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On the Perron-Bellman theorem for systems with constant coefficients. (English) Zbl 0536.34038
In the present paper we determine in a sense ”the smallest” class of functions f(t), for which if we have bounded solutions for the Cauchy problem: $$\dot x=Ax+f(t)$$, $$x(0)=0$$, then the homogeneous system $$\dot x=Ax$$ is exponentially stable.

##### MSC:
 34G10 Linear differential equations in abstract spaces 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 34D05 Asymptotic properties of solutions to ordinary differential equations
##### Keywords:
exponential stability; Cauchy problem