Multiplicity results for a class of fractional boundary value problems depending on two parameters

Abstract

We prove the existence of at least three solutions to the following fractional boundary value problem: { − d dt ( 1 2 0 D−σ t (u ′(t)) + 1 2 t D−σ T (u ′(t)) ) − λβ(t)f(u(t))− μγ(t)g(u(t)) = 0, a.e. t ∈ [0, T ], u(0) = u(T ) = 0, where 0D −σ t and tD −σ T are the left and right Riemann–Liouville fractional integrals of order 0 ≤ σ < 1 respectively. The approach is based on a recent three critical points theorem of Ricceri [B. Ricceri, A further refinement of a three critical points theorem, Nonlinear Anal. 74 (2011), 7446–7454].