Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-24T12:38:41.609Z Has data issue: false hasContentIssue false
  • English
  • Français

Positive functionals and oscillation criteria for second order differential systems

Published online by Cambridge University Press:  20 January 2009

Garret J. Etgen  and
Roger T. Lewis
Garret J. Etgen
Affiliation:
University of Houston
Roger T. Lewis
Affiliation:
University of Alabama in Birmingham
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.

Let ℋ be a Hilbert space, let ℬ = (ℋ, ℋ) be the B*-algebra of bounded linear operators from ℋ to ℋ with the uniform operator topology, and let ℒ be the subset of ℬ consisting of the self-adjoint operators. This article is concerned with the second order self-adjoint differential equation

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 22 , Issue 3 , October 1979 , pp. 277 - 290
Copyright
Copyright © Edinburgh Mathematical Society 1979

References

REFERENCES

(1) Ahlbrandt, C. D., Disconjugacy criteria for self-djoint differential systems, J. Differential Equations 6 (1969), 271–295 .CrossRefGoogle Scholar
(2) Ahmad, S. and Lazer, A. C., Component properties of second order linear systems, Bulletin Amer. Math. Soc. 82 (1979), 287–289.CrossRefGoogle Scholar
(3) Allegretto, W. and Erbe, L., Oscillation criteria for matrix differential inequalities, Canad. Math. Bull. 16 (1973), 5–10.CrossRefGoogle Scholar
(4) Atkinson, F. V., Discrete and Continuous Boundary Problems, (Mathematics in Science and Engineering, Volume 8, Academic Press, New York, 1964).Google Scholar
(5) Barrett, J. H., Oscillation theory of ordinary linear differential equations, Advances in Mathematics 3 (1969).CrossRefGoogle Scholar
(6) Coppel, W. A., Disconjugacy, (Lecture Notes in Mathematics No. 220, Springer-Verlag, Berlin-Heidelberg-New York, 1971).CrossRefGoogle Scholar
(7) Etgen, G. J., Oscillatory properties of certain nonlinear matrix differential systems, Trans. Amer. Math. Soc. 122 (1966), 289–310.CrossRefGoogle Scholar
(8) Etgen, G. J., TWO point boundary problems for second order matrix differential equations, Trans. Amer. Math. Soc. 147 (1970), 119–132.CrossRefGoogle Scholar
(9) Etgen, G. J., Oscillation criteria for nonlinear second order matrix differential equations, Proc. Amer. Math. Soc. 27 (1971), 259–267.CrossRefGoogle Scholar
(10) Etgen, G. J. and Pawlowski, J. F., Oscillation criteria for second order selfadjoint differential systems, Pacific J. Math. 66 (1976), 99–110.CrossRefGoogle Scholar
(11) Etgen, G. J. and Pawlowski, J. F., A comparison theorem and oscillation criteria for second order differential systems, Pacific J. Math. 72 (1977), 59–69.CrossRefGoogle Scholar
(12) Glazman, I. M., Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators, (Israel Program for Scientific Translation, Jerusalem, 1965).Google Scholar
(13) Hartman, P., Selfadjoint nonoscillatory systems of ordinary second order linear differential equations, Duke Math. J. 24 (1957), 25–35.CrossRefGoogle Scholar
(14) Hartman, P., Ordinary Differential Equations, (John Wiley and Sons, New York, (1964)Google Scholar
(15) Hayden, T. L. and Howard, H. C., Oscillation of differential equations in Banach spaces, Annali di Mathematica Pura ed Applicata 85 (1970), 383–394.CrossRefGoogle Scholar
(16) Hllle, E., Lectures on Ordinary Differential Equations, (Addison-Wesley Publishing Company, Reading, Massachusetts, 1969).Google Scholar
(17) Hinton, D. B., A criterion for n-n oscillation in differential equations of order 2n, Proc. Amer. Math. Soc. 19 (1968), 511–518.Google Scholar
(18) Howard, H. C., Oscillation criteria for matrix differential equations, Canad. J. Math. 19 (1967), 184–199.CrossRefGoogle Scholar
(19) Kartsatos, A. G., Oscillation of nonlinear systems of matrix differential equations, Proc. Amer. Math. Soc. 30 (1971), 97–101.CrossRefGoogle Scholar
(20) Kreith, K., Oscillation criteria for nonlinear matrix differential equations, Proc. Amer. Math. Soc. 26 (1970), 270–272.CrossRefGoogle Scholar
(21) Kreith, K., Oscillation Theory, (Lecture Notes in Mathematics, No. 324, Springer-Verlag, Berlin-Heidelberg-New York, 1973).CrossRefGoogle Scholar
(22) Lewis, R. T., Oscillation and nonoscillation criteria for some selfadjoint even order linear differential operators, Pacific J. Math. 51 (1974), 221–234.CrossRefGoogle Scholar
(23) Lewis, R. T., The existence of conjugate points for selfadjoint differential equations of even order, Proc. Amer. Math. Soc. 56 (1976), 162–166.CrossRefGoogle Scholar
(24) Lewis, R. T., Conjugate points of vector-matrix differential equations, Trans. Amer. Math. Soc., 231 (1977), 167–178.CrossRefGoogle Scholar
(25) Martynov, V. V., Conditions for discreteness and continuity of the spectrum in the case of a selfadjoint system of differential equations of even order, Dif. Uravneniya 1 (1965), 1578–1591.Google Scholar
(26) Morse, M., A generalization of the Sturm separation and comparison theorems in n-space, Math. Ann. 103 (1930), 52–69.CrossRefGoogle Scholar
(27) Morse, M., The Calculus of Variations in the Large, (Amer. Math. Soc. Colloq. Publ. 18, Providence, Rhode Island, 1934).Google Scholar
(28) Noussair, E. S. and Swanson, C. A., Oscillation criteria for differential systems, J. Math. Anal, and Appl. 36 (1971), 575–580.CrossRefGoogle Scholar
(29) Noussair, E. S., Differential equations in Banach spaces, Bull. Australian Math. Soc. 9 (1973), 219–226.CrossRefGoogle Scholar
(30) Reid, W. T., Riccati matrix differential equations and nonoscillation criteria for associated linear differential equations, Pacific J. Math. 13 (1963), 665–685.CrossRefGoogle Scholar
(31) Reid, W. T., Ordinary Differential Equations, (John Wiley and Sons, New York, 1971).Google Scholar
(32) Reid, W. T., Riccati Differential Equations, (Mathematics in Science and Engineering, Volume 86, Academic Press, New York, 1972).Google Scholar
(33) Rickart, C. E., General Theory of Banach Algebras, (Van Nostrand, New York, 1960).Google Scholar
(34) Swanson, C. A., Comparison and Oscillation Theory of Linear Differential Equations, (Mathematics in Science and Engineering, Volume 48, Academic Press, New York, 1968).CrossRefGoogle Scholar
(35) Swanso, C. A., Oscillation criteria for nonlinear matrix differential inequalities, Proc. Amer. Math. Soc. 24 (1970), 824–827.Google Scholar
(36) Swanson, C. A., Remarks on Picone's identities and related identities, Atti. Accad. Naz. Lincer 11 (1972), 1–15.Google Scholar
(37) Swanson, C. A., Picone's identity, Rendiconti di Matematica 8 Serie VI(1975), 373–397.Google Scholar
(38) Tomastik, E. C., Singular quadratic functionals of n dependent variables, Trans. Amer. Math. Soc. 124 (1966), 60–76.Google Scholar
(39) Tomastik, E. C., Oscillation of nonlinear matrix differential equations of second order, Proc. Amer. Math. Soc. 19 (1968), 1427–1431.Google Scholar
(40) Tomastik, E. C., Oscillation of systems of second order differential equations, J. Differential Equations 9 (1971), 436–442.CrossRefGoogle Scholar
(41) Tomastik, E. C., Principal quadratic functionals, Trans. Amer. Math. Soc. 218 (1976), 297–309.CrossRefGoogle Scholar
(42) Williams, C. M., Oscillation phenomena for linear differential systems in a B*-algebra (Ph.D. Dissertation, University of Oklahoma, 1971).Google Scholar
(43) Yakubovic, V. A., Oscillation properties of solutions of linear canonical systems of differential equations, Doklady Akad. Nauk. SSR 124 (1959), 533–536.Google Scholar
(44) Yakubovic, V. A., Oscillation and nonoscillation conditions for canonical linear sets of simultaneous differential equations, Doklady Akad. Nauk. SSR 124 (1959), 994–997.Google Scholar
(45) Yakubovic, V. A., Oscillation properties of solutions of canonical equations, Mat. Sb. 56 (1962), 3–42.Google Scholar