# A nonuniform Markus–Yamabe conjecture: Triangular case via uniformization

Álvaro Castañeda  ,  Ignacio Huerta  and  Gonzalo Robledo  Universidad de
Chile, Departamento de Matemáticas. Casilla 653, Santiago, Chile
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We introduce a nonautonomous nonuniform Markus-Yamabe Conjecture, namely, a
global problem on nonuniform asymptotic stability for nonautonomous
differential systems, whose restriction to the autonomous case is related to
the classical Markus–Yamabe Conjecture. Additionally, we prove that the
conjecture is verified in dimension one and for a family of triangular systems
of nonautonomous differential equations satisfying technical boundedness
assumptions. An essential tool to carry out the proof in the triangular case
is a necessary and sufficient condition ensuring the property of nonuniform
exponential dichotomy for upper block triangular linear differential systems,
which has been developed previously for the uniform case. We extend this
result for a specific nonuniform exponential dichotomy and we obtain some
byproducts having interest on itself, such as, the diagonal significance
property.

###### Key words and phrases:

Markus-Yamabe Problem; Nonuniform Bounded Growth; Nonuniform Exponential
Dichotomy; Nonuniform Asymptotic Stability; Nonuniform Spectrum; Diagonal
Significance

###### 2010 Mathematics Subject Classification:

34D09, 34C11

The first author was funded by FONDECYT Regular Grant 1200653. The second
author was funded by FONDECYT POSTDOCTORAL Grant 3210132. The third author was
funded by FONDECYT Regular Grant 1210733

##  1\. Introduction

###  1.1. State of art

In the last decade, the problem of global stability for ordinary differential
equations, also known as the Markus–Yamabe Conjecture, has been revisited from
different approaches, namely, the case of continuous and discontinuos
piecewise autonomous vector fields have been considered by J. Llibre $\&$ X.
Zhang [24], L. Menezes [25], and Y.Zhang $\&$ X-S. Yang [33]. A
infinite–dimensional perspective has been studied by H.M. Rodrigues et al.
[30]. On a nonautonomous dynamical systems context, D. Cheban [7] worked in
the framework of cocycles, while Á. Castañeda and G. Robledo [6] established a
version of this problem of global stability in terms of the uniform
exponential dichotomy.

Recall that the Markus-Yamabe Conjecture is a problem of global asymptotic
stability for continuous autonomous dynamical systems on finite dimension,
introduced in 1960 by L. Markus and H. Yamabe [27], which states that if the
differential system $\dot{x}=f(x)$, where
$f\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ of class $C^{1}$, $f(0)=0$ and it is
a Hurwitz vector field, that is, the eigenvalues of the Jacobian matrix of $f$
have negative real part at any $x\in\mathbb{R}^{n}$, or equivalently $Jf(x)$
is a Hurwitz matrix for any $x$, then the origin is globally uniformly
asymptotically stable.

It is known that this global stability problem is true when $n\leq 2.$ For
details about the proof in the planar case, see R. Feßler in [14], A.A.
Glutsyuk in [15] and C. Gutiérrez in [17]. When $n\geq 3$, the Markus–Yamabe
Conjecture is false due to A. Cima et al. [8] found a polynomial vector field
that satisfies the hypothesis of the problem, however the differential system
associated to this vector field has a solution which escape to infinity.

In spite that the conjecture is now completely resolved in its autonomous
classical version, many authors have dedicated to showing vector fields that
satisfy both the hypothesis and its conclusion (see [5], [9], [16]).
Remarkable examples of such setting is the case of triangular and gradient
vector fields. Indeed, in a triangular context, L. Markus and H. Yamabe proved
that the conjecture is true [27, Th.4] while P. Hartman [18, p. 539 Corollary
11.2] (see also [26]) showed that for the conjecture is also true gradient
vector fields.

The main idea of this article is to settle a global nonuniform stability
problem for a nonlinear nonautonomous system

(1.1) $\dot{x}=f(t,x).$

Notice that, in the linear case, the global nonuniform asymptotic stability is
consequence of the nonuniform exponential stability, which can be described in
terms of a particular nonuniform exponential dichotomy. We emphasize that this
property of dichotomy is associated to a spectral theory, which will allow us
to emulate the notion of Hurwitz vector fields to the nonuniform framework.

###  1.2. The Nonuniform conjecture

While the autonomous dynamical systems depend only on the time elapsed from
the initial time $t_{0}$, the nonautonomous dynamical systems are also
dependent on the initial time $t_{0}$ itself, which has several consequences
to characterize limiting objects. As it was pointed out in [21, Ch.2] and [22,
Sec.2.2], the dynamics arising from nonautonomous differential systems (1.1)
can be formally described by two approaches where the above mentioned $t_{0}$
plays a key role: the skew product semiflows and the process formalism, also
known as the two parameter$-(t,t_{0})-$semigroups.

Let us recall that the Markus–Yamabe Conjecture is stated in terms of the
negativeness of the real part of the eigenvalues of $Jf(x)$ and the global
attractiveness of the origin has a behavior described by the uniform
asymptotic stability. Contrarily, it is known that the local stability of a
linear nonautonomous system cannot be always determined by the eigenvalues,
see e.g. [27, p.310]. However, we point out about the existence of several
spectral theories based either on characteristic exponents (Lyapunov, Perron
and Bohl exponents) or dichotomies [12], which have associated a wide range of
asymptotic stabilities for nonautonomous linear systems, being the uniform
asymptotic stability only a particular case.

In this article, we will work with the global nonuniform asymptotic stability
to settle a nonautonomous and noununiform Markus–Yamabe conjecture (NNMYC). We
emulate the results obtained in [6] for a uniform version of the conjecture.
Namely, we prove the NNMYC for $n=1$ and give an example, in arbitrary
dimension, of a nonautonomous quasilinear vector field satisfying the
hypothesis of the nonuniform global stability problem and its conclusion. In
[6] it was also proved that the uniform conjecture is also true for triangular
systems. Nevertheless, we emphasize that, when we are working on the NNMYC,
the triangular case is not an extension of the uniform case and we require to
construct subtle and deeper methods to cope with this problem.

###  1.3. Triangular case setting

As we above stated, in this article we prove that this conjecture is also
verified for a family of triangular vector fields. We point out that the proof
is completely different to the ones made in [27, Th.4] in an autonomous
framework and [6, Cor.1] in a uniform nonautonomous case, respectively. In
both previous works, as the underlying stability is the uniform one, we can
use known results describing the stability of triangular differential systems
in terms of its diagonal properties. A tool used in the proof of the
triangular case in a nonautonomous uniform context is a result of F. Batelli
and K.J. Palmer in [3] providing necessary and sufficient conditions ensuring
that an upper triangular linear system has a uniform exponential dichotomy on
the half line whenever its diagonal subsystems also have this property.

In order to obtain a similar tool to tackle the nonuniform context, we
generalize the result of Batelli and Palmer by using the Lemma of
Uniformization introduced by L. Zhou et al. in [35, Lemma 1] and its
consequences. The generalization has a chain of byproducts, which allow us to
conclude that NNMYC is verified for a family of triangular systems.

###  1.4. Structure

The section 2 gives a general setting: the subsection 2.1 describe the
properties of nonuniform bounded growth, nonuniform contractions, nonuniform
exponential dichotomies, nonuniform spectrum and explore some relations
between them. The subsection 2.2 recalls the functional framework needed to
introduce the Uniformization Lemma. The section 3 recalls the property of
global nonuniform asymptotic stability for an equilibrium of (1.1) and
introduces the NNMYC, additionally we prove that this conjecture is true in
dimension one and for a family of quasilinear systems in arbitrary dimension.
Our main results are immersed in the section 4, where we prove –Theorem 4–
that the conjecture is verified for a family of triangular vector fields. The
proof of this theorem is based in Theorem 3 and its consequences. The Theorem
3 establishes a relation between upper triangular linear systems and its
diagonal subsystems associated with respect to the properties of nonuniform
exponential dichotomy and nonuniform bounded growth. In particular, Theorem 3
establishes that if the diagonal subsystems have the half nonuniform bounded
growth and nonuniform exponential dichotomy combined with the fact that the
non diagonal block is bounded in terms a of a parametrized norm, then the
nonuniform exponential dichotomy is preserved for the upper triangular system.
Finally, in the Appendix, we give the a detailed proof of Theorem 3.

###  1.5. Notations

Troughout this paper, $||\cdot||$ and $|\cdot|$ will denote matrix and vector
norms respectively. The set $[0,+\infty)$ is denoted by $\mathbb{R}_{0}^{+}$
and the set of square $n\times n$ matrices with real coefficients is denoted
by $M_{n}$, while $I_{n}$ is the identity matrix. A continuous function
$G\colon\mathbb{R}_{0}^{+}\to[1,+\infty)$ will be called a growth rate.

##  2\. Preliminar definitions and contextualization

###  2.1. Nonuniform bounded growth, nonuniform dichotomies and spectrum

We will recall basic properties about the nonautonomous linear differential
system

(2.1) $\dot{x}=A(t)x,$

where $A\colon\mathbb{R}_{0}^{+}\mapsto M_{n}(\mathbb{R})$ is a locally
integrable matrix function. A basis of solutions of (2.1) is denoted by
$T(t)$, which satisfies $\dot{T}(t)=A(t)T(t)$ and its corresponding evolution
operator is $T(t,s)=T(t)T^{-1}(s)$, then the solution of (2.1) with initial
condition $x_{0}$ at $t_{0}$ defined by $x(t,t_{0},x_{0})=T(t,t_{0})x_{0}$.

Similarly as in the uniform case, there exist two definitions of nonuniform
bounded growth in the literature:

######  Definition 1.

The evolution operator $T(t,s)$ of (2.1) has a:

  * a)

Full $(M(s),\nu)$–nonuniform bounded growth, namely, there exist a constant
$\nu>0$ and a growth rate $M\colon\mathbb{R}_{0}^{+}\to[1,+\infty)$ such that

$\left\|T(t,s)\right\|\leq M(s)e^{\nu\left|t-s\right|}\quad t,\
s\in\mathbb{R}_{0}^{+},$

  * b)

Half $(M(s),\nu)$–nonuniform bounded growth, namely, there exist a constant
$\nu>0$ and a growth rate $M\colon\mathbb{R}_{0}^{+}\to[1,+\infty)$ such that

$\left\|T(t,s)\right\|\leq M(s)e^{\nu(t-s)}\quad t\geq s\geq 0.$

We point out that there are no standard definition of bounded growth in the
current literature and we are proposing the previous ones in order
distinguished them and its consequences. Note that:

  * i)

The property of half $(M(s),\nu)$–nonuniform bounded growth is considered in
[35, p.686] under the name of nonuniform bounded growth.

  * ii)

The property of the full $(M(s),\nu)$–nonuniform bounded growth is considered
with the particular case of $M(s)=Me^{\varepsilon s}$ where $\varepsilon\geq
0$ and $M\geq 1$ by [11, p.547] and [34, p.1892], also under the name of
nonuniform bounded growth,

  * iii)

We propose to denote the particular case of $M(s)=M\geq 1$ as uniform bounded
growth, which has been considered respectively by S. Siegmund [31, p.253] and
W. Coppel [10, pp.8-9] in a full and half version respectively under the name
of bounded growth.

From now on, in this paper we will work with the property of half $(Me^{\delta
s},\nu)$–nonuniform bounded growth.

The property of exponential dichotomy plays an important role in the study of
nonautonomous linear systems and –roughly speaking– refers that the solutions
of (2.1) can be splitted as either exponential expansions or exponential
contractions. A formal definition for the nonuniform framework is given by:

######  Definition 2.

The linear system (2.1) has a $(K(s),\gamma)$–nonuniform exponential dichotomy
on $\mathbb{R}_{0}^{+}$ if there exist a family of invariant projections
$P(t)\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ for any $t\in\mathbb{R}_{0}^{+}$,
a positive constant $\gamma$ and a growth rate
$K\colon\mathbb{R}_{0}^{+}\to[1,+\infty)$ such that:

(2.2) $T(t,s)P(s)=P(t)T(t,s)\quad\textnormal{for $t\geq s\geq 0$},$ (2.3)
$\left\\{\begin{array}[]{rcl}||T(t,s)P(s)||&\leq&K(s)e^{-\gamma(t-s)}\quad\textnormal{for
$t\geq s\geq 0$},\\\ \\\
||T(t,s)[I-P(s)]||&\leq&K(s)e^{-\gamma(s-t)}\quad\textnormal{for $0\leq t\leq
s$}.\end{array}\right.$

The property (2.2) implies that the range of $P(\cdot)$ is invariant for any
$t$, which motivates the name invariant projectors.

The inequalities (2.3) imply that $t\mapsto T(t,t_{0})\xi$, the forward
solution of (2.1) passing through $\xi\neq 0$ at $t=t_{0}$, can be splitted by
$P(t_{0})$ in $t\mapsto T(t,t_{0})P(t_{0})\xi$ and $t\mapsto
T(t,t_{0})[I-P(t_{0})]\xi$, whose behavior for any $t\geq t_{0}$ verifies that

$\begin{array}[]{rcl}|T(t,t_{0})P(t_{0})\xi|&\leq&K(t_{0})e^{-\gamma(t-t_{0})}|P(t_{0})\xi|\\\
\\\
(1/K(t_{0}))e^{\gamma(t-t_{0})}|[I-P(t_{0})]\xi|&\leq&|T(t,t_{0})[I-P(t_{0})]\xi|,\end{array}$

that is, any solution is the sum of two solutions having a dichotomic
exponential behavior: the $(K(s),\gamma)$–nonuniform exponential contraction
$t\mapsto T(t,t_{0})P(t_{0})\xi$ and the $(K(s),\gamma)$–nonuniform
exponential expansion $t\mapsto T(t,t_{0})[I-P(t_{0})]$. In this context, we
will say that the system (2.1) is a $(K(s),\gamma)$–nonuniform exponential
contraction if it has a $(K(s),\gamma)$–nonuniform exponential dichotomy on
$\mathbb{R}_{0}^{+}$ with the projector $P(t)=I$ for any $t\geq 0$.

On the other hand, we emphasize that the growth rate $K(\cdot)$ can take a
wide range of possible behaviors. A distinguished case, which is older in the
literature, is given by the constant function $K(s):=K\geq 1$ and corresponds
to the uniform exponential dichotomy on $\mathbb{R}_{0}^{+}$ since the
exponential contractions (expansions) have an exponential upper (lower) bound
which are independent of $t_{0}$. Having in mind this noteworthy case, we can
see that the nonuniform exponential dichotomies have been residually defined
as dichotomies where the growth rate $K(\cdot)$ is not a constant function.

In this article, we will focus in the particular case of nonuniform
exponential dichotomy having a growth rate defined by $K(s)=Ke^{\varepsilon
s}$:

######  Definition 3.

The linear system (2.1) has a $(Ke^{\varepsilon s},\gamma)$–nonuniform
exponential dichotomy on $\mathbb{R}_{0}^{+}$ if there exist a family of
invariant projections $P(t)\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ for any
$t\in\mathbb{R}_{0}^{+}$, a constant $K\geq 1$, and a couple
$(\gamma,\varepsilon)$ of constants such that $\varepsilon\in[0,\gamma)$ and

$T(t,s)P(s)=P(t)T(t,s)\quad\textnormal{for $t,s\geq 0$},$

$\left\\{\begin{array}[]{rcl}||T(t,s)P(s)||&\leq&Ke^{-\gamma(t-s)}e^{\varepsilon
s}\quad\textnormal{for $t\geq s\geq 0$},\\\ \\\
||T(t,s)[I-P(s)]||&\leq&Ke^{-\gamma(s-t)}e^{\varepsilon s}\quad\textnormal{for
$0\leq t\leq s$}.\end{array}\right.$

The above dichotomy has been considered in several works as [1, 11, 34] where
is denoted as nonuniform exponential dichotomy and deserves some remarks:

$\bullet$ Note that if $\varepsilon=0$, we recover the uniform exponential
dichotomy on $\mathbb{R}_{0}^{+}$ and this prompts to denote the term
$e^{\varepsilon s}$ as the nonuniform part.

$\bullet$ In [1, Theorem 10.22], by using the Multiplicative Ergodic Theorem
[1, 10.27] and the Oseledets-Pesin Reduction result [1, Theorem 10.28], it is
shown that almost all variational equations obtained from a measure-preserving
flow admit nonuniform exponential dichotomy and furthermore, the nonuniformity
rate is arbitrarily small.

$\bullet$ There exists a spectral theory for the $(Ke^{\varepsilon
s},\gamma)$–nonuniform exponential dichotomy on $\mathbb{R}$, which has been
constructed in [11, 34] and adapted to the half line by [19] in the continuous
case and [2] in the discrete case.

######  Definition 4.

([11, 34, 37]) The $(Ke^{\varepsilon s},\gamma)$–nonuniform exponential
dichotomy spectrum of (2.1) is the set $\Sigma^{+}(A)$ of
$\lambda\in\mathbb{R}$ such that the system

(2.4) $\dot{x}=[A(t)-\lambda I]x$

does not have a $(Ke^{\varepsilon s},\gamma)$–nonuniform exponential dichotomy
on $\mathbb{R}_{0}^{+}.$ The resolvent $\rho(A)$ is defined as
$\mathbb{R}\setminus\Sigma^{+}(A)$, namely, the values of $\lambda$ such that
the system (2.4) have a $(Ke^{\varepsilon s},\gamma)$–nonuniform exponential
dichotomy on $\mathbb{R}_{0}^{+}$.

######  Proposition 1.

([11], [31], [34]),  If the evolution operator $T(t,s)$ of (2.1) has a half
$(Me^{\delta s},\nu)$–nonuniform bounded growth, then its nonuniform spectrum
$\Sigma^{+}(A)$ is the union of $m$ intervals where $0<m\leq n$, that is,

(2.5) $\Sigma^{+}(A)=\left\\{\begin{array}[]{c}[a_{1},b_{1}]\\\
\textnormal{or}\\\
(-\infty,b_{1}]\end{array}\right\\}\cup\;[a_{2},b_{2}]\;\cup\;\cdots\;\cup\;[a_{m-1},b_{m-1}]\;\cup\left\\{\begin{array}[]{c}[a_{m},b_{m}]\\\
\textnormal{or}\\\ \textnormal{$[a_{m},+\infty)$}\end{array}\right\\},$

with $-\infty<a_{1}\leq b_{1}<\ldots<a_{m}\leq b_{m}<+\infty$.

The intervals $[a_{i},b_{i}]$ are called spectral intervals for
$i=2,\ldots,m-1$ while the intervals $\rho_{i+1}(A)=(b_{i},a_{i+1})$ for
$i=1,\ldots,m-1$ are called spectral gaps. Additionally, if the first spectral
interval in (2.5) is $[a_{1},b_{1}]$, we can define
$\rho_{1}(A)=(-\infty,a_{1})$ and if the last spectral intervals in (2.5) is
$[a_{m},b_{m}]$, we can define $\rho_{m+1}(A)=(b_{m},+\infty)$. By the
definition of $\Sigma^{+}(A)$, it follows that for any
$\lambda\in\rho_{j}(A)$, the system (2.4) has a $(Ke^{\varepsilon
s},\gamma)$–nonuniform exponential dichotomy with $P_{j}:=P_{j}(\cdot)$. It
can be proved, see _e.g._ [11], that:

  * a)

If the first spectral interval is given by $[a_{1},b_{1}]$, then $P_{1}=0$,
$P_{m+1}=I_{n}$ and

$\dim\operatorname{Range}P_{i}<\dim\operatorname{Range}P_{i+1}\,\,\textnormal{for
any $i=1,\ldots,m$}.$

  * b)

If the first spectral interval is given by $(-\infty,b_{1}]$, then
$P_{m+1}=I_{n}$ and

$\dim\operatorname{Range}P_{i}<\dim\operatorname{Range}P_{i+1}\,\,\textnormal{for
any $i=2,\ldots,m$}.$

The nonuniform bounded growth definitions above stated allow to deduce
boundedness properties for $\Sigma^{+}(A)$. In fact, it is well known that if
(2.1) has a full $(Me^{\delta s},\nu)$–nonuniform bounded growth, then
$\Sigma^{+}(A)\subset[-\nu,\nu]$ and we refer to Lemma 2.10 from [11] for a
proof. In addition, it is easy to verify that if (2.1) has a half $(Me^{\delta
s},\nu)$–nonuniform bounded growth, then $\Sigma^{+}(A)\subset(-\infty,\nu]$.

The properties of $\Sigma^{+}(A)$ and its spectral gaps provides an
alternative characterization of the $(Ke^{\varepsilon s},\gamma)$–nonuniform
contractions.

######  Lemma 1.

The system (2.1) has a $(Ke^{\varepsilon s},\gamma)$–nonuniform exponential
contraction if and only if $\Sigma^{+}(A)\subset(-\infty,0)$.

###### Proof.

If $\Sigma^{+}(A)\subset(-\infty,0)$, it follows that $0\in\rho_{m+1}(A)$ and
(2.4) with $\lambda=0$ coincides with (2.1), which has a nonuniform
exponential dichotomy with projector $P_{m+1}=I_{n}$. Then, from Definition 3,
we obtain that (2.1) has a $(Ke^{\varepsilon s},\gamma)$–nonuniform
exponential contraction. On the other hand, if (2.1) has a $(Ke^{\varepsilon
s},\gamma)$–nonuniform exponential contraction, we know that this equivalent
to say that (2.1) has a $(Ke^{\varepsilon s},\gamma)$–nonuniform exponential
dichotomy with the identity as projector, which has full range, then we have
that $\lambda=0\in\rho_{m+1}(A)=(b_{m},+\infty)$ and implies that
$\Sigma^{+}(A)\subset(-\infty,b_{m})$ with $b_{m}<0$. ∎

###  2.2. Uniformization Lemma

Given a linear system (2.1) having a nonuniform exponential dichotomy and a
half nonuniform bounded growth, the Uniformization Lemma provides a way to
endow it with both an exponential dichotomy and uniform bounded growth.
Nevertheless, the price to pay is to work in a functional framework described
by parametrized vector and operator norms. This result was developed by L.
Zhou, K. Lu and W. Zhang in [35, p.697]. As a previous step to its statement,
we need to consider a family of norms in $\mathbb{R}^{n}$ parametrized by
$\mathbb{R}_{0}^{+}$ as
$\left\\{|\cdot|_{t}\right\\}_{t\in\mathbb{R}_{0}^{+}}$, that is,
$|\cdot|_{t}$ is a norm of $\mathbb{R}^{n}$ for any $t\in\mathbb{R}_{0}^{+}$.

Let $\\{|\cdot|_{t}\\}_{t\in\mathbb{R}_{0}^{+}}$ be a family of norms. By
following [35], we summarize basic facts about this family:

  * a)

By equivalence of norms, there exist two functions
$L_{i}\colon\mathbb{R}_{0}^{+}\to(0,+\infty)$ such that

(2.6) $L_{1}(t)|x|\leq|x|_{t}\leq L_{2}(t)|x|\quad\textnormal{for any
$t\in\mathbb{R}_{0}^{+}$}.$

  * b)

The family $\\{|\cdot|_{t}\\}_{t\in\mathbb{R}_{0}^{+}}$ is continuous if the
mapping $t\mapsto|x|_{t}$ is continuous on $\mathbb{R}_{0}^{+}$ for any fixed
$x\in\mathbb{R}^{n}$. In this case, it follows that the functions $L_{i}$ from
(2.6) are continuous on $\mathbb{R}_{0}^{+}$, [35, Prop.1].

  * c)

A continuous family $\\{|\cdot|_{t}\\}_{t\in\mathbb{R}_{0}^{+}}$ is called
uniformly lower bounded if $t\mapsto L_{1}(t)$ is uniformly bounded by a
positive constant $L_{1}>0$ and (2.6) can be replaced by

(2.7) $L_{1}|x|\leq|x|_{t}\leq L_{2}(t)|x|\quad\textnormal{for any
$t\in\mathbb{R}_{0}^{+}$}.$

In this article, we will assume that
$\left\\{|\cdot|_{t}\right\\}_{t\in\mathbb{R}_{0}^{+}}$ is a uniformly lower
bounded continuous family of norms. In consequence, we are implicitly assuming
that $t\mapsto L_{2}(t)$ is a unbounded and continuous function.

The following technical lemma will be useful

######  Lemma 2.

Given a couple of linear operators
$U\colon(\mathbb{R}^{n},|\cdot|_{s})\to(\mathbb{R}^{m},|\cdot|_{t})$ and
$U\colon(\mathbb{R}^{n},|\cdot|)\to(\mathbb{R}^{m},|\cdot|)$ with norms
defined by

$||U||_{s,t}=\sup\limits_{x\neq
0}\frac{|Ux|_{t}}{|x|_{s}}\quad\textnormal{and}\quad||U||=\sup\limits_{x\neq
0}\frac{|Ux|}{|x|},$

it follows that

(2.8)
$\frac{1}{\beta(t)}||U||_{s,t}\leq||U||\leq\beta(s)||U||_{s,t}\quad\textnormal{and}\quad\frac{1}{\beta(s)}||U||\leq||U||_{s,t}\leq\beta(t)||U||,$

where $\beta\colon\mathbb{R}_{0}^{+}\to[1,\infty)$ is the continuous and
upperly unbounded function:

$\beta(\tau)=\frac{L_{2}(\tau)}{L_{1}}.$

###### Proof.

By using (2.7) it can be proved that for any $x\in\mathbb{R}^{n}$ it follows
that

$\frac{|Ux|_{t}}{|x|_{s}}\geq\frac{L_{1}}{L_{2}(s)}\frac{|Ux|}{|x|}\quad\textnormal{and}\quad\frac{|Ux|_{t}}{|x|_{s}}\leq\frac{L_{2}(t)}{L_{1}}\frac{|Ux|}{|x|},$

and the Lemma follows. ∎

######  Remark 1.

Given a linear operator $U\colon\mathbb{R}^{p}\to\mathbb{R}^{q}$ and a couple
$(t,\tau)$ of positive real numbers. It will be useful to recall the
estimations

(2.9) $|U\xi|_{t}\leq||U||_{\tau,t}|\xi|_{\tau},$ (2.10)
$|U\xi|_{\tau}\leq||U||_{\tau,\tau}|\xi|_{\tau},$ (2.11)
$|U\xi|_{\tau}\leq||U||_{t,\tau}|\xi|_{t}.$

######  Lemma 3 (Uniformization Lemma).

The system (2.1) has both a $(K(s),\alpha)$ –nonuniform exponential dichotomy
and a half $(M(s),\nu)$–nonuniform bounded growth on $\mathbb{R}_{0}^{+}$ if
and only if there exists a continuous family
$\left\\{|\cdot|_{t}\right\\}_{t\in\mathbb{R}_{0}^{+}}$ of norms with a
uniform lower bound such that Eq.(2.1) has a half uniform exponential
dichotomy with respect to
$\left\\{|\cdot|_{t}\right\\}_{t\in\mathbb{R}_{0}^{+}}$, i.e., there are a
projection $P(t):\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ and a couple of
constants $\alpha>0$ and $\kappa\geq 1$ such that the invariant decomposition
condition (the last two inequalities in (2.3)) can be replaced by

$\begin{array}[]{rr}\left\|T(t,s)P(s)\right\|_{s,t}\leq\kappa
e^{-\alpha(t-s)}&\textnormal{ for }t\geq s\geq 0\\\
\left\|T(t,s)(I-P(s))\right\|_{s,t}\leq\kappa e^{-\alpha(s-t)}&\textnormal{
for }s\geq t\geq 0\end{array}$

and a half $(\mu,\nu)$–uniform bounded growth with respect to
$\left\\{|\cdot|_{t}\right\\}_{t\in\mathbb{R}_{0}^{+}}$, namely,

$\|T(t,s)\|_{s,t}\leq\mu e^{\nu(t-s)},\quad t\geq s\in\mathbb{R}_{0}^{+}.$

######  Remark 2.

A meticulous reading of the proof of the Uniformization Lemma [35, pp.697–700]
shows that the family of norms $\\{|\cdot|_{t}\\}$ verifies $|x|_{t}\leq
L_{2}(t)|x|$, where $L_{2}(t)=M(t)+K(t)$.

######  Remark 3.

A version of the Uniformization Lemma, where the linear system (2.1) has the
properties of $(Ke^{\varepsilon_{1}s},\gamma)$–nonuniform exponential
dichotomy and full $(Me^{\varepsilon_{2}s},\nu)$–nonuniform bounded growth,
has been used in [13]. Additionally, a linear system having the above
properties is said to have a strong nonuniform exponential dichotomy.

##  3\. nonuniform conjecture

In order to set the problem of nonuniform nonautonomous stability, we must to
recall a formal definition of nonuniform asymptotic stability. For this
purpose, let us consider the nonautonomous and nonlinear system of ordinary
differential equations:

(3.1) $\dot{x}=f(t,x)$

where $f\colon\mathbb{R}_{0}^{+}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ has
properties ensuring the existence, uniqueness and unbounded forward
continuation of solutions. The solution of (3.1) passing through $x_{0}$ at
$t_{0}$ will be denoted as $t\mapsto x(t,t_{0},x_{0})$. In addition, an
equilibrium of (3.1) is a vector $x^{*}$ such that $f(t,x^{*})=0$ for any
$t\in\mathbb{R}_{0}^{+}$.

######  Definition 5.

[36] A unique equilibrium of (3.1) is globally nonuniformly asymptotically
stable if, for any $\eta>0$, there exists a $\delta(t_{0},\eta)>0$ such that

$|x_{0}-x^{*}|<\delta(t_{0},\eta)\Rightarrow|x(t,t_{0},x_{0})-x^{*}|<\eta\quad\forall
t\geq t_{0}$

and for any $x_{0}\in\mathbb{R}^{n}$ it follows that
$\lim\limits_{t\to+\infty}x(t,t_{0},x_{0})=x^{*}$.

There exists several alternative characterizations for the above defined
stability, the most of them have made in terms of $\mathcal{KL}$ functions
(see [20] for details). In the particular case where (3.1) is the linear
system (2.1), it can be proved –with the help of $\mathcal{KL}$ functions–
that if the linear system has a $(Ke^{\varepsilon s},\gamma)$–nonuniform
exponential contraction, then the origin is globally nonuniformly
asymptotically stable.

Statement of the conjecture: As we have set forth the premises now we are able
to state our nonuniform global stability problem.

######  Conjecture 1 (Nonuniform Nonautonomous Markus–Yamabe Conjecture
(NNMYC)).

Let us consider the nonlinear system

(3.2) $\dot{x}=f(t,x)$

where $f\colon\mathbb{R}_{0}^{+}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$. If $f$
satisfies the following conditions

  * (G1)

$f$ is continuous in $\mathbb{R}_{0}^{+}\times\mathbb{R}^{n}$ and $C^{1}$ with
respect to $x$. Moreover, $f$ is such that the forward solutions are defined
in $[t_{0},+\infty)$ for any $t_{0}\geq 0$.

  * (G2)

$f(t,x)=0$ if $x=0$ for all $t\geq 0$.

  * (G3)

For any piecewise continuous function $t\mapsto\omega(t)$, the linear system

$\dot{\vartheta}=Jf(t,\omega(t))\vartheta,$

where $Jf(t,\cdot)$ is the jacobian matrix of $f(t,\cdot)$, has a
$(Ke^{\varepsilon s},\gamma)$–nonuniform exponential dichotomy spectrum
satisfying

$\Sigma^{+}(Jf(t,\omega(t)))\subset(-\infty,0).$

Then the trivial solution of the nonlinear system (3.2) is globally
nonuniformly asymptotically stable.

Let us recall that the autonomous version of the Markus–Yamabe conjecture is
stated in terms of the eigenvalues spectrum of the jacobian matrix
corresponding to the linearized vector field. In this context, the assumption
(G3) mimics the above fact and since both spectra (eigenvalues and nonuniform
spectrum) belong to $(-\infty,0)$.

A uniform version of the above conjecture has been recently stated in [6],
where the uniform conjecture was verified for scalar systems, triangular
systems and a family of quasilinear systems where the nonlinearity has
suitable properties.

In the rest of this section we will see that NNMYC is well posed. In fact, it
can be verified for the scalar and the quasilinear cases.

######  Theorem 1.

The NNMYC is verified for dimension $n=1$.

###### Proof.

The proof follows the lines of [6, Th.1] replacing the uniform context by the
nonuniform one. ∎

######  Theorem 2.

Let us consider the linear system (2.1) and its perturbation

(3.3) $\dot{x}=A(t)x+f(t,x),$

where $f\colon\mathbb{R}_{0}^{+}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ is
continuous in $\mathbb{R}_{0}^{+}\times\mathbb{R}^{n}$ and $C^{1}$ with
respect to $x$. Moreover, $f$ is such that the forward solutions are defined
in $[t_{0},+\infty)$ for any $t_{0}\geq 0$.

If we assume that

  * i)

The system (2.1) has a $(Ke^{\varepsilon s},\alpha)$–nonuniform exponential
dichotomy on $\mathbb{R}_{0}^{+}$ with projector $P(\cdot)=I$ and
$\alpha>\varepsilon\geq 0$.

  * ii)

The system (2.1) has a half nonuniform bounded growth on $\mathbb{R}_{0}^{+}$.

  * iii)

$f(t,x)=0$ if $x=0$ for all $t\in\mathbb{R}_{0}^{+}$.

  * iv)

There exists $\delta>0$ sufficiently small such that for any piecewise
continuous function $y\colon\mathbb{R}_{0}^{+}\to\mathbb{R}^{n}$ it follows
that

(3.4) $\displaystyle\|Jf(t,y(t))\|<\frac{\delta}{K^{2}}e^{-2\varepsilon
t}\quad\textnormal{for any $t\geq 0$},$

for some $\delta\in(0,\alpha K)$ sufficiently small.

Then the system (3.3) satisfies hypothesis of (NNMYC) and the trivial solution
is globally nonuniformly asymptotically stable.

###### Proof.

The proof follows the lines from [6, Th.2] and by using Theorem 2.5 from [23]
combined with the property (3.4). ∎

The Theorem 2 is related to the pioneer work of K.J. Palmer [28], which
studies the topological conjugacy between the solutions of the linear system
(2.1) and its perturbation (3.3). Roughly speaking, it shows that if (2.1) is
a $(Ke^{\varepsilon s},\alpha)$–nonuniform exponential contraction, then,
under suitable conditions, the trivial solution of (3.3) is globally
nonuniformly asymptotically stable, being an example verifying the nonuniform
Markus-Yamabe conjecture of dimension $n.$

We emphasize that the Palmer’s problem has been recently revisited due to lack
of regularity (see [13] for details) of the topological conjugacy between the
systems.

##  4\. Triangular Vector Fields

The aim of this section is to prove NNMYC for triangular vector fields, that
is, to show that the origin is a globally nonuniformly asymptotically stable
for (3.2) when $f(t,x)$ is triangular vector field.

As we say previously to Theorem 1, in [6] it was proved that the uniform
conjecture is verified for triangular systems and we emphasize that can be
seen as a consequence of the scalar case combined with a result of F. Batelli
and K.J. Palmer [3] for upper triangular systems whose diagonal subsystems
have the uniform exponential dichotomy. In consequence, if we intend to
emulate the ideas of the proof for the uniform triangular case, it is
necessary to generalize the Batelli–Palmer result to a nonuniform framework.

A first step to cope with this problem is to study the relation between the
nonuniform exponential dichotomy properties of the upper block triangular
systems

(4.1) $\dot{z}=\left[\begin{array}[]{cc}A(t)&C(t)\\\
0&B(t)\end{array}\right]z,$

with the nonuniform exponential dichotomy properties of the subsystems

(4.2) $\dot{x}=A(t)x\quad\textnormal{and}\quad\dot{y}=B(t)y,$

where $x\in\mathbb{R}^{n}$, $y\in\mathbb{R}^{m}$, $z\in\mathbb{R}^{n+m}$,
$A\in M_{n}(\mathbb{R})$, $B\in M_{m}(\mathbb{R}$) and $C\in
M_{nm}(\mathbb{R})$. As we said, this problem has been addressed by F. Batelli
and K.J. Palmer [3] in the context of the uniform exponential dichotomy on
$\mathbb{R}_{0}^{+}$ with an extension to the discrete case in [4].

In the case of the $(K(s),\gamma)$–nonuniform exponential dichotomy on
$\mathbb{R}_{0}^{+}$, a first approach was carried out by L. Tien, L. Niehn
and T. Chien in [32]. In particular, the first result of [32] is:

######  Proposition 2 (Theorem 2.1 in [32]).

If the system (4.1) has a $(K(s),\gamma)$–nonuniform exponential dichotomy on
$\mathbb{R}_{0}^{+}$ then the decoupled subsystems (4.2) also have a
$(K(s),\gamma)$–nonuniform exponential dichotomy on $\mathbb{R}_{0}^{+}$.

As the $(Ke^{\varepsilon s},\gamma)$–nonuniform exponential dichotomy is a
particular case of the $(K(s),\gamma)$ –nonuniform exponential dichotomy, the
following Corollary is immediate:

######  Corollary 1.

If the system (4.1) has a $(Ke^{\varepsilon s},\gamma)$–nonuniform exponential
dichotomy on $\mathbb{R}_{0}^{+}$ then the decoupled subsystems (4.2) also
have a $(Ke^{\varepsilon s},\gamma)$–nonuniform exponential dichotomy on
$\mathbb{R}_{0}^{+}$.

Our next result will show the converse of Proposition 2, which follows the
lines of the converse result proved by Batelli and Palmer in [3, Th.1] but
imposes boundedness conditions for $C$ in terms of matrix norms
$||\cdot||_{t,t}$ described in the subsection 2.2 and also incorporates a
creative use of the Uniformization Lemma.

Let us recall we are assuming that (2.1) has the property of half $(Me^{\delta
s},\nu)$–nonuniform bounded growth and a $(Ke^{\varepsilon
s},\gamma)$–nonuniform exponential dichotomy on $\mathbb{R}_{0}^{+}$. Then, by
using the Uniformization Lemma and Remark 2, we have that the continuous norms
$\\{|\cdot|_{t}\\}$ satisfy the inequalities

(4.3) $L_{1}|x|\leq|x|_{t}\leq\underbrace{(M+K)}_{:=L_{2}}e^{\theta
t}|x|\quad\textnormal{for any $t\in\mathbb{R}_{0}^{+}$},$

for some $L_{1}>0$, where $\theta:=\max\\{\delta,\varepsilon\\}$ . By
considering these specific norms, the inequalities (2.8) becomes

(4.4) $\frac{1}{L}e^{-\theta t}||U||_{s,t}\leq||U||\leq Le^{\theta
s}||U||_{s,t}\quad\textnormal{and}\quad\frac{1}{L}e^{-\theta
s}||U||\leq||U||_{s,t}\leq Le^{\theta t}||U||,$

where $L=L_{2}/L_{1}$. Now, the converse result of Corollary 1 is given by:

######  Theorem 3.

Let us consider the upper block triangular system (4.1). If

  * i)

The decoupled systems (4.2) have the property of half $(Me^{\ell
s},\omega)$–nonuniform bounded growth and half
$(\tilde{M}e^{\tilde{\ell}s},\tilde{\omega})$–nonuniform bounded growth
respectively on $\mathbb{R}_{0}^{+}$,

  * ii)

The decoupled systems (4.2) have the properties of $(Ke^{\varepsilon
s},\alpha)$–nonuniform exponential dichotomy and
$(\tilde{K}e^{\tilde{\varepsilon}s},\tilde{\alpha})$–nonuniform exponential
dichotomy respectively on $\mathbb{R}_{0}^{+}$,

  * iii)

The non diagonal block verifies

(4.5)
$||C||_{\tau,\infty}:=\sup\limits_{\tau\in\mathbb{R}_{0}^{+}}||C(\tau)||_{\tau,\tau}<\infty,$

where

$||C(\tau)||_{\tau,\tau}=\sup\limits_{x\neq
0}\frac{|C(\tau)x|_{\tau}}{|x|_{\tau}}\quad\textnormal{with $|\cdot|_{\tau}$
verifying (\ref{ineg1-BV})},$

then (4.1) also has a $(\bar{K}e^{\bar{\varepsilon}s},\bar{\alpha})$–
nonuniform exponential dichotomy $\mathbb{R}_{0}^{+}$, where $\bar{K}$ is a
constant dependent of $\alpha$ and $\tilde{\alpha}$, $\bar{\varepsilon}\geq 0$
and $\bar{\alpha}=\min\\{\alpha,\tilde{\alpha}\\}$, and a half
$(\bar{M}e^{\bar{\theta}s},\bar{\omega})$–nonuniform bounded growth

The proof of this result is plenty of bulky technicalities and the
Uniformization Lemma combined with the functional setting from [35] play a key
role. In order to give continuity to the reading of the article, the proof is
written in the Appendix.

######  Remark 4.

By using the left inequality (4.4), we have that $||C(\tau)||_{\tau,\tau}\leq
Le^{\theta\tau}||C(\tau)||$ for any $\tau\geq 0$. In consequence, a sufficient
condition ensuring (4.5) is given by

$\sup\limits_{\tau\in\mathbb{R}_{0}^{+}}Le^{\theta\tau}||C(\tau)||<\infty.$

######  Corollary 2.

The upper block system

(4.6)
$\dot{x}=\left[\begin{array}[]{cccc}A_{1}(t)&C_{12}(t)&\cdots&C_{1k}(t)\\\
0&A_{2}(t)&\cdots&C_{2k}(t)\\\ \vdots&&\ddots&\\\
0&\cdots&\cdots&A_{k}(t)\end{array}\right]x\quad$

has a $(Ke^{\varepsilon s},\alpha)$–nonuniform exponential dichotomy on
$\mathbb{R}_{0}^{+}$ if all the diagonal systems

$\dot{x}_{i}=A_{i}(t)x_{i}\quad\textnormal{for $i=1,\ldots,k$}$

have a half nonuniform bounded growth property and a
$(K_{i}e^{\varepsilon_{i}s},\alpha_{i})$–nonuniform exponential dichotomy on
$\mathbb{R}_{0}^{+}$ (with $i=1,\ldots,k$) provided that the upper diagonal
blocks verifies

(4.7)
$\sup\limits_{\tau\in\mathbb{R}_{0}^{+}}||C_{j}(\tau)||_{\tau,\tau}<\infty\quad\textnormal{for
any $j\in\\{2,\ldots,k\\}$},$

where $C_{j}(t)$ is defined by

$C_{j}(t)=\left[\begin{array}[]{c}C_{1j}(t)\\\ C_{2j}(t)\\\ \vdots\\\
C_{j-1\,j}(t)\end{array}\right].$

###### Proof.

The proof will be carried out recursively.

If $k=2$, the result is an immediate consequence of Theorem 3 since the matrix
of the system (4.6) has similar structure to (4.1) with $A_{1}(t)=A(t)$,
$A_{2}(t)=B(t)$ and $C_{12}(t)=C(t)$.

If $k=3$, the system (4.6) can be seen as having a similar structure that
(4.1) with

$A(t)=\left[\begin{array}[]{cc}A_{1}(t)&C_{12}(t)\\\
0&A_{2}(t)\end{array}\right],\quad B(t)=A_{3}(t)\quad\textnormal{and}\quad
C(t)=C_{3}(t)=\left[\begin{array}[]{c}C_{13}(t)\\\
C_{23}(t)\end{array}\right]$

Note that the subsystems $\dot{x}=A(t)x$ and $\dot{y}=B(t)y$ have a nonuniform
exponential dichotomy on $\mathbb{R}_{0}^{+}$. The first dichotomy property is
a consequence of the case $k=2$ while the second one is an hypothesis. As
(4.7) is verified for $j=3$, Theorem 3 implies that the system (4.6) with
$k=3$ has a nonuniform exponential dichotomy on $\mathbb{R}_{0}^{+}$ and the
proof is achieved by a recursive way for $k\geq 4$. ∎

A particular –but important– byproduct of the above Corollary is the following
result when all the diagonal terms $A_{i}(t)$ are scalar functions:

######  Corollary 3.

The upper triangular system

(4.8)
$\dot{x}=\left[\begin{array}[]{cccc}a_{1}(t)&c_{12}(t)&\cdots&c_{1n}(t)\\\
0&a_{2}(t)&\cdots&c_{2n}(t)\\\ \vdots&&\ddots&\\\
0&\cdots&\cdots&a_{n}(t)\end{array}\right]x\quad$

has a $(Ke^{\varepsilon s},\alpha)$–nonuniform exponential dichotomy on
$\mathbb{R}_{0}^{+}$ if all the scalar differential equations

(4.9) $\dot{x}_{i}=a_{i}(t)x_{i}\quad\textnormal{for $i=1,\ldots,k$}$

have a half ($M_{i}e^{\theta_{i}s},\omega_{i}$)–nonuniform bounded growth
property and a $(K_{i}e^{\varepsilon_{i}s},\alpha_{i})$–nonuniform exponential
dichotomy on $\mathbb{R}_{0}^{+}$ (with $i=1,\ldots,n$) provided that

(4.10) $\sup\limits_{\tau\geq 0}|C_{j}(\tau)|_{\tau}=\sup\limits_{\tau\geq
0}\left|\left(\begin{array}[]{c}c_{1j}(\tau)\\\ c_{2j}(\tau)\\\ \vdots\\\
c_{j-1\,j}(\tau)\end{array}\right)\right|_{\tau}<\infty\quad\textnormal{for
any $j\in\\{2,\ldots,n\\}$}.$

######  Remark 5.

Note that, when $j=2$ in (4.10), we have

$\sup\limits_{\tau\geq 0}|C_{2}(\tau)|_{\tau}=\sup\limits_{\tau\geq
0}|c_{12}(\tau)|_{\tau}<\infty,$

which propels to consider $|\cdot|_{\tau}$ as a family of norms and
$(\mathbb{R},|\cdot|_{\tau})$ as an $\mathbb{R}$–vector space of dimension
one. It can be proved that there exists a continous function
$h\colon\mathbb{R}_{0}^{+}\to(0,\infty)$ such that

$|x|_{\tau}=h(\tau)|x|\quad\textnormal{for any $x\in\mathbb{R}$}.$

######  Lemma 4.

Under the assumptions of Corollary 1 and Theorem 3, the $(Ke^{\varepsilon
s},\alpha)$–nonuniform exponential dichotomy spectrum of the upper block
system (4.1) verifies

$\Sigma^{+}(A)\cup\Sigma^{+}(B)=\Sigma^{+}(U)\quad\textnormal{with}\quad
U=\left(\begin{array}[]{cc}A&C\\\ 0&B\end{array}\right).$

###### Proof.

Firstly, if $\lambda\in\rho(A)\cap\rho(B)$ we have that the systems

(4.11) $\dot{x}=[A(t)-\lambda
I]x\quad\textnormal{and}\quad\dot{y}=[B(t)-\lambda I]y$

have a nonuniform exponential dichotomy and Theorem 3 implies that
$\lambda\in\rho(U)$ and consequently it follows that
$\rho(A)\cap\rho(B)\subset\rho(U)$.

Secondly, if $\lambda\in\rho(U)$, the Corollary 1 implies that the subsystems
(4.11) have a nonuniform exponential dichotomy on $\mathbb{R}_{0}^{+}$ which
is equivalent to $\lambda\in\rho(A)\cap\rho(B)$, which implies that
$\rho(U)\subset\rho(A)\cap\rho(B)$ and the result follows. ∎

Based on a recursive application of Lemma 4, we obtain the following
descriptions for the nonuniform exponential dichotomy spectrum for an upper
block system as in (4.6) and for an upper triangular system as in (4.8)
respectively.

######  Corollary 4.

Under the assumptions of Corollary 4, the nonuniform exponential dichotomy
spectrum of the upper block system (4.6) is described by

$\Sigma^{+}(A_{1})\cup\Sigma^{+}(A_{2})\cup\cdots\cup\Sigma^{+}(A_{k}).$

######  Corollary 5.

Under the assumptions of Corollary 3, the nonuniform exponential dichotomy
spectrum of the upper triangular system (4.8) is described by

$\Sigma^{+}(a_{1})\cup\Sigma^{+}(a_{2})\cup\cdots\cup\Sigma^{+}(a_{n}).$

######  Remark 6.

The previous Corollary says that the nonuniform exponential dichotomy spectrum
of an upper triangular system coincides with the union of the spectra of the
scalar equations (4.9). This property is known as diagonal significance and
was introduced for the discrete uniform exponential dichotomy spectrum by C.
Pötzsche in [29]. This fact is immediate in the autonomous case, while -
counterintuitively - in the nonautonomous framework is not always verified.

Theorem 3 and its consequences provide the framework to state and prove our
main result, namely, the nonautonomous nonuniform Markus–Yamabe conjecture is
verified for triangular system of nonautonomous ordinary differential
equations whose nondiagonal parts satisfy boundedness conditions described in
terms of parametrized norms.

######  Theorem 4.

Let us consider the triangular system

(4.12) $\begin{array}[]{rcr}\dot{x}_{1}&=&f_{1}(t,x_{1},x_{2},\ldots,x_{n})\\\
\dot{x}_{2}&=&f_{2}(t,x_{2},\ldots,x_{n})\\\ &\vdots&\\\
\dot{x}_{n}&=&f_{n}(t,x_{n}),\end{array}$

whose right part, namely $F(t,x)$, verifies (G1) and (G2). If for any
piecewise continuous function $t\mapsto\theta(t)$ it is verified that

  * (a)

There exist constants $k_{i}\geq 1$, $\alpha_{i}>0$ and $\varepsilon_{i}\geq
0$, with $\varepsilon_{i}<\alpha_{i}$ such that

$\int_{s}^{t}\frac{\partial f_{i}}{\partial
x_{i}}(\tau,\theta(\tau))\,d\tau\leq\ln(k_{i})-\alpha_{i}(t-s)+\varepsilon_{i}s\quad\textnormal{for
any $t\geq s\geq 0$ and $i=1,\ldots,n$}.$

  * (b)

For any $j\in\\{2,\ldots,n\\}$ and any piecewise continuous function
$t\mapsto\theta(t)$, the partial derivatives verify

$\displaystyle\sup\limits_{\tau\geq
0}\left|\left(\begin{array}[]{c}\frac{\partial
f_{1}(\tau,\theta(\tau))}{\partial x_{j}}\\\ \\\ \frac{\partial
f_{2}(\tau,\theta(\tau))}{\partial x_{j}}\\\ \vdots\\\ \frac{\partial
f_{j-1}(\tau,\theta(\tau))}{\partial
x_{j}}\end{array}\right)\right|_{\tau}<\infty.$

then the trivial solution of (4.12) is globally nonuniformly asymptotically
stable.

###### Proof.

Firstly, we will prove that (G3) is verified. In fact, the statement (a)
implies that, for any $i=1,\ldots,n$, the nonuniform exponential dichotomy
spectra of the differential equations

$\dot{x}=\frac{\partial f_{i}}{\partial x_{i}}(t,\theta(t))x$

verifies

(4.13) $\Sigma^{+}\left[\frac{\partial f_{i}}{\partial
x_{i}}(t,\theta(t))\right]\subset(-\infty,0).$

for any set of piecewise continuous functions
$t\mapsto\theta(t)=(\theta_{1}(t),\ldots,\theta_{n}(t))$.

On the other hand, let us recall that the jacobian matrix $JF$ is upper
triangular defined by

$JF(t,x_{1},\ldots,x_{n})_{ij}=\left\\{\begin{array}[]{ccl}\frac{\partial
f_{i}}{\partial x_{j}}(t,x_{i},\ldots,x_{n})&\textnormal{if}&i\leq j,\\\
0&\textnormal{if}&i>j.\end{array}\right.$

Then, the statement (b) combined with (4.13) and Corollary 5 imply that

$\displaystyle\Sigma^{+}\left[JF(t,\theta(t))\right]=\bigcup\limits_{i=1}^{n}\Sigma^{+}\left[\frac{\partial
f_{i}}{\partial x_{i}}(t,\theta(t))\right]\subset(-\infty,0)$

and (G3) follows.

Let $t\mapsto(x_{1}(t),x_{2}(t),\ldots,x_{n}(t))$ be a solution of (4.12)
passing through $(x_{1}^{0},x_{2}^{0},\ldots x_{n}^{0})$ at $t=t_{0}$. Note
that the scalar equation

$\dot{x}_{n}=f_{n}(t,x_{n})\quad\textnormal{with}\quad x_{n}(t_{0})=x_{n}^{0}$

is a subsystem of (4.12) whose solution is denoted by
$\phi_{n}(t):=x_{n}(t,t_{0},x_{n}^{0})$ and verifies
$\lim\limits_{t\to\infty}\phi_{n}(t)=0$ globally and nonuniformly, as we can
see by Theorem 1.

Now, we can see that the last two equations of (4.12) are

$\left\\{\begin{array}[]{rcl}\dot{x}_{n-1}&=&f_{n-1}(t,x_{n-1},x_{n})\\\
\dot{x}_{n}&=&f_{n}(t,x_{n})\end{array}\right.$

with initial conditions $(x_{n-1}^{0},x_{n}^{0})$ at $t=t_{0}$. The solution
of this system is denoted by $(\phi_{n-1}(t),\phi_{n}(t))$, where $\phi_{n}$
is defined above and $\phi_{n-1}$ is the solution of the scalar equation

$\dot{x}_{n-1}=f_{n}(t,x_{n-1},\phi_{n}(t))\quad\textnormal{with}\quad
x_{n-1}(t_{0})=x_{n-1}^{0}$

and, as before, also verifies $\lim\limits_{t\to\infty}\phi_{n-1}(t)=0$
globally and nonuniformly, as we can see again by Theorem 1. The rest of the
proof can be achieved in a recursive way. ∎

##  Appendix A Proof of Theorem 3

###  A.1. Preliminaries

By hypothesis, we know that the linear systems $\dot{x}=A(t)x$ and
$\dot{y}=B(t)y$ have a $(Ke^{\varepsilon s},\alpha)$ and
$(\tilde{K}e^{\tilde{\varepsilon}s},\tilde{\alpha})$–nonuniform exponential
dichotomy on $\mathbb{R}_{0}^{+}$ with projectors $P^{A}(\cdot)$ and
$P^{B}(\cdot)$ respectively:

(A.1)
$\begin{array}[]{rcl}||X(t,s)P^{A}(s)||&\leq&Ke^{-\alpha(t-s)+\varepsilon
s}\quad\textnormal{for $t\geq s\geq 0$},\\\
||X(t,s)[I_{n}-P^{A}(s)]||&\leq&Ke^{-\alpha(s-t)+\varepsilon
s}\quad\textnormal{for $s\geq t\geq 0$},\end{array}$

and

(A.2)
$\begin{array}[]{rcl}||Y(t,s)P^{B}(s)||&\leq&\tilde{K}e^{-\widetilde{\alpha}(t-s)+\tilde{\varepsilon}s}\quad\textnormal{for
$t\geq s\geq 0$},\\\
||Y(t,s)[I_{m}-P^{B}(s)]||&\leq&\tilde{K}e^{-\widetilde{\alpha}(s-t)+\tilde{\varepsilon}s}\quad\textnormal{for
$s\geq t\geq 0$},\end{array}$

where $X(t,s)$ and $Y(t,s)$ are the evolution operators corresponding to the
diagonal subsystems. Moreover, we also know that the above subsystems have the
properties of half $(Me^{\ell s},\omega)$ and half
$(\tilde{M}e^{\tilde{\ell}s},\tilde{\omega})$–nonuniform bounded growth on
$\mathbb{R}_{0}^{+}$ respectively, that is

(A.3) $||X(t,s)||\leq Me^{\omega(t-s)+\ell
s}\quad\textnormal{and}\quad||Y(t,s)||\leq\tilde{M}e^{\tilde{\omega}(t-s)+\tilde{\ell}s}\quad\textnormal{for
$t\geq s\geq 0$},$

then, the Uniformization Lemma can be applied to both subsystems. On one hand,
the dichotomy estimations become

$\begin{array}[]{rr}\left\|X(t,s)P^{A}(s)\right\|_{s,t}\leq\kappa
e^{-\alpha(t-s)}&\textnormal{ for }t\geq s\geq 0,\\\
\left\|X(t,s)[I-P^{A}(s)]\right\|_{s,t}\leq\kappa
e^{-\alpha(s-t)}&\textnormal{ for }s\geq t\geq 0,\end{array}$

and

$\begin{array}[]{rr}\left\|Y(t,s)P^{B}(s)\right\|_{s,t}\leq\tilde{\kappa}e^{-\tilde{\alpha}(t-s)}&\textnormal{
for }t\geq s\geq 0,\\\
\left\|Y(t,s)[I-P^{B}(s)]\right\|_{s,t}\leq\tilde{\kappa}e^{-\tilde{\alpha}(s-t)}&\textnormal{
for }s\geq t\geq 0,\end{array}$

while the half bounded growth properties become

(A.4) $\|X(t,s)\|_{s,t}\leq\mu
e^{\omega(t-s)}\quad\textnormal{and}\quad\|Y(t,s)\|_{s,t}\leq\tilde{\mu}e^{\tilde{\omega}(t-s)}\quad
t\geq s\geq 0.$

The Uniformization Lemma also ensures the existence of two family of norms: a
family $\\{|\cdot|_{t}^{A}\\}_{t}$ in $\mathbb{R}^{n}$ and a family
$\\{|\cdot|_{t}^{B}\\}_{t}$ in $\mathbb{R}^{m}$. Moreover, the inequalities
(2.7) are verified with $L_{1}^{A}$,$L_{1}^{B}$,$L_{1}^{A}(t)$ and
$L_{2}^{B}(t)$ respectively. By Remark 2, we know that

$L_{2}^{A}(t)=Me^{\ell t}+Ke^{\varepsilon t}\quad\textnormal{and}\quad
L_{2}^{B}(t)=\tilde{M}e^{\tilde{\ell}t}+\tilde{K}e^{\tilde{\varepsilon}t}$

and the inequalities (2.8) are verified with

(A.5) $\beta^{A}(t)=\frac{L_{2}^{A}(t)}{L_{1}^{A}}\leq Le^{\theta
t}\quad\textnormal{and}\quad\beta^{B}(t)=\frac{L_{2}^{B}(t)}{L_{1}^{B}}\leq
Le^{\theta t},$

where

(A.6)
$L=\max\left\\{\frac{M+K}{L_{1}^{A}},\frac{\tilde{M}+\tilde{K}}{L_{1}^{B}}\right\\}\quad\textnormal{and}\quad\theta=\max\\{\ell,\tilde{\ell},\varepsilon,\tilde{\varepsilon}\\}.$

These constants $L$ and $\theta$ are useful to state the following result:

######  Lemma 5.

For any $t,\tau,s\geq 0$, the evolution operators $X$ and $Y$ verify

$||X(t,\tau)Z(\tau)C(\tau)V(\tau)Y(\tau,s)||\leq Le^{\theta
s}||C||_{\tau,\infty}||X(t,\tau)Z(\tau)||_{\tau,t}||V(\tau)Y(\tau,s)||_{s,\tau},$

where $Z(\tau)$ is either $P^{A}(\tau)$ or $I_{n}-P^{A}(\tau)$ and $V(\tau)$
is either $P^{B}(\tau)$ or $I_{m}-P^{B}(\tau)$.

###### Proof.

Let $\xi\in\mathbb{R}^{m}\setminus\\{0\\}$. By using (4.3) followed by (2.9)
and recalling the dimensions of $C(\cdot)$, we have that

$\begin{array}[]{rcl}|X(t,\tau)Z(\tau)C(\tau)V(\tau)Y(\tau,s)\xi|&\leq&\displaystyle\frac{1}{L_{1}^{A}}|X(t,\tau)Z(\tau)C(\tau)V(\tau)Y(\tau,s)\xi|_{t}\\\
\\\
&\leq&\displaystyle\frac{1}{L_{1}^{A}}||X(t,\tau)Z(\tau)||_{\tau,t}|C(\tau)V(\tau)Y(\tau,s)\xi|_{\tau}.\end{array}$

By using (2.10) followed by (2.11), (4.5), (4.3) combined with (A.5)–(A.6), it
follows that

$\begin{array}[]{rcl}|X(t,\tau)Z(\tau)C(\tau)V(\tau)Y(\tau,s)\xi|&\leq&\displaystyle\frac{1}{L_{1}^{A}}||X(t,\tau)Z(\tau)||_{\tau,t}||C(\tau)||_{\tau,\tau}|V(\tau)Y(\tau,s)\xi|_{\tau}\\\
\\\
&\leq&\displaystyle\frac{1}{L_{1}^{A}}||X(t,\tau)Z(\tau)||_{\tau,t}||C(\tau)||_{\tau,\tau}||V(\tau)Y(\tau,s)||_{s,\tau}|\xi|_{s}\\\
\\\ &\leq&Le^{\theta
s}||C||_{\tau,\infty}||X(t,\tau)Z(\tau)||_{\tau,t}\,||V(\tau)Y(\tau,s)||_{s,\tau}|\xi|\end{array}$

and the Lemma follows. ∎

We will follow the lines of the work carried out by F. Batelli and K.J. Palmer
in [3], which proved that evolution operator of the triangular system (4.1) is
given by

$T(t,s)=\left[\begin{array}[]{cc}X(t,s)&W(t,s)\\\ 0&Y(t,s)\end{array}\right]$

where $W$ is a $n\times m$ matrix defined by

(A.7) $W(t,s):=\int_{s}^{t}X(t,\tau)C(\tau)Y(\tau,s)\,d\tau.$

In addition, as in [3], let us consider:

$P(t)=T(t,0)\left[\begin{array}[]{cc}P^{A}(0)&\mathcal{L}P^{B}(0)\\\
0&P^{B}(0)\end{array}\right]T(0,t)=\left[\begin{array}[]{cc}P^{A}(t)&R(t)\\\
0&P^{B}(t)\end{array}\right],$

where $\mathcal{L}\colon\mathcal{R}P^{B}(0)\to(\mathcal{R}P^{A}(0))^{\perp}$
is the linking operator defined by

$\mathcal{L}\eta=-\int_{0}^{\infty}[I_{m}-P^{A}(0)]X(0,\tau)C(\tau)Y(\tau,0)\eta\,d\tau,$

which plays an important role in the proof of Proposition 2. Moreover, $R(t)$
satisfies the matrix differential equation

$\dot{R}=A(t)R-RB(t)+C(t)P^{B}(t)-P^{A}(t)C(t)\quad\textnormal{with
$R(0)=\mathcal{L}P^{B}(0)$},$

whose solution is defined by

(A.8) $\begin{array}[]{rcl}R(t)&=&X(t,0)R(0)Y(0,t)\\\ \\\
&&\displaystyle+\int_{0}^{t}X(t,\tau)[C(\tau)P^{B}(\tau)-P^{A}(\tau)C(\tau)]Y(\tau,t)\,d\tau,\end{array}$

or alternatively as follows

$\begin{array}[]{rcl}R(t)&=&\displaystyle\underbrace{-\int_{t}^{+\infty}X(t,\tau)[I_{n}-P^{A}(\tau)]C(\tau)P^{B}(\tau)Y(\tau,t)\,d\tau}_{=R_{1}(t)}\\\
\\\
&&\displaystyle\underbrace{-\int_{0}^{t}X(t,\tau)P^{A}(\tau)C(\tau)[I_{m}-P^{B}(\tau)]Y(\tau,t)\,d\tau}_{=R_{2}(t)}.\end{array}$

The Lemma 5 will be helpful to provide an estimation for $R(t)$:

######  Lemma 6.

For any $t\geq 0$, the matrix $R(t)$ verifies

(A.9)
$||R(t)||\leq\frac{2\kappa\widetilde{\kappa}}{\widetilde{\alpha}+\alpha}Le^{\theta
t}||C||_{\tau,\infty}.$

###### Proof.

Let $\xi\in\mathbb{R}^{m}\setminus\\{0\\}$. By using Lemma 5 we can deduce

$\begin{array}[]{rcl}|R_{1}(t)\xi|&\leq&\displaystyle Le^{\theta
t}||C||_{\tau,\infty}\int_{t}^{\infty}||X(t,\tau)[I_{n}-P^{A}(\tau)]||_{\tau,t}\left\|P^{B}(\tau)Y(\tau,t)\right\|_{t,\tau}|\xi|\,d\tau.\end{array}$

Now, we apply the Uniformization Lemma to (A.1) and (A.2) respectively, and we
can deduce that

$\begin{array}[]{rcl}|R_{1}(t)\xi|&\leq&\displaystyle Le^{\theta
t}||C||_{\tau,\infty}\int_{t}^{\infty}\left\|X(t,\tau)[I_{n}-P^{A}(\tau)]\right\|_{\tau,t}||P^{B}(\tau)Y(\tau,t)||_{t,\tau}|\xi|\,d\tau\\\
\\\ &\leq&\displaystyle\kappa\,\widetilde{\kappa}Le^{\theta
t}||C||_{\tau,\infty}|\xi|\,e^{(\alpha+\widetilde{\alpha})t}\int_{t}^{\infty}e^{-(\alpha+\widetilde{\alpha})\tau}\,d\tau\\\
\\\
&\leq&\displaystyle\frac{\kappa\,\widetilde{\kappa}}{\alpha+\widetilde{\alpha}}Le^{\theta
t}||C||_{\tau,\infty}|\xi|.\end{array}$

Similarly, for the second term, by using Lemma 5 followed by the
Uniformization Lemma, we can deduce that

$\begin{array}[]{rcl}|R_{2}(t)\xi|&\leq&\displaystyle Le^{\theta
t}||C||_{\tau,\infty}\int_{0}^{t}||X(t,\tau)P^{A}(\tau)||_{\tau,t}\,||[I_{m}-P^{B}(\tau)]Y(\tau,t)||_{t,\tau}|\xi|\,d\tau\\\
\\\ &\leq&\displaystyle\kappa\,\widetilde{\kappa}Le^{\theta
t}||C||_{\tau,\infty}|\xi|\,e^{-(\alpha+\widetilde{\alpha})t}\int_{0}^{t}e^{(\alpha+\widetilde{\alpha})\tau}\,d\tau\\\
\\\
&\leq&\displaystyle\frac{\kappa\widetilde{\kappa}}{\alpha+\widetilde{\alpha}}Le^{\theta
t}||C||_{\tau,\infty}|\xi|,\end{array}$

and the inequality (A.9) follows. ∎

We will verify that $P(\cdot)$ is an invariant projector. In fact, the
property $P^{2}(t)=P(t)$ for any $t\geq 0$ is a consequence of its own
definition, while the next result proves its invariance. This last property
has not been proved in [3] and, in spite that can be deduced easily, we will
prove it.

######  Lemma 7.

The projector $P(\cdot)$ is invariant, namely, it verifies the property

$T(t,s)P(s)=P(t)T(t,s)\quad\textnormal{for any $t,s\geq 0$}.$

###### Proof.

Notice that

$T(t,s)P(s)=\left[\begin{array}[]{cc}X(t,s)P^{A}(s)&W(t,s)P^{B}(s)+X(t,s)R(s)\\\
0&Y(t,s)P^{B}(s)\end{array}\right].$

As $P^{A}(\cdot)$ and $P^{B}(\cdot)$ are invariant projectors, we can see that
the Lemma follows if and only if

$R(t)Y(t,s)+P^{A}(t)W(t,s)=W(t,s)P^{B}(s)+X(t,s)R(s)\quad\textnormal{for any
$t,s\geq 0$.}$

By defining $R_{t,s}(0):=X(t,0)R(0)Y(0,s)$, using (A.7) and (A.8) and
considering $t\geq s$, we can easily deduce that

$\begin{array}[]{rcl}R(t)Y(t,s)+P^{A}(t)W(t,s)&=&\displaystyle
R_{t,s}(0)+\int_{0}^{t}X(t,\tau)[C(\tau)P^{B}(\tau)-P^{A}(\tau)C(\tau)]Y(\tau,s)\,d\tau\\\
\\\ &&\displaystyle+\int_{s}^{t}X(t,\tau)P^{A}(\tau)C(\tau)Y(\tau,s)\,d\tau\\\
\\\ &=&\displaystyle
R_{t,s}(0)+\int_{0}^{s}X(t,\tau)[C(\tau)P^{B}(\tau)-P^{A}(\tau)C(\tau)]Y(\tau,s)\,d\tau\\\
\\\ &&\displaystyle+\int_{s}^{t}X(t,\tau)C(\tau)P^{B}(\tau)Y(\tau,s)\,d\tau\\\
\\\ &=&X(t,s)R(s)+W(t,s)P^{B}(s).\end{array}$

A similar identity can be deduced considering $t<s$ and the Lemma follows. ∎

Gathering the above results, it can be proved that the triangular system (4.1)
has a ($\bar{K}e^{\bar{\varepsilon}s},\bar{\alpha}$)–nonuniform exponential
dichotomy on $\mathbb{R}_{0}^{+}$ with the above defined invariant projector
$P(t)$.

######  Lemma 8.

There exist a constant $K_{3}\geq 1$, $\alpha_{3}>0$ and $\varepsilon_{3}\geq
0$, where $\varepsilon_{3}<\alpha_{3}$ such that

(A.10) $||W(t,s)P^{B}(s)+X(t,s)R(s)||\leq
K_{3}e^{-\alpha_{3}(t-s)+\varepsilon_{3}s}\quad\textnormal{for $t\geq s\geq
0$}.$

###### Proof.

In order to deduce this estimation, we will write

$\begin{array}[]{rcl}W(t,s)P^{B}(s)+X(t,s)R(s)&=&\displaystyle\underbrace{\int_{s}^{t}X(t,\tau)P^{A}(\tau)C(\tau)P^{B}(\tau)Y(\tau,s)d\tau}_{:=D_{1}}\\\
&-&\displaystyle\underbrace{\int_{t}^{+\infty}X(t,\tau)Q^{A}(\tau)C(\tau)P^{B}(\tau)Y(\tau,s)d\tau}_{:=D_{2}}\\\
&-&\displaystyle\underbrace{\int_{0}^{s}X(t,\tau)P^{A}(\tau)C(\tau)Q^{B}(\tau)Y(\tau,s)d\tau}_{:=D_{3}},\end{array}$

where $I_{n}-P^{A}(\tau)=Q^{A}(\tau)$ and $I_{m}-P^{B}(\tau)=Q^{B}(\tau)$.

By using again Lemma 5 followed by the Uniformization Lemma and recalling that
$s\leq t$, we have that:

(A.11) $\begin{array}[]{rcl}||D_{1}||&\leq&\displaystyle Le^{\theta
s}||C||_{\tau,\infty}\int_{s}^{t}||X(t,\tau)P^{A}(\tau)||_{\tau,t}\,||P^{B}(\tau)Y(\tau,s)||_{s,\tau}\,d\tau\\\
\\\ &\leq&\displaystyle Le^{\theta
s}||C||_{\tau,\infty}\kappa\widetilde{\kappa}\int_{s}^{t}e^{-\alpha(t-\tau)}e^{-\widetilde{\alpha}(\tau-s)}\,d\tau,\end{array}$
(A.12) $\begin{array}[]{rcl}||D_{2}||&\leq&\displaystyle Le^{\theta
s}||C||_{\tau,\infty}\int_{t}^{\infty}||X(t,\tau)Q^{A}(\tau)||_{t,\tau}\,||P^{B}(\tau)Y(\tau,s)||_{s,\tau}\,d\tau\\\
\\\ &\leq&\displaystyle Le^{\theta
s}||C||_{\tau,\infty}\kappa\widetilde{\kappa}\int_{t}^{\infty}e^{-\alpha(\tau-t)}e^{-\widetilde{\alpha}(\tau-s)}\,d\tau,\end{array}$
(A.13) $\begin{array}[]{rcl}||D_{3}||&\leq&\displaystyle Le^{\theta
s}||C||_{\tau,\infty}\int_{0}^{s}||X(t,\tau)P^{A}(\tau)||_{\tau,t}\,||Q^{B}(\tau)Y(\tau,s)||_{s,\tau}\,ds\\\
\\\ &\leq&\displaystyle Le^{\theta
s}||C||_{\tau,\infty}\kappa\widetilde{\kappa}\int_{0}^{s}e^{-\alpha(t-\tau)}e^{-\widetilde{\alpha}(s-\tau)}\,d\tau.\end{array}$

As a consequence of the estimates (A.11), (A.12), (A.13) and defining
$\kappa_{I}=\max\left\\{\kappa,\widetilde{\kappa}\right\\}$, we have:

$\begin{array}[]{rcl}&&||W(t,s)P^{B}(s)+X(t,s)R(s)||\leq\displaystyle
Le^{\theta
s}||C||_{\tau,\infty}\,\kappa_{I}^{2}\int_{s}^{t}e^{-\alpha(t-\tau)}e^{-\widetilde{\alpha}(\tau-s)}d\tau\\\
\\\ &+&\displaystyle Le^{\theta
s}||C||_{\tau,\infty}\,\kappa_{I}^{2}\int_{t}^{+\infty}e^{-\alpha(\tau-t)}e^{-\widetilde{\alpha}(\tau-s)}d\tau+\displaystyle
Le^{\theta
s}||C||_{\tau,\infty}\,\kappa_{I}^{2}\int_{0}^{s}e^{-\alpha(t-\tau)}e^{-\widetilde{\alpha}(s-\tau)}d\tau\\\
\\\ &\leq&\displaystyle Le^{\theta
s}||C||_{\tau,\infty}\,\kappa_{I}^{2}\cdot\left\\{\left(\frac{e^{-\widetilde{\alpha}(t-s)}-e^{-\alpha(t-s)}}{\alpha-\widetilde{\alpha}}\right)+\displaystyle\left(\frac{e^{-\widetilde{\alpha}(t-s)}}{\alpha+\widetilde{\alpha}}\right)+\displaystyle\left(\frac{e^{-\alpha(t-s)}-e^{-\alpha
t-\widetilde{\alpha}s}}{\alpha+\widetilde{\alpha}}\right)\right\\}\\\ \\\
&\leq&\displaystyle Le^{\theta
s}||C||_{\tau,\infty}\,\kappa_{I}^{2}\cdot\left[\frac{1}{|\alpha-\widetilde{\alpha}|}+\frac{1}{\alpha+\widetilde{\alpha}}+\frac{1}{\alpha+\widetilde{\alpha}}\right]e^{-\alpha_{1}(t-s)}\\\
\\\ &\leq&\displaystyle K_{1}e^{-\alpha_{1}(t-s)+\theta s},\end{array}$

and if $\alpha\neq\tilde{\alpha}$, then (A.10) is verified with
$\alpha_{1}=\min\\{\alpha,\widetilde{\alpha}\\}>\theta$ and

$K_{1}=\max\left\\{1,\;\displaystyle
L||C||_{\tau,\infty}\,\kappa_{I}^{2}\cdot\left[\frac{1}{|\alpha-\widetilde{\alpha}|}+\frac{1}{\alpha+\widetilde{\alpha}}+\frac{1}{\alpha+\widetilde{\alpha}}\right]\right\\}.$

Note that, similarly as done in [3], if $\alpha=\widetilde{\alpha}$, we can
see that only the first term in the above brackets must be replaced by a new
estimation of (A.11):

$\begin{array}[]{rcl}||D_{1}||&\leq&\displaystyle Le^{\theta
s}||C||_{\tau,\infty}\kappa\widetilde{\kappa}\int_{s}^{t}e^{-\alpha(t-\tau)}e^{-\alpha(\tau-s)}\,d\tau\\\
\\\ &\leq&\displaystyle Le^{\theta
s}||C||_{\tau,\infty}\kappa^{2}_{I}\int_{s}^{t}e^{-\alpha(t-s)}\,d\tau\\\ \\\
&\leq&\displaystyle Le^{\theta
s}||C||_{\tau,\infty}\kappa^{2}_{I}(t-s)e^{-\alpha(t-s)},\end{array}$

and since the estimation $(t-s)e^{-\gamma(t-s)}\leq\frac{1}{\gamma e}$, for a
positive $\gamma$, if we have that $\gamma<\alpha$ and $\theta<\alpha-\gamma$,
then the previous inequality becomes

$\begin{array}[]{rcl}||D_{1}||&\leq&\displaystyle Le^{\theta
s}||C||_{\tau,\infty}\kappa^{2}_{I}(t-s)e^{-\alpha(t-s)}\leq Le^{\theta
s}||C||_{\tau,\infty}\kappa^{2}_{I}\frac{1}{\gamma
e}e^{-(\alpha-\gamma)(t-s)},\\\ \end{array}$

thus we obtain that

$\begin{array}[]{rcl}&&||W(t,s)P^{B}(s)+X(t,s)R(s)||\\\ \\\
&\leq&\displaystyle Le^{\theta
s}||C||_{\tau,\infty}\kappa^{2}_{I}\frac{1}{\gamma
e}e^{-(\alpha-\gamma)(t-s)}+Le^{\theta
s}||C||_{\tau,\infty}\,\kappa_{I}^{2}\frac{1}{\alpha}e^{-\alpha(t-s)}\\\ \\\
&\leq&\displaystyle Le^{\theta
s}||C||_{\tau,\infty}\kappa^{2}_{I}\frac{1}{\gamma}e^{-(\alpha-\gamma)(t-s)}+Le^{\theta
s}||C||_{\tau,\infty}\,\kappa_{I}^{2}\frac{1}{\gamma}e^{-(\alpha-\gamma)(t-s)}\\\
\\\ &\leq&K_{2}e^{-\alpha_{2}(t-s)+\theta s},\end{array}$

where
$K_{2}=\max\left\\{1,\;2L||C||_{\tau,\infty}\kappa^{2}_{I}\frac{1}{\gamma}\right\\}$
and $\alpha_{2}=\alpha-\gamma>\theta$.

Furthermore, if we define $K_{3}:=\max\\{K_{1},K_{2}\\}$,
$\alpha_{3}:=\min\\{\alpha_{1},\alpha_{2}\\}$ and $\varepsilon_{3}:=\theta$,
we can conclude the estimate (A.10). ∎

######  Lemma 9.

The evolution operator of (4.1) and the projector $P(t)$ previously defined
verify:

$\begin{array}[]{rcl}\left\|T(t,s)P(s)\right\|&\leq&\bar{K}e^{-\bar{\alpha}(t-s)+\bar{\varepsilon}s}\quad\textnormal{for
any $t\geq s\geq 0$},\end{array}$

where $\bar{K}\geq 1$, $\bar{\alpha}>0$, $\bar{\varepsilon}\geq 0$, with
$\bar{\alpha}>\bar{\varepsilon}$.

###### Proof.

Let us consider
$(\xi_{1},\xi_{2})\in\mathbb{R}^{n}\times\mathbb{R}^{m}\setminus\\{(0,0)\\}$,
then we have that

$\begin{array}[]{rcl}\left|T(t,s)P(s)\begin{pmatrix}\xi_{1}\\\
\xi_{2}\end{pmatrix}\right|&=&\left|\left[\begin{array}[]{cc}X(t,s)P^{A}(s)&W(t,s)P^{B}(s)+X(t,s)R(s)\\\
0&Y(t,s)P^{B}(s)\end{array}\right]\begin{pmatrix}\xi_{1}\\\
\xi_{2}\end{pmatrix}\right|,\\\ \\\
&=&\left|\left[\begin{array}[]{cc}X(t,s)P^{A}(s)\xi_{1}+(W(t,s)P^{B}(s)+X(t,s)R(s))\xi_{2}\\\
Y(t,s)P^{B}(s)\xi_{2}\end{array}\right]\right|\\\ \\\
&\leq&||X(t,s)P^{A}(s)||\cdot|\xi_{1}|+\\\ \\\
&&||W(t,s)P^{B}(s)+X(t,s)R(s)||\cdot|\xi_{2}|+||Y(t,s)P^{B}(s)||\cdot|\xi_{2}|\end{array}$

and due to the estimates (A.10) for the second summand deduced in the above
Lemma, the fact that $|\xi_{i}|\leq|(\xi_{1},\xi_{2})|$ and the estimations
(A.1) and (A.2), the Lemma follows easily. ∎

The following two lemmas emulate the previous results considering the
complementary projectors and $t\leq s$. Allowing us to end the treatment and
study of the dichotomy properties.

######  Lemma 10.

There exist a constant $K_{3}\geq 1$, $\alpha_{3}>0$ and $\varepsilon_{3}\geq
0$, where $\varepsilon_{3}<\alpha_{3}$ such that

(A.14) $||(I_{n}-P^{A}(t))W(t,s)-R(t)Y(t,s)||\leq
K_{3}e^{-\alpha_{3}(s-t)+\varepsilon_{3}s}\quad\textnormal{for $s\geq t\geq
0$}.$

###### Proof.

The proof is a charbon copy of the proof of the Lemma 8 and is left for the
reader. ∎

######  Lemma 11.

The evolution operator of (4.1) and the projector $P(t)$ previously defined
verify:

$\begin{array}[]{rcl}\left\|T(t,s)[I-P(s)]\right\|&\leq&\bar{K}e^{-\bar{\alpha}(s-t)+\bar{\varepsilon}s}\quad\textnormal{for
any $s\geq t\geq 0$},\end{array}$

where $\bar{K}\geq 1$, $\bar{\alpha}>0$, $\bar{\varepsilon}\geq 0$, with
$\bar{\alpha}>\bar{\varepsilon}$.

###### Proof.

Let us consider
$(\xi_{1},\xi_{2})\in\mathbb{R}^{n}\times\mathbb{R}^{m}\setminus\\{(0,0)\\}$,
then we have that

$\begin{array}[]{rcl}\left|T(t,s)(I-P(s))\begin{pmatrix}\xi_{1}\\\
\xi_{2}\end{pmatrix}\right|&=&\left|(I-P(t))T(t,s)\begin{pmatrix}\xi_{1}\\\
\xi_{2}\end{pmatrix}\right|\\\ \\\
&=&\left|\left[\begin{array}[]{cc}Q^{A}(t)X(t,s)&Q^{A}(t)W(t,s)-R(t)Y(t,s)\\\
0&Q^{B}(t)Y(t,s)\end{array}\right]\begin{pmatrix}\xi_{1}\\\
\xi_{2}\end{pmatrix}\right|,\\\ \\\
&=&\left|\left[\begin{array}[]{cc}Q^{A}(t)X(t,s)\xi_{1}+(Q^{A}(t)W(t,s)-R(t)Y(t,s))\xi_{2}\\\
Q^{B}(t)Y(t,s)\xi_{2}\end{array}\right]\right|\\\ \\\
&\leq&||Q^{A}(t)X(t,s)||\cdot|\xi_{1}|+\\\ \\\
&&||Q^{A}(t)W(t,s)-R(t)Y(t,s)||\cdot|\xi_{2}|+||Q^{B}(t)Y(t,s)||\cdot|\xi_{2}|\end{array}$

and the Lemma is a consequence of (A.14), which estimates the second summand,
combined with the fact that $|\xi_{i}|\leq|(\xi_{1},\xi_{2})|$. ∎

The next result shows that the nondiagonal submatrix $W(t,s)$ of $T(t,s)$ has
a property reminiscent to the half nonuniform bounded growth.

######  Lemma 12.

There exist a constant $M_{3}\geq 1$, $\omega_{3}>0$ and $\theta\geq 0$ such
that the operator $W(t,s)$, defined in (A.7), verifies

$||W(t,s)||\leq M_{3}e^{\omega_{3}(t-s)+\theta s},\quad t\geq s.$

###### Proof.

Let us recall that the systems $\dot{x}=A(t)x$ and $\dot{y}=B(t)y$ have a half
nonuniform bounded growth on $\mathbb{R}_{0}^{+}$ described in (A.3). By using
Lemma 5, where $Z(\tau)=I_{n}$ and $V(\tau)=I_{m}$, combined with (A.4) which
arises from the Uniformization Lemma, we can see that when $t\geq s\geq 0$:

$\begin{array}[]{rcl}||W(t,s)||&\leq&\displaystyle Le^{\theta
s}||C||_{\tau,\infty}\int_{s}^{t}||X(t,\tau)||_{\tau,t}||Y(\tau,s)||_{s,\tau}\;d\tau\\\
\\\ &\leq&Le^{\theta
s}||C||_{\tau,\infty}\mu\tilde{\mu}\displaystyle\int_{s}^{t}e^{\omega(t-\tau)}e^{\tilde{\omega}(\tau-s)}\;d\tau.\end{array}$

Here we have two cases. The first one is when $\omega\neq\tilde{\omega}$, then
if $\mu_{I}=\max\\{\mu,\tilde{\mu}\\}$:

$\begin{array}[]{rcl}||W(t,s)||&\leq&Le^{\theta
s}||C||_{\tau,\infty}\mu_{I}^{2}e^{\omega
t-\tilde{\omega}s}\displaystyle\int_{s}^{t}e^{(\omega-\tilde{\omega})\tau}\;d\tau\\\
\\\ &=&\displaystyle Le^{\theta
s}||C||_{\tau,\infty}\mu_{I}^{2}\frac{1}{|\tilde{\omega}-\omega|}\left[e^{\tilde{\omega}(t-s)}-e^{\omega(t-s)}\right]\\\
\\\ &\leq&\displaystyle M_{1}e^{\omega_{1}(t-s)+\theta s},\end{array}$

where

$M_{1}=\max\left\\{1,\displaystyle
L||C||_{\tau,\infty}\mu_{I}^{2}\frac{2}{|\tilde{\omega}-\omega|}\right\\}\quad\textnormal{and}\quad\omega_{1}=\max\\{\omega,\tilde{\omega}\\}.$

The second case is when $\omega=\tilde{\omega}$, then

$\begin{array}[]{rcl}||W(t,s)||&\leq&Le^{\theta
s}||C||_{\tau,\infty}\mu_{I}^{2}\displaystyle\int_{s}^{t}e^{\omega(t-s)}\;d\tau\\\
\\\ &\leq&\displaystyle Le^{\theta
s}||C||_{\tau,\infty}\mu_{I}^{2}e^{\omega(t-s)}e^{(t-s)}\\\ \\\
&\leq&\displaystyle M_{2}e^{\omega_{2}(t-s)+\theta s},\end{array}$

where

$M_{2}=\max\left\\{1,L||C||_{\tau,\infty}\mu_{I}^{2}\right\\}\quad\textnormal{and}\quad\omega_{2}=\omega+1.$

Based on the two cases analyzed, we can conclude that

(A.15) $||W(t,s)||\leq Me^{\omega_{3}(t-s)+\theta s},\quad t\geq s,$

where

$M_{3}=\max\left\\{M_{1},M_{2}\right\\}\quad\textnormal{and}\quad\omega_{3}=\max\\{\omega_{1},\omega_{2}\\}.$

∎

The last result shows that the evolution operator associated to the upper
triangular system (2.1) has the property of half
$(\overline{M}e^{\overline{\theta}s},\overline{\omega})$–nonuniform bounded
growth.

######  Lemma 13.

The evolution operator of (4.1) verify:

$\begin{array}[]{rcl}\left\|T(t,s)\right\|&\leq&\bar{M}e^{\bar{\omega}(t-s)+\bar{\theta}s}\quad\textnormal{for
any $t\geq s\geq 0$},\end{array}$

where $\bar{M}\geq 1$, $\bar{\omega}>0$, $\bar{\theta}\geq 0$.

###### Proof.

If we consider
$(\xi_{1},\xi_{2})\in\mathbb{R}^{n}\times\mathbb{R}^{m}\setminus\\{(0,0)\\}$,
then we have that

$\begin{array}[]{rcl}\left|T(t,s)\begin{pmatrix}\xi_{1}\\\
\xi_{2}\end{pmatrix}\right|&=&\left|\left[\begin{array}[]{cc}X(t,s)&W(t,s)\\\
0&Y(t,s)\end{array}\right]\begin{pmatrix}\xi_{1}\\\
\xi_{2}\end{pmatrix}\right|,\\\ \\\
&=&\left|\left[\begin{array}[]{cc}X(t,s)\xi_{1}+W(t,s)\xi_{2}\\\
Y(t,s)\xi_{2}\end{array}\right]\right|\\\ \\\
&\leq&||X(t,s)||\cdot|\xi_{1}|+\\\ \\\
&&||W(t,s)||\cdot|\xi_{2}|+||Y(t,s)||\cdot|\xi_{2}|\end{array}$

and due to the estimation (A.15), the fact that
$|\xi_{i}|\leq|(\xi_{1},\xi_{2})|$ and both estimations in (A.3), we can
ensure that for $t\geq s$:

$||T(t,s)||\leq\bar{M}e^{\bar{\theta}s}e^{\bar{\omega}(t-s)}.$

∎

###  A.2. End of proof of Theorem 3

Firstly, the Lemmas 7, 9 and 11 imply that the triangular system (4.1) has a
$(\bar{K}e^{\bar{\varepsilon}s},\bar{\alpha})$–nonuniform exponential
dichotomy in $\mathbb{R}_{0}^{+}$.

Secondly, the Lemma 13 says that the system (4.1) has the property of half
$(\bar{M}e^{\bar{\theta}s},\bar{\omega})$–nonuniform bounded growth and the
Theorem follows.

######  Remark 7.

A meticulous reading of this Appendix shows that the property of half
nonuniform bounded growth is fundamental in several steps of the proof:

  * a)

Is a necessary condition in order to use the Uniformization Lemma, which
ensures the existence of a continuous family norms $\\{|\cdot|_{t}\\}$
verifying the inequality $|x|_{t}\leq L_{2}(t)|x|$. The half nonuniform
bounded growth property is a required tool to obtain explicit estimations for
$L_{2}(\cdot)$ and (A.5).

  * b)

The constants $\ell$ and $\tilde{\ell}$ are necessary to deduce (A.5) and
(A.6), these identities are immersed in Lemma 5, which is the main key to
deduce several estimations around the proof.

  * c)

The previous facts, also shows that the boundedness properties of
$||C(\tau)||_{\tau,\tau}$ involves estimations based in the half nonuniform
bounded growth property.

We point out that in [32], the property of half nonuniform bounded growth is
not considered neither in the statement of Uniformization Lemma (Lemma 2.2 in
[32]) nor in the statement of Theorem 2.3.

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