On a signchanging solution for some fractional differential equations
 Kemei Zhang^{1}Email authorView ORCID ID profile
Received: 4 February 2017
Accepted: 5 April 2017
Published: 20 April 2017
Abstract
In this paper, a kind of αth \((3<\alpha\leq4)\) order differential equation with twopoint boundary conditions is considered. The existence result of a signchanging solution is given by the topological degree theory and the fixed point index theory.
Keywords
MSC
1 Introduction
2 Preliminaries and some lemmas
Definition 2.1
Let E be a real Banach space and \(A:E\rightarrow E\) be a nonlinear operator. A nonzero solution to the equation \(x=\lambda Ax\) is called an eigenvector of the nonlinear operator A; the corresponding number λ is called a characteristic value of A, and \(\lambda^{1}\) is called a eigenvalue of A.
Definition 2.2
Lemma 2.1
[16]
Lemma 2.2
Lemma 2.3
[16]
Lemma 2.4
[14]
Remark 2.1
 (H_{1}):

\(f:[0,1]\times(\infty,+\infty)\rightarrow(\infty,+ \infty)\) is continuous and \(f(t,x)x>0\) for all \(x\in R\setminus \{0 \}\) and \(t\in[0,1]\).
 (H_{2}):

\(\lim_{x\rightarrow\infty}\frac{f(t,x)}{x}= \beta_{\infty}(t)\) uniformly with respect to \(t\in[0,1]\).
Lemma 2.5
The operator K defined by (2.3) satisfies \(K: X\rightarrow X_{e}\) and \(K: P\setminus\{\theta\}\rightarrow\stackrel{ \circ}{ P_{e}}\), where \(\stackrel{\circ}{P_{e}}=\{x \in X\vert \textit{ there exist } \tilde{\alpha}>0, \tilde{\beta}>0 \textit{ such that } \tilde{\alpha} e\leq x\leq\tilde{\beta} e\}\).
Proof
Lemma 2.6
Proof
In this paper, we always denote by \(\Omega_{r}=\{u\in X:\Vert u\Vert _{e}< r\} \) (\(r>0\)) the open ball of radius r and by θ the zero function in \(X_{e}\). For the concepts and properties on the cone and the topological degree, one can refer to [12, 14, 16]. □
3 Main results
Theorem 3.1
 (H_{3}):

There is \(p>0\) such that \(\vert u\vert \leq p, t\in [0,1]\) imply that \(\vert f(t,u)\vert <\eta p\), where \(\eta=\frac{\Gamma(\alpha)}{M_{0}}\).
 (H_{4}):

\(\sup_{t\in[0,1]}\beta_{\infty}(t)<\lambda_{1}\), where \(\lambda_{1}\) is the first characteristic value of K defined by (2.3).
If 1 is not the characteristic value of B and the sum of the algebraic multiplicities γ of the real eigenvalues of B in \((1,+ \infty)\) is odd, then the BVP (1.1) has at least a signchanging solution.
Proof
Declarations
Acknowledgements
This work was supported by National Natural Science Foundation of China (No. 11571197), and by Shandong Provincial Natural Science Foundation of China (No. 2016ZRB01076).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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