Abstract
Using fixed point theorems in ordered Banach spaces with the lattice structure, we consider the existence of nontrivial solutions under the condition that the nonlinear term can change sign and study the existence of sign-changing solutions for some second order three-point boundary value problems. Our results improve and generalize on those in the literatures.
1. Introduction
In this paper, we shall discuss the existence of nontrivial solutions for the following boundary value problem: where is continuous, .
Many problems of different areas of physics and applied mathematics can be changed into multipoint boundary value problems for ordinary differential equations (see [1]). In [2], Gupta firstly studied three-point boundary value problems for nonlinear second order ordinary differential equations in 1992. Since then, many authors have been concerned with second order multipoint boundary value problems (see [3–20] and references therein). For example, some authors have studied the existence and multiplicity of positive solutions for nonlinear multipoint boundary value problems under the condition that the nonlinear term may be nonnegative by applying Krasnosel’skii’s fixed point theorem, theory of fixed point index, and so on (see [3–8]). Meanwhile, some authors considered the existence of nontrivial solutions when the nonlinear term can be negative; for example, see [9–11] and references therein. For instance, in [10], under the assumption of non-well-ordered upper and lower solutions, some multiplicity results for solutions of three-point boundary value problems (1) have been obtained using the fixed point index theory. On the other hand, some authors have considered the existence of sign-changing solutions to some boundary value problems (see [12–16, 18] and references therein). For example, in [13], by using the fixed point index method, Xu and Sun have considered the existence of sign-changing solutions for the following three-point boundary value problem:
In [18], Rynne has considered the following second order -point boundary value problem: where is continuous. The author has used global bifurcation theorem to obtain sign-changing solutions of the boundary value problem (3) under some conditions on the asymptotic behavior of .
Inspired by [9–18, 21–25], we shall use fixed point theorems derived by Liu and Sun [21] to consider the existence of nontrivial solutions for BVP (1). There are two main features. Firstly, the used methods are fixed point theorems with respect to noncone mappings, which are different from those of [9–18]. Secondly, when we consider the existence of sign-changing solution to BVP (1), we generalize the nonlinear term , which is different from [13–15, 18].
The organization of this paper is as follows. In Section 2, some preliminaries and lemmas are given including some properties of the lattice and some lemmas that will be used to prove the main results. In Section 3, we shall give the main results. Finally, in Section 4, concrete examples are given to illustrate applications of obtained main results.
2. Preliminaries and Some Lemmas
Let be an ordered Banach space in which the partial ordering is induced by a cone . is called normal if there exists such that implies (see [26]).
Definition 1 (see [16, 21–23]). We call a lattice under the partial ordering , if and exist for arbitrary .
For , let and are called the positive part and the negative part of respectively, and clearly . Take , then , and is called the module of . For convenience, we use the notations , , and obviously , , (see [16, 21–23]).
Definition 2 (see [16, 21–23]). Let and be a nonlinear operator. is said to be quasi-additive on lattice, if there exists such that
Definition 3 (see [21]). Let be a Banach space with a cone and let be a nonlinear operator. We say that is a unilaterally asymptotically linear operator along , if there exists a bounded linear operator such that where is said to be the derived operator of along and will be denoted by .
Remark 4. The operator in Definition 3 is not assumed to be a cone mapping.
Let be a cone of Banach space . is said to be a positive fixed point of if is a fixed point of ; is said to be a negative fixed point of if is a fixed point of ; is said to be a sign-changing fixed point of if is a fixed point of (see [21–23]).
Lemma 5 (see [21]). Let be a Banach space with a lattice structure, let be a normal cone of , and let be completely continuous and quasi-additive on lattice. Suppose that there exist and a positive bounded linear operator with , such that
In addition, assume that ; the Fréchet derivative of at exists, 1 is not an eigenvalue of , the sum of the algebraic multiplicities for all eigenvalues of , lying in the interval , is an odd number, and exists, .
Then has at least one nontrivial fixed point.
Lemma 6 (see [21]). Let , , , and be as in Lemma 5. Suppose that there exist such that
In addition, assume that ; the Fréchet derivative of at exists; 1 is not an eigenvalue of , and exists; and 1 is not an eigenvalue of corresponding a positive eigenvector.
Then has at least one nontrivial fixed point.
Lemma 7 (see [21]). Suppose that is an ordered Banach space with a lattice structure, is a normal cone of , and is quasi-additive on the lattice. Assume that(i) is strongly increasing on and ;(ii)both and exist with and , and 1 is not an eigenvalue of or corresponding a positive eigenvector;(iii); the Fréchet derivative of at is strongly positive and ;(iv)the Fréchet derivative of at exists; 1 is not an eigenvalue of ; the sum of the algebraic multiplicities for all eigenvalues of , lying in the interval , is an even number.Then has at least three nontrivial fixed points containing one sign-changing fixed point.
Let . Define the norm ; then is an ordered Banach space. It is obvious that is a normal cone of and that is a lattice under the partial order which is induced by (see [16, 21–23]).
For convenience, we list the following conditions.(H1)The sequence of positive solutions of the equation is (H2) uniformly on .(H3) uniformly on .(H4), uniformly on .
By [4], it is well known that BVP (1) can be converted to the following nonlinear Hammerstein equation: where
Define the operators where is defined by (12), and obviously . It is obvious that fixed points of are solutions of BVP (1) (see [4]).
Lemma 8 (see [13]). Let be a positive number. The eigenvalues of the linear operator are and the algebraic multiplicity of each positive eigenvalue of the linear operator is equal to 1, where is defined by (10).
Lemma 9. Let and be defined as (14) and (15), respectively. Then (i) are completely continuous;(ii) is quasi-additive on the lattice;(iii), where is the spectral radius of the operator .
Proof. By [4], we know that (i) holds. The proof of (ii) is similar to that of [16, 21–25], so we omit it. We easily know that
Lemma 10. Let and be defined as (14) and (15), respectively. Then(i)if (H2) holds, then ;(ii)if (H3) holds, then ;(iii)if (H4) holds, then .
Proof. (i) By (H2), for any , there exists such that
Let . Then
and hence
which means
that is,
so by Definition 3, we have .
(ii) Similar to the proof of (i), we can prove that conclusion (ii) holds.
(iii) By (H4), since , . And for any , there exists such that
Then
and hence
that is,
which means .
3. Main Results
Theorem 11. Assume that (H1)–(H4) hold. In addition, suppose that (i), ;(ii)there exists an odd number such that
where , and are defined by (10).
Then BVP (1) has at least one nontrivial solution.
Proof. By (H2) and , there exists such that
and hence by (14) and (15), we have
And we easily know that . By , we know that there exists such that
By (H3) and (31), there exists such that
and therefore by (14) and (15), we get
And we easily know that . By (30) and (33), we know that (7) of Lemma 5 is satisfied.
Set ; by (31) and Lemma 9, we have .
By Lemma 10, we have .
By (H4), we easily know that . By Lemma 8, we know that are the eigenvalues of .
Since and is an odd number, 1 is not the eigenvalue of , and the sum of the algebraic multiplicities for all eigenvalues of , lying in the interval , is an odd number.
By and Lemma 9, .
So the conditions of Lemma 5 are satisfied. Lemma 5 assures that has at least one nontrivial fixed point. So BVP (1) has at least one nontrivial solution.
Theorem 12. Assume that (H1)–(H4) hold. In addition, suppose that (i);(ii)there exist natural numbers and such that
where are defined by (10).
Then BVP (1) has at least one nontrivial solution.
Proof. By (H4), we easily know that . By Lemma 10, we have , , .
From Lemma 8, are the eigenvalues of the linear operator . Since , 1 is not an eigenvalues of . Since and (10), we know that . So .
By (H3) and , there exists such that
hence by (14) and (15), we have
Obviously, . Since , there exists such that
By (H2), there exists such that
so by (14) and (15), we have
Set ; by (37) and Lemma 9, we have . Equations (36) and (39) show that (8)of Lemma 6 are satisfied. Therefore, Lemma 6 guarantees that Theorem 12 is valid.
Theorem 13. Suppose that (H1)–(H4) hold and . In addition, assume that (i) is strictly increasing on ;(ii)there exists an even number such that
(iii),
where , and are defined by (10).
Then BVP (1) has at least three nontrivial solutions, containing a sign-changing solution.
Proof. By (13), we know that
Since for any , by (12) and (41), we have
It follows from (12) that ; so by (42) we have
Form (13), we easily know that
Therefore, by (43) and (44), we get
where .
From (45) and (15), for any , we have that
which means
namely,
By (48) and the condition (i), we know that is strongly increasing; so condition (i) of Lemma 7 is satisfied.
By Lemma 10 we have
By Lemmas 8, 9, and (49), we know that are the eigenvalues of the linear operators . Since , we know that 1 is not an eigenvalue of and , and , so condition (ii) of Lemma 7 is also satisfied.
Since , . By (48), we know that is strongly positive. By (49), Lemma 9, and , we have . So condition (iii) of Lemma 7 is satisfied.
In the following, we prove that .
In fact, by (H2) (H3) and , for any , there exists such that
Let . Then
and hence
which means
that is,
so we have .
By condition (ii) and Lemma 8, we know that 1 is not the eigenvalue of , and the sum of the algebraic multiplicities for all eigenvalues of , lying in the interval , is an even number . So condition (iv) of Lemma 7 is satisfied. Therefore, Lemma 7 guarantees that Theorem 13 is valid.
Remark 14. In [13–15], the authors considered the boundary value problem (2). In this paper, it is obvious that we generalize and improve the nonlinear term , and we also obtain that BVP (1) has at least a sign-changing solution. The methods we use are different from those of [13–15]. In [18], the author has used global bifurcation theorem to obtain sign-changing solutions of the boundary value problem (3), so the methods we use are different from those of [18].
4. Applications
In this section, some examples are given to illustrate our main results obtained in Section 3.
Consider the following second order three-point boundary value problem:
By simple calculations, we know that , , and are solutions of the following equation:
Example 1. Choose
Obviously, the nonlinear term can be negative when and , , . It is easy to know that the conditions of Theorem 11 are satisfied. So BVP (55) has at least one nontrivial solution.
Example 2. Choose
It is easy to know that , , and the nonlinear term can be negative when . The conditions of Theorem 12 are satisfied. So BVP (55) has at least one nontrivial solution.
Example 3. Choose
Obviously, , . It is easy to know that the conditions of Theorem 13 are satisfied. Thus BVP (55) has at least three nontrivial solutions, containing a sign-changing solution.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author wishes to thank the referees for their valuable suggestions. The project is supported by the National Natural Science Foundation of China (10971179), Research Award Fund for Outstanding Young Scientists of Shandong Province (BS2012SF022), and A Project of Shandong Province Higher Educational Science and Technology Program (J11LA07).