THE PERIODIC FRACTIONAL STURM-LIOUVILLE PROBLEM Rachmat Okta Ariyanto, Sutrima, and Muslich Department of Mathematics Faculty of Mathematics and Natural Sciences Universitas Sebelas Maret Abstract. It is well known that the Sturm-Liouville problems are divided into regular, singular, and periodic types. By replacing the fractional derivatives in the operator Sturm-Liouville we can construct the fractional Sturm-Liouville problems. There are many research that investigate the fractional Sturm-Liouville problems of regular and singular type. The results of the both types parallel with the ones of the Sturm-Liouville problems. In this paper we investigate the periodic fractional Sturm-Liouville problems. The investigation is focused on the properties of selft-adjointness, eigenvalues, eigenfunctions, and simpleness of the periodic fractional Sturm-Liouville. The finally, we give some examples to simulate theory. Keywords: periodic, the fractional Sturm-Liouville problem, eigenvalue, eigenfunc