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# Article

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Keywords:
Self-adjoint equation; reciprocal equation; property BD; principal solution; minimal differential operator.Supported by the Grant No. 201/93/0452 of the Czech Grant Agency
Summary:
Let $L(y)=y^{(n)}+q_{n-1}(t)y^{(n-1)}+\dots +q_0(t)y,\,t\in [a,b)$, be an $n$-th order differential operator, $L^*$ be its adjoint and $p,w$ be positive functions. It is proved that the self-adjoint equation $L^*\bigl (p(t)L(y)\bigr ) =w(t)y$ is nonoscillatory at $b$ if and only if the equation $L\bigl (w^{-1}(t)L^*(y)\bigr )=p^{-1}(t)y$ is nonoscillatory at $b$. Using this result a new necessary condition for property BD of the self-adjoint differential operators with middle terms is obtained.
References:
[1] Ahlbrandt, C. D.: Principal and antiprincipal solutions of selfadjoint diferential systems and their reciprocals. Rocky Mountain J. Math. 2 (1972), 169–189. MR 0296388
[2] Ahlbrandt, C. D.: Equivalent boundary value problems for self-adjoint differential systems. J. Diff. Equations 9 (1971), 420–435. MR 0284636 | Zbl 0218.34020
[3] Ahlbrandt, C. D., Hinton, D. B., Lewis, R. T.: The effect of variable change on oscillation and disconjugacy criteria with applications to spectral theory and asymptotic theory. J. Math. Anal. Appl. 81 (1981), 234–277. MR 0618771
[4] Ahlbrandt, C. D., Hinton, D. B., Lewis, R. T.: Necessary and sufficient conditions for the discreteness of the spectrum of certain singular differential operators. Canad J. Math. 33 (1981), 229–246. MR 0608867
[5] Coppel, W. A.: Disconjugacy. Lectures Notes in Math., No. 220, Springer Verlag, Berlin-Heidelberg 1971. MR 0460785 | Zbl 0224.34003
[6] Došlý, O.: On transformation of self-adjoint linear diferential systems and their reciprocals. Annal. Pol. Math. 50 (1990), 223–234.
[7] Došlý, O.: Oscillation criteria and the discreteness of the spectrum of self-adjoint, even order, differential operators. Proc. Roy. Soc. Edinburgh 119A (1991), 219–232.
[8] Došlý, O.: Transformations of linear Hamiltonian systems preserving oscillatory behaviour. Arch. Math. 27 (1991), 211–219. MR 1189218
[9] Došlý, O.: Principal solutions and transformations of linear Hamiltonian systems. Arch. Math. 28 (1992), 113–120. MR 1201872
[10] Došlý, O., Osička, J.: Kneser type oscillation criteria for self-adjoint differential equations. Georgian Math. J. 2 (1995), 241–258. MR 1334880
[11] Dunford, N., Schwartz, J. T.: Linear Operators II, Spectral Theory. Interscience, New York 1982.
[12] Evans, W. D., Kwong, M. K., Zettl, A.: Lower bounds for spectrum of ordinary differential operators. J. Diff. Equations 48 (1983), 123–155. MR 0692847
[13] Glazman, I. M.: Direct Methods of Qualitative Analysis of Singular Differential Operators. Jerusalem 1965.
[14] Hinton, D. B., Lewis, R. T.: Discrete spectra criteria for singular differential operators with middle terms. Math. Proc. Cambridge Philos. Soc. 77 (1975), 337–347. MR 0367358
[15] Lewis, R. T.: The discreteness of the spectrum of self-adjoint, even order, differential operators. Proc. Amer. Mat. Soc. 42 (1974), 480–482. MR 0330608
[16] Müller-Pfeiffer, E.: Spectral Theory of Ordinary Differential Operators. Chelsea, 1981. MR 0606197
[17] Naimark, M. A.: Linear Differential Operators. Part II, Ungar, New York, 1968. Zbl 0227.34020
[18] Reid, W. T.: Sturmian Theory for Ordinary Differential Equations. Springer Verlag, New York 1980. MR 0606199 | Zbl 0459.34001
[19] Rasmussen, C. H.: Oscillation and asymptotic behaviour of systems of ordinary linear differential equations. Trans. Amer. Math. Soc. 256 (1979), 1–48. MR 0546906

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