Spectral and Oscillation Theory for an Unconventional Fractional Sturm–Liouville Problem

Abstract

Here, we investigate the spectral and oscillation theory for a class of fractional differential equations subject to specific boundary conditions. By transforming the problem into a modified version with a classical structure, we establish the orthogonality properties of eigenfunctions and some major comparison theorems for solutions. We also derive a new type of integration by using parts of formulas for modified fractional integrals and derivatives. Furthermore, we analyze the variational characterization of the first eigenvalue, revealing its non-zero first eigenfunction within the interior. Our findings demonstrate the potential for novel definitions of fractional derivatives to mirror the classical Sturm–Liouville theory through simple isospectral transformations.

Keywords

• Riemann–Liouville
• Caputo
• Sturm–Liouville
• eigenvalues
• fractional
• spectral theory