Volume 2009, Article ID 312058, 18 pagesdoi:10.1155/2009/312058 Research Article Existence of Positive Solutions for Multipoint Boundary Value Problem with p-Laplacian on Time Scales Men
Trang 1Volume 2009, Article ID 312058, 18 pages
doi:10.1155/2009/312058
Research Article
Existence of Positive Solutions for
Multipoint Boundary Value Problem with
p-Laplacian on Time Scales
Meng Zhang,1 Shurong Sun,1 and Zhenlai Han1, 2
1 School of Science, University of Jinan, Jinan, Shandong 250022, China
2 School 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
Copyrightq 2009 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 same time, motivated by the wide application of boundary value problems in physical and
applied mathematics, boundary value problems for dynamic equations with p-Laplacian on
time scales have received lots of interest7 16
Trang 2In7, Anderson et al considered the following three-point boundary value problem
with p-Laplacian on time scales:
ϕ puΔt∇ ctfut 0, t ∈ a, b,
ua − B0
uΔv 0, uΔb 0,
1.1
wherev ∈ a, b, f ∈ Cld0, ∞, 0, ∞, c ∈ Clda, b, 0, ∞, and Km x ≤ B0x ≤ KM x
for some positive constantsK m , K M 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 in8 using 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-dimensional p-Laplacian boundary value
problem on time scales:
ϕ puΔtΔ htfu σ t 0, t ∈ a, b,
ua − B0
uΔa 0, uΔσb 0,
1.2
whereht is a nonnegative rd-continuous function defined in a, b and satisfies that there
existst0 ∈ a, b such that ht0 > 0, fu is a nonnegative continuous function defined on
0, ∞, B1x ≤ B0x ≤ B2x for some positive constants B1, 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, in17 Ji and Ge investigated a
class of Sturm-Liouville-like four-point boundary value problem with p-Laplacian:
ϕ p
ut ft, ut 0, t ∈ 0, 1,
u0 − αuξ 0, u1 βu
η 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:
ϕ puΔtΔ htfut 0, t ∈ a, bT, αua − βuΔξ 0, γuσ2b δuΔ
η 0, uΔθ 0,
1.4
Trang 3whereT is a time scale, ϕpu |u| p−2 u, p > 1, α > 0, β ≥ 0, γ > 0, δ ≥ 0, a < ξ < θ < η < b,
and we denoteϕp−1 ϕqwith 1/p 1/q 1.
In the following, we denotea, b : a, bT a, b ∩ T for convenience And we list
the following hypotheses:
C1 fu is a nonnegative continuous function defined on 0, ∞;
C2 h : a, σ2b → 0, ∞ is rd-continuous with h · f /≡ 0.
2 Preliminaries
In this section, we provide some background material to facilitate analysis of problem1.4 Let the Banach spaceE {u : a, σ2b → R is rd-continuous} be endowed with the
norm u sup t∈a,σ2b |ut| and choose the cone P ⊂ E defined 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 BVP1.4 can be expressed as
ut
⎧
⎪
⎨
⎪
⎩
β
α ϕ q
θ
ξ hrfurΔr
t
a ϕ q
θ
s hrfurΔr
Δs, a ≤ t ≤ θ, δ
γ ϕ q
η
θ hrfurΔr
σ2b
t ϕ q
s
θ hrfurΔr
Δs, θ ≤ t ≤ σ2b.
2.2
IfV1 V2, where
V1 α β ϕ q
θ
ξ hrfurΔr
θ
a ϕ q
θ
s hrfurΔr
Δs,
V2 δ γ ϕ q
η
θ hrfurΔr
σ2b
θ ϕ q
s
θ hrfurΔr
Δs,
2.3
we define the operatorA : P → E by
Aut
⎧
⎪
⎨
⎪
⎩
β
α ϕ q
θ
ξ hrfurΔr
t
a ϕ q
θ
s hrfurΔr
Δs, a ≤ t ≤ θ, δ
γ ϕ q
η
θ hrfurΔr
σ2b
t ϕ q
s
θ hrfurΔr
Δs, θ ≤ t ≤ σ2b.
2.4
Trang 4It is easy to see u uθ, Aut ≥ 0 for t ∈ a, σ2b, and if Aut ut, then ut is
the positive solution of BVP1.4
From the definition ofA, for each u ∈ P, we have Au ∈ P, and Au Auθ.
In fact,
AuΔt
⎧
⎪
⎪
⎪
⎪
ϕ q
θ
t hrfurΔr
≥ 0, a ≤ t ≤ θ,
−ϕq
t
θ hrfurΔr
≤ 0, θ ≤ t ≤ σ2b
2.5
is continuous and nonincreasing in a, σ2b Moreover, ϕqx is a monotone increasing
continuously differentiable function,
θ
t hsfusΔs
Δ
−
t
θ hsfusΔs
Δ
−htfut ≤ 0, 2.6
then by the chain rule on time scales, we obtain
AuΔΔt ≤ 0, 2.7
so,A : P → P.
For the notational convenience, we denote
L1 β
α θ − a
ϕ q
θ
a hrΔr
,
L2
δ
γ σ2b − θ
ϕ q
σ2b
θ hrΔr
,
M1 β α ϕ q
θ
ξ hrΔr
θ
ξ ϕ q
θ
s hrΔr
Δs,
M2 δ γ ϕ q
η
θ hrΔr
η
θ ϕ q
s
θ hrΔr
Δs,
M3 min
ξ − a
θ − a ,
σ2b − η
σ2b − θ
,
M4 max
θ − a
ξ − a ,
σ2b − θ
σ2b − η
.
2.8
Trang 5Lemma 2.1 A : P → P is completely continuous.
Proof First, we show that A maps bounded set into bounded set.
Assume thatc > 0 is a constant and u ∈ P c Note that the continuity of f guarantees
that there existsK > 0 such that fu ≤ ϕ p K So
Au Auθ
β α ϕ q
θ
ξ hrfurΔr
θ
a ϕ q
θ
s hrfurΔr
Δs
≤ β
α ϕ q
θ
a hrϕ pKΔr
θ
a ϕ q
θ
a hrϕ pKΔr
Δs
K β
α θ − a
ϕ q
θ
a hrΔr
KL1,
Au Auθ
δ γ ϕ q
η
θ hrfurΔr
σ2b
θ ϕ q
s
θ hrfurΔr
Δs
≤ δ
γ ϕ q
σ2b
ξ hrϕ pKΔr
σ2b
θ ϕ q
σ2b
θ hrϕ pKΔr
Δs
K
δ
γ σ2b − θ
ϕ q
σ2b
θ hrΔr
KL2.
2.9
That is,AP cis uniformly bounded In addition, it is easy to see
|Aut1 − Aut2| ≤
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
C|t1− t2|ϕq
θ
a hrΔr
, t1, t2∈ a, θ,
C|t1− t2|ϕq
σ2b
a hrΔr
, t1∈ a, θ, t2∈θ, σ2b
or t2∈ a, θ, t1∈θ, σ2b, C|t1− t2|ϕq
σ2b
θ hrΔr
, t1, t2∈ a, θ.
2.10
Trang 6So, by applying Arzela-Ascoli Theorem on time scales, we obtain thatAP cis relatively compact
Second, we will show thatA : P c → P is continuous Suppose that {un}∞n 1 ⊂ Pcand
u n t converges to u0t uniformly on a, σ2b Hence, {Au n t}∞
n 1is 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{Aun l t}∞l 1be a subsequence which converges tovt uniformly on a, σ2b In addition,
0≤ Aunt ≤ min{KL1, KL2}. 2.11 Observe that
Au nt
⎧
⎪
⎪
⎪
⎪
β
α ϕ q
θ
ξ hrfu nrΔr
t
a ϕ q
θ
s hrfu nrΔr
Δs, a ≤ t ≤ θ, δ
γ ϕ q
η
θ hrfu nrΔr
σ2b
t ϕ qs
θ hrfu nrΔr
Δs, θ ≤ t ≤ σ2b.
2.12 Insertingu n l into the above and then lettingl → ∞, we obtain
vt
⎧
⎪
⎪
⎪
⎪
β
α ϕ q
θ
ξ hrfu0rΔr
t
a ϕ q
θ
s hrfu0rΔr
Δs, a ≤ t ≤ θ,
δ
γ ϕ q
η
θ hrfu0rΔr
σ2b
t ϕ q
s
θ hrfu0rΔr
Δs, θ ≤ t ≤ σ2b,
2.13
here we have used the Lebesgues dominated convergence theorem on time scales From the definition ofA, we know that vt Au0t on a, σ2b This shows that each subsequence
of{Au n t}∞n 1uniformly converges toAu0t Therefore, the sequence {Au n t}∞n 1uniformly converges toAu0t This means that A is continuous at u0 ∈ P c So,A is continuous on P c
sinceu0is arbitrary Thus,A is completely continuous.
The proof is complete
Lemma 2.2 Let u ∈ P, then ut ≥ t − a/θ − a u for t ∈ a, θ, and ut ≥ σ2b −
t/σ2b − θ u for t ∈ θ, σ2b.
Proof Since uΔΔt ≤ 0, it follows that uΔt is nonincreasing Hence, for a < t < θ,
ut − ua
t
a uΔsΔs ≥ t − auΔt,
uθ − ut
θ
t uΔsΔs ≤ θ − tuΔt,
2.14
Trang 7from which we have
ut ≥ uaθ − t t − auθ θ − a ≥ θ − a t − a uθ θ − a t − a u 2.15 Forθ ≤ t ≤ σ2b,
uσ2b− ut
σ2b
t uΔsΔs ≤σ2b − tuΔt,
ut − uθ
t
θ uΔsΔs ≥ t − θuΔt,
2.16
we know
ut ≥
σ2b − tuθ t − θuσ2b
σ2b − θ ≥
σ2b − t
σ2b − θ uθ σ σ22b − θ b − t u 2.17 The proof is complete
Lemma 2.3 18 Let P be a cone in a Banach space E Assum that Ω1, Ω2are open subsets of E with 0∈ Ω1, Ω1⊂ Ω2 If
A : P ∩Ω2\ Ω1
is a completely continuous operator such that either
i Ax ≤ x , ∀x ∈ P ∩ ∂Ω1 and Ax ≥ x , ∀x ∈ P ∩ ∂Ω2, or
ii Ax ≥ x , ∀x ∈ P ∩ ∂Ω1 and Ax ≤ x , ∀x ∈ P ∩ ∂Ω2.
Then A has a fixed point in P ∩ 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/ϕ p u, 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:
i i0 0 and i∞ 0;
ii i0 0 and i∞ 1;
iii i0 0 and i∞ 2;
Trang 8iv i0 1 and i∞ 0;
v i0 1 and i∞ 1;
vi i0 2 and i∞ 0.
Theorem 3.1 BVP 1.4 has at least one positive solution in the case i0 1 and i∞ 1.
Proof First, we consider the case f0 0 and f∞ ∞ Since f0 0, then there exists H1 > 0
such thatfu ≤ ϕ pεϕpu ϕpεu, for 0 < u ≤ H1, where ε satisfies
max{εL1, εL2} ≤ 1. 3.1
Ifu ∈ P, with u H1, then
Au Auθ
β
α ϕ q
θ
ξ hrfurΔr
θ
a ϕ q
θ
s hrfurΔr
Δs
≤ β
α ϕ q
θ
a hrfurΔr
θ
a ϕ q
θ
a hrfurΔr
Δs
≤ β
α ϕ q
θ
a hrϕ pε u Δr
θ
a ϕ q
θ
a hrϕ pε u Δr
Δs
u εL1
≤ u ,
Au Auθ
δ
γ ϕ q
η
θ hrfurΔr
σ2b
θ ϕ qs
θ hrfurΔrΔs
≤ δ γ ϕ q
σ2b
θ hrfurΔr
σ2b
θ ϕ q
σ2b
θ hrfurΔr
Δs
≤ δ γ ϕ q
σ2b
θ hrϕ pε u Δr
σ2b
θ ϕ q
σ2b
θ hrϕ pε u Δr
Δs
u εL2
≤ u
3.2
It follows that ifΩH {u ∈ E : u < H1}, then Au ≤ u for u ∈ P ∩ ∂Ω H
Trang 9Sincef∞ ∞, then there exists H
2 > 0 such that fu ≥ ϕ pkϕpu ϕpku, for
u ≥ H
2, where k > 0 is chosen such that
min
k θ − a ξ − a M1, k σ2b − η
σ2b − θ M2
SetH2 max{2H1, θ − a/ξ − aH
2, σ2b − θ/σ2b − ηH
2}, and ΩH2 {u ∈
E : u < H2}.
Ifu ∈ P with u H2, then
min
t∈ξ,θ ut uξ ≥ θ − a ξ − a u ≥ H
2,
min
t∈θ,η ut uη≥ σ2b − η
σ2b − θ u ≥ H2.
3.4
So that
Au Auθ
β α ϕ q
θ
ξ hrfurΔr
θ
a ϕ q
θ
s hrfurΔr
Δs
≥ β α ϕ q
θ
ξ hrϕ pkuΔr
θ
ξ ϕ q
θ
s hrϕ pkuΔr
Δs
≥ β α ϕ q
θ
ξ hrϕ p
k θ − a ξ − a u
Δr
θ
ξ ϕ q
θ
s hrϕ p
k θ − a ξ − a u
Δr
Δs
u k θ − a ξ − a M1
≥ u ,
Au Auθ δ γ ϕ q
η
θ hrfurΔr
σ2b
θ ϕ q
s
θ hrfurΔr
Δs
≥ δ γ ϕ q
η
θ hrϕ p
k σ σ22b − η b − θ u
Δr
η
θ ϕ q
s
θ hrϕ p
k σ σ22b − η b − θ u
Δr
Δs
u k σ2b − η
σ2b − θ M2
≥ u
3.5
Trang 10In other words, ifu ∈ P ∩ ∂Ω H2, then Au ≥ u Thus by i ofLemma 2.3, it follows thatA has a fixed point in P ∩ Ω H2\ ΩH1 with H1≤ u ≤ H2
Now we consider the casef0 ∞ and f∞ 0 Since f0 ∞, there exists H3 > 0, such
thatfu ≥ ϕ pmϕpu ϕpmu for 0 < u ≤ H3, wherem is such that
min
mM1ξ − a
θ − a , mM2
σ2b − η
σ2b − θ
Ifu ∈ P with u H3, then we have
Au Auθ
β α ϕ q
θ
ξ hrfurΔr
θ
a ϕ q
θ
s hrfurΔr
Δs
≥ β α ϕ q
θ
ξ hrϕ p
m θ − a ξ − a u
Δr
θ
ξ ϕ q
θ
s hrϕ p
m θ − a ξ − a u
Δr
Δs
u m ξ − a
θ − a M1
≥ u ,
Au Auθ
δ
γ ϕ q
η
θ hrfurΔr
σ2b
θ ϕ q
s
θ hrfurΔr
Δs
≥ δ
γ ϕ q
η
θ hrϕ p
m σ2b−η
σ2b−θ u
Δr
η
θ ϕ q
s
θ hrϕ p
m σ2b − η
σ2b − θ u
Δr
Δs
u m σ σ22b − η
b − θ M2
≥ u
3.7
Thus, we letΩH3 {u ∈ E : u < H3}, so that Au ≥ u for u ∈ P ∩ ∂ΩH3.
Next considerf∞ 0 By definition, there exists H
4 > 0 such that fu ≤ ϕ p εϕ p u
ϕ p εu for u ≥ H
4, whereε > 0 satisfies
max{εL1, εL2} ≤ 1. 3.8
Trang 11Supposef is bounded, then fu ≤ ϕ pK for all u ∈ 0, ∞, pick
H4 max{2H3, KL1, KL2}. 3.9
Ifu ∈ P with u H4, then
Au Auθ
β α ϕ q
θ
ξ hrfurΔr
θ
a ϕ q
θ
s hrfurΔr
Δs
≤ β α ϕ q
θ
a hrϕ pKΔr
θ
a ϕ q
θ
a hrϕ pKΔr
Δs
KL1
≤ H4
u ,
Au Auθ
δ γ ϕ q
η
θ hrfurΔr
σ2b
θ ϕ q
s
θ hrfurΔr
Δs
≤ δ γ ϕ q
σ2b
θ hrϕ p KΔr
σ2b
θ ϕ q
σ2b
θ hrϕ p KΔr
Δs
KL2
≤ H4
u
3.10
Now supposef is unbounded From condition C1, it is easy to know that there exists
H4≥ max{2H3, H4} such that fu ≤ fH4 for 0 ≤ u ≤ H4 If u ∈ P with u H4, then by
using3.8 we have
Au Auθ
β α ϕ q
θ
ξ hrfurΔr
θ
a ϕ q
θ
s hrfurΔr
Δs
≤ β α ϕ q
θ
a hrfH4Δr
θ
a ϕ q
θ
a hrfH4Δr
Δs
Trang 12≤ β α ϕ q
θ
a hrϕ pεH4Δr
θ
a ϕ q
θ
a hrϕ pεH4Δr
Δs
H4εL1
≤ H4
u ,
Au Auθ
δ γ ϕ q
η
θ hrfurΔr
σ2b
θ ϕ q
s
θ hrfurΔr
Δs
≤ δ
γ ϕ q
σ2b
θ hrfH4Δr
σ2b
θ ϕ q
σ2b
θ hrfH4Δr
Δs
≤ δ γ ϕ q
σ2b
θ hrϕ pεH4Δr
σ2b
θ ϕ q
σ2b
θ hrϕ pεH4Δr
Δs
H4εL2
≤ H4
u
3.11
Consequently, in either case we take
ΩH4 {u ∈ E : u < H4}, 3.12
so that foru ∈ P ∩ ∂Ω H4, we have Au ≥ u Thus by ii ofLemma 2.3, it follows thatA
has a fixed pointu in P ∩ Ω H4\ ΩH3 with H3≤ u ≤ H4.
The proof is complete
Theorem 3.2 Suppose i0 0, i∞ 1, and the following conditions hold,
C3: there exists constant p> 0 such that fu ≤ ϕ p pA1 for 0 ≤ u ≤ p, where
A1 minL−1
1 , L−1 2