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Comparison and oscillation theorems for singular Sturm-Liouville operators

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Języki publikacji
EN
Abstrakty
EN
We prove analogues of the classical Sturm comparison and oscillation theorems for Sturm-Liouville operators on a finite interval with real-valued distributional potentials.
Rocznik
Strony
97--113
Opis fizyczny
Bibliogr. 29 poz.
Twórcy
autor
  • University of Rzeszów Institute of Mathematics al. Rejtana 16A 35-959 Rzeszów, Poland
autor
  • Institute for Applied Problems of Mechanics and Mathematics 3b Naukova st., 79601 Lviv, Ukraine
  • University of Rzeszów Institute of Mathematics al. Rejtana 16A 35-959 Rzeszów, Poland
Bibliografia
  • [1] A. Alexiewicz, W. Orlicz, On a theorem of C. Carathéodory, Ann. Polon. Math. 1 (1955), 414–417.
  • [2] F.V. Atkinson, W.N. Everitt, A. Zettl, Regularization of a Sturm-Liouville problem with an interior singularity using quasi-derivatives, Diff. Integr. Equat. 1 (1988) 2, 213–221.
  • [3] J. Ben Amara, Sturm theory for the equation of vibrating beam, J. Math. Anal. Appl. 349 (2009) 1, 1–9.
  • [4] G. Berkolaiko, A lower bound for nodal count on discrete and metric graphs, Commun. Math. Phys. 278 (2008), 803–819.
  • [5] E.A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.
  • [6] J. Eckhardt, F. Gesztesy, R. Nichols, G. Teschl, Weyl-Titchmarsh theory for Sturm-Liouville operators with distributional potentials, Opuscula Math. 33 (2013) 3, 467–563.
  • [7] A.F. Filippov, Differential Equations with Discontinuous Righthand Sides, Nauka Publ., Moscow, 1985 [in Russian]; Engl. transl. in: Kluwer Academic Publishers, Dordrecht, 1988.
  • [8] A.S. Goriunov, V.A. Mikhailets, Regularization of singular Sturm-Liouville equations, Methods Funct. Anal. Topology 16 (2010) 2, 120–130.
  • [9] A.S. Goriunov, V.A. Mikhailets, Regularization of binomial differential equations with singular coefficients, Ukrainian Math. J. 63 (2011) 9, 1190–1205.
  • [10] P. Hartman, Ordinary Differential Equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964.
  • [11] D. Hinton, Sturm’s 1836 oscillation results. Evolution of the theory, [in:] W. O. Amrein, A. M. Hintz, D. B. Hinton (eds), Sturm-Liouville Theory: Past and Present, Birkhäuser Verlag, Basel, 2005; pp. 1–27.
  • [12] M. Homa, R. Hryniv, Oscillation properties of singular quantum trees (submitted).
  • [13] A. Kaminski, S. Mincheva-Kaminska, Equivalence of the Mikusinski-Shiraishi-Itano products in S0 for various classes of delta-sequences, Integral Transforms Spec. Funct. 20 (2009) 3–4, 207–214.
  • [14] T. Kappeler, P. Perry, M. Shubin, P. Topalov, The Miura map on the line, Int. Math. Res. Not. 2005 (2005) 50, 3091–3133.
  • [15] Y. Katznelson, An Introduction to Harmonic Analysis, Third edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2004.
  • [16] M. Nowaczyk, Inverse Problems for Graph Laplacians, PhD thesis, Lund, Sweden, 2007.
  • [17] M. Oberguggenberger, Multiplication of Distributions and Applications to Partial Differential Equations, Longman, J. Wiley, Essex, New York, 1992.
  • [18] Yu.V. Pokornyi, V.J. Pryadiev, A. Al’-Obied, On the oscillation of the spectrum of a boundary value problem on a graph, Matem. Zametki 60 (1996) 3, 468–470 [in Russian]; Engl. transl. in: Mathem. Notes, 60 (1996) 3, 351–353.
  • [19] Yu. V. Pokornyi, M.B. Zvereva, A.S. Ishchenko, S.A. Shabrov, On an irregular extension of the oscillation theory of the Sturm-Liouville spectral problem, Matem. Zametki 82 (2007) 4, 578–582 [in Russian]; Engl. transl. in: Mathem. Notes, 82 (2007) 3–4, 518–521.
  • [20] W.T. Reid, Ordinary Differential Equations, John Wiley & Sons, Inc., New York-London-Sydney, 1971.
  • [21] C. Sturm, Mémoire sur les Équations différentielles linéaires du second ordre, J. Math. Pures Appl. 1 (1836), 106–186.
  • [22] C. Sturm, Mémoire sur une classe d’Équations à différences partielles, J. Math. Pures Appl. 1 (1836), 373–444.
  • [23] A.M. Savchuk, A.A. Shkalikov, Sturm-Liouville operators with singular potentials, Matem. Zametki 66 (1999) 6, 897–912 [in Russian]; Engl. transl. in: Math. Notes 66 (1999) 5–6, 741–753.
  • [24] A.M. Savchuk, A.A. Shkalikov, The Sturm-Liouville operators with distributional potential, Trudy Mosk. Matem Ob-va 64 (2003) 159–212 [in Russian]; Engl. transl. in: Trans. Moscow Math. Soc. 2003 (2003), 143–192.
  • [25] Ph. Schapotschnikow, Eigenvalue and nodal properties on quantum graph trees, Waves Random Complex Media 16 (2006) 3, 167–178.
  • [26] B. Simon, Sturm oscillation and comparison theorems, [in:] W.O. Amrein, A.M. Hintz, D.B. Hinton (eds), Sturm-Liouville Theory: Past and Present, Birkhäuser Verlag, Basel, 2005; pp. 29–43.
  • [27] A.A. Shkalikov, J. Ben Amara, Oscillation theorems for Sturm-Liouville problems with distribution potentials, Vestnik Moskov. Univ. Ser. I Mat. Mekh. (2009) 3, 43–49 [In Russian]; Engl. transl. in: Moscow Univ. Math. Bull. 64 (2009) 3, 132–137.
  • [28] G. Teschl, Oscillation theory and renormalized oscillation theory for Jacobi operators, J. Differential Equations 129 (1996) 2, 532–558.
  • [29] A.A. Vladimirov, On the oscillation theory of the Sturm-Liouville problem with singular coefficients, Zh. Vychisl. Mat. Mat. Fiz. 49 (2009) 9, 1609–1621 [in Russian]; Engl. transl. in: Comput. Math. Math. Phys. 49 (2009) 9, 1535–1546.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-f6498e8e-cb2e-4462-bc3c-9dc01f244731
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