Eigenvalue comparison for a fractional boundary value problem with nonlocal boundary conditions

Abstract

Let n ϵ ℕ, n ≥ 3, n − 1 < α ≤  n, 1 ≤ β ≤ n − 2, and 0 ≤ γ ≤ β. We study the existence of and comparison of smallest eigenvalues of the fractional boundary value problems Dα0+u+λ1p(t)u = 0, Dα0+u+λ2q(t)u = 0, t (0, 1), satisfying the nonlocal boundary conditions u(i)(0) = 0, i = 0, 1,…, n − 2, Dβ0+u(1) = Σmi=1 aiDγ0+u(ξi), where p and q are nonnegative continuous functions on [0, 1] which do not vanish identically on any nondegenerate compact subinterval of [0, 1]. A weighted norm and sign conditions of the Green’s function are used along with the theory of u0-positive operators.