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A general oscillation theorem for self-adjoint differential systems with applications to Sturm-Liouville eigenvalue problems and quadratic functionals

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Abstract

For given 2n×2n matricesS 13,S 24 with rank(S 13,S 24)=2n \(S_{13} \bar S_{24}^T = S_{24} \bar S_{13}^T \) we consider the eigenvalue problem:u′=A(x)u+B(x)v,v′=C 1(x;λ)u-A T(x)v with

$$S_{13} \left( {_{u\left( b \right)}^{ - u\left( a \right)} } \right) + S_{24} \left( {_{\upsilon \left( b \right)}^{\upsilon \left( a \right)} } \right) = 0,{\text{ }}a< b;$$

where we assume that then×n matrices,A, B, C 1 satisfy:A, B, C 1, ∂/∂λC 1 are continuous on IR resp. IR2;B, C 1 are Hermitian;B, −∂/∂λC 1 are non-negative definite; and we assume the crucial normality-condition: for any solutionu, v (λ∈IR arbitrary) ∂/∂λC 1 u≡0 on some interval always impliesuv≡0. Then, the main result of the paper (Theorem 2) is the following oscillation result: For any conjoined basisU 1(x; λ),V 1(x; λ) of the differential system with fixed (with respect to λ) initial valuesU 1(a), V1(a), we haven 1(λ)+n 2(λ)=n 3(λ)+n 1+n 2 for λ ∈ IR with regularU 1(b; λ); where\(n_i = \mathop {lim}\limits_{\lambda \to \infty } \);n i(λ),i=1,2;n 1(λ) denotes number of focal points of 3U 1 in [a, b);n 3(λ) denotes the number of eigenvalues which are ≤λ; andn 2(λ) denotes the number of negative eigenvalues of a certain Hermitian 3n×3n matrixM(λ). Moreover, it is shown how classical results (e.g. Rayleigh's principle, existence theorem) can be derived from this oscillation theorem via a generalized Picone identity (which yields also the matrixM(λ) above).

Actually these eigenvalue problems in connection with an associated functional (the linear differential system above consists of the canonical form of the Euler-Lagrange equations of a corresponding Bolza problem) are very much related to the work of W.T. Reid (Wiley 1971). Many results of this paper, including the oscillation theorem above, are extensions of an earlier paper on Sturm-Liouville eigenvalue problems (Analysis 5(1985), 97–152).

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References

  1. Barret J.H.,Oscillation theory of ordinary linear differential equations, Advances in Math.3 (1969), 415–509.

    Article  MathSciNet  Google Scholar 

  2. Baur G.,Über selbstadjungierte Eigenwertprobleme bei Hamilton-Systemen, Dissertation, Ulm 1986.

  3. Birkhoff G.D.,Existence and oscillation theorem of a certain boundary value problem, Trans. Amer. Math. Soc.10 (1909), 259–270.

    Article  MathSciNet  MATH  Google Scholar 

  4. Cimmino G.,Autosoluzioni e autovalori nelle equazioni differenziali lineari ordinarie autoaggiunte di ordine superiore, Math. Z.32 (1930), 4–58.

    Article  MathSciNet  MATH  Google Scholar 

  5. Coppel W.A.,Stability and asymptotic behaviour of differential equations, Heath and Company 1965.

  6. Coppel W.A.,Comparison theorems for canonical systems of differential equations, J. Math. Anal. Appl.12 (1965), 306–315.

    Article  MathSciNet  MATH  Google Scholar 

  7. Coppel W.A.,Disconjugacy, Springer, Lecture Notes 1971.

  8. Eberhard W.,Über das asymptotische Verhalten von Lösungen der linearen Differentialgleichung M[y]=λN[y] für grosse Werte von |λ|, Math. Z.119 (1971), 160–170.

    Article  MathSciNet  MATH  Google Scholar 

  9. Hestenes M.R.,Calculus of variations and optimal control theory, R.E. Krieger Publishing Company 1980.

  10. Kamke E.,Über die definiten selbstadjungierten Eigenwertaufgaben bei gewöhnlichen linearen Differentialgleichungen I, Math. Z.45 (1939), 759–787.

    Article  MathSciNet  MATH  Google Scholar 

  11. Kratz W.,A substitute of l'Hospital's rule for matrices, Proc. Amer. Math. Soc.99 (1987), 395–402.

    MathSciNet  MATH  Google Scholar 

  12. Kratz W.,A limit theorem for matrix-solutions of Hamiltonian systems, Rend. Circ. Mat. Palermo (2)36 (1987), 457–473.

    Article  MathSciNet  MATH  Google Scholar 

  13. Kratz W., Peyerimhoff A.,Sturm-Liouville eigenvalue problems and Hilbert's invariant integral, Indian J. Math. 25 No.2 (1983), 201–222.

    MathSciNet  MATH  Google Scholar 

  14. Kratz W., Peyerimhoff A.,An elementary treatment of Sturmian eigenvalue problems, Analysis4 (1984), 73–85.

    Article  MathSciNet  MATH  Google Scholar 

  15. Kratz W., Peyerimhoff A.,A treatment of Sturm-Liouville eigenvalue problems via Picone's identity, Analysis5 (1985), 97–152.

    Article  MathSciNet  MATH  Google Scholar 

  16. Kreith K.,Oscillation Theory, Springer, Lecture Notes 1970.

  17. Lee E.B., Markus L.,Foundations of optimal control theory, Wiley 1967.

  18. Leighton W., Nehari Z.,On the oscillation of solutions of self-adjoint linear differential equations of fourth order, Trans. Amer. Math. Soc.89 (1958), 325–377.

    Article  MathSciNet  MATH  Google Scholar 

  19. Marcus M., Minc H.,A survey of matrix theory and matrix inequalities, Allyn and Bacon 1964.

  20. Morse M., Leighton W.,Singular quadratic functionals, Trans Amer. Math. Soc.40 (1936), 252–286.

    Article  MathSciNet  MATH  Google Scholar 

  21. Morse M.,The calculus of variations in the large, AMS Colloquium Publication18 (1934).

  22. Morse M.,Variational analysis: Critical extremals and Sturmian extensions, Wiley 1973.

  23. Neumark M.A.,Lineare Differentialoperatoren, Akademie Verlag 1967.

  24. Picone M.,Sulle autosoluzioni e sulle formule di maggiorazione per gli integrali delle equazioni differenziali lineari ordinarie autoaggiunte, Math. Z.28 (1928), 519–555.

    Article  MathSciNet  MATH  Google Scholar 

  25. Reid W.T.,Ordinary differential equations, Wiley 1971.

  26. Reid W.T.,Riccati differential equations, Academic Press 1972.

  27. Reid W.T.,Sturmian theory of ordinary differential equations, Springer, Lecture Notes 1980.

  28. Sagan H.,Calculus of variations, Mc Graw-Hill 1969.

  29. Swanson C.A.,Comparison and oscillation theory of linear differential equations, Academic Press 1968.

  30. Swanson C.A.,Picone's identity, Rend. Math. (6)8 (1975), 373–397.

    MathSciNet  MATH  Google Scholar 

  31. Tomastik E.C.,Singular quadratic functionals of n dependent variables, Trans. Amer. Math. Soc.124 (1966), 60–76.

    MathSciNet  MATH  Google Scholar 

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Baur, G., Kratz, W. A general oscillation theorem for self-adjoint differential systems with applications to Sturm-Liouville eigenvalue problems and quadratic functionals. Rend. Circ. Mat. Palermo 38, 329–370 (1989). https://doi.org/10.1007/BF02850019

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