Abstract
By using topological degree theory and fixed point index theory, we consider a discrete fourth-order Neumann boundary value problem. We provide sufficient conditions for the existence of sign-changing solutions, positive solutions, and negative solutions.
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1 Introduction
In this paper, we consider the existence of sign-changing solutions to the following discrete nonlinear fourth-order boundary value problem (BVP):
where Δ denotes the forward difference operator defined by , is an integer, is continuous, and α, β are real parameters and satisfy
Let a, b be two integers with . We employ to denote the discrete interval given by .
The theory of nonlinear difference equations has been widely used to study discrete models in many fields such as computer science, economics, neural network, ecology, cybernetics, etc. In recent years, a great deal of work has been done in the study of the existence and multiplicity of solutions for a discrete boundary value problem. For the background and recent results, we refer the reader to [1–5] and the references therein.
We may think of BVP (1.1), (1.2) as a discrete analogue of the fourth-order boundary value problem
The special case of BVP (1.3), (1.4) has been studied by many authors using various approaches; for example, see [6, 7].
However, it seems that there is no similar result in the literature on the existence of sign-changing solutions, positive solutions, and negative solutions for BVP (1.1), (1.2). Motivated by [8], our purpose is to apply some basic theorems in topological degree theory and fixed point index theory to establish some conditions for the nonlinear function f, which are able to guarantee the existence of sign-changing solutions, positive solutions, and negative solutions for the above discrete boundary value problem.
The organization of this paper is as follows. In Section 2, we state some notations and preliminary knowledge about the topological degree theory and fixed point index theory. In Section 3, we present the spectrum of second-order eigenvalue problems. In Section 4, we give the expression of Green’s function of second-order Neumann problems and consider the eigenvalue problem of fourth-order BVPs. In Section 5, by computing the topological degree and the fixed point index, we discuss the existence of multiple sign-changing solutions to BVP (1.1), (1.2).
2 Preliminaries
As we have mentioned, we will use the theory of the Leray-Schauder degree and the fixed point index in a cone to prove our main existence results. Let us collect some results that will be used below. One can refer to [9–12] for more details.
Let E be a Banach space and be a cone in E. Assume that Ω is a bounded open subset of E. Suppose that is a completely continuous operator. If there exists such that
then the fixed point index .
Let E be a Banach space and let be a cone in E. Assume that Ω is a bounded open subset of E, . Suppose that is a completely continuous operator. If
then the fixed point index .
Lemma 2.3 (see [11])
Let E be a Banach space, let Ω be a bounded open subset of E, , and be completely continuous. Suppose that
Then .
Lemma 2.4 (see [12])
Let A be a completely continuous operator which is defined on a Banach space E. Let be a fixed point of A and assume that A is defined in a neighborhood of and Fréchet differentiable at . If 1 is not an eigenvalue of the linear operator , then is an isolated singular point of the completely continuous vector field , and for small enough ,
where k is the sum of algebraic multiplicities of real eigenvalues of in .
Lemma 2.5 (see [12])
Let A be a completely continuous operator which is defined on a Banach space E. Assume that 1 is not an eigenvalue of the asymptotic derivative. Then the completely continuous vector field is nonsingular on spheres of sufficiently large radius ρ and
where k is the sum of algebraic multiplicities of real eigenvalues of in .
Lemma 2.6 (see [12])
Let X be a solid cone of a Banach space E ( is nonempty), let Ω be a relatively bounded open subset of X, and let be a completely continuous operator. If any fixed point of A in Ω is an interior point of X, there exists an open subset O of such that
Now, we will consider the space
equipped with the norm . Clearly, E is a -dimensional Banach space. Choose the cone defined by
Obviously, the interior of P is . For each , we write if for . A solution u of BVP (1.1), (1.2) is said to be a positive solution (a negative solution, resp.) if (, resp.). A solution u of BVP (1.1), (1.2) is said to be a sign-changing solution if .
3 Spectrum of second-order eigenvalue problems
Consider the second-order discrete linear eigenvalue problems
Lemma 3.1 (see [13] Theorem 1)
The eigenvalues of (3.1), (3.2) can be given by
and the corresponding eigenfunction is if , then . If , we have
(i) if , then ,
(ii) if , then .
4 Green’s function and eigenvalue problems to fourth-order BVPs
In this section, we construct Green’s function associated with BVP (1.1), (1.2).
Let , be roots of the polynomial , namely,
Then we have
where , , . Under the basic assumption on α, β, it is easy to see that .
Consider the two initial value problems:
By the direct computing, we get (4.1) has a unique solution
and (4.2) has a unique solution
Let
Lemma 4.1 Let and be fixed. Then the problem
has a unique solution
where is given by
Remark 4.1 Green’s function defined by Lemma 4.1 is positive on .
Define operators , respectively, by
Now, from Lemma 4.1, it is easy to see that BVP (1.1), (1.2) has a solution if and only if u is a fixed point of the operator A. It follows from the continuity of f that is completely continuous.
Lemma 4.2 Let . Then the linear discrete fourth-order boundary value problem
has a unique solution
and
Proof The conclusion is obvious, so we omit it. □
We will use the following assumptions.
(H1) α, β are real parameters and satisfy
(H2) is continuous, for any , ; for any and , .
(H3) There exists an even number such that
where uniformly for , is defined in Lemma 3.1 and .
(H4) There exists an even number such that
where uniformly for , and are given in the condition (H3).
(H5) There exists a constant such that for any ,
where .
Lemma 4.3 Suppose that (H2) holds and is a solution of BVP (1.1), (1.2). Then .
Proof According to (H2) and the positivity of Green’s function defined in Lemma 4.1, we can easily get the desired conclusion. □
Remark 4.2 Similarly to Lemma 4.3, we also know that if (H2) holds and is a solution of BVP (1.1), (1.2), then .
Lemma 4.4 Suppose that (H2)-(H4) hold. Then the operator A is Fréchet differentiable at θ and ∞, where the operator A is defined by (4.5). Moreover, and .
Proof By (H3), for any , there exists such that for any , . Hence, noticing that for any , we have
for any with , where . Consequently,
This means that the nonlinear operator A is Fréchet differentiable at θ and .
By (H4), for any , there exists such that for any , . Let . By the continuity of with respect to x, we have . Then, for any , . Thus
for any . Consequently,
which implies that operator A is Fréchet differentiable at ∞ and . The proof is completed. □
Lemma 4.5 Let M be given in the condition (H5). Suppose that (H1)-(H4) hold. Then , . Moreover, one has the following.
(i) There exists an such that for any ,
(ii) There exists an such that for any ,
Proof By (H2) and the fact that is positive on , we get that for any , , , and , . Then and .
We only need to prove conclusion (i). The proof of conclusion (ii) is similar and will be omitted here. Let . The condition (H3) yields . It follows from (4.12) that there exists such that
where . Setting , then is completely continuous. For any and , we obtain that
According to the homotopy invariance of the fixed point index, for any , we have
Let . Then and (see Lemma 3.1 and the proof of Lemma 4.2). We claim
Indeed, we assume that there exist and such that . Obviously, . Since , , then . Set . It is clear that and . Then
Since , then , which contradicts the definition of . This proves (4.21).
It follows from Lemma 2.1 and (4.20) that
Similarly to (4.22), we know also that
By (4.18), (4.19), (4.22), and (4.23), we conclude
□
5 Main results
Now, with the aid of the lemmas in Section 2, we are in a position to state and prove our main results.
Theorem 5.1 Assume that the conditions (H1)-(H5) hold. Then BVP (1.1), (1.2) has at least two sign-changing solutions. Moreover, BVP (1.1), (1.2) has at least two positive solutions and two negative solutions.
Proof Since is positive on , by (H5), we have for any with ,
This gives
By (5.2) and Lemmas 2.2 and 2.3, we have
From (H3) and Lemma 3.1, one has that the eigenvalues of the operator which are larger than 1 are
From (H4) and Lemma 3.1, one has that the eigenvalues of the operator which are larger than 1 are
It follows from Lemmas 2.4 and 2.5 that there exist and such that
where and are given in Lemma 4.5. Owing to Lemma 4.3, one has
According to the additivity of the fixed point index, by (5.4), (5.10), and (5.12), we have
Hence, the nonlinear operator A has at least two fixed points and , respectively. Then, and are positive solutions of BVP (1.1), (1.2). Using again the additivity of the fixed point index, by (5.5), (5.11), and (5.13), we get
Hence, the nonlinear operator A has at least two fixed points and , respectively. Then, and are negative solutions of BVP (1.1), (1.2).
Let
It follows from Lemmas 2.6, 4.3, Remark 4.2, and (5.14)-(5.17) that there exist open subsets , , , and of E such that
By (5.3), (5.20), (5.22), (5.8), and the additivity of the Leray-Schauder degree, we get
which implies that the nonlinear operator A has at least one fixed point .
Similarly, by (5.9), (5.21), (5.23), and (5.3), we get
which implies that the nonlinear operator A has at least one fixed point . Then, and are two distinct sign-changing solutions of BVP (1.1), (1.2). Thus, the proof of Theorem 5.1 is finished. □
Theorem 5.2 Assume that the conditions (H1)-(H5) hold, and that for and . Then BVP (1.1), (1.2) has at least four sign-changing solutions. Moreover, BVP (1.1), (1.2) has at least two positive solutions and two negative solutions.
Proof It follows from the proof of Theorem 5.1 that BVP (1.1), (1.2) has at least six different nontrivial solutions () satisfying
By the condition that for and , we know that and are also solutions of BVP (1.1), (1.2). Let , , then () are different nontrivial solutions of BVP (1.1), (1.2). The proof is completed. □
By the method used in the proof of Theorems 5.1 and 5.2, we can prove the following corollaries.
Corollary 5.3 Assume that the conditions (H1)-(H3) and (H5) or (H1), (H2), (H4), and (H5) hold. Then BVP (1.1), (1.2) has at least one sign-changing solution. Moreover, BVP (1.1), (1.2) has at least one positive solution and one negative solution.
Corollary 5.4 Assume that the conditions (H1)-(H3) and (H5) or (H1), (H2), (H4), and (H5) hold, and that for and . Then BVP (1.1), (1.2) has at least two sign-changing solutions. Moreover, BVP (1.1), (1.2) has at least one positive solution and one negative solution.
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Yang, J. Sign-changing solutions to discrete fourth-order Neumann boundary value problems. Adv Differ Equ 2013, 10 (2013). https://doi.org/10.1186/1687-1847-2013-10
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DOI: https://doi.org/10.1186/1687-1847-2013-10