Department of Mathematics
Guangdong University of Finance and Economics
Guangzhou 510000, China
email: liuyuji888@sohu.com
Yuji Liu
Abstract:
Firstly we prove existence and uniqueness of solutions of Cauchy
problems of linear fractional differential equations (LFDEs) with two
variable coefficients involving Caputo fractional derivative, Riemann-Liouville
derivative, Caputo type Hadamard derivative and Riemann-Liouville type Hadamard
fractional derivatives with order q in [n-1,n) by using the iterative method.
Secondly we obtain exact expressions for piecewise continuous solutions of
the linear fractional differential equations with a constant coefficient and
a variable one. These results provide new methods to transform an impulsive
fractional differential equation (IFDE) to a fractional integral equation (FIE).
Thirdly, we propose four classes of boundary value problems of singular
fractional differential equations with impulse effects. Sufficient conditions
are given for the existence of solutions of these problems. We allow the
nonlinearity p(t)f(t,x) in fractional differential equations to be singular
at t=0,1.
Finally, we point out some incorrect formulas of solutions in cited papers.
A new Banach space and the compact properties of subsets are proved. By
establishing a new framework to find the solutions for impulsive fractional
boundary value problems, the existence of solutions of three classes boundary
value problems of impulsive fractional differential equations with multi-term
fractional derivatives are established.
Submitted February 24, 2015. Published November 18, 2016.
Math Subject Classifications: 34A08, 26A33, 39B99, 45G10, 34B37, 34B15, 34B16.
Key Words: Higher order singular fractional differential system;
impulsive boundary value problem; Riemann-Liouville fractional derivative;
Caputo fractional derivative; Riemann-Liouville type Hadamard
fractional derivative; Caputo type Hadamard fractional derivative;
fixed point theorem.
An addendum was posted on February 13, 2017. It corrects some definitions and equations. See the last four pages of this article.
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Yuji Liu Department of Mathematics Guangdong University of Finance and Economics Guangzhou 510000, China email: liuyuji888@sohu.com |
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