Existence and Multiplicity of Positive Solutions for a Singular Third-Order Three-Point Boundary Value Problem with a Parameter ()
1. Introduction
In this paper, we are concerned with the existence, multiplicity and nonexistence of positive solutions for the following third-order boundary value problem (BVP for short):
(1)
(2)
where
,
are constants,
is a positive parameter,
,
are continuous and
may be singular at
and 1.
Third-order differential equations arise in a variety of different areas of applied mathematics and physics, e.g., in the deflection of curved beam having a constant or varying cross section, a three-layer beam, electromagnetic waves or gravity driven flows and so on [1]. In recent years, third-order boundary value problems have been studied by many methods [2] - [10], such as upper and lower solutions method, monotone iterative method and the different fixed point theory, etc.
In [11], Sun proved the existence of triple positive solutions to the following BVP by using a fixed-point theorem due to Avery and Peterson:
where
,
are constants.
and
are continuous.
By applying the Krasnoselskii’s fixed point theorem, Sun [12] established the existence of infinitely many solutions to the following BVP, which is the special case for
in BVP (1) and (2):
with
,
, where
is nonnegative continuous function defined on
and
is continuous,
may be singular at
and/or
.
Motivated by the above works, here we study the third order BVP (1) and (2). Under certain suitable conditions, we establish the results of existence, multiplicity and nonexistence of positive solutions for BVP (1) and (2) via the fixed point index theory.
2. Preliminaries
In this section, we present some notation and Lemmas that will be used in subsequent sections.
Lemma 2.1. [11] Let
,
, then the BVP
(3)
(4)
has a unique solution
where
(5)
Lemma 2.2. [11] Suppose
,
,
, then
Let
be equipped with norm
, then
is a real Banach space.
From Lemma 2.2, we know that if
,
, then for
, the unique solution
of BVP (2.1) and (2.2) is nonnegative and satisfies
Define the cone P by
then P is a non-empty closed and convex subset of E.
For
, we write
if
for any
. For any
, let
and
.
Define the operator
by
(6)
In view of the Lemma 2.1, it is easy to see that u is a positive solution BVP (1) and (2) if and only if u is a fixed point of the operator
.
In the following, we assume that:
(H1)
,
and
.
(H2)
,
is non-decreasing in
and
for any
,
.
Lemma 2.3. Assume (H1)-(H2) hold, then the operator
is completely continuous.
Proof. For
, according to the definition of T and Lemma 2.2, it is easy to prove that
. By the Ascoli-Arzela theorem, it is easy to show
is completely continuous.
The proofs of our main theorems are based on the fixed index theory. The following three well-known Lemmas in [13] [14].
Lemma 2.4. Let E be a Banach space and
be a cone in E. Assume that
is a bounded open subset of E. Suppose that
is a completely continuous operator. If there exists
such that
, for all
and
, then the fixed point index
.
Lemma 2.5. Let E be a Banach space and
be a cone in E. Assume that
is a bounded open subset of E. Suppose that
is a completely continuous operator. If
and
for
and
, then the fixed point index
.
Lemma 2.6. Let E be a Banach space and
be a cone in E. Assume that
is a bounded open subset of E with
. Suppose that
is a completely continuous operator. If
for all
and
, then the fixed point index
.
Now for convenience we use the following notations. Let
3. The Main Results and Proofs
Lemma 3.1. Suppose (H1) holds and
, then
.
Proof. Let
be fixed, then we can choose
small enough such that
. It is easy to see that
By Lemma 2.6, it follows that
(7)
From
, it follows that there exists
such that
(8)
We may suppose that
has no fixed point on
. Otherwise, the proof is finished. Let
for
, Then
. We claim that
(9)
In fact, if not, there exist
and
such that
, then
. For
and
, by Lemma 2.2 and (8), we have
we get
, which is a contradiction. Hence by Lemma 2.4, it follows that
(10)
By virtue of the additivity of the fixed point index, by (7) and (10), we have
which implies that the nonlinear operator
has one fixed point
. Therefore,
. The proof is complete.
Lemma 3.2. Suppose (H1) and (H2) hold,
, then
.
Proof. Let
be fixed. From (H2) and the definition of cone P, it follows that there exists
such that
for all
and
. Then for sufficiently large
with
and
, we have
This implies that
and
for
,
. By Lemma 2.5, it follows that
(11)
From
, there exists
such that
Then for
, by the definition of cone P, we get
, and so
We obtain
for
,
. It follows from Lemma 2.6 that
(12)
According to the additivity of the fixed point index, by (11) and (12), we have
which implies that the nonlinear operator
has at least one fixed point
. Therefore,
. The proof is complete.
Lemma 3.3. Suppose (H1) and (H2) hold,
, then
.
Proof. By Lemma 3.1, it is easy to see that
. It follows from (H2) and
that there exists
such that
for all
and
. Let
, by the definition of cone P and Lemma 2.1, we obtain that
so
, thus
. This completes the proof of Lemma 3.3.
Lemma 3.4. Suppose (H1) and (H2) hold, hold and
, then
.
Proof. By Lemma 3.2, it is easy to see that
. It follows from (H2) and
that there exists
such that
for all
and
. Let
, from the definition of cone P and Lemma 2.2, we have
so
, thus
. This completes the proof of Lemma 3.4.
Lemma 3.5. Suppose (H1) and (H2) hold,
, then
. Moreover, for any
, BVP (1) and (2) has at least two positive solutions.
Proof. For any fixed
, we prove that
. By the definition of
, there exists
, such that
and
. Let
be fixed. From the proof of Lemma 3.1, we see that there exist
and
such that
. It is easy to see that
for all
. Then we have
and
Consider now the modified BVP:
(13)
(14)
where
Clearly, the function
is bounded for
,
and is continuous in
. Define the operator
by
Then
is completely continuous and all the fixed points of operator
are the solutions for BVP (13) and (14). It is easy to see that there exists
such that
for any
. From Lemma 2.6, we have
(15)
Let
We claim that if
is a fixed point of operator
, then
. In fact, if
, then
and
From the excision property of the fixed point index and (15), we obtain that
From the definition of
, we know that
on
, then
(16)
Hence, the nonlinear operator
has at least fixed point
. Then
is one positive solution of BVP (1) and (2). This gives
,
and
.
We now find the second positive solution of BVP (1) and (2). By
and the continuity of
with respect to
, there exists
such that
(17)
For
, let
We claim that
is bounded in E. In fact, for any
, it follows from Lemma 2.2 and (17) that
This implies
. Thus
is bounded in E. Therefore there exists
such that
By Lemma 2.4, we get that
(18)
Using a similar argument as in deriving (10), we have that
(19)
where
. According to the additivity of the fixed point index, by (16), (18) and (19), we have
which implies that the nonlinear operator
has at least one fixed point
. Thus, BVP (1)-(2) has another positive solution. The proof is complete.
Lemma 3.6. Suppose (H1) and (H2) hold,
, then
. Moreover, for any
, BVP (1)-(2) has at least two positive solutions.
Proof. For any fixed
, we prove that
. By the definition of
, there exists
, such that
and
. Let
be fixed. From the proof of Lemma 3.2, we see that there exist
,
and
such that
. By the definition of cone P, it is easy to see that
for all
. Define
Using the method similar to get (16), we yield
(20)
Hence, the nonlinear operator
has at least fixed point
. Then
is one positive solution of BVP (1) and (2). This gives
,
and
.
We now find the second positive solution of BVP (1) and (2). From
, there exists
such that
Then for
, we have
This implies
for
,
. It follows from Lemma 2.6 that
(21)
Using a similar argument as in deriving (12), we have that
(22)
where
. According to the additivity of the fixed point index, by (20), (21) and (22), we have
which implies that the nonlinear operator
has at least one fixed point
. Thus, BVP (1)-(2) has another positive solution. The proof is complete.
Lemma 3.7. Suppose (H1) and (H2) hold,
, then
.
Proof. In view of Lemma 3.5, it suffices to prove that
. By the definition of
, we can choose
with
such that
as
. By the definition of
, there exists
such that
. We now show that
is bounded. Suppose the contrary, then there exists a subsequence of
(still denoted by
) such that
as
. It follows from
that
for all
. Choose sufficiently large
such that
By
, there exists
such that
for all
and
. Since
as
, there exists sufficiently large
such that
. Thus, we have
This gives
(23)
which contradicts the choice of
. Hence,
is bounded. It follows from the completely continuous of T that
is equicontinuous, i.e., for each
, there is a
such that
where
,
and
. Then
is equicontinuous. According to the Ascoli-Arzela theorem,
is relatively compact. Hence, there exists a subsequence of
(still denoted by
) and
such that
as
. By
, letting
, we obtain that
. If
, using a similar argument as in deriving (23), by
, we also get a contradiction. Then
, and so
. This completes the proof.
Lemma 3.8. Suppose (H1) and (H2) hold,
, then
.
Proof. In view of Lemma 3.6, it suffices to prove that
. By the definition of
, we can choose
with
such that
as
. By the definition of
, there exists
such that
. We now show that
is bounded. Suppose the contrary, then there exists a subsequence of
(still denoted by
) such that
as
. It follows from
that
for all
. Choose
small enough such that
By
, there exists
such that
for all
and
. Since
as
, there exists sufficiently large
such that
. Thus, we have
This gives
(24)
which contradicts the choice of
. Hence,
is bounded. It follows from the completely continuous of T that
is equicontinuous, i.e., for each
, there is a
such that
where
,
and
. Then
is equicontinuous. According to the Ascoli-Arzela theorem,
is relatively compact. Hence, there exists a subsequence of
(still denoted by
) and
such that
as
. By
, letting
, we obtain that
. If
, using a similar argument as in deriving (24), by
, we also get a contradiction. Then
, and so
. This completes the proof.
From Lemmas 3.1, 3.3, 3.5 and 3.7, we get the main result as follows.
Theorem 3.1. Let (H1), (H2) be fulfilled and suppose that
, then there exists
such that BVP (1)-(2) has at least two positive solutions for
, at least one positive solution for
and no positive solution for
.
By Lemmas 3.2, 3.4, 3.6 and 3.8, we obtain the main result as follows.
Theorem 3.2. Let (H1), (H2) be fulfilled and suppose that
, then there exists
such that BVP (1)-(2) has at least two positive solutions for
, at least one positive solution for
and no positive solution for
.
Founding
Supported by the Foundation for Basic Disciplines of Army Engineering University of PLA (KYSZJQZL2013).