Barbashin type characterizations for nonuniform h -dichotomy of evolution families

: The aim of this paper is to give some Barbashin type characterizations for the nonuniform h-dichotomy of evolution families in Banach spaces. Two necessary and sufficient conditions for the uniform h -dichotomy are pointed out using some important sets of growth functions. Additionally, as particular cases, we obtain a Barbashin type characterization for nonuniform exponential dichotomy and respectively a necessary and sufficient condition for the nonuniform polynomial dichotomy


Introduction
In the last decades, the qualitative theory for dynamical systems on Banach spaces is intensively investigated in the literature.Various results concerning this field have witnessed considerable development.Some concepts of the qualitative behaviors were defined and improved, such as exponential (in)stability, polynomial (in)stability, and h-(in)stability, based on the fact that the dynamical systems describing the process of science or engineering is extremely complex and it is difficult to determine an appropriate mathematical model.One of the most celebrated theorems in the qualitative theory of differential systems was given by Barbashin [1] in 1967.Barbashin


for all xX  , where X is a Banach space.Since then this theorem has inspired many extensions and generalizations along this line (see [2][3][4][5][6][7][8][9][10] and the references therein).For example, some Barbashin type conditions for uniform exponential stability of linear skewevolution semiflows were established by Hai [2] in terms of the existence of some functionals on certain function (sequence) spaces.In [6], Dragičević formulated Barbashin type conditions for (non)uniform exponential stability for linear cocycles over both maps and flows by making use of ergodic theory.In addition, the Barbashin type integral characterizations for uniform h-stability of evolution operators were investigated by Boruga, Megan and Toth in [8].Very recently, in [9], through the usage of Banach function (sequence) spaces, we obtained some discrete and continuous versions of Barbashin type theorem for uniform polynomial stability and respectively uniform polynomial instability of evolution families.
As a natural generalization of exponential (in)stability, exponential dichotomy is one of the most important asymptotic properties in the qualitative theory of evolution equations.To the best of our knowledge, the first study on the exponential dichotomy of differential equations was presented by Perron [11] in 1930.After the seminal work of Perron, many authors have made valuable contributions to this line of the research.For details and references we refer the reader to [12][13][14][15][16][17][18][19].
However, there are some situations where the notion of exponential dichotomy may look as too restrictive, therefore it is important to search for more general type of dichotomic behavior.In this sense we refer to the notion of polynomial dichotomy which was first considered in 2009 by Bento and Silva [20] for the discrete time case, Barreira and Valls [21] for the continuous time case and then it was discussed in the works of Dragičević [22,23], Boruga and Megan [24].In particular, in [24] the authors obtained two conditions of Datko type for the existence of the nonuniform polynomial dichotomy for evolution operators.In addition to the aforementioned references, we mention a recent and interesting paper of Megan et al. [25], where the authors proposed a more general notion, the socalled nonuniform h-dichotomy.Simultaneously, they established some Datko type characterizations for the nonuniform h-dichotomy of skew-evolution cocycles in Banach spaces.As is well known, in the Datko type theorems, the integration variable is the first parameter of the evolution family, while in the Barbashin type theorems, the integration variable is its second parameter.Naturally, the question arises whether Barbashin's theorem can be generalized to the case of a nonuniform h-dichotomy.
Inspired by [25], the main purpose of this paper is to obtain some Barbashin type conditions for the nonuniform h-dichotomy of evolution families in Banach spaces, by using some important sets of growth rates.The paper is organized as follows.In section 2, some notations, definitions and preliminary facts will be introduced.Section 3 is devoted to establishing the Barbashin type characterizations for the nonuniform h-dichotomy of evolution families.It should be noted that the growth rates considered in the main results of this paper are different from that used in [17].The growth rates used in this paper depends on Definition 2.7, while the growth rates used in [17] only require differentiability.Furthermore, in [17], the authors only established some Datko type conditions for the existence of nonuniform μ-dichotomy of evolution operators, and did not discuss its Barbashin type characterizations.for all ( , )  ts.

Notations and Preliminaries
is invariant to the evolution family U , then the pair ( , ) UP is called a dichotomic pair.Definition 2.4 (see [27]) We say that a nondecreasing function : In what follows, we suppose that : h + → is a growth rate.Definition 2. 5 The dichotomic pair ( , )  UP is called nonuniformly h-dichotomic (n.h.d) if there exist a nondecreasing function : It should be noted that the concept of nonuniform h-dichotomy considered in this paper is weaker than Definition 6 in [25].In Definition 2.5 above, the function N in condition (nhd1) depends on the first parameter t of the evolution family, while in [25], the function N in condition (nhd1) depends on the second parameter s.

Remark 2.2 In Definition 2.5, if we consider (i) ()
N t N = (a constant), then we obtain the property of uniform h-dichotomy (u.h.d) ; (ii) for all ( , , ) t s x X    .Thus Definition 2.5 is satisfied for ( ) 2 t Nt = and

If we suppose that ( , )
UP is u.h.d., then there exist two constants 1 ts.In particular, for 0 s = and t →, we obtain a contradiction and thus ( , )  UP is not u.h.d.6 We say that the dichotomic pair ( , )  UP has a nonuniform h-growth (n.h.g) if there exist a nondecreasing function : [1, ) M + → and a constant 0

Remark 2.3 The dichotomic pair
, for all ( , , )  t s x X    .

Remark 2.5
As particular cases of Definition 2.6, we give the following: (i) for () M t M = , we say that the pair ( , ) UP has a uniform h-growth; (ii) for ()  t h t e = , we say that the pair ts.In particular, for 0 s = and t →, we obtain a contradiction and thus ( , )  UP is not n.h.d.

Remark 2.7 The dichotomic pair ( , )
UP has a nonuniform h-growth if and only if there are a nondecreasing function : [1, ) M + → and 0   such that:  , for all ( , , ) t s x X    .Definition 2.7 (see [8]) We introduce the following classes of growth rates, which are very helpful for us to prove the main results: (i) 0 is the set of all growth rates : [1, ) is the set of all growth rates : [1, ) h + → with the property that there exists for all 0 t  ; (iii) 1 is the set of all growth rates : [1, ) h + → with the property that there exists 1 1  H  such that

( ( )) ( ) h h t H h t 
(2) for all 0 t  ; (iv) 2 is the set of all growth rates : [1, ) h + → with the property that for all 0   , there exists for all 0 t  ; (v) 3 is the set of all growth rates : [1, ) h + → with the property that for all 0   , there exists for all 0 t  .

The main results
In this section we will extend the Barbashin type results from [8] on uniform h-stability of evolution operators to the case of nonuniform h-dichotomy of evolution families.Throughout this section we suppose that U is reversible.We start with a Barbashin type characterization for the nonuniform h-dichotomy, by using growth rates in 2 .Theorem 3.1 Let 2 h  and ( , ) UP has a nonuniform h-growth.Then the pair ( , ) UP is nonuniformly h-dichotomic if and only if there are a nondecreasing function : [1, ) B + → and a constant 0 b  such that:

N t B t H h t h t P t x h t P t x
Analogously, by ( 2nhd ) and ( 3) we have Sufficiency.Let ( , )  ts.Firstly we prove that (  1) and ( 1nhD ), we have Thus we get ) and (1), we have that Based on ( 5) and ( 6) we obtained that there exist  1) and ( 2nhD ), we have ) and ( 4) we have ) and (4) we have From ( 7) and ( 8), it follows that there exist

1 nhD
. Subsequently, this result was extended to the case of evolution families, that is, an exponentially bounded evolution family   If the pair ( , ) UP is n.h.d., then it has n.h.g.But the converse is not necessarily valid.Example 2.2 (Dichotomic pair which has n.h.g. and is not n.h.d.) . Finally, by virtue of Remark 2.4, we conclude that the pair( , )UP is n.h.d.□ As a direct consequence of Theorem 3.1, we obtain the following corollary, which is a version of Barbashin's theorem for the case of the nonuniform exponential dichotomy concept. ,Theorem 3.1 does not include the particular case of nonuniform polynomial dichotomy.In order to make the conclusion include the case of nonuniform polynomial dichotomy, below we present another characterization of Barbashin type for the nonuniform h-dichotomy.