In this paper the notion of the derivative of the norm of a linear mapping in a normed vector space is introduced. The fundamental properties of the derivative of the norm are established. Using these properties, linear differential equations in a Banach space are studied and lower and upper estimates of the norms of their solutions are derived.
In the paper the concept of a controllable continuous flow in a metric space is introduced as a generalization of a controllable system of differential equations in a Banach space, and various kinds of stability and of boundedness of this flow are defined. Theorems stating necessary and sufficient conditions for particular kinds of stability and boundedness are formulated in terms of Ljapunov functions.
In the paper definitions of various kinds of stability and boundedness of solutions of linear controllable systems of partial differential equations are introduced and their interconnections are derived. By means of Ljapunov's functions theorems are proved which give necessary and sufficient conditions for particular kinds of stability and boundedness of the solutions.
The properties of solutions of the nonlinear differential equation in a Banach space and of the special case of the homogeneous linear differential equation are studied. Theorems and conditions guaranteeing boundedness of the solution of the nonlinear equation are given on the assumption that the solutions of the linear homogeneous equation have certain properties.
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