## Abstract

In this paper, the following nonlinear fractional ordinary differential boundary value problem

is considered, where \(\alpha (n-1 <\alpha \le n)\) is a real number. \(\lambda > 0\) is a parameter. \(D_{0+}^{\alpha }\) is the standard Caputo differentiation. Some sufficient conditions for the existence of positive solutions to this boundary value problem of nonlinear fractional differential equation are established by nonlinear alternative of Leray–Schauder type and Guo–Krasnoselskii fixed point theorem on cones. As applications, some examples are provided to illustrate our main results.

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## Acknowledgments

The author would like to thank the anonymous referees for their useful and valuable suggestions. This work is supported by the National Natural Sciences Foundation of Peoples Republic of China under Grant (No. 11161025), Yunnan Province natural scientific research fund project (No. 2011FZ058).

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Zhao, K., Gong, P. Existence of Positive Solutions for a Class of Higher-Order Caputo Fractional Differential Equation.
*Qual. Theory Dyn. Syst.* **14**, 157–171 (2015). https://doi.org/10.1007/s12346-014-0121-0

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DOI: https://doi.org/10.1007/s12346-014-0121-0

### Keywords

- Fractional differential equation
- Multiple positive solutions
- Boundary value problems
- Fixed point theorem