Positive solutions for an impulsive boundary value problem with Caputo fractional derivative

Volume 9, Issue 6, pp 4628--4638 http://dx.doi.org/10.22436/jnsa.009.06.101

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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, \(k = 1; 2; ... ;m, \mathbb{R}^+ := [0;+1)\).

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Keywords

• Caputo fractional derivative
• impulsive boundary value problem
• fixed point theorem
• positive solution.

MSC

•  34B37
•  34A08
•  34B15
•  34B18
•  47N20

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