Positive solutions for an impulsive boundary value problem with Caputo fractional derivative
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Authors
Keyu Zhang
- School of Mathematics, Shandong University, Jinan, Shandong, 250100, P. R. China.
- Department of Mathematics, Qilu Normal University, Jinan, Shandong, 250013, P. R. China.
Jiafa Xu
- School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, P. R. China.
Abstract
In this work we use fixed point theorem method to discuss the existence of positive solutions for the
impulsive boundary value problem with Caputo fractional derivative
\[
\begin{cases}
^cD^q_t u(t)=f(t,u(t)),\,\,\,\,\, \texttt{a.e.} t\in [0,1];\\
\Delta u(t_k)=I_k(u(t_k)), \Delta u'(t_k)=J_k(u(t_k)),\,\,\,\,\, k=1,2,...,m;\\
au(0)-bu(1)=0,\quad au'(0)-bu'(1)=0,
\end{cases}
\]
where \(q \in (1; 2)\) is a real number, \(a; b\) are real constants with \(a > b > 0\), and \(^cD^q_t\)
is the Caputo's fractional
derivative of order \(q, f : [0; 1] \times \mathbb{R}^+ \rightarrow \mathbb{R}^+\) and \(I_k; J_k : \mathbb{R}^+ \rightarrow \mathbb{R}^+\) are continuous functions, \(k = 1; 2; ... ;m,
\mathbb{R}^+ := [0;+1)\).
Share and Cite
ISRP Style
Keyu Zhang, Jiafa Xu, Positive solutions for an impulsive boundary value problem with Caputo fractional derivative, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 6, 4628--4638
AMA Style
Zhang Keyu, Xu Jiafa, Positive solutions for an impulsive boundary value problem with Caputo fractional derivative. J. Nonlinear Sci. Appl. (2016); 9(6):4628--4638
Chicago/Turabian Style
Zhang, Keyu, Xu, Jiafa. "Positive solutions for an impulsive boundary value problem with Caputo fractional derivative." Journal of Nonlinear Sciences and Applications, 9, no. 6 (2016): 4628--4638
Keywords
- Caputo fractional derivative
- impulsive boundary value problem
- fixed point theorem
- positive solution.
MSC
- 34B37
- 34A08
- 34B15
- 34B18
- 47N20
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