, ∞)
be defined by
h(y) = min ∆n−2 y(k), y ∈ K.
k∈Nξ,T
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Note that for y ∈ K, h(y) ≤ kyk.
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Remark 1. If y ∈ K, kyk ≤ c, then
0 ≤ y(k) ≤ qc,
where
q = q(n, T ) =
k ∈ NT +n−1 ,
(T + n − 1)(T + n) · · · (T + 2n − 4)
.
(n − 2)!
In fact, if y ∈ K, kyk ≤ c, then 0 ≤ ∆n−2 y(k) ≤ c, k ∈ NT +1 , i.e.,
0 ≤ ∆(∆n−3 y(k)) = ∆n−3 y(k + 1) − ∆n−3 y(k) ≤ c.
Then one has
Triple Solutions for a
Higher-order Difference
Equation
Zengji Du, Chunyan Xue and
Weigao Ge
0 ≤ ∆n−3 y(1) − ∆n−3 y(0) ≤ c,
0 ≤ ∆n−3 y(2) − ∆n−3 y(1) ≤ c,
..
.
n−3
0 ≤ ∆ y(k) − ∆n−3 y(k − 1) ≤ c.
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We sum the above inequalities to obtain
0 ≤ ∆n−3 y(k) ≤ kc,
k ∈ NT +2 .
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Similarly, we have
0 ≤ ∆n−4 y(k) ≤
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k
X
i=1
!
i c=
k(k + 1)
c,
2!
k ∈ NT +3 .
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By using the induction method, one has
0 ≤ y(k) ≤
k(k + 1) · · · (k + n − 3)
c,
(n − 2)!
k ∈ NT +n−1 .
Then
0 ≤ y(k) ≤
(T + n − 1)(T + n) · · · (T + 2n − 4)
c = qc,
(n − 2)!
k ∈ NT +n−1 .
Theorem
Assume that there exist constants a, b, c such that 0 < a < b <
3.1.
m
and satisfy
c · min σ, M
(H3 ) f (k, y) ≤
c
,
M
(k, y) ∈ [0, T + n − 1] × [0, qc],
(H4 ) f (k, y) <
a
,
M
(k, y) ∈ [0, T + n − 1] × [0, qa],
(H5 ) There exists some l0 ∈ [n−2, T +n−1], such that f (k, y) ≥ mb0 , (k, y) ∈
[n − 2, T + n − 1] × b, qb
, where m0 = min g(k, l) > 0.
σ
k,l∈NT
Then BVP (1.1) – (1.2) has at least three positive solutions y1 , y2 and y3 ,
such that
(3.1)
ky1 k < a, h(y2 ) > b,
and
(3.2)
ky3 k > a with h(y3 ) < b.
Proof. Let the operator S : K −→ E be defined by
(Sy)(k) =
T
X
l=1
G(k, l)f (l, y(l)),
Triple Solutions for a
Higher-order Difference
Equation
Zengji Du, Chunyan Xue and
Weigao Ge
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k ∈ NT +n−1 .
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It follows that
n−2
∆
(3.3)
(Sy)(k) =
T
X
g(k, l)f (l, y(l)), for
k ∈ NT +1 .
l=1
We shall now show that the operator S maps K into itself. For this, let y ∈ K,
from (H1 ), (H2 ), one has
T
X
(3.4)
∆n−2 (Sy)(k) =
g(k, l)f (l, y(l)) ≥ 0, for
k ∈ NT +1 ,
l=1
and
n−2
∆
(Sy)(k) =
T
X
Triple Solutions for a
Higher-order Difference
Equation
g(k, l)f (l, y(l))
l=1
≤ M2
T
X
Zengji Du, Chunyan Xue and
Weigao Ge
g(k, k)f (l, y(l))
l=1
f
≤ M2 M
T
X
Title Page
f (l, y(l)),
for
k ∈ NT +1 .
l=1
Thus
f
kSyk ≤ M2 M
T
X
f (l, y(l)).
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l=1
From (H1 ), (H2 ), and (3.3), for k ∈ Nξ,T , we have
T
X
n−2
g(k, k)f (l, y(l))
∆ (Sy)(k) ≥ M1
≥ M1 m
e
l=1
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l=1
T
X
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f (l, y(l)) ≥
M1 m
e
kSyk = σkSyk.
f
M2 M
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Subsequently
min ∆n−2 (Sy)(k) ≥ σkSyk.
(3.5)
k∈Nξ,T
From (3.4) and (3.5), we obtain Sy ∈ K. Hence S(K) ⊆ K. Also standard
arguments yield that S : K −→ K is completely continuous.
We now show that all of the conditions of Theorem 2.1 are fulfilled. For all
y ∈ Kc , we have kyk ≤ c. From assumption (H3 ), we get
n−2
kSyk = max |∆ (Sy)(k)|
k∈NT +1
T
X
g(k, l)f (l, y(l))
= max k∈NT +1 Triple Solutions for a
Higher-order Difference
Equation
Zengji Du, Chunyan Xue and
Weigao Ge
l=1
T
X
c
g(k, l) = c.
≤
max
M k∈NT +1 l=1
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Hence S : Kc −→ Kc .
Similarly, if y ∈ Ka , then assumption (H4 ) yields f (k, y) < Ma , for k ∈
NT +1 . As in the argument above, we can show S : Ka −→ Ka . Therefore,
condition (A2 ) of Theorem 2.1 is satisfied.
Now we prove that condition (A1 ) of Theorem 2.1 holds. Let
k(k + 1) · · · (k + n − 3)b
y (k) =
, for
(n − 2)!σ
∗
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k ∈ Nξ,T .
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Then we can show that y ∗ ∈ K h, b, qb
and h(y ∗ ) ≥ σb > b. So
σ
b
y ∈ K h, b,
: h(y) > b 6= ∅.
σ
From assumptions (H2 ) and (H5 ), one has
h(Sy) = min
k∈Nξ,T
> min
k∈Nξ,T
T
X
l=1
T
X
g(k, l)f (l, y(l))
Triple Solutions for a
Higher-order Difference
Equation
g(k, l)f (l, y(l))
l=ξ
Zengji Du, Chunyan Xue and
Weigao Ge
≥ min g(k, l0 )f (l0 , y(l0 ))
k∈Nξ,T
≥
b
min g(k, l) ≥ b.
m0 k∈Nξ,T
This shows that condition (A1 ) of Theorem 2.1 is satisfied.
Finally, suppose that y ∈ K(h, b, c) with kSyk > σb , then
h(Sy) = min ∆n−2 (Sy)(k) ≥ σkSyk > b.
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Thus, condition (A3 ) of Theorem 2.1 is also satisfied. Therefore, Theorem
2.1 implies that BVP (1.1) – (1.2) has at least three positive solutions y1 , y2 , y3
described by (3.1) and (3.2).
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Corollary 3.2. Suppose that there exist constants
n mo
n mo
0 < a1 < b1 < c1 · min σ,
< a2 < b2 < c2 · min σ,
< · · · < ap ,
M
M
p is a positive integer, such that the following conditions are satisfied:
(H7 ) f (k, y) <
ai
,
M
(k, y) ∈ [0, T + n − 1] × [0, qai ], i ∈ N1,p ;
(H8 ) There exist li0 ∈ [n − 2, T + n − 1], such that f (k, y) ≥
[n − 2, T + n − 1] × [bi , qbσi ], i ∈ N1,p−1 .
qbi
,
m0
(k, y) ∈
Then BVP (1.1) – (1.2) has at least 2p − 1 positive solutions.
Proof. When p = 1, from condition (H7 ), we show S : Ka1 −→ Ka1 ⊆ Ka1 .
By using the Schauder fixed point theorem, we show that BVP (1.1) – (1.2) has
at least one fixed point y1 ∈ Ka1 . When p = 2, it is clear that Theorem 3.1 holds
( with c1 = a2 ). Then we can obtain BVP (1.1) – (1.2) has at least three positive
solutions y1 , y2 and y3 , such that ky1 k < a1 , h(y2 ) > b1 , ky3 k > a1 , with
h(y3 ) < b1 . Following this way, we finish the proof by the induction method.
The proof is completed.
If the case n = 2, similar to the proof of Theorem 3.1, we obtain the following result.
Corollary
3.3.
Assume that there exist constants a, b, c such that 0 < a < b <
m
c · min σ, M
and satisfy
(H9 ) f (k, y) ≤
c
,
M
Triple Solutions for a
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Zengji Du, Chunyan Xue and
Weigao Ge
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(k, y) ∈ [0, T + n − 1] × [0, c],
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(H10 ) f (k, y) <
a
,
M
(k, y) ∈ [0, T + n − 1] × [0, a],
(H11 ) f (k, y) ≥
b
,
m
(k, y) ∈ [ξ, T + n − 1] × [b, σb ].
Then BVP (1.1) – (1.2) has at least three positive solutions y1 , y2 and y3 ,
satisfying (3.1) and (3.2).
Finally, we give an example to illustrate our main result.
Example 3.1. Consider the following second order third point boundary value
problem
∆2 y(k − 1) + f (k, y) = 0,
(3.6)
7
y(7) = y(3),
9
y(0) = 0,
(3.7)
where f (k, y) =
100
a(y),
k+100
a(y) =







k ∈ N1,6 ,
+ sin8 y,
1
720
+
+6 y−
89
5
+
1
30
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1
if y ∈ 0, 30
;
1 if y ∈ 30 , 3 ;
+ sin8 y,
sin2 (y−3)
2
+ sin8 y,
Zengji Du, Chunyan Xue and
Weigao Ge
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and
1
720
1
720
Triple Solutions for a
Higher-order Difference
Equation
if y ∈ [3, 360].
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Then T = 6, n = 3, α =
7
9
< 1, T + 1 − αn =
14
3
> 0. Then the conditions
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(H1 ), (H2 ) are satisfied, and the function
 l(42−2k)

,

9




3  3l(7−k)+7(k−l)
,
3
G(k, l) =
14 
k(42−2l)


,

9



k(7 − l),
l ∈ N1,k−1
T
N1,2 ;
l ∈ N3,k−1 ;
l ∈ Nk,2 ;
T
l ∈ Nk,6 N3,6 ,
is the Green’s function of the problem −∆2 y(k − 1) = 0, k ∈ N1,6 with (3.7).
f = 18 , M1 = 2 , M2 = 9,
Thus we can compute m = 27
, M = 18, m
e = 97 , M
2
7
9
1
m
1
1
σ = 81
<M
= 34 . We choose that a = 35
, b = 10
, c = 360, consequently,
100
a(y)
k + 100
1

+ 1 < 20 = Mc ,

 720
1
1
+
6
3
−
+ 1 < 20 =
≤ a(y) ≤
720
30


 1 + 89 + 3 < 20 = c ,
720
5
2
M
Triple Solutions for a
Higher-order Difference
Equation
Zengji Du, Chunyan Xue and
Weigao Ge
f (k, y) =
Thus
f (k, y) ≤
c
,
M
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1
(k, y) ∈ [0, 7] × 0, 30
;
c
1
, (k, y) ∈ [0, 7] × 30
,3 ;
M
(k, y) ∈ [0, 7] × [3, 360].
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(k, y) ∈ [0, 7] × [0, 360];
and
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f (k, y) ≤
1
1
a
+ sin8 y <
=
,
720
630
M
1
(k, y) ∈ [0, 7] × 0,
;
35
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100 1
1
1
1
b
8
f (k, y) ≥
+6
−
+ sin y ≥
= ,
107 720
10 30
135
m
1
(k, y) ∈ [3, 7] ×
,3 .
27
That is to say, all the conditions of Corollary 3.3 are satisfied. Then the boundary value problem (3.6), (3.7) has at least three positive solutions y1 , y2 and y3 ,
such that
1
1
y1 (k) < , for k ∈ N7 , y2 (k) > , for k ∈ N3,7 ,
35
27
and
1
1
max y3 (k) > , for k ∈ N7 with min y3 (k) < .
k∈N3,7
k∈N1,7
35
27
Triple Solutions for a
Higher-order Difference
Equation
Zengji Du, Chunyan Xue and
Weigao Ge
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References
[1] X.M. HE AND W.G. GE, Triple solutions for second order three-point
boundary value problems, J. Math. Anal. Appl., 268 (2002), 256–265.
[2] Y.P. GUO, W.G. GE AND Y. GAO, Two Positive solutions for higher-order
m-point boundary value problems with sign changing nonlinearities, Appl.
Math. Comput., 146 (2003), 299–311.
[3] R.P. AGARWAL, D. O’REGAN AND P.J.Y. WONG, Positive Solutions of
Differential, Difference and Integral Equations, Kluwer Academic Publishers, Boston, 1999.
[4] G.L. KARAKOSTAS, K.G. MAVRIDIS AND P.Ch. TSAMATOS, Multiple positive solutions for a functional second order boundary value problem, J. Math. Anal. Appl., 282 (2003), 567–577.
[5] H.B. THOMPSON, Existence of multiple solutions for finite difference
approximations to second-order boundary value problems, Nonlinear
Anal., 53 (2003), 97–110.
[6] R.M. ABU-SARIS AND Q.M. AL-HASSAN, On global periodicity of difference equations, J. Math. Anal. Appl., 283 (2003), 468–477.
[7] Z.J. DU, W.G. GE AND X.J. LIN, Existence of solution for a class of
third order nonlinear boundary value problems, J. Math. Anal. Appl., 294
(2004), 104–112.
Triple Solutions for a
Higher-order Difference
Equation
Zengji Du, Chunyan Xue and
Weigao Ge
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[8] A. CABADA, Extremal solutions for the difference p-Laplacian problem
with nonlinear functional boundary conditions, Comput. Math. Appl., 42
(2001), 593–601.
[9] R.Y. MA, Multiplicity of positive solutions for second order three-point
boundary value problems, Comput. Math. Appl., 40 (2000), 193–204.
[10] R.Y. MA, Positive solutions of a nonlinear three-point boundary value
problems, Electron J. Differential Equations, 34 (1999), 1–8.
[11] R.Y. MA, Positive solutions for second order three-point boundary value
problems, Appl. Math. Lett., 14 (2001), 1–5.
[12] R.W. LEGGETT AND L.R. WILLIAMS, Multiple positive fixed points of
nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J., 28
(1979), 673–688.
Triple Solutions for a
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Equation
Zengji Du, Chunyan Xue and
Weigao Ge
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