# Approaching the Discrete Dynamical Systems by means of Skew-Evolution Semiflows

Authors: Codruţa Stoica
Journal: Discrete Dynamics in Nature and Society
 (2016)

---

## Abstract

The aim of this paper is to highlight current developments and new trends in the stability theory. Due to the outstanding role played in the study of stable, instable, and, respectively, central manifolds, the properties of exponential dichotomy and trichotomy for evolution equations represent two domains of the stability theory with an impressive development. Hence, we intend to construct a framework for an asymptotic approach of these properties for discrete dynamical systems using the associated skew-evolution semiflows. To this aim, we give definitions and characterizations for the properties of exponential stability and instability, and we extend these techniques to obtain a unified study of the properties of exponential dichotomy and trichotomy. The results are underlined by several examples.


---

## Body

## 1. Introduction

The phenomena of the real world, in domains as economics, biology, or environmental sciences, do not take place continuously, but at certain moments in time. Therefore, a discrete-time approach is required. By means of skew-evolution semiflows, we intend to construct a framework that deepens the analysis of discrete dynamical systems.Playing an outstanding role in the study of stable and instable manifolds and in approaching several types of differential equations and difference equations, the exponential dichotomy for evolution equations is one of the domains of the stability theory with an impressive development. The dichotomy is a conditional stability, due to the fact that the asymptotic properties of the solutions of a given evolution equation depend on the location of the initial condition in a certain subspace of the phase space. Over the last decades, the classic techniques used to characterize asymptotic properties as stability and instability were generalized towards a natural generalization of the classic concept of dichotomy, the notion of trichotomy. The main idea in the study of trichotomy is to obtain, at any moment, a decomposition of the state space in three subspaces: a stable subspace, an instable one, and a third one called the central manifold. We intend to give several conditions in order to describe the behavior related to the third subspace.A relevant step in the study of evolution equations is due to Henry, who, in [1], studied the property of dichotomy in the discrete setting, in the spirit of the classic theory initiated by Perron in [2].A special interest is dedicated to the study of dynamic linear systems by means of associated difference equations, as emphasized by Chow and Leiva in [3] and Latushkin and Schnaubelt in [4].In [5], the uniform exponential dichotomy of discrete-time linear systems given by difference equations is presented, and the results are applied at the study of dichotomy of evolution families generated by evolution equations. In [6], a characterization of exponential dichotomy for evolution families associated with linear difference systems in terms of admissibility is given.In [7], characterizations for the uniform exponential stability of variational difference equations are obtained and, in [8], the uniform exponential dichotomy of semigroups of linear operators in terms of the solvability of discrete-time equations over N is characterized. In [9], new characterizations for the exponential dichotomy of evolution families in terms of solvability of associate difference and integral equations are deduced. In [10, 11], some dichotomous behaviors for variational difference equations are emphasized, such as a new method in the study of exponential dichotomy based on the convergence of some associated series of nonlinear trajectories or characterizations in terms of the admissibility of pairs of sequence spaces over N with respect to an associated control system.The notion of trichotomy was introduced in 1976 by Sacker and Sell and studied for the case of linear differential equations in the finite dimensional setting in [12]. For the first time, a sufficient condition for the existence of the trichotomy, in fact a continuous invariant decomposition of the state space R
n into three subspaces, was given. In the same study, the case of skew-product semiflows was as well approached.A stronger notion, but still in the finite dimensional case, was introduced by Elaydi and Hajek in [13], the exponential trichotomy for linear and nonlinear differential systems, by means of Lyapunov functions. They prove the fact that the property of exponential trichotomy of a differential system implies the Sacker-Sell type trichotomy. Meanwhile, the notion of invariance to perturbations of the property of trichotomy is given. Thus, in the case of a nonlinear perturbation of a linear exponential trichotomic system, the obtained system preserves the same qualitative behavior as the nonperturbed one.In [14], a relation between Lyapunov function and exponential trichotomy for the linear equation on time scales is given and, as application, the roughness of exponential trichotomy on time scales is proved. Several asymptotic properties for difference equations were studied in [15–17] and, recently, in [18, 19]. Other asymptotic properties for discrete-time dynamical systems were considered in [20, 21]. A new concept of  l
p-trichotomy for linear difference systems is given in [22], as an extension of the l
p-dichotomy and of the exponential trichotomy in  l
p spaces.The notion of skew-evolution semiflow considered in this paper and introduced by us in [23] generalizes the concepts of semigroups, evolution operators, and skew-product semiflows and seems to be more appropriate for the study of the asymptotic behavior of the solutions of evolution equations in the nonuniform case, as they depend on three variables. The applicability of the notion has been studied in [24–28].The case of stability for skew-evolution semiflows is emphasized in [29], and various concepts for trichotomy are studied in [30]. Some asymptotic properties, as stability, instability, and trichotomy for difference equations in a uniform as well in a nonuniform setting, were studied by us in [31–33].The following sections outline the structure of this paper. In Section2, the definitions for evolution semiflows, evolution cocycles, and skew-evolution semiflows are given, featured by examples. In Section 3, we present definitions and characterizations for the properties of exponential growth and decay, respectively, for the exponential stability and instability. The main results are stated in Sections 4 and 5, where we give definitions and characterizations for these asymptotic properties in discrete time for skew-evolution semiflows. Finally, some conclusions are emphasized in Section 6. The list of references allows us to build the overall context in which the discussed problem is placed.


## 2. Preparatory Notions

Let us consider(
X
,
d
) a metric space, V a real or complex Banach space, and B
(
V
) the family of linear V-valued bounded operators defined on V. The norm of vectors and operators is ·. In what follows, we will denote Y
=
X
×
V, T
=
(
t
,
t
0
)
∈
R
2
,
t
≥
t
0
≥
0, and Δ
=
(
m
,
n
)
∈
N
2
,
m
≥
n. By I we denoted the identity operator on V.Definition 1.
The mappingC
:
T
×
Y
→
Y defined by the relation (1)
C
t
,
s
,
x
,
v
=
φ
t
,
s
,
x
,
Φ
t
,
s
,
x
v
,where φ
:
T
×
X
→
X has the properties(
s
1
)
φ
(
t
,
t
,
x
)
=
x,  ∀
(
t
,
x
)
∈
R
+
×
X,
(
s
2
)
φ
(
t
,
s
,
φ
(
s
,
t
0
,
x
)
)
=
φ
(
t
,
t
0
,
x
),  ∀
(
t
,
s
,
t
0
)
∈
T,  x
∈
X and Φ
:
T
×
X
→
B
(
V
) satisfies(
c
1
)
Φ
(
t
,
t
,
x
)
=
I,  ∀
(
t
,
x
)
∈
R
+
×
X,
(
c
2
)
Φ
(
t
,
s
,
φ
(
s
,
t
0
,
x
)
)
Φ
(
s
,
t
0
,
x
)
=
Φ
(
t
,
t
0
,
x
),  ∀
(
t
,
s
,
t
0
)
∈
T, x
∈
X, is called skew-evolution semiflow on Y.Remark 2.
φ is calledevolution semiflow and Φ is anevolution cocycle.The approach of asymptotic properties in discrete time is of an obvious importance because the results obtained in this setting can easily be extended in continuous time.Example 3.
LetV
=
R be a Banach space and let X
=
C
(
N
,
V
) be the set of R-valued sequences (
x
n
)
n
≥
0. The mapping (2)
φ
:
Δ
×
X
⟶
X
,
φ
m
,
n
,
x
=
x
n
+
m
-
nis an evolution semiflow on X. We consider the linear system in discrete time:(3)
x
n
+
1
=
A
n
x
n
,
n
∈
N
,where A
:
N
→
B
(
V
). If we denote(4)
E
m
+
n
,
n
=
A
m
+
n
-
1
⋯
A
n
+
1
A
n
,
m
>
0
I
,
m
=
0
,where E
:
N
2
→
B
(
V
), then every solution of system (3) satisfies the relation(5)
x
n
+
p
=
E
n
+
p
,
n
x
n
,
∀
n
,
p
∈
N
.The pair C
E
=
(
φ
,
Φ
E
) is a skew-evolution semiflow, associated with system (3), where Φ
E is an evolution cocycle over the evolution semiflow φ, given by(6)
Φ
E
m
,
n
,
x
=
E
m
-
n
+
x
n
,
x
n
,
∀
m
,
n
∈
Δ
,
∀
x
∈
X
,where [
x
k
] denotes the integer part of the term of rank k.Example 4.
To every skew-evolution semiflowC
=
(
φ
,
Φ
), one can associate the mapping A
Φ
:
N
→
B
(
V
) given by(7)
A
Φ
n
=
Φ
n
+
1
,
n
,
x
,
n
∈
N
,
x
∈
X
,such that Φ
A
Φ
=
Φ.


## 3. Preliminary Results

This section aims to emphasize some asymptotic behaviors, as exponential growth and decay and exponential stability and instability, as a foundation for the main results. We give the definitions of these properties in continuous time and we underline the characterizations in discrete time, as results that play the role of equivalent definitions (see [33]).Definition 5.
A skew-evolution semiflowC hasexponential growth if there exist mappings M, ω
:
R
+
→
R
+
∗, such that(8)
Φ
t
,
t
0
,
x
v
≤
M
s
e
ω
s
t
-
s
Φ
s
,
t
0
,
x
v
,for all (
t
,
s
)
,
(
s
,
t
0
)
∈
T and all (
x
,
v
)
∈
Y.Definition 6.
A skew-evolution semiflowC is said to beexponentially stable if there exist a constant ν
>
0 and a mapping N
:
R
+
→
R
+
∗ such that(9)
Φ
t
,
t
0
,
x
v
≤
N
s
e
-
ν
t
-
s
Φ
s
,
t
0
,
x
v
, for all (
t
,
s
)
,
(
s
,
t
0
)
∈
T and all (
x
,
v
)
∈
Y.Proposition 7.
A skew-evolution semiflowC with exponential growth is exponentially stable if and only if there exist a constant μ
>
0 and a sequence of real numbers (
a
n
)
n
≥
0 with the property a
n
≥
1, ∀
n
≥
0, such that(10)
Φ
n
,
m
,
x
v
≤
a
m
e
-
μ
n
-
m
v
,for all (
n
,
m
)
∈
Δ and all (
x
,
v
)
∈
Y.Proof.
We have the following.
Necessity. It is obtained immediately if we consider in relation (9) t
=
n and s
=
t
0
=
m, and if we define(11)
a
m
=
N
m
,
m
∈
N
,
μ
=
ν
>
0
,where the existence of N
:
R
+
→
R
+
∗ and of ν is given by Definition 6.
Sufficiency. As a first step, if t
≥
t
0
+
1, we denote n
=
[
t
] and n
0
=
[
t
0
]. The following relations hold:(12)
n
≤
t
<
n
+
1
,
n
0
≤
t
0
<
n
0
+
1
,
n
0
+
1
≤
n
.We obtain(13)
Φ
t
,
t
0
,
x
v
≤
M
n
e
ω
n
t
-
n
Φ
n
,
n
0
+
1
,
φ
n
0
+
1
,
t
0
,
x
Φ
n
0
+
1
,
t
0
,
x
v
≤
a
n
M
2
n
e
2
ω
n
+
μ
e
-
μ
t
-
t
0
v
,for all (
x
,
v
)
∈
Y, where functions M and ω are given by Definition 5.
As a second step, fort
∈
[
t
0
,
t
0
+
1
), we have(14)
Φ
t
,
t
0
,
x
v
≤
M
t
0
e
ω
t
0
t
-
t
0
v
≤
M
t
0
e
ω
t
0
+
μ
e
-
μ
t
-
t
0
v
,for all (
x
,
v
)
∈
Y.
Hence,C is exponentially stable.Definition 8.
A skew-evolution semiflowC hasexponential decay if there exist mappings M, ω
:
R
+
→
R
+
∗, such that(15)
Φ
s
,
t
0
,
x
v
≤
M
t
e
ω
t
t
-
s
Φ
t
,
t
0
,
x
v
,for all (
t
,
s
)
,
(
s
,
t
0
)
∈
T and all (
x
,
v
)
∈
Y.Definition 9.
A skew-evolution semiflowC is said to beexponentially instable if there exist a mapping N
:
R
+
→
R
+
∗ and a constant ν
>
0 such that(16)
N
t
Φ
t
,
t
0
,
x
v
≥
e
ν
t
-
s
Φ
s
,
t
0
,
x
v
,for all (
t
,
s
)
,
(
s
,
t
0
)
∈
T and all (
x
,
v
)
∈
Y.Proposition 10.
A skew-evolution semiflow with exponential decayC is exponentially instable if and only if there exist a constant μ
>
0 and a sequence of real numbers (
a
n
)
n
≥
0 with the property a
n
≥
1, ∀
n
≥
0, such that(17)
Φ
n
,
n
0
,
x
v
≤
a
m
e
-
μ
m
-
n
0
Φ
m
,
n
0
,
x
v
,for all (
m
,
n
)
,
(
n
,
n
0
)
∈
Δ and all (
x
,
v
)
∈
Y.Proof.
We have the following.
Necessity. We take in relation (16) t
=
m, s
=
n, and t
0
=
n
0 and we define(18)
a
m
=
N
m
,
m
∈
N
,
μ
=
ν
>
0
,where the existence of function N
:
R
+
→
R
+
∗ and of constant ν is given by Definition 9.
Sufficiency. First step, let us take t
≥
t
0
+
1 and we denote n
=
[
t
] and, respectively, n
0
=
[
t
0
]. We obtain(19)
n
≤
t
<
n
+
1
,
n
0
≤
t
0
<
n
0
+
1
,
n
0
+
1
≤
n
.It follows that(20)
Φ
t
,
t
0
,
x
v
=
Φ
t
,
n
,
φ
n
,
n
0
+
1
,
x
Φ
n
,
n
0
+
1
,
φ
n
0
+
1
,
t
0
,
x
Φ
n
0
+
1
,
t
0
,
x
v
≥
M
t
-
1
e
-
ω
t
-
n
M
t
-
1
e
-
ω
n
0
+
1
-
t
0
Φ
n
,
n
0
+
1
,
x
v
≥
M
t
-
2
N
~
e
-
2
ω
e
ν
~
n
-
n
0
+
1
v
≥
e
ν
~
M
t
2
N
~
e
2
ω
e
ν
~
t
-
t
0
v
,for all (
x
,
v
)
∈
Y, where the existence of function M and of constant ω is assured by Definition 8.
As a second step, if we considert
∈
[
t
0
,
t
0
+
1
), we obtain(21)
M
Φ
t
,
t
0
,
x
v
≥
e
-
ω
t
-
t
0
v
≥
e
-
ν
~
+
ω
e
ν
~
t
-
t
0
v
,for all (
x
,
v
)
∈
Y.
Hence,(22)
N
Φ
t
,
t
0
,
x
v
≥
e
ν
t
-
t
0
v
,for all (
t
,
t
0
,
x
,
v
)
∈
T
×
Y, where we have denoted(23)
N
=
M
e
ν
~
+
ω
+
M
2
N
~
e
-
ν
~
+
2
ω
,
ν
=
ν
~
,which proves the exponential instability of C.


## 4. Nonuniform Discrete Dichotomic Behaviors

Definition 11.
A projectorP on Y is calledinvariant relative to a skew-evolution semiflow C
=
(
φ
,
Φ
) if the following relation(24)
P
φ
t
,
s
,
x
Φ
t
,
s
,
x
=
Φ
t
,
s
,
x
P
xholds for all (
t
,
s
)
∈
T and all x
∈
X.Definition 12.
Two projectorsP
1 and P
2 are said to becompatible with a skew-evolution semiflow C
=
(
φ
,
Φ
) if(
d
1
)
projectorsP
1 and P
2 are invariant on Y;
(
d
2
)
for allx
∈
X, the projections P
1
(
x
) and P
2
(
x
) verify the relations(25)
P
1
x
+
P
2
x
=
I
,
P
1
x
P
2
x
=
P
2
x
P
1
x
=
0
.Definition 13.
A skew-evolution semiflowC
=
(
φ
,
Φ
) is calledexponentially dichotomic if there exist functions N
1, N
2
:
R
+
→
R
+
∗, constants ν
1, ν
2
>
0, and two projectors P
1 and P
2 compatible with C such that(26)
e
ν
1
t
-
s
Φ
t
,
t
0
,
x
P
1
x
v
≤
N
1
s
Φ
s
,
t
0
,
x
P
1
x
v
(27)
e
ν
2
t
-
s
Φ
s
,
t
0
,
x
P
2
x
v
≤
N
2
t
Φ
t
,
t
0
,
x
P
2
x
vfor all (
t
,
s
)
,
(
s
,
t
0
)
∈
T and all (
x
,
v
)
∈
Y.Example 14.
We denote byC
=
C
(
R
+
,
R
+
) the set of all continuous functions x
:
R
+
→
R
+, endowed with the topology of uniform convergence on compact subsets of R
+, metrizable relative to the metric (28)
d
x
,
y
=
∑
n
=
1
∞
1
2
n
d
n
x
,
y
1
+
d
n
x
,
y
,
where
d
n
x
,
y
=
sup
t
∈
0
,
n
⁡
x
t
-
y
t
.If x
∈
C, then for all t
∈
R
+ we denote x
t
(
s
)
=
x
(
t
+
s
), x
t
∈
C. Let X be the closure in C of the set {
f
t
,
t
∈
R
+
}, where f
:
R
+
→
R
+
∗ is a nondecreasing function with the property lim
t
→
∞
⁡
f
(
t
)
=
l
>
0. Then, (
X
,
d
) is a metric space and the mapping (29)
φ
:
T
×
X
⟶
X
,
φ
t
,
s
,
x
τ
=
x
t
-
s
+
τis an evolution semiflow on X.
LetV
=
R
2 be endowed with the norm (30)
v
1
,
v
2
=
v
1
+
v
2
,
v
=
v
1
,
v
2
∈
V
.The mapping Φ
:
T
×
X
→
B
(
V
), given by (31)
Φ
t
,
s
,
x
v
1
,
v
2
=
e
t
sin
⁡
t
-
s
sin
⁡
s
-
2
t
+
2
s
v
1
,
e
2
t
-
2
s
-
3
t
cos
⁡
t
+
3
s
cos
⁡
s
v
2
,is an evolution cocycle over the evolution semiflow φ. We consider the projectors(32)
P
1
x
v
1
,
v
2
=
v
1
,
0
,
P
2
x
v
1
,
v
2
=
0
,
v
2
.As(33)
t
sin
⁡
t
-
s
sin
⁡
s
-
2
t
+
2
s
≤
-
t
+
3
s
,
∀
t
,
s
∈
T
,we obtain that(34)
Φ
t
,
s
,
x
P
1
x
v
≤
e
2
s
e
-
t
-
s
v
1
,
∀
t
,
s
,
x
,
v
∈
T
×
Y
.Similarly, as(35)
2
t
-
2
s
-
3
t
cos
⁡
t
+
3
s
cos
⁡
s
≥
-
t
-
5
s
,
∀
t
,
s
∈
Tit follows that(36)
e
6
t
Φ
t
,
s
,
x
P
2
x
v
≥
e
5
t
-
s
v
2
,
∀
t
,
s
,
x
,
v
∈
T
×
Y
.The skew-evolution semiflow C
=
(
φ
,
Φ
) is exponentially dichotomic with characteristics(37)
N
u
=
e
6
u
,
ν
=
1
.In what follows, let us denoteC
k
(
t
,
s
,
x
,
v
)
=
(
φ
(
t
,
s
,
x
)
,
Φ
(
t
,
t
0
,
x
)
P
k
(
x
)
v
), where (
t
,
t
0
,
x
,
v
)
∈
T
×
Y and k
∈
{
1,2
}
.In discrete time, we will describe the property of exponential dichotomy as given in the next proposition.Proposition 15.
A skew-evolution semiflowC
=
(
φ
,
Φ
) is exponentially dichotomic if and only if there exist two projectors {
P
k
}
k
∈
{
1,2
}, compatible with C, constants ν
1
≤
0
≤
ν
2, and a sequence of real positive numbers (
a
n
)
n
≥
0 such that(
d
1
′
)
(38)
Φ
m
,
n
,
x
P
1
x
v
≤
a
n
P
1
x
v
e
ν
1
m
-
n
(
d
2
′
)
(39)
P
2
x
v
≤
a
m
Φ
m
,
n
,
x
P
2
x
v
e
-
ν
2
m
-
n
for all(
m
,
n
)
∈
Δ and all (
x
,
v
)
∈
Y.Proof.
We have the following.
Necessity. If we consider for C
1 in relation (26) of Definition 13  t
=
m and s
=
t
0
=
n and if we define(40)
a
n
=
N
n
,
n
∈
N
,relation (
d
1
′
) is obtained.
Statementd
2
′ results from Definition 13 for C
2 if we consider in relation (27) t
=
m and s
=
t
0
=
n and(41)
a
m
=
N
m
,
m
∈
N
.
Sufficiency. It is obtained by means of Proposition 7 for C
1 and, respectively, of Proposition 10 for C
2.
Hence,C is exponentially dichotomic.Theorem 16.
A skew-evolution semiflowC
=
(
φ
,
Φ
) is exponentially dichotomic if and only if there exist two projectors {
P
k
}
k
∈
{
1,2
}, compatible with C such that(
e
d
1
′
)
there exist a constantρ
1
>
0 and a sequence of real positive numbers (
α
n
)
n
≥
0 such that(42)
∑
k
=
n
m
e
ρ
1
k
-
n
Φ
k
,
n
,
x
P
1
x
v
≤
α
n
P
1
x
v
(
e
d
2
′
)
there exist a constantρ
2
<
0 and a sequence of real positive numbers (
β
n
)
n
≥
0 such that(43)
∑
k
=
n
m
e
-
ρ
2
m
-
k
Φ
k
,
n
,
x
P
2
x
v
≤
β
m
Φ
m
,
n
,
x
P
2
x
v
for all(
m
,
p
)
,
(
p
,
n
)
∈
Δ and all (
x
,
v
)
∈
Y.Proof.
We have the following.
Necessity. According to Proposition 7, there exist a constant ν
>
0 and a sequence of real numbers (
a
n
)
n
≥
0 with the property a
n
≥
1, ∀
n
≥
0. We obtain for ρ
=
ν
/
2
>
0 and according to Proposition 7
(44)
∑
k
=
n
m
e
ρ
k
-
n
Φ
k
,
n
,
x
v
≤
a
n
∑
k
=
n
m
e
ρ
k
-
n
e
-
ν
k
-
n
Φ
n
,
n
,
x
v
=
a
n
v
∑
k
=
n
m
e
-
ν
/
2
k
-
n
≤
α
n
v
,
∀
m
,
n
∈
Δ
,
∀
x
,
v
∈
Y
,where we have denoted(45)
α
n
=
a
n
e
ν
/
2
,
n
∈
N
.
By Proposition10, there exist a constant ν
>
0 and a sequence of real numbers (
a
n
)
n
≥
0 with the property a
n
≥
1, ∀
n
≥
0. We obtain for ρ
=
-
ν
/
2
>
0
(46)
∑
k
=
n
m
e
-
ρ
m
-
k
Φ
k
,
n
,
x
v
≤
a
m
∑
k
=
n
m
e
-
ρ
m
-
k
e
-
ν
m
-
k
Φ
m
,
n
,
x
v
=
a
m
Φ
m
,
n
,
x
v
∑
k
=
n
m
e
-
ν
/
2
k
-
n
≤
α
m
Φ
m
,
n
,
x
v
,for all (
m
,
n
)
∈
Δ and all (
x
,
v
)
∈
Y, where we have denoted(47)
α
m
=
1
1
-
e
ρ
a
m
,
m
∈
N
.
Sufficiency. Let t
≥
t
0
+
1, t
0
≥
0. We define n
=
[
t
] and n
0
=
[
t
0
]. We consider C
-
ρ
=
(
φ
,
Φ
-
ρ
), where we define Φ
-
ρ
(
t
,
t
0
,
x
)
=
e
ρ
(
t
-
t
0
)
Φ
(
t
,
t
0
,
x
). We have that(48)
Φ
-
ρ
t
,
t
0
,
x
v
=
Φ
-
ρ
t
,
n
,
φ
n
,
t
0
,
x
Φ
-
ρ
n
,
t
0
,
x
v
≤
M
n
e
ω
t
-
n
Φ
-
ρ
n
,
t
0
,
x
v
≤
M
n
e
ω
Φ
-
ρ
n
,
t
0
,
x
v
,for all (
x
,
v
)
∈
Y, where M and ω are given by Definition 5. We obtain further for m
≥
n
(49)
∫
t
0
+
1
t
Φ
-
ρ
τ
,
t
0
,
x
v
d
τ
≤
∑
k
=
t
0
+
1
m
M
n
e
ω
Φ
-
ρ
k
,
t
0
,
x
v
≤
β
n
v
,for all (
x
,
v
)
∈
Y, where β
n
=
M
(
n
)
α
n
e
ω. Then, there exist N
≥
1 and ν
>
0 such that(50)
Φ
-
ρ
t
,
t
0
,
x
v
≤
N
e
-
ν
t
-
t
0
v
,
∀
t
≥
t
0
+
1
,
∀
x
,
v
∈
Y
.On the other hand, for t
∈
[
t
0
,
t
0
+
1
), we have(51)
Φ
-
ρ
t
,
t
0
,
x
v
≤
M
e
ω
t
-
t
0
v
≤
M
e
ω
v
,
∀
x
,
v
∈
Y
.We obtain that C
-
ρ is stable, where(52)
Φ
-
ρ
m
,
n
,
x
=
e
ρ
m
-
n
Φ
m
,
n
,
x
,
m
,
n
,
x
∈
Δ
×
X
.Hence, there exists a sequence (
a
n
)
n
≥
0 with the property a
n
≥
1, ∀
n
≥
0, such that(53)
e
ρ
m
-
n
Φ
m
,
n
,
x
v
≤
a
n
v
,
∀
m
,
n
,
x
,
v
∈
Δ
×
Y
,which implies the exponential stability of C
1 and ends the proof.
According to the hypothesis, if we considerk
=
n we obtain(54)
e
-
ρ
m
-
n
Φ
n
,
n
,
x
v
≤
a
m
Φ
m
,
n
,
x
v
,for all (
m
,
n
,
x
,
v
)
∈
Δ
×
Y, which implies the exponential instability of C
2 and ends the proof.


## 5. Nonuniform Discrete Trichotomic Behaviors

Definition 17.
Three projectors{
P
k
}
k
∈
{
1,2
,
3
} are said to becompatible with a skew-evolution semiflow C
=
(
φ
,
Φ
) if(
t
1
)
each projectorP
k, k
∈
{
1,2
,
3
} is invariant on Y;
(
t
2
)
for allx
∈
X, the projections P
0
(
x
), P
1
(
x
), and P
2
(
x
) verify the relations (55)
P
1
x
+
P
2
x
+
P
3
x
=
I
,
P
i
x
P
j
x
=
0
,
∀
i
,
j
∈
1,2
,
3
,
i
≠
j
.Definition 18.
A skew-evolution semiflowC
=
(
φ
,
Φ
) is calledexponentially trichotomic if there exist the functions N
1, N
2, N
3, N
4
:
R
+
→
R
+
∗, constants ν
1, ν
2, ν
3, and ν
4 with the properties(56)
ν
1
≤
ν
2
≤
0
≤
ν
3
≤
ν
4
,and three projectors P
1, P
2, and P
3 compatible with C such that(57)
Φ
t
,
t
0
,
x
P
1
x
v
≤
N
1
s
Φ
s
,
t
0
,
x
P
1
x
v
e
ν
1
t
-
s
(58)
N
4
t
Φ
t
,
t
0
,
x
P
2
x
v
≥
Φ
s
,
t
0
,
x
P
2
x
v
e
ν
4
t
-
s
(59)
N
2
t
Φ
t
,
t
0
,
x
P
3
x
v
≥
Φ
s
,
t
0
,
x
P
3
x
v
e
ν
2
t
-
s
(60)
Φ
t
,
t
0
,
x
P
3
x
v
≤
N
3
s
Φ
s
,
t
0
,
x
P
3
x
v
e
ν
3
t
-
sfor all (
t
,
s
)
,
(
s
,
t
0
)
∈
T and all (
x
,
v
)
∈
Y.Example 19.
We consider the evolution semiflowφ defined in Example 14. Let V
=
R
3 be endowed with the norm (61)
v
1
,
v
2
,
v
3
=
v
1
+
v
2
+
v
3
.The mapping Φ
:
T
×
X
→
B
(
V
), given by (62)
Φ
t
,
s
,
x
v
=
e
-
2
t
-
s
x
0
+
∫
0
t
x
τ
d
τ
v
1
,
e
t
-
s
+
∫
0
t
x
τ
d
τ
v
2
,
e
-
t
-
s
x
0
+
2
∫
0
t
x
τ
d
τ
v
3
,is an evolution cocycle. Then, C
=
(
φ
,
Φ
) is a skew-evolution semiflow.
We consider the projections(63)
P
1
x
v
=
v
1
,
0,0
,
P
2
x
v
=
0
,
v
2
,
0
,
P
3
x
v
=
0,0
,
v
3
.The skew-evolution semiflow C
=
(
φ
,
Φ
) is exponentially trichotomic with characteristics (64)
ν
1
=
ν
2
=
-
x
0
,
ν
3
=
x
0
,
ν
4
=
1
,
N
1
u
=
e
u
x
0
,
N
2
u
=
e
-
2
l
u
,
N
3
u
=
e
2
u
x
0
,
N
4
u
=
e
-
l
u
.As in the case of dichotomy, letC
k
(
t
,
s
,
x
,
v
)
=
(
φ
(
t
,
s
,
x
)
,
Φ
(
t
,
t
0
,
x
)
P
k
(
x
)
v
), where (
t
,
t
0
,
x
,
v
)
∈
T
×
Y and k
∈
{
1,2
,
3
}
.In discrete time, the trichotomy of a skew-evolution semiflow can be described as in the next proposition.Proposition 20.
A skew-evolution semiflowC
=
(
φ
,
Φ
) is exponentially trichotomic if and only if there exist three projectors {
P
k
}
k
∈
{
1,2
,
3
} compatible with C, constants ν
1, ν
2, ν
3, and ν
4 with the property ν
1
≤
ν
2
≤
0
≤
ν
3
≤
ν
4, and a sequence of positive real numbers (
a
n
)
n
≥
0 such that(
t
1
)
(65)
Φ
m
,
n
,
x
P
1
x
v
≤
a
p
Φ
p
,
n
,
x
P
1
x
v
e
ν
1
m
-
p
,
(
t
2
)
(66)
Φ
p
,
n
,
x
P
2
x
v
≤
a
m
Φ
m
,
n
,
x
P
2
x
v
e
-
ν
4
m
-
p
,
(
t
3
)
(67)
Φ
p
,
n
,
x
P
3
x
v
≤
a
m
Φ
m
,
n
,
x
P
3
x
v
e
-
ν
2
m
-
p
,
(
t
4
)
(68)
a
p
Φ
p
,
n
,
x
P
3
x
v
≥
Φ
m
,
n
,
x
P
3
x
v
e
-
ν
3
m
-
p
for all(
m
,
p
)
,
(
p
,
n
)
∈
Δ and all (
x
,
v
)
∈
Y.Proof.
We have the following.
Necessity. (
t
1
) is obtained if we consider for C
1 in relation (9) of Definition 6  t
=
n and s
=
t
0
=
m and if we define(69)
a
p
=
N
p
,
p
∈
N
,
ν
1
=
-
ν
<
0
.
(
t
2
) follows according to Definition 9 for C
2 if we consider in relation (16) t
=
m, s
=
n, and t
0
=
n
0 and(70)
a
m
=
N
m
,
m
∈
N
,
ν
4
=
ν
>
0
.
(
t
3
) is obtained for C
3 out of relation (8) of Definition 5 for t
=
m, s
=
p, and t
0
=
n and if we define(71)
a
m
=
M
m
,
ν
2
=
-
ω
m
<
0
,
m
∈
N
.
(
t
4
) follows for C
3 from relation (15) of Definition 8 for t
=
m, s
=
p, and t
0
=
n and if we consider(72)
a
p
=
M
p
,
ν
3
=
ω
p
>
0
,
p
∈
N
.
Sufficiency. Let t
≥
t
0
+
1. We denote n
=
[
t
] and n
0
=
[
t
0
] and we obtained the relations(73)
n
≤
t
<
n
+
1
,
n
0
≤
t
0
<
n
0
+
1
,
n
0
+
1
≤
n
.According to (
t
1
), we have(74)
Φ
t
,
t
0
,
x
P
1
x
v
≤
M
n
e
ω
n
t
-
n
Φ
n
,
n
0
+
1
,
φ
n
0
+
1
,
t
0
,
x
Φ
n
0
+
1
,
t
0
,
x
P
1
x
v
≤
a
n
M
2
n
e
2
ω
n
+
μ
e
-
μ
t
-
t
0
P
1
x
v
,for all (
x
,
v
)
∈
Y, where functions M and ω are given as in Definition 5.
Fort
∈
[
t
0
,
t
0
+
1
), we have(75)
Φ
t
,
t
0
,
x
P
1
x
v
≤
M
t
0
e
ω
t
0
t
-
t
0
P
1
x
v
≤
M
t
0
e
ω
t
0
+
μ
e
-
μ
t
-
t
0
P
1
x
v
,for all (
x
,
v
)
∈
Y. Hence, relation (57) is obtained.
Lett
≥
t
0
+
1 and n
=
[
t
] and, respectively, n
0
=
[
t
0
]. It follows that(76)
n
≤
t
<
n
+
1
,
n
0
≤
t
0
<
n
0
+
1
,
n
0
+
1
≤
n
.From (
t
2
), it is obtained that(77)
Φ
t
,
t
0
,
x
P
2
x
v
=
Φ
t
,
n
,
φ
n
,
n
0
+
1
,
x
Φ
n
,
n
0
+
1
,
φ
n
0
+
1
,
t
0
,
x
Φ
n
0
+
1
,
t
0
,
x
P
2
x
v
≥
M
t
-
1
e
-
ω
t
-
n
M
t
-
1
e
-
ω
n
0
+
1
-
t
0
Φ
n
,
n
0
+
1
,
x
P
2
x
v
≥
M
t
-
2
N
~
e
-
2
ω
e
ν
~
n
-
n
0
+
1
P
2
x
v
≥
e
ν
~
M
t
2
N
~
e
2
ω
e
ν
~
t
-
t
0
P
2
x
v
,for all (
x
,
v
)
∈
Y, where M and ω are given by Definition 8.
Fort
∈
[
t
0
,
t
0
+
1
), we have(78)
M
Φ
t
,
t
0
,
x
P
2
x
v
≥
e
-
ω
t
-
t
0
P
2
x
v
≥
e
-
ν
~
+
ω
e
ν
~
t
-
t
0
P
2
x
v
,for all (
x
,
v
)
∈
Y. It follows that(79)
N
Φ
t
,
t
0
,
x
P
2
x
v
≥
e
ν
t
-
t
0
P
2
x
v
,for all (
t
,
t
0
,
x
,
v
)
∈
T
×
Y, where we have denoted(80)
N
=
M
e
ν
~
+
ω
+
M
2
N
~
e
-
ν
~
+
2
ω
,
ν
=
ν
~and which implies relation (58).
By a similar reasoning, from(
t
3
) relation (59) is obtained, and from (
t
4
) relation (60) follows.
Hence, the skew-evolution semiflowC is exponentially trichotomic.Some characterizations in discrete time for the exponential trichotomy for skew-evolution semiflows are given in what follows.Theorem 21.
A skew-evolution semiflowC
=
(
φ
,
Φ
) is exponentially trichotomic if and only if there exist three projectors {
P
k
}
k
∈
{
1,2
,
3
} compatible with C such that C
1 has exponential growth, and C
2 has exponential decay and such that the following relations hold:(
t
1
′
)
there exist a constantρ
1
>
0 and a sequence of positive real numbers (
α
n
)
n
≥
0 such that(81)
∑
k
=
n
m
e
ρ
1
k
-
n
Φ
k
,
n
,
x
P
1
x
v
≤
α
n
P
1
x
v
,
(
t
2
′
)
there exist a constantρ
2
>
0 and a sequence of positive real numbers (
β
n
)
n
≥
0 such that(82)
∑
k
=
n
m
e
-
ρ
2
k
-
n
Φ
k
,
n
,
x
P
2
x
v
≤
β
m
e
-
ρ
2
m
-
n
Φ
m
,
n
,
x
P
2
x
v
,
(
t
3
′
)
there exist a constantρ
3
>
0 and a sequence of positive real numbers (
γ
n
)
n
≥
0 such that(83)
∑
k
=
p
m
e
-
ρ
3
k
-
n
Φ
k
,
n
,
x
P
3
x
v
≤
γ
n
e
-
ρ
3
p
-
n
Φ
p
,
n
,
x
P
3
x
v
,
(
t
4
′
)
there exist a constantρ
4
>
0 and a sequence of positive real numbers (
δ
n
)
n
≥
0 such that(84)
∑
k
=
p
m
e
ρ
4
k
-
n
Φ
k
,
n
,
x
P
3
x
v
≤
δ
m
e
ρ
4
m
-
n
Φ
m
,
n
,
x
P
3
x
v
for all(
m
,
p
)
,
(
p
,
n
)
∈
Δ and all (
x
,
v
)
∈
Y.Proof.
We have the following.
(
t
1
)
⇔
(
t
1
′
)
Necessity. As C is exponentially trichotomic, Proposition 20 assures the existence of a projector P, of a constant ν
1
≤
0, and of a sequence of positive real numbers (
a
n
)
n
≥
0 such (
t
1
) holds. Let P
1
=
P. If we consider(85)
ρ
1
=
-
ν
1
2
>
0
,we obtain (86)
∑
k
=
n
m
e
ρ
1
k
-
n
Φ
k
,
n
,
x
P
1
x
v
≤
a
n
∑
k
=
n
m
e
ρ
1
k
-
n
e
ν
1
k
-
n
Φ
n
,
n
,
x
P
1
x
v
=
a
n
P
1
x
v
∑
k
=
n
m
e
ρ
1
+
ν
1
k
-
n
≤
α
n
P
1
x
v
,
∀
m
,
n
∈
N
,
∀
x
,
v
∈
Y
,where we have denoted(87)
α
n
=
a
n
e
-
ν
1
/
2
,
n
∈
N
.
Sufficiency. As C
1 has exponential growth, there exist constants M
≥
1 and r
>
1 such that relation(88)
Φ
n
+
p
,
n
,
x
v
≤
M
r
p
v
,holds for all n
,
p
∈
N and all (
x
,
v
)
∈
Y. If we denote ω
=
ln
⁡
r
>
0, the inequality can be written as follows:(89)
Φ
m
,
n
,
x
≤
M
e
ω
m
-
k
Φ
k
,
n
,
x
,
∀
m
,
k
,
k
,
n
∈
Δ
,
x
∈
X
.We consider successively the m
-
n
+
1 relations. By denoting P
=
P
1, we obtain (90)
Φ
m
,
n
,
x
P
x
v
≤
M
e
ω
m
-
n
1
+
e
ρ
1
+
⋯
+
e
ρ
1
m
-
n
α
n
P
x
v
,for all (
m
,
n
)
∈
Δ and all (
x
,
v
)
∈
Y. If we define the constant(91)
ν
1
=
ω
-
ρ
1
for
ω
<
ρ
1and the sequence of nonnegative real numbers(92)
a
n
=
M
α
n
,
n
∈
N
,relation (
t
1
) is obtained.
Similarly, the other equivalences can also be proved.In order to characterize the exponential trichotomy by means of four projectors, we give the next definition.Definition 22.
Four invariant projectors{
R
k
}
k
∈
{
1,2
,
3,4
} that satisfy for all (
x
,
v
)
∈
Y the following relations(
p
c
1
′
)
R
1
(
x
)
+
R
3
(
x
)
=
R
2
(
x
)
+
R
4
(
x
)
=
I,
(
p
c
2
′
)
R
1
(
x
)
R
2
(
x
)
=
R
2
(
x
)
R
1
(
x
)
=
0 and  R
3
(
x
)
R
4
(
x
)
=
R
4
(
x
)
R
3
(
x
),
(
p
c
3
′
)
R
1
(
x
)
+
R
2
(
x
)
v
2
=
R
1
(
x
)
v
2
+
R
2
(
x
)
v
2,
(
p
c
4
′
)
R
1
(
x
)
+
R
3
(
x
)
R
4
(
x
)
v
2
=
R
1
(
x
)
v
2
+
R
3
(
x
)
R
4
(
x
)
v
2,
(
p
c
5
′
)
R
2
(
x
)
+
R
3
(
x
)
R
4
(
x
)
v
2
=
R
2
(
x
)
v
2
+
R
3
(
x
)
R
4
(
x
)
v
2,are calledcompatible with the skew-evolution semiflow C.Theorem 23.
A skew-evolution semiflowC
=
(
φ
,
Φ
) is exponentially trichotomic if and only if there exist four projectors {
R
k
}
k
∈
{
1,2
,
3,4
} compatible with C, constants μ
>
ν
>
0, and a sequence of positive real numbers (
α
n
)
n
≥
0 such that(
t
1
′
′
)
(93)
Φ
m
+
p
,
m
,
x
R
1
x
v
≤
α
m
R
1
x
v
e
-
ν
p
,
(
t
2
′
′
)
(94)
R
2
x
v
≤
α
p
Φ
m
+
p
,
m
,
x
R
2
x
v
e
-
μ
p
,
(
t
3
′
′
)
(95)
R
3
x
v
≤
α
p
Φ
m
+
p
,
m
,
x
R
3
x
v
e
ν
p
,
(
t
4
′
′
)
(96)
Φ
m
+
p
,
m
,
x
R
4
x
v
≤
α
m
R
4
x
v
e
μ
p
for allm
,
p
∈
N and all (
x
,
v
)
∈
Y.Proof.
We have the following.
Necessity. As C is exponentially trichotomic, according to Proposition 20 there exist three projectors {
P
k
}
k
∈
{
1,2
,
3
} compatible with C, constants ν
1
≤
ν
2
≤
0
≤
ν
3
≤
ν
4, and a sequence of positive real numbers (
a
n
)
n
≥
0 such that relations (65)–(68) hold.
We will define the projectors(97)
R
1
=
P
1
,
R
2
=
P
2
,
R
3
=
I
-
P
1
,
R
4
=
I
-
P
2
,such that(98)
R
3
R
4
=
R
4
R
3
=
P
3
.Projectors R
1, R
2, R
3, and R
4 are compatible with C. Let us define(99)
μ
=
ν
3
=
ν
4
>
0
,
ν
=
-
ν
1
=
-
ν
2
>
0
,
α
n
=
a
n
,
n
∈
N
.Hence, relations (93)–(96) hold.
Sufficiency. We consider the projectors(100)
P
1
=
R
1
,
P
2
=
R
2
,
P
3
=
R
3
R
4
.These are compatible with C.
The statements of Proposition20 follow if we consider(101)
ν
1
=
ν
2
=
-
ν
<
0
,
ν
3
=
ν
4
=
μ
>
0
,
a
n
=
α
n
,
n
∈
N
.Hence, C is exponentially trichotomic, which ends the proof.


## 6. Conclusions

The paper emphasizes a way to unify the analysis of continuous and discrete asymptotic properties for skew-evolution semiflows, such as the exponential dichotomy and trichotomy. Thus, we give characterizations for the asymptotic behaviors in discrete time, in order to gain necessary instruments in punctuating the properties of the solutions of difference equations.



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*Source: 4375069-2016-07-12.xml*