Geodesic
Existence theorem for a fractal Sturm–Liouville problem
Vladikavkazskij matematičeskij žurnal, Tome 26 (2024) no. 1, pp. 27-35 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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. 
@article{VMJ_2024_26_1_a2,
     author = {B. P. Allahverdiev and H. Tuna},
     title = {Existence theorem for a fractal {Sturm{\textendash}Liouville} problem},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
     pages = {27--35},
     year = {2024},
     volume = {26},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VMJ_2024_26_1_a2/}
}
TY  - JOUR
AU  - B. P. Allahverdiev
AU  - H. Tuna
TI  - Existence theorem for a fractal Sturm–Liouville problem
JO  - Vladikavkazskij matematičeskij žurnal
PY  - 2024
SP  - 27
EP  - 35
VL  - 26
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/VMJ_2024_26_1_a2/
LA  - en
ID  - VMJ_2024_26_1_a2
ER  - 
%0 Journal Article
%A B. P. Allahverdiev
%A H. Tuna
%T Existence theorem for a fractal Sturm–Liouville problem
%J Vladikavkazskij matematičeskij žurnal
%D 2024
%P 27-35
%V 26
%N 1
%U http://geodesic.mathdoc.fr/item/VMJ_2024_26_1_a2/
%G en
%F VMJ_2024_26_1_a2
B. P. Allahverdiev; H. Tuna. Existence theorem for a fractal Sturm–Liouville problem. Vladikavkazskij matematičeskij žurnal, Tome 26 (2024) no. 1, pp. 27-35. http://geodesic.mathdoc.fr/item/VMJ_2024_26_1_a2/

[1] Parvate, A. and Gangal, A. D., “Calculus on Fractal Subsets of Real Line — I: Formulation”, Fractals, 17:1 (2009), 53–81 | DOI | MR | Zbl

[2] Çetinkaya, F. A. and Golmankaneh, A. K., “General Characteristics of a Fractal Sturm–Liouville Problem”, Turkish Journal of Mathematics, 45:4 (2021), 1835–1846 | DOI | MR

[3] Golmankhaneh, A. K., Fractal Calculus and its Applications: $F^{\alpha}$-Calculus, World Scientific Publ. Co. Pte. Ltd, 2022 | DOI | MR

[4] Golmankhaneh, A. K. and Tunç, C., “Stochastic Differential Equations on Fractal Sets”, Stochastics, 92:8 (2020), 1244–1260 | DOI | MR | Zbl

[5] Golmankhaneh, A. K. and Tunç, C., “Sumudu Transform in Fractal Calculus”, Applied Mathematics and Computation, 350 (2019), 386–401 | DOI | MR | Zbl

[6] Golmankhaneh, A. K. and Tunc, C., “On the Lipschitz Condition in the Fractal Calculus”, Chaos, Solitons Fractals, 95 (2017), 140–147 | DOI | MR | Zbl

[7] Parvate, A. and Gangal, A. D., “Calculus on Fractal Subsets of Real Line – I: Conjugacy with Ordinary Calculus”, Fractals, 19:3 (2011), 271–290 | DOI | MR | Zbl

[8] Kolwankar, K. M. and Gangal, A. D., “Fractional Differentiability of Nowhere Differentiable Functions and Dimensions”, Chaos, 6:4 (1996), 505–513 | DOI | MR | Zbl

[9] Kolwankar, K. M. and Gangal, A. D., “Hölder Exponents of Irregular Signals and Local Fractional Derivatives”, Pramana — Journal of Physics, 48 (1997), 49–68 | DOI

[10] Kolwankar, K. M. and Gangal, A. D., “Local Fractional Fokker-Planck Equation”, Physical Review Letters, 80:2 (1998), 214–217 | DOI | MR | Zbl

[11] Kolwankar, K. M. and Gangal, A. D., “Local Fractional Derivatives and Fractal Functions of Several Variables”, Mathematical Physics, 1998, arXiv: physics/9801010 | DOI

[12] Aydemir, K. and Mukhtarov, O. Sh., “A New Type Sturm–Liouville Problem with an Abstract Linear Operator Contained in the Equation”, Quaestiones Mathematicae, 45:12 (2022), 1931–1948 | DOI | MR | Zbl

[13] Aydemir, K. and Mukhtarov, O. Sh., “Qualitative Analysis of Eigenvalues and Eigenfunctions of one Boundary Value-Transmission Problem”, Boundary Value Problems, 2016, 82 | DOI | MR | Zbl

[14] Levitan, B. M. and Sargsjan, I. S., Sturm–Liouville and Dirac Operators, Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers Group, Dordrecht, 1991 | MR

[15] Olgar, H. and Mukhtarov, O. Sh., “Weak Eigenfunctions of Two-Interval Sturm–Liouville Problems Together with Interaction Conditions”, Journal of Mathematical Physics, 58:4 (2017), 042201 | DOI | MR | Zbl

[16] Ozkan, A. S. and Adalar, I., “Inverse Nodal Problems for Sturm–Liouville Equation with Nonlocal Boundary Conditions”, Journal of Mathematical Analysis and Applications, 520:1 (2023), 126904 | DOI | MR | Zbl

[17] Koyunbakan, H., “Reconstruction of Potential in Discrete Sturm–Liouville Problem”, Qualitative Theory of Dynamical Systems, 21 (2022), 13 | DOI | MR | Zbl

[18] Karahan, D. and Mamedov, K. R., “On a $q$-Boundary Value Problem with Discontinuity Conditions”, Bulletin of the South Ural State University, Ser. Mathematics. Mechanics. Physics, 13:4 (2021), 5–12 | DOI | Zbl

[19] Karahan, D., “On a $q$-Analogue of the Sturm–Liouville Operator with Discontinuity Conditions”, Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, 26:3 (2022), 407–418 | DOI | MR | Zbl