Solvability of boundary-value problems for nonlinear fractional differential equations

Abstract

We consider the existence of nontrivial solutions of the boundary-value problems for nonlinear fractional differential equations $$D^{α}u(t)+λ[f(t,u(t))+q(t)]=0,\; 0 < t < 1, \; u(0) = 0,\; u(1) = βu(η),$$ where $λ > 0$ is a parameter, $1 < α ≤ 2,\; η ∈ (0, 1),\; β ∈ \mathbb{R} = (−∞,+∞),\; βη^{α−1} ≠ 1,\; D^{α}$ is a Riemann–Liouville differential operator of order $α$, $f: (0,1)×\mathbb{R}→\mathbb{R}$ is continuous, $f$ may be singular for $t = 0$ and/or $t = 1$, and $q(t) : [0, 1] → [0, +∞)$. We give some sufficient conditions for the existence of nontrivial solutions to the formulated boundary-value problems. Our approach is based on the Leray–Schauder nonlinear alternative. In particular, we do not use the assumption of nonnegativity and monotonicity of $f$ essential for the technique used in almost all available literature.