Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2009, Article ID 219251, 14 pages
doi:10.1155/2009/219251
Research Article
Multiple Positive Solutions for a Class of m-Point
Boundary Value Problems on Time Scales
Meiqiang Feng,1 Xuemei Zhang,2, 3 and Weigao Ge3
1
School of Science, Beijing Information Science & Technology University,
Beijing 100192, China
2
Department of Mathematics and Physics, North China Electric Power University,
Beijing 102206, China
3
Department of Applied Mathematics, Beijing Institute of Technology,
Beijing 100081, China
Correspondence should be addressed to Xuemei Zhang, zxm74@sina.com
Received 1 December 2008; Revised 15 April 2009; Accepted 10 June 2009
Recommended by Victoria Otero-Espinar
By constructing an available integral operator and combining Krasnosel’skii-Zabreiko fixed point
theorem with properties of Green’s function, this paper shows the existence of multiple positive
solutions for a class of m-point second-order Sturm-Liouville-like boundary value problems on
time scales with polynomial nonlinearity. The results significantly extend and improve many
known results for both the continuous case and more general time scales. We illustrate our results
by one example, which cannot be handled using the existing results.
Copyright q 2009 Meiqiang Feng 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
Recently, there have been many papers working on the existence of positive solutions to
boundary value problems for differential equations on time scales; see, for example, 1–
20. This has been mainly due to its unification of the theory of differential and difference
equations. An introduction to this unification is given in 11, 12, 18, 19. Now, this study
is still a new area of fairly theoretical exploration in mathematics. However, it has led to
several important applications, for example, in the study of insect population models, neural
networks, heat transfer, and epidemic models; see, for example, 10, 11. For some other
excellent results and applications of the case that boundary value problems on time scales
to a variety of problems from Khan et al. 21, Agarose et al. 22, Wang 23, Sun 24, Feng
et al. 25, Feng et al. 26 and Feng et al. 27.
Motivated by the works mentioned above, we intend in this paper to study the
existence of multiple positive solutions for the second-order m-point nonlinear dynamic
2
Advances in Difference Equations
equation on time scales with polynomial nonlinearity:
Δ
− ptx∇ t q t x t f t, x t ,
αx t1 − βp t1 x∇ t1 t1 < t < tm ,
m−1
ai x ti ,
i2
1.1
m−1
γx tm δp tm x∇ tm bi x ti ,
i2
where T is a time scale,
p, q : t1 , tm −→ 0, ∞ ,
p ∈ CΔ t1 , tm , q ∈ C t1 , tm ;
1.2
the points ti ∈ Tkk for i ∈ {1, 2, . . . , m} with t1 < t2 < · · · < tm ;
α, γ, β, δ ∈ 0, ∞
f t, x αγ αδ βγ > 0,
n
cj t xυj ,
ai , bi ∈ 0, ∞ , i ∈ {2, 3, . . . , m − 1} ;
1.3
cj ∈ C t1 , tm , 0, ∞ , υj ∈ 0, ∞ , j 1, 2, . . . , n.
1.4
j1
Recently, Xu 28 considered the following second-order two-point impulsive singular
differential equations boundary value problem:
y n
aj t xαj 0,
0 < t < 1, t / t1 ,
j1
1.5
Δy|tt1 I y t1 ,
y 0 y 1 0.
By means of fixed point index theory in a cone, the author established the existence of two
nonnegative solutions for problem 1.5.
More recently, by applying Guo-Krasnosel’skii fixed point theorem in a cone,
Anderson and Ma 6 established the existence of at least one positive solution to the
multipoint time-scale eigenvalue problem:
py∇
Δ
t − q t y t λh t f y 0,
αy t1 − βp t1 y∇ t1 n−1
t1 < t < tn ,
ai y ti ,
i2
γx tn δp tn x∇ tn n−1
i2
where f : 0, ∞ → 0, ∞ is continuous.
bi y ti ,
1.6
Advances in Difference Equations
3
As far as we know, there is no paper to study the existence of multiple positive
solutions to problem 1.1 on time scales with polynomial nonlinearity. The objective of the
present paper is to fill this gap. On the other hand, many difficulties occur when we study
BVPs on time scales. For example, basic tools from calculus such as Fermat’s theorem, Rolle’s
theorem and the intermediate value theorem may not necessarily hold. So it is interesting and
important to discuss the problem 1.1. The purpose of this paper is to prove that the problem
1.1 possesses at least two positive solutions. Moreover, the methods used in this paper are
different from 6, 28 and the results obtained in this paper generalize some results in 6, 28
to some degree.
The time scale related notations adopted in this paper can be found, if not explained
specifically, in almost all literature related to time scales. The readers who are unfamiliar with
this area can consult for example 11, 12, 18, 19 for details.
For convenience, we list the following well-known definitions.
Definition 1.1. A time scale T is a nonempty closed subset of R.
Definition 1.2. Define the forward backward jump operator σt at t for t < sup T ρt at t
for t > inf T by σt inf{τ > t : τ ∈ T} ρt sup{τ < t : τ ∈ T} for all t ∈ T.
We assume throughout that T has the topology that it inherits from the standard
topology on R and say t is right-scattered, left-scattered, right-dense and left-dense if σt >
t, ρt < t, σt t and ρt t, respectively. Finally, we introduce the sets Tk and Tk . which are
derived from the time scale T as follows. If T has a left-scattered maximum t∗1 , then Tk T−t∗1 ,
otherwise Tk T. If T has a right-scattered minimum t∗2 , then Tk T − t∗2 , otherwise Tk T.
Definition 1.3. Fix t ∈ T and let y : T → R. Define yΔ t to be the number if it exists with
the property that given ε > 0 there is a neighborhood U of t with
y σ t − y s − yΔ t σ t − s < ε |σ t − s|
1.7
for all s ∈ U, where yΔ denotes the delta derivative of y with respect to the first variable,
then
t
g t : ω t, τ Δτ
1.8
a
implies
Δ
g t t
ωΔ t, τ Δτ ω σ t , τ .
1.9
a
Definition 1.4. Fix t ∈ T and let y : T → R. Define y∇ t to be the number if it exists
with the property that given ε > 0 there is a neighborhood U of t with
y ρ t − y s − y∇ t ρ t − s < ε ρ t − s
for all s ∈ U. Call y∇ t the nabla derivative of yt at the point t.
1.10
4
Advances in Difference Equations
If T R then f Δ t f ∇ t f t. If T Z then f Δ t ft 1 − ft is the forward
difference operator while f ∇ t ft − ft − 1 is the backward difference operator.
Definition 1.5. A function f : T → R is called rd-continuous provided it is continuous at all
right dense points of T and its left sided limit exists finite at left dense points of T. We let
0
T denote the set of rd-continuous functions f : T → R.
Crd
Definition 1.6. A function f : T → R is called ld-continuous provided it is continuous at
all left dense points of T and its right sided limit exists finite at right dense points of T.
0
T denote the set of ld-continuous functions f : T → R.
We let Cld
Definition 1.7. A function F : Tk → R is called a delta-antiderivative of f : Tk → R provided
F Δ t ft holds for all t ∈ Tk . In this case we define the delta integral of f by
t
f s Δs F t − F a ,
1.11
a
for all a, t ∈ T.
Definition 1.8. A function Φ : Tk → R is called a nabla-antiderivative of f : Tk → R
provided Φ∇ t ft holds for all t ∈ Tk . In this case we define the nabla integral of f by
t
f s ∇s Φ t − Φ a ,
1.12
a
for all a, t ∈ T.
2. Preliminaries
In this section, we provide some necessary background. In particular, we state some
properties of Green’s function associated with problem 1.1, and we then state a fixed-point
theorem which is crucial to prove our main results.
The basic space used in this paper is E Cρt1 , tm . It is well known that E is a
Banach space with the norm || · || defined by ||x|| supt∈ρt1 ,tm |xt|. Let P be a cone of E,
Pr {x ∈ P : x ≤ r}, ∂Pr {x ∈ P : x r}, where r > 0.
In this paper, the Green’s function of the corresponding homogeneous BVP is defined
by
⎧
1 ⎨ψ t φ s ,
G t, s d ⎩ψ s φ t ,
if ρ t1 ≤ t ≤ s ≤ tm ,
if ρ t1 ≤ s ≤ t ≤ tm ,
2.1
where
d : αφ t1 − βp t1 φ∇ t1 γψ tm δp tm ψ ∇ tm ,
2.2
Advances in Difference Equations
5
and φ and ψ satisfy
Δ
− pψ ∇ t q t ψ t 0,
Δ
− pφ∇ t q t φ t 0,
ψ t1 β,
p t1 ψ ∇ t1 α,
2.3
φ tm δ,
p tm φ∇ tm −γ,
respectively.
Lemma 2.1 see 6. Assume that 1.2 and 1.3 hold. Then d > 0 and the functions ψ and φ
satisfy
t ∈ ρ t1 , tm ,
ψ t > 0, t ∈ ρ t1 , tm ,
φ t ≥ 0, t ∈ ρ t1 , tm ,
p t ψ ∇ t ≥ 0, t ∈ ρ t1 , tm ,
p t φ∇ t ≤ 0, t ∈ ρ t1 , tm .
φ t > 0, t ∈ ρ t1 , tm ,
ψ t ≥ 0,
2.4
From Lemma 2.1 and the definition of Gt, s, we can prove that Gt, s has the
following properties.
Proposition 2.2. For t, s ∈ ξ1 , ξ2 , one has
2.5
G t, s > 0,
where ξ1 , ξ2 ∈ Tkk , ρt1 < ξ1 < ξ2 < tm .
In fact, from Lemma 2.1, we have ψt > 0,
φt > 0 for t ∈ ξ1 , ξ2 . Therefore 2.5
holds.
Proposition 2.3. If 1.2 holds, then for t, s ∈ ρt1 , tm × ρt1 , tm , one has
0 ≤ G t, s ≤ G s, s .
2.6
Proof. In fact, from Lemma 2.1, we obtain ψt ≥ 0, φt ≥ 0 for t ∈ ρt1 , tm . So Gt, s ≥ 0.
On the other hand, from Lemma 2.1, we know that ptψ ∇ t ≥ 0, ptφ∇ t ≤ 0 for
t ∈ ρt1 , tm . This together with pt > 0 implies that ψ ∇ t ≥ 0, φ∇ t ≤ 0 for t ∈ ρt1 , tm .
Hence ψt is nondecreasing on ρt1 , tm , φ is nonincreasing on ρt1 , tm . So 2.6 holds.
Proposition 2.4. For all t ∈ ξ1 , ξ2 , s ∈ ρt1 , tm one has
G t, s ≥ σ t G s, s ,
2.7
6
Advances in Difference Equations
where
σ t : min
ψ t
φ t
, ψ tm φ ρ t1 2.8
.
Proof. In fact, for t ∈ ξ1 , ξ2 , we have
G t, s
≥ min
G s, s
ψ t φ t
,
ψ s φ s
≥ min
ψ t
φ t
, ψ tm φ ρ t1 : σ t .
2.9
Therefore 2.7 holds.
It is easy to see that 0 < σt < 1, for t ∈ ξ1 , ξ2 . Thus, there exists γ > 0 such that
Gt, s ≥ γGs, s for t ∈ ξ1 , ξ2 , where
γ min {σ t : t ∈ ξ1 , ξ2 } .
2.10
We remark that Proposition 2.2 implies that there exists τ > 0 such that for t, s ∈ ξ1 , ξ2 G t, s ≥ τ.
2.11
m−2
m−2
ai φ ti − ai ψ ti d −
i1
i1
.
D : m−2
m−2
bi ψ ti − bi φ ti d −
i1
i1
2.12
Set
Lemma 2.5 see 6. Assume that 1.2 and 1.3 hold. If D /
0 and u ∈ Crd t1 , tm , then the
nonhomogeneous boundary value problem
Δ
− p t x∇ t q t x t u t ,
αx t1 − βp t1 x∇ t1 t1 < t < tm ,
m−1
ai x ti ,
i2
γx tm δp tm x∇ tm 2.13
m−1
bi x ti i2
has a unique solution x for which the formula
x t tm
t1
G t, s u s Δs Γ u t ψ t Υ u t φ t
2.14
Advances in Difference Equations
7
holds, where
tm
m−1
m−1
G ti , s u s Δs d −
ai φ ti ai
1 i2
t1
i2
,
Γ u s : m−1
t
m−1
m
D G ti , s u s Δs − bi φ ti bi
i2
t1
i2
m−1
m−1
tm
ai
G ti , s u s Δs
− ai ψ ti 1 i2
t
i2
.
Υ u s : tm1
m−1
m−1
D
bi ψ ξi bi
G ti , s u s Δs d −
t1
i2
i2
2.15
2.16
By similar method, one can define
Γ0 f t, x0 t ,
Υ0 f t, x0 t ,
Γ1 f t, x1 t ,
Υ1 f t, x1 t ,
Γ2 f t, x2 t ,
Υ2 f t, x2 t ,
Γ∗ f t, x∗ t ,
Υ∗ f t, x∗ t .
2.17
The following lemma is crucial to prove our main results.
Lemma 2.6 see29, 30. Let Ω1 and Ω2 be two bounded open sets in a real Banach space E, such
that 0 ∈ Ω1 and Ω1 ⊂ Ω2 . Let the operator A : P ∩ Ω2 \ Ω1 → P be completely continuous, where
P is a cone in E. Suppose that one of the two conditions
i
Ax ≥
/ x,
∀x ∈ P ∩ ∂Ω1 ;
Ax ≤
/ x,
∀x ∈ P ∩ ∂Ω2 ,
2.18
ii
Ax ≤
/ x,
∀x ∈ P ∩ ∂Ω1 ;
Ax ≥
/ x,
∀x ∈ P ∩ ∂Ω2 ,
2.19
or
is satisfied. Then A has at least one fixed point in P ∩ Ω2 \ Ω1 .
3. Main Results
In this section, we apply Lemma 2.6 to establish the existence of at least two positive solutions
for BVP 1.1.
The following assumptions will stand throughout this paper.
H1 There exist υj1 < 1, υj2 > 1 such that
inf cj1 t τ1 > 0,
t∈ξ1 ,ξ2 inf cj2 t τ2 > 0,
t∈ξ1 ,ξ2 j 1, 2, . . . , n,
where υj1 , υj2 , cj1 t and cj2 t are defined in 1.4, respectively.
3.1
8
Advances in Difference Equations
H2 We have
d−
D < 0,
m−1
d−
ai φ ti > 0,
i2
m−1
bi ψ ti > 0
3.2
i2
for d and D given in 2.2 and 2.12, respectively.
If H2 holds, then we can show that Γft, x, Υft, x have the following
properties.
Proposition 3.1. If 1.2–1.4 and H2 hold, then from 2.15, for x ∈ Cρt1 , tm , one has
m−1
m−1
ai φ ti ai d −
n n 1 i2
i2
M cj xυj : ΓM
Γ f t, x ≤
cj xυj ,
m−1
m−1
L
L
D j1
j1
bi − bi φ ti i2
i2
where cj L :
tm
t1
3.3
|cj s|Δs, M maxt,s∈ρt1 ,tm ×ρt1 ,tm Gt, s.
Proof. Let
G
F
m−1
tm
i2
t1
m−1
tm
i2
t1
ai
bi
G ti , s f s, x s Δs,
H d−
m−1
ai φ ti ,
i2
G ti , s f s, x s Δs,
Q−
m−1
3.4
bi φ ti .
i2
Then from 1.2–1.4 and H2 , we obtain G ≥ 0, F ≥ 0, H > 0, Q ≤ 0. Therefore, GQ ≤
0, −FH ≤ 0.
On the other hand, since
tm
G ti , s f s, x s Δs ≤ M
t1
we have G ≤
m−1
i2
ai Λ, F ≤
n cj xυj : Λ,
L
3.5
j1
m−1
i2
bi Λ. So one has
m−1
m−1
i2
i2
ai ΛQ − H
This and D < 0 imply 3.3 holds.
bi Λ ≤ GQ − FH ≤ 0.
3.6
Advances in Difference Equations
9
Proposition 3.2. If 1.2–1.4 and H2 hold, then from 2.16, x ∈ Cρt1 , tm , one has
m−1
m−1
ai − ai ψ ti n n
i2 Υ f t, x ≤ 1 i2
cj xυj : ΥM
M
cj L xυj .
m−1
m−1
L
D j1
bi ψ ξi bi j1
d −
i2
i2
3.7
Proof. The proof is similar to that of Proposition 3.1. So we omit it.
For the sake of applying fixed point theorem on cone, we construct a cone in E Cρt1 , tm by
P
x ∈ E : x t ≥ 0, t ∈ ρ t1 , tm , min x t ≥ γx ,
3.8
t∈ξ1 ,ξ2 where γ is defined in 2.10.
Define A : P → P by
Ax t tm
G t, s f s, x s Δs Γ f t, x t ψ t Υ f t, x t φ t .
3.9
t1
By 2.14, it is well known that the problem 1.1 has a positive solution x if and only
if x ∈ P is a fixed point of A.
Lemma 3.3. Suppose that 1.2–1.4 and H1 -H2 hold. Then AP ⊂ P and A : P → P is
completely continuous.
Proof. For x ∈ P, by 2.14, we have Axt ≥ 0 and
Ax ≤
tm
G s, s f s, x s Δs Γ f t, x t ψ tm Υ f t, x t φ ρ t1 .
3.10
t1
On the other hand, for t ∈ ξ1 , ξ2 , by 3.9,3.10 and 2.7, we obtain
min Ax t min
t∈ξ1 ,ξ2 t
m
t∈ξ1 ,ξ2 ≥ σ t
G t, s f s, x s Δs Γ f t, x t ψ t Υ f t, x t φ t
t1
t
m
G s, s f s, x s ΔsΓ f t, x t ψ tm Υ f t, x t φ ρ t1 t1
≥ σ t Ax ≥ γAx.
3.11
Therefore Ax ∈ P , that is, AP ⊂ P .
Next by standard methods and the Ascoli-Arzela theorem one can prove that A : P →
P is completely continuous. So it is omitted.
10
Advances in Difference Equations
Theorem 3.4. Suppose that 1.2–1.4 and H1 -H2 hold. Then problem 1.1 has at least two
positive solutions provided
n cj 1 Γψ
tn Υφ
t1 < M−1 ,
L
3.12
j1
Υ
and M are defined in 3.3, 3.7 and in Proposition 3.1, respectively.
where Γ,
Proof. Let A be the cone preserving, completely continuous operator that was defined by
3.9.
Let Sl {x ∈ E : x < l}, where l > 0. Choosing r and r satisfy
0 < r < min 1, ττ1 ξ2 − ξ1 1/1−υj1 γ υj1 /1−υj1 ,
r > max 1, ττ2 ξ2 − ξ1 −1/υj2 −1 γ −υj2 /υj2 −1 .
3.13
Now we prove that
Ax /
≤ x,
∀x ∈ P ∩ ∂Sr ,
3.14
Ax /
≤ x,
∀x ∈ P ∩ ∂Sr .
3.15
In fact, if there exists x1 ∈ P ∩ ∂Sr such that Ax1 ≤ x1 , then for t ∈ ξ1 , ξ2 , we have
x1 t ≥ Ax1 t
tm
G t, s f s, x1 s Δs Γ1 f t, x1 t ψ t Υ1 f t, x1 t φ t
t1
≥
tm
G t, s f s, x1 s Δs
3.16
t1
≥
ξ2
G t, s cj1 s x1 sυj1 Δs
ξ1
≥ ττ1 ξ2 − ξ1 γ υj1 x1 υj1 ,
where Γ1 ft, xt, Υ1 ft, xt defined by 2.17.
Therefore r ≥ ττ1 ξ2 − ξ1 γ υj1 r υj1 , that is, r ≥ ττ1 ξ2 − ξ1 1/1−υj1 γ υj1 /1−υj1 , which is a
contradiction. Hence 3.14 holds.
Advances in Difference Equations
11
Next, turning to 3.15. If there exists x2 ∈ P ∩ ∂Sr such that Ax2 ≤ x2 , then for t ∈
ξ1 , ξ2 , we have
x2 t ≥ Ax2 t
tm
G t, s f s, x2 s Δs Γ2 f t, x2 t ψ t Υ2 f t, x2 t φ t
t1
≥
tm
G t, s f s, x2 s Δs
3.17
t1
≥
ξ2
G t, s cj2 s x2 sυj2 Δs
ξ1
≥ ττ2 ξ2 − ξ1 γ υj2 x2 υj2 ,
where Γ2 ft, xt, Υ2 ft, xt are defined by 2.17.
Therefore r ≥ ττ2 ξ2 − ξ1 γ υj2 r υj2 , that is, r ≤ ττ2 ξ2 − ξ1 −1/υj2 −1 γ −υj2 /υj2 −1 , which is a
contradiction. Hence 3.15 holds.
It remains to prove
Ax /
≥ x,
∀x ∈ P ∩ ∂S1 .
3.18
In fact, if there exists x0 ∈ P ∩ ∂S1 such that Ax0 ≥ x0 , then for t ∈ t1 , tm ∩ T, we have
1 x0 ≤ Ax0 ≤ M
n cj x0 υj 1 Γψ
tm Υφ
t1 L
3.19
j1
that is,
n cj 1 Γψ
tm Υφ
t1 ≥ M−1 ,
L
3.20
j1
which is a contradiction, where Γ0 ft, xt, Υ0 ft, xt are defined by 2.17. Hence
3.18 holds. From Lemma 2.6, 3.14, 3.15 and 3.18 yield that the problem 1.1 has at
least two solutions x∗ , x∗∗ and x∗ ∈ P ∩ Sr \ S1 , x∗∗ ∈ P ∩ S1 \ Sr . The proof is complete.
12
Advances in Difference Equations
4. Example
Example 4.1. To illustrate how our main results can be used in practice we present an example.
Let T {0, 1/2, 1/4, . . . , 1/2n , . . . , 1}. Take pt ≡ 1, qt ≡ 0, α 1, β 0, γ 1, δ 0, t1 0, t2 1/2, tm 1, a2 1/2, b2 1 in 1.1. Now we consider the following three
point boundary value problem
x∇Δ t f t, x t , 0 < t < 1,
1
1
1
x 0 x
x 1 x
,
2
2
2
4.1
where
f t, x 1 1/2 1 2
tx t x t − t2 x2 .
10
20
1 2
t ,
20
c3 t t − t2 ,
4.2
It is not difficult to see that
c1 t 1
t,
10
c2 t v1 1
,
2
v2 1,
v3 2.
On the other hand, by calculating we have ψt t, φt 1 − t, d 1, d −
1 − 1/2 × 1/2 3/4 > 0, d − m−1
i2 bi ψti 1 − 1 × 1/2 1/2 > 0,
1 3 −
1
D 4 4 − < 0,
1
1
4
− 2
2
G t, s m−1
i2
4.3
ai φti ⎧
⎨s 1 − t ,
4.4
⎩t 1 − s ,
7/2, Υ
2.
and M maxt,s∈0,1 Gt, s 1/4, Γ
Let υj1 1/2, υj2 2, cj1 t 1/10t, cj2 t t − t2 . Then υj1 < 1, υj2 > 1 and
inf cj1 t t∈ξ1 ,ξ2 1
ξ1 > 0,
10
inf cj2 t min {ξ1 1 − ξ1 , ξ2 1 − ξ2 } > 0,
t∈ξ1 ,ξ2 j 1, 2, 3. 4.5
It follows that H1 and H2 hold.
Finally, we prove that
n cj 1 Γψ
tn Υφ
t1 < M−1 .
L
4.6
j1
1 13/2, and
7/2, Υ
2, ψ1 1, φ0 1, we have 1 Γψt
n Υφt
In fact, from Γ
n cj 1 Γψ
tn Υφ
t1 2 × 13 13 < 4 M−1 .
L
5
2
5
j1
4.7
Advances in Difference Equations
13
Therefore, the conditions of Theorem 3.4 hold. Hence problem 4.1 has at least two positive
solutions.
Remark 4.2. Example 4.1 implies that there is a large number of functions that satisfy the
conditions of Theorem 3.4. In addition, the conditions of Theorem 3.4 are also easy to check.
Acknowledgments
This work is sponsored by the National Natural Science Foundation of China 10671012 and
the Scientific Creative Platform Foundation of Beijing Municipal Commission of Education
PXM2008-014224-067420. The authors thank the referee for his careful reading of the
manuscript and useful suggestions.
References
1 R. P. Agarwal, M. Bohner, and P. J. Y. Wong, “Sturm-Liouville eigenvalue problems on time scales,”
Applied Mathematics and Computation, vol. 99, no. 2-3, pp. 153–166, 1999.
2 R. I. Avery and D. R. Anderson, “Existence of three positive solutions to a second-order boundary
value problem on a measure chain,” Journal of Computational and Applied Mathematics, vol. 141, no. 1-2,
pp. 65–73, 2002.
3 J. Henderson, “Multiple solutions for 2mth order Sturm-Liouville boundary value problems on a
measure chain,” Journal of Difference Equations and Applications, vol. 6, no. 4, pp. 417–429, 2000.
4 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.
5 M. Bohner and H. Luo, “Singular second-order multipoint dynamic boundary value problems with
mixed derivatives,” Advances in Difference Equations, vol. 2006, Article ID 54989, 15 pages, 2006.
6 D. R. Anderson and R. Ma, “Second-order n-point eigenvalue problems on time scales,” Advances in
Difference Equations, vol. 2006, Article ID 59572, 17 pages, 2006.
7 A. C. Peterson, Y. N. Raffoul, and C. C. Tisdell, “Three point boundary value problems on time scales,”
Journal of Difference Equations and Applications, vol. 10, no. 9, pp. 843–849, 2004.
8 J. Henderson and C. C. Tisdell, “Topological transversality and boundary value problems on time
scales,” Journal of Mathematical Analysis and Applications, vol. 289, no. 1, pp. 110–125, 2004.
9 J. Henderson, A. Peterson, and C. C. Tisdell, “On the existence and uniqueness of solutions to
boundary value problems on time scales,” Advances in Difference Equations, vol. 2004, no. 2, pp. 93–109,
2004.
10 R. P. Agarwal, M. Bohner, and W.-T. Li, Nonoscillation and Oscillation: Theory for Functional Differential
Equations, vol. 267 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New
York, NY, USA, 2004.
11 M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications,
Birkhäuser, Boston, Mass, USA, 2001.
12 M. Bohner and A. Peterson, Eds., Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston,
Mass, USA, 2003.
13 D. R. Anderson, “Extension of a second-order multi-point problem to time scales,” Dynamic Systems
and Applications, vol. 12, no. 3-4, pp. 393–403, 2003.
14 L. Erbe and A. Peterson, “Green’s functions and comparison theorems for differential equations on
measure chains,” Dynamics of Continuous, Discrete and Impulsive Systems, vol. 6, no. 1, pp. 121–137,
1999.
15 D. R. Anderson, “Eigenvalue intervals for a two-point boundary value problem on a measure chain,”
Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 57–64, 2002.
16 Z. He and X. Jiang, “Triple positive solutions of boundary value problems for p-Laplacian dynamic
equations on time scales,” Journal of Mathematical Analysis and Applications, vol. 321, no. 2, pp. 911–920,
2006.
14
Advances in Difference Equations
17 C. J. Chyan and J. Henderson, “Eigenvalue problems for nonlinear differential equations on a measure
chain,” Journal of Mathematical Analysis and Applications, vol. 245, no. 2, pp. 547–559, 2000.
18 V. Lakshmikantham, S. Sivasundaram, and B. Kaymakcalan, Dynamic Systems on Measure Chains, vol.
370 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands,
1996.
19 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.
20 L. Erbe, A. Peterson, and S. H. Saker, “Hille-Kneser-type criteria for second-order dynamic equations
on time scales,” Advances in Difference Equations, vol. 2006, Article ID 51401, 18 pages, 2006.
21 R. A. Khan, J. J. Nieto, and V. Otero-Espinar, “Existence and approximation of solution of three-point
boundary value problems on time scales,” Journal of Difference Equations and Applications, vol. 14, no.
7, pp. 723–736, 2008.
22 R. P. Agarose, V. Other-Espial, K. Parera, and D. R. Vivre, “Multiple positive solutions in the sense of
distributions of singular BAPS on time scales and an application to Eden-Fowler equations,” Advances
in Difference Equations, vol. 2008, Article ID 796851, 13 pages, 2008.
23 D.-B. Wang, “Three positive solutions of three-point boundary value problems for p-Laplacian
dynamic equations on time scales,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no.
8, pp. 2172–2180, 2008.
24 J.-P. Sun, “A new existence theorem for right focal boundary value problems on a measure chain,”
Applied Mathematics Letters, vol. 18, no. 1, pp. 41–47, 2005.
25 M. Feng, X. Zhang, and W. Ge, “Positive solutions for a class of boundary value problems on time
scales,” Computers & Mathematics with Applications, vol. 54, no. 4, pp. 467–475, 2007.
26 M.-Q. Feng, X.-G. Li, and W.-G. Ge, “Triple positive solutions of fourth-order four-point boundary
value problems of p-Laplacian dynamic equations on time scales,” Advances in Difference Equations,
vol. 2008, Article ID 496078, 9 pages, 2008.
27 M. Feng, H. Feng, X. Zhang, and W. Ge, “Triple positive solutions for a class of m-point dynamic
equations on time scales with p-Laplacian,” Mathematical and Computer Modelling, vol. 48, no. 7-8, pp.
1213–1226, 2008.
28 X. Xu, Multiple solutions for impulsive singular differential equations boundary value problems, Doctorial
thesis, Shandong University, Shandong, China, 2001.
29 K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.
30 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, Boston, Mass, USA, 1988.