Existence of positive solutions of nonlinear fractional differential equations

Existence of positive solutions for the nonlinear fractional differential equation D(s)u(x) = f(x, u(x)), 0 < s < 1, has been studied (S. Zhang, J. Math. Anal. Appl. 252 (2000) 804-812), where D-s denotes Riemann-Liouville fractional derivative. In the present work we study existence of positive solutions in case of 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)D(sn-1) ... - a(1)D(s1), 0 < s(1) < s(2) < ••• s(n) < 1, and a(j) > 0, For All(j). We give further conditions on f and a(j)'s under which the equation has unique solution which is positive. Further, the condition a(j) > 0 is relaxed and conditions on f and a(j)'s are given under which the equation has unique solution, which may not necessarily be positive. (C) 2003 Elsevier Science (USA). All rights reserved.