On the Existence of Positive Solutions for a Boundary Valued Problem of Fractional Order

In this paper, we investigate the problem of existence and nonexistence of positive solutions for the nonlinear boundary value problem of fractional order: D(alpha)u(t) + lambda a(t) f(u(t)) = 0, 0 < t < 1, 2 < alpha <= 3, u(0) = u"(0) = 0, gamma u'(1) + beta u"(1) = 0, where D-alpha is the Caputo's fractional derivative and lambda is a positive parameter. By using Krasnoesel'skii's fixed-point theorem of cone preserving operators, we establish various results on the existence of positive solutions of the boundary value problem. Under various assumptions on a(t) and f (u(t)), we give the intervals of the parameter lambda which yield the existence of the positive solutions.