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


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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 · doi:10.1215/S0012-7094-43-01059-2
[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.
[4] M. Nagumo, Eine hinreichende Bedingung für die Unität der Lösung von Differentialgleichungen erster Ordnung,Japanese Journal of Math.,3 (1926), pp. 107–112.
[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. · doi:10.1007/BF01181159
[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.
[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. · doi:10.1007/BF01206598
[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.
[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. · doi:10.1007/BF01458218
[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. · doi:10.1007/BF01175682
[18] M. Lavrentiev, Sur une équation différentielle du premier ordre,Math. Zeitschrift,23 (1925), pp. 197–198. · doi:10.1007/BF01506227
[19] Kamke, loc. cit.Differentialgleichungen reeller Funktionen, p. 82, Satz 1.
[20] Kamke, loc. cit.,Differentialgleichungen reeller Funktionen, p. 83.
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