## A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations.(English. Russian summary)Zbl 0070.08201

### Keywords:

ordinary differential equations
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 [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. · Zbl 0061.18502 [2] E. g. В. В. Немыцкий и В. В. Степанов, Качественная теория дифферен-пиальных уравнений (Москва, 1947), p. 19. · Zbl 1152.17301 [3] E. g.E. Kamke,Differentialgleichungen reeller Funktionen, p. 93. · JFM 56.0375.03 [4] M. Nagumo, Eine hinreichende Bedingung für die Unität der Lösung von Differentialgleichungen erster Ordnung,Japanese Journal of Math.,3 (1926), pp. 107–112. · JFM 52.0438.01 [5] This procedure may be generalized: If in (1)k=0,F(t) is continuous ina<x 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)K(x). [6] 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. · JFM 54.0451.03 [7] 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. · JFM 29.0260.03 [8] We make use of the procedure applied to prove the generalized Bellman lemma. [9] Here {$$\omega$$}(u) is subjected to the same conditions as in 3 and {$$\Omega$$}(u) is also the same function as in 3. [10] A similar formula holds for x {$$\xi$$}2. [11] Kamke, loc. cit.,Differentialgleichungen reeller Funktionen, p. 87. [12] Kamke, loc. cit,Differentialgleichungen reeler Funktionen, p. 78. [13] O. Perron, Über Ein- und Mehrdeutigkeit des Integrals eines systems von Differentialgleichungen,Math. Ann.,95 (1926), pp. 98–101. · JFM 51.0331.07 [14] E. Bompiani, Un teorema di confronto ed un teorema di unicità per l’equazione differenzialey’=f(x, y), Rendiconti dell’Accad. dei Lincei, Classe di Scienze Fisiche, (6),1, (1925), pp. 298–302. · JFM 51.0331.06 [15] 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. · JFM 45.0469.01 [16] Kamke, loc. cit.,Differentialgleichungen reeller Funktionen, p. 83. [17] J. Tamarkine, Sur le théorème d’unicité des solutions des équations différentielles ordinaires,Math. Zeitschrift,16 (1923), pp. 207–212. · JFM 49.0302.01 [18] M. Lavrentiev, Sur une équation différentielle du premier ordre,Math. Zeitschrift,23 (1925), pp. 197–198. · JFM 51.0332.04 [19] Kamke, loc. cit.Differentialgleichungen reeller Funktionen, p. 82, Satz 1. [20] Kamke, loc. cit.,Differentialgleichungen reeller Funktionen, p. 83.
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