Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 234, pp. 1-11.
Title: Existence and uniqueness of positive solutions to higher-order
nonlinear fractional differential equation with integral
boundary conditions
Author: Chenxing Zhou (Changchun Normal Univ., Jilin, China)
Abstract:
In this article, we consider the nonlinear fractional order
three-point boundary-value problem
$$\displaylines{
D_{0+}^{\alpha} u(t) + f(t,u(t))=0, \quad 0 < t < 1,\cr
u(0) = u'(0) = \dots = u^{(n-2)}(0)=0, \quad u^{(n-2)}(1) =
\int_0^\eta u(s)ds,
}$$
where $D_{0+}^{\alpha}$ is the standard Riemann-Liouville fractional
derivative, $n-1 < \alpha \leq n$, $n \geq 3$. By using a fixed-point
theorem in partially ordered sets, we obtain sufficient conditions
for the existence and uniqueness of positive and nondecreasing
solutions to the above boundary value problem.
Submitted September 19, 2012. Published December 21, 2012.
Math Subject Classifications: 26A33, 34B18, 34B27.
Key Words: Partially ordered sets; fixed-point theorem; positive solution.