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