Solvability of boundary-value problems for nonlinear fractional differential equations - Guo, Y. (Ukrains’kyi Matematychnyi Zhurnal, 2010)
We consider the existence of nontrivial solutions of the boundary-value problems for nonlinear fractional differential equations D^αu(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 ∈ R = (−∞,+∞), βη^α−1 ≠ 1, D^α is a Riemann–Liouville differential operator of order α, f: (0,1)×R→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.