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The relation between the notions of nonuniform asymptotic stability and admissibility is considered. Using appropriate Lyapunov norms, it is showed that if any of their associated

The study of the admissibility property has a fairly long history, and it goes back to the pioneering work of Perron [

Over the last decades an increasing interest can be seen in the study of the asymptotic behavior of evolution equations in abstract spaces. In [

In the case of nonuniform exponential dichotomies, Preda and Megan [

In the present paper, inspired by Barreira and Valls [

We first concentrate on the simpler case of admissibility for nonuniform

We say that an increasing function

We say that a family of linear operators

In this section, we also assume that

there exist

If

We have

We say that an evolution process

In the following, we introduce several Banach spaces that are used throughout the paper. We first set

For each Banach space

Repeating arguments in the proof of Theorem 3 in [

For each

We say that a Banach space

By Lemma

There exists

We define a linear operator

If for some

We follow arguments in [

Now given

We consider the spaces

If the evolution process

We first take

Now we take

We consider in this second part admissibility for nonuniform

We consider an evolution process

We also consider a function

We will refer to

the map

is invertible for every

We also assume that

there exist

We note that due to the invertibility assumption in condition (

We always consider in the paper an evolution process

We say that an evolution process

In the following, we still consider several spaces

We also obtain easily the same statement in Lemma

We say that a Banach space

the function

is in

the function

is in

We note that since

If for some

We follow arguments in [

We define a linear operator

Using the similar proof of Lemma

It follows from (

Therefore, for each

If for some

We first consider the space

Now we consider the space

On the other hand, for each

Now given

We consider the spaces

If the evolution process

We first take

Now we take

This work is partly supported by the National Natural Science Foundation of China under Grant no. 11171090 and the Fundamental Research Funds for the Central Universities. The authors would like to show their great thanks to Professor Jifeng Chu for his useful suggestions and comments.