Positive and Nondecreasing Solutions to an m-Point Boundary Value Problem for Nonlinear Fractional Differential Equation. (3rd January 2012) - Cabrera, I. J.; Harjani, J.; Sadarangani, K. B.
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: D 0 + α u ( t ) + f ( t, u ( t ) ) = 0, 0 < t < 1, 2 < α ≤ 3, u ( 0 ) = u ' ( 0 ) = 0, u ' ( 1 ) = ∑ i = 1 m - 2 a i u ' ( ξ i ), where D 0 + α denotes the standard Riemann-Liouville fractional derivative, f : [ 0, 1 ] × [ 0, ∞ ) → [ 0, ∞ ) is a continuous function, a i ≥ 0 for i = 1, 2, …, m - 2, and 0 < ξ 1 < ξ 2 < ⋯ < ξ m - 2 < 1 . Our analysis relies on a fixed point theorem in partially ordered sets. Some examples are also presented to illustrate the main results.