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

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Literatur

  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.

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  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).

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  8. We make use of the procedure applied to prove the generalized Bellman lemma.

  9. Here ω(u) is subjected to the same conditions as in 3 and Ω(u) is also the same function as in 3.

  10. A similar formula holds for x ≦ ξ2.

  11. Kamke, loc. cit.,Differentialgleichungen reeller Funktionen, p. 87.

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  19. Kamke, loc. cit.Differentialgleichungen reeller Funktionen, p. 82, Satz 1.

  20. Kamke, loc. cit.,Differentialgleichungen reeller Funktionen, p. 83.

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  1. Budapest

    I. Bihari

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  1. I. Bihari
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Bihari, I. A generalization of a lemma of bellman and its application to uniqueness problems of differential equations. Acta Mathematica Academiae Scientiarum Hungaricae 7, 81–94 (1956). https://doi.org/10.1007/BF02022967

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