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Left-Definite Theory for Fractal Sturm--Liouville Equations

Year 2025, Volume: 13 Issue: 2, 241 - 249, 31.10.2025

Abstract

In this paper, the left-definite theory of fractal Sturm--Liouville problems in the regular case is studied.

References

  • [1] B. P. Allahverdiev and H. Tuna, Existence theorem for a fractal Sturm–Liouville problem, Vladikavkaz Math. J., 26 (1) (2024), 27-35.
  • [2] R. Amirov, A. Ergun and S. Durak, Half-inverse problems for the quadratic pencil of the Sturm–Liouville equations with impulse, Numer. Meth. Partial Differ. Equat., 37 (2020), 915-924.
  • [3] R. K. Amirov and A. S. Ozkan, Discontinuous Sturm–Liouville problems with eigenvalue dependent boundary condition, Math. Phys. Anal. Geom., 17 (2014), 483-491.
  • [4] K. Aydemir, H. Ol˘gar and O. Sh. Mukhtarov, The principal eigenvalue and the principal eigenfunction of a boundary-value-transmission problem, Turkish J. Math. Comput. Sci., 11 (2) (2019), 97-100.
  • [5] K. Aydemir, H. Olgar, O. Sh. Mukhtarov and F. Muhtarov, Differential operator equations with interface conditions in modified direct sum spaces, Filomat, 32 (3) (2018), 921-931.
  • [6] S. Cebesoy, E. Bairamov and Y. Aygar, Scattering problems of impulsive Schr¨odinger equations with matrix coefficients, Ric. Mat., 72 (1) (2023), 399-415.
  • [7] C. Bennewitz and W. N. Everitt, On the second-order left-definite boundary value problems, Lecture Notes in Mathematics, vol. 1032, Springer, Heidelberg, 31-67, 1983.
  • [8] W. N. Everitt, On certain regular ordinary differential expressions and related differential operators, Editor(s): Ian W. Knowles, Roger T. Lewis, North-Holland Mathematics Studies, North-Holland, vol 55, 115-167, 1981
  • [9] F. A. C¸ etinkaya and A. K. Golmankaneh, General characteristics of a fractal Sturm–Liouville problem, Turkish J. Math., 45 (4) (2021), 1835-1846.
  • [10] A. K. Golmankhaneh, Fractal Calculus and its Applications, World Scientific, 2022.
  • [11] A. K. Golmankhaneh and C. Tunc¸, Stochastic differential equations on fractal sets, Stochastics, 92 (8) (2020), 1244-1260.
  • [12] A. K. Golmankhaneh and C. Tunc¸, Sumudu transform in fractal calculus, Appl. Math. Comput., 350 (2019), 386-401.
  • [13] A. K. Golmankhaneh, Z. Vidovi´c, H. Tuna and B. P. Allahverdiev, Fractal Sturm–Liouville Theory. Fractal and Fractional, ; 9 (5) (2025), 268.
  • [14] A. K. Golmankhaneh, P.E.T. Jørgensen, C. Serpa and K. Welch, About Sobolev spaces on fractals: fractal gradians and Laplacians, Aequat. Math., 99 (2025), 465-490.
  • [15] D. Karahan and Kh. R. Mamedov, On a q-boundary value problem with discontinuity conditions, Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz., 13 (4) (2021), 5-12.
  • [16] D. Karahan and Kh. R. Mamedov, On a q-analogue of the Sturm–Liouville operator with discontinuity conditions, Vestn. Samar. Gos. Tekh. Univ., Ser. Fiz.-Mat. Nauk., 26 (3) (2022), 407-418.
  • [17] K. M. Kolwankar and A. D. Gangal, Fractional differentiability of nowhere differentiable functions and dimensions, Chaos, 6 (4) (1996), 505-513.
  • [18] K. M. Kolwankar and A. D. Gangal, Local fractional derivatives and fractal functions of several variables, arXiv preprint Physics, 9801010, (1998), 1-4.
  • [19] Q. Kong, H. Wu and A. Zettl, Left-definite Sturm–Liouville problems, J. Different. Equat., 177 (2001), 1-26.
  • [20] A. M. Krall, Left definite theory for second order differential operators with mixed boundary conditions, J. Different. Equat., 118 (1) (1995), 153-165.
  • [21] A. M. Krall, Hilbert space, boundary value problems and orthogonal polynomials, Operator Theory: Advances and Applications, Birkhauser Verlag, Basel, 2002.
  • [22] T. K¨opr¨ubas¸ı and Y. Aygar K¨uc¸ ¨ukevcilio˘glu, Discrete impulsive Sturm–Liouville equation with hyperbolic eigenparameter, Turkish J. Math., 46 (1) (2022), 387-396.
  • [23] O. Sh. Mukhtarov, K. Aydemir, Basis properties of the eigenfunctions of two-interval Sturm–Liouville problems, Analys. Math. Phys., 9 (2019), 1363–1382.
  • [24] A. Parvate and A. D. Gangal, Calculus on fractal subsets of real line I: formulation, Fractals, 17 (1) (2009), 53-81.
  • [25] A. Parvate and A. D. Gangal, Calculus on fractal subsets of real line II: conjugacy with ordinary calculus, Fractals, 19 (3) (2011), 271-290.
  • [26] A° . Pleijel, Some remarks about the limit point and limit circle theory, Ark. Mat., 7 (1969), 543-550.
  • [27] A° . Pleijel, Complementary remarks about the limit point and limit circle theory, Ark. Mat., 8 (1969),45-47.
  • [28] M. Uc, Spectral and algebraic analysis of the fractal Volterra operator on Ck(F), Chaos, Solitons & Fractals, 200 (3), 117061 (2025), 1-18.

Year 2025, Volume: 13 Issue: 2, 241 - 249, 31.10.2025

Abstract

References

  • [1] B. P. Allahverdiev and H. Tuna, Existence theorem for a fractal Sturm–Liouville problem, Vladikavkaz Math. J., 26 (1) (2024), 27-35.
  • [2] R. Amirov, A. Ergun and S. Durak, Half-inverse problems for the quadratic pencil of the Sturm–Liouville equations with impulse, Numer. Meth. Partial Differ. Equat., 37 (2020), 915-924.
  • [3] R. K. Amirov and A. S. Ozkan, Discontinuous Sturm–Liouville problems with eigenvalue dependent boundary condition, Math. Phys. Anal. Geom., 17 (2014), 483-491.
  • [4] K. Aydemir, H. Ol˘gar and O. Sh. Mukhtarov, The principal eigenvalue and the principal eigenfunction of a boundary-value-transmission problem, Turkish J. Math. Comput. Sci., 11 (2) (2019), 97-100.
  • [5] K. Aydemir, H. Olgar, O. Sh. Mukhtarov and F. Muhtarov, Differential operator equations with interface conditions in modified direct sum spaces, Filomat, 32 (3) (2018), 921-931.
  • [6] S. Cebesoy, E. Bairamov and Y. Aygar, Scattering problems of impulsive Schr¨odinger equations with matrix coefficients, Ric. Mat., 72 (1) (2023), 399-415.
  • [7] C. Bennewitz and W. N. Everitt, On the second-order left-definite boundary value problems, Lecture Notes in Mathematics, vol. 1032, Springer, Heidelberg, 31-67, 1983.
  • [8] W. N. Everitt, On certain regular ordinary differential expressions and related differential operators, Editor(s): Ian W. Knowles, Roger T. Lewis, North-Holland Mathematics Studies, North-Holland, vol 55, 115-167, 1981
  • [9] F. A. C¸ etinkaya and A. K. Golmankaneh, General characteristics of a fractal Sturm–Liouville problem, Turkish J. Math., 45 (4) (2021), 1835-1846.
  • [10] A. K. Golmankhaneh, Fractal Calculus and its Applications, World Scientific, 2022.
  • [11] A. K. Golmankhaneh and C. Tunc¸, Stochastic differential equations on fractal sets, Stochastics, 92 (8) (2020), 1244-1260.
  • [12] A. K. Golmankhaneh and C. Tunc¸, Sumudu transform in fractal calculus, Appl. Math. Comput., 350 (2019), 386-401.
  • [13] A. K. Golmankhaneh, Z. Vidovi´c, H. Tuna and B. P. Allahverdiev, Fractal Sturm–Liouville Theory. Fractal and Fractional, ; 9 (5) (2025), 268.
  • [14] A. K. Golmankhaneh, P.E.T. Jørgensen, C. Serpa and K. Welch, About Sobolev spaces on fractals: fractal gradians and Laplacians, Aequat. Math., 99 (2025), 465-490.
  • [15] D. Karahan and Kh. R. Mamedov, On a q-boundary value problem with discontinuity conditions, Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz., 13 (4) (2021), 5-12.
  • [16] D. Karahan and Kh. R. Mamedov, On a q-analogue of the Sturm–Liouville operator with discontinuity conditions, Vestn. Samar. Gos. Tekh. Univ., Ser. Fiz.-Mat. Nauk., 26 (3) (2022), 407-418.
  • [17] K. M. Kolwankar and A. D. Gangal, Fractional differentiability of nowhere differentiable functions and dimensions, Chaos, 6 (4) (1996), 505-513.
  • [18] K. M. Kolwankar and A. D. Gangal, Local fractional derivatives and fractal functions of several variables, arXiv preprint Physics, 9801010, (1998), 1-4.
  • [19] Q. Kong, H. Wu and A. Zettl, Left-definite Sturm–Liouville problems, J. Different. Equat., 177 (2001), 1-26.
  • [20] A. M. Krall, Left definite theory for second order differential operators with mixed boundary conditions, J. Different. Equat., 118 (1) (1995), 153-165.
  • [21] A. M. Krall, Hilbert space, boundary value problems and orthogonal polynomials, Operator Theory: Advances and Applications, Birkhauser Verlag, Basel, 2002.
  • [22] T. K¨opr¨ubas¸ı and Y. Aygar K¨uc¸ ¨ukevcilio˘glu, Discrete impulsive Sturm–Liouville equation with hyperbolic eigenparameter, Turkish J. Math., 46 (1) (2022), 387-396.
  • [23] O. Sh. Mukhtarov, K. Aydemir, Basis properties of the eigenfunctions of two-interval Sturm–Liouville problems, Analys. Math. Phys., 9 (2019), 1363–1382.
  • [24] A. Parvate and A. D. Gangal, Calculus on fractal subsets of real line I: formulation, Fractals, 17 (1) (2009), 53-81.
  • [25] A. Parvate and A. D. Gangal, Calculus on fractal subsets of real line II: conjugacy with ordinary calculus, Fractals, 19 (3) (2011), 271-290.
  • [26] A° . Pleijel, Some remarks about the limit point and limit circle theory, Ark. Mat., 7 (1969), 543-550.
  • [27] A° . Pleijel, Complementary remarks about the limit point and limit circle theory, Ark. Mat., 8 (1969),45-47.
  • [28] M. Uc, Spectral and algebraic analysis of the fractal Volterra operator on Ck(F), Chaos, Solitons & Fractals, 200 (3), 117061 (2025), 1-18.
There are 28 citations in total.

Details

Primary Language English
Subjects Mathematical Methods and Special Functions
Journal Section Research Article
Authors

Bilender P. Allahverdiev

Hüseyin Tuna

Publication Date October 31, 2025
Submission Date May 9, 2025
Acceptance Date September 11, 2025
Published in Issue Year 2025 Volume: 13 Issue: 2

Cite

APA Allahverdiev, B. P., & Tuna, H. (2025). Left-Definite Theory for Fractal Sturm--Liouville Equations. Konuralp Journal of Mathematics, 13(2), 241-249.
AMA Allahverdiev BP, Tuna H. Left-Definite Theory for Fractal Sturm--Liouville Equations. Konuralp J. Math. October 2025;13(2):241-249.
Chicago Allahverdiev, Bilender P., and Hüseyin Tuna. “Left-Definite Theory for Fractal Sturm--Liouville Equations”. Konuralp Journal of Mathematics 13, no. 2 (October 2025): 241-49.
EndNote Allahverdiev BP, Tuna H (October 1, 2025) Left-Definite Theory for Fractal Sturm--Liouville Equations. Konuralp Journal of Mathematics 13 2 241–249.
IEEE B. P. Allahverdiev and H. Tuna, “Left-Definite Theory for Fractal Sturm--Liouville Equations”, Konuralp J. Math., vol. 13, no. 2, pp. 241–249, 2025.
ISNAD Allahverdiev, Bilender P. - Tuna, Hüseyin. “Left-Definite Theory for Fractal Sturm--Liouville Equations”. Konuralp Journal of Mathematics 13/2 (October2025), 241-249.
JAMA Allahverdiev BP, Tuna H. Left-Definite Theory for Fractal Sturm--Liouville Equations. Konuralp J. Math. 2025;13:241–249.
MLA Allahverdiev, Bilender P. and Hüseyin Tuna. “Left-Definite Theory for Fractal Sturm--Liouville Equations”. Konuralp Journal of Mathematics, vol. 13, no. 2, 2025, pp. 241-9.
Vancouver Allahverdiev BP, Tuna H. Left-Definite Theory for Fractal Sturm--Liouville Equations. Konuralp J. Math. 2025;13(2):241-9.
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