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

Source abstract

Abstract Recent studies have shown that the Sturm-Liouville problem $$-Au=\lambda u$$ - 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 $$\alpha$$ α is sufficiently close to 1. It has been shown that the principal eigenvalue, denoted by $$\lambda (\alpha ),$$ λ ( α ) , always exists when A is the Riemann-Liouville fractional derivative $${\mathcal {D}_{a+}^{\alpha }}$$ D a + α for any $$1<\alpha <2$$ 1 &lt; α &lt; 2 , highlighting a striking contrast with the Caputo case. We introduce appropriate spaces to develop a systematic study of $$\lambda (\alpha )$$ λ ( α ) as a function of the real parameter $$\alpha$$ α and show, for example, that although $$\lambda (\alpha )$$ λ ( α ) is non-monotonic, the quotient $$\frac{\Gamma (\alpha -1)}{(b-a)^{\alpha -1}}\frac{1}{\lambda (\alpha )}$$ Γ ( α - 1 ) ( b - a ) α - 1 1 λ ( α ) is strictly decreasing with respect to $$\alpha$$ α . For the sake of completeness, we shall again show the existence of a principal eigenvalue for each $$\alpha \in (1,2)$$ α ∈ ( 1 , 2 ) using the techniques developed here. Finally, we extend our techniques and results to the more general weighted spectral problem $$-Au=\lambda p(x) u$$ - A u = λ p ( x ) u for a class of coefficient functions p .