Title | POSITIVE SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR HIGHER-ORDER NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS |
Publication Type | Journal Article |
Year of Publication | 2015 |
Authors | KONG, QINGKAI, MCCABE, MICHAEL |
Volume | 19 |
Issue | 4 |
Start Page | 527 |
Pagination | 15 |
Date Published | 2015 |
ISSN | 1083-2564 |
AMS | 34B15, 34B18 |
Abstract | In this paper, we study the boundary value problem consisting of the higher-order fractional differential equation$${ (−1)^m (D^α_{0+})^m u = f(t, u), \ \ 0 < t < 1, }$$ and the boundary conditions $${ \left( (D^α_{0+})^i u \right) (0) = \left( (D^α_{0+})^i u\right) (1) = 0, \ \ i = 0, 1, . . . , m − 1, }$$ where ${ 1 < α < 2, m ∈ \mathbb{N},\ \ D^α_{0+} }$ is the Riemann-Liouville fractional differential operator, and ${ \ {(}D^α_{0+})^{j+1} = D^α_{0+}\ {(} D^α_{0+}\ {)}^j}$ for ${ j = 0, . . . , m − 1, }$ with ${ \ {(}D^α_{0+}\ {)}^0 = I, }$ the identity operator. By finding the Green’s function using the iteration method and applying the Krasnosel’skii fixed point theorem, we establish the existence of one, two, any finite number, and even a countably infinite number of positive solutions. Criteria for the nonexistence of positive solutions are also obtained. Our results cover, improve, and complement those by Jiang and Yuan for the case |
URL | http://www.acadsol.eu/en/articles/19/4/5.pdf |
Refereed Designation | Refereed |
Full Text | REFERENCES |