doi:10.1155/2009/169321
Research Article
The Existence of Positive Solutions for Third-Order
p
-Laplacian
m
-Point Boundary Value Problems with
Sign Changing Nonlinearity on Time Scales
Fuyi Xu and Zhaowei Meng
School of Science, Shandong University of Technology, Zibo, Shandong 255049, China
Correspondence should be addressed to Fuyi Xu,xfy [email protected]
Received 25 February 2009; Revised 10 April 2009; Accepted 2 June 2009
Recommended by Alberto Cabada
We study the following third-orderp-Laplacianm-point boundary value problems on time scales
φpuΔ∇∇ atft, ut 0, t ∈ 0, T Tκ, u0 mi−21 biuξi, uΔT 0, φpuΔ∇0
m−2
i1 ciφpuΔ∇ξi, whereφpsis p-Laplacian operator, that is,φps |s|p−2s, p > 1,φ−1p
φq,1/p1/q 1, 0 < ξ1 < · · · < ξm−2 < ρT. We obtain the existence of positive solutions by
using fixed-point theorem in cones. In particular, the nonlinear termft, uis allowed to change sign. The conclusions in this paper essentially extend and improve the known results.
Copyrightq2009 F. Xu and Z. Meng. 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 mean 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 withp-Laplacian operator have also been discussed extensively in literature; for example, see 7–18 . However, to the best of our knowledge, there are not many results concerning the higher-orderp-Laplacian mutilpoint boundary value problem on time scales. A time scaleTis a nonempty closed subset ofR. We make the blanket assumption that 0, T are points inT. By an interval0, TT, we always mean the intersection of the real interval 0, Twith the given time scale; that is0, T∩T.
In 19 , Anderson considered the following third-order nonlinear boundary value problemBVP:
xt ft, xt, t1≤t≤t3,
author studied the existence of solutions for the nonlinear boundary value problem by using Krasnoselskii’s fixed point theorem and Leggett and Williams fixed point theorem, respectively.
In9,10 , He considered the existence of positive solutions of thep-Laplacian dynamic equations on time scales
φpuΔ
∇
atfut 0, t∈0, T T, 1.2
satisfying the boundary conditions
u0−B0
uΔη0, uΔT 0, 1.3
or
uΔ0 0, uT−B1
uΔη0, 1.4
whereη∈0, ρT. He obtained the existence of at least double and triple positive solutions of the problems by using a new double fixed point theorem and triple fixed point theorem, respectively.
In18, Zhou and Ma firstly studied the existence and iteration of positive solutions for the following third-order generalized right-focal boundary value problem withp-Laplacian operator
φp
ut qtft, ut, 0≤t≤1,
u0 m
i1
αiuξi, un 0, u1 n
i1
βiuθi.
1.5
They established a corresponding iterative scheme for the problem by using the monotone iterative technique.
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:
φpuΔ∇
∇
atft, ut 0, t∈0, T Tκ,
u0 m−2
i1
biuξi, uΔT 0, φp
uΔ∇0 m−2
i1
ciφp
uΔ∇ξi
,
1.6
whereφpsisp-Laplacian operator, that is,φps |s|p−2s,p > 1,φ−p1 φq, 1/p1/q 1, andbi,ci,a,fsatisfy
H1 bi, ci∈0,∞, 0< ξ1<· · ·< ξm−2< ρT, 0<
m−2
i1 bi<1, 0<
m−2
i1 ci<1;
H2f : 0, T Tκ ×0,∞ → −∞,∞is continuous,a ∈ Cld0, T Tκ,0,∞, and
there existst0∈0, TTκ such thatat0>0.
2. Preliminaries and Lemmas
For convenience, we list the following definitions which can be found in1–5 .
Definition 2.1. A time scaleTis a nonempty closed subset of real numbersR. Fort <supTand r > inf T, define the forward jump operatorσand backward jump operatorρ, respectively, by
σt inf{τ∈T|τ > t} ∈T,
ρr sup{τ∈T|τ < r} ∈T 2.1
for allt, r ∈T. Ifσt> t,tis said to be right scattered, ifρr< r,ris said to be left scattered; ifσt t,tis said to be right dense, and ifρr r,ris said to be left dense. IfThas a right scattered minimumm, defineTk T− {m}; otherwise setTk T. IfThas a left scattered maximumM, defineTkT− {M}; otherwise setTkT.
Definition 2.2. Forf :T → Randt∈Tk, the delta derivative offat the pointtis defined to be the numberfΔt provided that it exists, with the property that for each >0, there is a neighborhoodUoftsuch that
fσt−fs−fΔtσt−s≤|σt−s| 2.2
for alls∈U.
Forf:T → Randt∈Tk, the nabla derivative offatt, denoted byf∇t provided it existswith the property that for each >0, there is a neighborhoodUoftsuch that
fρt−fs−f∇tρt−s≤ρt−s 2.3
Definition 2.3. A functionfis left-dense continuousi.e.,ld-continuous, iffis continuous at each left-dense point inTand its right-sided limit exists at each right-dense point inT.
Definition 2.4. IfφΔt ft, then we define the delta integral by
b
a
ftΔtφb−φa. 2.4
IfF∇t ft, then we define the nabla integral by
b
a
ft∇tFb−Fa. 2.5
Lemma 2.5. If conditionH1holds, then forh∈Cld0, T Tκ, the boundary value problem (BVP)
uΔ∇ht 0, t∈0, T Tκ,
u0 m−2
i1
biuξi, uΔT 0
2.6
has the unique solution
ut t
0
T−shs∇s
m−2 i1 bi
ξi
0T−shs∇s
1−mi−12bi
. 2.7
Proof. By caculating, we can easily get2.7. So we omit it.
Lemma 2.6. If conditionH1holds, then forh∈Cld0, T Tκ, the boundary value problem (BVP)
φp
uΔ∇∇ht 0, t∈0, T Tκ,
u0 m−2
i1
biuξi, uΔT 0, φp
uΔ∇0 m−2
i1
ciφp
uΔ∇ξi
2.8
has the unique solution
ut t
0
T−sφq s
0
hr∇rC
∇s
m−2
i1 bi
ξi
0T−sφq
s
0hr∇rC
∇s 1−mi−12bi
, 2.9
whereCmi−12ci
ξi
0hr∇r/1−
Proof. Integrating both sides of equation in2.8on0, t , we have
This together withLemma 2.5implies that
ut together with the assumption that the boundary conditionuΔT 0 implies thatuΔt ≥0 fort∈0, T Tκ. This implies that
min
t∈0,TTκut u
So we only proveu0≥0.By conditionH1we have
u0
m−2 i1 bi
ξi
0T−sφq
s
0hr∇rC
∇s 1−mi−12bi
≥0. 2.17
The proof is completed.
Lemma 2.8. Let condition H1hold. If h ∈ Cld0, T Tκ and ht ≥ 0, then the unique positive solutionutof (BVP)2.8satisfies
inf t∈0,TTκut
≥σ1u, 2.18
whereσ1
m−2
i1 biξi/T−
m−2
i1 biT−ξi,usupt∈0,TTκ|ut|.
Proof. ByuΔ∇t −φqmi−12ci
ξi
0hr∇r/1−
m−2 i1 ci
t
0hr∇r ≤ 0, we can know that
the graph ofutis concave down on0, TTκ, and uΔt is nonincreasing on0, T Tκ. This together with the assumption that the boundary conditionuΔT 0 implies thatuΔt ≥0 fort∈0, T Tκ. This implies that
uuT, min
t∈0,TTκut u
0. 2.19
For alli∈ {1,2, . . . , m−2}, we have from the concavity ofuthat
uξi−u0
ξi ≥
uT−u0
T , 2.20
that is,
uξi−u0 ξi Tu0≥
ξi
TuT. 2.21
This together with the boundary conditionu0 im−12biuξiimplies that
min t∈0,T Tκut≥
m−2 i1 biξi T−im1−2biT−ξi
uT. 2.22
This completes the proof.
LetECld0, T Tκbe endowed with the orderingx≤yifxt≤ytfor allt∈0, T Tκ, and u maxt∈0,T Tκ|ut|is defined as usual by maximum norm. Clearly, it follows that
For the convenience, let
We define two cones by
P {u:u∈E, ut≥0, t∈0, TTκ}, fixed point of operatorF.
Lemma 2.9. S:K → Kis completely continuous.
Lemma 2.10see20,21 . LetKbe a cone in a Banach spaceX. LetDbe an open bounded subset
Lemma 2.11see20 . Ωρdefined above has the following properties: aKσρ⊂Ωρ⊂Kρ;
b Ωρis open relative to K;
cu∈∂Ωρif and only ifmint∈0,T Tκut σρ;
difu∈∂Ωρ, thenσρ≤ut≤ρfort∈0, T Tκ.
For the convenience, we introduce the following notations:
1
Lemma 2.13. Iffsatisfies the following condition :
So that
ϕs φq s
0
arfr, ur∇rA
≤mρψT. 2.33
Therefore,
Sut≤ T
0
T−sϕs∇s
m−2 i1 bi
ξi
0ϕs∇s
1−mi−12bi
≤mρ
⎛ ⎝ T
0
T−sψT∇s
m−2 i1 bi
ξi
0ψT∇s
1−mi−12bi
⎞ ⎠ρ.
2.34
This implies thatSu ≤ uforu∈∂Kρ. Hence byLemma 2.101it follows thatiKS, Kρ 1.
Lemma 2.14. Iffsatisfies the following condition:
ft, u≥φp
Mσρ, t, u∈0, T Tκ×
σρ, ρ, u /Su, u∈∂Ωρ, 2.35
then
iK
S,Ωρ
0. 2.36
Proof. Letet≡1 fort∈0, T Tκ. Thene∈∂K1. We claim that
u /Suλe, u∈∂Ωρ, λ >0. 2.37
In fact, if not, there existu0∈∂Ωρandλ0>0 such thatu0Su0λ0e. Byft, u0≥φpMσρ, we have
s
0
arfr, u0r∇rA s
0
arfr, u0r∇r
m−2
i1 ci
ξi
0arfr, u0r∇r
1−mi−12ci
≥φp
Mσρ
⎛ ⎝ s
0
ar∇r
m−2
i1 ci
ξi
0ar∇r
1−mi−12ci
⎞ ⎠.
So that
ϕs φq s
0
arfr, u0r∇rA
≥Mσρφq
⎛ ⎝ s
0
ar∇r
m−2 i1 ci
ξi
0ar∇r
1−mi−12ci
⎞ ⎠
Mσρψs.
2.39
Fort∈0, T Tκ, then
u0t Su0t λ0et
≥Su00 λ0
m−2 i1 bi
ξi
0ϕs∇s
1−mi−12bi
λ0
≥ Mσρ
1−mi−12bi m−2
i1
bi ξi
0
ψs∇sλ0
σρλ0.
2.40
This together withLemma 2.11cimplies that
σρ≥σρλ0, 2.41
a contradiction. Hence byLemma 2.102it follows thatiKS,Ωρ 0.
3. Main Results
We now give our results on the existence of positive solutions of BVP1.6.
Theorem 3.1. Suppose that conditionsH1andH2hold, and assume that one of the following conditions holds.
H3There existρ1, ρ2∈0,∞withρ1< σρ2such that ift, u≤φpmρ1,t, u∈0, T Tκ×0, ρ1 ;
iift, u ≥ 0,t, u ∈ 0, T Tκ ×σρ1, ρ2 , moreover ft, u ≥ φpMσρ2,t, u ∈
0, T Tκ ×σρ2, ρ2 .
H4There existρ1, ρ2∈0,∞withρ1< ρ2such that ift, u≤φpmρ2,t, u∈0, T Tκ×0, ρ2 ;
iift, u≥φpMσρ1,t, u∈0, T Tκ×σ2ρ1, ρ2 .
Proof. Assume thatH3holds, we show thatShas a fixed pointu1inΩρ2\Kρ1. Byft, u≤
φpmρ1andLemma 2.13, we have that
iK
S, Kρ1
1. 3.1
Byft, u≥φpMσρ2andLemma 2.14, we have that
iK
S,Ωρ2
0. 3.2
ByLemma 2.11aandρ1 < σρ2, we haveKρ1 ⊂Kσρ2 ⊂Ωρ2. It follows fromLemma 2.103
thatShas a fixed pointu1inΩρ2\Kρ1. Clearly,
u1> ρ1, min
t∈0,TTκu1t≥σu1> σρ1, 3.3
which implies thatσρ1≤u1t≤ρ2,t∈0, T Tκ. By conditionH3ii, we haveft, u1t≥0, t∈0, T Tκ, that is,ft, u1t ft, u1t. Hence,
Fu1Su1. 3.4
This means thatu1is a fixed point of operatorF.
When conditionH4holds, byft, u≤φpmρ2andLemma 2.13, we have that
iK
S, Kρ2
1. 3.5
Byft, u≥φpMσρ1andLemma 2.14, we have that
iK
S,Ωρ1
0. 3.6
ByLemma 2.11aandρ1 < ρ2, we haveKσρ1 ⊂ Ωρ1 ⊂ Kρ2. It follows fromLemma 2.103
thatShas a fixed pointu2inKρ2\Ωρ1. Obviously,
u2> σρ1, min t∈0,TTκ
u2t≥σu2> σ2ρ
1, 3.7
which implies thatσ2ρ
1≤u2t≤ρ2,t∈0, T Tκ. By conditionH4ii, we haveft, u2t≥ 0,t∈0, T Tκ, that is,ft, u2t ft, u2t. Hence,
Fu2Su2. 3.8
This means thatu2 is a fixed point of operator F. Therefore, the BVP1.6has at least one
Theorem 3.2. Assume that conditionsH1andH2hold, and suppose that one of the following conditions holds.
H5There existρ1, ρ2, andρ3∈0,∞withρ1< σρ2, andρ2< ρ3such that ift, u≤φpmρ1,t, u∈0, T Tκ×0, ρ1 ;
iift, u ≥ 0,t, u ∈ 0, T Tκ ×σρ1, ρ3 , moreover ft, u ≥ φpMσρ2,t, u ∈ 0, T Tκ ×σρ2, ρ2 ,u /Su,∀u∈∂Ωρ2;
iiift, u≤φpmρ3,t, u∈0, T Tκ×0, ρ3 .
H6There existρ1, ρ2, andρ3∈0,∞withρ1< ρ2< σρ3such that ift, u≥φpMσρ1,t, u∈0, T Tκ×σ2ρ1, ρ2 ;
iift, u≤φpm1ρ2,t, u∈0, T Tκ×0, ρ2 ,u /Su,∀u∈∂Kρ2;
iiift, u ≥ 0,t, u ∈ 0, T Tκ ×σρ2, ρ3 , moreover,ft, u ≥ φpMσρ3,t, u ∈
0, T Tκ ×σρ3, ρ3 .
Then, the BVP1.6has at least two positive solutions.
Proof. Assume that conditionH5holds, we show thatShas a fixed pointu1either in∂Kρ1
or inΩρ2\Kρ1. Ifu /Suforu∈∂Kρ1∪∂Kρ3. by Lemmas2.13and2.14, we have that
iK
S, Kρ1
1,
iK
S, Kρ3
1,
iK
S,Ωρ2
0.
3.9
ByLemma 2.11aandρ1 < σρ2, we haveKρ1 ⊂Kσρ2 ⊂Ωρ2. It follows fromLemma 2.103
thatShas a fixed pointu1inΩρ2\Kρ1. Similarly,Shas a fixed pointu2inKρ3\Ωρ2. Clearly,
u1> ρ1, min
t∈0,TTκu1t≥σu1> σρ1, 3.10
which implies thatσρ1≤u1t≤ρ2,t∈0, T Tκ. By conditionH5ii, we haveft, u1t≥0, t∈0, T Tκ, that is,ft, u1t ft, u1t. Hence,
Fu1Su1. 3.11
This means thatu1is a fixed point of operatorF. On the other hand, fromu2 ∈Kρ3\Ωρ2, ρ2<
ρ3andLemma 2.11a, we haveKσρ2 ⊂Ωρ2⊂Kρ3. Clearly,
u2> σρ2, min
t∈0,TTκu2t≥σu2> σ
2ρ
2, 3.12
which implies thatσ2ρ
2 ≤ u2t ≤ ρ3, t ∈ 0, T Tκ. Byρ1 < σρ2 and conditionH5ii, we
haveft, u2t≥0, t∈0, T Tκ, that is,ft, u2t ft, u2t. Hence,
This means thatu2is a fixed point of operatorF. Then, the BVP1.6has at least two positive
solutions.
When conditionH6holds, the proof is similar to the above, and so we omit it here.
4. An Example
In the section, we present some simple examples to explain our results.
Example 4.1. LetT 0,1/2 {1},T 1. Consider the following three-point boundary value problem withp-Laplacian
Then, by the definition offwe have
ift, u≤φpmρ1 48/35,t, u∈0,1 ×0, ρ1 ;
iift, u ≥ 0, t, u ∈ 0,1 ×σρ1, ρ2 , moreoverft, u ≥ φpMσρ2 3744/175, t, u∈0,1 ×σρ2, ρ2 .
Acknowledgment
This project was supported by the National Natural Science Foundation of China10471075, 10771117.
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