Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 918082, 17 pages
doi:10.1155/2012/918082
Research Article
Existence and Multiplicity of Positive
Solutions of a Nonlinear Discrete Fourth-Order
Boundary Value Problem
Ruyun Ma and Yanqiong Lu
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
Correspondence should be addressed to Ruyun Ma, ruyun ma@126.com
Received 16 September 2011; Revised 14 December 2011; Accepted 25 December 2011
Academic Editor: Yuriy Rogovchenko
Copyright q 2012 R. Ma and Y. Lu. 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.
we show the existence and multiplicity of positive solutions of the nonlinear discrete fourth-order
boundary value problem Δ4 ut − 2 λhtf ut, t ∈ Ì2, u1 uT 1 Δ2 u0 Δ2 uT 0,
where λ > 0, h : Ì2 → 0, ∞ is continuous, and f : Ê → 0, ∞ is continuous, T > 4, Ì2 {2, 3, . . . , T}. The main tool is the Dancer’s global bifurcation theorem.
1. Introduction
It’s well known that the fourth order boundary value problem
ut ft, ut,
t ∈ 0, 1,
u0 u1 u 0 u 1 0
1.1
can describe the stationary states of the deflection of an elastic beam with both ends hinged,
it also models a rotating shaft. The existence and multiplicity of positive solutions of the
boundary value problem 1.1 have been considered extensively in the literature, see 1–10.
The existence and multiplicity of positive solutions of the parameterized boundary value
problem
ut λhtfut,
t ∈ 0, 1,
u0 u1 u 0 u 1 0
1.2
2
Abstract and Applied Analysis
have also been studied by several authors, see Bai and Wang 11, Cid et al. 12, and the references therein.
However, relatively little is known about the corresponding discrete fourth-order problems. Let
T > 4,
Ì0 {0, 1, . . . , T 2},
Ì1 {1, 2,
. . . , T 1},
Ì2 {2, 3,
. . . , T}.
1.3
Zhang et al. 13, and He and Yu 14 used the fixed point index theory in cones to study the
following discrete analogue
Δ4 ut − 2 λhtfut,
t ∈ Ì2 ,
1.4
1.5
u0 uT 2 Δ2 u0 Δ2 uT 0,
where Δ4 ut − 2 denote the fourth forward difference operator and Δut ut 1 − ut. It
has been pointed out in 13, 14 that 1.4, 1.5 are equivalent to the equation of the form:
ut λ
T 1
T
Gt, s G1 s, j h j f u j : A0 ut,
s1
j2
t ∈ Ì0 ,
1.6
where
⎧
1 ⎨sT 2 − t,
Gt, s T 2 ⎩tT 2 − s,
⎧
1 ⎨T 1 − s j − 1 ,
G1 s, j T ⎩T 1 − j s − 1,
1 ≤ s ≤ t ≤ T 2,
0 ≤ t ≤ s ≤ T 1,
2 ≤ j ≤ s ≤ T 1,
1.7
1 ≤ s ≤ j ≤ T.
Notice that two distinct Green’s functions used in 1.6 make the construction of cones and
the verification of strong positivity of A0 become more complex and difficult. Therefore, Ma
and Xu 15 considered 1.4 with the boundary condition
u1 uT 1 Δ2 u0 Δ2 uT 0,
1.8
and introduced the definition of generalized positive solutions:
Definition 1.1. A function y : Ì0 → Ê is called a generalized positive solution of 1.4, 1.8, if
y satisfies 1.4, 1.8, and yt ≥ 0 on Ì1 and yt > 0 on Ì2 .
Remark 1.2. Notice that the fact y : Ì0 → Ê is a generalized positive solution of 1.4, 1.8
does not means that yt ≥ 0 on Ì0 . In fact, y satisfies
1 yt ≥ 0 for t ∈ Ì2 ;
2 y1 yT 1 0;
3 y0 −y2, yT 2 −yT.
Abstract and Applied Analysis
3
Ma and Xu 15 also applied the fixed point theorem in cones to obtain some results
on the existence of generalized positive solutions.
It is the purpose of this paper to show some new results on the existence and multiplicity of generalized positive solutions of 1.4, 1.8 by Dancer’s global bifurcation theorem.
To wit, we get the following.
Theorem 1.3. Let h : Ì2 → 0, ∞, f ∈ CÊ, 0, ∞, and
lim
s→0
fs
f0 ∈ 0, ∞,
s
fs
s→∞ s
lim
f∞ ∞.
1.9
Assume that there exists B ∈ 0, ∞ such that f is nondecreasing on 0, B. Then
i 1.4, 1.8 have at least one generalized positive solution if 0 < λ < λ1 /f0 ;
ii 1.4, 1.8 have at least two generalized positive solutions if
s
λ1
,
< λ < sup ∗
f0
γ
fs
s∈0,B
where γ ∗ maxt∈Ì1
value of
T
s2
1.10
Kt, shs, Kt, s is defined as 2.3 and λ1 is the first eigen-
t ∈ Ì2 ,
Δ4 ut − 2 λhtut,
u1 uT 1 Δ2 u0 Δ2 uT 0.
1.11
The “dual” of Theorem 1.3 is as follows.
Theorem 1.4. Let h : Ì2 → 0, ∞, f ∈ CÊ, 0, ∞, and
lim
s→0
fs
f0 ∈ 0, ∞,
s
lim
s→∞
fs
f∞ 0.
s
1.12
Assume that there exists B ∈ 0, ∞ such that f is nondecreasing on 0, B. Then
i 1.4, 1.8 have at least a generalized positive solution provided
λ>
where γ∗ mint∈Ì2
T
s2
s
,
s∈0,c1 B c1 γ∗ fs
inf
Kt, shs;
1.13
4
Abstract and Applied Analysis
ii 1.4, 1.8 have at least two generalized positive solutions provided
inf
s
s∈0,c1 B c1 γ∗ fs
<λ<
λ1
.
f0
1.14
The rest of the paper is organized as follows: in Section 2, we present the form of
the Green’s function of 1.4, 1.8 and its properties, and we enunciate the Dancer’s global
bifurcation theorem 16, Corollary 15.2. In Section 3, we use the Dancer’s bifurcation
theorem to prove Theorems 1.3 and 1.4 and in Section 4, we finish the paper presenting a
couple of illustrative examples.
Remark 1.5. For other results on the existence and multiplicity of positive solutions and nodal
solutions for fourth-order boundary value problems based on bifurcation techniques, see 17–
21.
2. Preliminaries and Dancer’s Global Bifurcation Theorem
Lemma 2.1. Let h : Ì2 →
Ê. Then the linear boundary value problem
Δ4 ut − 2 ht,
t ∈ Ì2 ,
u1 uT 1 Δ2 u0 Δ2 uT 0
2.1
has a solution
ut T
s2
Kt, shs,
t ∈ Ì1 ,
2.2
where
⎧
2
⎪
−
−
2s
−
1T
1
−
t
2Tt
−
1
−
−
1
s
s
t
⎪
⎪
⎪
⎪
, 2 ≤ s ≤ t ≤ T 1,
⎨
6T
Kt, s ⎪
⎪
⎪
t − 1T 1 − t 2Ts − 1 − s − 12 − t − 2t
⎪
⎪
⎩
,
1 ≤ t ≤ s ≤ T.
6T
2.3
Proof. By a simple summing computation and u1 Δ2 u0 0, we can obtain
ut Δu0t − 1 t−1
tt − 1t − 2 3
t − st − s − 1t − s 1
Δ u0 hs.
6
6
s2
2.4
Abstract and Applied Analysis
5
This together with uT 1 Δ2 uT 0, it follows that
ut 2
T T 1 − st − 1 2Ts − 1 − s − 1 − tt − 2
6T
s2
t−1
t − st − s − 1t − s 1
6T
s2
hs
2
T T 1 − st − 1 2Ts − 1 − s − 1 − tt − 2
6T
st
hs
2.5
hs
2
t−1 T 1 − ts − 1 2Tt − 1 − t − 1 − ss − 2
6T
s2
hs.
Therefore, 2.2 holds.
Remark 2.2. It has been pointed out in 15 that 2.1 is equivalent to the summation equation
of the form
ut T
T
G1 t, s G1 s, j h j ,
s2
j2
t ∈ Ì1 .
2.6
It is easy to verify that 2.2 and 2.6 are equivalent.
By a similar method in 9, it follows that Kt, s satisfies
Kt, s ≤ Φs for s ∈ Ì1 ,
t ∈ Ì1 ,
Kt, s ≥ ctΦs for s ∈ Ì1 ,
2.7
t ∈ Ì1 ,
where
Φs ⎧
⎪
⎪
⎪
⎨
√
3/2
3
,
s − 1 T 2 − s − 2s
27T
√
⎪
⎪
⎪
⎩ 3 T 1 − s2Ts − 1 − s − 2s3/2 ,
27T
ct ⎧
⎪
⎪
⎪
⎪
⎪
⎨
√ 3 3 T 2 − tt − 2 t − 1
3/2
,
1
√
⎪
⎪
3 3T 1 − t2Tt − 1 − tt − 2
⎪
⎪
⎪
,
⎩
2T 2 13/2
2T 2
1≤s≤
T
1,
2
T
1 < s ≤ T 1,
2
1≤t≤
T
1,
2
T
1 < t ≤ T 1.
2
2.8
2.9
6
Abstract and Applied Analysis
Moreover, we have that
Kt, s ≥ c1 Φs,
for s ∈ Ì1 , t ∈ Ì2 .
2.10
√
3/2
here c1 3 3T 2 /2T 2 1 .
Let X be a real Banach space with a cone K such that X K − K. Let us consider the
equation:
x μLx Nx, μ ∈ Ê, x ∈ X
2.11
under the assumptions:
A1 The operators L, N : X → X are compact. Furthermore, L is linear, Nx
X /
x
X →
0 as x
X → 0, and L NK ⊆ K.
A2 The spectral radius rL of L is positive. Denote μ0 rL−1 .
A3 L is strongly positive.
Dancer’s global bifurcation theorem is the following.
Theorem 2.3 see 16, Corollary 15.2. Let
S :
μ, x ∈ Ê × X | μ, x is a solution of 2.11 with x > 0 and μ > 0 .
2.12
If (A1) and (A2) are satisfied, then μ0 , 0 is a bifurcation point of 2.11 and S has an unbounded
solution component C which passes through μ0 , 0. Additionally, if (A3) is satisfied, then μ, x ∈ C
and μ / μ0 always implies x > 0 and μ > 0.
3. Proof of the Main Results
Before proving Theorem 1.3, we state some preliminary results and notations. Let
ρ : 4 sin2
π
,
2T
et : sin
πt − 1
,
T
t ∈ Ì1 ,
X : u | u : Ì0 −→ Ê, u1 uT 1 Δ2 u0 Δ2 uT 0 .
Then X is a Banach space under the normal:
γ
| −γ et ≤ −Δ2 ut − 1 ≤ γ et, t ∈ Ì1 .
u
X : inf
ρ
See 22 for the detail.
Let
K : u ∈ X | Δ2 ut − 1 ≤ 0, ut ≥ 0, t ∈ Ì1 .
Then K is normal and has a nonempty interior and X K − K.
3.1
3.2
3.3
3.4
Abstract and Applied Analysis
Let Y {u | u : Ì2 →
7
Ê}. Then Y is a Banach space under the norm:
u
∞ max|ut|.
t∈
3.5
Lu : Δ4 ut − 2, u ∈ X.
3.6
Ì2
Define L : X → Y by setting
It is easy to check that L−1 : Y → X is compact.
Lemma 3.1. Let h ∈ Y with h ≥ 0 and ht0 > 0 for some t0 ∈ Ì2 , and
Lu − h 0.
3.7
Then u ∈ int K.
Proof. It is enough to show that there exist two constants r1 , r2 ∈ 0, ∞ such that
r1 et ≤ −Δ2 ut − 1 ≤ r2 et,
t ∈ Ì1 .
3.8
t ∈ Ì1 .
3.9
In fact, we have from 3.7 that
−Δ2 ut − 1 T
G1 t, shs,
s2
This together with the relation t − 1T 1 − t/TG1 s, s ≤ G1 t, s ≤ t − 1T 1 − t/T
implies that
T
T
t − 1T 1 − t
t − 1T 1 − t ≤
.
G1 s, shs
G1 t, shs ≤ h
∞
T
T
s2
s2
3.10
Combining 3.9 with 3.10 and the fact that
c1 sin
πt − 1 t − 1T 1 − t
πt − 1
≤
≤ c2 sin
,
T
T
T
t ∈ Ì1
3.11
for some constants c1 , c2 ∈ 0, ∞, we conclude that 3.8 is true.
Let ζ ∈ CÊ, Ê be such that
fu f0 u ζu,
3.12
8
Abstract and Applied Analysis
clearly
lim
|u| → 0
ζu
0.
u
3.13
Let us consider
Lu λ h·f0 u h·ζu
3.14
as a bifurcation problem from the trivial solution u ≡ 0.
By 1.4, 1.8, it follows that if ut ∈ X is one solution of 1.4, 1.8, then ut satisfies
u0 −u2, uT 2 −uT. So, u0, 0, u2, . . . , uT, 0, uT 2 is a solution of 1.4,
1.8, if and only if, 0, u2, . . . , uT, 0 solves the operator equation
ut λ
T
Gt, shsfus,
s2
t ∈ Ì1 .
3.15
Now, let J : Y → X be the linear operator:
Ju2, u3, . . . , uT −u2, 0, u2, u3, . . . , uT, 0, −uT,
u ∈ Y.
3.16
Let L, N : X → X be the operators:
Lu : J ◦ L−1 h·f0 u ,
3.17
Nu : J ◦ L−1 h·ζu,
3.18
respectively. Then Lemma 3.1 yields that L : X → X is strongly positive. Moreover, 16,
Theorem 7.c implies rL > 0.
Now, it follows from Theorem 2.3 that there exists a continuum
C ⊆
μ, x ∈ Ê × X | μ, x is a solution of 1.4, 1.8 with x > 0 and μ > 0 ,
3.19
which joins rL−1 , 0 with infinity in 0, ∞ × K and
μ, x ∈ C ,
−1
μ
/ rL ⇒ x > 0,
μ > 0.
3.20
It is easy to check that
rL−1 λ1
.
f0
3.21
Abstract and Applied Analysis
9
Lemma 3.2. Let h1 , h2 ∈ Y with h1 ≥ h2 > 0. Then the eigenvalue problems
Lut λhi tut,
t ∈ Ì2 , i 1, 2
3.22
have the principal eigenvalue λi , i 1, 2 such that λ1 ≤ λ2 . Moreover, the corresponding eigenfunctions ψi are positive in Ì2 .
Proof . Let Li : X → X be the operator
Li u : λJ ◦ L−1 hi ·u,
3.23
i 1, 2.
Then Lemma 3.1 yields that Li : X → X is strongly positive. By Krein-Rutman
theorem 16, Theorem 7.c the spectral radius rLi > 0 and there exist ψi ∈ X with ψi > 0 on
Ì2 such that
Li ψi t rLi ψi t,
i 1, 2.
3.24
That is, the eigenvalue problems 3.22 have the principal eigenvalues λi 1/rLi , and ψi t
is the corresponding eigenfunctions of λi , i 1, 2.
Next, we prove λ1 ≤ λ2 . Since Tt2 Δ4 ψ1 t − 2 ψ 2 t Tt2 ψ1 tΔ4 ψ2 t − 2, it follows
that
T
T
T
λ2
1 4
h1 tψ1 tψ2 t ≥
h2 tψ2 tψ1 t Δ ψ2 t − 2ψ1 t
t2
t2 λ2
t2 λ2
T
1
t2
λ2
T
λ1 λ2 t2
ψ2 tΔ4 ψ1 t − 2 T
ψ2 t
t2
λ2
λ1 h1 tψ1 t
3.25
h1 tψ1 tψ2 t.
Therefore, λ1 ≤ λ2 .
Suppose that Ìa {a 1, a 2, . . . , b − 1} is a strict subset of
restriction of h on Ìa . Consider the linear eigenvalue problems:
Δ4 ut − 2 λha tf0 ut,
t ∈ Ìa ,
ua ub Δ ua − 1 Δ ub − 1 0.
2
2
Ì2 and ha denote the
3.26
Then we get the following result.
Lemma 3.3. Let λ1 is the principal eigenvalue of 3.17, then 3.26 has only one principal eigenvalue
λa such that 0 < λ1 < λa .
10
Abstract and Applied Analysis
Proof. It is not difficult to prove that 3.26 has only one principal eigenvalue λa > 0 by
Lemma 3.1, and the corresponding eigenfunction ψa > 0 on Ìa . So we only to verify that
0 < λ1 < λa .
Let ψ1 be the corresponding eigenfunction of λ1 , we have that
b−1
Δ4 ψa t − 2ψ1 t ta1
b−1
Δ4 ψ1 t − 2ψa t − ψ1 bΔ2 ψa b − 2 − ψa b − 1Δ2 ψ1 b − 1
ta1
− Δ2 ψa aψ1 a − ψa a 1Δ2 ψ1 a − 1
>
b−1
Δ4 ψ1 t − 2ψa t.
ta1
3.27
So
b−1
b−1
1 4
Δ ψa t − 2ψ1 t
λ
ta1 a
htψa tψ1 t ta1
b−1
1 4
Δ ψ1 t − 2ψa t
λ
ta1 a
>
b−1
ψa t λ1 htψ1 t
λa
ta1
3.28
b−1
λ1 ht ψa tψ1 t.
λa ta1
Thus 0 < λ1 < λa .
Proof of Theorem 1.3. We divide the proof into three steps.
Let {μn , yn } ⊂ C be such that
μn yn X
→ ∞,
n → ∞.
3.29
Then
Δ4 yn t − 2 μn htf yn t ,
t ∈ Ì2 ,
yn 1 yn T 1 Δ2 yn 0 Δ2 yn T 0.
3.30
Step 1. We show that there exists a constant M such that μn ∈ 0, M for all n.
Suppose on the contrary that
lim μn ∞.
n→∞
3.31
Abstract and Applied Analysis
11
Let vn yn /
yn X . Then it follows from 3.30 that
f yn t
Δ vn t − 2 μn ht
vn t,
yn t
4
t ∈ Ì2 ,
3.32
vn 1 vn T 1 Δ vn 0 Δ vn T 0.
2
2
Since
inf
fs
|s>0
s
: M0 > 0,
3.33
there exists a constant M0 > 0, such that
f yn t
> M0 > 0.
yn t
3.34
Let λ∗ be the principal eigenvalue of the linear eigenvalue problems:
Δ4 vt − 2 λht M0 vt,
t ∈ Ì2 ,
v1 vT 1 Δ2 v0 Δ2 vT 0.
3.35
Combining 3.31 and 3.34 with the relation 3.32, using Lemma 3.2, we get
0 < μn ≤ λ∗ .
3.36
This contradicts 3.31. So μn ∈ 0, M for all n.
Step 2. We show that C joins λ1 /f0 , 0 with 0, ∞.
Assume that there exist δ > 0 and {μn , yn } ⊂ C such that
0 < δ ≤ μn ≤ M; yn X −→ ∞, n −→ ∞.
3.37
First, we show that
yn −→ ∞ ⇒ yn −→ ∞.
X
∞
3.38
Suppose on the contrary that
yn ≤ M1
∞
3.39
for some M1 > 0 independent on n. Then it follows from 3.30 and 0 < δ ≤ |μn | ≤ M that
4 3.40
Δ yn ≤ M
h
∞ sup fs | 0 < s ≤ M1 ,
∞
and subsequently, {
yn X } is bounded. This is a contradiction. So, 3.38 holds.
12
Abstract and Applied Analysis
Next, we show that
yn −→ ∞ ⇒ min yn t | t ∈ Ì2 −→ ∞.
∞
3.41
In fact,
yn t μn
T
Kt, shsf yn s , t ∈ Ì1 .
3.42
s2
This together with 2.7 imply that 3.41 is valid.
Finally, we have from the facts that min{yn t | t ∈
that
f yn t
μn
−→ ∞,
yn t
Ì2 }
→ ∞ and 0 < δ ≤ |μn | ≤ M
n −→ ∞ for any t ∈ Ìa .
3.43
Consider the following linear eigenvalue problems:
t ∈ Ìa ,
Δ4 vt − 2 λha t vt,
va vb Δ2 va − 1 Δ2 vb − 1 0.
3.44
By Lemma 3.3 and 3.32, 3.44 has a positive principal eigenvalue λa , and
f yn t
μn
≤ λa ,
yn t
3.45
which contradicts 3.43. Thus limn → ∞ μn 0.
Step 3. Fixed λ such that
0 < λ < sup
s∈0,B
s
γ ∗ fs
.
3.46
Then there exists b ∈ 0, B such that
0<λ<
b
γ ∗ fb
.
3.47
We show that there is no μ, u ∈ C such that
u
∞ b,
0<μ<
b
.
γ ∗ fb
3.48
Abstract and Applied Analysis
13
In fact, if there exists η, y ∈ C satisfying 3.48, then
yt η
T
Kt, shsf ys
s2
3.49
≤ ηγ ∗ fb
ηγ ∗
fb
·b
b
for t ∈ Ì1 , and subsequently, η ≥ b/γ ∗ fb. Therefore, no μ, u ∈ C satisfies 3.48.
Now, combining the conclusions in Steps 2 and 3, using the fact that no μ, u ∈ C
satisfies 3.48, it concludes that for every λ ∈ λ1 /f0 , b/γ ∗ fb, 1.4, 1.8 has at least two
generalized positive solutions in C . For arbitrary λ ∈ 0, sups∈0,B s/γ ∗ fs, we may find
b bλ satisfying 3.47. So, for every λ ∈ λ1 /f0 , sups∈0,B s/γ ∗ fs, 1.4, 1.8 has at
least two generalized positive solutions in C .
Proof of Theorem 1.4. We divide the proof into three steps.
Step 1. We show that there exists a positive constant β > 0 such that
inf μ | μ, u ∈ C : β > 0.
3.50
Suppose on the contrary that there exists {μn , yn } ⊂ C such that
μn → 0 ,
as n → ∞.
3.51
Then we have from 3.32, 3.51, f0 ∈ 0, ∞ and f∞ 0 that
vn X → 0,
as n → ∞.
3.52
However, this contradicts with the fact that vn X 1 for all n ∈ Æ . Therefore, 3.50 holds.
Step 2. We show that for any closed interval I ⊂ β, ∞, there exists MI > 0 such that
sup u
| μ, u ∈ C ≤ MI .
3.53
Suppose on the contrary that there exists {μn , yn } ⊂ C with
μn ⊂ I,
yn −→ ∞ as n −→ ∞.
X
3.54
Then by 3.38,
yn −→ ∞,
∞
n −→ ∞.
3.55
14
Abstract and Applied Analysis
and subsequently
minyn t ≥ c1 yn ∞ −→ ∞.
Ì2
t∈
3.56
This together with 3.32 and f0 ∈ 0, ∞ and f∞ 0 that
vn |Ì2
∞ −→ 0,
n −→ ∞.
3.57
However, this contradicts with the fact that
n ∈ Æ.
minvn t≥ c1 ,
Ì2
t∈
3.58
Therefore, 3.53 holds.
Step 3. Fixed λ such that
λ>
s
inf
s∈0,c1 B c1 γ∗ fs
.
3.59
Then there exists l ∈ 0, c1 B such that
λ>
l
.
γ∗ c1 fl
3.60
We show that there is no η, y ∈ C such that
y l
∞
c1
η>
l
.
γ∗ c1 fl
3.61
Suppose on the contrary that there exists η, y ∈ C satisfying 3.61. Then for t ∈ Ì2 ,
yt η
T
Kt, shsf ys
s2
≥η
T
Kt, shsf c1 y∞
s2
η
3.62
T
Kt, shsfl
s2
≥ ηγ∗ fl ηγ∗
fl
· l,
l
and subsequently, η ≤ l/c1 γ∗ fl. Therefore, there is no η, y ∈ C such that 3.61 holds.
Abstract and Applied Analysis
15
Now, combining the conclusions in Steps 2 and 3, using the fact that no μ, u ∈ C
satisfying 3.61, it concludes that for every λ ∈ l/c1 γ∗ fl, λ1 /f0 , 1.4, 1.8 has at least
two generalized positive solutions in C . For arbitrary λ ∈ infs∈0,c1 B s/c1 γ∗ fs, ∞, we
may find l lλ satisfying 3.60. So, for every λ ∈ infs∈0,c1 B s/γ∗ fs, λ1 /f0 , 1.4, 1.8
has at least two generalized positive solutions in C .
4. Some Examples
In this section, we will apply our results to two examples.
For convenience, set T 12, then Ì1 {1, 2, . . . , 13},
Ì2 {2, 3, . . . , 12}.
Example 4.1. Let us consider the boundary value problem
Δ4 ut − 2 λfut,
t ∈ Ì2 ,
u1 u13 Δ2 u0 Δ2 u12 0,
4.1
where
fu ⎧
⎨
arctan u,
u ∈ 0, 1000,
⎩ u − 10002 arctan 1000,
u ∈ 1000, ∞.
4.2
Clearly, fu is nondecreasing, f0 1, f∞ ∞. Take B 1000. By a simple computation, it
follows that infs∈0,1000 fs/s arctan 1000/1000 ≈ 0.00157, λ1 16 sin4 π/24 ≈ 0.0048
and γ ∗ maxt∈Ì1 12
s2 Kt, s 1629/6, then
0.0048 ≈ 16 sin4
λ1
π
s
6
< sup
≈ 2.34631.
24 f0 s∈0,1000 fsγ ∗ 1629 infs∈0,1000 fs/s
4.3
So, Theorem 1.3i implies that 4.1 has at least one generalized positive solution for
0<λ<
λ1
≈ 0.0048;
f0
4.4
Theorem 1.3ii implies that 4.1 has at least two generalized positive solutions for
1
λ1
<λ< ≈ 2.34631.
4
f0
γ ∗ 16sin π/24 − 1
4.5
Example 4.2. Let us consider the boundary value problem:
Δ4 ut − 2 λfut,
t ∈ Ì2 ,
u1 u13 Δ2 u0 Δ2 u12 0,
4.6
16
Abstract and Applied Analysis
where
⎧
⎪
⎪
⎨
eu − 1
,
u
fu 30
√
⎪
e −1
⎪
⎩ u − 30 ,
30
u ∈ 0, 30,
4.7
u ∈ 30, ∞.
Obviously, fu
is nondecreasing in 0, ∞, so f0 limu → 0 fu/u
1, f∞ 0. By a simple computation,
that λ1 16 sin4 π/24 and
limu → 0 fu/u
√ it follows
√
12
γ∗ mint∈Ì2 s2 Kt, s 143/2, c1 216 3/145 145 ≈ 0.214. Take B 50. Since
e10.715 − 1/10.715 ≈ 4202.0726, it follows that
sups∈0,10.715 fs/s
0.0000155 ≈
1
s
π
λ1
≈ 0.0048.
<
16 sin4
s∈0,c1 B c γ fs
24
c1 γ∗ f0
sups∈0,10.715 fs/s
1 ∗
inf
4.8
Therefore, i of Theorem 1.4 implies that 4.6 has at least one generalized positive solution
for
λ>
inf
s
s∈0,c1 B c γ fs
1 ∗
≈ 0.0000155;
4.9
ii of Theorem 1.4 implies that 4.6 has at least two generalized positive solutions for
0.0000155 < λ < 16 sin4
π
.
24
4.10
Acknowledgments
This paper was written when the first author visited Tabuk University, Tabuk, during May 16June 13, 2011 and he is very thankful to the administration of Tabuk University for providing
him the hospitalities during the stay. The second author gratefully acknowledges the
partial financial support from the Deanship of Scientific Research DSR at King Abdulaziz
University, Jeddah. The authors thank the referees for their valuable comments.
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