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Published in: Journal of Applied Mathematics and Computing 1-2/2017

23-12-2015 | Original Research

Existence and uniqueness for a class of multi-term fractional differential equations

Authors: Qiuping Li, Chuanxia Hou, Liying Sun, Zhenlai Han

Published in: Journal of Applied Mathematics and Computing | Issue 1-2/2017

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Abstract

In this paper, we consider the initial value problem for the nonlinear fractional differential equations
$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} D^\alpha u(t)=f(t,u(t),D^{\beta _1}u(t),\ldots ,D^{\beta _N}u(t)), \quad &{}t\in (0,1],\\ D^{\alpha -k}u(0)=0, \quad &{}k=1,2,\ldots ,n, \end{array} \right. \end{aligned}$$
where \(\alpha >\beta _1>\beta _2>\cdots \beta _N>0\), \(n=[\alpha ]+1\) for \(\alpha \notin \mathbb {N}\) and \(\alpha =n\) for \(\alpha \in \mathbb {N}\), \(\beta _j<1\) for any \(j\in \{1,2,\ldots ,N\}\), D is the standard Riemann–Liouville derivative and \(f:[0,1]\times \mathbb {R}^{N+1}\rightarrow \mathbb {R}\) is a given function. By means of Schauder fixed point theorem and Banach contraction principle, an existence result and a unique result for the solution are obtained,respectively. As an application, some examples are presented to illustrate the main results.

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Metadata
Title
Existence and uniqueness for a class of multi-term fractional differential equations
Authors
Qiuping Li
Chuanxia Hou
Liying Sun
Zhenlai Han
Publication date
23-12-2015
Publisher
Springer Berlin Heidelberg
Published in
Journal of Applied Mathematics and Computing / Issue 1-2/2017
Print ISSN: 1598-5865
Electronic ISSN: 1865-2085
DOI
https://doi.org/10.1007/s12190-015-0973-8

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