Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-13T22:35:41.356Z Has data issue: false hasContentIssue false
  • English
  • Français

Quadratic Functionals of nth Order

Published online by Cambridge University Press:  20 November 2018

S. Easwaran
S. Easwaran*
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is well known that disconjugacy theorems for self-adjoint differential equations are very closely related to the study of the positivity of certain associated quadratic functionals. In this paper, this relationship is closely examined for quadratic functionals of nth order associated with selfadjoint differential equations of order 2n. The motivation of this work comes from a recent paper of W. Leighton [8] on quadratic functionals of second order. It is shown that the method used by W. Leighton extends in a straightforward manner to the study of quadratic functionals of nth order, provided one makes use of an identity due to G. Cimmino [1]. Several authors have considered related problems. In particular we wish to mention the recent papers of G. Ladas [6], W. Simons [13], D. B. Hinton [5]. The book by C. A. Swanson [14] also contains many other related results. As a consequence of our results we can obtain several known and some new criteria for oscillation of 2nth order self-adjoint differential equations.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 19 , Issue 2 , 01 June 1976 , pp. 159 - 167
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Cimmino, G., Estensione dell’identita di Picone all piu generale equazioni lineari autogaggiunta, R. Ace. Naz. Lincei 28 (1938) 354–364.Google Scholar
2. Cimmino, G., Autosoluzioni e autovalori nelle equazioni differenziali lineari ordinaire autogaggiunta di ordine Superiore, Math. Z. 32 (1930) 4058.CrossRefGoogle Scholar
3. Coppel, W. A., “Disconjugacy”, Lecture notes in mathematics, Springer-Verlag, New York, 1971.Google Scholar
4. Glazman, I. M., Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators, (Israel Program for Scientific Translation), Davey, New York, 1965.Google Scholar
5. Hinton, D. B., Clamped end boundary conditions for fourth order self-adjoint differential equations, Duke Math. Jour. 34 (1967) 131–138.CrossRefGoogle Scholar
6. Ladas, G., Connections between oscillation of self-adjoint linear differential operators of order 2n. Comm. Pure. Appl. Math. 22 (1969) 561–585.CrossRefGoogle Scholar
7. Leighton, W. and Nehari, Z., On the oscillation of solutions of self-adjoint linear differential equations of the fourth order, Trans. Amer. Math. Soc. 89 (1958) 309–322.CrossRefGoogle Scholar
8. Leighton, W., Quadratic functionals of second order, Trans. Amer. Math. Soc. 151 (1970) 309–322.CrossRefGoogle Scholar
9. Levin, A. Ju., Nonoscillation of solutions of the equations x (n)+p 1(t)x (n-1)+…+p n (t)x = 0, Russian Math. Surveys 24 (1969) 43–100.CrossRefGoogle Scholar
10. Noussair, E. S., Oscillation theory of elliptic equations of second order 2n, J. Diff. Equs. 10 (1971) 100–111.CrossRefGoogle Scholar
11. Reid, W. T., Oscillation Criteria for self-adjoint differential system, Trans. Amer. Math. Soc, 101 (1961) 91–106.CrossRefGoogle Scholar
12. Reid, W. T., Oscillation Criteria for linear differential systems with complex coefficients, Pacific J. Math. 6 (1956) 733–751.CrossRefGoogle Scholar
13. Simons, W., Some disconjugacy criteria for self-adjoint linear differential equations, J. Math. Anal. Appl. 34 (1971) 445–463.CrossRefGoogle Scholar
14. Swanson, C. A., Comparison and oscillation theory of linear differential equations, Academic Press, New York, 1968.Google Scholar