Investigation of the spectrum of singular Sturm–Liouville operators on unbounded time scales

Hilger, S.: Analysis on measure chains-a unified approach to continuous and discrete calculus. Results Math. 18(1–2), 18–56 (1990)

Jones, M.A., Song, B., Thomas, D.M.: Controlling wound healing through debridement. Math. Comput. Model. 40(9–10), 1057–1064 (2004)

Spedding, V.: Taming nature’s numbers. New Sci. 179, 28–31 (2003)

Thomas, D.M., Vandemuelebroeke, L., Yamaguchi, K.: A mathematical evolution model for phytoremediation of metals, discrete and continuous dynamical systems. Ser. B 5(2), 411–422 (2005)

Allahverdiev, B.P., Eryilmaz, A., Tuna, H.: Dissipative Sturm-Liouville operators with a spectral parameter in the boundary condition on bounded time scales. Electron J. Differ. Equ. 95, 1–13 (2017)

Guseinov, GSh: An expansion theorem for a Sturm–Liouville operator on semi-unbounded time scales. Adv. Dyn. Syst. Appl. 3, 147–160 (2008)

Allakhverdiev, B.P.: Extensions of symmetric singular second-order dynamic operators on time scales. Filomat 30(6), 1475–1484 (2016)

Allahverdiev, B.P.: Non-self-adjoint singular second-order dynamic operators on time scale. Math. Meth. Appl. Sci. 42, 229–236 (2019)

Zemánek, P.: Krein-von Neumann and Friedrichs extensions for second order operators on time scales. Int. J. Dyn. Syst. Differ. Equ. 3(1–2), 132–144 (2011)

Tuna, H.: Completeness of the root vectors of a dissipative Sturm–Liouville operators in time scales. Appl. Math. Comput. 228, 108–115 (2014)

Agarwal, R.P., Bohner, M., Wong, J.P.Y.: Sturm-Liouville eigenvalue problems on time scales. Appl. Math. Comput. 99, 2(3), 153–166 (1999)

Huseynov, A.: Weyl’s Limit Point and Limit Circle for a Dynamic Systems, Dynamical Systems and Methods, 215–225. Springer, New York (2012)

Davidson, F.A., Rynne, B.P.: Eigenfunction expansions in L2 spaces for boundary value problems on time-scales. J. Math. Anal. Appl. 335(2), 1038–1051 (2007)

Hoffacker, J.: Green’s functions and eigenvalue comparisons for a focal problem on time scales. Comput. Math. Appl. 45(6–9), 1339–1368 (2003)

Allahverdiev, B.P., Tuna, H.: Spectral analysis of singular Sturm–Liouville operators on time scales. Ann. Univ. Mariae Curie-Sklodowska Sectio A Math. 72(1), 1–11 (2018)

Weyl, H.: Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen. Math. Ann. 68(2), 220–269 (1910)

Glazman, I.M.: Direct Methods of the Qualitative Spectral Analysis of Singular Differential Operators. Israel Program of Scientific Translations, Jeruzalem (1965)

Berkowitz, J.: On the discreteness of spectra of singular Sturm–Liouville problems. Commun. Pure Appl. Math. 12, 523–542 (1959)

Friedrics, K.: Criteria for the Discrete Character of the Spectra of Ordinary Differential Equations. Courant Anniversary Volume. Interscience, New York (1948)

Friedrics, K.: Criteria for discrete spectra. Commun. Pure. Appl. Math. 3, 439–449 (1950)

Hinton, D.B., Lewis, R.T.: Discrete spectra critaria for singular differential operators with middle terms. Math. Proc. Cambridge Philos. Soc. 77, 337–347 (1975)

Ismagilov, R.S.: Conditions for semiboundedness and discreteness of the spectrum for one-dimensional differential equations (Russian). Dokl. Akad. Nauk SSSR 140, 33–36 (1961)

Molchanov, A.M.: Conditions for the discreteness of the spectrum of self-adjoint second-order differential equations (Russian). Trudy Moskov. Mat. Obs. 2, 169–200 (1953)

Allahverdiev, B.P., Tuna, H.: Qualitative spectral analysis of singular $$q-$$Sturm–Liouville operators. Bull. Malays. Math. Sci. Soc. (2019). https://doi.org/10.1007/s40840-019-00747-3

Rollins, L.W.: Criteria for discrete spectrum of singular self-adjoint differential perators. Proc. Am. Math. Soc. 34, 195–200 (1972)

Bohner, M., Peterson, A.: Dynamic Equations on Time Scales. Birkhäuser, Boston (2001)

Bohner, M., Peterson, A. (eds.): Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston (2003)

Lakshmikantham, V., Sivasundaram, S., Kaymakcalan, B.: Dynamic Systems on Measure Chains. Kluwer Academic Publishers, Dordrecht (1996)

Atici, M.F., Guseinov, G.S.: On Green’s functions and positive solutions for boundary value problems on time scales. J. Comput. Appl. Math. 141(1–2), 75–99 (2002)

Agarwal, R.P., Bohner, M., Li, W.-T.: Nonoscillation and Oscillation Theory for Functional Differential Equations, vol. 267 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York (2004)

Guseinov, GSh: Self-adjoint boundary value problems on time scales and symmetric Green’s functions. Turkish J. Math. 29(4), 365–380 (2005)

Anderson, D.R., Guseinov, GSh, Hoffacker, J.: Higher-order self-adjoint boundary-value problems on time scales. J. Comput. Appl. Math. 194(2), 309–342 (2006)

Rynne, B.P.: $$L^{2}$$ spaces and boundary value problems on time-scales. J. Math. Anal. Appl. 328, 1217–1236 (2007)

Naimark, M. A.: Linear Differential Operators, 2nd edn.,1968, Nauka, Moscow, English transl. of 1st. edn., 1, 2, New York (1969)

Kreyszig, E.: Introductory Functional Analysis with Applications. Wiley, New York (1989)

Dunford, N., Schwartz, J.T.: Linear Operators, Part II: Spectral Theory. Interscience, New York (1963)

Titchmarsh, E.C.: Eigenfunction Expansions Associated with Second-Order Differential Equations Part I. Second Edition Clarendon Press, Oxford (1962)