Positive and nondecreasing solutions to an m-point boundary value problem for nonlinear fractional differential equation

Abstract

We are concerned with the existence and uniqueness of a positive and nondecreasing solution for the following nonlinear fractional m-point boundary value problem: Dα 0+ u (t) + f (t, u (t)) =0, 0 <t < 1, 2 < α ≤ 3, u(0) = u’(0) = 0, u’ (1) = ∑m−2 i=1 aiu’ (ξi), where Dα 0+ denotes the standard Riemann-Liouville fractional derivative, f : [0, 1] × [0,∞) → [0,∞) is a continuous function, ai ≥ 0 for i = 1, 2, . . . , m − 2, and 0 < ξ1 < ξ2 < · · · < ξm−2 < 1. Our analysis relies on a fixed point theorem in partially ordered sets. Some examples are also presented to illustrate the main results.