Existence of positive solutions of nonlinear fractional differential equations

Existence of positive solutions for the nonlinear fractional differential equation Dsu(x) = f (x,u(x)), 0 < s < 1, has been studied (S. Zhang, J. Math. Anal. Appl. 252 (2000) 804–812), where Ds 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 0, ∀j. We give further conditions on f and aj ’s under which the equation has unique solution which is positive. Further, the condition aj > 0 is relaxed and conditions on f and aj ’s are given under which the equation has unique solution, which may not necessarily be positive.  2003 Elsevier Science (USA). All rights reserved.