Local Fractional Sturm-Liouville Theory on Fractal Sets with Integration by Parts and Equivalence of Local Fractional Derivatives

An approach to creating a fractional version of the Sturm and Liouville theorem for an inter val by means of integration by parts as a structural axiom will be described. The relationship between some properties and definitions of this development will be proven formally as having equivalents under appropriate admissibility conditions; creating an adjointly unique structure. This theory leads to developing self adjoint local fractional Sturm and Liouville operators with orthogonal eigenfunctions having real spectra. The stair step variable is provided as an explicit example of the creation of a spectral structure on fractals. 2020 Mathematics Subject Classification. Primary 34L10; Secondary 26A33, 28A80, 47A75.