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On quadratic integral inequalities associated with second-order symmetric differential expressions

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Ordinary Differential Equations and Operators

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References

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

    MATH  Google Scholar 

  2. Akhiezer, N. I. and Glazman, I. M.: Theory of linear operators in Hilbert space (Pitman Publishing, London, 1981).

    MATH  Google Scholar 

  3. Amos, R. J.: On some problems concerned with integral inequalities associated with symmetric ordinary differential expressions (Ph.D. thesis, University of Dundee, Scotland, 1977).

    Google Scholar 

  4. Amos, R. J.: On a Dirichlet and limit-circle criterion for second-order ordinary differential expressions, Quaestiones Mathematicae 3 (1978), 53–65.

    Article  MathSciNet  MATH  Google Scholar 

  5. Amos, R. J. and Everitt, W. N.: On a quadratic integral inequality, Proc. Royal Soc. Edinburgh (A) 78 (1978),241–256.

    Article  MathSciNet  MATH  Google Scholar 

  6. Amos, R. J. and Everitt, W. N.: On integral inequalities associated with ordinary regular differential expressions, Differential Equations and Applications, 237–255 (North Holland, 1978; Mathematical Studies 31; Edited by W. Eckhaus and E. M. de Jager).

    Google Scholar 

  7. Amos, R. J. and Everitt, W. N.: On integral inequalities and compact embeddings associated with ordinary differential expressions, Arch. Rat. Mech. Anal. 71 (1979), 15–40.

    Article  MathSciNet  MATH  Google Scholar 

  8. Atkinson, F. V.: Limit-n criteria of integral type, Proc. Royal Soc. Edinburgh (A) 73 (1975), 167–198.

    MathSciNet  MATH  Google Scholar 

  9. Beesack, P. R.: Math. Reviews 80 (1980), 34017.

    Google Scholar 

  10. Beesack, P. R.: Minimum properties of eigenvalues-elementary proofs, General Inequalities 2, 109–120 (Birkhäuser Verlag, Basel, 1980; Edited by E. F. Beckenbach).

    Chapter  Google Scholar 

  11. Bradley, J. S. and Everitt, W. N.: Inequalities associated with regular and singular problems in the calculus of variations, Trans. Amer. Math. Soc. 182 (1973), 303–321.

    Article  MathSciNet  MATH  Google Scholar 

  12. Bradley, J. S. and Everitt, W. N.: A singular integral inequality on a bounded interval, Proc. Amer. Math. Soc. 61 (1976), 29–35.

    Article  MathSciNet  MATH  Google Scholar 

  13. Bradley, J. S., Hinton, D. B. and Kauffman, R. M.: On the minimization of singular quadratic functionals, Proc. Royal Soc. Edinburgh (A) 87 (1981), 193–208.

    Article  MathSciNet  MATH  Google Scholar 

  14. Evans, W. D.: On limit-point and Dirichlet type results for second-order differential expressions, Lecture Notes in Mathematics 564 (1976), 78–92 (Edited by W. N. Everitt and B. D. Sleeman; Springer Verlag, Heidelberg, 1976).

    Google Scholar 

  15. Everitt, W. N.: On the strong limit-point condition of second-order differential expressions, Proceedings International Conference on Differential Equations, Los Angeles, 1974; pages 287–307 (Edited by W. A. Harris; Academic Press, New York, 1975).

    Google Scholar 

  16. Everitt, W. N.: A note on the Dirichlet condition for second-order differential expressions, Canadian J. Math. 28 (1976), 312–320.

    Article  MathSciNet  MATH  Google Scholar 

  17. Everitt, W. N.: An integral inequality with an application to ordinary differential operators, Proc. Royal Soc. Edinburgh (A) 80 (1979), 35–44.

    Article  MathSciNet  MATH  Google Scholar 

  18. Everitt, W. N.: A note on an integral inequality, Quaestiones Mathematicae 2 (1978), 461–478.

    Article  MathSciNet  MATH  Google Scholar 

  19. Everitt, W. N.: A general integral inequality associated with certain ordinary differential operators, Quaestiones Mathematicae 2 (1978), 479–494.

    Article  MathSciNet  MATH  Google Scholar 

  20. Everitt, W. N. and Giertz, M.: A Dirichlet type result for ordinary differential operators, Math. Ann. 203 (1973), 119–218.

    Article  MathSciNet  MATH  Google Scholar 

  21. Everitt, W. N., Giertz, M. and McLeod, J. B.: On the strong and weak limit-point classification of second-order differential expressions, Proc. London Math. Soc. (3) 29 (1974), 142–158.

    Article  MathSciNet  MATH  Google Scholar 

  22. Everitt, W. N., Giertz, M. and Weidmann, J.: Some remarks on a separation and limit-point criterion of second-order, ordinary differential expressions, Math. Ann. 200 (1973), 335–346.

    Article  MathSciNet  MATH  Google Scholar 

  23. Everitt, W. N., Kwong, M. K. and Zettl, A.: Differential operators and quadratic inequalities with a degenerate weight, (to appear in J. of Math. Analysis and Applications).

    Google Scholar 

  24. Everitt, W. N. and Wray, S. D.: A singular spectral identity involving the Dirichlet integral of an ordinary differential expression, (to appear in the Czechoslovak Mathematical Journal).

    Google Scholar 

  25. Everitt, W. N. and Zettl, A.: Generalized symmetric ordinary differential expressions I: the general theory, Nieuw Archief voor Wiskunde (3) 27 (1979), 363–397.

    MathSciNet  MATH  Google Scholar 

  26. Florkiewicz, B. and Rybarski, A.: Some integral inequalities of Sturm-Liouville type, Colloq. Math. 36 (1976), 127–141.

    MathSciNet  MATH  Google Scholar 

  27. Gregory, J.: Quadratic form theory and differential equations (Academic Press, New York, 1981).

    Google Scholar 

  28. Hardy, G. H., Littlewood, J. E. and Pólya, G.: Inequalities (Cambridge University Press, 1934).

    Google Scholar 

  29. Hellwig, G.: Differential operators of mathematical physics (Addison-Wesley, London, 1967).

    MATH  Google Scholar 

  30. Hildebrandt, S.: Rand-und Eigenwertaufgaben bei stark elliptischen Systemen linearer Differentialgleichungen, Math. Ann. 148 (1962), 411–429.

    Article  MathSciNet  MATH  Google Scholar 

  31. Hinton, D. B.: On the eigenfunction expansions of singular ordinary differential equations, J. Differential Equations 24 (1977), 282–308.

    Article  MathSciNet  MATH  Google Scholar 

  32. Hinton, D. B.: Eigenfunction expansions and spectral matrices of singular differential operators, Proc. Royal Soc. Edinburgh (A) 80 (1978), 289–308.

    Article  MathSciNet  MATH  Google Scholar 

  33. Kalf, H.: Remarks on some Dirichlet type results for semibounded Sturm-Liouville operators, Math. Ann. 210 (1974), 197–205.

    Article  MathSciNet  MATH  Google Scholar 

  34. Kato, T.: Perturbation theory for linear operators (first/second edition; Springer Verlag, Heidelberg, 1966/1976).

    Book  MATH  Google Scholar 

  35. Kwong, M. K.: Note on the strong limit-point condition of second-order differential expressions, Quart. J. Math. (Oxford) (2) 28 (1977), 201–20.

    Article  MathSciNet  MATH  Google Scholar 

  36. Kwong, M. K.: Conditional Dirichlet property of second-order differential expressions, Quart. J. Math. (Oxford) (2) 28 (1977), 329–338.

    Article  MathSciNet  MATH  Google Scholar 

  37. Mitrinović, D. S.: Analytic inequalities (Springer Verlag, Heidelberg, 1970).

    Book  MATH  Google Scholar 

  38. Namark, M. A.: Linear differential operators Part II (Ungar, New York, 1968).

    Google Scholar 

  39. Penning, F. and Sauer, N.: Note on the minimization of \(\int_0^\infty {\{ p(x)|f'(x)|^2 + {\mathbf{ }}q(x)|f(x)|^2 \} dx}\), Research Report UP TW 2, 1976; Department of Applied Mathematics, University of Pretoria, South Africa.

    Google Scholar 

  40. Putnam, C. R.: An application of spectral theory to a singular calculus of variations problem, Amer. J. Math. 70 (1948), 780–803.

    Article  MathSciNet  MATH  Google Scholar 

  41. Sears, D. B. and Wray, S. D.: An inequality of C. R. Putnam involving a Dirichlet functional, Proc. Royal Soc. Edinburgh (A) 75 (1976), 199–207.

    MathSciNet  MATH  Google Scholar 

  42. Titchmarsh, E. C.: Eigenfunction expansions Part I (second edition; Oxford University Press, 1962).

    Google Scholar 

  43. Walker, P.: A vector-matrix formulation for formally symmetric ordinary differential equations with applications to solutions of integrable-square, J. London Math. Soc. 9 (1974), 151–159.

    Article  MathSciNet  MATH  Google Scholar 

  44. Weidmann, J.: Linear operators in Hilbert spaces (Springer Verlag, Heidelberg, 1980).

    Book  MATH  Google Scholar 

  45. Wray, S. D.: An inequality involving a conditionally convergent Dirichlet integral of an ordinary differential expression Utilitas Mathematica), 22 (1982), 161–184.

    MathSciNet  MATH  Google Scholar 

  46. Wray, S. D.: On a second-order differential expression and its Dirichlet integral (to appear).

    Google Scholar 

  47. Brown, R.G.: A von Neumann factorization of some self-adjoint extensions of positive symmetric differential operators and its application to inequalities, Lecture Notes in Mathematics, vol. 1032 (Edited by W.N. Everitt and R.T. Lewis Springer-Verlag, Heidelberg, 1983).

    Google Scholar 

  48. Brown, R.G.: A factorization method for symmetric differential operators and its applications to Dirichlet inequalities and to the Dirichlet index (to appear in the Proceedings of the 1983 International Conference on Differential Equations held at the University of Alabama in Birmingham, U.S.A).

    Google Scholar 

  49. Brown, R.G. and Hinton, D.B.: Sufficient conditions for weighted inequalities of sum and product form (in preparation).

    Google Scholar 

  50. Krzywicki, A. and Rybarski A.: On some integral inequalities involving Chebyshev weight function, Colloq. Math. 18 (1967), 147–50.

    MathSciNet  MATH  Google Scholar 

  51. Krzywicki, A. Rybarski, A.: On an integral inequality connected with Hardy's inequality, Applicationes Mathematicae (Hugo Steinhaus Memorial Volume)X (1969), 37–41.

    MathSciNet  MATH  Google Scholar 

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W. N. Everitt R. T. Lewis

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Everitt, W.N., Wray, S.D. (1983). On quadratic integral inequalities associated with second-order symmetric differential expressions. In: Everitt, W.N., Lewis, R.T. (eds) Ordinary Differential Equations and Operators. Lecture Notes in Mathematics, vol 1032. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0076798

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  • DOI: https://doi.org/10.1007/BFb0076798

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