Abstract
The purpose of this paper is to extend a well-known theorem of Lyapunov on matrices to unbounded linear operators in an infinite-dimensional Banach space by using semigroup theory. This extension is used for the study of stability problems of nonlinear differential equations. In the special case of a Hilbert space, a direct extension of Lyapunov’s theorem for a bounded operator is obtained; and in the general case of a Banach space the idea of equivalent semi-inner products is introduced. Applications are given to a class of initial boundary value problems in a bounded domain in $R^n $ under nonlinear perturbations.
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