**Triple positive solutions for a boundary value problem of nonlinear fractional differential equation**

**Chuanzhi Bai**, Huaiyin Teachers College, Huaian, Jiangsu, P. R. China

E. J. Qualitative Theory of Diff. Equ., No. 24. (2008), pp. 1-10.

Communicated by S. K. Ntouyas. | Received on 2008-04-13 Appeared on 2008-07-25 |

**Abstract: **In this paper, we investigate the existence of three positive solutions for the nonlinear fractional boundary value problem

$D_{0+}^{\alpha} u(t) + a(t)f(t,u(t), u^{\prime \prime}(t))=0, \quad 0 < t < 1, \quad 3 < \alpha \leq 4,$

$u(0) = u^{\prime}(0) = u^{\prime \prime}(0)= u^{\prime \prime}(1)=0 $,

where $D_{0+}^{\alpha}$ is the standard Riemann-Liouville fractional derivative. The method involves applications of a new fixed-point theorem due to Bai and Ge. The interesting point lies in the fact that the nonlinear term is allowed to depend on the second order derivative $u^{\prime \prime}$.

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