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

Full Text: DOI


[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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.