Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2008, Article ID 143040, 16 pages
doi:10.1155/2008/143040
Research Article
Positive Solutions for Third-Order Nonlinear
p-Laplacian m-Point Boundary Value Problems
on Time Scales
Fuyi Xu
School of Mathematics and Information Science, Shandong University of Technology,
Zibo, Shandong 255049, China
Correspondence should be addressed to Fuyi Xu, zbxufuyi@163.com
Received 7 August 2008; Revised 8 October 2008; Accepted 9 November 2008
Recommended by Leonid Shaikhet
We study the following third-order p-Laplacian m-point boundary value problems on time scales:
∇
Δ∇
φp uΔ∇ atft, ut 0, t ∈ 0, T T , βu0 − γuΔ 0 0, uT m−2
0 i1 ai uξi , φp u
m−2
p−2
Δ∇
−1
b
φ
u
ξ
,
where
φ
s
is
p-Laplacian
operator,
that
is,
φ
s
|s|
s,
p
>
1,
φ
i
p
i
p
p
p φq ,
i1
1/p 1/q 1, 0 < ξ1 < · · · < ξm−2 < ρT . We obtain the existence of positive solutions by using
fixed-point theorem in cones. The conclusions in this paper essentially extend and improve the
known results.
Copyright q 2008 Fuyi Xu. 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
The theory of time scales was initiated by Hilger 1 as a means of unifying and extending
theories from differential and difference equations. The study of time scales has lead to
several important applications in the study of insect population models, neural networks,
heat transfer, and epidemic models, see, for example 2–6. Recently, the boundary value
problems with p-Laplacian operator have also been discussed extensively in the literature,
for example, see 7–13.
A time scale T is a nonempty closed subset of R. We make the blanket assumption that
0, 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 14, Anderson considered the the following third-order nonlinear boundary value
problem BVP:
x t f t, xt , t1 ≤ t ≤ t3 ,
x t1 x t2 0,
γx t3 δx t3 0.
1.1
2
Discrete Dynamics in Nature and Society
Author studied the existence of solutions for the nonlinear boundary value problem by
using the Krasnoselskii’s fixed point theorem and Leggett and Williams fixed point theorem,
respectively.
In 8, 9, He considered the existence of positive solutions of the p-Laplacian dynamic
equations on time scales
∇
atf ut 0,
φp uΔ
t ∈ 0, T T ,
1.2
satisfying the boundary conditions
u0 − B0 uΔ η 0,
uΔ T 0,
1.3
uT − B1 uΔ η 0,
1.4
or
uΔ 0 0,
where η ∈ 0, ρT . He obtained the existence of at least double and triple positive solutions
of the boundary value problems by using a new double fixed point theorem and triple fixed
point theorem, respectively.
In 13, Zhou and Ma firstly studied the existence and iteration of positive solutions for
the following third-order generalized right-focal boundary value problem with p-Laplacian
operator:
u0 φp u t qtf t, ut ,
m
αi u ξi ,
u η 0,
0 ≤ t ≤ 1,
u 1 i1
n
βi u θi .
1.5
i1
They established a corresponding iterative scheme for the problem by using the monotone
iterative technique.
However, to the best of our knowledge, little work has been done on the existence
of positive solutions for third-order p-Laplacian m-point boundary value problems on time
scales. This paper attempts to fill this gap in the literature.
In this paper, by using different method, we are concerned with the existence
of positive solutions for the following third-order p-Laplacian m-point boundary value
problems on time scales:
∇
atf t, ut 0,
φp uΔ∇
βu0 − γuΔ 0 0,
uT t ∈ 0, T T ,
m−2
m−2
bi φp uΔ∇ ξi ,
φp uΔ∇ 0 i1
i1
ai u ξi ,
1.6
where φp s is p-Laplacian operator, that is, φp s |s|p−2 s, p > 1, φp−1 φq , 1/p1/q 1,
Fuyi Xu
3
and ai , bi , a, f satisfy
H1 β, γ ≥ 0, β γ > 0, ai ∈ 0, ∞, i 1, 2, . . . , m − 3, am−2 > 0, 0 < ξ1 < · · · < ξm−2 <
m−2
m−2
m−2
ρT , 0 < m−2
i1 bi < 1, 0 <
i1 ai ξi < T , d βT −
i1 ai ξi γ1 −
i1 ai > 0;
H2 f : 0, T T × 0, ∞ → R is continuous, a ∈ Cld 0, T T , R and there exists
t0 ∈ ξm−2 , T T such that at0 > 0, where R 0, ∞.
2. Preliminaries and lemmas
For convenience, we list the following definitions which can be found in 1–5.
Definition 2.1. A time scale T is a nonempty closed subset of real numbers R. For t < sup T and
r > inf T, define the forward jump operator σ and 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; if ρr < r, r is said to be left scattered;
if σt t, t is said to be right dense; 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 it exists, with the property that for each > 0, there is a
neighborhood U of t such that
f σt − fs − f Δ t σt − s ≤ σt − s,
2.2
for all s ∈ U.
For f : T → R and t ∈ Tk , the nabla derivative of f at t, denoted by f ∇ t provided it
exists, with the property that for each > 0, there is a neighborhood U of t such that
f ρt − fs − f ∇ t ρt − 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.
Definition 2.4. If φΔ t ft, then one defines the delta integral by
b
a
ftΔt φb − φa.
2.4
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Discrete Dynamics in Nature and Society
If F ∇ t ft, then one defines the nabla integral by
b
ft∇t Fb − Fa.
2.5
a
Lemma 2.5. If d βT −
value problem (BVP)
m−2
i1
ai ξi γ1 −
m−2
i1
ai > 0, then for h ∈ Cld 0, T T , the boundary
uΔ∇ ht 0,
t ∈ 0, T T ,
βu0 − γuΔ 0 0,
m−2
ai u ξi
uT 2.6
i1
has the unique solution
ut −
t
t − shs∇s 0
βt γ m−2
−
ai
d i1
ξi
βt γ
d
T
T − shs∇s
0
2.7
ξi − s hs∇s.
0
Proof. By direct computation, we can easily get 2.7. So, we omit it.
m−2
m−2
m−2
Lemma 2.6. If 0 < m−2
i1 bi < 1, 0 <
i1 ai ξi < T , d βT −
i1 ai ξi γ1 −
i1 ai > 0, then
for h ∈ Cld 0, T T , the boundary value problem (BVP)
∇
ht 0,
φp uΔ∇
βu0 − γuΔ 0 0,
uT t ∈ 0, T T ,
m−2
m−2
bi φp uΔ∇ ξi
φp uΔ∇ 0 i1
i1
ai u ξi ,
2.8
has the unique solution
t
s
βt γ T
hr∇r B ∇s T − sφq
hr∇r B ∇s
d
0
0
0
0
ξ
s
m−2
i
βt γ −
ai
hr∇r B ∇s,
ξi − s φq
d i1
0
0
ut −
where B t − sφq
s
2.9
m−2 ξi
m−2
i1 bi 0 hr∇r/1 −
i1 bi .
Proof. Integrating both sides of 1.6 on 0, t, we have
φp u
Δ∇
t φp u
Δ∇
0 −
t
hr∇r.
0
2.10
Fuyi Xu
5
So
φp u
Δ∇
ξi
ξi φp uΔ∇ 0 −
2.11
hr∇r.
0
By boundary value condition φp uΔ∇ 0 m−2
i1
bi φp uΔ∇ ξi , we have
m−2 ξi
i1 bi 0 hr∇r
.
φp uΔ∇ 0 −
1 − m−2
i1 bi
2.12
By 2.10 and 2.12, we know
Δ∇
u
t
m−2 ξi
i1 bi 0 hr∇r
t −φq
hr∇r .
m−2
1 − i1 bi
0
2.13
This together with Lemma 2.5 implies that
t
s
βt γ T
ut − t − sφq
hr∇r B ∇s T − sφq
hr∇r B ∇s
d
0
0
0
0
ξi
s
βt γ m−2
−
ai
hr∇r B ∇s,
ξi − s φq
d i1
0
0
where B s
2.14
m−2 ξi
m−2
i1 bi 0 hr∇r/1 −
i1 bi . The proof is complete.
m−2
Lemma 2.7. Let 0 <
u of 2.8 satisfies
i1
ai ξi < 1, d > 0. If h ∈ Cld 0, T T and ht ≥ 0, then the unique solution
ut ≥ 0.
2.15
t
ξi
m−2
Proof. By uΔ∇ t −φq m−2
i1 bi 0 hr∇r/1 −
i1 bi 0 hr∇r ≤ 0, we can know that the
graph of ut is concave down on 0, T T . So we only prove u0 ≥ 0, uT ≥ 0.
Firstly, we will prove u0 ≥ 0 by the following two perspectives.
i If 0 < m−2
i1 ai ≤ 1, we have
γ
u0 d
γ
≥
d
γ
d
T
T − sφq
s
0
0
T
s
T − sφq
0
1−
hr∇r B ∇s −
hr∇r B ∇s −
0
m−2
T
i1
0
ai
T − sφq
s
0
m−2
ξi
i1
0
m−2
T
i1
0
ai
ai
ξi − s φq
s
hr∇r B ∇s
0
T − sφq
hr∇r B ∇s ≥ 0.
s
0
hr∇r B ∇s
2.16
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Discrete Dynamics in Nature and Society
ii If
m−2
i1
γ
u0 d
γ
≥
d
γ
d
T
ai > 1, by 2.8, we have
s
T − sφq
0
0
T
s
T − sφq
0
hr∇r B ∇s −
hr∇r B ∇s −
0
T T−
0
m−2
ai ξi
i1
m−2
m−2
ξi
i1
0
m−2
T
i1
0
ai
ai
ai − 1 s φq
s
ξi − s φq
s
hr∇r B ∇s
0
ξi − s φq
s
hr∇r B ∇s
0
hr∇r B ∇s ≥ 0.
0
i1
2.17
On the other hand, we have
s
βγ T
hr∇r B ∇s T − sφq
hr∇r B ∇s
d
0
0
0
0
ξ
s
m−2
i
βγ −
ai
hr∇r B ∇s
ξi − s φq
d i1
0
0
s
ξi β m−2
≥
ai
hr∇r B ∇s
ξi T − s − T ξi − s φq
d i1
0
0
s
T
m−2
ai ξi T − sφq
hr∇r B ∇s
uT −
T
T − sφq
i1
s
ξi
0
T
s
s
T
γ m−2
ai
T − sφq
hr∇r B ∇s −
hr∇r B ∇s
ξi − s φq
d i1
0
0
0
0
T
s
s
ξi β m−2
ai
hr∇r B ∇s ξi T − sφq
hr∇r B ∇s
T − ξi sφq
d i1
0
ξi
0
0
s
T
γ m−2
T − ξi φq
ai
hr∇r B ∇s ≥ 0.
d i1
0
0
2.18
The proof is completed.
Lemma 2.8. Let ai ≥ 0, i 1, . . . , m − 2, 0 < m−2
i1 ai ξi < T, d > 0. If h ∈ Cld 0, T T and ht ≥ 0,
then the unique positive solution ut of (BVP) 2.8 satisfies
inf
t∈ξm−2 ,T T
ut ≥ σ||u||,
2.19
where σ min{am−2 T − ξm−2 /T − am−2 ξm−2 , am−2 ξm−2 /T, ξm−2 /T }, ||u|| supt∈0,T T |ut|.
Fuyi Xu
7
Proof. Let ut maxt∈0,T T ut ||u||, we shall discuss it from the following two
perspectives.
Case 1. If 0 < m−2
i1 ai < 1.
Firstly, assume t < ξm−2 < T , then mint∈ξm−2 ,T T ut uT . By uT m−2
i1 ai uξi ≥
am−2 uξm−2 , we have
uT − u ξm−2
T
T
0 − T uT −
uT u ξm−2
ut ≤ uT T − ξm−2
T − ξm−2
T − ξm−2
T
T
T − am−2 ξm−2
≤ uT 1 −
uT .
T − ξm−2 am−2 T − ξm−2
am−2 T − ξm−2
2.20
So
am−2 T − ξm−2
min ut ≥
||u||.
t∈ξm−2 ,T T
T − am−2 ξm−2
2.21
Secondly, assume ξm−2 < t < T , then mint∈ξm−2 ,T T ut uT . Otherwise, we have
mint∈ξm−2 ,T T ut uξm−2 , then t ∈ ξm−2 , T T , uξm−2 ≥ uξm−1 ≥ · · · ≥ uξ2 ≥ uξ1 .
By 0 < m−2
i1 ai < 1, we have
uT m−2
m−2
ai u ξi ≤
ai u ξm−2 < u ξm−2 ≤ uT ,
i1
2.22
i1
a contradiction.
By concave of ut, we get uξm−2 /ξm−2 ≥ ut/t ≥ ut/T . In fact, since uT ≥
am−2 uξm−2 , then uT /am−2 ξm−2 ≥ ut/T , which implies
min ut ≥
t∈ξm−2 ,T T
am−2 ξm−2
||u||.
T
2.23
Case 2. If m−2
i1 ai > 1.
Firstly, assume uξm−2 ≤ uT , then mint∈ξm−2 ,T T ut uξm−2 . By concave of ut, we
have t ∈ ξm−2 , tT , which implies uξm−2 /ξm−2 ≥ ut/t ≥ ut/T , then
min ut ≥
t∈ξm−2 ,T T
ξm−2
||u||.
T
2.24
Secondly, assume uξm−2 > ut, then mint∈ξm−2 ,T T ut uT , and t ∈ ξ1 , T T . If not,
t ∈ 0, ξ1 , then uξ1 ≥ · · · ≥ uξm−2 > uT . So, we have
uT m−2
m−2
ai u ξi > uT ai ≥ uT ,
i1
i1
2.25
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Discrete Dynamics in Nature and Society
a contradiction. By m−2
i1 ai > 1, there exists ξ ∈ {ξ1 , ξ2 , . . . , ξm−2 } such that uξ ≤ uT , then
uξ1 ≤ uξ2 ≤ · · · ≤ uξm−2 ≤ u1. By concave of ut, we have u1/ξ1 ≥ uξ1 /ξ1 ≥ ut/t ≥
ut/T , then
min ut ≥ ξ1 ||u||.
2.26
ut ≥ σ||u||,
2.27
t∈ξm−2 ,T T
Therefore, by 2.21–2.26, we have
inf
t∈ξm−2 ,T T
where σ min{am−2 T − ξm−2 /T − am−2 ξm−2 , am−2 ξm−2 /T, ξm−2 /T }. The proof is complete.
Let E Cld 0, T T be endowed with the ordering x ≤ y if xt ≤ yt, for all t ∈ 0, T T ,
and u
maxt∈0,T T |ut| is defined as usual by maximum norm. Clearly, it follows that
E, u
is a Banach space.
We define a cone by
K u : u ∈ E, ut is concave, nonnegative on 0, T T ,
inf
t∈ξm−2 ,T T
ut ≥ σ||u|| .
2.28
Define an operator S : K → E by setting
Sut −
t
t − sφq
0
βt γ
d
−
T
0
s
arf r, ur ∇r A ∇s
0
T − sφq
s
arf r, ur ∇r A ∇s
0
2.29
s
ξi βt γ m−2
ai
arf r, ur ∇r A ∇s,
ξi − s φq
d i1
0
0
ξi
m−2
where A m−2
i1 bi 0 arfr, ur∇r/1 −
i1 bi . Obviously, u is a solution of boundary
value problem 1.6 if and only if u is a fixed point of operator S.
Lemma 2.9. S : K → K is completely continuous.
Proof. By H2 and Lemmas 2.7-2.8, we easily get SK ⊂ K. By Arzela-Ascoli theorem and
Lebesgue dominated convergence theorem, we can easily prove S is completely continuous.
Lemma 2.10 see 15. Let K be a cone in a Banach space X. Let D be an open bounded subset of X
∅ and DK /
Ax
K. Assume that A : DK → K is a compact map such that x /
with DK D ∩ K /
for x ∈ ∂DK . Then the following results hold.
Fuyi Xu
9
1 If Ax
≤ x
, x ∈ ∂DK , then iK A, DK 1.
Ax λx0, for all x ∈ ∂DK and all λ > 0, then
2 If there exists x0 ∈ K \ {0} such that x /
iK A, DK 0.
3 Let U be open in X such that U ⊂ DK . If iK A, DK 1 and iK A, UK 0, then A has a
fixed point in DK \ UK . The same result holds if iK A, DK 0 and iK A, UK 1, where
iK A, DK denotes fixed point index.
One defines
Kρ ut ∈ K : u
< ρ ,
Ωρ ut ∈ K : min xt < σρ .
ξm−2 ≤t≤T
2.30
Lemma 2.11 see 15. Ωρ defined above has the following properties:
a Kσρ ⊂ Ωρ ⊂ Kρ ;
b Ωρ is open relative to K;
c x ∈ ∂Ωρ if and only if minξm−2 ≤t≤T xt σρ;
d if x ∈ ∂Ωρ , then σρ ≤ xt ≤ ρ, for t ∈ ξm−2 , T T .
For the convenience, we introduce the following notations:
ρ
fσρ
ft, u
: u ∈ σρ, ρ ,
min min
ξm−2 ≤t≤T φp ρ
u→α
1
1
M d
T
ξm−2
ft, u
: u ∈ 0, ρ ,
max max
0≤t≤T φp ρ
ft, u
α : ∞ or 0 ,
φp u
m−2 ξi
T
βT γ T
1
i1 bi 0 ar∇r
T − s∇sφq
ar∇r ,
m
d
1 − m−2
0
0
i1 bi
s
m−2
m−2
T −sφq
ar∇r ∇s min βξm−2 γ, β max
ai ξ1 , am−2 ξm−2 γ ai .
f α lim sup max
0≤t≤T
ft, u
,
φp u
ρ
f0
fα lim inf max
u→α
ξm−2 ≤t≤T
ξm−2
i1
i1
2.31
Lemma 2.12. If f satisfies the following condition:
ρ
f0 ≤ φp m,
u/
Su, u ∈ ∂Kρ ,
2.32
then
iK S, Kρ 1.
2.33
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Discrete Dynamics in Nature and Society
Proof. For u ∈ ∂Kρ , then from 2.32, we have
m−2 ξi
i1 bi 0 arfr, ur∇r
arf r, ur ∇r A arf r, ur ∇r 1 − m−2
0
0
i1 bi
m−2 ξi
T
i1 bi 0 arfr, ur∇r
≤ arf r, ur ∇r 1 − m−2
0
i1 bi
ξi
T
m−2
i1 bi 0 ar∇r
≤ φp mρ
ar∇r .
1 − m−2
0
i1 bi
s
s
2.34
So that
s
φq
arf r, ur ∇r A
≤ mρφq
m−2 ξi
i1 bi 0 ar∇r
ar∇r .
m−2
1 − i1 bi
0
T
0
2.35
Therefore,
Sut ≤
βt γ
d
T
T − sφq
0
s
arf r, ur ∇r A ∇s
0
βT γmρ
≤
d
T
T − s∇sφq
0
m−2 ξi
i1 bi 0 ar∇r
ar∇r 1 − m−2
0
i1 bi
T
2.36
ρ.
This implies that ||Su|| ≤ ||u|| for u ∈ ∂Kρ . Hence, by Lemma 2.101 it follows that
iK S, Kρ 1.
Lemma 2.13. If f satisfies the following condition:
ρ
fσρ ≥ φp Mσ,
u/
Su, u ∈ ∂Ωρ ,
2.37
then
iK S, Ωρ 0.
2.38
Proof. Let et ≡ 1 for t ∈ 0, T T . Then e ∈ ∂K1 . We claim that
u/
Su λe,
u ∈ ∂Ωρ , λ > 0.
2.39
Fuyi Xu
11
ρ
In fact, if not, there exist u0 ∈ ∂Ωρ and λ0 > 0 such that u0 Su0 λ0 e. By fσρ ≥ φp Mσ, we
have
m−2 ξi
i1 bi 0 arfr, u0 r∇r
arf r, u0 r ∇r A arf r, u0 r ∇r 1 − m−2
0
0
i1 bi
s
≥
arf r, ur ∇r
s
s
ξm−2
≥ φp Mσρ
s
2.40
ar∇r .
ξm−2
So that
s
s
arf r, ur ∇r A ≥ Mσρφq
φq
0
ar∇r .
2.41
ξm−2
By 16, Theorem 2.2iv, for t > 0, we have
t
0
t − sφq
s
0
Δ
arfr, u0 r∇r A ∇s
t
t
0
sφq
s
0
arfr, u0 r∇r A ∇s
tσt
≥ 0.
2.42
So, for i 1, 2, . . . , m − 2, we have
ξm−2 0
s
ξm−2 − s φq 0 arfr, u0 r∇r A ∇s
ξm−2
s
ξi − s φq 0 arfr, u0 r∇r A ∇s
ξi 0
≥
ξi
2.43
Therefore,
Su0 ξm−2
T
s
β
ξm−2 t − sφq
arf r, u0 r ∇r A ∇s
≥
d
0
0
ξm−2
s
arf r, u0 r ∇r A ∇s
−T
ξm−2 − s φq
γ
d
T
0
t − sφq
0
−
s
0
arf r, u0 r ∇r A ∇s
0
ξm−2
0
ξm−2 − s φq
s
0
.
arf r, u0 r ∇r A ∇s
12
Discrete Dynamics in Nature and Society
≥
≥
βξm−2 γ
d
T
T − sφq
s
ξm−2
arf r, u0 r ∇r A ∇s
0
βξm−2 γMσρ
d
T
T − sφq
s
ξm−2
ar∇r ∇s,
ξm−2
2.44
s
ξi β m−2
Su0 T ≥
ai
arf r, u0 r ∇r A ∇s
t − ξi sφq
d i1
0
0
ξi
T
T − sφq
s
ξi
arf r, u0 r ∇r A ∇s
0
s
T
γ m−2
t − sφq
arf r, u0 r ∇r A ∇s
d i1 0
0
−
ξi
T − sφq
s
0
0
≥
β m−2
ai ξi
d i1
arf r, u0 r ∇r A ∇s
T
t − sφq
s
ξi
arf r, u0 r ∇r A ∇s
0
s
T
γ m−2
t − sφq
arf r, u0 r ∇r A ∇s
d i1 ξi
0
s
m−2
m−2
T
Mσρ
β max
≥
ai ξ1 , am−2 ξm−2 γ
T − sφq
ar∇r ∇s.
d
ξm−2
ξm−2
i1
i1
2.45
Obviously, we can know
min Su0 t min Su0 ξm−2 , Su0 T t∈ξm−2 ,T T
Mσρ
≥
d
T
T − sφq
s
ξm−2
ar∇r ∇s
ξm−2
× min βξm−2 γ, β max
m−2
ai ξ1 , am−2 ξm−2
i1
γ
m−2
2.46
i1
≥ σρ.
For t ∈ ξm−2 , T T , then
u0 t Su0 t λ0 et ≥ min Su0 t λ0
t∈ξm−2 ,T min Su0 ξm−2 , Su0 T λ0 ≥ σρ λ0 .
2.47
Fuyi Xu
13
This together with Lemma 2.11c implies that
σρ ≥ σρ λ0 ,
2.48
a contradiction. Hence, by Lemma 2.102, it follows that iK S, Ωρ 0.
3. Main results
We now give our results on the existence of positive solutions of BVP 1.6.
Theorem 3.1. Suppose conditions H1 and H2 hold, and assume that one of the following
conditions holds.
ρ
ρ
H3 There exist ρ1 , ρ2 ∈ 0, ∞ with ρ1 < σρ2 such that f0 1 ≤ φp m, fσρ2 2 ≥ φp Mσ.
ρ
ρ
H4 There exist ρ1 , ρ2 ∈ 0, ∞ with ρ1 < ρ2 such that f0 2 ≤ φp m, fσρ1 1 ≥ φp Mσ.
Then, the boundary value problem 1.6 has at least one positive solution.
ρ
Proof. Assume that H3 holds, we show that S has a fixed point u1 in Ωρ2 \K ρ1 . By f0 1 ≤ φp m
and Lemma 2.12, we have that
iK S, Kρ1 1.
3.1
ρ
By fσρ2 2 ≥ φp Mσ and Lemma 2.13, we have that
iK S, Kρ2 0.
3.2
By Lemma 2.11a and ρ1 < σρ2 , we have K ρ1 ⊂ Kσρ2 ⊂ Ωρ2 . It follows from Lemma 2.103
that S has a fixed point u1 in Ωρ2 \ K ρ1 . When condition H4 holds, the proof is similar to the
above, so we omit it here.
As a special case of Theorem 3.1, we obtain the following result.
Corollary 3.2. Suppose conditions H1 and H2 hold, and assume that one of the following
conditions holds.
H5 0 ≤ f 0 < φp m and φp M < f∞ ≤ ∞.
H6 0 ≤ f ∞ < φp m and φp M < f0 ≤ ∞.
Then, the boundary value problem 1.6 has at least one positive solution.
Theorem 3.3. Assume conditions H1 and H2 hold, and suppose that one of the following
conditions holds.
H7 There exist ρ1 , ρ2 , and ρ3 ∈ 0, ∞ with ρ1 < σρ2 and ρ2 < ρ3 such that
ρ
f0 1 ≤ φp m,
ρ
fσρ2 2 ≥ φp Mσ,
u/
Su,
∀ u ∈ ∂Ωρ2 ,
ρ
f0 3 ≤ φp m.
3.3
14
Discrete Dynamics in Nature and Society
H8 There exist ρ1 , ρ2 , and ρ3 ∈ 0, ∞ with ρ1 < ρ2 < σρ3 such that
ρ
f0 2 ≤ φp m,
ρ
fσρ1 1 ≥ φp Mσ,
u/
Su,
∀ u ∈ ∂Kρ2 ,
ρ
fσρ3 3 ≥ φp Mσ.
3.4
ρ
Then, the boundary value problem 1.6 has at least two positive solutions. Moreover, if in H7 f0 1 ≤
ρ
φp m is replaced by f0 1 < φp m, then the BVP 1.6 has a third positive solution u3 ∈ Kρ1 .
Proof. Assume that condition H7 holds, we show that either S has a fixed point u1 in ∂Kρ1
or Ωρ2 \ K ρ1 . If u /
Su for u ∈ ∂Kρ1 ∪ ∂Kρ3 . By Lemma 2.12 and Lemma 2.13, we have that
iK S, Kρ1 1,
iK S, Kρ3 1,
iK S, Kρ2 0.
3.5
By Lemma 2.11a and ρ1 < σρ2 , we have K ρ1 ⊂ Kσρ2 ⊂ Ωρ2 . It follows from Lemma 2.103
that S has a fixed point u1 in Ωρ2 \ K ρ1 . Similarly, S has a fixed point u2 in Kρ3 \ Ωρ2 . When
condition H8 holds, the proof is similar to the above, so we omit it here.
As a special case of Theorem 3.3, we obtain the following result.
Corollary 3.4. Assume conditions H1 and H2 hold, if there exists ρ > 0 such that one of the
following conditions holds.
ρ
Su, ∀u ∈ ∂Ωρ and 0 ≤ f ∞ < φp m.
H9 0 ≤ f 0 < φp m, fσρ ≥ φp Mσ, u /
ρ
H10 φp M < f0 ≤ ∞, f0 ≤ φp m, u / Su, ∀u ∈ ∂Kρ and φp M < f∞ ≤ ∞.
Then, the boundary value problem 1.6 has at least two positive solutions.
4. Some examples
In this section, we present some simple examples to explain our results. We only study the
case T R, 0, T T 0, 1.
Example 4.1. Consider the following three-point boundary value problem with p-Laplacian:
u 0 0,
φp u atft, u 0, 0 < t < 1,
1
1
1
1
,
φp u 0 φp u
,
u1 u
2
3
4
3
4.1
where β 0, γ 1, a1 1/2, b1 1/4, ξ1 1/3, at 1, p q 2. By computing, we can
know σ 1/6, M 819/16, m 9/10. Let ρ1 1, ρ2 208, then σρ1 < ρ1 < σρ2 < ρ2 . We
Fuyi Xu
15
define a nonlinearity f as follows:
⎧ 3
⎪
9t3 1
⎪
⎪
−u ,
⎪
⎪
⎪
10 6
⎪
⎪
⎪
⎪
⎪
⎪
⎪
9t3
6π
1π
⎪
⎪
sin
u
−
,
⎪
⎪
⎪ 10
52
52
⎪
⎪
⎪
⎪
⎪
⎨ 9t3 208
6
819 6
6
ft, u −
u u−
,
⎪
10 202 202
96 202
202
⎪
⎪
⎪
⎪
⎪
2
⎪
⎪
⎪
⎪ 819 t3 u − 208 ,
⎪
⎪
⎪
6
⎪ 96
⎪
⎪
⎪
⎪
⎪
⎪ 819
⎪
208 2 ⎪
3
⎪
1 u − 208 ,
t
208
−
⎩
96
6
1
0 < t < 1, u ∈ 0, ,
6
1
0 < t < 1, u ∈
,1 ,
6
208
0 < t < 1, u ∈ 1,
,
6
4.2
208
, 208 ,
0 < t < 1, u ∈
6
0 < t < 1, u ∈ 208, ∞ .
Then, by the definition of f, we have
ρ
i f0 1 ≤ φp m 9/10;
ρ
ii fσρ2 2 ≥ φp Mσ 819/19968.
So condition H3 holds, by Theorem 3.1, boundary value problem 4.1 has at least
one positive solution.
Acknowledgment
Project supported by the National Natural Science Foundation of China 10471075.
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