Connections between Some Concepts of Polynomial Trichotomy for Noninvertible Evolution Operators in Banach Spaces

Abstract

The present paper treats three concepts of nonuniform polynomial trichotomies for noninvertible evolution operators acting on Banach spaces. The connections between these concepts are established through numerous examples and counterexamples for systems defined on the Banach space of square-summable sequences.

1. Introduction

In the theory of asymptotic behavior of first-order differential equations, one of the main problems is to decompose the state space into a direct sum of subspaces on which the solutions of the given system have prescribed behavior. One of these behaviors can be modelled by the notion of exponential dichotomy, in which the state space is decomposed into a direct sum of two subspaces (the stable and unstable subspace) such that on the stable subspace the norm of the solution tends to zero (exponentially, polynomially, or with the aid of a general function) and on the unstable subspace the norm of the solution tends to infinity (usually with the same type of growth rate—exponential, polynomial, etc.—as the stable one). The notion of exponential dichotomy has its origins from the work of Perron in 1930 [1]. This field has seen a rich development in the last decades, as it can be seen from [2–11].

Another behavior given by the above-mentioned problem is the decomposition of the state space into three subspaces: a stable subspace, an unstable subspace, and a central manifold. The behavior on the stable and unstable subspaces is dichotomic, and, in addition, the solution of the system must be bounded (or have a growth property). This behavior is known in the literature as the trichotomy property. The trichotomy property was first defined by Sacker and Sell in [12], and, later on, the study was widely spread and many results were obtained (see [13–18] and the references therein).

This paper extends the above-mentioned study of the property of trichotomy in the case in which the decay, expansion, and growth on the stable, unstable, and central manifold, respectively, are described by a polynomial behavior. We study three concepts of polynomial trichotomy (both uniform and nonuniform) defined in the general case of noninvertible evolution operators: polynomial trichotomy, strong polynomial trichotomy, and weak polynomial trichotomy. We establish the connections between the three concepts and, with the aid of the examples and counterexamples from Section 5, on one hand we point out the existence of systems which possess the above-defined properties, and, on the other hand, we delimit the behaviors presented in this paper.

2. Supplementary Families of Projections

Throughout this paper, we will consider the following framework:(i) will be the Banach space of all real valued sequences satisfying  endowed with the norm .(ii) will be be a real or complex Banach space and will be the Banach space of all bounded linear operators on .(iii)The norms on and will be denoted by .(iv)The identity operator on is denoted by .(v) will be the set defined by

Definition 1. A mapping is called a family of projections on if

Definition 2. A family of projections is called (i) polynomially bounded if there exist and such that (ii) bounded if there exists such that

Definition 3. Three families of projections are called supplementary if for all one has that

In what follows, we present two leading examples of families of projections which will be used in Section 5.

Example 4. Let and be a nondecreasing function. For every we define by where Let . One can see that is linear and if , we have that from where it follows that and .
Moreover, let and given by From it follows that From here we get thatMoreover, for we define the family of projections by , where Moreover, for one can see that henceFinally, define by , where We have that is bounded with and moreover the families of projections , , and are supplementary.

Example 5. Let and define by , , and , where, for , We have that , and are three supplementary families of projections with

3. Evolution Operators

Definition 6. A mapping is called an evolution operator on if (), ;(), .

Definition 7. A family of projections is said to be invariant for the evolution operator if

Given three supplementary families of projections , , and which are invariant for a given evolution operator , we will name the quadruple a trichotomy quadruple.

Two important examples of trichotomy quadruples are given below, which will serve as a milestone in our examples and counterexamples.

Example 8. On consider the families of projections , , and from Example 4. Consider and given by for all .
Taking into account that for all the following relations hold, it follows that is an evolution operator. It is easy to check that , , and are invariant for ; hence is a trichotomy quadruple. Moreover we have thatfor all .

Example 9. On consider , , and to be the families of projections defined in Example 5. For define by where for all .
It is easy to see that is a trichotomy quadruple and for one has thatwhich is given by and , where

In what follows, we will present the main trichotomy concepts that will be studied in the present paper.

4. Polynomial Trichotomies

Definition 10. A trichotomy quadruple is said to be polynomially trichotomic (p.t.) if there exist , , and such that );););) for all

If from the above definition is equal to 0, then we say that is uniformly polynomially trichotomic (u.p.t.).

Remark 11. The following assertions hold: (i)If a trichotomy quadruple is (p.t.) then and are polynomially bounded, and hence is also polynomially bounded.(ii)If a trichotomy quadruple is (u.p.t.) then and are bounded, and hence is also bounded. In other words, if is (p.t.) with constants , , and then

Remark 12. If is (u.p.t.) then it is (p.t.). The converse is not generally true. Take, for example, the trichotomy quadruple from Example 8 with . It is easy to check that is (p.t.), but it cannot be (u.p.t.), because is not bounded.

Definition 13. A trichotomy quadruple is said to be strongly polynomially trichotomic (s.p.t.) if there exist , , and such that );););) for all

If from the above definition is equal to 0, then we say that is uniformly strongly polynomially trichotomic (u.s.p.t.).

Remark 14. If is (u.s.p.t.) then it is (s.p.t.). The converse is not generally true, as shown in Example 1.

Remark 15. If is (s.p.t.) then the following condition holds: In other words, for all ,

Remark 16. Under the same assumption as in Remark 15, we also have that

Definition 17. A trichotomy quadruple is said to be weakly polynomially trichotomic (w.p.t.) if there exist , , and such that );););) for all

If then we say that is uniformly weakly polynomially trichotomic (u.w.p.t.).

Remark 18. If is (u.w.p.t.) then it is (w.p.t.). The converse is not generally true, as shown in Example 2.

In what follows we will study the connections between these three trichotomy concepts.

Remark 19. If a trichotomy quadruple is (s.p.t.) then it is also (w.p.t.). Moreover, if is (u.s.p.t.), then it is (u.w.p.t.).

Proposition 20. Let be a trichotomy quadruple. If is (p.t.) then it is also (w.p.t.).

Proof. Let , , and be given by Definition 10. By Remark 11, we have that Let now . From the estimations it follows that is (w.p.t.) with constants , , and .

Remark 21. From the proof of the above proposition, we can easily see that, by setting , we obtain the implication .

Other connections are given by the following.

Remark 22. (i) (s.p.t.) does not imply (p.t.) and (u.s.p.t.) does not imply (u.p.t.) as shown by Example 3.
(ii) The concepts of (p.t.) and (w.p.t.) do not coincide, as we can see from Example 4.
(iii) (p.t.) does not imply (s.p.t.) and (u.p.t.) does not imply (u.s.p.t.), as shown by Example 5.
(iv) (w.p.t.) does not imply (s.p.t.) and (u.w.p.t.) does not imply (u.s.p.t.) as shown in Example 6.

Remark 23. The connection between the presented concepts is given by the following diagram:

5. Examples and Counterexamples

Example 1 (trichotomy quadruple that is (s.p.t.) but not (u.s.p.t.)). Let be the trichotomy quadruple from Example 8 with and , . Then we have that for all ; hence is (s.p.t.).
Assume, by a contradiction, that is (u.s.p.t.). Then there exist and such that for all and for and we have, from , that which leads us to the contradiction

Example 2 (trichotomy quadruple that is (w.p.t.) but not (u.w.p.t.)). Let be as in Example 1. By Remark 19 we have that is (w.p.t.). The same contradiction is obtained as in Example 1, by assuming that is (u.w.p.t.).

Example 3 (trichotomy quadruple which is (s.p.t.) but fails to be (p.t.)). Let be the trichotomy quadruple from Example 8 with and .
Fromfor all , we can see that is (u.s.p.t.) and hence (s.p.t.).
Assume, by a contradiction, that is (p.t.). Then, by Remark 11, we have that there exist , , such that This leads us to the contradiction It follows that is not (p.t.) and hence not (u.p.t.).

Example 4 (trichotomy quadruple which is (w.p.t.) but not (p.t.)). Let the trichotomy quadruple from Example 3. By Remark 19, we have that is (u.w.p.t.) and hence (w.p.t.). But, by Example 3, it is not (p.t.) and hence not (u.p.t.).

Example 5 (trichotomy quadruple which is (p.t.) but fails to be (s.p.t.)). Let be the trichotomy quadruple from Example 9 with . First of all we will show that is (u.p.t.). Let . We have that hence If , consider given by We have that and hence Having in mind that , it follows that () and () hold for with . The case in which obviously leads us to the above estimation, and so the conclusion follows.
In what follows, we will show that is not (s.p.t.), and hence it is not (u.s.p.t.). Assume by a contradiction that is (s.p.t.). We will disprove the result from Remark 15. Let given by Obviously and denote, for every , , where hence is a nonzero sequence.
Consider now with . By denoting , with it follows that , which contradicts Remark 15; hence is not (s.p.t.).

Example 6 (trichotomy quadruple that is (w.p.t.) but fails to be (s.p.t.)). Let be the trichotomy quadruple from Example 5. Taking into account that, for all , , it follows that is (u.w.p.t.) and hence (w.p.t.).
Again, from Example 5, we get that is not (s.p.t.) and hence not (u.s.p.t.).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.