Multiplicity of positive solutions of a class of nonlinear fractional differential equations

This paper is concerned with the nonlinear fractional differential equation L (D) u = f (x,u), u(0) = 0, 0 < x < 1, where L(D) = D-sn - a(n-1) Dsn-1 - (...) - 0 < s(1) < s(2) < (...) < s(n) < 1, and a(j) > 0, j = 1, 2,..., n - 1. Some results are obtained for the existence, nonexistence, and multiplicity of positive solutions of the above equation by using Krasnoselskii's fixed-point theorem in a cone. In particular, it is proved that the above equation has N positive solutions under suitable conditions, where N is an arbitrary positive integer. (C) 2005 Elsevier Ltd. All rights reserved.