By Professor Masaya Yamaguti, Professor Louis Nirenberg (auth.), Professor Masaya Yamaguti, Professor Louis Nirenberg, Professor Sigeru Mizohata, Professor Yasutaka Sibuya (eds.)
In particular between jap mathematicians Mitio Nagumo (b.1905) is considered one of many maximum pioneers in learn on differential equations. notwithstanding, to date so much of his papers have simply been released in jap journals and have been unavailable within the West. This Collected Papers quantity includes essentially all mathematical papers Nagumo wrote in languages except eastern and may be a uncomplicated reference quantity and crucial operating software for each library and for lots of energetic mathematicians in differential equations, topology, and differential geometry. additionally, papers that have been initially released in eastern have been translated particularly for this variation. There are 3 major sections during this e-book, dedicated to usual differential equations, partial differential equations and different equations. every one part is followed via an in depth remark supplied by way of the editors.
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Die Gesamtheiten del' Kurven Xl=~(t), (i=l, 2, ... , IC), von ~'" die durch Po gehen und in M' bzw. M" munden, bezeichnen wir mit St' bzw. ) im B mit M/ bzw. M/,. M/ und M/' sind auch abgeschlossen. Dann konnen wir einen Wert t=t2 finden, so dass fiir aIle 7' zwischen tl und t2 die abgeschlossene Mengen M/ und M/' 30 SYSTEM DER GEWOHNLICHEN J)JFFERENTIALGLEIGHUNGEN. 22t1 von einander fremd sind. Del' Bequemlichkeit hal bel' setzen wir voraus, dass tJ < tl> also auch to < t2 < t1• Die untere Grenze von solchem t2 sei t".
Die durch xl=x/,(t) dargesteUten Kurven laufen fur 7"-0 < t < 7"+0 ganz im B. D. , fiir 7"-0 < t :S 7"+0 sind stets: / Xln(t)-XIO / J~("') fn(x)dx a fiir aIle x in ~(a) < a, b >. n (ep(x)) ep' (x) dx = a F(ep(x)) ep'(x)dx. a I fn(x) I in < a, b > gleichmassig beschrankt, und die b> summierbar ist, so ist die Folge I fn(ep(x))ep'(x) I in < a, b > gleichmassig summierber. Aus fn(ep(x)) = rJn(ep(x))ep' (x)J ep'~X) und del' gleichmassigen Beschranktheit von I fn(ep(x)) I in < a, b > folgt Da die Folge Funktion ep'(x) in < a, dann nach Satz 3 J'" F(ep(x))dx. tz 5. ~n: I fn(x) I in < a, b > gleichmassig I '"fn(x)dx= J'" F(x)dx a a fur alle x in < a, b >.