Existence theorem for a fractal Sturm-Liouville problem
Автор: Allahverdiev B.P., Tuna H.
Журнал: Владикавказский математический журнал @vmj-ru
Статья в выпуске: 1 т.26, 2024 года.
Бесплатный доступ
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 Fα-calculus defined on fractal subsets of real numbers are given, i.e., Fα-continuity, Fα-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.
Fractal sturm-liouville problems, existence problems
Короткий адрес: https://sciup.org/143182365
IDR: 143182365 | DOI: 10.46698/h4206-1961-4981-h
Список литературы Existence theorem for a fractal Sturm-Liouville problem
- Parvate, A. and Gangal, A. D. Calculus on Fractal Subsets of Real Line - I: Formulation, Fractals, 2009, vol. 17, no. 1, pp. 53-81. DOI: 10.1142/S0218348X09004181.
- Cetinkaya, F. A. and Golmankaneh, A. K. General Characteristics of a Fractal Sturm-Liouville Problem, Turkish Journal of Mathematics, 2021, vol. 45, no. 4, pp. 1835-1846. DOI: 10.3906/mat-2101-38.
- Golmankhaneh, A. K. Fractal Calculus and its Applications: Fα-Calculus, World Scientific Publ. Co. Pte. Ltd., 2022. DOI: 10.1142/12988.
- Golmankhaneh, A. K. and Tunc, C. Stochastic Differential Equations on Fractal Sets, Stochastics, 2020, vol. 92, no. 8, pp. 1244-1260. DOI: 10.1080/17442508.2019.1697268.
- Golmankhaneh, A. K. and Tunc, C. Sumudu Transform in Fractal Calculus, Applied Mathematics and Computation, 2019, vol. 350, pp. 386-401. DOI: 10.1016/j.amc.2019.01.
- Golmankhaneh, A. K. and Tunc, C. On the Lipschitz Condition in the Fractal Calculus, Chaos, Solitons & Fractals, 2017, vol. 95, pp. 140-147. DOI: 10.1016/j.chaos.2016.12.001.
- Parvate, A. and Gangal, A. D. Calculus on Fractal Subsets of Real Line - I: Conjugacy with Ordinary Calculus, Fractals, 2011, vol. 19, no. 3, pp. 271-290. DOI: 10.1142/S0218348X11005440.
- Kolwankar, K. M. and Gangal, A. D. Fractional Differentiability of Nowhere Differentiable Functions and Dimensions, 1996, Chaos, vol. 6, no. 4, pp. 505-513. DOI: 10.1063/1.166197.
- Kolwankar, K. M. and Gangal, A. D. Holder Exponents of Irregular Signals and Local Fractional Derivatives, Pramana - Journal of Physics, 1997, vol. 48, pp. 49-68. DOI: 10.1007/BF02845622.
- Kolwankar, K. M. and Gangal, A. D. Local Fractional Fokker-Planck Equation, Physical Review Letters, 1998, vol. 80, no. 2, pp. 214-217. DOI: 10.1103/PhysRevLett.80.214.
- Kolwankar, K. M. and Gangal, A. D. Local Fractional Derivatives and Fractal Functions of Several Variables, Mathematical Physics, 1998, arXiv:physics/9801010. DOI: 10.48550/arXiv.physics/9801010.
- Aydemir, K. and Mukhtarov, O. Sh. A New Type Sturm-Liouville Problem with an Abstract Linear Operator Contained in the Equation, Quaestiones Mathematicae, 2022, vol. 45, no. 12, pp. 1931-1948. DOI: 10.2989/16073606.2021.1979681.
- Aydemir, K. and Mukhtarov, O. Sh. Qualitative Analysis of Eigenvalues and Eigenfunctions of one Boundary Value-Transmission Problem, Boundary Value Problems, 2016, article no. 82. DOI: 10.1186/s13661-016-0589-4.
- Levitan, B. M. and Sargsjan, I. S. Sturm-Liouville and Dirac Operators, Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers Group, Dordrecht, 1991.
- Olgar, H. and Mukhtarov, O. Sh. Weak Eigenfunctions of Two-Interval Sturm-Liouville Problems Together with Interaction Conditions, Journal of Mathematical Physics, 2017, vol. 58, no. 4, 042201. DOI: 10.1063/1.4979615.
- Ozkan, A. S. and Adalar, I. Inverse Nodal Problems for Sturm-Liouville Equation with Nonlocal Boundary Conditions, Journal of Mathematical Analysis and Applications, 2023, vol. 520, no. 1, 126904. DOI: 10.1016/j.jmaa.2022.126904.
- Koyunbakan, H. Reconstruction of Potential in Discrete Sturm-Liouville Problem, Qualitative Theory of Dynamical Systems, 2022, vol. 21, article no. 13. DOI: 10.1007/s12346-021-00548-9.
- 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, 2021, vol. 13, no. 4, pp. 5-12. DOI: 10.14529/mmph210401.
- 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, 2022, vol. 26, no. 3, pp. 407-418. DOI: 10.14498/vsgtu1934.
