Hindawi Publishing Corporation
Advances in Difference Equations
Volume 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
2
School of Science, University of Jinan, Jinan, Shandong 250022, China
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 with p-Laplacian on time scales. By using the well-known Krasnosel’ski’s fixedpoint 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.
Copyright q 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 Hilger 1. 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 breaks 2. By using the theory on time scales we can also study insect
population, biology, heat transfer, stock market, epidemic models 2–6, 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 interest 7–16.
2
Advances in Difference Equations
In 7, Anderson et al. considered the following three-point boundary value problem
with p-Laplacian on time scales:
ϕp uΔ t
∇
ctfut 0,
ua − B0 uΔ v 0,
t ∈ a, b,
1.1
uΔ b 0,
where v ∈ a, b, f ∈ Cld 0, ∞, 0, ∞, c ∈ Cld a, b, 0, ∞, and Km x ≤ B0 x ≤ KM x
for some positive constants Km , 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 in 8 using a new fixed point theorem due
to Avery and Henderson obtained the existence results for at least two positive solutions.
In 9, Sun and Li studied the following one-dimensional p-Laplacian boundary value
problem on time scales:
ϕp uΔ t
Δ
htfuσ t 0,
Δ
ua − B0 u a 0,
t ∈ a, b,
Δ
1.2
u σb 0,
where ht is a nonnegative rd-continuous function defined in a, b and satisfies that there
exists t0 ∈ a, b such that ht0 > 0, fu is a nonnegative continuous function defined on
0, ∞, B1 x ≤ B0 x ≤ B2 x 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, in 17 Ji and Ge investigated a
class of Sturm-Liouville-like four-point boundary value problem with p-Laplacian:
ϕp u t ft, ut 0,
u0 − αu ξ 0,
t ∈ 0, 1,
u1 βu η 0,
1.3
where ξ < 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:
Δ
htfut 0,
ϕp uΔ t
αua − βuΔ ξ 0,
t ∈ a, bT ,
γu σ 2 b δuΔ η 0,
uΔ θ 0,
1.4
Advances in Difference Equations
3
where T is a time scale, ϕp u |u|p−2 u, p > 1, α > 0, β ≥ 0, γ > 0, δ ≥ 0, a < ξ < θ < η < b,
and we denote ϕp −1 ϕq with 1/p 1/q 1.
In the following, we denote a, 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, σ 2 b → 0, ∞ is rd-continuous with h · f /
≡ 0.
2. Preliminaries
In this section, we provide some background material to facilitate analysis of problem 1.4.
Let the Banach space E {u : a, σ 2 b → R is rd-continuous} be endowed with the
norm u supt∈a,σ 2 b |ut| and choose the cone P ⊂ E defined by
P u ∈ E : ut ≥ 0, t ∈ a, σ 2 b , uΔΔ t ≤ 0, t ∈ a, b .
2.1
It is easy to see that the solution of BVP 1.4 can be expressed as
θ
t θ
⎧
β
⎪
⎪
⎪ ϕq
hrfurΔr ϕq
hrfurΔr Δs,
⎨
α
ξ
a
s
ut σ 2 b s
⎪ δ η
⎪
⎪
⎩ ϕq
hrfurΔr ϕq
hrfurΔr Δs,
γ
θ
θ
t
a ≤ t ≤ θ,
2.2
θ ≤ t ≤ σ b.
2
If V1 V2 , where
β
V1 ϕq
α
δ
V2 ϕq
γ
θ
ξ
η
θ
hrfurΔr
hrfurΔr
θ
a
ϕq
σ 2 b
θ
θ
ϕq
s
hrfurΔr Δs,
s
θ
2.3
hrfurΔr Δs,
we define the operator A : P → E by
θ
t θ
⎧
β
⎪
⎪
⎪
hrfurΔr ϕq
hrfurΔr Δs,
⎨ α ϕq
ξ
a
s
Aut η
σ 2 b s
⎪
δ
⎪
⎪
⎩ ϕq
hrfurΔr ϕq
hrfurΔr Δs,
γ
θ
t
θ
a ≤ t ≤ θ,
θ ≤ t ≤ σ 2 b.
2.4
4
Advances in Difference Equations
It is easy to see u uθ, Aut ≥ 0 for t ∈ a, σ 2 b, and if Aut ut, then ut is
the positive solution of BVP 1.4.
From the definition of A, for each u ∈ P, we have Au ∈ P, and Au Auθ.
In fact,
⎧ θ
⎪
⎪
⎪
hrfurΔr ≥ 0,
⎪
⎨ϕq
t
Δ
Au t t
⎪
⎪
⎪
hrfurΔr ≤ 0,
⎪
⎩−ϕq
θ
a ≤ t ≤ θ,
2.5
θ ≤ t ≤ σ 2 b
is continuous and nonincreasing in a, σ 2 b. Moreover, ϕq x is a monotone increasing
continuously differentiable function,
θ
t
Δ
hsfusΔs
t
Δ
− hsfusΔs
−htfut ≤ 0,
θ
2.6
then by the chain rule on time scales, we obtain
AuΔΔ t ≤ 0,
2.7
so, A : P → P.
For the notational convenience, we denote
L1 θ
β
θ − a ϕq
hrΔr ,
α
a
σ 2 b
δ
σ 2 b − θ ϕq
hrΔr ,
γ
θ
θ
θ θ
β
M1 ϕq
hrΔr ϕq
hrΔr Δs,
α
ξ
ξ
s
η s
η
δ
hrΔr ϕq
hrΔr Δs,
M2 ϕq
γ
θ
θ
θ
ξ − a σ 2 b − η
M3 min
,
,
θ − a σ 2 b − θ
θ − a σ 2 b − θ
,
M4 max
.
ξ − a σ 2 b − η
L2 2.8
Advances in Difference Equations
5
Lemma 2.1. A : P → P is completely continuous.
Proof. First, we show that A maps bounded set into bounded set.
Assume that c > 0 is a constant and u ∈ Pc . Note that the continuity of f guarantees
that there exists K > 0 such that fu ≤ ϕp K. So
Au Auθ
θ
θ θ
β
ϕq
hrfurΔr ϕq
hrfurΔr Δs
α
ξ
a
s
β
≤ ϕq
α
θ
a
hrϕp KΔr
θ
a
ϕq
θ
a
hrϕp KΔr Δs
θ
β
K
θ − a ϕq
hrΔr
α
a
KL1 ,
2.9
Au Auθ
δ
ϕq
γ
δ
≤ ϕq
γ
η
θ
hrfurΔr
σ 2 b
ξ
σ 2 b
θ
hrϕp KΔr
ϕq
σ 2 b
θ
s
θ
ϕq
hrfurΔr Δs
σ 2 b
θ
hrϕp KΔr Δs
σ 2 b
δ
2
σ b − θ ϕq
K
hrΔr
γ
θ
KL2 .
That is, APc is uniformly bounded. In addition, it is easy to see
|Aut1 − Aut2 | ≤
θ
⎧
⎪
⎪
⎪
hrΔr ,
⎪C|t1 − t2 |ϕq
⎪
⎪
a
⎪
⎪
⎪
⎪
⎪
⎪
σ 2 b
⎪
⎪
⎪
⎪
⎨C|t1 − t2 |ϕq
hrΔr ,
a
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
σ 2
⎪
⎪
b
⎪
⎪
⎪
⎪
hrΔr ,
⎩C|t1 − t2 |ϕq
θ
t1 , t2 ∈ a, θ,
t1 ∈ a, θ, t2 ∈ θ, σ 2 b
or t2 ∈ a, θ, t1 ∈ θ, σ 2 b ,
t1 , t2 ∈ a, θ.
2.10
6
Advances in Difference Equations
So, by applying Arzela-Ascoli Theorem on time scales, we obtain that APc is relatively
compact.
Second, we will show that A : Pc → P is continuous. Suppose that {un }∞
n1 ⊂ Pc and
∞
2
un t converges to u0 t uniformly on a, σ b. Hence, {Aun t}n1 is uniformly bounded
and equicontinuous on a, σ 2 b. The Arzela-Ascoli Theorem on time scales tells us that there
∞
exists uniformly convergent subsequence in {Aun t}∞
n1 . Let {Aunl t}l1 be a subsequence
2
which converges to vt uniformly on a, σ b. In addition,
0 ≤ Aun t ≤ min{KL1 , KL2 }.
2.11
Observe that
θ
t θ
⎧
β
⎪
⎪
⎪
hrfun rΔr ϕq
hrfun rΔr Δs,
⎪
⎨ α ϕq
ξ
a
s
Aun t σ 2 b s
⎪
⎪δ η
⎪
⎪
hrfun rΔr ϕq
hrfun rΔr Δs,
⎩ ϕq
γ
θ
t
θ
a ≤ t ≤ θ,
θ ≤ t ≤ σ 2 b.
2.12
Inserting unl into the above and then letting l → ∞, we obtain
vt ⎧
θ
t θ
⎪
β
⎪
⎪ ϕq
hrfu0 rΔr ϕq
hrfu0 rΔr Δs,
⎪
⎪
⎨α
ξ
a
s
a ≤ t ≤ θ,
η
σ 2 b s
⎪
⎪
⎪
δ
⎪
⎪
ϕ
hrfu
ϕ
hrfu
Δs,
rΔr
rΔr
⎩ q
0
q
0
γ
θ
t
θ
θ ≤ t ≤ σ 2 b,
2.13
here we have used the Lebesgues dominated convergence theorem on time scales. From the
definition of A, we know that vt Au0 t on a, σ 2 b. This shows that each subsequence
∞
of {Aun t}∞
n1 uniformly converges to Au0 t. Therefore, the sequence {Aun t}n1 uniformly
converges to Au0 t. This means that A is continuous at u0 ∈ Pc . So, A is continuous on Pc
since u0 is arbitrary. Thus, A is completely continuous.
The proof is complete.
Lemma 2.2. Let u ∈ P, then ut ≥ t − a/θ − au for t ∈ a, θ, and ut ≥ σ 2 b −
t/σ 2 b − θu for t ∈ θ, σ 2 b.
Proof. Since uΔΔ t ≤ 0, it follows that uΔ t is nonincreasing. Hence, for a < t < θ,
ut − ua uθ − ut t
a
θ
t
uΔ sΔs ≥ t − auΔ t,
2.14
uΔ sΔs ≤ θ − tuΔ t,
Advances in Difference Equations
7
from which we have
ut ≥
t−a
t−a
uaθ − t t − auθ
≥
uθ u.
θ−a
θ−a
θ−a
2.15
For θ ≤ t ≤ σ 2 b,
σ 2 b
uΔ sΔs ≤ σ 2 b − t uΔ t,
u σ 2 b − ut t
ut − uθ t
θ
2.16
uΔ sΔs ≥ t − θuΔ t,
we know
σ 2 b − t uθ t − θu σ 2 b
σ 2 b − t
σ 2 b − t
ut ≥
≥
uθ
u.
σ 2 b − θ
σ 2 b − θ
σ 2 b − θ
2.17
The proof is complete.
Lemma 2.3 18. Let P be a cone in a Banach space E. Assum that Ω1 , Ω2 are open subsets of E
with 0 ∈ Ω1 , Ω1 ⊂ Ω2 . If
A : P ∩ Ω2 \ Ω1 −→ P
2.18
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 BVP 1.4.
For the sake of convenience, we define f0 limu → 0 fu/ϕp u, f∞ limu → ∞ fu/ϕp u, 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;
8
Advances in Difference Equations
iv 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 that fu ≤ ϕp εϕp u ϕp εu, for 0 < u ≤ H1 , where ε satisfies
max{εL1 , εL2 } ≤ 1.
3.1
If u ∈ P, with u H1 , then
Au Auθ
θ
θ θ
β
ϕq
hrfurΔr ϕq
hrfurΔr Δs
α
ξ
a
s
θ
θ θ
β
≤ ϕq
hrfurΔr ϕq
hrfurΔr Δs
α
a
a
a
θ
θ θ
β
≤ ϕq
hrϕp εuΔr ϕq
hrϕp εuΔr Δs
α
a
a
a
uεL1
≤ u,
3.2
Au Auθ
δ
ϕq
γ
δ
≤ ϕq
γ
δ
≤ ϕq
γ
η
θ
hrfurΔr
σ 2 b
θ
σ 2 b
θ
σ 2 b
hrfurΔr
θ
s
σ 2 b
hrϕp εuΔr
ϕq
θ
θ
ϕq
σ 2 b
θ
hrfurΔr Δs
σ 2 b
ϕq
θ
hrfurΔr Δs
σ 2 b
θ
hrϕp εuΔr Δs
uεL2
≤ u.
It follows that if ΩH1 {u ∈ E : u < H1 }, then Au ≤ u for u ∈ P ∩ ∂ΩH1 .
Advances in Difference Equations
u≥
9
Since f∞ ∞, then there exists H2 > 0 such that fu ≥ ϕp kϕp u ϕp ku, for
where k > 0 is chosen such that
H2 ,
σ 2 b − η
ξ−a
min k
M1 , k 2
M2
θ−a
σ b − θ
≥ 1.
3.3
Set H2 max{2H1 , θ − a/ξ − aH2 , σ 2 b − θ/σ 2 b − ηH2 }, and ΩH2 {u ∈
E : u < H2 }.
If u ∈ P with u H2 , then
min ut uξ ≥
t∈ξ,θ
ξ−a
u ≥ H2 ,
θ−a
σ 2 b − η
min ut u η ≥ 2
u ≥ H2 .
t∈θ,η
σ b − θ
3.4
So that
Au Auθ
θ
θ θ
β
ϕq
hrfurΔr ϕq
hrfurΔr Δs
α
ξ
a
s
θ
θ θ
β
≥ ϕq
hrϕp kuΔr ϕq
hrϕp kuΔr Δs
α
ξ
ξ
s
θ
θ θ
β
ξ−a
ξ−a
u Δr ϕq
u Δr Δs
hrϕp k
hrϕp k
≥ ϕq
α
θ−a
θ−a
ξ
ξ
s
uk
ξ−a
M1
θ−a
≥ u,
σ 2 b s
η
δ
hrfurΔr ϕq
hrfurΔr Δs
Au Auθ ϕq
γ
θ
θ
θ
η s
η
σ 2 b − η
σ 2 b − η
δ
≥ ϕq
u Δr ϕq
u Δr Δs
hrϕp k 2
hrϕp k 2
γ
σ b − θ
σ b − θ
θ
θ
θ
uk
σ 2 b − η
M2
σ 2 b − θ
≥ u.
3.5
10
Advances in Difference Equations
In other words, if u ∈ P ∩ ∂ΩH2 , then Au ≥ u. Thus by i of Lemma 2.3, it follows
that A has a fixed point in P ∩ ΩH2 \ ΩH1 with H1 ≤ u ≤ H2 .
Now we consider the case f0 ∞ and f∞ 0. Since f0 ∞, there exists H3 > 0, such
that fu ≥ ϕp mϕp u ϕp mu for 0 < u ≤ H3 , where m is such that
σ 2 b − η
ξ−a
min mM1
, mM2 2
θ−a
σ b − θ
≥ 1.
3.6
If u ∈ P with u H3 , then we have
Au Auθ
θ
θ θ
β
ϕq
hrfurΔr ϕq
hrfurΔr Δs
α
ξ
a
s
β
≥ ϕq
α
θ
θ θ
ξ−a
ξ−a
u Δr ϕq
u Δr Δs
hrϕp m
hrϕp m
θ−a
θ−a
ξ
ξ
s
um
ξ−a
M1
θ−a
≥ u,
Au Auθ
δ
ϕq
γ
δ
≥ ϕq
γ
η
θ
hrfurΔr
η
um
θ
σ 2 b
θ
ϕq
s
θ
hrfurΔr Δs
hrϕp
η s
σ 2 b−η
σ 2 b − η
m 2
u Δr ϕq
u Δr Δs
hrϕp m 2
σ b−θ
σ b − θ
θ
θ
σ 2 b − η
M2
σ 2 b − θ
≥ u.
3.7
Thus, we let ΩH3 {u ∈ E : u < H3 }, so that Au ≥ u for u ∈ P ∩ ∂ΩH3 .
Next consider f∞ 0. By definition, there exists H4 > 0 such that fu ≤ ϕp εϕp u ϕp εu for u ≥ H4 , where ε > 0 satisfies
max{εL1 , εL2 } ≤ 1.
3.8
Advances in Difference Equations
11
Suppose f is bounded, then fu ≤ ϕp K for all u ∈ 0, ∞, pick
H4 max{2H3 , KL1 , KL2 }.
3.9
If u ∈ P with u H4 , then
Au Auθ
θ
θ θ
β
ϕq
hrfurΔr ϕq
hrfurΔr Δs
α
ξ
a
s
β
≤ ϕq
α
θ
a
hrϕp KΔr
θ
a
ϕq
θ
a
hrϕp KΔr Δs
KL1
≤ H4
u,
3.10
Au Auθ
δ
ϕq
γ
δ
≤ ϕq
γ
η
θ
σ 2 b
hrfurΔr
σ 2 b
θ
θ
hrϕp KΔr
ϕq
σ 2 b
θ
s
θ
ϕq
hrfurΔr Δs
σ 2 b
θ
hrϕp KΔr Δs
KL2
≤ H4
u.
Now suppose f 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
using 3.8 we have
Au Auθ
β
ϕq
α
β
≤ ϕq
α
θ
ξ
θ
a
hrfurΔr
hrfH4 Δr
θ
a
θ
a
ϕq
ϕq
θ
s
θ
a
hrfurΔr Δs
hrfH4 Δr Δs
12
β
≤ ϕq
α
θ
a
hrϕp εH4 Δr
θ
a
ϕq
θ
a
Advances in Difference Equations
hrϕp εH4 Δr Δs
H4 εL1
≤ H4
u,
Au Auθ
δ
ϕq
γ
δ
≤ ϕq
γ
δ
≤ ϕq
γ
η
θ
hrfurΔr
σ 2 b
θ
σ 2 b
θ
σ 2 b
θ
hrfH4 Δr
σ 2 b
θ
hrϕp εH4 Δr
ϕq
s
θ
ϕq
σ 2 b
θ
hrfurΔr Δs
σ 2 b
ϕq
θ
hrfH4 Δr Δs
σ 2 b
θ
hrϕp εH4 Δr Δs
H4 εL2
≤ H4
u.
3.11
Consequently, in either case we take
ΩH4 {u ∈ E : u < H4 },
3.12
so that for u ∈ P ∩ ∂ΩH4 , we have Au ≥ u. Thus by ii of Lemma 2.3, it follows that A
has a fixed point u in P ∩ ΩH4 \ 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 ≤ ϕp p A1 for 0 ≤ u ≤ p , where
−1
A1 min L−1
,
1 , L2
3.13
Advances in Difference Equations
13
C4 : there exists constant q > 0 such that fu ≥ ϕp q A2 for u ∈ M3 q , M3 , where
A2 max M1−1 , M2−1 ,
3.14
furthermore, p /
q . Then BVP 1.4 has at least one positive solution u, such that u lies between p
and q .
Proof. Without loss of generality, we may assume that p < q .
Let Ωp {u ∈ E : u < p }, for any u ∈ P ∩ ∂Ωp . In view of C3 we have
Au Auθ
θ
θ θ
β
ϕq
hrfurΔr ϕq
hrfurΔr Δs
α
ξ
a
s
θ
θ θ
β
≤ ϕq
hrϕp p A1 Δr ϕq
hrϕp p A1 Δr Δs
α
a
a
a
p A1 L 1
≤ p ,
3.15
Au Auθ
δ
ϕq
γ
δ
≤ ϕq
γ
η
θ
hrfurΔr
σ 2 b
θ
σ 2 b
hrϕp p A1 Δr
θ
ϕq
s
θ
σ 2 b
θ
hrfurΔr Δs
ϕq
σ 2 b
θ
hrϕp p A1 Δr Δs
p A1 L 2
≤ p ,
which yields
Au ≤ u for u ∈ P ∩ ∂Ωp .
3.16
Now set Ωq {u ∈ E : u < q } for u ∈ P ∩ ∂Ωq , we have
ξ−a q ≤ ut ≤ q
θ−a
σ 2 b − η q ≤ ut ≤ q
σ 2 b − θ
for t ∈ ξ, θ,
for t ∈ θ, η .
3.17
14
Advances in Difference Equations
Hence by condition C4 , we can get
Au Auθ
θ
θ θ
β
ϕq
hrfurΔr ϕq
hrfurΔr Δs
α
ξ
a
s
θ θ
θ
β
≥ ϕq
hrϕp q A2 Δr ϕq
hrϕp q A2 Δr Δs
α
ξ
ξ
s
q A2 M1
≥ q ,
3.18
Au Auθ
≥
δ
ϕq
γ
δ
ϕq
γ
η
θ
η
θ
hrfurΔr
hrϕp q A2 Δr
σ 2 b
θ
η
θ
ϕq
ϕq
s
θ
s
θ
hrfurΔr Δs
hrϕp q A2 Δs
q A2 M2
≥ q .
So if we take Ωq {u ∈ E : u < q }, then
Au ≥ u,
u ∈ P ∩ ∂Ωq .
3.19
Consequently, in view of p < q , 3.16, and 3.19, it follows from Lemma 2.3 that A has a
fixed point u in P ∩ Ωq \ Ωp . Moreover, it is a positive solution of 1.4 and p < u < q .
The proof is complete.
For the case i0 1, i∞ 0 or i0 0, i∞ 1 we have the following results.
Theorem 3.3. Suppose that f0 ∈ 0, ϕp A1 and f∞ ∈ ϕp M4 A2 , ∞ hold. Then BVP 1.4 has
at least one positive solution.
Proof. It is easy to see that under the assumptions, the conditions C3 and C4 in Theorem 3.2
are satisfied. So the proof is easy and we omit it here.
Theorem 3.4. Suppose that f0 ∈ ϕp M4 A2 , ∞ and f∞ ∈ 0, ϕp A1 hold. Then BVP 1.4 has
at least one positive solution.
Proof. Since f0 ∈ ϕp M4 A2 , ∞, for ε f0 − ϕp θ − a/ξ − aA2 , there exists a sufficiently
small q1 such that
fu
θ−a
A2 ,
≥ f0 − ε ϕp
ϕp u
ξ−a
u ∈ 0, q1 .
3.20
Advances in Difference Equations
15
Thus, if u ∈ ξ − a/θ − aq1 , q1 , then we have
fu ≥ ϕp uϕp
θ−a
A2
ξ−a
≥ ϕp q1 A2 ;
3.21
by the similar method, one can get if u ∈ σ 2 b − η/σ 2 b − θq2 , q2 , then
fu ≥ ϕp uϕp
σ 2 b − θ
A2
σ 2 b − η
≥ ϕp q2 A2 .
3.22
So, if we choose q min{q1 , q2 }, then for u ∈ M3 q , q , we have fu ≥ ϕp q A2 ,
which yields condition C4 in Theorem 3.2.
Next, by f∞ ∈ 0, ϕp A1 , for ε ϕp A1 − f∞ , there exists a sufficiently large p > q such that
fu
≤ f∞ ε ϕp A1 ,
ϕp u
u ∈ p , ∞ ,
3.23
where we consider two cases.
Case 1. Suppose that f is bounded, say
fu ≤ ϕp K,
u ∈ 0, ∞.
3.24
In this case, take sufficiently large p such that p ≥ max{K/A1 , p }, then from 3.24, we know
fu ≤ ϕp K ≤ ϕp A1 p for u ∈ 0, p , which yields condition C3 in Theorem 3.2.
Case 2. Suppose that f is unbounded. it is easy to know that there is p > p such that
fu ≤ f p ,
u ∈ 0, p .
3.25
Since p > p then from 3.23 and 3.25, we get
fu ≤ f p ≤ ϕp p A1 ,
u ∈ 0, p .
3.26
Thus, the condition C3 of Theorem 3.2 is satisfied.
Hence, from Theorem 3.2, BVP 1.4 has at least one positive solution.
The proof is complete.
From Theorems 3.3 and 3.4, we have the following two results.
Corollary 3.5. Suppose that f0 0 and the condition C4 in Theorem 3.2 hold. Then BVP 1.4 has
at least one positive solution.
16
Advances in Difference Equations
Corollary 3.6. Suppose that f∞ 0 and the condition C4 in Theorem 3.2 hold. Then BVP 1.4
has at least one positive solution.
Theorem 3.7. Suppose that f0 ∈ 0, ϕp A1 and f∞ ∞ hold. Then BVP 1.4 has at least one
positive solution.
Proof. In view of f∞ ∞, similar to the first part of Theorem 3.1, we have
Au ≥ u,
u ∈ P ∩ ∂ΩH2 .
3.27
Since f0 ∈ 0, ϕp A1 , for ε ϕp A1 − f0 > 0, there exists a sufficiently small p ∈ 0, H2 such
that
fu ≤ f0 ε ϕp u ϕp A1 u ≤ ϕp A1 p ,
u ∈ 0, p .
3.28
Similar to the proof of Theorem 3.2, we obtain
Au ≤ u,
u ∈ P ∩ ∂Ωp .
3.29
The result is obtained, and the proof is complete.
Theorem 3.8. Suppose that f∞ ∈ 0, ϕp A1 and f0 ∞ hold. Then BVP 1.4 has at least one
positive solution.
Proof. Since f0 ∞, similar to the second part of Theorem 3.1, we have Au ≥ u for
u ∈ P ∩ ∂ΩH3 .
By f∞ ∈ 0, ϕp A1 , 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 Theorems 3.7 and 3.8, we can get the following corollaries.
Corollary 3.9. Suppose that f∞ ∞ and the condition C3 in Theorem 3.2 hold. Then BVP 1.4
has at least one positive solution.
Corollary 3.10. Suppose that f0 ∞ and the condition C3 in Theorem 3.2 hold. Then BVP 1.4
has at least one positive solution.
Theorem 3.11. Suppose that i0 0, i∞ 2, and the condition C3 of Theorem 3.2 hold. Then BVP
1.4 has at least two positive solutions u1 , u2 ∈ P such that 0 < u1 < p < u2 .
Proof. By using the method of proving Theorems 3.1 and 3.2, we can deduce the conclusion
easily, so we omit it here.
Theorem 3.12. Suppose that i0 2, i∞ 0, and the condition C4 of Theorem 3.2 hold. Then BVP
1.4 has at least two positive solutions u1 , u2 ∈ P such that 0 < u1 < q < u2 .
Advances in Difference Equations
17
Proof. Combining the proofs of Theorems 3.1 and 3.2, 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. When T R,
u u 1 − t4 − arctan u, 0 < t < 1,
1
1
u0 u
,
u1 −u
,
4
2
4.1
where, p 3, α β γ δ 1, ht 1 − t, fu 4 − arctan u.
It is easy to see that the condition C1 and C2 are satisfied and
f0 lim
u→0
fu
∞,
ϕp u
f∞ lim
u→∞
fu
0.
ϕp u
4.2
So, by Theorem 3.1, the BVP 4.1 has at least one positive solution.
Acknowledgments
This research is supported by the Natural Science Foundation of China 60774004, 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.
References
1 S. Hilger, “Analysis on measure chains—a unified approach to continuous and discrete calculus,”
Results in Mathematics, vol. 18, no. 1-2, pp. 18–56, 1990.
2 V. Spedding, “Taming nature’s numbers,” New Scientist, pp. 28–32, July 2003.
3 M. A. Jones, B. Song, and D. M. Thomas, “Controlling wound healing through debridement,”
Mathematical and Computer Modelling, vol. 40, no. 9-10, pp. 1057–1064, 2004.
4 D. M. Thomas, L. Vandemuelebroeke, and K. Yamaguchi, “A mathematical evolution model for
phytoremediation of metals,” Discrete and Continuous Dynamical Systems. Series B, vol. 5, no. 2, pp.
411–422, 2005.
5 M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications,
Birkhäuser, Boston, Mass, USA, 2001.
6 M. Bohner and A. Peterson, Eds., Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston,
Mass, USA, 2003.
7 D. Anderson, R. Avery, and J. Henderson, “Existence of solutions for a one dimensional p-Laplacian
on time-scales,” Journal of Difference Equations and Applications, vol. 10, no. 10, pp. 889–896, 2004.
8 Z. He, “Double positive solutions of three-point boundary value problems for p-Laplacian dynamic
equations on time scales,” Journal of Computational and Applied Mathematics, vol. 182, no. 2, pp. 304–315,
2005.
9 H.-R. Sun and W.-T. Li, “Existence theory for positive solutions to one-dimensional p-Laplacian
boundary value problems on time scales,” Journal of Differential Equations, vol. 240, no. 2, pp. 217–
248, 2007.
18
Advances in Difference Equations
10 W.-T. Li and H. R. Sun, “Positive solutions for second-order m-point boundary value problems on
time scales,” Acta Mathematica Sinica, vol. 22, no. 6, pp. 1797–1804, 2006.
11 H. R. Sun, Boundary value problems for dynamic equations on measure chains, Ph. D. thesis, Lanzhou
University, Lanzhou, China, 2004.
12 D. R. Anderson, “Existence of solutions for a first-order p-Laplacian BVP on time scales,” Nonlinear
Analysis: Theory, Methods & Applications, vol. 69, no. 12, pp. 4521–4525, 2008.
13 Y.-H. Su, W.-T. Li, and H.-R. Sun, “Positive solutions of singular p-Laplacian BVPs with sign changing
nonlinearity on time scales,” Mathematical and Computer Modelling, vol. 48, no. 5-6, pp. 845–858, 2008.
14 X. Zhang and W. Ge, “Existence of positive solutions for a class of m-point boundary value problems,”
Advances in Difference Equations, vol. 2008, Article ID 845121, 9 pages, 2008.
15 M. Feng, X. Li, and W. Ge, “Triple positive solutions of fourth-order four-point boundary value
problems for p-Laplacian dynamic equations on time scales,” Advances in Difference Equations, vol.
2008, Article ID 496078, 9 pages, 2008.
16 C. Song, “Eigenvalue problems for p-Laplacian functional dynamic equations on time scales,”
Advances in Difference Equations, vol. 2008, Article ID 879140, 9 pages, 2008.
17 D. Ji and W. Ge, “Existence of multiple positive solutions for Sturm-Liouville-like four-point
boundary value problem with p-Laplacian,” Nonlinear Analysis: Theory, Methods & Applications, vol.
68, no. 9, pp. 2638–2646, 2008.
18 D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5 of Notes and Reports in
Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1988.