# Positive solutions for singular coupled integral boundary value problems of nonlinear Hadamard fractional differential equations

Volume 8, Issue 2, pp 110--129 Publication Date: March 28, 2015
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### Authors

Wengui Yang - Ministry of Public Education, Sanmenxia Polytechnic, Sanmenxia, Henan 472000, China.

### Abstract

In this paper, we study the existence of positive solutions for a class of coupled integral boundary value problems of nonlinear semipositone Hadamard fractional differential equations $D^\alpha u(t) + \lambda f(t, u(t), v(t)) = 0,\quad D^\alpha v(t) + \lambda g(t, u(t), v(t)) = 0,\quad t \in (1, e),\quad \lambda > 0$ $u^{(j)}(1) = v^{(j)}(1) = 0, 0 \leq j \leq n - 2; u(e) = \mu\int^e_1 v(s) \frac{ds}{ s} , v(e) = \nu\int^e_1 u(s) \frac{ds}{ s},$ where $\lambda,\mu,\nu$ are three parameters with $0<\mu<\beta$ and $0<\nu<\alpha,\quad \alpha,\beta\in (n - 1; n]$ are two real numbers and $n\geq 3, D^\alpha, D^\beta$ are the Hadamard fractional derivative of fractional order, and $f; g$ are sign-changing continuous functions and may be singular at $t = 1$ or/and $t = e$. First of all, we obtain the corresponding Green's function for the boundary value problem and some of its properties. Furthermore, by means of the nonlinear alternative of Leray-Schauder type and Krasnoselskii's fixed point theorems, we derive an interval of $\lambda$ such that the semipositone boundary value problem has one or multiple positive solutions for any $\lambda$ lying in this interval. At last, several illustrative examples were given to illustrate the main results.

### Keywords

• Hadamard fractional differential equations
• coupled integral boundary conditions
• positive solutions
• Green's function
• fixed point theorems.

•  34A08
•  34B16
•  34B18

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