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The Scientific World Journal
Volume 2013 (2013), Article ID 793813, 8 pages
Trichotomy for Dynamical Systems in Banach Spaces
Department of Mathematics and Computer Science, Aurel Vlaicu University of Arad, 2 Elena Drăgoi Str, 310330 Arad, Romania
Received 25 July 2013; Accepted 13 August 2013
Academic Editors: G. Bonanno, F. Minhós, and G.-Q. Xu
Copyright © 2013 Codruţa Stoica. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We construct a framework for the study of dynamical systems that describe phenomena from physics and engineering in infinite dimensions and whose state evolution is set out by skew-evolution semiflows. Therefore, we introduce the concept of -trichotomy. Characterizations in a uniform setting are proved, using techniques from the domain of nonautonomous evolution equations with unbounded coefficients, and connections with the classic notion of trichotomy are given. The statements are sustained by several examples.
The possibility of reducing the nonautonomous case in the study of associated evolution operators to the autonomous case of evolution semigroups on various Banach function spaces can be considered as an important way towards applications issued from the real world. Of great importance in the study the solution of differential equations is the approach by evolution families, as the techniques from the domain of non autonomous equations with unbounded coefficients in infinite dimensions can be extended in this direction.
Appropriate for the study of evolution equations in infinite dimensions are also the skew-evolution semiflows, introduced by us in , as generalizations for evolution operators and skew-product semiflows, and whose applicability is pointed out, for example, by Bento and Silva in  and by Hai in [3, 4]. Other asymptotic properties for skew-evolution semiflows are defined and characterized in [5, 6].
The techniques used in the investigation of exponential stability and exponential instability were generalized for the case of exponential dichotomy in [7, 8] and for the case of exponential trichotomy in [9, 10]. The main idea in the study of trichotomy, introduced for the finite dimensional case by Sacker and Sell in , as a natural generalization of the concept of dichotomy is to obtain, at any moment, a decomposition of the state space into three subspaces: a stable subspace, an instable one, and a third one, the central manifold. The aim of the present study is to emphasize the property of -trichotomy for skew-evolution semiflows in Banach spaces and to give several conditions in order to describe the behavior related to the third subspace.
2. Skew-Evolution Semiflows
Let be a metric space, a Banach space, and its topological dual. Let be the space of all -valued bounded operators defined on . The norm of vectors on and on and of operators on is denoted by . Let us consider that and . is the identity operator.
Definition 1. A mapping is called evolution semiflow on if the following properties are satisfied:
Definition 2. A mapping is called evolution cocycle over an evolution semiflow if it satisfies the following properties:
Definition 3. The mapping defined by where is an evolution cocycle over an evolution semiflow , is called skew-evolution semiflow on .
Example 4. Let be a decreasing function with the property that there exists . We denote by the set of all continuous functions , endowed with the topology of uniform convergence on compact subsets of , metrizable by means of the distance
If , then, for all , we denote that , . Let be the closure in of the set . It follows that is a metric space, and the mapping , is an evolution semiflow on .
We consider that , with the norm , . If , then the mapping defined by is an evolution cocycle over , and is a skew-evolution semiflow.
A particular class of skew-evolution semiflows is emphasized in the following.
Remark 5. Let us consider a skew-evolution semiflow and a parameter . We define the mapping
We observe that is also a skew-evolution semiflow, called -shifted skew-evolution semiflow on . We will call the -shifted evolution cocycle.
Example 6. Let be the metric space defined in the first Example. We define the mapping , , where , for all , which is a classic semiflow on . Let us consider for every the parabolic system with Neumann's boundary conditions as follows:
Let be a separable Hilbert space with the orthonormal basis , , , where , .
We denote that , and we define the operator
which generates a -semigroup , defined by where denotes the scalar product in . For every , let us define an operator , , which allows us to rewrite system (5) in as
is a classic cocycle over the semiflow , and is a linear skew-product semiflow strongly continuous on . Also, for all , we have obtained that , , is a strongly solution of system (8). As is a skew-product semiflow on , then the mapping , , where
is a skew-evolution semiflow on . Hence, the skew-evolution semiflows generalize the notion of skew-product semiflows.
More directly, if is the solution of the Cauchy problem
then is a linear skew-evolution semiflow.
Let us recall the definition of a semigroup of linear operators, and let us give an example which shows that this is generating a skew-evolution semiflow.
Definition 7. A mapping is called semigroup of linear operators on if the following relations hold:
Example 8. One can naturally associate to every semigroup of operators the mapping , defined by , which is an evolution cocycle on over evolution semiflows given, for example, by (see Example 4).
Other examples of skew-evolution semiflows are given in . The asymptotic properties, as well as their characterizations, are given by means of norms of the trajectories or orbits of , given by , which are considered measurable. A particular case of skew-evolution semiflows is given by the following.
Definition 9. A skew-evolution semiflow is -strongly measurable if, for every , the mapping given by is measurable on .
3. On Trichotomy Issues
We intend to give a new approach for the property of trichotomy for skew-evolution semiflows, the -trichotomy. Some examples and connections with the classic concept of exponential trichotomy are also provided.
Let , be a skew-evolution semiflow on . We recall that a mapping with the property
is called projections family on .
Definition 10. A projections family is said to be invariant relative to the skew-evolution semiflow if
The splitting of the state space into three subspaces will be assured by the following.
Definition 11. Three projections families are said to be compatible with a skew-evolution semiflow if
each of , is invariant relative to ;
for all , the projections verify the relations
In what follows we introduce the elements which will allow us to introduce a new concept of trichotomy for skew-evolution semiflows. We consider a mapping , and we define
Remark 12. For every function , the following statements hold:(i) and ;(ii) implies the existence of a constant such that
(iii) implies that there exists a constant such that
Let us denote that
Definition 13. A skew-evolution semiflow is called -trichotomic if there exist three projections families compatible with and some functions with the properties
such that();();() and , for all and all .
Example 14. Let be a decreasing function with the property that there exists . Let us denote that . Let us consider the Banach space with the norm , . The mapping
where , , is an evolution cocycle over the evolution semiflow given in Example 4. We define the projections families by , , , for all and all . The following inequalities
hold for all , and all . The mappings , , , defined by
satisfy relations (19). For and according to Definition 2, we obtain relations (i)–(iii) in Definition 13. Hence, is -trichotomic.
Remark 15. For , the property of -dichotomy is obtained. If we consider , we obtain the property of -stability and for , the property of -instability.
In what follows, if is a given projections family, we will denote that
for every and . We remark that the following relations hold:(i), ;(ii), , , .
Proposition 16. A skew-evolution semiflow is -trichotomic if and only if there exist some constants , , , , , and three projections families compatible with such that
for all and all .
Proof. Necessity. Let be -trichotomic. Then there exist three projections families compatible with and three functions with the properties
such that relations (i)–(iii) of Definition 13 are verified. For , there exists a constant such that , for all , and for there exists a constant with the property , for all .
As , there exist and , such that , for all . Hence, relation is obtained. As , it follows that there exist and , such that , for all , which implies . From and , it follows that there exist , with the property
Hence, relation is satisfied.
Sufficiency. Let us assume that there exist three projections families compatible with and , , , , , such that relations – hold. Let us define the mappings , , by
We obtain the relations
Hence, relations (i)–(iii) of Definition 13 are verified, and, therefore, is -trichotomic, which ends the proof.
4. Main Results
We obtain a characterization for the property of trichotomy, by means of the shifted skew-evolution semiflow.
Theorem 18. A skew-evolution semiflow is -trichotomic if and only if there exist three projections families compatible with and(i)the evolution cocycle is exponentially stable;(ii)the evolution cocycle is exponentially instable;(iii)there exists a constant such that the -shifted evolution cocycle is exponentially stable and the -shifted evolution cocycle is exponentially instable.
Proof. Necessity. Statements (i) and (ii) are obtained immediately from the necessity of Proposition 16. According to , there exist and such that
Let us consider that . We obtain successively
for all and all , which shows that is exponentially stable. Also, we have
If we consider that , we obtain, according also to Definition 2 ,
for all and all , which proves that is exponentially instable.
Sufficiency. Relations (i) and (ii) in Definition 13 are obtained from the sufficiency of Proposition 16. As there exists a constant such that the skew-evolution semiflow is exponentially stable, then there exists some constants and such that
Further, we obtain
for all and all . Denoting
the first relation in is obtained.
If there exists a constant such that the -shifted skew-evolution semiflow is exponentially instable, then there exist some constants and such that
for all and all . If we consider defined as in (34), the second relation in is obtained. Hence, also (iii) in Definition 13 is true and is -trichotomic.
Another characterization for the property of -trichotomy is given relative to the dual space of the Banach space . To this aim, let us consider three projections families compatible with such that the evolution cocycle has exponential growth and the evolution cocycle has exponential decay.
Theorem 19. A -strongly measurable skew-evolution semiflow is -trichotomic if the and only if the following statements hold:(i)there exists such that for all , all , and all ;(ii)there exists such that for all , all , and all ;(iii)there exist and such that for all and all and there exist and such that for all and all .
Proof. Necessity. (i) As is -trichotomic, according to Definition 13, there exists a mapping , with the property , such that
An equivalent relation is obtained, if we consider Definition 2 , given by
for all and all . According to the properties of function , there exist and such that the following relations hold:
for all and all , where we have denoted that .
(ii) According to Definition 13, there exists a mapping , with the property , such that
The property of function assures the existence of some constants and , such that, for all , the following relations hold:
for all and all , where we have denoted that .
(iii) Both relations are obtained by a similar proof as in (i) and (ii), according to Theorem 18.
Sufficiency. (i) As the evolution cocycle has exponential growth, there exist and such that
Let . We consider that
For every there exists such that . We obtain
Hence, we have that
For we obtain
which implies that where we have denoted
Further, the following relations hold:
By integrating on , it follows that
which implies that where we have considered , which satisfies . Statement of Definition 13 is hence obtained.
(ii) Let . Let us denote that
According to the fact that , for all , we obtain, for all ,
As the evolution cocycle has exponential decay, there exist some constants and such that Let . We obtain
For , we obtain
and, if ,
for all . Hence, where we have considered defined by
with the property . Statement of Definition 13 is proved.
(iii) A similar proof for the -shifted evolution cocycle in and for the -shifted evolution cocycle in (ii) leads to the existence of two functions , with the properties and , such that
If we consider the definition of the shifted evolution cocycle, the previous relations are equivalent with
We define by and , which have the properties and . Hence, property of Definition 13 is obtained.
Remark 20. The property described by relation (36) is also called -strong integral stability for a skew-evolution semiflow. Relation (36) is a characterization of Barbashin type in the strong topology, and relations (37), (38), and (39) are characterizations of Datko type for the asymptotic properties of skew-evolution semiflows involved in the definition of -trichotomy.
This paper was conceived during the visit at the Institute of Mathematics of Bordeaux, France. The author wishes to express her profound and respectful gratitude to Professor Bernard Chevreau and Professor Mihail Megan. Also, the author gratefully acknowledges helpful comments and suggestions from the referees.
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