doi:10.1155/2009/312058
Research Article
Existence of Positive Solutions for
Multipoint Boundary Value Problem with
p
-Laplacian on Time Scales
Meng Zhang,
1Shurong Sun,
1and Zhenlai Han
1, 21School of Science, University of Jinan, Jinan, Shandong 250022, China
2School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China
Correspondence should be addressed to Shurong Sun,sshrong@163.com
Received 11 March 2009; Accepted 8 May 2009
Recommended by Victoria Otero-Espinar
We consider the existence of positive solutions for a class of second-order multi-point boundary value problem withp-Laplacian on time scales. By using the well-known Krasnosel’ski’s fixed-point theorem, some new existence criteria for positive solutions of the boundary value problem are presented. As an application, an example is given to illustrate the main results.
Copyrightq2009 Meng Zhang 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.
1. Introduction
The theory of time scales has become a new important mathematical branch since it was introduced by Hilger1. Theoretically, the time scales approach not only unifies calculus
of differential and difference equations, but also solves other problems that are a mix of stop start and continuous behavior. Practically, the time scales calculus has a tremendous potential for application, for example, Thomas believes that time scales calculus is the best way to understand Thomas models populations of mosquitoes that carry West Nile virus 2. In addition, Spedding have used this theory to model how students suffering from the
eating disorder bulimia are influenced by their college friends; with the theory on time scales, they can model how the number of sufferers changes during the continuous college term as well as during long breaks2. By using the theory on time scales we can also study insect
population, biology, heat transfer, stock market, epidemic models2–6, and so forth. At the
In7, Anderson et al. considered the following three-point boundary value problem
withp-Laplacian on time scales:
ϕpuΔt
∇
ctfut 0, t∈a, b,
ua−B0
uΔv 0, uΔb 0, 1.1
wherev ∈ a, b, f ∈ Cld0,∞,0,∞, c ∈ Clda, b,0,∞, andKmx ≤ B0x ≤ KMx for some positive constantsKm, KM.They established the existence results for at least one positive solution by using a fixed point theorem of cone expansion and compression of functional type.
For the same boundary value problem, He in8using a new fixed point theorem due to Avery and Henderson obtained the existence results for at least two positive solutions.
In9, Sun and Li studied the following one-dimensionalp-Laplacian boundary value problem on time scales:
ϕpuΔt
Δ
htfuσt 0, t∈a, b,
ua−B0
uΔa 0, uΔσb 0,
1.2
wherehtis a nonnegative rd-continuous function defined ina, band satisfies that there existst0 ∈ a, bsuch that ht0 > 0, fuis a nonnegative continuous function defined on 0,∞, B1x≤B0x≤B2xfor some positive constantsB1, B2.They established the existence results for at least single, twin, or triple positive solutions of the above problem by using Krasnosel’skii’s fixed point theorem, new fixed point theorem due to Avery and Henderson and Leggett-Williams fixed point theorem.
For the Sturm-Liouville-like boundary value problem, in17Ji and Ge investigated a class of Sturm-Liouville-like four-point boundary value problem withp-Laplacian:
ϕputft, ut 0, t∈0,1,
u0−αuξ 0, u1 βuη 0, 1.3
whereξ < η, f ∈ C0,1×0,∞,0,∞.By using fixed-point theorem for operators on a cone, they obtained some existence of at least three positive solutions for the above problem. However, to the best of our knowledge, there has not any results concerning the similar problems on time scales.
Motivated by the above works, in this paper we consider the following multi-point boundary value problem on time scales:
ϕp
uΔtΔhtfut 0, t∈a, b
T,
αua−βuΔξ 0, γuσ2bδuΔη 0, uΔθ 0,
whereTis a time scale, ϕpu |u|p−2u, p >1, α >0, β≥0, γ >0, δ≥0, a < ξ < θ < η < b,
In this section, we provide some background material to facilitate analysis of problem1.4.
Let the Banach spaceE {u:a, σ2b → Ris rd-continuous}be endowed with the normu supt∈a,σ2b|ut|and choose the coneP⊂Edefined by
P u∈E:ut≥0, t∈a, σ2b, uΔΔt≤0, t∈a, b . 2.1
It is easy to see that the solution of BVP1.4can be expressed as
It is easy to seeu uθ,Aut≥0 fort∈a, σ2b,and ifAut ut,thenutis the positive solution of BVP1.4.
From the definition ofA,for eachu∈P,we haveAu∈P,andAu Auθ.
then by the chain rule on time scales, we obtain
AuΔΔt≤0, 2.7
so,A:P → P.
For the notational convenience, we denote
Lemma 2.1. A:P → Pis completely continuous.
Proof. First, we show thatAmaps bounded set into bounded set.
Assume thatc > 0 is a constant andu∈Pc.Note that the continuity offguarantees that there existsK >0 such thatfu≤ϕpK. So
That is,APcis uniformly bounded. In addition, it is easy to see
So, by applying Arzela-Ascoli Theorem on time scales, we obtain thatAPcis relatively compact.
Second, we will show thatA :Pc → Pis continuous. Suppose that{un}∞n 1 ⊂ Pcand
untconverges tou0tuniformly ona, σ2b. Hence,{Aunt}∞n 1 is uniformly bounded and equicontinuous ona, σ2b. The Arzela-Ascoli Theorem on time scales tells us that there exists uniformly convergent subsequence in{Aunt}∞n 1. Let{Aunlt}∞l 1 be a subsequence
Insertingunl into the above and then lettingl → ∞, we obtain
vt
here we have used the Lebesgues dominated convergence theorem on time scales. From the definition ofA, we know thatvt Au0tona, σ2b. This shows that each subsequence of{Aunt}∞n 1uniformly converges toAu0t. Therefore, the sequence{Aunt}∞n 1uniformly converges toAu0t. This means thatAis continuous atu0 ∈ Pc. So,Ais continuous onPc sinceu0is arbitrary. Thus,Ais completely continuous.
The proof is complete.
from which we have
ut≥ uaθ−tθ −at−auθ ≥ θt−−aauθ θt−−aau. 2.15
Forθ≤t≤σ2b,
uσ2b−ut
σ2b
t u
ΔsΔs≤σ2b−tuΔt,
ut−uθ
t
θu
ΔsΔs≥t−θuΔt,
2.16
we know
ut≥
σ2b−tuθ t−θuσ2b
σ2b−θ ≥
σ2b−t
σ2b−θuθ
σ2b−t
σ2b−θu. 2.17
The proof is complete.
Lemma 2.318. LetP be a cone in a Banach spaceE.Assum thatΩ1,Ω2are open subsets ofE
with0∈Ω1,Ω1⊂Ω2.If
A:P∩Ω2\Ω1
−→P 2.18
is a completely continuous operator such that either
iAx ≤ x,∀x∈P∩∂Ω1 and Ax ≥ x,∀x∈P∩∂Ω2,or iiAx ≥ x,∀x∈P∩∂Ω1 and Ax ≤ x,∀x∈P∩∂Ω2.
ThenAhas a fixed point inP∩Ω2\Ω1.
3. Main Results
In this section, we present our main results with respect to BVP1.4.
For the sake of convenience, we define f0 limu→0fu/ϕpu, f∞
limu→ ∞fu/ϕpu, i0 number of zeros in the set {f0, f∞}, and i∞ number of ∞ in
the set{f0, f∞}.
Clearly,i0, i∞ 0,1,or 2 and there are six possible cases:
ii0 0 andi∞ 0; iii0 0 andi∞ 1;
ivi0 1 andi∞ 0;
vi0 1 andi∞ 1;
vii0 2 andi∞ 0.
Theorem 3.1. BVP1.4has at least one positive solution in the casei0 1andi∞ 1.
Proof. First, we consider the casef0 0 andf∞ ∞.Sincef0 0,then there existsH1 > 0
such thatfu≤ϕpεϕpu ϕpεu,for 0< u≤H1,whereεsatisfies
max{εL1, εL2} ≤1. 3.1
Ifu∈P,withu H1,then
Au Auθ
β αϕq
θ
ξhrfurΔr
θ
aϕq
θ
shrfurΔr
Δs
≤ β
αϕq
θ
ahrfurΔr
θ
aϕq
θ
ahrfurΔr
Δs
≤ β
αϕq
θ
ahrϕpεuΔr
θ
aϕq
θ
ahrϕpεuΔr
Δs
uεL1 ≤ u,
Au Auθ
δ γϕq
η
θhrfurΔr
σ2b
θ ϕq
s
θhrfurΔr
Δs
≤ δγϕq
σ2b
θ hrfurΔr
σ2b
θ ϕq
σ2b
θ hrfurΔr
Δs
≤ δγϕq
σ2b
θ hrϕpεuΔr
σ2b
θ ϕq
σ2b
θ hrϕpεuΔr
Δs
uεL2 ≤ u.
3.2
Supposefis bounded, thenfu≤ϕpKfor allu∈0,∞,pick
H4 max{2H3, KL1, KL2}. 3.9
Ifu∈Pwithu H4,then
Au Auθ
β αϕq
θ
ξhrfurΔr
θ
aϕq
θ
shrfurΔr
Δs
≤ βαϕq
θ
ahrϕpKΔr
θ
aϕq
θ
ahrϕpKΔr
Δs
KL1 ≤H4
u,
Au Auθ
δ γϕq
η
θhrfurΔr
σ2b
θ ϕq
s
θhrfurΔr
Δs
≤ δγϕq
σ2b
θ hrϕpKΔr
σ2b
θ ϕq
σ2b
θ hrϕpKΔr
Δs
KL2 ≤H4
u.
3.10
Now supposefis unbounded. From conditionC1,it is easy to know that there exists
H4≥max{2H3, H4}such thatfu≤fH4for 0≤u≤H4.Ifu∈P withu H4,then by using3.8we have
Au Auθ
β αϕq
θ
ξhrfurΔr
θ
aϕq
θ
shrfurΔr
Δs
≤ βαϕq
θ
ahrfH4Δr
θ
aϕq
θ
ahrfH4Δr
≤ βαϕq
θ
ahrϕpεH4Δr
θ
aϕq
θ
ahrϕpεH4Δr
Δs
H4εL1
≤H4
u,
Au Auθ
δ γϕq
η
θhrfurΔr
σ2b
θ ϕq
s
θhrfurΔr
Δs
≤ δ
γϕq
σ2b
θ hrfH4Δr
σ2b
θ ϕq
σ2b
θ hrfH4Δr
Δs
≤ δγϕq
σ2b
θ hrϕpεH4Δr
σ2b
θ ϕq
σ2b
θ hrϕpεH4Δr
Δs
H4εL2
≤H4
u.
3.11
Consequently, in either case we take
ΩH4 {u∈E:u< H4}, 3.12
so that foru∈P ∩∂ΩH4,we haveAu ≥ u.Thus byiiofLemma 2.3, it follows thatA
has a fixed pointuinP∩ΩH4\ΩH3withH3≤ u ≤H4.
The proof is complete.
Theorem 3.2. Supposei0 0, i∞ 1, and the following conditions hold,
C3:there exists constantp>0such thatfu≤ϕppA1for0≤u≤p,where
A1 min
L−1
C4:there exists constantq>0such thatfu≥ϕpqA2foru∈M3q, M3,where
Proof. Without loss of generality, we may assume thatp< q.
Hence by conditionC4,we can get
The proof is complete.
For the casei0 1, i∞ 0 ori0 0, i∞ 1 we have the following results.
Theorem 3.3. Suppose thatf0 ∈0, ϕpA1andf∞ ∈ϕpM4A2,∞hold. Then BVP1.4has
at least one positive solution.
Proof. It is easy to see that under the assumptions, the conditionsC3andC4inTheorem 3.2
are satisfied. So the proof is easy and we omit it here.
Theorem 3.4. Suppose thatf0 ∈ϕpM4A2,∞andf∞ ∈0, ϕpA1hold. Then BVP1.4has
at least one positive solution.
Thus, ifu∈ξ−a/θ−aq1, q1, then we have
fu≥ϕpuϕp
θ−a
ξ−aA2
≥ϕpq1A2
; 3.21
by the similar method, one can get ifu∈σ2b−η/σ2b−θq2, q2,then
fu≥ϕpuϕp
σ2b−θ
σ2b−ηA2
≥ϕpq2A2
. 3.22
So, if we chooseq min{q1, q2},then foru ∈ M3q, q,we havefu ≥ ϕpqA2, which yields conditionC4inTheorem 3.2.
Next, byf∞∈0, ϕpA1,forε ϕpA1−f∞,there exists a sufficiently largep> q
such that
fu
ϕpu ≤f∞ε ϕpA1, u∈
p,∞, 3.23
where we consider two cases.
Case 1. Suppose thatfis bounded, say
fu≤ϕpK, u∈0,∞. 3.24
In this case, take sufficiently largepsuch thatp≥max{K/A
1, p},then from3.24, we know
fu≤ϕpK≤ϕpA1pforu∈0, p,which yields conditionC3inTheorem 3.2.
Case 2. Suppose thatfis unbounded. it is easy to know that there isp> psuch that
fu≤fp, u∈0, p. 3.25
Sincep> pthen from3.23and3.25, we get
fu≤fp≤ϕ ppA1
, u∈0, p. 3.26
Thus, the conditionC3ofTheorem 3.2is satisfied.
Hence, fromTheorem 3.2, BVP1.4has at least one positive solution. The proof is complete.
From Theorems3.3and3.4, we have the following two results.
Corollary 3.5. Suppose thatf0 0and the conditionC4inTheorem 3.2hold. Then BVP1.4has
Corollary 3.6. Suppose thatf∞ 0and the conditionC4inTheorem 3.2hold. Then BVP1.4
has at least one positive solution.
Theorem 3.7. Suppose thatf0 ∈ 0, ϕpA1andf∞ ∞hold. Then BVP1.4has at least one
positive solution.
Proof. In view off∞ ∞,similar to the first part ofTheorem 3.1, we have
Au ≥ u, u∈P∩∂ΩH2. 3.27
Sincef0∈0, ϕpA1,forε ϕpA1−f0>0,there exists a sufficiently smallp∈0, H2such that
fu≤f0ε
ϕpu ϕpA1u≤ϕp
A1p
, u∈0, p. 3.28
Similar to the proof ofTheorem 3.2, we obtain
Au ≤ u, u∈P∩∂Ωp. 3.29
The result is obtained, and the proof is complete.
Theorem 3.8. Suppose thatf∞ ∈ 0, ϕpA1andf0 ∞hold. Then BVP1.4has at least one
positive solution.
Proof. Sincef0 ∞,similar to the second part of Theorem 3.1, we have Au ≥ u for
u∈P∩∂ΩH3.
By f∞ ∈ 0, ϕpA1, similar to the second part of proof of Theorem 3.4, we have Au ≤ u for u ∈ P ∩∂Ωp,where p > H3.Thus BVP 1.4 has at least one positive solution.
The proof is complete.
From Theorems3.7and3.8, we can get the following corollaries.
Corollary 3.9. Suppose thatf∞ ∞and the conditionC3inTheorem 3.2hold. Then BVP1.4
has at least one positive solution.
Corollary 3.10. Suppose thatf0 ∞and the conditionC3inTheorem 3.2hold. Then BVP1.4
has at least one positive solution.
Theorem 3.11. Suppose thati0 0, i∞ 2,and the conditionC3ofTheorem 3.2hold. Then BVP 1.4has at least two positive solutionsu1, u2∈Psuch that0<u1< p<u2.
Proof. By using the method of proving Theorems3.1and3.2, we can deduce the conclusion
easily, so we omit it here.
Proof. Combining the proofs of Theorems3.1and3.2, the conclusion is easy to see, and we omit it here.
4. Applications and Examples
In this section, we present a simple example to explain our result. WhenT R,
uu 1−t4−arctanu, 0< t <1,
u0 u1
4
, u1 −u1
2
, 4.1
where,p 3, α β γ δ 1, ht 1−t, fu 4−arctanu.
It is easy to see that the conditionC1andC2are satisfied and
f0 lim u→0
fu
ϕpu ∞, f∞ ulim→ ∞
fu
ϕpu 0. 4.2
So, byTheorem 3.1, the BVP4.1has at least one positive solution.
Acknowledgments
This research is supported by the Natural Science Foundation of China60774004, China Postdoctoral Science Foundation Funded Project 20080441126, Shandong Postdoctoral Funded Project 200802018, the Natural Science Foundation of Shandong Y2007A27, Y2008A28, and the Fund of Doctoral Program Research of University of Jinan B0621, XBS0843.
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