Abstract
The general form of a quasilinear nonstationaryk-step method for solving of the Cauchy problem for ordinary differential equations is discussed. The convergence theorem is stated under rather weak conditions. It is not assumed that the increment function is Lipschitz-continuous but only that it satisfies the Perron type condition appearing in the uniqueness theory for the Cauchy problem with a nondecreasing comparison function. The result established in the paper is an extension of the theory given by G. Dahlquist and the recent result of K. Taubert.
Zusammenfassung
Es wird ein allgemeines, quasilineares, nichtstationäresk-Schrittverfahren für die Lösung des Cauchy-Problems für gewöhnliche Differentialgleichungen untersucht. Ein Konvergenzsatz mit ziemlich schwachen Bedingungen wird angegeben. Die Inkrementfunktion muß nicht Lipschitz-stetig sein; es genügt, wenn diese Funktion die Perron-Bedingung aus der Eindeutigkeitstheorie für das Cauchy-Problem mit nichtabnehmender Vergleichfunktion erfüllt. Das Ergebnis ist eine Erweiterung der Theorie von G. Dahlquist und des letzten Resultats von K. Taubert.
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Jackiewicz, Z., Kwapisz, M. On the convergence of multistep methods for the Cauchy problem for ordinary differential equations. Computing 20, 351–361 (1978). https://doi.org/10.1007/BF02252383
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DOI: https://doi.org/10.1007/BF02252383