TY - JOUR T1 - Left-Definite Theory for Fractal Sturm--Liouville Equations AU - Tuna, Hüseyin AU - Allahverdiev, Bilender P. PY - 2025 DA - October Y2 - 2025 JF - Konuralp Journal of Mathematics JO - Konuralp J. Math. PB - Mehmet Zeki SARIKAYA WT - DergiPark SN - 2147-625X SP - 241 EP - 249 VL - 13 IS - 2 LA - en AB - In this paper, the left-definite theory of fractal Sturm--Liouville problems in the regular case is studied. KW - left-definiteness. KW - Fractals KW - Sturm-Liouville problems CR - [1] B. P. Allahverdiev and H. Tuna, Existence theorem for a fractal Sturm–Liouville problem, Vladikavkaz Math. J., 26 (1) (2024), 27-35. CR - [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. CR - [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. CR - [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. CR - [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. CR - [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. CR - [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. CR - [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 CR - [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. CR - [10] A. K. Golmankhaneh, Fractal Calculus and its Applications, World Scientific, 2022. CR - [11] A. K. Golmankhaneh and C. Tunc¸, Stochastic differential equations on fractal sets, Stochastics, 92 (8) (2020), 1244-1260. CR - [12] A. K. Golmankhaneh and C. Tunc¸, Sumudu transform in fractal calculus, Appl. Math. Comput., 350 (2019), 386-401. CR - [13] A. K. Golmankhaneh, Z. Vidovi´c, H. Tuna and B. P. Allahverdiev, Fractal Sturm–Liouville Theory. Fractal and Fractional, ; 9 (5) (2025), 268. CR - [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. CR - [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. CR - [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. CR - [17] K. M. Kolwankar and A. D. Gangal, Fractional differentiability of nowhere differentiable functions and dimensions, Chaos, 6 (4) (1996), 505-513. CR - [18] K. M. Kolwankar and A. D. Gangal, Local fractional derivatives and fractal functions of several variables, arXiv preprint Physics, 9801010, (1998), 1-4. CR - [19] Q. Kong, H. Wu and A. Zettl, Left-definite Sturm–Liouville problems, J. Different. Equat., 177 (2001), 1-26. CR - [20] A. M. Krall, Left definite theory for second order differential operators with mixed boundary conditions, J. Different. Equat., 118 (1) (1995), 153-165. CR - [21] A. M. Krall, Hilbert space, boundary value problems and orthogonal polynomials, Operator Theory: Advances and Applications, Birkhauser Verlag, Basel, 2002. CR - [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. CR - [23] O. Sh. Mukhtarov, K. Aydemir, Basis properties of the eigenfunctions of two-interval Sturm–Liouville problems, Analys. Math. Phys., 9 (2019), 1363–1382. CR - [24] A. Parvate and A. D. Gangal, Calculus on fractal subsets of real line I: formulation, Fractals, 17 (1) (2009), 53-81. CR - [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. CR - [26] A° . Pleijel, Some remarks about the limit point and limit circle theory, Ark. Mat., 7 (1969), 543-550. CR - [27] A° . Pleijel, Complementary remarks about the limit point and limit circle theory, Ark. Mat., 8 (1969),45-47. CR - [28] M. Uc, Spectral and algebraic analysis of the fractal Volterra operator on Ck(F), Chaos, Solitons & Fractals, 200 (3), 117061 (2025), 1-18. UR - https://dergipark.org.tr/en/pub/konuralpjournalmath/issue//1696348 L1 - https://dergipark.org.tr/en/download/article-file/4856780 ER -