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On the completeness and minimality of the derived chains of eigen- and associated functions of boundary eigenvalue problems nonlinearly dependant on the parameter

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Literature

  1. R.A. Adams. Sobolev spaces. Academic Press, New York, (1975).

    MATH  Google Scholar 

  2. H.E. Benzinger, Green’s function for ordinary differential operators. J. Differential Equations 7, 478–496 (1970).

    Article  MathSciNet  MATH  Google Scholar 

  3. G.D. Birkhoff, On the asymptotic character of the solutions of certain linear differential equations containing a parameter. Trans. Amer. Math. Soc. 9, 219–231 (1908).

    Article  MathSciNet  Google Scholar 

  4. G.D. Birkhoff, Boundary value and expansion problems of ordinary linear differential equations. Trans. Amer. Math. Soc. 9, 373–395 (1908).

    Article  MathSciNet  MATH  Google Scholar 

  5. N. Dunford and J.T. Schwartz. Linear differential operators, Part I, II, III. Interscience, New York.

  6. W. Eberhard and G. Freiling. Stone-reguläre Eigenwertprobleme. Math. Z. 160, 139–161 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  7. G. Freiling. Reguläre Eigenwertprobleme mit Mehrpunkt-Integral-Randbedingungen. Math. Z. 171, 113–131 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  8. G. Freiling. Boundary value problems associated with multiple roots of the characteristic equation. Results in Mathematics 11, 44–62 (1987).

    MathSciNet  MATH  Google Scholar 

  9. G. Freiling. General boundary eigenvalue problems for systems of differential equations with multiple roots of the characteristic equation. Proc. Royal. Soc. Edinburgh, to appear.

  10. G. Heisecke. Rand-Eigenwertprobleme N(y) = μP(y) bei λ-abhängigen Randbedingungen. Mitt. Math. Sem. Gie\en No. 145, 1–74 (1980).

    MathSciNet  Google Scholar 

  11. J. Horn. Über eine lineare Differentialgleichung zweiter Ordnung mit einem willkürlichen Parameter. Math. Annalen 52, 271–292 und 340-362 (1899).

    Article  MathSciNet  MATH  Google Scholar 

  12. M.V. Keldysh. Completeness of characteristic functions of certain classes of non-selfadjoint operators. Russ. Math. Surveys, 15-44 (1971); short version: Dokl. Akad. Nauk SSSR 77, 11-14 (1951).

  13. R.E. Langer. The boundary problem of an ordinary linear differential system in the complex domain. Trans. Amer. Math. Soc. 46, 151–190 (1939).

    MathSciNet  Google Scholar 

  14. R. Mennicken and M. Möller. Boundary eigenvalue problems. Notas de Algebra y Analysis 14, 100 pages (1986), Bahia Bianca, Argentina.

  15. R. Mennicken and M. Möler. Boundary eigenvalue problems. In: Operator Theory and Systems. Proceedings Workshop Amsterdam, ed.: I. Gohberg, H. Bart, M.A. Kaashoek; Birkhäuser, Stuttgart (1986)

    Google Scholar 

  16. M.A. Naimark. Linear differential operators, Part I and Part II. F. Ungar New York, 1967.

    MATH  Google Scholar 

  17. P. Noailion. Développements asymptotiques dans les équations differentielles linéaires à paramètre variable. Mémoires de la Soc. des Sc. de Liège, 3. Série, IX, 197 pages (1912).

  18. A.S. Pechentsov. Boundary- value problems for differential equations with a parameter, having characteristic equations with multiple root. Diff. Equ. 20, 206–215 (1984).

    MathSciNet  MATH  Google Scholar 

  19. O. Perron. Über die Abhängigkeit der Integrale linearer Differentialgleichungen mit Parameter I, II, III. Sitzungsberichte der Heidelberger Akad., Math.-Naturw. Klasse 1918 (A. 13, A. 15), 1919 (A.6).

  20. H. Poincaré. Sur les intégrales irregulières des équations linéaires. Acta Math. 8, 295–344 (1886).

    Article  MathSciNet  MATH  Google Scholar 

  21. L. Schlesinger. Über asymptotische Darstellungen der Lösungen linearer Differentialsysteme als Funktionen eines Parameters. Matn. nn. 63, 277–300 (1907).

    MathSciNet  MATH  Google Scholar 

  22. A.A. Shkalikov. Boundary value problems for ordinary differential equations with a parameter in the boundary conditions. Journal of Soviet Math. 33, 1311–1342 (1986), Transl. of: Trudy Sem. imeni Petrovskogo 9, 190-229 (1983).

    Article  MATH  Google Scholar 

  23. J.D. Tamarkin. Some general problems of the theory of ordinary linear differential equations and expansions of an arbitrary function in series of fundamental functions (Russion).. Thesis, Petrograd (1917), Short version: Math. Z. 27, 1-54 (1927).

  24. W.J. Trjitzinsky. Theory of linear differential equations containing a parameter. Acta. Math 67 (1936), 1–50 (1936).

    Article  MathSciNet  Google Scholar 

  25. H.L. Turrittin. Asymptotic solutions of certain ordinary differential equations associated with multiple roots of the characteristic equtaion. Am. J. of Math. 58, 364–376 (1936).

    Article  MathSciNet  Google Scholar 

  26. H.L. Turrittin. Asymptotic distribution of zeros for certain exponential sums. Amer. J. of Math. 66, 199–228 (1944).

    Article  MathSciNet  MATH  Google Scholar 

  27. H.L. Turrittin. Asymptotic expansions of solutions of systems of ordinary linear differential equations containing a parameter. Contributions to the theory of nonlinear oscillations, Vol. 2, Princeton Univ. Press, 81-116 (1952).

  28. C.E. Wilder. Expansion problems of ordinary linear differential equations with auxiliary conditions at more than two points. Trans. Amer. Math. Soc. 18, 415–442 (1917).

    Article  MathSciNet  MATH  Google Scholar 

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  1. Lehrstuhl II f. Mathematik für Ingenieure, RWTH-Aachen, Templergraben 5 5, D-5100, Aachen

    Gerhard Freiling

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Freiling, G. On the completeness and minimality of the derived chains of eigen- and associated functions of boundary eigenvalue problems nonlinearly dependant on the parameter. Results. Math. 14, 64–83 (1988). https://doi.org/10.1007/BF03323217

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