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Existence results for boundary value problems of fractional functional differential equations with delay

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Abstract

In this paper, we study boundary value problems of the following fractional functional differential equations involving the Caputo fractional derivative with delay

$$\begin{aligned} \left\{ \begin{array}{ll} ^CD^{\alpha }u(t)+f(t,u_{t})=0,\; t \in [0,T],\\ u_{0}=\varphi ,\ \ u(T)=A, \ \ u''(0)=0 \end{array} \right. \end{aligned}$$

where \(f:[0,T]\times C[-r,0]\rightarrow {\mathbb {R}}\) is continuous function, \(\varphi \in C[-r,0]\) and \(A \in {\mathbb {R}},\) \(2<\alpha \le 3\), \(0\le r\le T.\) We use Green function to reformulate the boundary value problems into an abstract operator equation. By means of Guo–Krasnosel’skii fixed point theorem on cone, Banach contraction principle and Schaefer’s fixed point theorem, some existence results of solutions and positive solutions are obtained, respectively. As applications, some examples are presented to illustrate the main results.

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Acknowledgments

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have lead to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (61374074), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119) and supported by Shandong Provincial Natural Science Foundation (ZR2012AM009, ZR2013AL003).

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Jinan, Jinan, 250022, Shandong, People’s Republic of China

    Zhenlai Han & Yanan Li

  2. School of Sciences, Shandong Jianzhu University, Jinan, 250101, Shandong, People’s Republic of China

    Meizhen Sui

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  1. Zhenlai Han
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  2. Yanan Li
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  3. Meizhen Sui
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Correspondence to Zhenlai Han.

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Han, Z., Li, Y. & Sui, M. Existence results for boundary value problems of fractional functional differential equations with delay. J. Appl. Math. Comput. 51, 367–381 (2016). https://doi.org/10.1007/s12190-015-0910-x

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  • DOI: https://doi.org/10.1007/s12190-015-0910-x

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