A generalization of a lemma of bellman and its application to uniqueness problems of differential equations

  • I. Bihari


Differential Equation Uniqueness Problem 
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  1. 1.
    R. Bellman, The stability of solutions of linear differential equations,Duke Math. Journal,10 (1943), pp. 643–647. However, the lemma holds for arbitrary continuousY(x) and non-negative continuousF(x). The valuek=0 is also possible.MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
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  5. 5.
    This procedure may be generalized: If in (1)k=0,F(t) is continuous ina<x≦b and\(\mathop {\lim }\limits_{x = a + 0} F(x)Y(x) = A\) exists and\(\mathop {\lim }\limits_{\delta = a + 0} \delta e^{\int\limits_{a + \delta }^x {F\left( t \right)dt} } \leqq K(x)\), thenY(x)≦AK(x).Google Scholar
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    O. Perron, Eine hinreichende Bedingung für Unität der Lösung von Differentialgleichungen erster Ordnung,Math. Zeitschrift,28 (1928), pp. 216–219.Perron has shown thatM=1 cannot be increased at all.MathSciNetCrossRefMATHGoogle Scholar
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    W. F. Osgood, Beweis der Existenz einer Lösung der Differentialgleichungy′=f(x,y) ohne Hinzuname der Cauchy-Lipschitz Bedingung,Monatschefte f. Math. u. Phys.,9 (1898), pp. 331–345.MathSciNetMATHGoogle Scholar
  8. 8.
    We make use of the procedure applied to prove the generalized Bellman lemma.Google Scholar
  9. 9.
    Here ω(u) is subjected to the same conditions as in 3 and Ω(u) is also the same function as in 3.Google Scholar
  10. 10.
    A similar formula holds for x ≦ ξ2.Google Scholar
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    Kamke, loc. cit.,Differentialgleichungen reeller Funktionen, p. 87.Google Scholar
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    O. Perron Ein neuer Existenzbeweis für die Integrale der Differentialgleichungy′=f(x, y), Math. Ann.,76 (1915), pp. 471–484, especially pp. 473 and 479.MathSciNetCrossRefMATHGoogle Scholar
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    Kamke, loc. cit.,Differentialgleichungen reeller Funktionen, p. 83.Google Scholar
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    Kamke, loc. cit.Differentialgleichungen reeller Funktionen, p. 82, Satz 1.Google Scholar
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    Kamke, loc. cit.,Differentialgleichungen reeller Funktionen, p. 83.Google Scholar

Copyright information

© Magyar Tudományos Akadémia 1956

Authors and Affiliations

  • I. Bihari
    • 1
  1. 1.Budapest

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