Abstract
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, n - 1 < alpha <= n, n >= 3,
u(0) = u ''(0) = u'''(0) = ... = u((n - 1))(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 Krasnoeselskii'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. An example is also given to illustrate the main results.