Smallest eigenvalues for boundary value problems of two term fractional differential operators depending on fractional boundary conditions

Abstract

Let n ≥ 2 be an integer, and let n - 1 < α ≤ n. We consider eigenvalue problems for two point n - 1, 1 boundary value problems Dα0+ u + α(t)u + λp(t)u = 0, 0 < t < 1, u(i)(0) = 0, i = 0, 1,..., n - 2, Dβ0+ u(1) = 0, where 0 ≤ β ≤ n - 1 and Dα0+ and Dβ0+ denote standard Riemann-Liouville differential operators. We prove the existence of smallest positive eigenvalues and then obtain comparisons of these smallest eigenvalues as functions of both p and β.