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Differential equations in Banach spaces and the extension of Lyapunov's method

Published online by Cambridge University Press:  24 October 2008

V. Lakshmikantham
V. Lakshmikantham
Affiliation:
Mathematics Research Center, United States Army, University of Wisconsin
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Extract

The concept of Lyapunov's function is an important tool in studying various problems of ordinary differential equations. In the present paper we shall extend the Lyapunov's method to study some problems of differential equations in Banach spaces. Continuing the theory of one parameter semi-groups of linear and bounded operators founded by Hille and Yoshida, Kato(4) presented some uniqueness and existence theorems for the solutions of linear differential equations of the type

where A(t) is a given function whose values are linear operators in Banach space. Krasnoselskii, Krein and Soboleveskii (5,6) also considered such equations including non-linear differential equations of the type

Mlak (9) obtained some results concerning the limitations of solutions of the latter equation.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 59 , Issue 2 , April 1963 , pp. 373 - 381
Copyright
Copyright © Cambridge Philosophical Society 1963

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References

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