Electron. J. Diff. Equ., Vol. 2012 (2012), No. 234, pp. 1-11.

Existence and uniqueness of positive solutions to higher-order nonlinear fractional differential equation with integral boundary conditions

Chenxing Zhou

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.

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Chenxing Zhou
College of Mathematics
Changchun Normal University
Changchun 130032, Jilin, China

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