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Inequalities Involving Functions and Their Integrals and Derivatives pp 216–239Cite as

Inequalities of Lyapunov and of De La Vallée Poussin

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Part of the book series: Mathematics and Its Applications ((MAEE,volume 53))

Abstract

Let p be a real continuous function of the real variable x on [a,b]. If the differential equation

$$y'' + p(x)y = 0$$

has a nontrivial solution y that vanishes at two points of [a,b], then, according to A. M. Lyapunov [1], p is subject to the inequality

$$(b - a)\int\limits_a^b {|p(x)|dx > 4.}$$
(1.1)

This inequality is sharp in the sense that the constant 4 cannot be replaced by a larger number.

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References

  1. Lyapunov, A. M., Problème général de la stabilité du mouvement, Ann de la Faculté de Toulouse (2) 9 (1907), 406.

    Google Scholar 

  2. Hartman, P. H., “Ordinary Differential Equations,” New York-London-Sydney, 1964.

    Google Scholar 

  3. De La VallÉe Poussin, Ch., Sur l’équation différentielle linéaire du second ordre, J. Math. Pures Appl. 89 (1929), 126–144.

    Google Scholar 

  4. Levin, A. Yu., Nauc. Dokl. Vysš. Školy Fiz.-Mat. Nauki 5 (1958), 34.

    Google Scholar 

  5. Levin, A. Yu. —, Some estimates for a differentiate function, Soviet Math. Dokl 2 (1961), 523–524.

    MathSciNet  MATH  Google Scholar 

  6. Levin, A. Yu. —, An estimate for a function with monotonically distributed zeros of successive derivatives, Mat. Sb. 64 (1964), 396–409, (Russian).

    MathSciNet  Google Scholar 

  7. Bessmertnyh, G. A. and A. Yu. Levin, Some estimates of differentiable function of one variable, Dokl. Akad. Nauk. SSSR 144 (1963), 471–474.

    MathSciNet  Google Scholar 

  8. Hartman, P., On disconjugacy criteria, Proc. Amer. Math. Soc. 24 (1970), 374–381.

    Article  MathSciNet  MATH  Google Scholar 

  9. Willett, D., Generalized de la Vallée Poussin disconjugacy tests for linear differential equations, Canad. Math. Bull. 143 (1971), 419–428.

    Article  MathSciNet  MATH  Google Scholar 

  10. Opial, Z., Sur une inégalité de C. de la Vallée Poussin, Ann. Polon. Math. 6 (1959), 87–91.

    MathSciNet  MATH  Google Scholar 

  11. AramĂ, O. and D. Ripianu, Quelques recherches actuelles concernant l’équation de Ch. J. de la Vallée Poussin, relative au probléme polylocal dans la théorei des équations différentielles, Mathematica 831 (1966), 1, 18-28.

    Google Scholar 

  12. AramĂ, O. —, Sur l’équation de Ch. J. de la Vallée Poussin relative au probléme polylocal de la théorie des équation différentielles, Colloque sur la théorie de l’approximation des fonctions, Cluj 15-20 IX (1967).

    Google Scholar 

  13. Ripianu, D., Asupra inegalitátii lui de la Vallée Poussin in ca ecuatiilor diferentiale de ordinul al doilea, Stuii si cercet. matem. (Cluj), XIV (1963), 123–150, 399-403.

    MathSciNet  Google Scholar 

  14. Ripianu, D. —, O problemá de minimum in teoria interpolárii, Studii si cercet de matematică (Cluj), XIV (1963), 365–397.

    Google Scholar 

  15. Szmanda, B., Oszacowanie przedzialu istnienia i jednoznacznosci rozwiaza’nm pewnego zagadnienia brzegowego, Fasciculi Mathematici Nr 574 (1970), 23–31.

    MathSciNet  MATH  Google Scholar 

  16. AramĂ, O., Sur un problème d’interpolation relatif aux solutions des équations différentielles linéaires du quatrième ordre, Mathematica (Cluj) 1033 (1968), 1, 5-15.

    Google Scholar 

  17. Bobrowski, D., O odleglosci miedzy zerami pewnych równania rózn zkowego liniowego czwatego rzedu, Zeszyty Naukowe, Politichniki Poznanskiej 2 (Matematyka) (1965), 55–60.

    Google Scholar 

  18. LupaŞ, A., An interpolation problem for the solutions of a class of linear differential equations, Mathematica 1235 (1970), 87–104.

    Google Scholar 

  19. LupaŞ, A. —, On the length of the interpolation interval for a class of linear differential operators, Proc. Intern. Conf. Construc. Funct. Theory, Varna (1870), 209–215.

    Google Scholar 

  20. Bailey, P. and P. Waltman, On the distance between constructive zeros for second order differential equations, J. Math. Anal. Appl. 1 (1966), 23–30.

    Article  MathSciNet  Google Scholar 

  21. Lasota, A., Sur les problèmes linéaires aux limites pour un système d’équations différentilles ordinaires, Bull. Acad. Polon. Sci. Sér. Sei. Math., Ast. et Phys. 10 (1962), 565–570.

    MathSciNet  MATH  Google Scholar 

  22. Lasota, A. —, Sur la distance entre les zéros de l’équation différtielle linéare du troisième orde, Ann. Polon. Math. 132 (1963), 129–132.

    MathSciNet  MATH  Google Scholar 

  23. Lasota, A. and Z. Opial, L’application du principe de Pontriaguin à l’évaluation de l’intervalle d’existence et d’unicité des solutions d’un problème aux limites, Bull. Acad. Polon. Sci., Sér Sci. Math., Ast. et Phys. 112 (1963), 41–46.

    MathSciNet  MATH  Google Scholar 

  24. Lettenmeyer, F., Über die von einem punktausgehenden Integralkunven einer Differentialgleichung 2. Ordnung, Deut. Math. 7 (1944), 56–74.

    MathSciNet  Google Scholar 

  25. Szmanda, B., O odleglości miedzu zerami pewnych rozwiazan rpwnania rózniczkowego liniowego n-tego rzedu., Zesz. Nauk. Pol. Pozn. Matern. 4 (1969), 65–70.

    MathSciNet  MATH  Google Scholar 

  26. Richard, U., Sullz diseguaglianza di De La Vallée Poussin per le equasioni differenzialli di ordine n, Rend. Mat. 10 (1977), 583–596.

    MathSciNet  Google Scholar 

  27. Levin, A. Yu, A fredholm equation with a smooth kernel and boundary value problems for a linear differential equation, Dokl. Akad. Nauk SSSR 159 (1964), (Soviet Math. Dokl. 5 (1964), 1415–1419).

    Google Scholar 

  28. Levin, A. Yu —, Non—oscillation of solutions of the equation Uspehi Mat. Nauk 24 (1969), 43–96, (Russian Math. Surveys 24 (1969), 43-100).

    Google Scholar 

  29. Willett, D., Asymptotic behavior of dis conjugate nth order differential equations, Canad. J. Math. 23 (1971), 293–314.

    Article  MathSciNet  MATH  Google Scholar 

  30. Richard, U., Sur un théorème d’unicité de M. C. De La Vallée Poussin pour l’équation différentielle du second ordre, Congrès Int. Math. Moscou 1966, Résumés 6 (Equations Différentielles Ordinaires).

    Google Scholar 

  31. Richard, U. —, Metodi diversi per ottenere diseguaglianze alla de la Vallée Poussin nelle equazioni differenziali ordinarie del secondo e terzo ordine, Rend. Sem. Mat. Univ. Pol. Torino 27 (1967/68), 35–68.

    MathSciNet  Google Scholar 

  32. Casadei, G., Sul teorema di unicita di De La Vallée Poussin per equzioni differenziali del terzo ordine, Rend. Sem. Mat. Univ. di Padova.

    Google Scholar 

  33. Brink, J., Inequalities involving for f with n-zeros, Pacific J. Math. 422 (1972), 289–311.

    MathSciNet  MATH  Google Scholar 

  34. Fink, A. M., Conjugate inequalities for functions and their derivatives, SIAM J. Math. Anal. 53 (1974), 399–411.

    Article  MathSciNet  MATH  Google Scholar 

  35. Fink, A. M. —, Differential inequalities and disconjugacy, J. Math. Anal. Appl. 49 (1975), 758–772.

    Article  MathSciNet  MATH  Google Scholar 

  36. Zaiceva, G. S., A multipoint boundary value problem, Soviet. Math. Dokl. 8 (1967), 1183–1185.

    Google Scholar 

  37. Martelli, M., Sul criteria di univitd di de la Vallée Poussin, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 458 (1968), 7–12.

    MathSciNet  MATH  Google Scholar 

  38. Willett, D., Oscillation on finite or infinite intervals of second order linear differential equations, Canad. Math. Bull. 144 (1971), 539–550.

    Article  MathSciNet  MATH  Google Scholar 

  39. Kim, W. J., Disconjugacy and nonoscillation criteria for linear differential equations, J. Diff. Eq. 8 (1970), 163–172.

    Article  MATH  Google Scholar 

  40. Kim, W. J. —, On the zeros of solutions of y (n) + py = 0, J. Math. Anal. Appl. 25 (1969), 189–208.

    Article  MathSciNet  MATH  Google Scholar 

  41. Makai, E., Über Eigenwertabschätzungen bei gewissen homogenen linearen Differentialgleichungen zweiter Ordnung, Composito Math. 6 (1939), 368–374.

    MathSciNet  MATH  Google Scholar 

  42. Makai, E. —, Über die Nullstellen von Funktionen, die Lösung Sturm-Liouville’ scher Differentialgleichungen sind, Comment. Math. Helv. 16 (1943-4), 153–199.

    Article  MathSciNet  Google Scholar 

  43. Hartman, P. H., and A. Wintner, On an oscillation criterion of de la Vallée Poussin, Quart. Appl. Math. 13 (1955), 330–332.

    MathSciNet  MATH  Google Scholar 

  44. Sansone, G., “Equazioni differenziali nel campo reale,” Vol. I, 2nd ed., Bologna, 1948.

    Google Scholar 

  45. Fink, A. M. and D. S. St. Mary, On an inequality of Nehari, Notices Amer. Math. Soc. 16 (1969), 91.

    Google Scholar 

  46. Levin, A. Yu, On linear second order differential equations, Dokl. Akad. Nauk SSSR 153 (1963), 1257–1260, (Soviet. Math. 4 (1963), 1814-1817).

    MathSciNet  Google Scholar 

  47. Hochstadt, H., On an inequality of Lyapunov, Proc. Amer. Math. Soc. 22 (1969), 282–284.

    Article  MathSciNet  MATH  Google Scholar 

  48. Fink, A. M., The functional and the zeroes of a second order linear differential equation, J. Math. Pures et Appl. 45 (1966), 387–394.

    MathSciNet  MATH  Google Scholar 

  49. Leighton, W., On the zeroes of solutions of a second-order linear differential equation, J. Math. Pures et Appl. 44 (1965), 296–310.

    MathSciNet  Google Scholar 

  50. Banks, D., Bounds for the eigenvalues of some vibrating systems, Pacific J. Math. 10 (1960), 439–474.

    MathSciNet  MATH  Google Scholar 

  51. Barnes, D. C., Bounds for the eigenvalues of some vibrating systems, Pacific J. Math. 29 (1969), 43–61.

    MathSciNet  MATH  Google Scholar 

  52. Galbraith, A., On the zeros of solutions of ordinary differential equations of the second order, Proc. Amer. Math. Soc. 17 (1966), 333–337.

    Article  MathSciNet  MATH  Google Scholar 

  53. Fink, A. M. On the zeroes of y″+py = 0 with linear, convex and concave p, J. Math. Pures et Appl. 46 (1967), 1–10.

    MathSciNet  MATH  Google Scholar 

  54. Elbert, A., On the zeros of the solutions of the differential equation y″ + q(x)y = 0, where [q(x)]v is concave, Studia Sci. Math. Hungarica 2 (1967), 293–298.

    MathSciNet  MATH  Google Scholar 

  55. Elbert, A. —, On a functional connected with the zeros of the solutions of the differential equation y″ + q(t)y = 0, where q(t) is a nonnegative, monotonic, concave function, Ann. Univ. Sci. Budapest. Rol. Eot. nom. sect. Math. 11 (1968), 129–136.

    MathSciNet  MATH  Google Scholar 

  56. Elbert, A. —, On the zeros of solutions of ordinary second order differential equations, Publ. Math. (Debrecen) 15 (1968), 13–17.

    MathSciNet  MATH  Google Scholar 

  57. Elbert, A. —, On the solutions of the differential equation y″ + q(x)y = 0, where [q(x)]v is concave, I. Acta Math. Acad. Sci. Hung. 201–2 (1969), 1–11.

    Article  MathSciNet  MATH  Google Scholar 

  58. Elbert, A. —, On the solutions of the differential equation y″ + q(x)y = 0, where [q(x)]v is concave, II, Acta Math. Acad. Sci. Hung. 4 (1969), 257–266.

    MathSciNet  MATH  Google Scholar 

  59. Schwarz, B. On the extrema of the frequencies of nonhomogeneous strings with equimeasurable density, J. Math, and Mech. 10 (1961), 401–422.

    MathSciNet  MATH  Google Scholar 

  60. Beesack, P. R. and B. Schwarz, On the zeros of solutions of second-order linear differential equations, Canad. J. of Math. 8 (1956), 504–515.

    Article  MathSciNet  MATH  Google Scholar 

  61. Eliason, S. B., The integral and the boundary value problem, J. Diff. Eq. 4 (1968), 646–659.

    Article  MathSciNet  MATH  Google Scholar 

  62. St. Mary, D. F. and S. B. Eliason, Upper bounds of and the differential equation x″ + p(t)x = 0, J. Diff. Eq. 6 (1969), 154–160.

    Article  MathSciNet  Google Scholar 

  63. Levin, V., On some integral inequalities involving periodic functions, J. London Math. Soc. 10 (1935), 45–48.

    Article  Google Scholar 

  64. SatÔ, T., Seizi Senkei Bibunhôteisiki no Zero-ten no Bunpu ni tuite, Kansû Hôteisiki 22 (1940), 39–43.

    Google Scholar 

  65. Tumura, M., Kôkai Zyôbibunhôteisiki ni tuite, Kansû Hôteisiki 30 (1941), 20–35.

    Google Scholar 

  66. Borg, G., Über die Stabilität geisser Klassen von linearen Differentialgleichungen, Ark. Mat. Ast. Fys. 31 A (1944),No. 1, 31 pp.

    MathSciNet  Google Scholar 

  67. Borg, G. —, On a Liapounoff crieterion of stability, Amer.J. Math. 71 (1949), 67–70.

    Article  MathSciNet  MATH  Google Scholar 

  68. Cordes, H. O., An inequality of G. Borg, Amer. Math. Monthly 63 (1956), 27–29.

    Article  MathSciNet  MATH  Google Scholar 

  69. Cohen, H. J., Problem E 1205, Amer. Math. Monthly 63 (1956), 582–583.

    Article  Google Scholar 

  70. Nehari, Z., Univalent functions and linear differential equations, in “Lectures on Functions of a Complex Variable,” Ann Arbor, 1955, pp. 148–151.

    Google Scholar 

  71. Beesack, P. R., Nonoscillation and disconjugacy in the complex domain, Trans. Amer. Math. Soc. 81 (1956), 211–242.

    Article  MathSciNet  MATH  Google Scholar 

  72. Elbert, Á., Oboscenie odnogo neravenstva Ljapunova dlja differencial’nyh uravneiĭ s zapazdyvajuscim argumentom, Ann. Univ. Sci. Budapest. Rol. Eöt. nom. sect. Mat. 12 (1969), 107–111.

    MathSciNet  Google Scholar 

  73. Eliason, S. B., A Lypanov inequality for a certain second order nonlinear differential equation, J. Lond. Math. Soc. 22 (1970), 461–466.

    MathSciNet  MATH  Google Scholar 

  74. Eliason, S. B. —, Lypanov inequalities and bounds on solutions of certain second order equations, Canad. Math. Bull. 17 (1974), 499–504.

    Article  MathSciNet  MATH  Google Scholar 

  75. St. Mary, D. F., Some oscillation and comparison theorems for (r(t)y’’ + p(t)y = 0, J. Differential Eq. 5 (1969), 314–323.

    Article  MathSciNet  Google Scholar 

  76. Kwong, M. K., On Lyapunov’s inequality for disfocality, J. Math. Anal. Appl. 83 (1981), 486–494.

    Article  MathSciNet  MATH  Google Scholar 

  77. Harris, B. J., On an inequality of Lyapunov for disfocality, J. Math. Anal. Appl. 146 (1990), 495–500.

    Article  MathSciNet  MATH  Google Scholar 

  78. Agarwal, R. P. and G. V. MilovanoviĆ, On an inequality of Bogar and Gustafson, J. Math. Anal. Appl. 146 (1990), 207–216.

    Article  MathSciNet  MATH  Google Scholar 

  79. Bogar, G. A. and G. B. Gustafson, Effective estimates of invertibility intervals for linear multipoint boundary value problems, J. Differential Eq. 29 (1978), 180–204.

    Article  MathSciNet  MATH  Google Scholar 

  80. Leighton, W., On Liapunov’s inequality, Proc. Amer. Math. Soc. 33 (1972), 627–628.

    MathSciNet  Google Scholar 

  81. Kreïn, M. G., Ob odnom predpoloženii A. M. Ljapunova, Funk. Anal. Prilož. 7 (1973), 45–54.

    Google Scholar 

  82. Elbert, A., Nekotorye zadači kačestvennoĭ teorii differencial’nyh uravneniĭ vtorogo porjadka, Univ. Družbi narodov im. P. Lumumby, Moskva, 1971.

    Google Scholar 

  83. Beesack, P. R., On the Green’s function of an N-point boundary value problem, Pacific J. Math. 12 (1962), 801–812.

    MathSciNet  MATH  Google Scholar 

  84. Cheng, S. S., A discrete analogue of the inequality of Lyapunov, Hokkaido Math. J. 121 (1983), 105–112.

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. University of Belgrade, Yugoslavia

    D. S. Mitrinović

  2. University of Zagreb, Yugoslavia

    J. E. Pečarić

  3. Iowa State University, Ames, USA

    A. M. Fink

Authors
  1. D. S. Mitrinović
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  2. J. E. Pečarić
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  3. A. M. Fink
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Mitrinović, D.S., Pečarić, J.E., Fink, A.M. (1991). Inequalities of Lyapunov and of De La Vallée Poussin. In: Inequalities Involving Functions and Their Integrals and Derivatives. Mathematics and Its Applications, vol 53. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3562-7_6

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