NONTRIVIAL SOLUTIONS FOR BOUNDARY-VALUE PROBLEMS OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS

In this paper, we consider the existence of nontrivial solutions for the nonlinear fractional differential equation boundary-value problem (BVP) -D(0)(alpha)+u(t) = lambda [f (t, u(t)) + q(t)], 0 < t < 1 u(0) = u(1) = 0, where lambda > 0 is a parameter, 1 < alpha <= 2, D(0+)(alpha) is the standard Riemann-Liouville differentiation, f : [0, 1] x R -> R is continuous, and q(t): (0, 1) -> [0, +infinity) is Lebesgue integrable. We obtain serval sufficient conditions of the existence and uniqueness of nontrivial solution of BVP when lambda in some interval. Our approach is based on Leray-Schauder nonlinear alternative. Particularly, we do not use the nonnegative assumption and monotonicity which was essential for the technique used in almost all existed literature on f.