Existence Theorem for a Fractal Sturm-Liouville Problem

Abstract

In this article, using a new calculus defined on fractal subsets of the set of real numbers, a Sturm--Lioville type problem is discussed, namely the fractal Sturm--Liouville problem. The existence and uniqueness theorem has been proved for such equations. In this context, the historical development of the subject is discussed in the introduction. In Section 2, the basic concepts of $F^{\alpha}$-calculus defined on fractal subsets of real numbers are given, i. e., $F^{\alpha}$-continuity, $F^{\alpha}$-derivative and fractal integral definitions are given and some theorems to be used in the article are given. In Section 3, the existence and uniqueness of the solutions for the fractal Sturm--Liouville problem are obtained by using the successive approximations method. Thus, the well-known existence and uniqueness problem for Sturm--Liouville equations in ordinary calculus is handled on the fractal calculus axis, and the existing results are generalized.