## The Cauchy-Kovalevsky Theorem and Noncompactness Measures

J. Math. Sci. Univ. Tokyo
Vol. 4 (1997), No. 3, Page 627--647.

Ghisi, Marina
The Cauchy-Kovalevsky Theorem and Noncompactness Measures
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Abstract:
We give an abstract version of the Cauchy-Kovalevsky Theorem for the equation $u' = A(t,u)$ where $A$ is a Caratheodory operator having properties based on noncompactness measures, including Lipschitz and compactness conditions. We give an application of this result to the equation $\partial_{t}^n u + \sum_{i=1,n} f_{i}(u)B^{(n - i + 1)} \partial_{t}^{i - 1}u = 0$ that generalizes the Kirchhoff equation for the vibrating string, when $B$ is {\em not} a compact operator. Our technique is based on Nagumo's weights and on Tonelli delayed problems.

Mathematics Subject Classification (1991): 35A10
Mathematical Reviews Number: MR1484605