Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 939162, 19 pages
doi:10.1155/2012/939162
Research Article
Higher-Order Dynamic Delay Differential
Equations on Time Scales
Hua Su,1 Lishan Liu,2 and Xinjun Wang3
1
School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics,
Jinan 250014, China
2
School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
3
School of Economics, Shandong University, Jinan 250014, China
Correspondence should be addressed to Hua Su, jnsuhua@163.com
Received 11 October 2011; Accepted 12 February 2012
Academic Editor: Yansheng Liu
Copyright q 2012 Hua Su 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.
We study the existence of positive solutions for the nonlinear four-point singular boundary
value problem with higher-order p-Laplacian dynamic delay differential equations on time scales,
subject to some boundary conditions. By using the fixed-point index theory, the existence of
positive solution and many positive solutions for nonlinear four-point singular boundary value
problem with p-Laplacian operator are obtained.
1. Introduction
The study of dynamic equations on time scales goes back to its founder Stefan Hilger 1 and
is a new area of still fairly theoretical exploration in mathematics. Boundary value problems
for delay differential equations arise in a variety of areas of applied mathematics, physics, and
variational problems of control theory see 2, 3. In recent years, many authors have begun
to pay attention to the study of boundary value problems or with p-Laplacian equations or
with p-Laplacian dynamic equations on time scales see 4–19 and the references therein.
In 7, Sun and Li considered the existence of positive solution of the following
dynamic equations on time scales:
uΔ∇ t atft, ut 0,
βu0 − γuΔ 0 0,
t ∈ 0, T ,
αu η uT ,
1.1
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Journal of Applied Mathematics
where β, γ ≥ 0, β γ > 0, η ∈ 0, ρT , 0 < α < T/η. They obtained the existence of single and
multiple positive solutions of the problem 1.1 by using fixed point theorem and LeggettWilliams fixed point theorem, respectively.
In 10, Avery and Anderson discussed the following dynamic equation on time scales:
uΔ∇ t atfut 0, t ∈ 0, T ,
u0 0,
αu η uT .
1.2
He obtained some results for the existence of one positive solution of the problem 1.2 based
on the limits f0 limu → 0 fu/u and f∞ limu → ∞ fu/u.
In 11, Wang et al. discussed the following dynamic equation by using Avery-Peterson
fixed theorem see 10:
φp u qtft, ut, ut − 1, u t 0,
t ∈ 0, 1,
1.3
ut ξt,
−1 ≤ t ≤ 0, u1 0,
1.4
ut ξt,
−1 ≤ t ≤ 0, u 1 0.
1.5
They obtained some results for the existence, three positive solutions of the problem 1.3,
1.4 and 1.3, 1.5, respectively.
However, there are not many concerning the p-Laplacian problems on time scales.
Especially, for the singular multi point boundary value problems for higher-order p-Laplacian
dynamic delay differential equations on time scales, with the author’s acknowledg, no one
has studied the existence of positive solutions in this case.
Recently, in 16, we study the existence of positive solutions for the following
nonlinear two-point singular boundary value problem with p-Laplacian operator
φp u atfut 0, 0 < t < 1,
αφp u0 − βφp u 0 0,
γφp u1 δφp u 1 0,
1.6
by using the fixed point theorem of cone expansion and compression of norm type, the
existence of positive solution and infinitely many positive solutions for nonlinear singular
boundary value problem 1.6 with p-Laplacian operator are obtained.
Now, motivated by the results mentioned above, in this paper, we study the existence
of positive solutions for the following nonlinear four-point singular boundary value problem
Journal of Applied Mathematics
3
with higher-order p-Laplacian dynamic delay differential equations operator on time scales
SBVP:
n−1 ∇
n−2
gtf ut, ut − τ, uΔ t, . . . , uΔ t 0,
φp uΔ t
ut ζt,
uΔ 0 0,
i
0 < t < T,
1.7
−τ ≤ t ≤ 0,
1 ≤ i ≤ n − 3,
n−2 n−1 αφp uΔ 0 − βφp uΔ ξ 0,
1.8
n ≥ 3,
n−2
n−1 γφp uΔ T δφp uΔ η 0,
where φp s is p-Laplacian operator, that is, φp s |s|p−2 s, p > 1, φq φp−1 , 1/p 1/q 1;
ξ, η ∈ 0, T , τ ∈ 0, T is prescribed and ξ < η, g : 0, T → 0, ∞, α > 0, β ≥ 0, γ > 0, δ ≥ 0.
In this paper, by constructing one integral equation which is equivalent to the problem
1.7, 1.8, we research the existence of positive solutions for nonlinear singular boundary
value problem 1.7, 1.8 when g and f satisfy some suitable conditions.
Our main tool of this paper is the following fixed point index theory.
Theorem 1.1 see 18. Suppose E is a real Banach space, K ⊂ E is a cone, let Ωr {u ∈ K : u
≤
x, for all x ∈ ∂Ωr . Then
r}. Let operator T : Ωr → K be completely continuous and satisfy T x /
i if T x
≤ x
, for all x ∈ ∂Ωr , then iT, Ωr , K 1;
ii if T x
≥ x
, for all x ∈ ∂Ωr , then iT, Ωr , K 0.
This paper is organized as follows. In Section 2, we present some preliminaries and
lemmas that will be used to prove our main results. In Section 3, we study the existence of
at least two solutions of the systems 1.7, 1.8. In Section 4, we give an examples as the
application.
2. Preliminaries and Lemmas
A time scale T is an arbitrary nonempty closed subset of real numbers R . In 1, 14, 20, we
can find some basic definitions about time scale. The operators σ and ρ from T to T:
σt inf{τ ∈ T | τ > t} ∈ T,
ρt sup{τ ∈ T | τ < t} ∈ T
2.1
are called the forward jump operator and the backward jump operator, respectively.
If T R, then xΔ t x∇ t x t. If T Z, then xΔ t xt 1 − xt is the forward
difference operator, while x∇ t xt − xt − 1 is the backward difference operator.
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Journal of Applied Mathematics
A function f is left-dense continuous i.e., ld-continuous, if f is continuous at each
left-dense point in T and its right-sided limit exists at each right-dense point in T. It is well
known that f is ld-continuous.
If F ∇ t ft, then we define the nabla integral by
b
ft∇t Fb − Fa.
2.2
a
If F Δ t ft, then we define the delta integral by
b
ftΔt Fb − Fa.
2.3
a
In the rest of this paper, T is closed subset of R with 0 ∈ Tk , T ∈ Tk . And let
i
n−2
B u ∈ C−τ, 0 ∩ Cld
0, T : uΔ 0 0, 0 ≤ i ≤ n − 3 .
2.4
Here,
n−2
Cld
0, T u : 0, T → R | ut is left − dense n − 2 order continuously differentiable .
2.5
Then B is a Banach space with the norm u
maxt∈0,T |uΔ t|. And let
n−2
n−2
n−2
K u ∈ B : uΔ t ≥ 0, uΔ t is concave function, t ∈ 0, T .
2.6
Obviously, K is a cone in B. Set Kr {u ∈ K : u
≤ r}.
Definition 2.1. ut is called a solution of SBVP 1.7 and 1.8 if it satisfies the following:
n−1
0, T ;
1 u ∈ C−τ, 0 ∩ Cld
2 ut > 0 for all t ∈ 0, T and satisfy conditions 1.8;
∇
3 φp uΔ t −gtfut, ut − τ, uΔ t, . . . , uΔ t hold for t ∈ 0, T .
n−1
n−2
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5
In the rest of the paper, we also make the following assumptions:
H1 f ∈ Cld 0, ∞n , 0, ∞;
H2 gt ∈ Cld 0, T , 0, ∞ and there exists t0 ∈ 0, T , such that
T
gt0 > 0,
0<
gs∇s < ∞;
2.7
0
H3 ζt ∈ C−τ, 0, ζt > 0 on −τ, 0 and ζ0 0.
It is easy to check that condition H2 implies that
s
T
φq
0<
0
gs1 ∇s1 Δs < ∞.
2.8
0
We can easily get the following lemmas.
Lemma 2.2. Suppose condition H2 holds. Then there exists a constant θ ∈ 0, 1/2 satisfing
T −θ
0<
gt∇t < ∞.
2.9
θ
Furthermore, the function
At t
t
φq
θ
gs1 ∇s1 Δs T −θ
s
s
φq
t
gs1 ∇s1 ∇s,
t ∈ θ, T − θ,
2.10
t
is positive continuous functions on θ, T − θ; therefore, At has minimum on θ, T − θ. Hence we
suppose, that there exists L > 0 such that At ≥ L, t ∈ θ, T − θ.
Proof. At first, it is easily seen that At is continuous on θ, T − θ. Next, let
A1 t t
t
φq
θ
gs1 ∇s1 Δs,
A2 t s
s
T −θ
φq
t
gs1 ∇s1 Δs.
2.11
t
Then, from condition H2 , we have that the function A1 t is strictly monotone nondecreasing on θ, T − θ and A1 θ 0, the function A2 t is strictly monotone nonincreasing on
θ, T − θ and A2 T − θ 0, which implies L mint∈θ,T −θ At > 0. The proof is complete.
Lemma 2.3. Let u ∈ K and θ of Lemma 2.2, then
ut ≥ θ
u
,
t ∈ θ, T − θ.
The proof of the above lemma is similar to the proof in [17, Lemma 2.2], so we omit it.
2.12
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Journal of Applied Mathematics
n−1
0, 1 is a solution of
Lemma 2.4. Suppose that conditions H1 , H2 , H3 hold, ut ∈ B ∩ Cld
the following boundary value problems:
n−1 ∇
n−2
gtf ut, ut − τ ht − τ, uΔ t, . . . , uΔ t 0,
φp uΔ t
ut 0,
0 < t < T, 2.13
−τ ≤ t ≤ 0,
uΔ 0 0, 1 ≤ i ≤ n − 3,
n−2 n−1 αφp uΔ 0 − βφp uΔ ξ 0,
n−2
n−1 n ≥ 3,
γφp uΔ T δφp uΔ η 0,
i
2.14
where
ht ⎧
⎨ζt,
−τ ≤ t ≤ 0,
⎩0,
0 ≤ t ≤ T.
2.15
Then, ut ut ht, −τ ≤ t ≤ T is a positive solution to the SBVP 1.7 and 1.8.
Proof. It is easy to check that ut satisfies 1.7 and 1.8.
So in the rest of the sections of this paper, we focus on SBVP 2.13 and 2.14.
n−1
Lemma 2.5. Suppose that conditions H1 , H2 , H3 hold, ut ∈ B ∩ Cld
0, 1 is a solution of
boundary value problems 2.13, 2.14 if and only if ut ∈ B is a solution of the following integral
equation:
⎧
⎪
⎨ζt,
t s1
sn−3
ut ⎪
···
wsn−2 Δsn−2 Δsn−3 · · · Δs1 ,
⎩
0
0
−τ ≤ t ≤ 0,
0 ≤ t ≤ T,
2.16
0
where
⎧ σ
β
⎪
n−2
⎪
⎪
gsf us, us − τ hs − τ, uΔ s, . . . , uΔ s ∇s
φq
⎪
⎪
α ξ
⎪
⎪
⎪
t σ
⎪
⎪
⎪
Δ
Δn−2
⎪
⎪
∇r Δs,
φ
grf
ur,
ur
−
τ
hr
−
τ,
u
.
.
.
,
u
r,
r
q
⎪
⎪
⎪
0
s
⎪
⎨
0 ≤ t ≤ σ,
wt η
⎪
δ
n−2
⎪
⎪
φ
gsf us, us − τ hs − τ, uΔ s, . . . , uΔ s ∇s
⎪
⎪
⎪ q γ σ
⎪
T s
⎪
⎪
⎪
⎪
Δ
Δn−2
⎪
∇r Δs,
φ
grf
ur,
ur
−
τ
hr
−
τ,
u
.
.
.
,
u
r,
r
⎪
q
⎪
⎪
⎪
t
σ
⎪
⎩
σ ≤ t ≤ T.
2.17
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7
Proof. Necessity. Obviously, for t ∈ −τ, 0, we have ut ζt. If t ∈ 0, 1, by the equation
n−1
n−1
of the boundary condition, we have uΔ ξ ≥ 0, uΔ η ≤ 0, then there exists a constant
Δn−1
σ ∈ ξ, η ⊂ 0, T such that u σ 0.
Firstly, by integrating the equation of the problems 2.13 on σ, T , we have
φp u
Δn−1
t φp u
Δn−1
σ −
t
n−2
gsf us, us − τ hs − τ, uΔ s, . . . , uΔ s ∇s,
σ
2.18
then
Δn−1
u
t −φq
t
Δ
Δn−2
gsf us, us − τ hs − τ, u s, . . . , u
s ∇s ,
2.19
σ
thus
Δn−2
u
Δn−2
t u
σ −
s
t
φq
σ
n−2
grf ur, ur − τ hr − τ, uΔ r, . . . , uΔ r ∇r Δs.
σ
2.20
By uΔ σ 0 and condition 2.18, let t η on 2.18, we have
n−1
η
n−1 n−2
φp uΔ η − gsf us, us − τ hs − τ, uΔ s, . . . , uΔ s ∇s.
2.21
σ
By the equation of the boundary condition 2.14, we have
n−2
δ n−1 φp uΔ T − φp uΔ η ,
γ
2.22
then
uΔ T φq
n−2
η
δ
n−2
gsf us, us − τ hs − τ, uΔ s, . . . , uΔ s ∇s .
γ σ
2.23
Then, by 2.20 and leting t T on 2.20, we have
Δn−2
u
η
δ
n−2
gsf us, us − τ hs − τ, uΔ s, . . . , uΔ s ∇s
σ φq
γ σ
T s
Δ
Δn−2
φq
grf ur, ur − τ hr − τ, u r, . . . , u r ∇r Δs.
σ
σ
2.24
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Journal of Applied Mathematics
Then
η
δ
n−2
gsf us, us − τ hs − τ, uΔ s, . . . , uΔ s ∇s
γ σ
T s
n−2
φq
grf ur, ur − τ hr − τ, uΔ r, . . . , uΔ r ∇r Δs.
uΔ t φq
n−2
t
2.25
σ
Then, by integrating 2.25 for n − 2 times on 0, T , we have
ut t s1
0
···
sn−3
0
0
t s1
0
φq
···
0
σ
γ
η
gsf us, us − τ
δ
n−2
hs − τ, uΔ s, . . . , uΔ s ∇s Δssn−2 · · · Δs2 Δs1
sn−3 T
s
φq
sn−2
0
n−2
grf ur, ur − τ hr − τ, uΔ r, . . . , uΔ r
σ
×∇r Δs Δssn−2 · · · Δs2 Δs1 .
2.26
Similarly, for t ∈ 0, σ, by integrating the equation of problems 2.13 on 0, σ, we have
ut t s1
0
···
0
sn−3
0
σ
β
φq
gsf us, us − τ
α ξ
Δ
Δn−2
hs − τ, u s, . . . , u
t s1
0
0
···
sn−3 sn−2
σ
φq
0
s
0
s ∇s Δssn−2 · · · Δs2 Δs1
n−2
grf ur, ur − τ hr − τ, uΔ r, . . . , uΔ r
×∇r Δs Δssn−2 · · · Δs2 Δs1 .
2.27
Therefore, for any t ∈ 0, T , ut can be expressed as equation
⎧
⎪
⎪
⎨ζt,
sn−3
ut t s1
⎪
⎪
···
wsn−2 Δsn−2 Δsn−3 · · · Δs1 ,
⎩
0
0
−τ ≤ t ≤ 0,
0 ≤ t ≤ T,
0
where wt is expressed as 2.17. Then the results of Lemma 2.3 hold.
2.28
Journal of Applied Mathematics
9
Sufficiency. Suppose that ut by 2.17, we have
t s1
0 0
···
sn−3
0
wsn−2 Δsn−2 Δsn−3 · · · Δs1 , 0 ≤ t ≤ T . Then
⎧ σ
⎪
Δ
Δn−2
⎪
∇s ≥ 0,
gsf
us,
us
−
τ
hs
−
τ,
u
.
.
.
,
u
φ
s,
s
⎪ q
⎪
⎪
⎪
t
⎪
⎪
⎪
⎪
⎨
0 ≤ t ≤ σ,
n−1
Δ
t
u t ⎪
⎪
Δ
Δn−2
⎪
gsf us, us − τ hs − τ, u s, . . . , u s ∇s ≤ 0,
⎪−φq
⎪
⎪
⎪
σ
⎪
⎪
⎪
⎩
σ ≤ t ≤ T.
2.29
So, φp uΔ ∇ gtfut, ut − τ ht − τ, uΔ t, . . . , uΔ t 0, 0 < t < T . These imply
that 2.13 holds. Furthermore, by letting t 0 and t T on 2.17 and 2.29, we can obtain
the boundary value equations of 2.14. The proof is complete.
n−1
n−2
Now, we define an operator equation T given by
⎧
⎪
⎨ζt,
sn−3
T ut t s1
⎪
⎩
···
wsn−2 Δsn−2 Δsn−3 · · · Δs1 ,
0
0
−τ ≤ t ≤ 0,
0 ≤ t ≤ T,
2.30
0
where wt is given by 2.17.
From the definition of T and the previous discussion, we deduce that, for each u ∈ K,
T u ∈ K. Moreover, we have the following lemmas.
Lemma 2.6. T : K → K is completely continuous.
Proof. Because
⎧ σ
⎪
Δ
Δn−2
⎪
gsf us, us− τ hs− τ, u s, . . . , u s ∇s ≥ 0,
φq
⎪
⎪
⎪
⎪
t
⎪
⎪
⎪
⎪
⎨
0 ≤ t ≤ σ,
Δn−1
Δ
t
T u t w t ⎪
⎪
Δ
Δn−2
⎪
−φq
gsf us, us− τhs − τ, u s, . . . , u s ∇s ≤ 0,
⎪
⎪
⎪
⎪
σ
⎪
⎪
⎪
⎩
σ ≤ t ≤ T,
2.31
is continuous, decreasing on 0, T and satisfies T uΔ σ 0, then, T u ∈ K for each u ∈ K
n−2
n−2
and T uΔ σ maxt∈0,T T uΔ t. This shows that T K ⊂ K. Furthermore, it is easy to
check by Arzela-ascoli Theorem that T : K → K is completely continuous.
n−1
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Journal of Applied Mathematics
Lemma 2.7. Suppose that conditions H1 , H2 , H3 hold, the solution ut of problem 2.13,
2.14 satisfies
max|ut − τ ht − τ| ≤ max |ζt|,
−τ≤t≤0
0≤t≤T
Δ
ut ≤ T u t ≤ · · · ≤ T
n−3 Δ
u
n−3
2.32
t ∈ 0, T ,
t,
and for θ ∈ 0, T/2 in Lemma 2.2, one has
T Δn−2
u t,
θ
uΔ t ≤
n−3
t ∈ θ, T − θ.
2.33
Proof. Firstly, we can have
max|ut − τ ht − τ| ≤ max|ut − τ| max|ht − τ|
0≤t≤T
0≤t≤T
0≤t≤T
max |ut| max |ht|
−τ≤t≤T −τ
−τ≤t≤T −τ
2.34
max |ζt|.
−τ≤t≤0
Next, if ut is the solution of problem 2.13, 2.14, then uΔ t is concave function,
Δi
and u t ≥ 0 i 0, 1, . . . , n − 2, t ∈ 0, T . Thus, we have
n−2
uΔ t i
t
uΔ sΔs ≤ tuΔ t ≤ T uΔ t,
i1
i1
i1
i 0, 1, . . . , n − 4,
2.35
0
that is, ut ≤ T uΔ t ≤ · · · ≤ T n−3 uΔ t, t ∈ 0, T .
n−2
n−2
n−3
Finally, by Lemma 2.3, for t ∈ θ, T − θ, we have uΔ t ≥ θ
uΔ . By uΔ t t Δn−2
n−2
u sΔs ≤ T uΔ , we have
0
n−3
uΔ t ≤
n−3
T Δn−2
u t,
θ
t ∈ θ, T − θ.
2.36
1
,
T
T φq β/α φq 0 gr∇r
2.37
The proof is complete.
For convenience, we set
H max |ζt|,
−τ≤t≤0
θ∗ 2
,
L
θ∗ m ∈ θ∗ , ∞, M ∈ 0, θ∗ ,
Journal of Applied Mathematics
11
where L is the constant from Lemma 2.2. By Lemma 2.5, we can also set
f0 lim
max
f∞ lim
min
fu1 , u2 , . . . , un p−1
un → 0u1 ,u2 ,...,un ∈ℵ
un
,
fu1 , u2 , . . . , un p−1
un → ∞u1 ,u2 ,...,un ∈ℵ
un
2.38
,
where ℵ {u1 , u2 , . . . , un | 0 ≤ u1 ≤ T u3 · · · ≤ T n−3 un−1 ≤ T n−2 /θun , u2 ≤ H}.
3. The Existence of Multiple Positive Solutions
In this section, we also make the following conditions:
A1 fu1 , u2 , . . . , un ≥ mrp−1 , for θr ≤ un ≤ r, u1 , u2 , . . . , un ∈ ℵ;
A2 fu1 , u2 , . . . , un ≤ MRp−1 , for 0 ≤ un ≤ R, u1 , u2 , . . . , un ∈ ℵ.
Next, we will discuss the existence of multiple positive solutions.
Theorem 3.1. Suppose that conditions (H1 ), (H2 ), (H3 ), and (A2 ) hold. Assume that f also satisfies
A3 f0 ∞;
A4 f∞ ∞.
Then, the SBVP 2.13, 2.14 hase at last two solutions u1 , u2 such that
3.1
0 < u1 < R < u2 .
Proof. For any u ∈ K, by Lemma 2.3, we have
uΔ t ≥ θ
u
,
n−2
t ∈ θ, T − θ.
3.2
First, by condition A3 , for any N > 2/θL, there exists a constant ρ∗ ∈ 0, R such that
fu1 , u2 , . . . , un ≥ Nun p−1 ,
0 < u n ≤ ρ ∗ , un /
0.
3.3
Set Ωρ∗ {u ∈ K : u
< ρ∗ }. For any u ∈ ∂Ωρ∗ , by 3.2 we have
ρ∗ u
≥ uΔ t ≥ θ
u
θρ∗ ,
n−2
t ∈ θ, T − θ.
3.4
For any u ∈ ∂Ωρ∗ , by 3.3 and Lemmas 2.3–2.6, we will discuss it from three perspectives.
12
Journal of Applied Mathematics
i If σ ∈ θ, T − θ, we have
2
T u
2T uΔ σ
σ σ
n−2
≥
φq
grf ur, ur − τ hr − τ, uΔ r, . . . , uΔ r ∇r Δs
n−2
0
≥
s
T
s
φq
σ
σ
σ
φq
θ
s
T −θ
Δ
Δn−2
grf ur, ur − τ hr − τ, u r, . . . , u
σ
Δ
Δn−2
grf ur, ur − τ hr − τ, u r, . . . , u
s
φq
σ
r ∇r Δs
3.5
r ∇r Δs
n−2
grf ur, ur − τ hr − τ, uΔ r, . . . , uΔ r ∇r Δs
σ
≥ NθAσ
u
≥ 2
u
.
ii If σ ∈ T − θ, T , we have
T u
T uΔ σ
σ
β
Δ
Δn−2
gsf us, us − τ hs − τ, u s, . . . , u s ∇s
≥ φq
α ξ
σ σ
n−2
φq
grf ur, ur − τ hr − τ, uΔ r, . . . , uΔ r ∇r Δs
n−2
0
≥
s
T −θ
T −θ
φq
θ
Δ
Δn−2
grf ur, ur − τ hr − τ, u r, . . . , u
3.6
r ∇r Δs
s
≥ NθAT − θ
u
> u
.
iii If σ ∈ 0, θ, we have
T u
T uΔ σ
η
δ
n−2
≥ φq
gsf us, us − τ hr − τ, uΔ s, . . . , uΔ s ∇s
γ σ
T s
n−2
φq
grf ur, ur − τ hr − τ, uΔ r, . . . , uΔ r ∇r Δs
n−2
σ
≥
σ
T −θ
s
φq
θ
3.7
n−2
grf ur, ur − τ hr − τ, uΔ r, . . . , uΔ r ∇r Δs
θ
≥ NθAθ
u
> u
.
Therefore, no matter under which condition, we all have
T u
≥ u
,
∀u ∈ ∂Ωρ∗ .
3.8
Journal of Applied Mathematics
13
Then, by Theorem 1.1, we have
i T, Ωρ∗ , K 0.
3.9
Next, by condition A4 , for any N > 2/θL, there exists a constant ρ0 > 0 such that
p−1
,
fu1 , u2 , . . . , un ≥ Nun
un > ρ 0 .
3.10
We choose a constant ρ∗ > max{R, ρ0 /θ}, obviously ρ∗ < R < ρ∗ . Set Ωρ∗ {u ∈ K : u
< ρ∗ }.
For any u ∈ ∂Ωρ∗ , by Lemma 2.3, we have
ut ≥ θ
u
θρ∗ > ρ0 ,
t ∈ θ, T − θ.
3.11
Then, by 3.10, Lemmas 2.3–2.6 and also similar to the previous proof, we can also have from
three perspectives that
T u
≥ u
,
∀u ∈ ∂Ωρ∗ .
3.12
Then, by Theorem 1.1, we have
i T, Ωρ∗ , K 0.
3.13
Finally, set ΩR {u ∈ K : u
< R}. For any u ∈ ∂ΩR , we have ut ≤ u
R, by A2 we
know
T u
T uΔ σ
σ
β
Δ
Δn−2
gsf us, us − τ hs − τ, u s, . . . , u s ∇s
≤ φq
α ξ
T σ
Δ
Δn−2
φq
grf ur, ur − τ hr − τ, u r, . . . , u r ∇r Δs
n−2
0
3.14
s
T
β
≤ T φq
gr∇r ≤ R u
.
MRφq
α
0
Thus,
T u
≤ u
,
∀u ∈ ∂ΩR .
3.15
Then, by Theorem 1.1, we have
iT, ΩR , K 1.
3.16
14
Journal of Applied Mathematics
Therefore, by 3.9, 3.13, 3.16, ρ∗ < R < ρ∗ we have
i T, ΩR \ Ωρ∗ , K 1,
i T, Ωρ∗ \ ΩR , K −1.
3.17
Then T has fixed point u1 ∈ ΩR \ Ωρ∗ and fixed point u2 ∈ Ωρ∗ \ ΩR . Obviously, u1 , u2 are
all positive solutions of problem 2.13, 2.14 and ρ∗ < u1 < R < u2 < ρ∗ . Proof of
Theorem 3.1 is complete.
Theorem 3.2. Suppose that conditions (H1 ), (H2 ), (H3 ), (A1 ) hold. Assume that f also satisfies
A5 f0 0;
A6 f∞ 0.
Then, the SBVP 2.13, 2.14 has at last two solutions u1 , u2 such that 0 < u1 < r < u2 .
Proof. First, by f0 0, for 1 ∈ 0, θ∗ , there exists a constant ρ∗ ∈ 0, r such that
fu1 , u2 , . . . , un ≤ 1 un p−1 ,
0 < un ≤ ρ ∗ .
3.18
Set Ωρ∗ {u ∈ K : u
< ρ∗ }, for any u ∈ ∂Ωρ∗ , by 3.18, we have
T u
T uΔ σ
σ
β
Δ
Δn−2
gsf us, us − τ hs − τ, u s, . . . , u s ∇s
≤ φq
α ξ
T σ
n−2
φq
grf ur, ur − τ hr − τ, uΔ r, . . . , uΔ r ∇r Δs
n−2
0
s
σ
β
Δ
Δn−2
≤ φq
gsf us, us − τ hs − τ, u s, . . . , u s ∇s
α ξ
T
Δ
Δn−2
T φq
grf ur, ur − τ hr − τ, u r, . . . , u r ∇r
3.19
0
T
β
≤ T φq
gr∇r ≤ ρ∗ u
,
1 ρ∗ φq
α
0
that is,
T u
≤ u
,
∀u ∈ ∂Ωρ∗
3.20
Then, by Theorem 1.1, we have
i T, Ωρ∗ , K 1.
3.21
Journal of Applied Mathematics
15
Next, let f ∗ x max0≤un−1 ≤x fu1 , u2 , . . . , un−1 ; note that f ∗ x is monotone increasing
with respect to x ≥ 0. Then, from f∞ 0, it is easy to see that
f ∗ x
0.
x → ∞ xp−1
lim
3.22
Therefore, for any 2 ∈ 0, θ∗ , there exists a constant ρ∗ > r such that
f ∗ x ≤ 2 xp−1 ,
x ≥ ρ∗ .
3.23
Set Ωρ∗ {u ∈ K : u
< ρ∗ }, for any u ∈ ∂Ωρ∗ , by 3.23, we have
T u
T uΔ σ
σ
β
Δ
Δn−2
gsf us, us − τ hs − τ, u s, . . . , u s ∇s
≤ φq
α ξ
T σ
n−2
φq
grf ur, ur − τ hr − τ, uΔ r, . . . , uΔ r ∇r Δs
n−2
0
s
σ
β
Δ
Δn−2
≤ φq
gsf us, us − τ hs − τ, u s, . . . , u s ∇s
α ξ
T
Δ
Δn−2
T φq
grf ur, ur − τ hr − τ, u r, . . . , u r ∇r
3.24
0
T
∗
β
∗
≤ T φq
grf ρ ∇r
φq
α
0
T
β
∗
2 ρ φq
gr∇r ≤ r ∗ u
,
≤ T φq
α
0
that is,
T u
≤ u
,
∀u ∈ ∂Ωρ∗ .
3.25
Then, by Theorem 1.1, we have
i T, Ωρ∗ , K 1.
3.26
Finally, set Ωr {u ∈ K : u
< r}. For any u ∈ ∂Ωr , by A1 , Lemma 2.3 and also
similar to the previous proof of Theorem 3.1, we can also have
T u
≥ u
,
∀u ∈ ∂Ωr .
3.27
Then, by Theorem 1.1, we have
iT, Ωr , K 0.
3.28
16
Journal of Applied Mathematics
Therefore, by 3.21, 3.28, 3.26, ρ∗ < r < ρ∗ , we have
i T, Ωr \ Ωρ∗ , K −1,
i T, Ωρ∗ \ Ωr , K 1.
3.29
Then T has fixed point u1 ∈ Ωr \ Ωρ∗ and fixed point u2 ∈ Ωρ∗ \ Ωr . Obviously, u1 , u2 are
all positive solutions of problem 2.13, 2.14 and ρ∗ < u1 < r < u2 < ρ∗ . The proof of
Theorem 3.2 is complete.
Similar to Theorems 3.1 and 3.2, we also obtain the following theorems.
Theorem 3.3. Suppose that conditions (H1 ), (H2 ), (H3 ), and (A2 ) hold and
A7 f∞ λ ∈ 2θ∗ /θp−1 , ∞,
A8 f0 ϕ ∈ 2θ∗ /θp−1 , ∞.
Then, the SBVP 2.13, 2.14 has at last two solutions u1 , u2 such that 0 < u1 < R < u2 .
Theorem 3.4. Suppose that conditions (H1 ), (H2 ), (H3 ), and (A1 ) hold and
A9 f0 ϕ ∈ 0, θ∗ /4p−1 ;
A10 f∞ λ ∈ 0, θ∗ /4p−1 .
Then, the SBVP 2.13, 2.14 has at last two solutions u1 , u2 such that 0 < u1 < r < u2 .
4. An Example
Example 4.1. Consider the following 3-order singular boundary value problem SBVP with
p-Laplacian:
φp u
ΔΔ
∇
2
4 1 −1/2
Δ
t
t 1 − t ut ut − 1 u
t uΔ t 0,
64π 4
ut −tet ,
1
2φp uΔ 0 − φp uΔΔ
0,
4
0 < t < 1,
−1 ≤ t ≤ 0,
1
φp uΔ 1 δφp uΔΔ
0,
2
4.1
where
β γ 1,
α 2,
1
η ,
3
p 4,
1
θ ,
4
δ ≥ 0,
p 4,
τ T 1.
ξ
1
,
4
4.2
Journal of Applied Mathematics
17
So, by Lemma 2.4, we discuss the following SBVP:
∇
φp uΔΔ
t 2φp uΔ 0 − φp
1 −1/2
t
−
t
ut ut − 1 ht − 1
1
64π 4
2
4 uΔ t uΔ t 0,
ut 0,
1
0,
uΔΔ
4
0 < t < 1,
−1 ≤ t ≤ 0,
1
φp uΔ 1 δφp uΔΔ
0,
2
4.3
where
ht gt ⎧
⎨ ζt, −1 ≤ t ≤ 0,
⎩
0 ≤ t ≤ 1,
0,
1 −1/2
t
1 − t,
64π 4
ζt −tet ,
4.4
fu1 , u2 , u3 u1 u2 u23 u43 .
Then, obviously,
4
q ,
3
1
gt∇t 0
1
,
64π 3
H max |ζt| e,
f∞ ∞,
−1≤t≤0
f0 ∞,
4.5
so conditions H1 , H2 , H3 , A2 , and A3 hold.
Next,
1
at∇t
φq
0
1
,
4π
θ∗ 4π
,
√
3
4
4.6
1
we choose R 3, M 2 and for θ 1/4, because of the monotone increasing of fu1 , u2 , u3 on 0, ∞3 , then
3
3
, e, 3 e 90,
fu1 , u2 , u3 ≤ f
4
4
0 ≤ u3 ≤ 3, 0 ≤ u1 ≤
1
u3 , 0 ≤ u2 ≤ e.
4
4.7
Therefore, by
M ∈ 0, θ∗ ,
MRp−1 63 216,
4.8
we know
fu1 , u2 , u3 ≤ MRp−1 ,
0 ≤ u3 ≤ 3, 0 ≤ u1 ≤
1
u3 , 0 ≤ u2 ≤ e,
4
4.9
18
Journal of Applied Mathematics
so condition A2 holds. Then, by Theorem 3.1, SBVP 4.3 has at least two positive solutions
v1 , v2 and 0 < v1 < 3 < v2 . Then, by Lemma 2.4, v1 t v1 t ht, v2 t v2 t ht,
t ∈ −1, 1 are the positive solutions of the SBVP 4.1.
Acknowldgments
The first and second authors were supported financially by Shandong Province Natural
Science Foundation ZR2009AQ004, and National Natural Science Foundation of China
11071141, and the third author was supported by Shandong Province planning Foundation
of Social Science 09BJGJ14.
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