1. Introduction
It is well known that the subject of the existence of solutions to numerous boundary value problems (BVP) for differential equations such as second-order [
1,
2,
3], fourth-order [
4,
5,
6], even fractional order BVP [
7,
8,
9,
10,
11] has gained considerable attention and popularity. A growing number of outstanding progress has been made in the theory of such BVP in the last decades due mainly to their extensive applications in the fields of hydrodynamics, nuclear physics, biomathematics, chemistry, and control theory. For further details, please see References [
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29] and references therein.
It is noted that fourth-order boundary value problems have an important application in practical problems, that is, they can be used to describe the deformation of elastic beam, see References [
30,
31,
32,
33] and references therein. For example, in Reference [
32], by means of the theory of fixed point index on cone, Y. Li investigated the following boundary value problems of fourth-order ordinary differential equation
where
,
and satisfy
. By constructing a special cone, the existence of at least one positive solution was obtained under some suitable assumptions.
Recently, in Reference [
33], Q. Wang and L. Yang studied the following boundary value problems
where
, and
satisfy the following conditions:
These conditions involve a two-parameter non-resonance condition. By constructing two classes of cones and using the fixed point theory, the existence of at least one positive solution was obtained. It is remarkable that the premise of this establishment of the result in Reference [
33] is that the nonlinear term
must be positive.
We point out that there are some limitations in those existing results of fourth-order boundary value problems. All solutions obtained in the above references are positive, and moreover, the corresponding conclusions in them are not valid when the nonlinear term is allowed to be non-positive. Considering that two variables
u and
v in the nonlinear term usually have some connections in many practical problems, there is no description of the relationship between them in the aforementioned papers. It is an interesting problem to seek such solutions for BVP (
1) that one variable is positive and the other may be non-positive under the assumptions that nonlinearity may be semipositoned, and some connection will be added between these two variables. As far as we know, there is no paper considering such problem for BVP (
1). The purpose of the present paper is to fill this gap.
This paper, motivated by all the above mentioned discussions, investigates the multiple solutions for BVP (
1) under the more different conditions compared with Reference [
33]. By constructing a very special cone and using the fixed point index theory, the existence and multiplicity results of solutions to (
1) are obtained when
satisfy the conditions (
2),
, and
.
The nonlinear term
is allowed to change sign by contrast,
. A relationship is imposed between two variables
in nonlinear terms, which is that the variable
v is controlled by
u. In obtained solution
, the component
u is positive, but the component
v is allowed to be negative in comparison with Reference [
33].
The rest of this paper is organized as follows—
Section 2 contains some background materials and preliminaries. The main results will be given and proved in
Section 3. Finally, in
Section 4, two examples are given to support our results.
2. Background Materials and Preliminaries
The basic space used in this paper is
. It is a Banach space endowed with the norm
for
, where
,
. Under the condition (
2), as in Reference [
32], let
and let
be the Green’s function of the linear boundary value problem
Then for
, the solution of the following nonlinear boundary value problem
can be expressed as
Lemma 1. The function has the following properties:
(1) for ;
(2) for , where is a constant;
(3) for , where is a constant;
(4) for , where is a constant.
Proof of Lemma 1. (1)–(3) can be seen from Reference [
32]. In addition, by careful calculation and Lemma 2.1 in Reference [
32], it is not difficult to prove that
. Immediately, (4) is derived. □
The main tool used here is the following fixed-point index theory.
Lemma 2 ([
34])
. Let be a Banach space and P be a cone in . Denote and . Let be a complete continuous mapping, then the following conclusions are valid.(1) If for and , then
(2) If and for and , then
3. Main Results
In this section, we shall establish the existence and multiplicity results, which is based on the fixed point index theory. For this matter, first we define the mappings
, and
by
Then, BVP (
1) in operator forms becomes
By (
3), one can easily see that the existence of solutions for BVP (1) is equivalent to the existence of nontrivial fixed point of
T. Therefore, we need to find only the nontrivial fixed point of
T in the following work.
Subsequently, for simplicity and convenience, set
Then, , and are positive numbers.
Now let us list the following assumptions satisfied throughout the paper.
(H1) , , and there exists such that for
(H2) .
(H3) .
In addition, for the sake of obtaining the nontrivial fixed point of operator
T, let
where
and
.
,
, and
are defined in Lemma 1 and (H1), respectively.
Obviously, P is a nonempty, convex, and closed subset of E. Furthermore, one can prove that P is a cone of Banach space E.
It is not difficult to see that is a relatively open and bounded set of P for each .
Lemma 3. To calculate the fixed point index of T in , we first need to prove the following result. Assume that hold. Then is completely continuous, and .
Proof of Lemma 3. For
, by virtue of Lemma 1, one can easily obtain that
Moreover, (H1) together with Lemma 1 guarantees that
Therefore, , namely, . In addition, since , , and are continuous, one can deduce that T is completely continuous by using normal methods such as Arscoli-Arzela theorem, and so forth. □
Now we are in a position to prove our main results in the following.
Theorem 1. Under the assumptions (H1) and (H2), the BVP (1) admits at least one nontrivial solution. Proof of Theorem 1. To obtain the nontrivial solution for BVP (
1), we will choose a bounded open set
in cone
P and calculate the fixed point index
. For this, the proof of Theorem 1 will be carried out in three steps.
First, notice that by (H2), there exist
and
such that
To this end, suppose on the contrary that there exist
and
such that
Therefore,
satisfies the following differential equation
It follows from (
4) and (
6) that
Multiplying the above inequality by
and then integrating from 0 to 1, one can easily get
Noticing that , we obtain a contradiction.
Second, from (H2), there exist
and
such that
Set
. Then one can easily find that
Now, we will show that there exists
such that
Suppose, on the contrary, that there exist
and
such that
. Combining (
6) with (
8), we immediately get
On the other hand, in view of the definition of cone
P, one can easily obtain that
which means
Therefore, if , immediately, one can get for and
In addition, if
, then by the definition of cone
P, one can get that for any
and
,
So, by (
7), (
11), and Lemma 1, one can get that for all
,
That is,
. So, we can ultimately choose
such that (
9) holds.
Based on (
5), (
9), Lemma 2, and Lemma 3, we have
As a result, the conclusion of this theorem follows. □
Theorem 2. Assume that (H1) and (H3) hold. Then the BVP (1) has at least one nontrivial solution. Proof of Theorem 2. In the following, we divide the proof of Theorem 2 into three steps.
From condition (H3), there exist
and
such that
Subsequently, we claim that
In fact, if there exist
and
such that
, then by (
6) and (
12), one can obtain immediately
Noticing that we get a contradiction.
In addition, it follows from Lemma 1 and (
12) that for
,
which yields
.
The assumption (H3) implies that there exist
and
such that
Moreover, by the continuity of
and
, there exists
such that
We claim that there exists a large enough
such that
Suppose, on the contrary, there exist
and
such that
. Then (
6) together with (
15) guarantees
Moreover, based on the definition of cone
P, we can immediately get
which means
So, one can choose
such that (
16) holds.
From (
13), (
16), Lemma 2, and Lemma 3, we deduce that
As a result, BVP(
1) has at least one nontrivial solution. □
Up to now, some existence results of BVP(
1) have been obtained by applying the fixed point index theory. In the following, the multiple solutions will be considered for BVP (
1).
Theorem 3. Assume that (H1) holds. In addition, suppose that
(1) , ;
(2) There exists and a continuous nonnegative function such thatand Then the BVP (1) has at least two nontrivial solutions. Proof of Theorem 3. In order to obtain this conclusion, we firstly claim that
Suppose, on the contrary, there exist
and
such that
. Then,
Taking the maximum for both sides of the above inequality in
, we get that
This means
, which is a contradiction. Moreover, one can easily see that
holds from (
19) and (
20).
Next, similar to the process of proving (
5) and (
16), there exist
and
such that
Thus, by (
18), (
21), (
22), Lemma 2, and Lemma 3, one can immediately obtain that
Namely, there exist
and
satisfying
, that is,
is the solution of BVP(
1).
Finally, we show
. To see this we need only to prove BVP(
1) has no solution on
. Suppose on the contrary, there exists
being a solution of BVP(
1). Then
. By a similar process of obtaining (
20), one can get
, which is a contradiction. To sum up, Theorem 3 is proved. □
From a process similar to the above, the following conclusion can be obtained.
Theorem 4. Suppose that (H1) holds. In addition, suppose that
(1) , ;
(2) There exists , and a continuous nonnegative function such that
and
Then the BVP (1) has at least two nontrivial solutions. Proof of Theorem 4. To this end, suppose on the contrary that there exist
and
such that
. Hence, we get
, that is
Noticing that , this is a contradiction.
Next, from a process similar to (
9) and (
13), there exist
and
such that
So, by (
23)–(
26), Lemma 2, and Lemma 3, one can get
Finally, from a process similar to the end of proof of Theorem 3, BVP(
1) has at least two nontrivial solutions. As a result, the conclusion of this theorem follows. □
4. Examples
In this section, two illustrative examples are worked out to show the effectiveness of the obtained results.
Example 1. Consider the following BVP of fourth-order ordinary differential systems
Then, BVP (
27) has at least two nontrivial solutions.
Proof of Example 1. BVP (27) can be regarded as a BVP of the form (1). Choosing
,
, and
, then we have
Clearly,
and
satisfy the condition (
2). Moreover, by careful calculation and Lemma 2.1 in Reference [
32], one can obtain that
where
Now, , , and . Thus, one can easily get that (H1) holds by choosing , where .
In addition, by calculation, we get that
Then, it is not difficult to obtain that the condition (2) in Theorem 3 holds. Hence, our conclusion follows from Theorem 3. □
Example 2. Consider the following BVP of fourth-order ordinary differential systems
Then, BVP (
28) has at least two nontrivial solutions.
Proof of Example 2. BVP (28) can be regarded as a BVP of the form (1). Using a similar process of the proof of Example 1, one can easily obtain that
In addition, it is obvious that (H1) holds by choosing
. In the following, set
and
Then, it is trivial to verify that assumption (2) of Theorem 3 is true.
As a result, by Theorem 4, system (
28) has at least two nontrivial solutions. □