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.

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 [

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 [

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

Throughout this paper, we will consider the following framework:

endowed with the norm

The norms on

The identity operator on

A mapping

A family of projections

Three families of projections

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

Let

Moreover, let

Moreover, for

hence

Finally, define

Let

We have that

A mapping

A family of projections

Given three supplementary families of projections

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

On

Taking into account that for all

On

It is easy to see that

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

A trichotomy quadruple

for all

If

The following assertions hold:

If a trichotomy quadruple

If a trichotomy quadruple

In other words, if

If

A trichotomy quadruple

for all

If

If

If

Under the same assumption as in Remark

A trichotomy quadruple

for all

If

If

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

If a trichotomy quadruple

Let

Let

From the proof of the above proposition, we can easily see that, by setting

Other connections are given by the following.

(i) (s.p.t.) does not imply (p.t.) and (u.s.p.t.) does not imply (u.p.t.) as shown by Example

(ii) The concepts of (p.t.) and (w.p.t.) do not coincide, as we can see from Example

(iii) (p.t.) does not imply (s.p.t.) and (u.p.t.) does not imply (u.s.p.t.), as shown by Example

(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

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

Let

Assume, by a contradiction, that

Let

Let

From

Assume, by a contradiction, that

Let

Let

In what follows, we will show that

Consider now

Let

Again, from Example

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