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) -D₀α+u(t)=λ[f(t, u(t))+q(t)] , 0 < t < 1 u(0) = u(1) = 0, where λ > 0 is a parameter, 1 < α ≤ 2, D₀₊α is the standard Riemann-Liouville differentiation, f : [0, 1] ×R→R is continuous, and q(t) : (0, 1) → [0, +∞ ] is Lebesgue integrable. We obtain serval sufficient conditions of the existence and uniqueness of nontrivial solution of BVP when λ 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.