Existence, comparison, and monotonicity of principal eigenvalues of fractional Sturm-Liouville problems involving Riemann-Liouville derivatives

Recent studies have shown that the Sturm-Liouville problem -Au=λ u - A u = λ u with zero boundary conditions on an interval has no principal eigenvalue when A is the Caputo fractional derivative "Equation missing" , provided that α α is sufficiently close to 1. It has been shown that the principal eigenvalue, denoted by λ (α ), λ ( α ) , always exists when A is the Riemann-Liouville fractional derivative Dₐ₊α D a + α for any 1<α <2 1 < α < 2 , highlighting a striking contrast with the Caputo case. We introduce appropriate spaces to develop a systematic study of λ (α ) λ ( α ) as a function of the real parameter α α and show, for example, that although λ (α ) λ ( α ) is non-monotonic, the quotient Γ (α -1)(b-a)α -11/λ (α ) Γ ( α - 1 ) ( b - a ) α - 1 1 λ ( α ) is strictly decreasing with respect to α α . For the sake of completeness, we shall again show the existence of a principal eigenvalue for each α ∈ (1,2) α ∈ ( 1 , 2 ) using the techniques developed here. Finally, we extend our techniques and results to the more general weighted spectral problem -Au=λ p(x) u - A u = λ p