 Research
 Open Access
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
 signchanging solution
 topological degree
 fixed point index
MSC
 34B15
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
References
 Podlubny, I: Fractional Differential Equations. Mathematics in Science and Engineering, vol. 198. Academic Press, New York (1999) MATHGoogle Scholar
 Kilbas, AA, Srivastava, HM, Nieto, JJ: Theory and Applicational Differential Equations. Elsevier, Amsterdam (2006) Google Scholar
 Lakshmikantham, V, Vatsala, AS: General uniqueness and monotone iterative technique for fractional differential equations. Appl. Math. Lett. 21, 828834 (2008) MathSciNetView ArticleMATHGoogle Scholar
 Jankowski, T: Boundary problems for fractional differential equations. Appl. Math. Lett. 28, 1419 (2014) MathSciNetView ArticleMATHGoogle Scholar
 Zhang, XG, Liu, LS, Wu, YH, Wiwatanapataphee, B: The spectral analysis for a singular fractional differential equation with a signed measure. Appl. Math. Comput. 257, 252263 (2015) MathSciNetMATHGoogle Scholar
 Zhang, XG, Liu, LS, Wu, YH: The eigenvalue for a class of singular pLaplacian fractional differential equations involving the RiemannStieltjes integral boundary condition. Appl. Math. Comput. 235, 412422 (2014) MathSciNetMATHGoogle Scholar
 Zhang, XQ: Positive solutions for a class of singular fractional differential equation with infinitepoint boundary value conditions. Appl. Math. Lett. 39, 2227 (2015) MathSciNetView ArticleMATHGoogle Scholar
 Xu, XJ, Jiang, DQ, Yuan, CJ: Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation. Nonlinear Anal. 71, 46764685 (2009) MathSciNetView ArticleMATHGoogle Scholar
 Xu, XJ, Hu, WW: A new existence results of positive solution for a class of nonlinear fractional differential equation boundary value problems. J. Syst. Sci. Math. Sci. 32(5), 580590 (2012) MathSciNetMATHGoogle Scholar
 Zhang, XQ, Wang, LQ, Wang, S: Existence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions. Appl. Math. Comput. 226, 708718 (2014) MathSciNetMATHGoogle Scholar
 Zhai, CB, Hao, MR: Fixed point theorems for mixed monotone operators with perturbation and applications to fractional differential equation boundary value problems. Nonlinear Anal. TMA 75, 25422555 (2012) MathSciNetView ArticleMATHGoogle Scholar
 Liu, YS: Multiple positive solutions of nonlinear singular boundary value problem for fourthorder equations. Appl. Math. Lett. 17, 747757 (2004) MathSciNetView ArticleMATHGoogle Scholar
 Krasnoselskii, MA, Zabreiko, PP: Geometrical Methods of Nonlinear Analysis. Springer, New York (1984) View ArticleGoogle Scholar
 Guo, DJ, Laksmikantham, V: Nonlinear Problems in Abstract Cones. Academic Press, Boston (1988) Google Scholar
 Zhang, KM: Nontrivial solutions of fourthorder singular boundary value problems with signchanging nonlinear terms. Topol. Methods Nonlinear Anal. 40, 5370 (2012) MathSciNetMATHGoogle Scholar
 Deimling, K: Nonlinear Functional Analysis. Springer, Berlin (1985) View ArticleMATHGoogle Scholar
 Zhang, KM, Xie, XJ: Existence of signchanging solutions for some asymptotically linear threepoint boundary value problems. Nonlinear Anal. TMA 70(7), 27962805 (2009) MathSciNetView ArticleMATHGoogle Scholar
 Cui, YJ: Uniqueness of solution for boundary value problems for fractional differential equations. Appl. Math. Lett. 51, 4854 (2016) MathSciNetView ArticleMATHGoogle Scholar