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On an Inequality of Peano(1)

  • James S. Muldowney (a1)

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Let f be a real valued function on an open subset of R2 . It is assumed that f satisfies Carathéodory's conditions: f (t,x) is continuous in x for each t, Lebesgue measurable in t for each x and there is a locally integrable function m(t) such that |f(t, x)| ≤ m(t) uniformly in x. A proof will be given of the following theorem.

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Copyright

Footnotes

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(1)

This work was supported by Defence Research Board Grant DRB-9540-28.

Footnotes

References

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1. Cafiero, F., Su un problema ai limiti relativo al’equazione y' = f(x, y, λ), Giorn. Mat. Battaglini 77 (1947), 145163.
2. Kamke, E., Differentialgleichungen reeler Funktionen, Akademische Verlagsgesellschaft, Leipzig, 1930; Chelsea, New York, 1947.
3. Lakshmikantham, V., On the boundedness of solutions of nonlinear differential equations, Proc. Amer. Math. Soc. 8 (1957), 10441048.
4. Natanson, I. P., Theory of functions of a real variable, Ungar, New York, 1961.
5. Olech, C. and Opial, Z., Sur une inégalité différentielle, Ann. Polon. Math. 7 (1960), 247264.
6. Peano, G., Sul'integrabilità délie equazione differenziali di primo ordine, Atti. Accad. Sci. Torino, Atti. R.|Accad. Torino 21 (1885/86), 677685.
7. Perron, O.,Ein neuer Existenzbeweis für die Intégrale der Differentialgleichung y' = f(x,y), Math. Ann. 76 (1915), 471484.
8. Reid, W. T., Ordinary differential equations, Wiley-Interscience, New York, 1971.
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On an Inequality of Peano(1)

  • James S. Muldowney (a1)

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