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JOURNALS // Vladikavkazskii Matematicheskii Zhurnal // Archive

Vladikavkaz. Mat. Zh., 2024 Volume 26, Number 1, Pages 27–35 (Mi vmj895)

Existence theorem for a fractal Sturm–Liouville problem

B. P. Allahverdieva, H. Tunab

a Department of Mathematics, Khazar University, 11 Mehseti St., Baku AZ1096, Azerbaijan
b Department of Mathematics, Burdur Mehmet Akif Ersoy University, Antalya Burdur Yolu, 15030 Burdur, Turkey

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.

Key words: fractal Sturm–Liouville problems, existence problems.

UDC: 517.91+517.538.72

MSC: 28A80, 34A08, 35A01

Received: 10.01.2023

Language: English

DOI: 10.46698/h4206-1961-4981-h



© Steklov Math. Inst. of RAS, 2024