Some problems which includes diamond type derivative on time scale

In classical terms, the Sturm-Liouville equation is one of the most important equations of mathematical physics. Initially, the Sturm-Liouville theory, applied extensively in heat conduction problems, is now applied in di¤erent physical problems. Different generalizations have been made in the classical Sturm-Liouville problem due to the need of mathematical physics and accordingly linear differential operators theory has been formed. In this study, this equation is considered on a time scale and the diamond alpha type Sturm-Liouville problem is defined. The simplicity of the eigenvalues of this problem, the real property, the orthogonality of the eigenvalues corresponding to the di¤erent eigenvalues, and the Lagrange equality and Green formula in the alpha type are given. In addition, an equality was found for the eigenfunctions of the problem. Diamond alpha-type derivative and integral on time scales are the most trouble operators in terms of processing. This is because the derivative and integral of diamond alpha type is a combination of delta and nabla derivative and integral.In this respect, obtained eigenfunction is very important.