Existence of positive solution for some class of nonlinear fractional differential equations

In this paper, we investigate the multiple and infinitely solvability of positive solutions for nonlinear fractional differential equation Du(t) = t(nu) f (u), 0 < t < 1, where D = t(-betadelta)D(beta)(gamma-delta,delta), beta > 0, gamma greater than or equal to 0, 0 < delta < 1, nu > -beta(gamma + 1). Our main work is to deal with limit case of f (s)/s as s --> 0 or s --> infinity and Phi(s)/s, psi(s)/s as s --> 0 or s --> infinity, where Phi(s), psi(s) are functions connected with function f. In J. Math. Appl. 252 (2000) 804-812, we consider the existence of a positive solution for the particular case of Eq. (1.1), i.e., the Riemann-Liouville type (beta = 1, gamma = 0) nonlinear fractional differential equation, using the super-lower solutions method. Here, we devote to the existence of positive solution and multi-positive solutions for Eq. (1.1) by means of the fixed point theorems for the cone. (C) 2003 Elsevier Science (USA). All rights reserved.