Abstract
This paper deals with the positive solutions of a fourthorder boundary value problem in Banach spaces. By using the fixedpoint theorem of strictsetcontractions, some sufficient conditions for the existence of at least one or two positive solutions to a fourthorder boundary value problem in Banach spaces are obtained. An example illustrating the main results is given.
MSC: 34B15.
Keywords:
1 Introduction
In this paper, we consider the existence of multiple positive solutions for the fourthorder ordinary differential equation boundary value problem in a Banach space E
where
However, the above works in a Banach space were carried out under the assumption that
the secondorder derivative
The main features of this paper are as follows. First, we discuss the existence results
in an abstract space E, not
The paper is organized as follows. In Section 2, we present some preliminaries and lemmas that will be used to prove our main results. In Section 3, various conditions on the existence of positive solutions to BVP (1.1) are discussed. In Section 4, we give an example to demonstrate our result.
2 Preliminaries
Let the real Banach space E with norm
For a bounded set V in a Banach space, we denote by
Lemma 2.1[6]
Let
The key tool in our approach is the following fixedpoint theorem of strictsetcontractions:
Theorem 2.1[8]
LetPbe a cone of a real Banach spaceEand
(i) for
(ii)
then the operatorAhas at least one fixed point
The following concept is due to Krasnosel’skii [12,13], with a slightly more general definition in [12].
Definition 2.1 We say that a bounded linear operator
Lemma 2.2LetTbe
In the following, the closed balls in spaces E and
For convenience, let us list the following assumptions:
(
(
for all
(
for all
(
and
(
for all
(
(
where
Now, let
Obviously,
Set
Obviously,
Let
Using the above transformation and (2.2), BVP (1.1) becomes
with
From (2.3) and (2.4), we have
Now, define an operator A on Q by
The following Lemma 2.3 can be easily obtained.
Lemma 2.3Assume that (
(i)
(ii) BVP (1.1) has a solution in
Let
It is easy to see that
Lemma 2.4Suppose that (
Proof For any
and thus
Since f is uniformly continuous and bounded on
where
Using the obvious formula
and observing
where
From the fact of [9], we know
It follows from (2.6) and (2.7) that
and consequently, A is a strict set contraction on
3 Main results
Theorem 3.1Let a conePbe normal and condition (
Proof Set
It is clear that
We first assume that (
In the following, we prove that W is bounded.
For any
For
Since
It follows from
Taking
Next, we are going to verify that for any
If this is false, then there exists
For
It is easy to see that
In fact,
which is a contradiction to
By Lemma 2.4, A is a strict set contraction on
Next, in the case that (
Let
It is clear that
Let
where
Take
By [12], we have
In the following, we prove that
Let
satisfying
that is,
By the above argument, it is easy to see that there exists a
Choose
Now, we assert that
If this is not true, then there exists
Thus,
and
So, by (
It is easy to see that
In fact,
which is a contradiction to
By Lemma 2.4, A is a strict set contraction on
Theorem 3.2Let a conePbe normal. Suppose that conditions (
Proof We can take the same
On the other hand, it is easy to see that
In fact, if there exists
and so
where, by virtue of (
It follows from (3.12) and (3.13) that
a contradiction. Thus (3.11) is true.
By Lemma 2.4, A is a strict set contraction on
Theorem 3.3Let a conePbe normal. Suppose that conditions (
Proof We can take the same
On the other hand, it is easy to see that
In fact, if there exists
Observing
which is a contradiction. Hence, (3.16) holds.
By Lemma 2.4, A is a strict set contraction on
4 One example
Now, we consider an example to illustrate our results.
Example 4.1 Consider the following boundary value problem of the finite system of scalar differential equations:
where
Claim (4.1) has at least two positive solutions
Proof Let
Evidently,
Noticing
for all
for all
Choosing
So, condition (
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors typed, read and approved the final manuscript.
Acknowledgements
The authors would like to thank the referees for carefully reading this article and making valuable comments and suggestions. This work is supported by the Foundation items: NSFC (10971179), NSF (BS2010SF023, BS2012SF022) of Shandong Province.
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