Discrete Dynamics in Nature and Society
Volume 2016 (2016), Article ID 3760262, 8 pages
http://dx.doi.org/10.1155/2016/3760262

Research Article

The Equivalence of Datko and Lyapunov Properties for ()-Trichotomic Linear Discrete-Time Systems

Claudia-Luminiţa Mihiţ,¹ Mihail Megan,1,2 and Traian Ceauşu¹

¹Department of Mathematics, Faculty of Mathematics and Computer Science, West University of Timişoara, Vasile Pârvan Boulevard No. 4, 300223 Timişoara, Romania
²Academy of Romanian Scientists, Independenţei 54, 050094 Bucharest, Romania

Received 22 July 2015; Accepted 26 January 2016

Academic Editor: Allan C. Peterson

Copyright © 2016 Claudia-Luminiţa Mihiţ et al. 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.

Abstract.

The aim of this paper is to characterize a general property of -trichotomy through some Lyapunov functions for linear discrete-time systems in infinite dimensional spaces. Also, we apply the results to illustrate necessary and sufficient conditions for nonuniform exponential trichotomy and nonuniform polynomial trichotomy.

1. Introduction

In the last few years an important development has been made in the field of the asymptotic behaviors of dynamical systems. Among the most important asymptotic behaviors studied, we mention the properties of stability, dichotomy, and trichotomy (see [1–14] and the references therein).

A remarkable characterization for the stability property of continuous dynamical systems was proved by Datko in 1972 (see [8]) and later, Przyłuski and Rolewicz obtain in [15] a similar result for discrete-time systems. This was a starting point for the development of the area and the results were extended to the dichotomy case in [16, 17].

An important generalization of the dichotomy concept (approached in various manners in [2, 3, 6, 18]) is the notion of trichotomy, the most complex asymptotic property of dynamical systems. The trichotomy supposes the splitting of the state space, at any moment, into three subspaces: the stable subspace, the unstable subspace, and the central subspace.

The concept of (exponential) trichotomy was introduced by Elaydi and Hajek (see [9, 10]) for nonlinear differential equations and later, the case of difference equations is treated by Elaydi and Janglajew in [11]. Also, important contributions on the line of trichotomy in discrete-time are due to Cuevas and Vidal [7], López-Fenner and Pinto [13], Megan and Stoica [19, 20], Papaschinopoulos [21], and Popa et al. [22].

In [23], A. L. Sasu and B. Sasu propose an interesting technique for exponential trichotomy of difference equations, the admissibility technique, and in [24] the authors obtain for the first time nonlinear conditions for the exponential trichotomy in infinite dimensional spaces.

The Lyapunov functions represent an important tool in the study of the asymptotic properties of dynamical systems (see, e.g., [4, 5, 25, 26]).

The objective of this paper is to approach the general concept of -trichotomy, where and are growth rates, for linear discrete-time systems in Banach spaces and as particular cases we deduce the results for (nonuniform) exponential trichotomy and (nonuniform) polynomial trichotomy.

Also, we obtain necessary and sufficient conditions for a general concept of -trichotomy (called -trichotomy of Datko type) and the main result is the characterization of this concept of trichotomy in terms of Lyapunov functions.

The results are applied to illustrate criteria through the Lyapunov functions for nonuniform exponential trichotomy and nonuniform polynomial trichotomy.

2. Growth Rates

Definition 1. An increasing sequence , is called a growth rate if

Definition 2. One says that the growth rate satisfies hypothesis if there exist a growth rate and such that

(H ₁) 

(H ₂) 

for all , .
Now, we present some examples of growth rates which satisfy hypothesis

Example 3. Let , , and ; is a growth rate with

(H ₁)    

(H ₂)  

for all ,

Example 4. If with , then with is a growth rate which satisfies the following:

(H ₁)

(H ₂)

for all ,

Example 5. Let with and Then with , is a growth rate with the following properties:

(H ₁)

(H ₂)

for all ,

3. -Trichotomy

Let be a real or complex Banach space and the Banach algebra of all bounded linear operators on . represents the identity operator on and the norms on and on will be denoted by . Also,

where is the set of nonnegative integers.

We consider the linear discrete-time system

with ,

Every solution of is given by

for all , where

Remark 6. We observe that

for all

Definition 7. A sequence , is called a projections sequence on if

for all

Definition 8. One says that is a family of

(i)supplementary projections sequences if

(s ₁)

(s ₂) 

(ii)invariant projections sequences for if

In what follows, we consider two growth rates and a pair , where is a family of supplementary and invariant projections sequences for .

Definition 9. The pair is called -trichotomic if there exists a nondecreasing sequence , , such that

(ht ₁)

(ht ₂)

(kt ₃)

(kt ₄) 

for all .
In the particular case when is a constant sequence, is called uniformly -trichotomic.

As particular cases of -trichotomy we remark the following:

(i)if , with we obtain the concept of (nonuniform) exponential trichotomy and if is constant it results in the property of uniform exponential trichotomy;(ii)if , with we recover the concept of (nonuniform) polynomial trichotomy and if is constant it results in the property of uniform polynomial trichotomy;(iii)if for all it results in the notion of -dichotomy, nonuniform exponential dichotomy (for , , uniform exponential dichotomy (for , , and constant), nonuniform polynomial dichotomy (for , ), and uniform polynomial dichotomy (for , , and constant).

We give a general example of a pair which is -trichotomic.

Example 10. Let and be two growth rates and a nondecreasing sequence of positive real numbers, .
Let be a family of supplementary projections sequences with

Linear discrete-time system , defined by

verifies the relation

Then

for all .
For all the following properties hold:

(i)

(ii)

(iii)

(iv)

and we deduce that the pair is -trichotomic.

Remark 11. It is obvious that if the pair is uniformly -trichotomic then it is also -trichotomic. In the following example we show that the converse implication is not valid.

Example 12. We consider a family of supplementary projections sequences with the property

Linear discrete-time system is given by

where and and is a periodic sequence.
We have that is invariant for and

for all .
A simple computation shows that for , with , , and the pair is -trichotomic.
If we suppose that is uniformly -trichotomic, then, for , , , , and , we obtain

which is a contradiction.

4. -Trichotomy of Datko Type

Let be a growth rate which satisfies hypothesis and let be a growth rate given by Definition 2.

We consider a pair , where is a family of supplementary and invariant projections sequences for .

The following result emphasizes a necessary condition for -trichotomy.

Theorem 13. If is -trichotomic and satisfies hypothesis then there exist a growth rate and a nondecreasing sequence , , such that

Proof. It is easy to see that, for

where , are given by Definition 2, and , are given by Definition 9, relations , , , and are satisfied.

A necessary condition for polynomial trichotomy is represented by the following.

Corollary 14. If the pair is polynomially trichotomic, then there are a nondecreasing sequence , , and two constants such that

Proof. It results from Theorem 13.

Definition 15. One says that the pair admits a -trichotomy of Datko type if there exists a nondecreasing sequence with such that

As particular cases, we mention the following:

(i)if with and with we obtain the notion of exponential trichotomy of Datko type;(ii)if with and with we recover the concept of polynomial trichotomy of Datko type.

Remark 16. Theorem 13 emphasizes that if , are two growth rates, satisfies hypothesis , and the pair is -trichotomic, then admits a -trichotomy of Datko type.

Theorem 17. If the pair admits a -trichotomy of Datko type, then is -trichotomic.

Proof. Using condition we obtain

Similarly, by for we have

Obviously, the relation is valid for .
Inequality implies that

By (for ) we deduce

and the inequality is verified for
So, the pair is -trichotomic.

Corollary 18. If there exist two constants and a nondecreasing sequence , , such that conditions and ( and , resp.) from Corollary 14 are fulfilled for all (for all , resp.) with then is polynomially trichotomic.

Proof. It results from Theorem 17 for with and , with .

The following result represents a characterization for the exponential trichotomy.

Corollary 19. The pair is exponentially trichotomic if and only if there exist the constants and a nondecreasing sequence , , with the following properties:

Proof.
Necessity. It is a particular case of Theorem 13 for

Sufficiency. Using Theorem 17, for and we obtain that is exponentially trichotomic.

Remark 20. The previous result shows that the exponential trichotomy and the exponential trichotomy of Datko type are equivalent.

5. Lyapunov Functions for -Trichotomy

Throughout this section, , represent two growth rates, is a linear discrete-time system, and is a family of supplementary and invariant projections sequences for

Definition 21. Two mappings are called -Lyapunov functions for the pair if there exists a nondecreasing sequence , , such that

for all with and for all .

In particular, if

(i), with , , with then the -Lyapunov functions are called exponential Lyapunov functions;(ii), with , , with then the -Lyapunov functions are called polynomial Lyapunov functions.

Example 22. On , the Banach space of bounded real-valued sequences, endowed with the norm

we consider , with

where represents the characteristic function of set
Also, linear discrete-time system is defined by

and we have that

for all
For the growth rates and we define the exponential Lyapunov functions:

After some computations, we obtain that for the mappings and are exponential Lyapunov functions for the pair

In the following, we give a characterization for the -trichotomy of Datko type in terms of -Lyapunov functions.

Theorem 23. The pair admits a -trichotomy of Datko type if and only if there exist two -Lyapunov functions for .

Proof.
Necessity. Let , defined by

respectively. Then

for all with and for all .

Sufficiency. From , for we obtain

and for it results in condition .

For in we deduce

for all .

Putting in we obtain

and for it results in .

By condition it follows

for all .

Thus, it results in that admits a -trichotomy of Datko type.

A sufficient condition for -trichotomy given through the Lyapunov functions is as follows.

Corollary 24. If there exist two -Lyapunov functions for the pair , then is -trichotomic.

Proof. It is obtained from Theorems 17 and 23.

Corollary 25. Let , be two growth rates such that satisfies hypothesis . If is -trichotomic then there exist two -Lyapunov functions for , where , are the growth rates given by Definition 2 (Theorem 13, resp.).

Proof. It is immediate by Remark 16 and Theorem 23.

An important characterization for the exponential trichotomy in terms of Lyapunov functions is represented by the following.

Corollary 26. The pair is exponentially trichotomic if and only if there exist , two exponential Lyapunov functions for .

Proof. It results by Remark 20 and Theorem 23.

Corollary 27. If is polynomially trichotomic, then there exist , polynomial Lyapunov functions for .

Proof. It is a consequence of Corollary 25.

Corollary 28. If there exist two -Lyapunov functions for the pair , where

then is polynomially trichotomic.

Proof. It results from Theorems 17 and 23 for

Conflict of Interests

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

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