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

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.

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.
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 (ℎ, )-trichotomy, where ℎ 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 (ℎ, )-trichotomy (called (ℎ, )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.

Growth Rates
Definition 2. One says that the growth rate (ℎ  ) satisfies hypothesis (H) if there exist a growth rate (  ) and  ∈ (1, +∞) such that for all (, ) ∈ N × N,  > .Now, we present some examples of growth rates which satisfy hypothesis (H).

(ℎ, 𝑘)-Trichotomy
Let  be a real or complex Banach space and B() the Banach algebra of all bounded linear operators on . represents the identity operator on  and the norms on  and on B() will be denoted by ‖ ⋅ ‖.Also, where N is the set of nonnegative integers.We consider the linear discrete-time system (A) with  : N → B(), () =   .Every solution of (A) is given by for all (, ) ∈ Δ, where Remark 6.We observe that for all (, ), (, ) ∈ Δ.
for all  ∈ N.
In the particular case when (  ) is a constant sequence, (A, P) is called uniformly (ℎ, )-trichotomic.
Example 10.Let (ℎ  ) and (  ) be two growth rates and (  ) a nondecreasing sequence of positive real numbers,   ≥ 1.
Let P = { 1  ,  2  ,  3  } be a family of supplementary projections sequences with Linear discrete-time system (A), defined by verifies the relation Then for all (, ) ∈ Δ.For all (, , ) ∈ Δ ×  the following properties hold: and we deduce that the pair (A, P) is (ℎ, )-trichotomic.
Remark 11.It is obvious that if the pair (A, P) is uniformly (ℎ, )-trichotomic then it is also (ℎ, )-trichotomic.In the following example we show that the converse implication is not valid.
We have that P is invariant for (A) and for all (, ) ∈ Δ.

(ℎ, 𝑘)-Trichotomy of Datko Type
Let (ℎ  ) be a growth rate which satisfies hypothesis (H) and let (  ) be a growth rate given by Definition 2.
We consider a pair (A, P), where P = { 1  ,  2  ,  3  } is a family of supplementary and invariant projections sequences for (A).
The following result emphasizes a necessary condition for (ℎ, )-trichotomy. Proof.
Necessity.It is a particular case of Theorem 13 for Sufficiency.Using Theorem 17, for ℎ  =   and   =   we obtain that (A, P) is exponentially trichotomic.
Remark 20.The previous result shows that the exponential trichotomy and the exponential trichotomy of Datko type are equivalent.
Example 22. On  =  ∞ (N, R), the Banach space of bounded real-valued sequences, endowed with the norm we consider P = { 1  ,  2  ,  3  }, with where   represents the characteristic function of set . Also, linear discrete-time system (A) is defined by and we have that for all (, ) ∈ Δ.
For the growth rates ℎ  =  2 and   =  4 we define the exponential Lyapunov functions: After some computations, we obtain that for   =  4(+1) /( 4 − 1) the mappings  1 and  2 are exponential Lyapunov functions for the pair (A, P).
In the following, we give a characterization for the (ℎ, )-trichotomy of Datko type in terms of (ℎ, )-Lyapunov functions.
A sufficient condition for (ℎ, )-trichotomy given through the Lyapunov functions is as follows.
Proof.It is obtained from Theorems 17 and 23.

Proof.Corollary 26 .
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.The pair (A, P) is exponentially trichotomic if and only if there exist  1 ,  2 two exponential Lyapunov functions for (A, P).

Proof.
It results by Remark 20 and Theorem 23.