Existence of Positive Solutions for a Fourth-Order Three-Point BVP with Sign-Changing Green’s Function ()
1. Introduction
The boundary value problem of fourth-order ordinary differential equations (BVP for short) has attracted much attention due to its amazing application in engineering, physics, material mechanics, fluid mechanics and so on. Many authors use Banach contraction to study the existence of single or multiple positive solutions for certain third-order BVP-Guo (Orem), Guo-Krasnoselsky (Krasnoselsky) Fixed point theorem, Leray-Schauder nonlinear substitution, fixed point index theory of viewing cone, monotonic iterative technique, upper and lower solution method, degree theory, the Critical point theorem in a conical shell, etc. see [1] - [6].
However, it is necessary to point out that, in most of the existing literature, the Greens functions involved are nonnegative, which is an important condition in the study on BVP Positive Solution.
Recently, when the corresponding Green’s function is changing signs, some work has been done on the positive solution of the second or third order BVP. For example, Zhong and An [7] studied the existence of at least one positive solution of the following second-order periodic BVP with positive and negative transformation Green’s function
where
. The main tool used is the fixed point index theory of cone
mapping 2008, for a singular third-order three-point BVP of Green’s function with infinite signature
where
. Palamide and Smirlis [8] discussed the existence of at least
one positive solution. Their technique is a combination of Guo-Krasnosel’sski fixed point theory and the corresponding vector field characteristics. In 2012, Sun and Zhao [9] [10] obtained single or multiple positive solutions with three-point positive and negative BVP by applying the fixed point theory of Guo-Krasnosel’skii and Leggett-Williams.
where
. For relevant results, one can refer to [11] - [18]. It is worth
mentioning that there are other types of achievements on either sign-changing or vanishing Green’s functions which prove the existence of sign-changing solutions, positive in some cases, see [11] [19] [20] [21] [22].
Inspired and inspired by the above works, this article focuses on the following fourth-order three-point BVP with the iconic Green’s function.
Throughout this paper, we always assume that
and
.
Obviously, the BVP (2.1) is a special case of the BVP (2.2). However, it is necessary to point out that this paper is not a simple extension of [23], which is different from the restriction in [23]. On the other hand, compared with [23], we can only prove that the obtained solution is concave on
.
Our main tool is the following well-known Guo-Krasnoselskii fixed point theorem [24] [25]:
Let K be a cone in a real Banach space E.
Definition 1.1. A functional
is said to be increasing on K provided
for all
with
, where
if and only if
.
Definition 1.2. Let
be continuous. For each
, one defines the set
Theorem 1.1. Let
and
be increasing, nonnegative, and continuous functionals on K, and let
be a nonnegative continuous functional on K with
such that, for some
and
,
for all
. Suppose there exist a completely continuous operator
and
such that
and
(H1)
for all
;
(H1)
for all
;
(H3)
and
for all
.
Then T has at least two fixed points
and
in
such that
with
,
with
.
2. Preliminaries
The remainder of this paper, we assume that Banach space
is equipped with the norm
.
For the following BVP:
(2.1)
then we have the following lemma.
Lemma 2.1. The BVP (2.1) has only trivial solution.
Proof. Easy to check.
Now, for any
, we consider the boundary value problems
(2.2)
After a direct computation, one may obtain the expression of Green’s function
of the BVP (2) as following:
For
and
Lemma 2.2. It is not difficult to verify that
has the following characteristics:
1) If
, then
is nonincreasing with respect to
.
2)
changes its sign on
. In details, if
, then
. If
, then
.
3) If
, then
such that
for
and
for
.
Moreover, if
, then
if
, then
Now, let
is nonnegative and decreasing on
.
Then
is a cone in C [0, 1].
Lemma 2.3. Let
and
. Then u is the unique solution of the BVP (1.2) and
. Moreover,
is concave on
Proof. For
, we have
since
we get
At the same time,
shows that
For
, we have
In view of
and
, we get
Obviously,
for
,
,
. This shows that u is a solution of the BVP (2.2). The uniqueness follows immediately from Lemma 2.1. Since
for
and
, we have
for
. So,
. In view of
for
, we know that
is concave on
.
Lemma 2.4. Assume
then the unique solution
of the BVP (2.2) satisfies
where
and
.
Proof. From Lemma 2.2, we know that
is concave on
, thus for
,
(2.3)
In view of
, we know that
, which together with (2.3) implies that
according to that
3. Main Results
In this section, we are concerned with the existence of at least one positive solution of the problem (2.1). Assume that
(C1) For each
, the mapping
is decreasing;
(C2) For each
, the mapping
is increasing.
Let
Then it is easy to see that K is a cone in
.
Now, we define an operator
by
distinctly, if u is a fixed point of A in K, then u is a positive and nondecreasing solution of the BVP (2.2), by lemma 2.3 and lemma 2.4 we know,
although
is not continuous, it follows from known textbook results, for example, see [26], that
, is completely continuous. Set
Lemma 3.1. Suppose that (C1) and (C2) hold. Moreover, If there exist three constants a, b and c with
such that
(F1)
,
(F2)
,
(F3)
then boundary value problem (1.1) has at least two positive solutions
Proof. First, we define the increasing, nonnegative, and continuous functionals
,
and
on K as follows:
Obviously, for any
,
. At the same time, for each
, in view of
, we have
Furthermore, we also note that
for
,
.
Next, for any
, we claim that
(3.1)
In fact, it follows from (C1), (C2), and
Now, we assert that
for all
. To prove this, let
; that is,
and
. Then
(3.2)
Since
is decreasing on
, it follows from (3.1), (3.2), (C2), (C1) and (F1) that
Then, we assert that
for all
. To see this, suppose that
; that is,
and
. Since
, we have
(3.3)
In view of the properties of
, (F2), (3.3), (C1) and (C2), we get
Finally, we assert that
and
for all
.
In fact, the constant function
. Moreover, for
, that is
and
. Then
(3.4)
Since
is decreasing on
, it follows from (F3), (3.1), (3.4), (C1) and (C2) that
To sum up, all the hypotheses of Theorem 1.1 are satisfied. Consequently A has at least two fixed points; that is, the BVP (1.1) has at least two positive solutions
and
such that
Acknowledgements
The author expresses gratitude to the referees for their valuable comments and suggestions.
The Term of the Foundation
This work is partly supported by the National Natural Science Foundation of China (11561064) and partly supported by NWNU-LKQN-14-6.