- Meiqiang Feng1,
- Xuemei Zhang2, 3Email author and
- Weigao Ge3
https://doi.org/10.1155/2009/219251
© Meiqiang Feng et al. 2009
Received: 1 December 2008
Accepted: 10 June 2009
Published: 19 July 2009
Abstract
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.
Keywords
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].
By means of fixed point index theory in a cone, the author established the existence of two nonnegative solutions for problem (1.5).
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
is a nonempty closed subset of
.
Definition 1.2.
Define the forward (backward) jump operator
at
for
at
for
by
for all
.
We assume throughout that
has the topology that it inherits from the standard topology on
and say
is right-scattered, left-scattered, right-dense and left-dense if
and
, respectively. Finally, we introduce the sets
and
. which are derived from the time scale
as follows. If
has a left-scattered maximum
, then
, otherwise
. If
has a right-scattered minimum
, then
,
.
Definition 1.3.






Definition 1.4.





for all
. Call
the nabla derivative of
at the point
.
If
then
. If
then
is the forward difference operator while
is the backward difference operator.
Definition 1.5.
A function
is called rd-continuous provided it is continuous at all right dense points of
and its left sided limit exists (finite) at left dense
. We let
denote the set of rd-continuous functions
.
Definition 1.6.
is called ld-continuous provided it is continuous at all left
and its right sided limit exists (finite) at right dense
. We let
denote the set of ld-continuous
.
Definition 1.7.





Definition 1.8.
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
. It is well known that
is a Banach space with the norm
defined by
. Let
be a cone of
,
, where
.
respectively.
Lemma 2.1 (see [6]).
From Lemma 2.1 and the definition of
, we can prove that
has the following properties.
Proposition 2.2.
In fact, from Lemma 2.1, we have
for
Therefore (2.5) holds.
Proposition 2.3.
Proof.
In fact, from Lemma 2.1, we obtain
for
So
.
On the other hand, from Lemma 2.1, we know that
for
. This together with
implies that
for
. Hence
is nondecreasing on
,
is nonincreasing on
. So (2.6) holds.
Proposition 2.4.
Proof.
Therefore (2.7) holds.
Lemma 2.5 (see [6]).
The following lemma is crucial to prove our main results.
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.
where
and
are defined in (1.4), respectively.
for
and
given in (2.2) and (2.12), respectively.
If
holds, then we can show that
have the following properties.
Proposition 3.1.
Proof.
Then from (1.2)–(1.4) and
, we obtain
Therefore,
On the other hand, since
Proposition 3.2.
Proof.
The proof is similar to that of Proposition 3.1. So we omit it.
By (2.14), it is well known that the problem (1.1) has a positive solution
if and only if
is a fixed point of
.
Lemma 3.3.
Suppose that (1.2)–(1.4) and
-
hold. Then
and
is completely continuous.
Proof.
On the other hand, for
, by (3.9),(3.10) and (2.7), we obtain
Next by standard methods and the Ascoli-Arzela theorem one can prove that
is completely continuous. So it is omitted.
Theorem 3.4.


where
and
are defined in (3.3), (3.7) and in Proposition 3.1, respectively.
Proof.
Let
be the cone preserving, completely continuous operator that was defined by (3.9).
Let
, where
. Choosing
and
satisfy
Now we prove that
In fact, if there exists
such that
, then for
, we have
Therefore
, that is,
, which is a contradiction. Hence (3.14) holds.
Next, turning to (3.15). If there exists
such that
, then for
, we have
Therefore
, that is,
, which is a contradiction. Hence (3.15) holds.
It remains to prove
In fact, if there exists
such that
, then for
, we have
which is a contradiction, where
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
and
. The proof is complete.
4. Example
Example 4.1.
To illustrate how our main results can be used in practice we present an example.
Let
. Take
in (1.1). Now we consider the following three point boundary value problem
Finally, we prove that
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.
Declarations
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.
Authors’ Affiliations
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