Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2012, Article ID 293734, 19 pages
doi:10.1155/2012/293734
Research Article
Positive Solutions for p-Laplacian
Fourth-Order Differential System with Integral
Boundary Conditions
Jiqiang Jiang,1 Lishan Liu,1, 2 and Yonghong Wu2
1
2
School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia
Correspondence should be addressed to Jiqiang Jiang, qfjjq@163.com
and Lishan Liu, lls@mail.qfnu.edu.cn
Received 25 March 2012; Accepted 15 May 2012
Academic Editor: Cengiz Çinar
Copyright q 2012 Jiqiang Jiang 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 paper investigates the existence of positive solutions for a class of singular p-Laplacian fourthorder differential equations with integral boundary conditions. By using the fixed point theory in
cones, explicit range for λ and μ is derived such that for any λ and μ lie in their respective interval,
the existence of at least one positive solution to the boundary value system is guaranteed.
1. Introduction
Boundary value problems for ordinary differential equations arise in different areas of
applied mathematics and physics and so on. Fourth-order differential equations boundary
value problems, including those with the p-Laplacian operator, have their origin in beam
theory 1, 2, ice formation 3, 4, fluids on lungs 5, brain warping 6, 7, designing special
curves on surfaces 8, and so forth. In beam theory, more specifically, a beam with a small
deformation, a beam of a material that satisfies a nonlinear power-like stress and strain law,
and a beam with two-sided links that satisfies a nonlinear powerlike elasticity law can be
described by fourth order differential equations along with their boundary value conditions.
For more background and applications, we refer the reader to the work by Timoshenko 9
on elasticity, the monograph by Soedel 10 on deformation of structure, and the work by
Dulcska 11 on the effects of soil settlement. Due to their wide applications, the existence and
multiplicity of positive solutions for fourth-order including p-Laplacian operator boundary
value problems has also attracted increasing attention over the last decades; see 12–33 and
2
Discrete Dynamics in Nature and Society
references therein. In 28, Zhang and Liu studied the following singular fourth-order fourpoint boundary value problem
ft, ut,
φp u t
0 < t < 1,
u0 u1 − auξ u 0 u 1 − bu η 0,
1.1
where φp x |x|p−2 x, p > 1, 0 < ξ, η < 1, 0 ≤ a, b < 1, f ∈ C0, 1 × 0, ∞, 0, ∞, ft, x
may be singular at t 0 and/or t 1 and x 0. The authors gave sufficient conditions for the
existence of one positive solution by using the upper and lower solution method, fixed point
theorems, and the properties of the Green function.
In 32, Zhang et al. discussed the existence and nonexistence of symmetric positive
solutions of the following fourth-order boundary value problem with integral boundary
conditions:
wtft, ut, 0 < t < 1,
φp u t
1
gsusds,
u0 u1 0
φp u 0 φp u 1 1
1.2
hsφp u s ds,
0
where φp x |x|p−2 x, p > 1, w ∈ L1 0, 1 is nonnegative, symmetric on the interval 0, 1, f :
0, 1 × 0, ∞ → 0, ∞ is continuous, f1 − t, x ft, x for all t, x ∈ 0, 1 × 0, ∞,
and g, h ∈ L1 0, 1 are nonnegative, symmetric on 0, 1.
Motivated by the work of the above papers, in this paper, we study the existence of
positive solutions of the following singular fourth-order boundary value system with integral
boundary conditions:
λp1 −1 a1 tf1 t, ut, vt, 0 < t < 1,
φp1 u t
μp2 −1 a2 tf2 t, ut, vt,
φp2 v t
1
usdξ1 s,
u0 u1 0
φp1 u 0 φp1 u 1 v0 v1 1
0
φp2 v 0 φp2 v 1 1
φp1 u s dη1 s,
1.3
0
vsdξ2 s,
1
φp2 v s dη2 s,
0
where λ and μ are positive parameters, φpi x |x|pi −2 x, pi > 1, φqi φp−1i , 1/pi 1/qi 1,
ξi , ηi : 0, 1 → R i 1, 2 are nondecreasing functions of bounded variation, and
Discrete Dynamics in Nature and Society
3
the integrals in 1.3 are Riemann-Stieltjes integrals, f1 : 0, 1 × R0 × R → R and
f2 : 0, 1 × R × R0 → R are two continuous functions, and f1 t, x, y may be singular
at x 0 while f2 t, x, y may be singular at y 0; a1 , a2 : 0, 1 → R are continuous and
may be singular at t 0 and/or t 1, in which R 0, ∞, R0 0, ∞.
Compared to previous results, our work presented in this paper has the following new
features. Firstly, our study is on singular nonlinear differential systems, that is, a1 and a2 in
1.3 are allowed to be singular at t 0 and/or t 1, meanwhile f1 t, x, y is allowed to be
singular at x 0 while f2 t, x, y is allowed to be singular at y 0, which bring about many
difficulties. Secondly, the main tools used in this paper is a fixed-point theorem in cones,
and the results obtained are the conditions for the existence of solutions to the more general
system 1.3. Thirdly, the techniques used in this paper are approximation methods, and a
special cone has been developed to overcome the difficulties due to the singularity and to
apply the fixed-point theorem. Finally, we discuss the boundary value problem with integral
boundary conditions, that is, system 1.3 including fourth-order three-point, multipoint and
nonlocal boundary value problems as special cases. To our knowledge, very few authors
studied the existence of positive solutions for p-Laplacian fourth-order differential equation
with boundary conditions involving Riemann-Stieltjes integrals. Hence we improve and
generalize the results of previous papers to some degree, and so it is interesting and important
to study the existence of positive solutions for system 1.3.
The rest of this paper is organized as follows. In Section 2, we present some lemmas
that are used to prove our main results. In Section 3, the existence of positive solution for
system 1.3 is established by using the fixed point theory in cone. Finally, in Section 4, one
example is also included to illustrate the main results.
Definition 1.1. A vector u, v ∈ C2 0, 1∩C4 0, 1×C2 0, 1∩C4 0, 1 is said to be a positive
solution of system 1.3 if and only if u, v satisfies 1.3 and ut > 0, vt ≥ 0 or ut ≥ 0,
vt > 0 for any t ∈ 0, 1.
Let K be a cone in a Banach space E. For 0 < r < R < ∞, let Kr {x ∈ K : x < r},
∂Kr {x ∈ K : x r}, and K r,R {x ∈ K : r ≤ x ≤ R}. The proof of the main theorem
of this paper is based on the fixed point theory in cone. We list one lemma 34, 35 which is
needed in our following argument.
Lemma 1.2. Let K be a positive cone in real Banach space E and T : K r,R → K a completely
continuous operator. If the following conditions hold
i T x ≤ x for x ∈ ∂KR ;
ii there exists e ∈ ∂K1 such that x /
T x me for any x ∈ ∂Kr and m > 0. Then T has a fixed
point in K r,R .
Remark 1.3. If i and ii are satisfied for x ∈ ∂Kr and x ∈ ∂KR , respectively. Then Lemma 1.2
is still true.
2. Preliminaries and Lemmas
The basic space used in this paper is E C0, 1 × C0, 1. Obviously, the space E is a Banach
space if it is endowed with the norm as follows:
u, v : u v,
u max |ut|,
0≤t≤1
v max |vt|
0≤t≤1
2.1
4
Discrete Dynamics in Nature and Society
for any u, v ∈ E. Denote C 0, 1 {u ∈ C0, 1 : ut ≥ 0, 0 ≤ t ≤ 1}. For convenience, we
list the following assumptions:
H1 a1 , a2 : 0, 1 → R are continuous and
1
0 < Li :
qi −1
< ∞,
esai sds
i 1, 2,
2.2
0
where es s1 − s, s ∈ 0, 1.
H2 ξi , ηi : 0, 1 → R i 1, 2 are nondecreasing functions of bounded variation, and
αi ∈ 0, 1, βi ∈ 0, 1, where
αi 1
dξi s, βi 0
1
dηi s,
i 1, 2.
2.3
0
H3 f1 : 0, 1 × R0 × R → R , f2 : 0, 1 × R × R0 → R are continuous and satisfy
f1 t, x, y ≤ g1 t, x h1 t, y , ∀ t, x, y ∈ 0, 1 × R0 × R ,
2.4
f2 t, x, y ≤ g2 t, x h2 t, y , ∀ t, x, y ∈ 0, 1 × R × R0 ,
where g1 , h2 : 0, 1 × R0 → R are continuous and nonincreasing in the second variable, and
g2 , h1 : 0, 1 × R → R are continuous and for any constant r > 0,
1
0<
esa1 sg1 s, rds < ∞,
1
0<
0
esa2 sh2 s, rds < ∞.
2.5
0
Similar to the proof of Lemmas 2.1 and 2.2 in 32, the following two lemmas are valid.
Lemma 2.1. If H2 holds, then for any y ∈ L0, 1, the boundary value problem
−x t φqi yt , 0 < t < 1,
1
xsdξi s
x0 x1 2.6
0
has a unique solution
xt 1
Hi t, sφqi ys ds,
2.7
0
where
1
1
Hi t, s Gt, s Gτ, sdξi τ, i 1, 2,
1 − αi 0
s1 − t, 0 ≤ s ≤ t ≤ 1,
Gt, s t1 − s, 0 ≤ t ≤ s ≤ 1.
2.8
Discrete Dynamics in Nature and Society
5
Lemma 2.2. If H2 holds, then for any z ∈ L0, 1, the boundary value problem
−y t zt, 0 < t < 1,
1
ysdηi s
y0 y1 2.9
0
has a unique solution
yt 1
2.10
Ki t, szsds,
0
where
1
Ki t, s Gt, s 1 − βi
1
Gτ, sdηi τ,
i 1, 2.
2.11
1
.
4
2.12
0
Remark 2.3. For t, s ∈ 0, 1, we have
etes ≤ Gt, s ≤ es
or et ≤ max et t∈0,1
Remark 2.4. If H2 holds, it is easy to testify Hi t, s defined by 2.8 that:
ρi es ≤ Hi t, s ≤ γi es ≤
1
γi < γi ,
4
t, s ∈ 0, 1, i 1, 2,
2.13
where
1
γi ,
1 − αi
1
ρi 0
eτdξi τ
1 − αi
,
i 1, 2.
2.14
Remark 2.5. From 2.11, we can prove that the properties of Ki t, s i 1, 2 are similar to
those of Hi t, s i 1, 2.
Lemma 2.6. For x > 0, y > 0, we have
2qi −1 φqi x φqi y , qi ≥ 2
,
1 < qi < 2
φqi x φqi y ,
≤ 2qi −1 φqi x φqi y ,
qi > 1, i 1, 2,
φqi x > φqi y > φqi 0 0, x > y > 0, qi > 1, i 1, 2.
φqi x y ≤
Proof. The proof of this lemma is easy, and we omit it.
2.15
2.16
6
Discrete Dynamics in Nature and Society
Let
K u, v ∈ C 0, 1 × C 0, 1 : u, v are concave on 0, 1,
2.17
min ut ≥ Λu, min vt ≥ Λv ,
t∈0,1
t∈0,1
where
⎧
⎫
⎨ ρ1 σ q1 −1 ρ2 σ q2 −1 ⎬
2
1
Λ min
,
,
⎩ γ νq1 −1 γ νq2 −1 ⎭
1 1
1
0
σi esdηi s
1 − βi
2 2
, νi 1
, i 1, 2.
1 − βi
2.18
It is easy to see that K is a cone of E. For any 0 < r < R, let Kr,R {u, v ∈ K : r <
u < R, r < v < R}.
Remark 2.7. By the definition of ρi , σi , γi , νi i 1, 2, we have 0 < Λ < 1.
To overcome singularity, we consider the following approximate problem of 1.3:
λp1 −1 a1 tf1n t, ut, vt,
φp1 u t
0 < t < 1,
μp2 −1 a2 tf2n t, ut, vt,
φp2 v t
u0 u1 1
usdξ1 s,
0
φp1 u 0 φp1 u 1 1
φp1 u s dη1 s,
2.19
0
v0 v1 1
vsdξ2 s,
0
φp2 v 0 φp2 v 1 1
φp2 v s dη2 s,
0
where n is a positive integer and
f1n t, u, v f1 t, max u, n−1 , v ,
f2n t, u, v f2 t, u, max v, n−1 .
Clearly, fin ∈ C0, 1 × R × R , R i 1, 2.
2.20
Discrete Dynamics in Nature and Society
7
By Lemmas 2.1 and 2.2, for each n ∈ N, λ > 0, μ > 0, let us define operators Aλn : K →
μ
C0, 1, Bn : K → C0, 1, and Tn : K → E by
Aλn u, vt
μ
Bn u, vt
1
λ
μ
1
H1 t, sφq1
0
0
1
1
H2 t, sφq2
0
K1 s, τa1 τf1n τ, uτ, vτdτ ds,
2.21
K2 s, τa2 τf2n τ, uτ, vτdτ ds,
2.22
0
μ
and Tn u, v Aλn u, v, Bn u, v, respectively.
Lemma 2.8. Assume that H1 –H3 hold, then for each λ > 0, μ > 0, n ∈ N, Tn : K r,R → K is a
completely continuous operator.
Proof. Let λ > 0, μ > 0, and n ∈ N be fixed. For any u, v ∈ K, by 2.21, we have
Aλn u, v
t −λφq1
1
K1 t, τa1 τf1n τ, uτ, vτdτ
≤ 0,
0
2.23
Aλn u, v0 Aλn u, v1
λ
1
H1 0, sφq1
1
0
K1 s, τa1 τf1n τ, uτ, vτdτ ds ≥ 0,
0
which implies that Aλn is nonnegative and concave on 0, 1. Similarly, by 2.22 we can obtain
μ
that Bn is nonnegative and concave on 0, 1. For any u, v ∈ K and t ∈ 0, 1, it follows from
2.13 that
Aλn u, vt
λ
1
H1 t, sφq1
1
0
K1 s, τa1 τf1n τ, uτ, vτdτ ds
0
q −1
≤λγ1 ν1 1
1
1
esφq1
0
2.24
eτa1 τf1n τ, uτ, vτdτ ds.
0
Thus
1
q1 −1
λ
esφq1
An u, v ≤ λγ1 ν1
0
1
0
eτa1 τf1n τ, uτ, vτdτ ds.
2.25
8
Discrete Dynamics in Nature and Society
On the other hand, by 2.13 and 2.18, we have
Aλn u, vt
λ
1
H1 t, sφq1
1
0
q −1
≥λρ1 σ1 1
K1 s, τa1 τf1n τ, uτ, vτdτ ds
0
1
1
esφq1
0
eτa1 τf1n τ, uτ, vτdτ ds
0
2.26
q −1
ρ1 σ1 1 λ
λ
≥
Λ
≥
v
v
u,
u,
A
.
A
n
n
q1 −1
γ1 ν1
This implies that
min Aλn u, vt ≥ ΛAλn u, v.
2.27
μ
μ
min Bn u, vt ≥ ΛBn u, v.
2.28
t∈0,1
Similar to 2.27, we also have
t∈0,1
Therefore, Tn K ⊂ K.
Next, we prove that Tn : K r,R → K is completely continuous. Suppose um , vm ∈
K r,R and u0 , v0 ∈ K r,R with um , vm − u0 , v0 → 0 m → ∞. We notice that t ∈
0, 1 fin t, um t, vm t − fin t, u0 t, v0 t → 0 m → ∞. Using the Lebesgue dominated
convergence theorem, we have
1
p1 −1
K1 s, τa1 τf1n τ, um τ, vm τdτ
φq1
0
p −1
−φq11
1
≤ ν1
0
1
K1 s, τa1 τf1n τ, u0 τ, v0 τdτ eτa1 τf1n τ, um τ, vm τ − f1n τ, u0 τ, v0 τdτ −→ 0,
0
m −→ ∞.
2.29
Therefore,
λ
An um , vm − Aλn u0 , v0 1
1
≤ λγ1 esφq1
K1 s, τa1 τf1n τ, um τ, vm τdτ
0
0
1
K1 s, τa1 τf1n τ, u0 τ, v0 τdτ ds −→ 0, m −→ ∞.
−φq1
0
2.30
Discrete Dynamics in Nature and Society
9
Similarly, we also have
μ
μ
Bn um , vm − Bn u0 , v0 −→ 0,
m −→ ∞.
2.31
β
So Aλn : K r,R → C0, 1 and Bn : K r,R → C0, 1 are continuous. Therefore, Tn : K r,R → K is
also continuous.
Let D ⊂ K r,R be any bounded set, then for any u, v ∈ D, we have u, v ∈ K, r ≤
u ≤ R, r ≤ v ≤ R, and then 0 < Λr ≤ uτ ≤ R, 0 < Λr ≤ vτ ≤ R for any τ ∈ 0, 1. By
H3 , we have
1
Lr :
q1 −1
2.32
< ∞.
eτa1 τg1 s, rΛdτ
0
It is easy to show that Aλn D is uniformly bounded. In order to show that Tn is a compact
operator, we only need to show that Aλn D is equicontinuous. By the uniformly continuity
of H1 t, s on 0, 1 × 0, 1, for all ε > 0, there is δ > 0 such that for any t1 , t2 , s ∈ 0, 1 and
|t1 − t2 | < δ, we have
|H1 t1 , s − H1 t2 , s| < ε.
2.33
This together with 2.15 and 2.32 implies
λ
An u, vt1 − Aλn u, vt2 ≤λ
1
|H1 t1 , s − H1 t2 , s|φq1
0
<
≤
q −1
ελν1 1 φq1
q −1
ελν1 1 φq1
K1 s, τa1 τf1n τ, uτ, vτdτ ds
0
1
eτa1 τ g1 τ, max uτ, n−1 h1 τ, vτ dτ
0
1
eτa1 τ g1 τ, rΛ h1 τ, vτ dτ
q −1
ελν1 1 2q1 −1
0
1
≤
1
φq1
eτa1 τg1 τ, rΛdτ
0
⎡
φq1
1
eτa1 τh1 τ, vτdτ
0
⎞q1 −1 ⎤
q −1
⎢
⎥
≤ ελν1 1 2q1 −1 ⎣Lr L1 ⎝ max h1 τ, y ⎠ ⎦,
τ∈0,1
⎛
|t1 − t2 | < δ, u, v ∈ D.
y∈rΛ,R
2.34
This means that Aλn D is equicontinuous. By the Arzela-Ascoli theorem, Aλn D is a relatively
compact set and that Aλn : K r,R → C0, 1 is a completely continuous operator.
10
Discrete Dynamics in Nature and Society
μ
In the same way, we can show that Bn : K r,R → C0, 1 is also completely continuous,
and so Tn : K r,R → K is completely continuous. Now since λ, μ, and n are given arbitrarily,
the conclusion of this lemma is valid.
3. Main Results
For notational convenience, we denote by
−1
Mi 6 ρi σ qi −1 ΛLi ,
q −1 −1
Ni γi νi i Li ,
⎞
q1 −1
f
t,
x,
y
1
⎟
⎜
f1α ⎝lim sup sup
⎠ ,
φp1 x
t∈0,1
x→α
⎛
y∈R
⎞q1 −1
f1 t, x, y
⎠ ,
⎝lim inf inf
x → α t∈0,1 φp1 x
⎛
⎞q2 −1
f
t,
x,
y
2
f2α ⎝lim sup sup
⎠ ,
y → α t∈0,1 φp2 y
x∈R
⎛
f1α
i 1, 2,
3.1
q2 −1
f2 t, x, y
lim inf inf
,
y → α t∈0,1 φp y
2
x∈R
f2α y∈R
where α denotes 0 or ∞. The main results of this paper are the following.
Theorem 3.1. Assume that H1 –H3 hold. Then we have:
C1 If f10 , f1∞ , f20 ∈ 0, ∞ and M1 /f1∞ < N1 /f10 , then for each λ ∈ M1 /f1∞ , N1 /f10 ,
μ ∈ 0, N2 /f20 , the system 1.3 has at least one positive solution.
C2 If f10 , f20 , f2∞ ∈ 0, ∞ and M2 /f2∞ < N2 /f20 , then for each λ ∈ 0, N1 /f10 , μ ∈
M2 /f2∞ , N2 /f20 , the system 1.3 has at least one positive solution.
C3 If f10 0, f1∞ ∞, 0 < f20 < ∞, then for each λ ∈ 0, ∞, μ ∈ 0, N2 /f20 , the system
1.3 has at least one positive solution.
C4 If 0 < f10 < ∞, f20 0, f2∞ ∞, then for each λ ∈ 0, N1 /f10 , μ ∈ 0, ∞, the system
1.3 has at least one positive solution.
C5 If fi0 0, fi∞ ∞ i 1, 2, then for each λ ∈ 0, ∞, μ ∈ 0, ∞, the system 1.3 has at
least one positive solution.
C6 If 0 < f10 < ∞, f1∞ ∞ or f2∞ ∞, 0 < f20 < ∞, then for each λ ∈ 0, N1 /f10 ,
μ ∈ 0, N2 /f20 , the system 1.3 has at least one positive solution.
C7 If f10 0, 0 < f1∞ < ∞, and f10 0, 0 < f2∞ < ∞, then for each λ ∈ M1 /f1∞ , ∞,
μ ∈ 0, ∞ or λ ∈ 0, ∞, μ ∈ M2 /f2∞ , ∞, the system 1.3 has at least one positive
solution.
Proof. We only prove the condition in which C1 holds. The other cases can be proved
similarly.
Let λ ∈ M1 /f1∞ , N1 /f10 , μ ∈ 0, N2 /f20 , choose ε1 > 0 such that f1∞ − ε1 > 0 and
M1
N1
≤λ≤ 0
,
f1∞ − ε1
f1 ε1
0<μ≤
N2
.
ε1
f20
3.2
Discrete Dynamics in Nature and Society
11
It follows from fi0 ∈ 0, ∞ of C1 that there exists r1 > 0 such that for any t ∈ 0, 1,
p1 −1
p1 −1
f1 t, x, y ≤ f10 ε1
φp1 x ≤ r1 f10 ε1
,
p2 −1 p2 −1
f2 t, x, y ≤ f20 ε1
φp2 y ≤ r1 f20 ε1
,
0 < x ≤ r1 , y ≥ 0,
3.3
x ≥ 0, 0 < y ≤ r1 .
3.4
Let Kr1 {u, v ∈ K : u < r1 , v < r1 }. For any u, v ∈ ∂Kr1 , n > 1/r1 , by 2.13, 3.3, we
have
1
λ
max
λ
H1 t, sφq1
v
u,
An
t∈0,1
≤
0
q −1
λγ1 ν1 1 φq1
q
≤ λγ1 ν1 1
1
−1
1
0
K1 s, τa1 τf1n τ, uτ, vτdτ ds
0
p −1
eτa1 τ f10 ε1 1 φp1 uτdτ
3.5
f10 ε1 L1 r1
λN1−1 f10 ε1 r1 .
Similarly, we also have
μ
Bn u, v ≤ μN2−1 f20 ε1 r1 .
3.6
Therefore, we have
μ
Tn u, v Aλn u, v Bn u, v
≤ λN1−1 f10 ε1 μN2−1 f20 ε1 r1
3.7
≤ 2r1 u, v.
On the other hand, by f1∞ > f1∞ − ε1 > 0, there exists R0 > 0 such that
p −1
f1 t, x, y ≥ f1∞ − ε1 1 φp1 x,
t ∈ 0, 1, x ≥ R0 , y ≥ 0.
3.8
Let R1 > max{2r1 , Λ−1 R0 }, KR1 {u, v ∈ K : u < R1 , v < R1 }. Next, we take ϕ1 , ϕ2 1, 1 ∈ ∂K1 , and for any u, v ∈ ∂KR1 , m > 0, n ∈ N, we will show
Aλn u, v m ϕ1 , ϕ2 .
u, v /
3.9
Otherwise, there exist u0 , v0 ∈ ∂KR1 and m0 > 0 such that
u0 , v0 Aλn u0 , v0 m0 ϕ1 , ϕ2 .
3.10
12
Discrete Dynamics in Nature and Society
From u0 , v0 ∈ ∂KR1 , we know that u0 R1 or v0 R1 . Without loss of generality, we
may suppose that u0 R1 , then u0 τ ≥ Λu0 ΛR1 > R0 for any τ ∈ 0, 1. So, by 2.13,
3.8, we have
u0 t λ
1
H1 t, sφq1
1
0
K1 s, τa1 τf1n τ, u0 τ, v0 τdτ ds m0
0
q −1
≥λρ1 σ1 1
q −1
≥λρ1 σ1 1
q −1
≥λρ1 σ1 1
1
1
esφq1
0
0
1
1
esφq1
0
0
1
1
esφq1
0
eτa1 τf1n τ, u0 τ, v0 τdτ ds m0
eτa1 τ f1∞ − ε1
p1 −1
φp1 u0 τ dτ ds m0
3.11
p −1
eτa1 τ f1∞ − ε1 1 ΛR1 p1 −1 dτ ds m0
0
1
q −1 λρ1 σ1 1 f1∞ − ε1 ΛR1 L1 m0
6
λM1−1 f1∞ − ε1 R1 m0 > R1 .
This implies that R1 > R1 , which is a contradiction. This yields that 3.9 holds. By 3.7, 3.9,
and Lemma 1.2, for any n > 1/r1 and λ ∈ M1 /f1∞ , N1 /f10 , μ ∈ 0, N2 /f20 , we obtain that
Tn has a fixed point un , vn in K r1 ,R1 satisfying r1 < un < R1 ,r1 < vn < R1 .
Let {un , vn }n≥n1 be the sequence of solutions of boundary value problems 2.19,
where n1 > 1/r1 is a fixed integer. It is easy to see that they are uniformly bounded. Next
we show that {un }n≥n1 are equicontinuous on 0, 1. From un , vn ∈ K r1 ,R1 , we know that
R1 ≥ un τ ≥ Λun ≥ Λr1 , R1 ≥ vn τ ≥ Λvn ≥ Λr1 , τ ∈ 0, 1. For any ε > 0, by the
continuous of H1 t, s in 0, 1 × 0, 1, there exists δ1 > 0 such that for any t1 , t2 , s ∈ 0, 1 and
|t1 − t2 | < δ1 , we have
|H1 t1 , s − H1 t2 , s| < ε.
3.12
This combining with 2.15, 2.32 implies that for any t1 , t2 ∈ 0, 1 and |t1 − t2 | < δ1 , we have
|un t1 − un t2 |
≤λ
1
|H1 t1 , s − H1 t2 , s|φq1
0
<
q −1
ελν1 1 φq1
1
K1 s, τa1 τf1n τ, un τ, vn τdτ ds
0
1
0
eτa1 τ g1 τ, max un τ, n−1 h1 τ, vn τ dτ
Discrete Dynamics in Nature and Society
1
≤
q −1
ελν1 1 2q1 −1
eτa1 τg1 τ, rΛdτ
φq1
13
φq1
0
⎡
1
eτa1 τh1 τ, vτdτ
0
⎞q1 −1 ⎤
q −1
⎢
⎥
≤ ελν1 1 2q1 −1 ⎣Lr L1 ⎝ max h1 τ, y ⎠ ⎦.
τ∈0,1
⎛
y∈r1 Λ,R1 3.13
Similarly, {vn }n≥n1 are also equicontinuous on 0, 1. By the Ascoli-Arzela theorem,
the sequence {un , vn }n≥n1 has a subsequence being uniformly convergent on 0, 1. From
Lemma 2.2, we know that
un s λp1 −1
vn s
μ
p2 −1
1
K1 s, τa1 τf1n τ, un τ, vn τdτ,
0
1
3.14
K2 s, τa2 τf2n τ, un τ, vn τdτ.
0
Since the properties of Ki t, s i 1, 2 are similar to those of Hi t, s i 1, 2, so un , vn have the similar properties of un , vn , that is, un , vn also has a subsequence being uniformly
convergent on 0, 1. Without loss of generality, we still assume that {un , vn }n≥n1 itself
uniformly converges to u, v on 0, 1 and {un , vn }n≥n1 itself uniformly converges to u , v on 0, 1, respectively. Since {un , vn }n≥n1 ∈ K r1 ,R1 ⊂ K, so we have un ≥ 0, vn ≥ 0. By 2.19,
we have
* ) *
) * )
1 1
1
t−
u
2
2 n 2
t
s
)
) * )
) *
*
1 p1 −1 1
p1 −1 1
−
ds
φq1 un
s2 −
un
2
2
2
1/2
1/2
s2
s1
−
ds1
λp1 −1 a1 τf1n τ, un τ, vn τdτ ds2 , t ∈ 0, 1,
un t un
1/2
1/2
) * )
1
vn t vn
t−
2
s
t
ds
−
* ) *
1 1
v
2 n 2
*
)
) * )
) *
1 p2 −1 1
p2 −1 1
s2 −
vn
φq2 vn
2
2
2
1/2
1/2
s2
s1
*
−
ds1
μp2 −1 a2 τf2n τ, un τ, vn τdτ ds2 ,
1/2
3.15
3.16
t ∈ 0, 1.
1/2
From 3.15 and 3.16, we know that {un 1/2}n≥n1 , {vn 1/2}n≥n1 , {un 1/2}n≥n1 ,
{vn 1/2}n≥n1 , {u
n 1/2}n≥n1 , {vn 1/2}n≥n1 are bounded sets. Without loss of gener
ality, we may assume un 1/2, vn 1/2 → c1 , d1 , un 1/2, vn 1/2 → c2 , d2 ,
14
Discrete Dynamics in Nature and Society
u
n 1/2, vn 1/2 → c3 , d3 as n → ∞. Then by 3.15, 3.16, and the Lebesgue dominated
convergence theorem, we have
s
* t
*
) *
)
)
)
1
1
1
p −1
p −1
c1 t −
−
ut u
ds
φq1 c2 1 c3 1
s2 −
2
2
2
1/2
1/2
s2
s1
−
ds1
λp1 −1 a1 τf1 τ, uτ, vτdτ ds2 , t ∈ 0, 1,
1/2
3.17
1/2
) *
)
*
1
1
vt v
d1 t −
2
2
s
t
*
)
1
p −1
p −1
ds
φq2 d2 2 d3 2
s2 −
−
2
1/2
1/2
s2
s1
−
ds1
μp2 −1 a2 τf2 τ, uτ, vτdτds2
1/2
3.18
t ∈ 0, 1.
1/2
By 3.17 and 3.18, direct computation shows that
φp1 u t
λp1 −1 a1 tf1 t, ut, vt,
μp2 −1 a2 tf2 t, ut, vt,
φp2 v t
3.19
0 < t < 1.
On the other hand, u, v satisfies the boundary condition of 1.3. In fact, un 0 1
1
un 1 0 un sdξ1 s, vn 0 vn 1 0 vn sdξ2 s, φp1 un 0 φp1 un 1 1
1
φ un sdη1 s, φp2 vn 0 φp2 vn 1 0 φp2 vn sdη2 s, and so the conclusion
0 p1
holds by letting n → ∞.
Theorem 3.2. Assume that H1 –H3 hold. Then we have:
D1 If f10 , f1∞ , f2∞ ∈ 0, ∞ and M1 /f10 < N1 /f1∞ , then for each λ ∈ M1 /f10 , N1 /f1∞ ,
μ ∈ 0, N2 / f2∞ , the system 1.3 has at least one positive solution.
D2 If f1∞ , f20 , f2∞ ∈ 0, ∞ and M2 /f20 < N2 /f2∞ , then for each λ ∈ 0, N1 /f1∞ , μ ∈
M2 /f20 , N2 /f2∞ , the system 1.3 has at least one positive solution.
D3 If f10 ∞, f1∞ 0, 0 < f2∞ < ∞, then for each λ ∈ 0, ∞, μ ∈ 0, N2 /f2∞ , the system
1.3 has at least one positive solution.
D4 If 0 < f1∞ < ∞, f20 ∞, f2∞ 0, then for each λ ∈ 0, N1 /f1∞ , μ ∈ 0, ∞, the system
1.3 has at least one positive solution.
D5 If fi0 ∞, fi∞ 0 i 1, 2, then for each λ ∈ 0, ∞, μ ∈ 0, ∞, the system 1.3 has at
least one positive solution.
D6 If 0 < f1∞ < ∞, f10 ∞ or f20 ∞, 0 < f2∞ < ∞, then for each λ ∈ 0, N1 /f1∞ ,
μ ∈ 0, N2 /f2∞ , the system 1.3 has at least one positive solution.
D7 If f1∞ 0, 0 < f10 < ∞, and f2∞ 0, 0 < f20 < ∞, then for each λ ∈ M1 /f10 , ∞,
μ ∈ 0, ∞ or λ ∈ 0, ∞, μ ∈ M2 /f20 , ∞, the system 1.3 has at least one positive
solution.
Discrete Dynamics in Nature and Society
15
Proof. We may suppose that condition D1 holds. Similarly, we can prove the other cases.
Let λ ∈ M1 /f10 , N1 /f1∞ , μ ∈ 0, N2 /f2∞ . We can choose ε2 > 0 such that N1 − ε2 > 0,
N2 − ε2 > 0 and
λf1∞ < N1 − ε2 ,
μf2∞ < N2 − ε2 .
3.20
It follows from D1 and 2.16 that there exists R∗2 > 0 such that for any t ∈ 0, 1
)
*p1 −1
1
f1 t, x, y ≤
φp1 x,
N1 − ε2 λ
*p2 −1
)
1
f2 t, x, y ≤
φp2 y ,
N2 − ε2 λ
x ≥ R∗2 , y ≥ 0,
3.21
x ≥ 0, y ≥ R∗2 .
3.22
Let R2 Λ−1 R∗2 , KR2 {u, v ∈ K : u < R2 , v < R2 }. For any u, v ∈ ∂KR2 , n ∈ N, by
2.13, 3.21, we have
1
λ
An u, v max λ H1 t, sφq1
t∈0,1
≤
0
q −1
λγ1 ν1 1 φq1
q −1 1
≤ λγ1 ν1 1
λ
1
0
1
K1 s, τa1 τf1n τ, uτ, vτdτ ds
0
)
*p1 −1
1
eτa1 τ
φp1 uτdτ
N1 − ε2 λ
3.23
N1 − ε2 L1 R2 < R2 .
μ
Similarly, by 3.22 we have Bn u, v < R2 . Therefore,
μ
Tn u, v Aλn u, v Bn u, v ≤ 2R2 u, v,
u, v ∈ ∂KR2 , n ∈ N.
3.24
On the other hand, choose ε3 > 0 such that M1 ε3 < λf10 . By the condition f10 ∈ 0, ∞
of D1 and 2.16, there exists r2∗ > 0 such that
f1 t, x, y ≥
)
1
M1 ε3 λ
*p1 −1
φp1 x,
t ∈ 0, 1, 0 < x ≤ r2∗ , y ≥ 0.
3.25
Let 0 < r2 < min{R2 , r2∗ }, Kr2 {u, v ∈ K : u < r2 , v < r2 }. Next, we take ϕ1 , ϕ2 1, 1 ∈ ∂K1 , n > 1/r2 , and for any u, v ∈ ∂Kr2 , m > 0, we will show
Aλn u, v m ϕ1 , ϕ2 .
u, v /
3.26
16
Discrete Dynamics in Nature and Society
Otherwise, there exist u0 , v0 ∈ ∂Kr2 and m0 > 0 such that
u0 , v0 Aλn u0 , v0 m0 ϕ1 , ϕ2 .
3.27
From u0 , v0 ∈ ∂Kr2 , we know that u0 r2 or v0 r2 . Without loss of generality, we may
suppose that u0 r2 , then u0 τ ≥ Λu0 ≥ Λr2 for any τ ∈ 0, 1. So, we have
u0 t λ
1
H1 t, sφq1
1
0
≥
≥
≥
q −1
λρ1 σ1 1
q −1
λρ1 σ1 1
q −1
λρ1 σ1 1
K1 s, τa1 τf1n τ, u0 τ, v0 τdτ ds m0
0
1
1
esφq1
0
0
1
1
esφq1
0
0
1
1
esφq1
0
eτa1 τf1n τ, u0 τ, v0 τdτ ds m0
)
1
eτa1 τ
M1 ε3 λ
*p1 −1
φp1 u0 τdτ ds m0
3.28
eτa1 τM1 ε3 p1 −1
Λr2 p1 −1
dτ ds m0
0
1
q −1 1
λρ1 σ1 1
M1 ε3 Λr2 L1 m0 > r2 .
6
λ
This implies that r2 > r2 , which is a contradiction. This yields that 3.26 holds. By 3.24,
3.26, and Lemma 1.2, for any n > 1/r2 and λ ∈ M1 /f10 , N1 /f1∞ , μ ∈ 0, N2 /f2∞ , we
obtain that Tn has a fixed point un , vn in K r2 ,R2 and r2 < un < R2 , r2 < vn < R2 . The rest
of proof is similar to Theorem 3.1.
4. An Example
+
Example 4.1.
√ We consider system 1.3 with p1 3/2, p2 7/3, a1 t 1/t 1 − t , a2 t 1/1 − t t,
t2 1
f1 t, u, v √ 1 sin v2 v t ,
u
t, u, v ∈ 0, 1 × R0 × R ,
t4 t 3
f2 t, u, v 2 sinu lnt 1 √
,
v
4.1
t, u, v ∈ 0, 1 × R × R0 .
Discrete Dynamics in Nature and Society
17
Obviously, a1 , a2 are singular at t 0 and t 1, f1 t, u, v is singular at u 0 and f2 t, u, v
√
is singular at v 0. Choose g1 t, u t2 1/ u, h1 t, v 1 sinv2 v t, g2 t, u √
2 sinu lnt 1, and h2 t, v t4 t 3/ v. Let
⎧
*
1
⎪
⎪
,
0, s ∈ 0,
⎪
⎪
⎪
3
⎪
⎪
⎪
*
⎨1
1 2
ξ1 s , s∈
,
,
⎪
5
3 3
⎪
⎪
⎪
.
⎪
⎪
1
2
⎪
⎪
,1 ,
⎩ , s∈
4
3
⎧
*
1
⎪
⎪
⎪
⎨0, s ∈ 0, 2 ,
η1 s .
⎪
3
1
⎪
⎪
,1 ,
⎩ , s∈
5
2
⎧
*
1
⎪
⎪
,
0, s ∈ 0,
⎪
⎪
⎪
2
⎪
⎪
⎪
*
⎨1
1 3
ξ2 s , s∈
,
,
⎪
7
2 4
⎪
⎪
⎪
.
⎪
⎪
1
3
⎪
⎪
,1 ,
⎩ , s∈
3
4
⎧
*
1
⎪
⎪
⎪
⎨0, s ∈ 0, 2 ,
η2 s .
⎪
4
1
⎪
⎪
,1 .
⎩ , s∈
7
2
4.2
1
By direct calculation, we have α1 1/4, α2 1/3, β1 3/5, β2 4/7, 0 esai sds 2/3 i 1, 2. It is easy to check that f10 f20 ∞, f1∞ f2∞ 0, and the conditions H1 –
H3 and D5 are satisfied. By Theorem 3.2, system 1.3 has at least one positive solution
provided λ, μ ∈ 0, ∞.
Remark 4.2. Example 4.1 not only implies that f1 t, u, v, f2 t, u, v can be singular at u 0
and v 0, respectively, but also indicates that there is a large number of functions that satisfy
the conditions of Theorem 3.2. In addition, the condition D5 is also easy to check.
Acknowledgments
The first and second authors were supported financially by the National Natural Science
Foundation of China 11071141, 11126231 and the Natural Science Foundation of Shandong
Province of China ZR2009AL014, ZR2011AQ008. The third author was supported financially by the Australia Research Council through an ARC Discovery Project Grant.
References
1 F. Bernis, “Compactness of the support in convex and nonconvex fourth order elasticity problems,”
Nonlinear Analysis, vol. 6, no. 11, pp. 1221–1243, 1982.
2 D. G. Zill and M. R. Cullen, Differential Equations and Boundary Value Problems: Computing and Modeling,
Brooks Cole, 5th edition, 2001.
3 T. G. Myers and J. P. F. Charpin, “A mathematical model for atmospheric ice accretion and water flow
on a cold surface,” International Journal of Heat and Mass Transfer, vol. 47, no. 25, pp. 5483–5500, 2004.
4 T. G. Myers, J. P. F. Charpin, and S. J. Chapman, “The flow and solidification of a thin fluid film on an
arbitrary three-dimensional surface,” Physics of Fluids, vol. 14, no. 8, pp. 2788–2803, 2002.
5 D. Halpern, O. E. Jensen, and J. B. Grotberg, “A theoretical study of surfactant and liquid delivery
into the lung,” Journal of Applied Physiology, vol. 85, no. 1, pp. 333–352, 1998.
6 F. Mémoli, G. Sapiro, and P. Thompson, “Implicit brain imaging,” NeuroImage, vol. 23, supplement 1,
pp. S179–S188, 2004.
7 A. Toga, Brain Warping, Academic Press, New York, NY, USA, 1998.
18
Discrete Dynamics in Nature and Society
8 M. Hofer and H. Pottmann, “Energy-minimizing splines in manifolds,” in Proceedings of the ACM Transactions on Graphics (SIGGRAPH ’04), pp. 284–293, August 2004.
9 S. P. Timoshenko, Theory of Elastic Stability, McGraw-Hill, New York, NY, USA, 1961.
10 W. Soedel, Vibrations of Shells and Plates, Marcel Dekker, New York, NY, USA, 1993.
11 E. Dulcska, Soil Settlement Effects on Buildings, vol. 69 of Developments in Geotechnical Engineering, Elsevier, Amsterdam, The Netherlands, 1992.
12 Z. Bai, “Existence and multiplicity of positive for a fourth-order p-Laplacian equation,” Applied Mathematics and Mechanics, vol. 22, no. 12, pp. 1476–1480, 2001.
13 Z. Bai, B. Huang, and W. Ge, “The iterative solutions for some fourth-order p-Laplace equation
boundary value problems,” Applied Mathematics Letters, vol. 19, no. 1, pp. 8–14, 2006.
14 Z. Bai and H. Wang, “On positive solutions of some nonlinear fourth-order beam equations,” Journal
of Mathematical Analysis and Applications, vol. 270, no. 2, pp. 357–368, 2002.
15 Z. Cheng and J. Ren, “Periodic solutions for a fourth-order Rayleigh type p-Laplacian delay
equation,” Nonlinear Analysis, vol. 70, no. 1, pp. 516–523, 2009.
16 M. Feng, “Multiple positive solutions of fourth-order impulsive differential equations with integral
boundary conditions and one-dimensional p-laplacian,” Boundary Value Problems, vol. 2011, Article
ID 654871, 26 pages, 2011.
17 J. R. Graef and L. Kong, “Necessary and sufficient conditions for the existence of symmetric positive
solutions of singular boundary value problems,” Journal of Mathematical Analysis and Applications, vol.
331, no. 2, pp. 1467–1484, 2007.
18 J. R. Graef, C. Qian, and B. Yang, “A three point boundary value problem for nonlinear fourth order
differential equations,” Journal of Mathematical Analysis and Applications, vol. 287, no. 1, pp. 217–233,
2003.
19 Z. Guo, J. Yin, and Y. Ke, “Multiplicity of positive solutions for a fourth-order quasilinear singular
differential equation,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 27, pp. 1–15,
2010.
20 S. Lu and S. Jin, “Existence of periodic solutions for a fourth-order p-Laplacian equation with a
deviating argument,” Journal of Computational and Applied Mathematics, vol. 230, no. 2, pp. 513–520,
2009.
21 Y. Luo and Z. Luo, “Symmetric positive solutions for nonlinear boundary value problems with pLaplacian operator,” Applied Mathematics Letters, vol. 23, no. 6, pp. 657–664, 2010.
22 S. Jin and S. Lu, “Periodic solutions for a fourth-order p-Laplacian differential equation with a
deviating argument,” Nonlinear Analysis, vol. 69, no. 5-6, pp. 1710–1718, 2008.
23 G. Shi and X. Meng, “Monotone iterative for fourth-order p-Laplacian boundary value problems with
impulsive effects,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 1243–1248, 2006.
24 Z. Wang, L. Qian, and S. Lu, “On the existence of periodic solutions to a fourth-order p-Laplacian
differential equation with a deviating argument,” Nonlinear Analysis, vol. 11, no. 3, pp. 1660–1669,
2010.
25 J. Xu and Z. Yang, “Positive solutions for a fourth order p-Laplacian boundary value problem,” Nonlinear Analysis, vol. 74, no. 7, pp. 2612–2623, 2011.
26 J. Yang and Z. Wei, “Existence of positive solutions for fourth-order m-point boundary value problems
with a one-dimensional p-Laplacian operator,” Nonlinear Analysis, vol. 71, no. 7-8, pp. 2985–2996, 2009.
27 J. Zhang and G. Shi, “Positive solutions for fourth-order singular p-Laplacian boundary value
problems,” Applicable Analysis, vol. 85, no. 11, pp. 1373–1382, 2006.
28 X. Zhang and L. Liu, “Positive solutions of fourth-order four-point boundary value problems with
p-Laplacian operator,” Journal of Mathematical Analysis and Applications, vol. 336, no. 2, pp. 1414–1423,
2007.
29 X. Zhang and L. Liu, “A necessary and sufficient condition for positive solutions for fourth-order
multi-point boundary value problems with p-Laplacian,” Nonlinear Analysis, vol. 68, no. 10, pp. 3127–
3137, 2008.
30 X. Zhang and Y. Cui, “Positive solutions for fourth-order singular p-Laplacian differential equations
with integral boundary conditions,” Boundary Value Problems, vol. 2010, Article ID 862079, 23 pages,
2010.
31 X. Zhang, M. Feng, and W. Ge, “Existence results for nonlinear boundary-value problems with integral boundary conditions in Banach spaces,” Nonlinear Analysis, vol. 69, no. 10, pp. 3310–3321, 2008.
Discrete Dynamics in Nature and Society
19
32 X. Zhang, M. Feng, and W. Ge, “Symmetric positive solutions for p-Laplacian fourth-order differential
equations with integral boundary conditions,” Journal of Computational and Applied Mathematics, vol.
222, no. 2, pp. 561–573, 2008.
33 X. Zhang and W. Ge, “Positive solutions for a class of boundary-value problems with integral
boundary conditions,” Computers & Mathematics with Applications, vol. 58, no. 2, pp. 203–215, 2009.
34 D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cone, Academic Press, New York, NY,
USA, 1988.
35 K. Deimling, Nonlinear Functional Analysis, Springer, New York, NY, USA, 1985.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014