On uniform exponential N -dichotomy

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and hence E X~ .Let JI~ be the set of strictly increasing real functions N defined on ~0, oo~ wich satisfies : = 0 and N(t.to) N(t)N(to) for all t, to > o.
The case of uniform exponential-N-stable processes has been considered by Ichikawa in [2 j.The particular case when the process is a strongly continuous semigroup of bounded linear operators has been studied by Zabczyk in [5 ]and Rolewicz in [4 j.
In this paper we shall extend these results in two directions.First, we shall give a characterization of u.e.-N-dichotomy, which can be considered as a generalization of , Datko's theorem.Second, we shall not assume the linearity and boundedness of the pro- cess ~(., .).The obtained results are applicable for a large class of nonlinear differential equations described in [2 ].
Proof.: Is obvious from Proposition 2,I for N(t) = t.

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The relation between u, e-N-d.and u.e.d.properties is given by Proposition 2.2 : The evolutionary process W(., .) is u.e.d. if and only if there is N e M such that W(., .) is u.e-N-d.

Proof :
The necessity is obvious from Definition 1.2.
Then for.s > to > 0 and z.i e X[ we have the other hand, for s > to > 0 and x2 E X a we have On = ss 2u/s2.