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

Authors: Mihai-Gabriel Babuţia; Nicolae Marian Seimeanu
Journal: International Journal of Differential Equations
 (2015)

---

## 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.


---

## Body

## 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 Section5, 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)
l
2
(
N
,
R
) will be the Banach space of all real valued sequences x
=
(
x
n
)
n
≥
0 satisfying (1)
∑
n
=
0
∞
x
n
2
<
∞
endowed with the normx
2
=
∑
n
=
0
∞
x
n
2
1
/
2.
(ii)
X will be be a real or complex Banach space and B
(
X
) will be the Banach space of all bounded linear operators on X.
(iii)
The norms onX and B
(
X
) will be denoted by ·.
(iv)
The identity operator onX is denoted by I.
(v)
Δ will be the set defined by Δ
=
{
(
t
,
s
)
∈
R
+
2
:
t
≥
s
≥
0
}
.Definition 1.
A mappingP
:
R
+
→
B
(
X
) is called afamily of projections on X if (2)
P
t
P
t
=
P
t
,
∀
t
≥
0
.Definition 2.
A family of projectionsP
:
R
+
→
B
(
X
) is called (i)
polynomially bounded if there exist M
≥
1 and γ
≥
0 such that (3)
P
t
≤
M
t
+
1
γ
,
∀
t
≥
0
;
(ii)
bounded if there exists M
≥
1 such that (4)
P
t
≤
M
,
∀
t
≥
0
.Definition 3.
Three families of projectionsP
,
Q
,
R
:
R
+
→
B
(
X
) are calledsupplementary if for all t
≥
0 one has that(5)
P
t
+
Q
t
+
R
t
=
I
.In what follows, we present two leading examples of families of projections which will be used in Section5.Example 4.
LetX
=
l
2
(
N
,
R
) and p
:
R
+
→
R
+ be a nondecreasing function. For every t
≥
0 we define P
1
(
t
)
:
l
2
(
N
,
R
)
→
l
2
(
N
,
R
) by (6)
P
1
t
x
=
y
n
t
n
≥
0
,where(7)
y
3
n
t
=
x
3
n
+
p
t
·
x
3
n
+
1
,
y
3
n
+
1
t
=
0
,
y
3
n
+
2
t
=
0
,
n
∈
N
. Let t
≥
0. One can see that P
1
(
t
) is linear and if x
=
(
x
n
)
n
≥
0
∈
l
2
(
N
,
R
), we have that(8)
P
1
t
x
2
2
=
∑
n
=
0
∞
y
3
n
t
2
≤
∑
n
=
0
∞
x
n
2
1
/
2
+
p
t
∑
n
=
0
∞
x
n
2
1
/
2
2
=
1
+
p
t
2
·
x
2
2 from where it follows that P
1
(
t
)
∈
B
(
l
2
(
N
,
R
)
) and P
1
(
t
)
≤
1
+
p
(
t
).
Moreover, lett
≥
0 and x
~
=
(
x
~
n
)
n
≥
0 given by (9)
x
~
3
n
=
x
~
3
n
+
2
=
0
,
x
~
3
n
+
1
=
1
3
n
+
2
,
n
∈
N
.From (10)
P
1
t
x
~
2
=
∑
n
=
0
∞
y
3
n
t
2
1
/
2
=
p
t
·
∑
n
=
0
∞
1
3
n
+
2
2
1
/
2
=
p
t
·
x
~
2 it follows that P
1
(
t
)
≥
p
(
t
)
. From here we get that(11)
m
a
x
1
,
p
t
≤
P
1
t
≤
1
+
p
t
∀
t
∈
R
+
.
Moreover, for(
t
,
x
)
∈
R
+
×
l
2
(
N
,
R
) we define the family of projections Q
1
:
R
+
→
B
(
X
) by Q
1
(
t
)
x
=
(
z
n
(
t
)
)
n
≥
0, where (12)
z
3
n
t
=
-
p
t
x
3
n
+
1
,
z
3
n
+
1
t
=
x
3
n
+
1
,
z
3
n
+
2
t
=
0
,
n
∈
N
,
t
≥
0
. Moreover, for (
t
,
s
,
x
)
∈
Δ
×
l
2
(
N
,
R
) one can see that (13)
Q
1
s
x
2
2
=
∑
n
=
0
∞
z
n
s
2
=
∑
n
=
0
∞
z
3
n
s
2
+
∑
n
=
0
∞
z
3
n
+
1
s
2
=
1
+
p
s
2
∑
n
=
0
∞
x
3
n
+
1
2
≤
1
+
p
t
2
∑
n
=
0
∞
x
3
n
+
1
2
=
Q
1
t
x
2
2
;
hence(14)
Q
1
s
x
2
≤
Q
1
t
x
2
,
Q
1
t
x
2
=
1
+
p
t
2
·
∑
n
=
0
∞
x
3
n
+
1
2
1
/
2
.
Finally, defineR
1
:
R
+
→
B
(
X
) by R
1
(
t
)
=
(
w
n
(
t
)
)
n
≥
0, where (15)
w
3
n
t
=
0
,
w
3
n
+
1
t
=
0
,
w
3
n
+
2
t
=
x
3
n
+
2
,
n
∈
N
,
t
≥
0
.We have that R
1 is bounded with (16)
R
1
t
=
1
,
∀
t
≥
0and moreover the families of projections P
1, Q
1, and R
1 are supplementary.Example 5.
LetX
=
l
2
(
N
,
R
) and define P
2
,
Q
2
,
R
2
:
R
+
→
B
(
l
2
(
N
,
R
)
) by P
2
(
t
)
x
=
(
y
n
(
t
)
)
n
≥
0, Q
2
(
t
)
x
=
(
z
n
(
t
)
)
t
≥
0, and R
2
(
t
)
x
=
(
w
n
(
t
)
)
n
≥
0, where, for n
∈
N, (17)
y
4
n
t
=
x
4
n
,
y
4
n
+
1
t
=
0
,
y
4
n
+
2
t
=
0
,
y
4
n
+
3
t
=
0
,
z
4
n
t
=
0
,
z
4
n
+
1
t
=
x
4
n
+
1
,
z
4
n
+
2
t
=
x
4
n
+
2
,
z
4
n
+
3
t
=
0
,
w
4
n
t
=
0
,
w
4
n
+
1
t
=
0
,
w
4
n
+
2
t
=
0
,
w
4
n
+
3
t
=
x
4
n
+
3
.
We have thatP
2
,
Q
2, and R
2 are three supplementary families of projections with (18)
P
2
t
=
Q
2
t
=
R
2
t
=
1
∀
t
≥
0
.


## 3. Evolution Operators

Definition 6.
A mappingΦ
:
Δ
→
B
(
X
) is called anevolution operator on X if (e
1)
Φ
(
t
,
t
)
=
I, ∀
t
≥
0;
(e
2)
Φ
(
t
,
s
)
Φ
(
s
,
t
0
)
=
Φ
(
t
,
t
0
), ∀
(
t
,
s
)
,
(
s
,
t
0
)
∈
Δ.Definition 7.
A family of projectionsP
:
R
+
→
B
(
X
) is said to beinvariant for the evolution operator Φ
:
Δ
→
B
(
X
) if (19)
Φ
t
,
s
P
s
=
P
t
Φ
t
,
s
∀
t
,
s
∈
Δ
.Given three supplementary families of projectionsP, Q, and R which are invariant for a given evolution operator Φ, we will name the quadruple (
Φ
,
P
,
Q
,
R
) atrichotomy quadruple.Two important examples of trichotomy quadruples are given below, which will serve as a milestone in our examples and counterexamples.Example 8.
OnX
=
l
2
(
N
,
R
) consider the families of projections P
1, Q
1, and R
1 from Example 4. Consider φ
:
R
+
→
(
0
,
∞
) and Φ
1
:
Δ
→
B
(
l
2
(
N
,
R
)
) given by (20)
Φ
1
t
,
s
=
φ
s
φ
t
·
P
1
s
+
φ
t
φ
s
·
Q
1
t
+
R
1
s for all (
t
,
s
)
∈
Δ.
Taking into account that for allt
,
s
∈
R
+ the following relations hold, (21)
P
1
t
P
1
s
=
P
1
s
,
Q
1
t
Q
1
s
=
Q
1
t
,it follows that Φ
1 is an evolution operator. It is easy to check that P
1, Q
1, and R
1 are invariant for Φ
1; hence (
Φ
1
,
P
1
,
Q
1
,
R
1
) is a trichotomy quadruple. Moreover we have that(22)
Φ
1
t
,
s
P
1
s
=
φ
s
φ
t
P
1
s
,
Φ
1
t
,
s
Q
1
s
=
φ
t
φ
s
Q
1
t
,
Φ
1
t
,
s
R
1
s
=
R
1
sfor all (
t
,
s
)
∈
Δ.Example 9.
OnX
=
l
2
(
N
,
R
) consider P
2, Q
2, and R
2 to be the families of projections defined in Example 5. For ψ
:
R
+
→
(
0
,
∞
) define Φ
2
:
Δ
→
B
(
l
2
(
N
,
R
)
) by (23)
Φ
2
t
,
s
x
=
y
n
t
,
s
n
≥
0
if
t
>
s
x
,
if
t
=
s
,where (24)
y
4
n
t
,
s
=
ψ
s
ψ
t
x
4
n
,
y
4
n
+
1
t
,
s
=
ψ
t
ψ
s
x
4
n
+
1
,
y
4
n
+
2
t
,
s
=
0
,
y
4
n
+
3
t
,
s
=
x
4
n
+
3
,
n
∈
Nfor all (
t
,
s
,
x
)
∈
Δ
×
l
2
(
N
,
R
).
It is easy to see that(
Φ
2
,
P
2
,
Q
2
,
R
2
) is a trichotomy quadruple and for (
t
,
s
,
x
)
∈
Δ
×
l
2
(
N
,
R
) one has that(25)
Φ
2
t
,
s
P
2
s
x
=
p
n
t
,
s
n
≥
0
,
p
4
n
t
,
s
=
ψ
s
ψ
t
x
4
n
,
p
4
n
+
1
t
,
s
=
p
4
n
+
2
t
,
s
=
p
4
n
+
3
t
,
s
=
0
.
Φ
2
t
,
s
Q
2
s
x
=
q
n
t
,
s
n
≥
0
,
t
>
s
ρ
n
t
,
s
n
≥
0
,
t
=
swhich is given by (26)
q
4
n
t
,
s
=
q
4
n
+
3
t
,
s
=
0
,
q
4
n
+
1
t
,
s
=
ψ
t
ψ
s
x
4
n
+
1
,
q
4
n
+
2
t
,
s
=
0
,
ρ
4
n
t
,
s
=
ρ
4
n
+
3
t
,
s
=
0
,
ρ
4
n
+
1
t
,
s
=
x
4
n
+
1
,
ρ
4
n
+
2
t
,
s
=
x
4
n
+
2
,
n
∈
Nand Φ
2
(
t
,
s
)
R
2
(
s
)
x
=
(
r
n
(
t
,
s
)
)
n
≥
0, where (27)
r
4
n
t
,
s
=
r
4
n
+
1
t
,
s
=
r
4
n
+
2
t
,
s
=
0
,
r
4
n
+
3
t
,
s
=
x
4
n
+
3
.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(
Φ
,
P
,
Q
,
R
) is said to bepolynomially trichotomic (p.t.) if there exist N
≥
1, α
>
0, and β
≥
0 such that (
p
t
1)
(
t
+
1
)
α
Φ
(
t
,
s
)
P
(
s
)
≤
N
(
s
+
1
)
α
+
β;
(
p
t
2)
(
t
+
1
)
α
≤
N
(
t
+
1
)
β
(
s
+
1
)
α
Φ
(
t
,
s
)
Q
(
s
);
(
p
t
3)
(
s
+
1
)
α
Φ
(
t
,
s
)
R
(
s
)
≤
N
(
t
+
1
)
α
(
s
+
1
)
β;
(
p
t
4)
(
s
+
1
)
α
≤
N
(
t
+
1
)
α
+
β
Φ
t
,
s
R
s
for all(
t
,
s
)
∈
Δ
.Ifβ from the above definition is equal to 0, then we say that (
Φ
,
P
,
Q
,
R
) isuniformly polynomially trichotomic (u.p.t.).Remark 11.
The following assertions hold:(i)
If a trichotomy quadruple(
Φ
,
P
,
Q
,
R
) is (p.t.) then P and R are polynomially bounded, and hence Q is also polynomially bounded.
(ii)
If a trichotomy quadruple(
Φ
,
P
,
Q
,
R
) is (u.p.t.) then P and R are bounded, and hence Q is also bounded.
In other words, if(
Φ
,
P
,
Q
,
R
) is (p.t.) with constants N, α, and β then (28)
max
⁡
P
t
,
Q
t
,
R
t
≤
3
N
t
+
1
β
,
∀
t
≥
0
.Remark 12.
If(
Φ
,
P
,
Q
,
R
) is (u.p.t.) then it is (p.t.). The converse is not generally true. Take, for example, the trichotomy quadruple (
Φ
1
,
P
1
,
Q
1
,
R
1
) from Example 8 with p
(
t
)
=
φ
(
t
)
=
t
+
1. It is easy to check that (
Φ
1
,
P
1
,
Q
1
,
R
1
) is (p.t.), but it cannot be (u.p.t.), because P is not bounded.Definition 13.
A trichotomy quadruple(
Φ
,
P
,
Q
,
R
) is said to bestrongly polynomially trichotomic (s.p.t.) if there exist N
≥
1, α
>
0, and β
≥
0 such that (
s
p
t
1)
(
t
+
1
)
α
Φ
(
t
,
s
)
P
(
s
)
x
≤
N
(
s
+
1
)
α
+
β
P
(
s
)
x;
(
s
p
t
2)
(
t
+
1
)
α
Q
(
s
)
x
≤
N
(
t
+
1
)
β
(
s
+
1
)
α
Φ
(
t
,
s
)
Q
(
s
)
x;
(
s
p
t
3)
(
s
+
1
)
α
Φ
(
t
,
s
)
R
(
s
)
x
≤
N
(
t
+
1
)
α
(
s
+
1
)
β
R
(
s
)
x;
(
s
p
t
4)
(
s
+
1
)
α
R
(
s
)
x
≤
N
(
t
+
1
)
α
+
β
Φ
(
t
,
s
)
R
(
s
)
x
for all(
t
,
s
,
x
)
∈
Δ
×
X
.Ifβ from the above definition is equal to 0, then we say that (
Φ
,
P
,
Q
,
R
) isuniformly strongly polynomially trichotomic (u.s.p.t.).Remark 14.
If(
Φ
,
P
,
Q
,
R
) 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(
Φ
,
P
,
Q
,
R
) is (s.p.t.) then the following condition holds: (29)
Range
⁡
Q
s
∩
Ker
⁡
Φ
t
,
s
=
0
∀
t
,
s
∈
Δ
.In other words, for all (
t
,
s
)
∈
Δ, (30)
x
∈
Range
⁡
Q
s
,
Φ
t
,
s
x
=
0
⟹
x
=
0
.Remark 16.
Under the same assumption as in Remark15, we also have that (31)
Range
⁡
R
s
∩
Ker
⁡
Φ
t
,
s
=
0
∀
t
,
s
∈
Δ
.Definition 17.
A trichotomy quadruple(
Φ
,
P
,
Q
,
R
) is said to beweakly polynomially trichotomic (w.p.t.) if there exist N
≥
1, α
>
0, and β
≥
0 such that (
w
p
t
1)
(
t
+
1
)
α
Φ
(
t
,
s
)
P
(
s
)
≤
N
(
s
+
1
)
α
+
β
P
(
s
);
(
w
p
t
2)
(
t
+
1
)
α
Q
(
s
)
≤
N
(
t
+
1
)
β
(
s
+
1
)
α
Φ
(
t
,
s
)
Q
(
s
);
(
w
p
t
3)
(
s
+
1
)
α
Φ
(
t
,
s
)
R
(
s
)
≤
N
(
t
+
1
)
α
(
s
+
1
)
β
R
(
s
);
(
w
p
t
4)
(
s
+
1
)
α
R
(
s
)
≤
N
(
t
+
1
)
α
+
β
Φ
(
t
,
s
)
R
(
s
)
for all(
t
,
s
)
∈
Δ
.Ifβ
=
0 then we say that (
Φ
,
P
,
Q
,
R
) isuniformly weakly polynomially trichotomic (u.w.p.t.).Remark 18.
If(
Φ
,
P
,
Q
,
R
) 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(
Φ
,
P
,
Q
,
R
) is (s.p.t.) then it is also (w.p.t.). Moreover, if (
Φ
,
P
,
Q
,
R
) is (u.s.p.t.), then it is (u.w.p.t.).Proposition 20.
Let(
Φ
,
P
,
Q
,
R
) be a trichotomy quadruple. If (
Φ
,
P
,
Q
,
R
) is (p.t.) then it is also (w.p.t.).Proof.
LetN
≥
1, α
>
0, and β
≥
0 be given by Definition 10. By Remark 11, we have that (32)
1
≤
m
a
x
P
t
,
Q
t
,
R
t
≤
3
N
t
+
1
β
∀
t
≥
0
.Let now (
t
,
s
)
∈
Δ. From the estimations (33)
t
+
1
α
Φ
t
,
s
P
s
≤
N
s
+
1
α
+
β
≤
3
N
2
s
+
1
α
+
2
β
P
s
,
t
+
1
α
Q
s
≤
N
t
+
1
β
s
+
1
α
Φ
t
,
s
Q
s
·
Q
s
≤
3
N
2
s
+
1
α
t
+
1
2
β
Φ
t
,
s
Q
s
s
+
1
α
Φ
t
,
s
R
s
≤
N
t
+
1
α
s
+
1
β
≤
N
t
+
1
α
s
+
1
β
R
s
≤
3
N
2
t
+
1
α
s
+
1
2
β
R
s
,
s
+
1
α
R
s
≤
N
t
+
1
α
+
β
Φ
t
,
s
R
s
·
R
s
≤
3
N
2
t
+
1
α
+
β
s
+
1
β
Φ
t
,
s
R
s
≤
3
N
2
t
+
1
α
+
2
β
Φ
t
,
s
R
s
,it follows that (
Φ
,
P
,
Q
,
R
) is (w.p.t.) with constants 3
N
2
≥
1, α
>
0, and 2
β
≥
0.Remark 21.
From the proof of the above proposition, we can easily see that, by settingβ
=
0, we obtain the implication (
u
.
p
.
t
.
)
⇒
(
u
.
w
.
p
.
t
.
).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 Example3.
(ii) The concepts of (p.t.) and (w.p.t.) do not coincide, as we can see from Example4.
(iii) (p.t.) does not imply (s.p.t.) and (u.p.t.) does not imply (u.s.p.t.), as shown by Example5.
(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 Example6.Remark 23.
The connection between the presented concepts is given by the following diagram:(34)
u
.
p
.
t
.
⇏
⇍
u
.
s
.
p
.
t
.
⇒
⇍
u
.
w
.
p
.
t
.
⇏
⇐
u
.
p
.
t
.
⇑
⇓
⇑
⇓
⇑
⇓
⇑
⇓
p
.
t
.
⇏
⇍
s
.
p
.
t
.
⇒
⇍
w
.
p
.
t
.
⇏
⇐
p
.
t
.


## 5. Examples and Counterexamples

Example 1 (trichotomy quadruple that is (s.p.t.) but not (u.s.p.t.)).
Let(
Φ
1
,
P
1
,
Q
1
,
R
1
) be the trichotomy quadruple from Example 8 with φ
(
t
)
=
(
t
+
1
)
3
-
cos
⁡
ln
⁡
(
t
+
1
) and p
(
t
)
=
0, t
≥
0. Then we have that (35)
t
+
1
2
Φ
1
t
,
s
P
1
s
x
2
≤
s
+
1
4
P
1
s
x
2
,
t
+
1
2
Q
1
s
x
2
≤
s
+
1
4
Φ
1
t
,
s
Q
1
s
x
2
,
Φ
t
,
s
R
s
x
2
=
R
s
x
2 for all (
t
,
s
,
x
)
∈
Δ
×
X; hence (
Φ
1
,
P
1
,
Q
1
,
R
1
) is (s.p.t.).
Assume, by a contradiction, that(
Φ
,
P
,
Q
,
R
) is (u.s.p.t.). Then there exist N
≥
1 and α
>
0 such that for all n
∈
N and for t
n
=
e
2
n
π
-
1 and s
n
=
e
2
n
π
-
π
/
2
-
1 we have, from (
s
p
t
1
), that (36)
e
2
n
π
-
π
/
2
4
e
4
n
π
≤
N
e
2
n
π
-
π
/
2
2
2
n
π
α which leads us to the contradiction (37)
e
2
n
π
≤
N
e
-
π
/
2
,
∀
n
∈
N
.Example 2 (trichotomy quadruple that is (w.p.t.) but not (u.w.p.t.)).
Let(
Φ
1
,
P
1
,
Q
1
,
R
1
) be as in Example 1. By Remark 19 we have that (
Φ
1
,
P
1
,
Q
1
,
R
1
) is (w.p.t.). The same contradiction is obtained as in Example 1, by assuming that (
Φ
1
,
P
1
,
Q
1
,
R
1
) is (u.w.p.t.).Example 3 (trichotomy quadruple which is (s.p.t.) but fails to be (p.t.)).
Let(
Φ
1
,
P
1
,
Q
1
,
R
1
) be the trichotomy quadruple from Example 8 with p
(
t
)
=
(
t
+
1
)
t
+
1 and φ
(
t
)
=
t
+
1.
From(38)
t
+
1
Φ
1
t
,
s
P
1
s
x
2
=
s
+
1
P
1
s
x
2
,
t
+
1
Q
1
s
x
2
≤
φ
t
Q
1
t
x
2
=
s
+
1
Φ
1
t
,
s
Q
1
s
x
2
,
s
+
1
Φ
t
,
s
R
s
x
2
≤
s
+
1
t
+
1
R
s
x
2
,
s
+
1
R
s
x
2
≤
N
t
+
1
2
Φ
t
,
s
R
s
x
2for all (
t
,
s
,
x
)
∈
Δ
×
X, we can see that (
Φ
1
,
P
1
,
Q
1
,
R
1
) is (u.s.p.t.) and hence (s.p.t.).
Assume, by a contradiction, that(
Φ
1
,
P
1
,
Q
1
,
R
1
) is (p.t.). Then, by Remark 11, we have that there exist M
≥
1, γ
≥
0, such that (39)
P
1
t
≤
M
t
+
1
γ
∀
t
≥
0
.This leads us to the contradiction (40)
t
+
1
t
+
1
=
p
t
≤
P
1
t
≤
M
t
+
1
γ
∀
t
≥
0
.It follows that (
Φ
1
,
P
1
,
Q
1
,
R
1
) is not (p.t.) and hence not (u.p.t.).Example 4 (trichotomy quadruple which is (w.p.t.) but not (p.t.)).
Let(
Φ
1
,
P
1
,
Q
1
,
R
1
) the trichotomy quadruple from Example 3. By Remark 19, we have that (
Φ
1
,
P
1
,
Q
1
,
R
1
) 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(
Φ
2
,
P
2
,
Q
2
,
R
2
) be the trichotomy quadruple from Example 9 with ψ
(
t
)
=
t
+
1. First of all we will show that (
Φ
2
,
P
2
,
Q
2
,
R
2
) is (u.p.t.). Let (
t
,
s
,
x
)
∈
Δ
×
l
2
(
N
,
R
). We have that (41)
t
+
1
2
Φ
2
t
,
s
P
2
s
x
2
2
=
∑
n
=
0
∞
p
n
t
,
s
2
=
s
+
1
2
∑
n
=
0
∞
x
4
n
2
=
s
+
1
2
P
2
s
x
2
2
;hence (42)
t
+
1
Φ
2
t
,
s
P
2
s
≤
s
+
1
.If t
>
s, consider x
=
(
x
n
)
n
≥
0 given by (43)
x
n
=
1
n
,
n
=
4
k
+
1
0
,
otherwise.We have that x
2
=
∑
n
=
0
∞
(
1
/
(
4
n
+
1
)
2
)
1
/
2 and (44)
s
+
1
Φ
2
t
,
s
Q
2
s
x
2
=
∑
n
=
0
∞
q
n
t
,
s
2
1
/
2
=
t
+
1
x
2
;hence (45)
t
+
1
≤
s
+
1
Φ
2
t
,
s
Q
2
s
.Having in mind that Φ
2
(
t
,
s
)
R
(
s
)
x
2
=
R
(
s
)
x
2, it follows that (s
p
t
3) and (s
p
t
4) hold for (
t
,
s
,
x
)
∈
Δ
×
X with t
>
s. The case in which t
=
s obviously leads us to the above estimation, and so the conclusion follows.
In what follows, we will show that(
Φ
2
,
P
2
,
Q
2
,
R
2
) is not (s.p.t.), and hence it is not (u.s.p.t.). Assume by a contradiction that (
Φ
2
,
P
2
,
Q
2
,
R
2
) is (s.p.t.). We will disprove the result from Remark 15. Let x
=
(
x
n
)
n
≥
0 given by (46)
x
4
n
+
2
=
1
4
n
+
2
,
x
4
n
+
3
=
x
4
n
+
1
=
x
4
n
=
0
n
∈
N
.Obviously x
∈
l
2
(
N
,
R
) and denote, for every s
≥
0, Q
2
(
s
)
x
=
(
z
n
(
s
)
)
n
≥
0, where (47)
z
4
n
s
=
0
,
z
4
n
+
1
s
=
x
4
n
+
1
=
0
,
z
4
n
+
2
s
=
x
4
n
+
2
=
1
4
n
+
2
,
z
4
n
+
3
s
=
0
;hence (
z
n
(
s
)
)
n
≥
0 is a nonzero sequence.
Consider now(
t
,
s
)
∈
Δ with t
>
s. By denoting Φ
2
(
t
,
s
)
Q
2
(
s
)
x
=
(
q
n
(
t
,
s
)
)
n
≥
0, with (48)
q
4
n
t
,
s
=
0
,
q
4
n
+
1
t
,
s
=
t
+
1
s
+
1
x
4
n
+
1
=
0
,
q
4
n
+
2
t
,
s
=
0
,
q
4
n
+
3
t
,
s
=
0
,it follows that Φ
2
(
t
,
s
)
Q
2
(
s
)
x
=
0, which contradicts Remark 15; hence (
Φ
2
,
P
2
,
Q
2
,
R
2
) is not (s.p.t.).Example 6 (trichotomy quadruple that is (w.p.t.) but fails to be (s.p.t.)).
Let(
Φ
2
,
P
2
,
Q
2
,
R
2
) be the trichotomy quadruple from Example 5. Taking into account that, for all s
≥
0, P
2
(
s
)
=
Q
2
(
s
)
=
R
2
(
s
)
=
1, it follows that (
Φ
2
,
P
2
,
Q
2
,
R
2
) is (u.w.p.t.) and hence (w.p.t.).
Again, from Example5, we get that (
Φ
2
,
P
2
,
Q
2
,
R
2
) is not (s.p.t.) and hence not (u.s.p.t.).



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*Source: 241402-2015-10-29.xml*