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General boundary eigenvalue problems for systems of differential equations with multiple roots of the characteristic equation

Published online by Cambridge University Press:  14 November 2011

Gerhard Freiling
Gerhard Freiling
Affiliation:
Fachbereich Mathematik, Universität-GH Duisburg, D-4100 Duisburg 1, West Germany
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Synopsis

We consider a class of non-self adjoint multipoint eigenvalue problems. Using necessary conditions for the regularity of these problems, we obtain a theorem on the expansion of certain functions into a series of eigen- and associated functions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 108 , Issue 1-2 , 1988 , pp. 45 - 56
Copyright
Copyright © Royal Society of Edinburgh 1988

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References

1Boese, G.Einschluβ sämtlicher Nullstellen von Exponentialpolynomen mit konstanten Koeffizienten. Z. Angew. Math. Mech. 62 (1982), 317–320.Google Scholar
2Birkhoff, G. D. and Langer, R. E.. The boundary problems and developments associated with a system of ordinary linear differential equations of the first order. Proc. Amer. Acad. Arts and Sciences 58 (1923), 51–128.Google Scholar
3Cole, R. H.. General boundary conditions for an ordinary linear differential system. Trans. Amer. Math. Soc. 111 (1964), 521–550.Google Scholar
4Eberhard, W. and Freiling, G.. Stone-regulare Eigenwertprobleme. Math. Z. 160 (1978), 139–161.Google Scholar
5Freiling, G.. Regulare Eigenwertprobleme mit Mehrpunkt-Integral-Randbedingungen. Math. Z. 171 (1980), 113–131.CrossRefGoogle Scholar
6Freiling, G.. Boundary value problems associated with multiple roots of the characteristic equation. Resultate Math. 11 (1987), 44–62.CrossRefGoogle Scholar
7Friedman, B. and Mishoe, L. I.. Eigenfunction expansion associated with a nonselfad joint differential equation. Pacific J. Math. 6 (1956), 249–270.Google Scholar
8Gasymov, M. G. and Magerramov, A. M.. Investigation of a class of differential operator pencils of even order. Soviet Math. Dokl. 26 (1982), 55–58.Google Scholar
9Heisecke, G.. Rand-Eigenwertprobleme N(y) = XP(y) bei A-abhangigen Randbedingungen. Mitt. Math. Sem. Giessen 145 (1980), 1–74.Google Scholar
10Langer, R. E.. The boundary problem of an ordinary linear differential system in the complex domain. Trans. Amer. Math. Soc. 46 (1939), 151–190.Google Scholar
11Mennicken, R.. Spectral theory for certain operator polynomials. In Functional Analysis. Holomorphy and Approximation Theory II, ed. Zapata, G., pp. 203–243. North-Holland Math. Studies 86 (Notas de Mathematics 92) (Amsterdam: North-Holland, 1984).CrossRefGoogle Scholar
12Mennicken, R. and Möller, M.. Root functions, eigenvectors, associated vectors and the inverse of a holomorphic function. Arch. Math. (Basel) 42 (1984), 455–463.Google Scholar
13Mennicken, R. and Moller, M.. Root functions of boundary eigenvalue operator functions. Integral Equations Operator Theory 9 (1986), 237–265.CrossRefGoogle Scholar
14Mennicken, R. and Moller, M.. Boundary eigenvalue problems. In Operator Theory and Systems. Proceedings Workshop Amsterdam, eds. Gohberg, I., Bart, H., & Kasshoek, M. A., pp. 279–331 (Stuttgart: Birkhauser, 1986).Google Scholar
15Meyer, G.. Inversion von Exponentialpolynomen in der Na'he von transzendent kritischen Stellen. Habilitationsschrift, Universitat Wiirzburg, 1981.Google Scholar
16Pechentsov, A. S.. Asymptotic expansions of linear differential equations containing a parameter. Differential Equations 17 (1981), 1023–1031.Google Scholar
17Pechentsov, A. S.. Boundary-value problems for differential equations with a parameter, having characteristic equations with multiple roots. Differential Equations 20 (1984), 206–215.Google Scholar
18Shkalikov, A. A.. Boundary value problems for ordinary differential equations with a parameter in the boundary conditions. J. Soviet Math. 33 (1986) 1311–1342; Trudy Sem. imeni Petrovskogo 9 (1983) 190-229.Google Scholar
19Steinmetz, N.. Zur Wertverteilung von Exponentialpolynomen. Manuscripta Math. 26 (1978), 155–167.Google Scholar
20Stone, M. H.. A comparison of the series of Fourier and Birkhoff. Trans. Amer. Math. Soc. 28 (1926), 695–761.Google Scholar
21Tamarkin, J. D.. Some general problems of the theory of ordinary linear differential equations and expansions of an arbitrary function in series of fundamental functions (Russian). Thesis, Petrograd, 1917.Google Scholar
22Tamarkin, J. D.. Some general problems of the theory of ordinary linear differential equations and expansions of an arbitrary function in series of fundamental functions. Math. Z. 27 (1927), 1–54.Google Scholar
23Tijdeman, R.. On the number of zeros of general exponential polynomials. Indag. Math. 33 (1971), 1–7.Google Scholar
24Trjitzinsky, W. J.. Theory of linear differential equations containing a parameter. Ada. Math. 67 (1936).Google Scholar
25Turrittin, H. L.. Asymptotic solutions of certain ordinary differential equations associated with multiple roots of the characteristic equation. Amer. J. Math. 58 (1936), 364–376.Google Scholar
26Turrittin, H. L.. Asymptotic distribution of zeroes for certain exponential sums. Amer. J. Math. 66 (1944), 199–228.Google Scholar
27Turrittin, H. L.. Asymptotic expansions of solutions of systems of ordinary linear differential equations containing a parameter. In The theory of nonlinear oscillations, Vol. 2, pp. 81–116 (Princeton: Princeton University Press, 1952).Google Scholar
28Vagabov, A. I.. On equiconvergence in expansions in trogonometric Fourier series and in principal functions of ordinary differential operators. Math. USSR Izv. 24 (1985), 567–582.CrossRefGoogle Scholar
29Voorhoeve, M.. Zeros of exponential polynomials. Dissertation, University of Leiden, 1976.CrossRefGoogle Scholar