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Global theory of ordinary linear homogeneous differential equations in the real domain—II

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References

  1. Aczél, J.,Lectures on functional equations and their applications. Acad. Press, New York-London, 1966.

    Google Scholar 

  2. Azbelev, N. V. andCaljuk, Z. B.,K voprosu o raspredelenii nulej rešenij linejnogo differencial'nogo uravnenija tret'ego porjadka. Mat. Sb. (N.S.)51 (1960), 475–486.

    Google Scholar 

  3. Barrett, J. H.,Oscillation theory of ordinary linear differential equations. Adv. in Math3 (1986), 415–509.

    Article  Google Scholar 

  4. Barvínek, E.,O rozložení nulových bodů řešení lineární diferenciální rovnice y″ = Q(t)y a jejich derivací. Acta Math. Univ. Comenian5 (1961), 465–474.

    Google Scholar 

  5. Berkovič, L. M.,O faktorizacii obyknovennych linějnych differencial'nych operatorov, preobrazuemych v operatory s postojannymi koefficientami I + II, Izv. Vyssh. Uchebn Zaved. Aviatsion Tekhn.4 (1965), 8–16;12 (1967), 3–14.

    Google Scholar 

  6. Besse, A. L.,Manifolds all of whose geodesics are closed. Ergebnisse Math., 93. Springer, Berlin—New York, 1978.

    Google Scholar 

  7. Birkhoff, G. D.,On the solutions of ordinary linear homogeneous differential equations of the third order. Ann of Math.12 (1910/11), 103–127.

    Google Scholar 

  8. Brioschi, F.,Sur les équations différentielles linéaires. Bull. Soc. Math. France.7 (1879), 105–108.

    Google Scholar 

  9. Blanton, G. andBaker, J. A.,Iteration groups generated by C n functions. Arch. Math. (Brno)19 (1982), 121–127.

    Google Scholar 

  10. Borůvka, O.,Lineare Differentialtransformationen 2. Ordnung. VEB, Berlin 1967.Linear Differential Transformations of the Second Order. The English Univ. Press, London, 1971.

    Google Scholar 

  11. Borůvka, O.,Teorija global'nych svojstv obyknovennych linejnych differencial'nych uravněnij vtorogo porjadka. Differencial'nyje Uravněnija12 (1976), 1347–1383.

    Google Scholar 

  12. Borůvka, O.,Algebraic methods in the theory of global properties of the oscillatory equations Y″ = Q(t)Y. In:Equadiff IV Proceedings, Prague 1977, Lecture Notes in Mathematics, 703. Springer, Berlin—New York, 1979, pp. 35–45.

    Google Scholar 

  13. Borůvka, O.,Sur une classe des groupes continus à un paramètre formés des functions réelles d'une variable. Ann. Polon. Math.42 (1982), 27–39.

    Google Scholar 

  14. Burton, T. A. andPatula, W. T.,Limit circle results for second order equations. Monatsh. Math.81 (1976), 185–194.

    Google Scholar 

  15. Čadek, M.,Form of general pointwise transformations of linear differential equations. Czechoslovak Math. J.35 (110) (1985), 617–624.

    Google Scholar 

  16. Čadek, M.,Pointwise transformations of linear differential equations (to appear).

  17. Cartan, É.,La théorie des groupes finis et la géometrie différentielle traitées par la méthode du repère mobile, Gauthier-Villars, 1937.

  18. Chanturija, T. A.,On the oscillation of solutions of higher order linear differential equations (in Russian). Reports of Seminar Vekua Inst. Appl. Math. Univ. Tbilissi 16 (1982), 1–74.

    Google Scholar 

  19. Chrastina, J.,On dispersions of the 1st and 2nd kind of differential equation y″ =q(x)y. Publ. Fac. Sci. Univ. J. E. Purkyně, Brno, No.508 (1969), 353–377.

    Google Scholar 

  20. Coppel, W. A.,Disconjugacy, Lecture notes in mathematics,220. Springer, Berlin 1971.

    Google Scholar 

  21. Došlý, O.,Phase matrix of linear differential systems. Časopis Pěst. Mat.110 (1985), 183–192.

    Google Scholar 

  22. Everitt, W. N.,Integrable-square, analytic solutions of odd-order, formally symmetric, ordinary differential equations. Proc. London Math. Soc.25 (1972), 156–182.

    Google Scholar 

  23. Everitt, W. N.,On the transformation theory of ordinary second-order linear symmetric differential expressions. Czechoslovak Math. J.32 (1982), 275–306.

    Google Scholar 

  24. Everitt, W. N. andNeuman, F.,A concept of adjointness and symmetry of differential expressions based on the generalised Lagrange identity and Green's formula. In:Ordinary differential equations and operators. Proceedings, Dundee 1982. Lecture Notes in Mathematics, 1032. Springer, Berlin—New York, 1983, pp. 161–169.

    Google Scholar 

  25. Forsyth, A. R.,Invariants, covariants and quotient-derivatives associated with linear differential equations. Philos. Trans. Roy. Soc. London Ser. A,179 (1899), 377–489.

    Google Scholar 

  26. Greguš, M.,Lineárna differenciálna rovnica tretieho rádu, Veda, Bratislava 1981, extended form in English:Linear Differential Equations of the Third Order. North Holland & Reidel Co., Dordrecht—Boston—Lancaster, 1986.

    Google Scholar 

  27. Guggenheimer, H.,Distribution of zeros and limit behavior of solutions of differential equations. Proc. Amer. Math. Soc.,61 (1976), 275–279.

    Google Scholar 

  28. Gustafson, G. B.,Higher order separation and comparison theorems, with applications to solution space problems. Ann. Mat. Pura Appl. (4)95 (1973), 245–254.

    Google Scholar 

  29. Halphen, G. H.,Mémoire sur la réduction des équations différentielles linéaires aux formes intégrables. Mémoires présentés par divers savants à l'académie des sciences de l'Institut de France28 (1884), 1–307.

    Google Scholar 

  30. Hanan, M.,Oscillation criteria for third-order linear differential equations. Pacific J. Math.11 (1961), 919–944.

    Google Scholar 

  31. Hartman, P.,On differential equations and the function J 2μ + Y 2μ . Amer. J. Math.83 (1961), 154–188.

    Google Scholar 

  32. Hasse, M. andMichler, L.,Theorie der Kategorien. VEB, Berlin 1966.

    Google Scholar 

  33. Hustý, Z.,Die Iteration homogener linearer Differentialgleichungen. Publ. Fac. Sci. Univ. J. E. Purknyě (Brno)449 (1964), 23–56.

    Google Scholar 

  34. Hustý, Z.,Über die Transformation und Äquivalenz homogener linearer Differentialgleichungen von höherer als der zweiten Ordnung I + II + III. Czechoslovak Math. J.15 (90) (1965), 479–502;16 (91) (1966), 1–13;16 (91) (1966), 161–185.

    Google Scholar 

  35. Kiguradze, I. T.,Někotoryje singularnyje krajevyje zadači dlja obyknovennych differencial'nych uravnenij. Univ. Press, Tbilisi, 1975.

    Google Scholar 

  36. Kondratjev, V. A.,O koleblemosti rešenij linejnych uravněnij trat'ego i četvertogo porjadka. Trudy Moskov. Mat. Obshch.8 (1959), 259–282.

    Google Scholar 

  37. Kuczma, M.,Functional equations in a single variable. Podstawy Sterowania, Warszawa, 1968.

    Google Scholar 

  38. Kummer, E. E.,De generali quadam aequatione differentiali tertii ordinis. Progr. Evang. Königl. Stadtgymnasium Liegnitz 1834 (reprinted in J. Reine Angew. Math.100 (1887), 1–10).

  39. Kwong, M. K.,On boundedness of solutions of second order differential equations in the limit circle case. Proc. Amer. Math. Soc.52 (1975), 242–246.

    Google Scholar 

  40. Laguerre, E.,Sur les équations différentielles linéaires du troisième ordre. C.R. Acad. Sci., Paris88 (1879), 116–118.

    Google Scholar 

  41. Laitoch, M.,L'équation associée dans la théorie des transformation des équations différentielles du second ordre. Palack. Olomucensis Fac. Rerum. Natur. Chem.12 (1963), 45–62.

    Google Scholar 

  42. Lazer, A. C.,The behavior of solutions of the differential equation y‴ + p(x)y′ + q(x)y = 0. Pacific J. Math.17 (1966), 435–466.

    Google Scholar 

  43. Levin, A. Ju.,O raspredelenii nulej rešenij linejnogo differencial'nogo uravnenija. Dokl. Akad. Nauk. SSSR156 (1964), 1281–1284.

    Google Scholar 

  44. Lie, S. andEngel, F.,Theorie der Transformationsgruppe. Teubner, Leipzig, 1930.

    Google Scholar 

  45. Lorch, L., Muldoon, M. E. andSzegö, P.,Higher monotonicity properties of certain Sturm-Liouville functions IV. Canad. J. Math.24 (1972), 349–368.

    Google Scholar 

  46. Mac Lane, S. andBirkhoff, G.,Algebra. Macmillan, New York, 1965.

    Google Scholar 

  47. Mammana, G.,Decomposizione delle espressioni differenziali lineari omogenee in prodotti di fattori simbolici e applicazione rellativa allo studio delle equazioni differenziali lineari. Math. Z.33 (1931), 186–231.

    Google Scholar 

  48. Moravčík, J.,Über die Äquivalenz linearer gewöhnlicher Differentialgleichungen n-ter Ordnung (n = 3,4). Mat. Časopis19 (1969), 17–42.

    Google Scholar 

  49. Neuman, F.,Linear differential equations of the second order and their applications. Rend. Mat.4 (1971), 559–617.

    Google Scholar 

  50. Neuman, F.,Geometrical approach to linear differential equations of the n-th order. Rend. Mat.5 (1972), 579–602 (Abstract:Some results on geometrical approach to linear differential equations of the n-th order. Comment. Math. Univ. Carolin.12 (1971), 307–315).

    Google Scholar 

  51. Neuman, F.,Distribution of zeros of solutions of y″ = q(t)y in relation to their behaviour in large. Studia Sci. Math. Hungar.8 (1973), 177–185.

    Google Scholar 

  52. Neuman, F.,On two problems about oscillation of linear differential equations of the third order. J. Differential Equations15 (1974), 589–596.

    Google Scholar 

  53. Neuman, F.,On solutions of the vector functional equation y(ξ(x)) = f(x).Ay(x). Aequationes Math.16 (1977), 245–257.

    Google Scholar 

  54. Neuman, F.,Limit circle classification and boundedness of solutions. Proc. Roy. Soc. Edinburgh,81A (1978), 31–34, also as preprint: Základní knihovna ČSAV, Praha, No. 15.688 (April 1977).

    Google Scholar 

  55. Neuman, F.,Invarianty linejnych differencial'nych uravněnij 3-go porjadka i metod podvižnogo repera E. Kartana. Differencial'nyje Uravněnija14 (1979), 398–404.

    Google Scholar 

  56. Neuman, F.,Second order linear differential systems. Ann. Sci, École Norm. Sup.13 (1980), 437–449.

    Google Scholar 

  57. Neuman, F.,Factorizations of matrices and functions of two variables. Czechoslovak Math. J.,32 (107), 1982, 582–588.

    Google Scholar 

  58. Neuman, F.,Simultaneous solutions of a system of Abel equations and differential equations with several deviations. Czechoslovak Math. J.,32 (107), 1982, 488–494.

    Google Scholar 

  59. Neuman, F.,A survey of global properties of linear differential equations of the n-th order. In:Ordinary and Partial Differential Equations. Proceedings, Dundee 1982. Lecture Notes in Mathematics, 964. Springer, Boston—New York, 1982, pp. 548–563.

    Google Scholar 

  60. Neuman, F.,Criterion of global equivalence of linear differential equations. Proc. Roy. Soc. Edinburgh, Sect. A97 (1984), 217–221.

    Google Scholar 

  61. Neuman, F.,Stationary groups of linear differential equations. Czechoslovak Math. J.34 (109) (1984), 645–663. (C.R. Acad. Sci. Paris Sér I Math.299 (1984), 319–322).

    Google Scholar 

  62. Neuman, F.,A note on smoothness of the Stäckel transformation. Prace Mat. WSP Krakow11 (1985), 147–151.

    Google Scholar 

  63. Neuman, F.,Covariant constructions in the theory of linear differential equations. Časopis Pěst. Mat.111 (1986), 201–207.

    Google Scholar 

  64. Neuman, F.,Another proof of Borůvka's criterion on global equivalence of the second order linear differential equations (to appear).

  65. Neuman, F.,Non-smoothability as an invariant property of coefficients of linear differential equations (to appear).

  66. Posluszny, J. andRubel, L. A.,The motion of an ordinary differential equation. J. Differential Equations34 (1979), 291–302.

    Google Scholar 

  67. Ráb, M.,Asymptotische Formeln für die Lösungen der Differentialgleichung y″ + q(x)y = 0. Math. J.14 (89) (1964), 203–221.

    Google Scholar 

  68. Sansone, G.,Studi sulle equazioni differenziali lineari omogenee di terzo ordine nel campo reale. Revista Mat. Fis. Teor. Tucuman6 (1948), 195–253.

    Google Scholar 

  69. Schlesinger, L.,Handbuch der Theorie der linearen Differentialgleichungen. Leipzig, 1897.

  70. Šeda, V.,Über die Transformation der linearen Differentialgleichungen n-ter Ordnung I + II. Časopis Pěst. Mat.90 (1965), 385–412;92 (1967), 418–435.

    Google Scholar 

  71. Stäckel, P.,Über Transformationen von Differentialgleichungen. J. Reine Angew. Math.111 (1893), 290–302.

    Google Scholar 

  72. Staněk, S.,On an algebraic structure of a group of phases. Math. Slovaca27 (1977), 423–430.

    Google Scholar 

  73. Suchomel, J.,Preobrazovanije linejnych odnorodnych differencial'nych uravnenij vysšego porjadka. Arch. Math. (Brno)13 (1977), 41–43.

    Google Scholar 

  74. Švec, M.,Einige asymptotische und oszillatorische Eigenschaften der Differentialgleichung y‴ + A(x)y′ + B(x)y = 0. Czechoslovak Math. J.15 (1965) 90, 378–393.

    Google Scholar 

  75. Swanson, C. A.,Comparison and oscillation theory of linear differential equations. Acad. Press, New York—London, 1968.

    Google Scholar 

  76. Tryhuk, V.,The most general transformation of homogeneous retarded linear differential equations of the n-th order. Math. Slovaka33 (1983), 15–21.

    Google Scholar 

  77. Vosmanský, J.,Distribution of zeros of solutions of linear second order differential equations. J. Differential Equations15 (1979), 1511–1519.

    Google Scholar 

  78. Wilczynski, E. J.,Projective differential geometry of curves and ruled surfaces. Teubner, Leipzig, 1906.

    Google Scholar 

  79. Zlámal, M.,Über asymptotische Eigenschaften der Lösengen der linearen Differentialgleichung zweiter Ordnung. Czechoslovak Math. J.6 (81) (1956), 75–93.

    Google Scholar 

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Dedicated to Professor Otto Haupt with best wishes on his 100th birthday

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Neuman, F. Global theory of ordinary linear homogeneous differential equations in the real domain—II. Aeq. Math. 34, 1–22 (1987). https://doi.org/10.1007/BF01840120

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