Positive solutions for singular coupled integral boundary value problems of nonlinear Hadamard fractional differential equations
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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.
Share and Cite
ISRP Style
Wengui Yang, Positive solutions for singular coupled integral boundary value problems of nonlinear Hadamard fractional differential equations, Journal of Nonlinear Sciences and Applications, 8 (2015), no. 2, 110--129
AMA Style
Yang Wengui, Positive solutions for singular coupled integral boundary value problems of nonlinear Hadamard fractional differential equations. J. Nonlinear Sci. Appl. (2015); 8(2):110--129
Chicago/Turabian Style
Yang, Wengui. "Positive solutions for singular coupled integral boundary value problems of nonlinear Hadamard fractional differential equations." Journal of Nonlinear Sciences and Applications, 8, no. 2 (2015): 110--129
Keywords
- Hadamard fractional differential equations
- coupled integral boundary conditions
- positive solutions
- Green's function
- fixed point theorems.
MSC
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