Positive solutions for singular coupled integral boundary value problems of nonlinear Hadamard fractional differential equations
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 signchanging
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 LeraySchauder 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.
MSC
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