Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2008, Article ID 257680, 19 pages
doi:10.1155/2008/257680
Research Article
Twin Positive Solutions of a Nonlinear
m-Point Boundary Value Problem for Third-Order
p-Laplacian Dynamic Equations on Time Scales
Wei Han1 and Guang Zhang2
1
2
Department of Mathematics, North University of China, Taiyuan 030051, China
Department of Mathematics, School of Science, Tianjin University of Commerce, Tianjin 300134, China
Correspondence should be addressed to Wei Han, qd hanweiwei1@126.com
Received 20 April 2008; Accepted 20 June 2008
Recommended by Binggen Zhang
Several existence theorems of twin positive solutions are established for a nonlinear m-point
boundary value problem of third-order p-Laplacian dynamic equations on time scales by using
a fixed point theorem. We present two theorems and four corollaries which generalize the results of
related literature. As an application, an example to demonstrate our results is given. The obtained
conditions are different from some known results.
Copyright q 2008 W. Han and G. Zhang. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
A time scale T is a nonempty closed subset of R. We make the blanket assumption that 0 and T
are points in T. By an interval 0, T , we always mean the intersection of the real interval 0, T with the given time scale, that is, 0, T ∩ T.
In this paper, we will be concerned with the existence of positive solutions of the pLaplacian dynamic equations on time scales:
∇
φp uΔ∇ atft, ut 0,
φp uΔ∇ 0 m−2
ai φp uΔ∇ ξi ,
i1
uΔ 0 0,
t ∈ 0, T ,
uT 1.1
m−2
bi uξi ,
1.2
i1
where φp s is p-Laplacian operator; that is, φp s |s|p−2 s, p > 1, φp−1 φq , 1/p 1/q 1, 0 <
ξ1 < · · · < ξm−2 < ρT , and
m−2
H1 ai , bi ∈ 0, ∞, i 1, 2, . . . , satisfy 0 < m−2
i1 ai < 1 and
i1 bi < 1;
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Discrete Dynamics in Nature and Society
H2 at ∈ Cld 0, T , 0, ∞ and there exists t0 ∈ ξm−2 , T such that at0 > 0;
H3 f ∈ C0, T × 0, ∞, 0, ∞.
We point out that the Δ-derivative and the ∇-derivative in 1.2 and the Cld space in H2 are
defined in Section 2.
Recently, there has been much attention paid to the existence of positive solutions for
third-order nonlinear boundary value problems of differential equations. For example, see
1–10 and the listed references. Anderson 2 considered the following third-order nonlinear
problem:
x t ft, xt,
t1 ≤ t ≤ t3 ,
xt1 x t2 0,
γxt3 δx t3 0.
1.3
He used the Krasnoselskii and the Leggett and Williams fixed-point theorems to prove the
existence of solutions to the nonlinear problem 1.3. Li 6 considered the existence of single
and multiple positive solutions to the nonlinear singular third-order two-point boundary value
problem:
u t λatfut 0,
0 < t < 1,
u0 u 0 u 1 0.
1.4
Under various assumptions on a and f, they established intervals of the parameter λ which
yield the existence of at least two and infinitely many positive solutions of the boundary value
problem by using Krasnoselski’s fixed-point theorem of cone expansion-compression type. Liu
et al. 7 discussed the existence of at least one or two nondecreasing positive solutions for the
following singular nonlinear third-order differential equations:
x t λαtft, xt 0,
a < t < b,
xa x a x b 0.
1.5
Green’s function and the fixed-point theorem of cone expansion-compression type are utilized
in their paper. In 8, Sun considered the following nonlinear singular third-order three-point
boundary value problem:
u t − λatFt, ut 0,
0 < t < 1,
u0 u η u 1 0.
1.6
He obtained various results on the existence of single and multiple positive solutions to the
boundary value problem 1.6 by using a fixed-point theorem of cone expansion-compression
type due to Krasnosel’skii. In 10, Zhou and Ma studied the existence and iteration of positive
solutions for the following third-order generalized right-focal boundary value problem with
p-Laplacian operator:
φp u t qtft, ut, 0 ≤ t ≤ 1,
m
n
u0 αi uξi ,
u η 0,
u 1 βi u θi .
i1
i1
1.7
W. Han and G. Zhang
3
They established a corresponding iterative scheme for 1.7 by using the monotone iterative
technique.
On the other hand, the existence of positive solutions for third-order nonlinear boundary
value problems of difference equations is also extensively studied by a number of authors see
1, 3, 5, 9 and the listed references. The present work is motivated by a recent paper 4. In
4, Henderson and Yin considered the existence of solutions for a third-order boundary value
problem on a time-scale equation of the form
uΔ ft, u, uΔ , uΔΔ ,
3
t ∈ T,
1.8
which is uniform for the third-order difference equation and the third-order differential
equation.
2. Preliminaries and lemmas
For convenience, we list the following definitions which can be found in 4, 11–15.
Definition 2.1. Let T be a time scale. For t < sup T and r > inf T, define the forward jump
operator σ and the backward jump operator ρ, respectively, by
σt inf{τ ∈ T | τ > t} ∈ T,
ρr sup{τ ∈ T | τ < r} ∈ T
2.1
for all t, r ∈ T. If σt > t, t is said to be right-scattered, and if ρr < r, r is said to be leftscattered; if σt t, t is said to be right-dense, and if ρr r, r is said to be left-dense. If
T has a right-scattered minimum m, define Tk T − {m}; otherwise set Tk T. If T has a
left-scattered maximum M, define Tk T − {M}; otherwise set Tk T.
Definition 2.2. For f : T→R and t ∈ Tk , the delta derivative of f at the point t is defined to
be the number f Δ t provided that it exists, with the property that for each > 0 there is a
neighborhood U of t such that
|fσt − fs − f Δ tσt − s| ≤ |σt − s|
2.2
for all s ∈ U.
For f : T→R and t ∈ Tk , the nabla derivative of f at t is denoted by f ∇ t provided that
it exists, with the property that for each > 0 there is a neighborhood U of t such that
|fρt − fs − f ∇ tρt − s| ≤ |ρt − s|
2.3
for all s ∈ U.
Definition 2.3. 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.
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Discrete Dynamics in Nature and Society
Definition 2.4. If φΔ t ft, then one defines the delta integral by
b
ftΔt φb − φa.
2.4
a
If F ∇ t ft, then one defines the nabla integral by
b
ft∇t Fb − Fa.
2.5
a
To prove the main results in this paper, we will employ several lemmas. These lemmas
are based on the linear BVP
∇
φp uΔ∇ ht 0,
φp uΔ∇ 0 m−2
ai φp uΔ∇ ξi ,
t ∈ 0, T ,
uΔ 0 0,
uT i1
Lemma 2.5. If
solution
m−2
i1
1 and
ai /
2.6
m−2
bi uξi .
2.7
i1
m−2
i1
ut −
t
1, then for h ∈ Cld 0, T the BVP 2.6-2.7 has the unique
bi /
t − sφq
s
0
hτ∇τ − A ∇s C,
2.8
0
where
ξ
ai 0i hτ∇τ
,
A−
1 − m−2
i1 ai
T
s
s
ξi
T − sφq 0 hτ∇τ − A∇s − m−2
i1 bi 0 ξi − sφq 0 hτ∇τ − A∇s
0
.
C
1 − m−2
i1 bi
m−2
i1
2.9
Proof. i Let u be a solution, then we will show that 2.8 holds. By taking the nabla integral of
problem 2.6 on 0, t, we have
φp uΔ∇ t −
t
hτ∇τ A
2.10
t
hτ∇τ A −φq
hτ∇τ − A .
2.11
0
then
Δ∇
u
t φq
−
t
0
0
By taking the nabla integral of 2.11 on 0, t, we can get
uΔ t −
s
t
φq
0
0
hτ∇τ − A ∇s B.
2.12
W. Han and G. Zhang
5
By taking the delta integral of 2.12 on 0, t, we can get
ut −
t
t − sφq
s
0
hτ∇τ − A ∇s Bt C.
2.13
0
Similarly, let t 0 on 2.10, then we have φp uΔ∇ 0 A; let t ξi on 2.10, then we have
φp uΔ∇ ξi −
ξi
hτ∇τ A.
2.14
0
Let t 0 on 2.12, then we have
uΔ 0 B.
2.15
Let t T on 2.13, then we have
uT −
T
T − sφq
s
0
hτ∇τ − A ∇s BT C.
2.16
hτ∇τ − A ∇s Bξi C.
2.17
0
Similarly, let t ξi on 2.13, then we have
uξi −
ξi
ξi − sφq
s
0
0
By the boundary condition 2.7, we can get
A
m−2
ai
B 0,
ξi
− hτ∇τ A .
2.18
2.19
0
i1
Solving 2.19, we get
ξ
ai 0i hτ∇τ
.
1 − m−2
i1 ai
m−2
A−
i1
2.20
By the boundary condition 2.7, we can obtain
−
T
0
T − sφq
s
0
s
m−2
ξi
hτ∇τ − A ∇s C bi − ξi − sφq
hτ∇τ − A ∇s C .
i1
0
0
2.21
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Discrete Dynamics in Nature and Society
Substituting 2.20 in the above expression, one has
T
C
0
ξi
s
s
T − sφq 0 hτ∇τ − A∇s − m−2
i1 bi 0 ξi − sφq 0 hτ∇τ − A∇s
.
1 − m−2
i1 bi
2.22
ii We show that the function u given in 2.8 is a solution.
Let u be as in 2.8. By 12, Theorem 2.10iii and taking the delta derivative of 2.8, we
have
uΔ t −
s
t
φq
0
hτ∇τ − A ∇s;
2.23
0
moreover, we get
u
Δ∇
t −φq
t
hτ∇τ − A ,
0
φp uΔ∇ −
t
2.24
hτ∇τ − A .
0
∇
Taking the nabla derivative of this expression yields φp uΔ∇ −ht. Also, routine
calculation verifies that u satisfies the boundary value conditions in 2.7 so that u given in
2.8 is a solution of 2.6 and 2.7. The proof is complete.
Lemma 2.6. Assume H1 holds. For h ∈ Cld 0, T and h ≥ 0, the unique solution u of 2.6 and 2.7
satisfies
ut ≥ 0
for t ∈ 0, T .
2.25
Proof. Let
ϕ0 s φq
s
hτ∇τ − A .
2.26
0
Since
s
0
then ϕ0 s ≥ 0.
hτ∇τ − A s
0
ξ
ai 0i hτ∇τ
≥ 0,
1 − m−2
i1 ai
m−2
hτ∇τ i1
2.27
W. Han and G. Zhang
7
According to Lemma 2.5, we get
T
ξi
T − sϕ0 s∇s − m−2
i1 bi 0 ξi − sϕ0 s∇s
0
u0 C 1 − m−2
i1 bi
T
m−2 ξi
T − sϕ0 s∇s − i1 bi 0 T − sϕ0 s∇s
≥ 0
1 − m−2
i1 bi
T
T
m−2 T
T − sϕ0 s∇s − i1 bi 0 T − sϕ0 s∇s − ξi T − sϕ0 s∇s
0
1 − m−2
i1 bi
m−2 T
T
i1 bi ξi T − sϕ0 s∇s
T − sϕ0 s∇s ≥ 0,
1 − m−2
0
i1 bi
T
uT − T − sϕ0 s∇s C
0
2.28
m−2 ξi
i1 bi 0 ξi − sϕ0 s∇s
− T − sϕ0 s∇s m−2
1 − i1 bi
0
ξi
T
T
T − sϕ0 s∇s − m−2
i1 bi 0 T − sϕ0 s∇s
0
≥ − T − sϕ0 s∇s 1 − m−2
0
i1 bi
m−2 T
i1 bi ξi T − sϕ0 s∇s
≥ 0.
1 − m−2
i1 bi
T
T
0
If t ∈ 0, T , we have
t
ut − t − sϕ0 s∇s 0
T
≥−
T − sϕ0 s∇s 0
1−
1−
1−
1
m−2
i1
1
m−2
i1
bi
bi
1
m−2
1−
i1
T
bi
1
m−2
i1
T − sϕ0 s∇s −
T − sϕ0 s∇s −
m−2
ξi
i1
0
0
T
bi
T − sϕ0 s∇s −
0
bi
ξi − sϕ0 s∇s
m−2
ξi
i1
0
bi
T − sϕ0 s∇s
T
T
m−2
m−2
ξi
− 1−
bi
T −sϕ0 s∇s T −sϕ0 s∇s− bi T −sϕ0 s∇s
i1
m−2
T
i1
ξi
bi
0
0
i1
0
T − sϕ0 s∇s ≥ 0.
2.29
So ut ≥ 0, t ∈ 0, T .
Lemma 2.7. Assume H1 holds. If h ∈ Cld 0, T and h ≥ 0, then the unique solution u of 2.6 and
2.7 satisfies
inf ut ≥ γ
u
,
t∈0,T 2.30
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Discrete Dynamics in Nature and Society
where
m−2
bi T − ξi ,
u
max |ut|.
m−2
t∈0,T T − i1 bi ξi
t
Proof. It is easy to check that uΔ t − 0 ϕ0 s∇s ≤ 0; this implies that
i1
γ
u
u0,
min ut uT .
t∈0,T 2.31
2.32
It is easy to see that uΔ t2 ≤ uΔ t1 for any t1 , t2 ∈ 0, T with t1 ≤ t2 . Hence, uΔ t is a
decreasing function on 0, T . This means that the graph of ut is concave down on 0, T .
For each i ∈ {1, 2, . . . , m − 2}, we have
uT − u0 uT − uξi ,
≥
T −0
T − ξi
2.33
T uξi − ξi uT ≥ T − ξi u0,
2.34
that is,
so that
T
m−2
m−2
m−2
i1
i1
i1
bi uξi −
bi ξi uT ≥
m−2
bi T − ξi u0.
bi uξi , we have
m−2
bi T − ξi u0.
uT ≥ i1 m−2
T − i1 bi ξi
With the boundary condition uT 2.35
i1
2.36
This completes the proof.
Let the norm on Cld 0, T be the maximum norm. Then, the Cld 0, T is a Banach space.
It is easy to see that BVP 1.1-1.2 has a solution u ut if and only if u is a fixed point of the
operator
t
s
∇s C,
Aut − t − sφq
aτfτ, uτ∇τ − A
2.37
0
where
0
m−2
Denote
i1
ai
ξi
aτfτ, uτ∇τ
,
1 − m−2
i1 ai
s
s
T
m−2 ξi
T −sφq 0 aτfτ, uτ∇τ − A∇s−
bi 0 ξi − sφq 0 aτfτ, uτ∇τ − A∇s
i1
0
.
C
m−2
1 − i1 bi
2.38
−
A
0
K u | u ∈ Cld 0, T , ut ≥ 0, inf ut ≥ γ
u
,
t∈0,T 2.39
where γ is the same as in Lemma 2.7. It is obvious that K is a cone in Cld 0, T . By Lemma 2.7,
AK ⊂ K. So by applying Arzela-Ascoli theorem on time scales 16, we can obtain that AK
is relatively compact. In view of Lebesgue’s dominated convergence theorem on time scales
13, it is easy to prove that A is continuous. Hence, A : K→K is completely continuous.
W. Han and G. Zhang
9
Lemma 2.8. A : K→K is completely continuous.
Proof. First, we show that A maps bounded set into bounded set.
Assume c > 0 is a constant and u ∈ Kc {u ∈ K : u
≤ c}. Note that the continuity of f
guarantees that there is c > 0 such that ft, ut ≤ φp c for t ∈ 0, T . So
Au
max |Aut| ≤ C
t∈0,T T
s
T − sφq 0 aτfτ, uτ∇τ − A∇s
≤
m−2
1 − i1 bi
ξi
m−2
T
s
c 0 T − sφq 0 aτ∇τ m−2
i1 ai 0 aτ∇τ/1 −
i1 ai ∇s
≤
.
m−2
1 − i1 bi
0
2.40
That is, AKc is uniformly bounded.
In addition, notice that for any t1 , t2 ∈ 0, T , we have
|Aut1 − Aut2 |
t2
t1
s
s
∇s t2 −sφq
∇s
t2 −t1 φq
aτfτ, uτ∇τ − A
aτfτ, uτ∇τ − A
0
0
ξi
t1
0
m−2
T s
i1 ai 0 aτ∇τ
≤ c |t1 − t2 |
φq
aτ∇τ ∇s
1 − m−2
0
0
i1 ai
m−2 ξi
s
i1 ai 0 aτ∇τ
.
aτ∇τ T max φq
s∈0,T 1 − m−2
0
i1 ai
2.41
So, by applying Arzela-Ascoli theorem on time scales 16, we obtain that AKc is relatively
compact.
Finally, we prove that A : Kc →K is continuous. Suppose that {un }∞
n1 ⊂ Kc and
∞
∗
un t converges to u t uniformly on 0, T . Hence, {Aun t}n1 is uniformly bounded and
equicontinuous on 0, T . The Arzela-Ascoli theorem on time scales 16 tells us that there
∞
exists uniformly convergent subsequence in {Aun t}∞
n1 . Let {Aunm t}m1 be a subsequence
which converges to vt uniformly on 0, T . In addition,
T
ξi
s
m−2
c 0 T − sφq 0 aτ∇τ m−2
i1 ai 0 aτ∇τ/1 −
i1 ai ∇s
0 ≤ Aun t ≤
.
m−2
1 − i1 bi
2.42
Observe the expression of {Aunm t}, and then letting m→∞, we obtain
vt −
t
0
t − sφq
s
0
∗,
∗ ∇s C
aτfτ, u∗ τ∇τ − A
2.43
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Discrete Dynamics in Nature and Society
where
m−2
ξi
aτfτ, u∗ τ∇τ
,
1 − m−2
i1 ai
T
s
1
∗
∗
∗
T − sφq
aτfτ, u τ∇τ − A ∇s
C 1 − m−2
0
0
i1 bi
s
m−2
ξi
∗
∗
bi ξi − sφq
aτfτ, u τ∇τ − A ∇s .
−
∗ −
A
i1
ai
0
0
i1
2.44
0
Here, we have used the Lebesgue dominated convergence theorem on time scales 13. From
the definition of A, we know that vt Au∗ t on 0, T . This shows that each subsequence
∞
∗
of {Aun t}∞
n1 uniformly converges to Au t. Therefore, the sequence {Aun t}n1 uniformly
∗
∗
converges to Au t. This means that A is continuous at u ∈ Kc . So, A is continuous on Kc
since u∗ is arbitrary. Thus, A is completely continuous. This proof is complete.
Lemma 2.9. Let
ϕs φq
s
.
aτfτ, uτ∇τ − A
2.45
0
For ξi i 1, . . . , m − 2,
ξi
ξi − sϕs∇s ≤
0
ξi
T
T
T − sϕs∇s.
2.46
0
Proof. Since
s
aτfτ, uτ∇τ − A
s
0
m−2
aτfτ, uτ∇τ i1
0
ai
ξi
aτfτ, uτ∇τ
1 − m−2
i1 ai
0
2.47
≥ 0,
then ϕs ≥ 0. For all t ∈ 0, T , we have
t
0
t − sϕs∇s
t
∇
t
t
t 0 ϕs∇s − 0 t − sϕs∇s
tρt
≥ 0.
2.48
t
t
In fact, let ψt t 0 ϕs∇s − 0 t − sϕs∇s; taking the nabla derivative of this expression, we
have
t
t
ψ ∇ t ϕs∇s tϕt − ϕs∇s tϕt ≥ 0.
2.49
0
0
Hence, ψt is a nondecreasing function on 0, T . That is,
ψt ≥ 0.
2.50
W. Han and G. Zhang
11
For all t ∈ 0, T ,
t
0
t − sϕs∇s
t
T
0
≤
T − sϕs∇s
T
.
2.51
By 2.51, for ξi i 1, . . . , m − 2, we have
ξi
ξi − sϕs∇s ≤
0
ξi
T
T
T − sϕs∇s.
2.52
0
Lemma 2.10 see 17. Let E be a Banach space, and let K ⊂ E be a cone. Assume Ω1 , Ω2 are open
bounded subsets of E with 0 ∈ Ω1 , Ω1 ⊂ Ω2 , and let
F : K ∩ Ω2 \ Ω1 →K be a completely continuous operator such that
i Fu
≤ u
, u ∈ K ∩ ∂Ω1 , and Fu
≥ u
, u ∈ K ∩ ∂Ω2 , or
ii Fu
≥ u
, u ∈ K ∩ ∂Ω1 , and Fu
≤ u
, u ∈ K ∩ ∂Ω2 .
Then, F has a fixed point in K ∩ Ω2 \ Ω1 .
Now, we introduce the following notations. Let
A0 1−
1
m−2
i1
T
bi
T − sφq
0
s
ξ
−1
ai 0i aτ∇τ
,
∇s
1 − m−2
i1 ai
m−2
aτ∇τ 0
i1
m−2 ξi
T
s
−1
m−2
T i1 bi − m−2
i1 ai 0 aτ∇τ
i1 bi ξi
∇s
T − sφq
aτ∇τ .
B0 T 1 − m−2
1 − m−2
0
0
i1 bi i1 ai
2.53
For l > 0, Ωl {u ∈ K : u
< l}, and ∂Ωl {u ∈ K : u
l},
αl sup{
Au
: u ∈ ∂Ωl },
βl inf{
Au
: u ∈ ∂Ωl },
2.54
by Lemma 2.6, where α and β are well defined.
3. Main results
Theorem 3.1. Assume H1 , H2 , and H3 hold, and assume that the following conditions hold:
A1 pi ∈ C0, ∞, 0, ∞, i 1, 2, and
lim
l→0
p1 l
p−1
< A0 ,
lp−1
p2 l
p−1
< A0 ;
l→∞ lp−1
lim
3.1
A2 ki ∈ L1 0, T , 0, ∞, i 1, 2;
A3 there exist 0 < c1 ≤ c2 and 0 ≤ λ2 < p − 1 < λ1 such that
ft, l ≤ p1 l k1 tlλ1 ,
t, l ∈ 0, T × 0, c1 ,
ft, l ≤ p2 l k2 tl ,
t, l ∈ 0, T × c2 , ∞;
λ2
3.2
12
Discrete Dynamics in Nature and Society
A4 there exists b > 0 such that
min{ft, l : t, l ∈ 0, T × γb, b} ≥ bB0 p−1 .
3.3
Then, problem 1.1-1.2 has at least two positive solutions u∗1 , u∗2 satisfying 0 < u∗1 < b < u∗2 .
Theorem 3.2. Assume H1 , H2 , and H3 hold, and assume that the following conditions hold:
B1 pi ∈ C0, ∞, 0, ∞, i 3, 4, and
lim
l→0
p3 l
>
lp−1
B0
γ
p−1
,
lim
l→∞
p4 l
>
lp−1
B0
γ
p−1
;
3.4
B2 ki ∈ L1 0, T , 0, ∞, i 3, 4;
B3 there exist 0 < c3 ≤ c4 and 0 ≤ λ4 < p − 1 < λ3 such that
ft, l ≥ p3 l − k3 tlλ3 ,
t, l ∈ 0, T × 0, c3 ,
ft, l ≥ p4 l − k4 tl ,
t, l ∈ 0, T × c4 , ∞;
λ4
3.5
B4 there exists a > 0 such that
max{ft, l : t, l ∈ 0, T × 0, a} ≤ aA0 p−1 .
3.6
Then, problem 1.1-1.2 has at least two positive solutions u∗3 , u∗4 satisfying 0 < u∗3 < a < u∗4 .
Proof of Theorem 3.1. Let
p1 l
p2 l
1
p−1
p−1
min A0 − lim p−1 , A0 − lim p−1 ,
l→0 l
l→∞ l
2
3.7
then there exist 0 < a1 ≤ c1 and c2 ≤ a2 < ∞ such that
p−1
− lp−1 ,
0 ≤ l ≤ a1 ,
p−1
A0
− lp−1 ,
a2 ≤ l ≤ ∞.
p1 l ≤ A0
p2 l ≤
3.8
If 0 ≤ l ≤ a1 , u ∈ ∂Ωl , then 0 ≤ ut ≤ l, 0 ≤ t ≤ T . By condition A3 , we have
ft, ut ≤ p1 ut k1 tuλ1 t
p−1
− up−1 t k1 tuλ1 t
p−1
− u
p−1 k1 t
u
λ1
p−1
− lp−1 k1 tlλ1
≤ A0
≤ A0
A0
3.9
W. Han and G. Zhang
13
so that
s
aτfτ, uτ∇τ − A
0
s
m−2
aτfτ, uτ∇τ Therefore,
Au
≤ C
≤
≤
1−
1
m−2
i1
bi
T
1
m−2
1−
T
bi
i1
ai
ξi
aτfτ, uτ∇τ
1 − m−2
0
i1 ai
m−2 ξi
s
p−1
− lp−1 k1 τlλ1 ∇τ
p−1
i1 ai 0 aτA0
p−1
λ1
≤ aτA0 − l k1 τl ∇τ .
m−2
1 − i1 ai
0
3.10
i1
0
T − sϕs∇s −
m−2
ξi
i1
0
0
bi
ξi − sϕs∇s
T − sϕs∇s
0
1
m−2
1 − i1 bi
T
s
p−1
× T − sφq
aτA0 − lp−1 k1 τlλ1 ∇τ
0
0
m−2
i1
ai
ξi
0
− lp−1 k1 τlλ1 ∇τ
∇s.
1 − m−2
i1 ai
p−1
aτA0
It follows that
αl
1
≤
m−2
l
1 − i1 bi
T
s
p−1
× T − sφq
aτA0 − k1 τlλ1 −p1 ∇τ
0
0
m−2
i1
ai
ξi
0
3.11
3.12
− k1 τlλ1 −p1 ∇τ
∇s.
1 − m−2
i1 ai
p−1
aτA0
Noticing λ1 − p 1 > 0, we have
m−2 ξi
T
p−1
s
− ∇τ
αl
1
p−1
i1 ai 0 aτA0
∇s
lim
T
−
sφ
aτA
−
∇τ
≤
q
0
l→0
l
1 − m−2
1 − m−2
0
i1 bi 0
i1 ai
p−1
A0
1−
p−1
A0
1/p−1 − m−2
i1
bi
T
T − sφq
0
s
0
ξ
ai 0i aτ∇τ
∇s
m−2
1 − i1 ai
m−2
aτ∇τ i1
− 1/p−1 A−1
0
−p−1
1 − A0
1/p−1 < 1.
3.13
Therefore, there exist 0 < a1 < a1 such that αa1 < a1 . It implies that Au
< u
, u ∈ ∂Ωa1 .
14
Discrete Dynamics in Nature and Society
If a2 ≤ l < ∞ and u ∈ ∂Ωl , then 0 ≤ ut ≤ l. Similar to the above argument, noticing
that λ2 − p 1 < 0, we can get liml→∞ αl/l < 1. Therefore, there exist 0 < a2 < a2 such that
αa2 < a2 . It implies that Au
< u
, u ∈ ∂Ωa2 .
On the other hand, since f : 0, T × 0, ∞→0, ∞ is continuous, by condition A4 ,
there exist a1 < b1 < b < b2 < a2 such that
min{ft, l : t, l ∈ 0, T × γbi , bi } ≥ bi B0 p−1 ,
i 1, 2.
3.14
If u ∈ ∂Ωb1 , then γb1 ≤ ut ≤ b1 , 0 ≤ t ≤ T . Applying Lemma 2.9, it follows that
Au
max|Aut|
0≤t≤T
T
≥−
T − sϕs∇s 0
≥
≥
m−2
1−
i1 bi
m−2
i1 bi
m−2
1−
T
T
T
i1 bi
m−2
i1 bi
1−
1
m−2
i1
T
bi
T − sϕs∇s −
0
m−2
ξi
i1
0
bi
ξi − sϕs∇s
m−2 ξi
i1 bi 0 ξi − sϕs∇s
T − sϕs∇s −
1 − m−2
0
i1 bi
T
T
T − sϕs∇s −
0
m−2
i1
T 1 −
T
bi ξi
m−2
i1
bi T − sϕs∇s
0
m−2
T
bi − m−2
i1 bi ξi
T − sϕs∇s
T 1 − m−2
0
i1 bi i1
m−2 ξi
T
s
p−1
∇τ
bi − m−2
i1 ai 0 aτb1 B0 p−1
i1 bi ξi
∇s
T
−
sφ
aτb
B
∇τ
q
1 0
T 1 − m−2
1 − m−2
0
0
i1 bi i1 ai
m−2
i1
m−2 ξi
T
s
bi − m−2
i1 ai 0 aτ∇τ
i1 bi ξi
∇s
aτ∇τ b1 B0 T − sφq
T 1 − m−2
1 − m−2
0
0
i1 bi i1 ai
m−2
i1
b1 B0 B0−1 b1 u
.
3.15
In the same way, we can prove that if u ∈ ∂Ωb2 , then Au
≥ u
.
Now, we consider the operator A on Ωb1 \Ωa1 and Ωa2 \Ωb2 , respectively. By Lemma 2.10,
we assert that the operator A has two fixed points u∗1 , u∗2 ∈ K such that a1 ≤ u∗1 ≤ b1 and
b2 ≤ u∗2 ≤ a2 . Therefore, u∗i , i 1, 2, are positive solutions of problem 1.1-1.2.
Proof of Theorem 3.2. Let
p−1
p−1 p3 l
p4 l
1
B0
B0
,
, lim p−1 −
min lim p−1 −
2
γ
γ
l→0 l
l→∞ l
3.16
W. Han and G. Zhang
15
then there exist 0 < b3 ≤ c3 and c4 ≤ b4 < ∞ such that
p3 l ≥
p4 l ≥
B0
γ
B0
γ
p−1
lp−1 ,
p−1
l
0 ≤ l ≤ b3 ,
3.17
p−1
,
b4 ≤ l ≤ ∞.
If 0 ≤ l ≤ b3 , u ∈ ∂Ωl , then γl ≤ ut ≤ l, 0 ≤ t ≤ T . By Lemma 2.9 and condition B3 , we have
Au
max|Aut|
0≤t≤T
≥
≥
T
T
×
m−2
T
bi − m−2
i1 bi ξi
T − sϕs∇s
m−2
T 1 − i1 bi 0
i1
m−2
bi − m−2
i1 bi ξi
m−2
T 1 − i1 bi i1
T
T − sφq
s
0
aτp3 uτ − k3 τuλ3 ∇τ
0
m−2
i1
≥
T
×
ai
ξi
0
aτp3 uτ − k3 τuλ3 ∇τ
∇s
1 − m−2
i1 ai
3.18
m−2
bi − m−2
i1 bi ξi
m−2
T 1 − i1 bi i1
T
T − sφq
s
p−1
B0
p−1
λ3
aτ
γl − k3 τl ∇τ
γ
0
0
m−2
i1
ai
ξi
0
aτB0 /γp−1 γlp−1 − k3 τlλ3 ∇τ
∇s.
1 − m−2
i1 ai
It follows that
m−2
βl T m−2
i1 bi −
i1 bi ξi
≥
m−2
l
T 1 − i1 bi ×
T
0
T − sφq
s
0
p−1
aτB0
m−2
i1
ai
ξi
0
γ p−1 − k3 τlλ3 −p1 ∇τ
γ p−1 − k3 τlλ3 −p1 ∇τ
∇s.
1 − m−2
i1 ai
p−1
aτB0
3.19
16
Discrete Dynamics in Nature and Society
Noticing λ3 − p 1 > 0, we get
m−2
βl T m−2
i1 bi −
i1 bi ξi
lim
≥
m−2
l
l→0
T 1 − i1 bi ×
T
T − sφq
s
0
0
p−1
B0
p−1
aτB0
m−2
γ
p−1
∇τ i1
ai
ξi
0
p−1
aτB0 γ p−1 ∇τ
∇s
1 − m−2
i1 ai
γ p−1 1/p−1 B0−1
−p−1
1 γ p−1 B0
1/p−1 > 1.
3.20
Therefore, there exists b3 with 0 < b3 < a such that βb3 > b3 . It implies that Au
> u
for
u ∈ ∂Ωb3 .
If b4 ≤ γl < ∞ and u ∈ ∂Ωl , then b4 ≤ γl ≤ ut ≤ l, 0 ≤ t ≤ T . Similar to the above
argument, noticing that λ4 − p 1 < 0, we can get liml→∞ βl/l > 1.
Therefore, there exist b4 with 0 < b4 < ∞ such that βb4 > b4 . It implies that Au
> u
for u ∈ ∂Ωb4 .
By condition B4 , we can see that there exist b3 < a3 < a < a4 < b4 such that
max{ft, l : t, l ∈ 0, T × 0, ai } ≤ ai A0 p−1 ,
i 3, 4.
3.21
If u ∈ ∂Ωa3 , then 0 ≤ ut ≤ a3 , 0 ≤ t ≤ T , and ft, ut ≤ a3 A0 p−1 . It follows that
Au
≤
≤
1−
1−
1
m−2
i1
1
m−2
i1
T
bi
T − sϕs∇s
0
T
bi
a3 A0
T − sφq
0
s
0
ξ
ai 0i aτ∇τ
∇s
1 − m−2
i1 ai
m−2
aτ∇τ i1
3.22
a3 u
.
Similarly, if u ∈ ∂Ωa4 , then Au
≤ u
.
Now, we study the operator A on Ωa3 \ Ωb3 and Ωb4 \ Ωa4 , respectively. By Lemma 2.10,
we assert that the operator A has two fixed points u∗3 , u∗4 ∈ K such that b3 ≤ u∗3 ≤ a3 and
a4 ≤ u∗4 ≤ b4 . Therefore, u∗i , i 3, 4, are positive solutions of problem 1.1-1.2.
4. Further discussion
If the conditions of Theorems 3.1 and 3.2 are weakened, we will get the existence of single
positive solution of problem 1.1-1.2.
Corollary 4.1. Assume H1 , H2 , and H3 hold, and assume that the following conditions hold:
p−1
C1 p1 ∈ C0, ∞, 0, ∞, and liml→0 p1 l/lp−1 < A0 ;
C2 k1 ∈ L1 0, T , 0, ∞;
W. Han and G. Zhang
17
C3 there exist c1 > 0 and λ1 > p − 1 such that
ft, l ≤ p1 l k1 tlλ1 ,
t, l ∈ 0, T × 0, c1 ;
4.1
min{ft, l : t, l ∈ 0, T × γb, b} ≥ bB0 p−1 .
4.2
C4 there exists b > 0 such that
Then, problem 1.1-1.2 has at least one positive solution.
Corollary 4.2. Assume H1 , H2 , and H3 hold, and assume that the following conditions hold:
p−1
D1 p2 ∈ C0, ∞, 0, ∞, and liml→∞ p2 l/lp−1 < A0 ;
D2 k2 ∈ L1 0, T , 0, ∞;
C3 there exist c2 > 0 and 0 ≤ λ2 < p − 1 such that
ft, l ≤ p2 l k2 tlλ2 ,
t, l ∈ 0, T × c2 , ∞;
4.3
D4 there exists b > 0 such that
min{ft, l : t, l ∈ 0, T × γb, b} ≥ bB0 p−1 .
4.4
Then, problem 1.1-1.2 has at least one positive solution.
Corollary 4.3. Assume H1 , H2 , and H3 hold, and assume that the following conditions hold:
E1 p3 ∈ C0, ∞, 0, ∞, and liml→0 p3 l/lp−1 > B0 /γp−1 ;
E2 k3 ∈ L1 0, T , 0, ∞;
E3 there exist c3 > 0 and λ3 > p − 1 such that
ft, l ≥ p3 l − k3 tlλ3 ,
t, l ∈ 0, T × 0, c3 ;
4.5
max{ft, l : t, l ∈ 0, T × 0, a} ≤ aA0 p−1 .
4.6
E4 there exists a > 0 such that
Then, problem 1.1-1.2 has at least one positive solution.
Corollary 4.4. Assume H1 , H2 , and H3 hold, and assume that the following conditions hold:
F1 p4 ∈ C0, ∞, 0, ∞, and liml→∞ p4 l/lp−1 > B0 /γp−1 ;
F2 k4 ∈ L1 0, T , 0, ∞;
F3 there exist c4 > 0 and 0 ≤ λ4 < p − 1 such that
ft, l ≥ p4 l − k4 tlλ4 ,
t, l ∈ 0, T × c4 , ∞;
4.7
F4 there exists a > 0 such that
max{ft, l : t, l ∈ 0, T × 0, a} ≤ aA0 p−1 .
4.8
Then, problem 1.1-1.2 has at least one positive solution.
The proof of the above results is similar to those of Theorems 3.1 and 3.2; thus we omit
it.
18
Discrete Dynamics in Nature and Society
5. Some examples
In this section, we present a simple example to explain our results. We only study the case
T R, 0, T 0, 1.
Let ft, 0 ≡ 0. Consider the following BVP:
where
φ3 u ft, ut 0, t ∈ 0, 1,
1
1
1
1
φ3 u 0 φ3 u
u1 u
,
u 0 0,
,
2
2
2
2
5.1
⎧
1
2 √ 5
⎪
3
⎪
,
min
u,
t, u ∈ 0, 1 × 0, 1,
223u
⎪
⎪
⎪
t1 − t u
⎪
⎪
⎨
t, u ∈ 0, 1 × 1, 3,
ft, u 225,
⎪
√
⎪
⎪
⎪
2u √ 3
3
1
⎪
⎪
⎪
,
u , t, u ∈ 0, 1 × 3, ∞.
⎩74u 6 min t1 − t 3
5.2
It is easy to check that f : 0, 1 × 0, ∞→0, ∞ is continuous. In this case, p 3, at ≡
1, m 3, a1 b1 1/2, and ξ1 1/2, and it follows from a direct calculation that
1
A0 1 − 1/2
1
−1
1/2·1/2
ds
1 − sφq s 1.1062,
1 − 1/2
0
1
b1 1 − ξ1 1/21 − 1/2
.
γ
1 − b1 ξ1
1 − 1/2·1/2 3
5.3
We have
b1 − b1 ξ1
B0 1 − b1
T
−1
a1 ξ1
ds
1 − sφq s 1 − a1
0
1/2 − 1/2·1/2
1 − 1/2
1
−1
1/2·1/2 1/2
1 − s s ds
1 − 1/2
0
5.4
4.4248 < 5.
Choosing c1 1, c2 3, b 3, λ1 5/2, λ2 3/2, p1 u 223u3 , p2 u 74u, and k1 t k2 t 1/ t1 − t, it is easy to check that
ft, u ≤ p1 u k1 tu5/2 ,
t, u ∈ 0, 1 × 0, 1,
ft, u ≤ p2 u k2 tu3/2 ,
t, u ∈ 0, 1 × 3, ∞,
p1 u
223u3
lim
0 < A20 1.10622 ,
u→0 u2
l→0
u2
lim
lim
u→∞
p2 u
74u
lim 2 0 < A20 1.10622 ,
2
l→∞ u
u
min{ft, u : t, u ∈ 0, 1 × 1, 3} 225 > 13.2742 bB0 2 .
5.5
W. Han and G. Zhang
19
It follows that f satisfies the conditions A1 –A4 of Theorem 3.1; then problem 5.1 has at
least two positive solutions.
Acknowledgment
Project supported by the National Natural Science Foundation of China Grant NO. 10471040.
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