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A CHARACTERIZATION OF SELF-ADJOINT OPERATORS DETERMINED BY THE WEAK FORMULATION OF SECOND-ORDER SINGULAR DIFFERENTIAL EXPRESSIONS

Published online by Cambridge University Press:  01 May 2009

MOHAMED EL-GEBEILY  and
DONAL O'REGAN
MOHAMED EL-GEBEILY
Affiliation:
Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia e-mail: mgebeily@kfupm.edu.sa
DONAL O'REGAN
Affiliation:
Department of Mathematics, National University of Ireland Galway, Ireland e-mail: donal.oregan@nuigalway.ie
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Abstract

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In this paper we describe a special class of self-adjoint operators associated with the singular self-adjoint second-order differential expression ℓ. This class is defined by the requirement that the sesquilinear form q(u, v) obtained from ℓ by integration by parts once agrees with the inner product 〈ℓu, v〉. We call this class Type I operators. The Friedrichs Extension is a special case of these operators. A complete characterization of these operators is given, for the various values of the deficiency index, in terms of their domains and the boundary conditions they satisfy (separated or coupled).

Keywords

34B1534A34
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 51 , Issue 2 , May 2009 , pp. 385 - 404
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

References

REFERENCES

1.El-Gebeily, M. A. and Furati, K. M., Real self-adjoint Sturm–Liouville problems, Appl. Anal. 83 (4) (2004), 377–387.CrossRefGoogle Scholar
2.Everitt, W. N., Giertz, M. and McLeod, J. B., On the strong and weak limit-point classification of second-order differential expressions, Proc. London Math. Soc. 29 (1974), 142–158.CrossRefGoogle Scholar
3.Kato, T., Perturbation in Theory for linear operators (Springer Verlag, Heidelberg, Berlin, 1995).CrossRefGoogle Scholar
4.Krall, A. M. and Zettl, A., Singular self-adjoint Sturm–Liouville problems, Diff. Integral Eq. 1 (4) (1988), 423–432.Google Scholar
5.Marletta, M. and Zettl, A., The Friedrichs extension of singular differential operators, JDE 160 (2001), 404–421.CrossRefGoogle Scholar
6.Naimark, M. A., Linear differential operators: Part II (Ungar, New York, 1968).Google Scholar
7.Niessen, H.-D. and Zettl, A., The Friedrichs extension of regular ordinary differential operators, Proc. Roy. Soc. Edinburgh Sect. A 114 (1990), 229–236.CrossRefGoogle Scholar
8.O'Regan, D. and El-Gebeily, M., Existence, upper and lower solutions and quasilinearization for singular differential equations, IMA J. Appl. Math. 73 (2008), 323–344.CrossRefGoogle Scholar
9.Weidmann, J., Spectral theory of ordinary differential operators, Lecture Notes in Mathematics, vol. 1258 (Springer, Heidelberg, Germany, 1987).CrossRefGoogle Scholar
10.Zettl, A., Sturm–Liouville problems, in Spectral theory and computational methods of Sturm–Liouville problems (Hinton, D. and Schaefer, P., Editors), Pure and Applied Mathematics (Marcel Dekker, New York, 1997), 1–104.Google Scholar