# Poincaré-Perron problem for high order differential equations in the class
of almost periodic type functions

H. Bustos  H. Bustos, Centro de Docencia de Ciencias Básicas para Ingeniería,
Facultad de Ciencias de la Ingeniería, Universidad Austral, Campus Miraflores,
Valdivia, Chile<EMAIL_ADDRESS>,  P. Figueroa  P. Figueroa, Instituto
de Ciencias Físicas y Matemáticas, Facultad de Ciencias, Universidad Austral,
Campus Isla Teja, Valdivia, Chile<EMAIL_ADDRESS>and  M. Pinto  M.
Pinto, Departamento de Matemáticas, Universidad de Chile, Las Palmeras 3425,
Casilla 653, Santiago, Chile<EMAIL_ADDRESS>
###### Abstract.

We address the Poincaré-Perron’s classical problem of approximation for high
order linear differential equations in the class of almost periodic type
functions, extending the results for a second order linear differential
equation in [23]. We obtain explicit formulae for solutions of these
equations, for any fixed order $n\geq 3$, by studying a Riccati type equation
associated with the logarithmic derivative of a solution. Moreover, we provide
sufficient conditions to ensure the existence of a fundamental system of
solutions. The fixed point Banach argument allows us to find almost periodic
and asymptotically almost periodic solutions to this Riccati type equation. A
decomposition property of the perturbations induces a decomposition on the
Riccati type equation and its solutions. In particular, by using this
decomposition we obtain asymptotically almost periodic and also $p$-almost
periodic solutions to the Riccati type equation. We illustrate our results for
a third order linear differential equation.

###### Key words and phrases:

Perturbed high order linear differential equation, almost periodic solutions,
Riccati type equation

###### 2020 Mathematics Subject Classification:

34A05, 34A30, 34E10, 34E05

##  1\. Introduction

Several problems of differential equations, stability theory, dynamical
systems and many others has been studied in connection with the development of
the theory of almost periodic functions, see for instance [10, 16, 41].
Precisely, almost periodic functions have been useful when one looks over long
enough time scales, since they come arbitrarily close to being periodic and
they play an important role in describing the phenomena that are similar to
the periodic oscillations which can be observed frequently in many fields such
as celestial mechanics, nonlinear vibration, electromagnetic theory, plasma
physics, engineering, ecosphere, life sciences and so on [2, 11, 19, 28, 31,
32, 40]. Furthermore, the concept of asymptotic almost periodicity was
introduced in the literature [24, 25] by Fréchet in the early 1940s, as a
natural extension of almost periodicity. Since then, this notion has generated
lots of developments and applications. In particular, many authors apply the
asymptotic property of asymptotically almost periodic functions to determine
the existence of almost periodic solutions to various ordinary differential
equations, partial differential equations, differential equations in Banach
spaces, integro-differential equations as well as fractional differential
equations, see for instance [5, 7, 14, 45, 48] and the references therein.

On the other hand, high order differential equations have been intensively and
extensively studied during the last decades. They have been attracted great
attention due to their possible applications to different areas such as
theoretical physics, population dynamics, biology, ecology, etc., see for
example [18, 35, 36, 37]. In this context, the classical Poincaré-Perron’s
problem of approximation for high order linear differential equations consists
in

$y^{(n)}(t)+\sum_{i=0}^{n-1}(a_{i}+r_{i}(t))y^{(i)}(t)=0,$ (1.1)

where $a_{0},\dots,a_{n-1}\in\mathbb{C}$ are constants and functions
$r_{0}(t),\dots,r_{n-1}(t)$ are small in some sense with $n\geq 2$. Precisely,
Poincaré [44] proved in 1885 the existence of a solution $y$ to (1.1) such
that $y^{\prime}(t)/y(t)$ converges as $t\to+\infty$, assuming that the roots
$\lambda_{1},\dots,\lambda_{n}$ of the equation
$x^{n}+\sum_{i=0}^{n-1}a_{i}x^{i}=0$ have distinct real part, $r_{i}$’s are
continuous on $[t_{0},+\infty)$ and $r_{i}\in C_{0}$, i.e., $r_{i}(t)\to 0$ as
$t\to+\infty$ for all $i=0,\dots,n-1$. At the beginning of the twentieth
century, Perron [38] improved this result, under the same assumptions,
assuring the existence of $n$ linearly independent solutions
$y_{1},\dots,y_{n}$ with the same asymptotic behaviour, namely, logarithmic
derivative $y_{i}^{\prime}(t)/y_{i}(t)$ converges to $\lambda_{i}$ as
$t\to+\infty$. However, they do not provide a more precise formula for the
asymptotic behaviour of solutions in either case.

Another perturbations have been also studied, for instance $r_{i}\in L^{p}$
for some $p\geq 1$, in systems of linear differential equations, which can be
applied to this equation. These results are due to Levinson [17, Th. 1.3.1]
for $p=1$, to Hartman-Wintner [17, Th. 1.5.1] for $p\in(1,2]$ and to Harris-
Lutz [29] for $p>2$. Related results have been obtained by Hartman [30, Th.
17.2], S̆ims̆a [46] and Trench [47] assuming mild integrable conditions:
$|r_{i}(t)|t^{q}$ or $r_{i}(t)t^{q}$ are integrable. In several works have
been also studied cases $n=2$ in [20], $n=3$ in [21, 22] and $n=4$ in [12,
13], recovering Poincaré and Perron’s results and providing precise asymptotic
formulae for solutions to problem (1.1) and estimated their errors, in the
class of $C_{0}$ and $L^{p}$ functions. In almost diagonal linear systems,
perhaps the first results in this sense were obtained in Pinto et al. [41, 42,
43]. Recently, in [4] we have recovered Poincaré and Perron’s results and
extended results in [12, 13, 20, 21, 22] for any $n\geq 5$, namely, we have
obtained a precise asymptotic formulae for solutions to problem (1.1) and
estimated their errors, in the class of $C_{0}$ and $L^{p}$ functions.

To the best of our knowledge, concerning different type of perturbations
$r_{i}$’s in (1.1) such as almost periodic type functions (see Definitions 2.1
and 2.3) few results are available. In particular, in case $n=2$ the authors
in [23] addressed equation

$y^{\prime\prime}+(a_{1}+r_{1}(t))y^{\prime}+(a_{0}+r_{0}(t))y=0,\qquad
t\in\mathbb{R}$

in the class of almost periodic type functions:
$r_{i}\in\mathcal{AP}(\mathbb{R},\mathbb{C})$ or
$r_{i}\in\mathcal{AAP}(\mathbb{R},\mathbb{C})$ or
$r_{i}\in\mathcal{AP}(\mathbb{R},\mathbb{C},p)$, obtaining explicit formulae
for solutions by studying a Riccati equation associated with the logarithmic
derivative of a solution.

In this paper, inspired by results in [4, 23] we are interested in generalize
Poincaré’s and Perron’s classical problem of approximation (1.1) to the class
of almost periodic type functions for any $n\geq 3$, namely, equation (1.1)
with either $r_{i}\in\mathcal{AP}(\mathbb{R},\mathbb{C})$ or
$r_{i}\in\mathcal{AAP}(\mathbb{R},\mathbb{C})$ or
$r_{i}\in\mathcal{AP}(\mathbb{R},\mathbb{C},p)$. Precisely, we have obtained
explicit formulae for solutions of these equations under sufficient conditions
by studying a Riccati type equation associated with the logarithmic derivative
of a solution. In particular case $n=3$, if $\lambda$, $\lambda_{1}$ and
$\lambda_{2}$ are roots of $P(a;x)=x^{3}+a_{2}x^{2}+a_{1}x+a_{0}$ and have
distinct real parts, and $r_{i}$, $i=0,1,2$ are sufficiently small in
$L^{\infty}$-sense and almost periodic, then there is a solution
$y:\mathbb{R}\to\mathbb{C}$ satisfying (3.27) and

$\begin{split}y_{\lambda}(t)=&\,e^{\lambda
t}\exp\left(-\gamma_{1}^{-1}\gamma_{2}^{-1}\int_{0}^{t}[P(r;\lambda)(s)+\mathcal{L}(s,z(s))+\mathcal{F}(s,Z(s))]\,ds\right)\\\
&\times\exp\left(-\sum_{j=1}^{2}\frac{1}{\Gamma_{j}\gamma_{j}}G_{\gamma_{j}}[P(r;\lambda)+\mathcal{L}(\cdot,z)+\mathcal{F}(\cdot,Z)](t)\right).\end{split}$
(1.2)

where

$P(r(t);\lambda)=\lambda^{2}r_{2}(t)+\lambda r_{1}(t)+r_{0}(t),$ (1.3)
$\mathcal{L}(t,z)=r_{2}(t)z^{\prime}+(2\lambda r_{2}(t)+r_{1}(t))z\qquad\text{
and }$ (1.4)
$\mathcal{F}(t,z,z^{\prime})=(a_{2}+r_{2}(t)+3\lambda)z^{2}+3zz^{\prime}+z^{3}$
(1.5)

and $z$ satisfies a Riccati type equation, is an almost periodic complex value
function. Some conditions (see (3.11)) insure the estimate
$z^{(i)}=O\big{(}(I_{\beta}+I_{-\beta})[P(r;\lambda)]\big{)}$, $i=0,1$, where
$0<\beta<\min\\{\Re(\lambda_{j}-\lambda)\ :\ j=1,2\\}$ of $z$. We have denoted
$\gamma_{i}=\lambda_{i}-\lambda$ and
$\Gamma_{i}=(-1)^{i}(\lambda_{2}-\lambda_{2})$ for $i=1,2$ and we have used
Green’s operators acting on locally integrable functions
$f:\mathbb{R}\to\mathbb{C}$ given by

$G_{\omega}\big{[}f\big{]}(t):=\int_{-\infty}^{\infty}g_{\omega}(t,s)f(s)\,ds\quad\text{and}\quad
I_{\omega}\big{[}f\big{]}(t):=\int_{-\infty}^{\infty}|g_{\omega}(t,s)f(s)|\,ds\,,$
(1.6)

where

$g_{\omega}(t,s):=\begin{cases}-{\rm sgn}(\Re{\omega})e^{\omega(t-s)}&{\rm
sgn}(\Re{\omega})(t-s)<0\\\ 0&\text{otherwise}\end{cases}\,,$ (1.7)

for a given $\omega\in\mathbb{C}$ with $\Re\omega\neq 0$, and $\Re{\omega}$
denotes the real part of $\omega$. See Theorem 3.2 for the precise statement
for any $n\geq 3$. Furthermore, we have obtained sufficient conditions to
ensure the existence of a fundamental system of solutions of the form (1.2),
see Theorem 3.4.

In general, given a $(n+1)$-tuple $c=(c_{0},\dots,c_{n})$ of complex numbers,
we will denote $P(c;x):=\sum_{i=0}^{n}c_{i}x^{i}$, so that
$P(a;x)=x^{n}+\sum_{i=0}^{n-1}a_{i}x^{i}$ is the characteristic polynomial of
(1.1) with $r_{i}\equiv 0$, $i=0,\dots,n-1$, where
$a=(a_{0},\dots,a_{n-1},1)$, $a_{i}$’s are the complex numbers associated to
the equation (1.1) and $a_{n}=1$. Hence, as in [38, 44], we will also assume
that $P(a;x)$ has $n$ roots, $\lambda,\lambda_{1},\dots,\lambda_{n-1}$, with
different real parts, so that the characteristic polynomial of (2.8)
$P_{D}(a,x)$ (see (2.12)) will have roots with different and non zero real
parts. Therefore an exponential dichotomy can be considered a scalar
exponential dichotomy with scalar Green’s functions (1.7). These operators
$G_{\omega}$ and $I_{\omega}$ allows us to define the Green’s function
$\mathcal{G}$ and Green’s operator $G$ for equation $\mathcal{D}z=f$ (see
equations (2.14) and (2.15)), so that an integral equation for $z$ is
obtained, see (3.10).

On the other hand, these type of results (Theorems 3.2 and 3.4) are also true
for $r_{i}$, $i=0,\dots,n-1$ in either $C_{0}$ or $L^{p}$, see [3, 4, 8, 17].
In general, $z$ is related with the logarithmic derivative of $y$ and
represent the error function belonging to spaces under consideration $C_{0}$,
$L^{p}$ or almost periodic type functions, and is small with respect to the
norm in the space; $L^{\infty}$ in our case. The fixed point Banach argument
allows us to find such an almost periodic or asymptotically almost periodic
solution $z$ to a Riccati type equation, see (2.7). Furthermore, this
procedure allows us to address $p$-almost periodic perturbations, see
Definition 2.3. In other words, there exist several classes of almost periodic
type functions verifying these results. This is the case of those with a
summand $C_{0}$ or $L^{p}$ (asymptotically almost periodic functions or
$p$-almost periodic functions), namely, $r_{i}=\mu_{i}+\nu_{i}$,
$i=0,\dots,n-1$ with $\mu_{i}\in\mathcal{AP}(\mathbb{R},\mathbb{C})$ and
either $\nu_{i}\in C_{0}$ or $\nu_{i}\in L^{p}$ for $i=0,\dots,n-1$, see
Theorems 3.5 and 3.7. This decomposition of the coefficients $r_{i}$,
$i=0,\dots,n-1$ induces the decomposition $z=\theta+\psi$, where $\theta$ is
the almost periodic part and $\psi$ is $C_{0}$ or $L^{p}$, respectively. These
functions $\theta$ and $\psi$ satisfy their own equations, which can be
treated and solved separately, implying the existence of new solutions. See
equations (3.23)-(3.24) and Theorems 3.5 and 3.7. In section 4, we will
discuss these results through an illustrative example. Finally, we point out
that the used method is scalar [20, 21, 22], reducing the order of the
equation and avoiding the usual diagonalization process [8, 17, 27].

##  2\. Preliminaries

Following ideas presented in [23], here we present first almost periodic type
functions, then Green’s type operators, complete Bell polynomials and we end
up this section with the Riccati equation and almost periodic functions.

###  2.1. Almost periodic type functions

According to [8, 33, 48], the notion of almost periodic function we shall use
is the following.

######  Definition 2.1.

  * $a$)

A continuous function $r:\mathbb{R}\to\mathbb{C}$ satisfies
$r\in\mathcal{AP}(\mathbb{R},\mathbb{C})$ if for every sequence $\\{b_{k}\\}$
of real numbers there exists a subsequence $\\{\tilde{b}_{m}\\}$ such that
$\lim_{m\to\infty}r(t+\tilde{b}_{m})$ exists uniformly for $t\in\mathbb{R}$.

  * $b$)

A continuous function $F:\mathbb{R}\times\mathbb{C}^{n}\to\mathbb{C}$
satisfies $F\in\mathcal{AP}(\mathbb{R}\times\mathbb{C}^{n},\mathbb{C})$ if for
every sequence $\\{b_{k}\\}$ of real numbers there exists a subsequence
$\\{\tilde{b}_{m}\\}$ such that $\lim_{m\to\infty}F(t+\tilde{b}_{m},Z)$ exists
uniformly for $t\in\mathbb{R}$ and for $Z$ in compact subsets of
$\mathbb{C}^{n}$ .

In order to perturb almost periodic functions we present the following result,
see [23, Lemma 1] for a proof.

######  Lemma 2.2.

Let $f\in\mathcal{AP}(\mathbb{R},\mathbb{C})$ and $p\geq 1$. If $f(t)\to c$ as
$t\to+\infty$ then $f\equiv c$. If $f\in L^{p}[0,+\infty)$ then $f\equiv 0$.

Let us introduce the following function spaces: $BC(\mathbb{R},\mathbb{C})$ is
the set of all bounded continuous functions from $\mathbb{R}$ to $\mathbb{C}$.
In addition, we define $BC_{0}(\mathbb{R},\mathbb{C})=\\{f\in
BC(\mathbb{R},\mathbb{C})\mid f(t)\to 0\text{ as }t\to\infty\\}$,
$C_{00}(\mathbb{R},\mathbb{C})=\\{f\in BC_{0}(\mathbb{R},\mathbb{C})\mid
f(t)\to 0\text{ as }t\to-\infty\\},$ and
$L^{p}_{0}=\\{f:\mathbb{R}\to\mathbb{C}\mid f\text{ is bounded on}$
$\text{$(-\infty,0]$ and }\int_{0}^{\infty}|f|^{p}<\infty\\}$. In other words,
from Lemma 2.2 we have that $\mathcal{AP}(\mathbb{R},\mathbb{C})\cap
BC_{0}(\mathbb{R},\mathbb{C})=\\{0\\}$ and
$\mathcal{AP}(\mathbb{R},\mathbb{C})\cap L^{p}_{0}=\\{0\\}$. Thus, we set

######  Definition 2.3.

  1. $c$)

A bounded continuous function $f:\mathbb{R}\to\mathbb{C}$ is _asymptotically
almost periodic_ if $f=\phi+g$ with
$\phi\in\mathcal{AP}(\mathbb{R},\mathbb{C})$ and $\lim_{t\to+\infty}g(t)=0$.
The set of asymptotically almost periodic functions from $\mathbb{R}$ to
$\mathbb{C}$ will be denoted by $\mathcal{AAP}(\mathbb{R},\mathbb{C})$. In
addition, if $\lim_{t\to-\infty}g(t)=0$, we will say
$f\in\mathcal{AAP}_{0}(\mathbb{R},\mathbb{C})$. See [7, 16].

  2. $d$)

A bounded continuous function $f:\mathbb{R}\to\mathbb{C}$ is _p-almost
periodic_ with $p\geq 1$, if $f=\phi+g$ with
$\phi\in\mathcal{AP}(\mathbb{R},\mathbb{C})$ and $g\in L^{p}_{0}$. The set of
all $p$-almost periodic functions from $\mathbb{R}$ to $\mathbb{C}$ will be
denoted by $\mathcal{AP}(\mathbb{R},\mathbb{C},p)$. In addition, if $g\in
L^{p}(\mathbb{R})$, we will say
$f\in\mathcal{AP}_{0}(\mathbb{R},\mathbb{C},p)$. See [16, section 4.3; page
46].

  3. $e$)

A bounded continuous function $f:\mathbb{R}\times\mathbb{C}^{n}\to\mathbb{C}$
satisfied $f\in\mathcal{AAP}(\mathbb{R}\times\mathbb{C}^{n},\mathbb{C})$ if
$f=\phi+g$ with $\phi\in\mathcal{AP}(\mathbb{R}\times\mathbb{C}^{n})$ and
$\lim_{t\to+\infty}g(t,x)=0$ uniformly for $x$ on compact subsets of
$\mathbb{C}^{n}$. In addition, if $\lim_{t\to-\infty}g(t,x)=0$ uniformly for
$x$ on compact subsets of $\mathbb{C}$, we will say
$f\in\mathcal{AAP}_{0}(\mathbb{R}\times\mathbb{C}^{n},\mathbb{C})$.

  4. $f$)

A bounded continuous function $f:\mathbb{R}\times\mathbb{C}^{n}\to\mathbb{C}$
satisfied $f\in\mathcal{AP}(\mathbb{R}\times\mathbb{C}^{n},\mathbb{C},p)$ if
$f=\phi+g$ with
$\phi\in\mathcal{AP}(\mathbb{R}\times\mathbb{C}^{n},\mathbb{C})$ and
$g(\cdot,x)\in L^{p}_{0}$ uniformly for $x$ on compact subsets of
$\mathbb{C}^{n}$. In addition, if $g(\cdot,x)\in L^{p}(\mathbb{R})$ uniformly
for $x$ on compact subsets of $\mathbb{C}$, we will say
$f\in\mathcal{AP}_{0}(\mathbb{R}\times\mathbb{C}^{n},\mathbb{C},p)$.

######  Remark 2.4.

It follows that
$\mathcal{AP}(\mathbb{R},\mathbb{C})\subset\mathcal{AAP}_{0}(\mathbb{R},\mathbb{C})\subset\mathcal{AAP}(\mathbb{R},\mathbb{C})\subset
BC(\mathbb{R},\mathbb{C})$ and the direct sums
$\mathcal{AAP}(\mathbb{R},\mathbb{C})=\mathcal{AP}(\mathbb{R},\mathbb{C})\oplus
BC_{0}(\mathbb{R},\mathbb{C})$ and
$\mathcal{AAP}_{0}(\mathbb{R},\mathbb{C})=\mathcal{AP}(\mathbb{R},\mathbb{C})\oplus
C_{00}(\mathbb{R},\mathbb{C})$. Furthermore,
$\mathcal{AP}(\mathbb{R},\mathbb{C})$,
$\mathcal{AAP}_{0}(\mathbb{R},\mathbb{C})$ and
$\mathcal{AAP}(\mathbb{R},\mathbb{C})$ are closed subspaces of the Banach
space $BC(\mathbb{R},\mathbb{C})$ endowed with supremum norm
$\|\cdot\|_{\infty}$. Also,
$\mathcal{AP}(\mathbb{R},\mathbb{C},p)=\mathcal{AP}(\mathbb{R},\mathbb{C})\oplus[L^{p}_{0}\cap
BC(\mathbb{R},\mathbb{C})]$ and
$\mathcal{AP}_{0}(\mathbb{R},\mathbb{C},p)=\mathcal{AP}(\mathbb{R},\mathbb{C})\oplus[L^{p}(\mathbb{R})\cap
BC(\mathbb{R},\mathbb{C})]$. Notice that $f=\phi+g$ with
$\phi\in\mathcal{AP}(\mathbb{R},\mathbb{C})$ and $g\in L^{p}_{0}$ then
$g|_{[t_{0},+\infty)}\in L^{p}[t_{0},+\infty)$ for any $t_{0}\in\mathbb{R}$.

######  Remark 2.5.

Since all the functions we will consider are from $\mathbb{R}$ into
$\mathbb{C}$ for simplicity we shall omit set $\mathbb{R}$ and $\mathbb{C}$ so
that we shall denote $\mathcal{AP}(\mathbb{R},\mathbb{C})=\mathcal{AP}$,
$\mathcal{AAP}_{0}(\mathbb{R},\mathbb{C})=\mathcal{AAP}_{0}$,
$BC(\mathbb{R},\mathbb{C})=BC$, etc.

###  2.2. Green’s type operators

It is readily checked that $I_{\omega}=I_{\Re\omega}$

$\big{|}G_{\omega}\big{[}f\big{]}(t)\big{|}\leq
I_{\Re{\omega}}\big{[}f\big{]}(t).$ (2.1)

Operators $G_{\omega}$ and $I_{\omega}$ have been very useful in asymptotic
integration, see for instance [3, 12, 17, 21, 22, 23]. They also satisfy the
following inequalities, see [21] for a proof.

######  Lemma 2.6.

Let $\alpha$ be a non zero real number and $0<\beta<|\alpha|$ . Then

$I_{\alpha}\big{[}aI_{\operatorname{{sgn}}(\alpha)\beta}[b]\big{]}(t)\leq
I_{\operatorname{{sgn}}(\alpha)\beta}[b](t)I_{\alpha-\operatorname{{sgn}}(\alpha)\beta}[a](t)\,,$
$I_{\alpha}[a](t)\leq I_{\operatorname{{sgn}}(\alpha)\beta}[a]\,,$
$I_{\alpha}\big{[}a\big{(}I_{\beta}+I_{-\beta}\big{)}[b]\big{)}\big{]}(t)\leq
2\big{(}I_{\beta}+I_{-\beta}\big{)}[b](t)I_{\alpha-{\rm
sgn}(\alpha)\beta}[a](t)\,.$

Let us recall the following result concerning the Green’s operators which has
been useful in asymptotic integration, see [3, 8, 9, 17, 30].

######  Lemma 2.7.

Let $\gamma\in\mathbb{C}$, $\alpha=\Re\gamma\neq 0$ and let $r\in
BC(\mathbb{R},\mathbb{C})$ be a bounded continuous function. If $r\in
BC_{0}(\mathbb{R},\mathbb{C})$ (resp. $C_{00}(\mathbb{R},\mathbb{C})$) then
$I_{\alpha}[r]\in BC_{0}(\mathbb{R},\mathbb{C})$ (resp.
$C_{00}(\mathbb{R},\mathbb{C})$). If $r\in L^{p}_{0}$ (resp.
$L^{p}(\mathbb{R})$) for some $p\geq 1$ then $I_{\alpha}[r]\in
BC_{0}(\mathbb{R},\mathbb{C})\cap L^{p}_{0}$ (resp.
$C_{00}(\mathbb{R},\mathbb{C})\cap L^{p}(\mathbb{R})$). Similarly, if $r\in
L^{p}(-\infty,0]$ for some $p\geq 1$ then $I_{\alpha}[r](t)\to 0$ as
$t\to-\infty$ and $I_{\alpha}[r]\in L^{p}(-\infty,0]$.

Notice that $G_{\omega}[r]$ is a solution of the linear equation
$y^{\prime}=\omega y+r$. In order to study linear equations with
$r\in\mathcal{AP}(\mathbb{R},\mathbb{C})$ and nonlinear perturbations of them
we have the following invariant property of the Green’s operator, which is
straightforward from (1.6), the Definition 2.1 and Lemma 2.7, see [23, Lemma
5] for a proof.

######  Lemma 2.8.

Let $\omega\in\mathbb{C}$, $\Re\omega\neq 0$. It holds $G_{\omega}:E\to E$,
where either $E=\mathcal{AP}(\mathbb{R},\mathbb{C})$,
$\mathcal{AAP}(\mathbb{R},\mathbb{C})$ or
$\mathcal{AAP}_{0}(\mathbb{R},\mathbb{C})$. Similarly, if
$r\in\mathcal{AP}(\mathbb{R},\mathbb{C},p)$ (resp.
$r\in\mathcal{AP}_{0}(\mathbb{R},\mathbb{C},p)$) then
$G_{\omega}[r]\in\mathcal{AP}(\mathbb{R},\mathbb{C},p)\cap\mathcal{AAP}(\mathbb{R},\mathbb{C})$
(resp.
$G_{\omega}[r]\in\mathcal{AP}_{0}(\mathbb{R},\mathbb{C},p)\cap\mathcal{AAP}_{0}(\mathbb{R},\mathbb{C})$.

######  Remark 2.9.

Notice that it is also possible to consider
$E=\mathcal{PAP}(\mathbb{R},\mathbb{C})$,
$\mathcal{AA}(\mathbb{R},\mathbb{C})$, $\mathcal{AAA}(\mathbb{R},\mathbb{C})$
or $\mathcal{PAA}(\mathbb{R},\mathbb{C})$, namely, the sets of pseudo-almost
periodic functions, almost automorphic functions, asymptotically almost
automorphic functions and pseudo-almost automorphic functions respectively,
see for instance [6, 7, 15, 16, 34, 39, 42, 48].

###  2.3. The Riccati type equation and almost periodic functions

In order to study (1.1) a key step is to make the following change of
variables $z=y^{\prime}/y-\lambda$ , where $\lambda$ denotes a fixed root of
$P(a;x)$. Then, the aim is to finding such a function $z$ depending on
$r_{i}$’s with property $z\in\mathcal{AP}$. This change is equivalent to
consider

$y(t)=\exp\left(\int_{0}^{t}[\lambda+z(s)]\,ds\right)\,,\quad t\in\mathbb{R}.$
(2.2)

Hence, replacing $y$ into equation (1.1) an equation for $z$ is obtained.
Indeed, one obtains (see [4, Section 2.3])

$\frac{y^{(i)}(t)}{y(t)}=\lambda^{i}+i\lambda^{i-1}z(t)+\sum_{j=2}^{i}{i\choose
j}\lambda^{i-j}B_{j}\big{(}z(t),z^{\prime}(t),\dots,z^{(j-1)}(t)\big{)}\,,$
(2.3)

where $B_{j}$’s are the complete Bell polynomials. Those polynomials can be
defined recursively by the following formula, by setting $B_{0}=1$ and for
$i\in\mathbb{N}$

$B_{i+1}(x_{1},\dots,x_{i+1})=\sum_{j=0}^{i}{i\choose
j}B_{i-j}(x_{1},\dots,x_{i-j})x_{j+1}\,,$ (2.4)

and satisfy the binomial type relation

$B_{i}(x_{1}+y_{1},\dots,x_{i}+y_{i})=\sum_{j=0}^{i}{i\choose
j}B_{i-j}(x_{1},\dots,x_{i-j})B_{j}(y_{1},\dots,y_{j})\,.$ (2.5)

The unique linear term of the complete Bell polynomial $B_{i}$ is $x_{i}$,
$i\geq 1$. Moreover, define polynomials $f_{i}$, $i\geq 0$ by

$\begin{split}f_{i}(x_{1},\dots,x_{i})&=B_{i+1}(x_{1},\dots,x_{i+1})-x_{i+1}=\sum_{j=0}^{i-1}{i\choose
j}B_{i-j}(x_{1},\dots,x_{i-j})x_{j+1},\end{split}$ (2.6)

in view of (2.4). Notice that $f_{0}\equiv 0$. These type of polynomials and
related ones have been useful in combinatorial mathematics. They also occur in
many applications, such as in the Faá di Bruno’s formula, and also they are
related to Stirling and Bell numbers, see [1].

Summarising, we will use the following Theorem.

######  Theorem 2.10.

[4, Theorem 1.1] Function $y$ given by (2.2) is a solution to (1.1) in
$\mathbb{R}$ if and only if $z$ is a solution to

$\mathcal{D}z(t)+P(r(t);\lambda)+\mathcal{L}(t,z)+\mathcal{F}(t,z,\dots,z^{(n-2)})=0\,,\
\ \text{in $\mathbb{R}$,}$ (2.7)

where $\lambda$ is a fixed root of $P(a;x)$, the differential linear part with
constant coefficients is the $n-1$ order differential operator

$\mathcal{D}z=\sum_{j=1}^{n}\frac{1}{j!}\frac{\partial^{j}}{\partial
x^{j}}P(a;\lambda)z^{(j-1)},$ (2.8)

the function depending only on perturbations $r=(r_{0},\dots,r_{n})$ with
$r_{n}\equiv 0$ is

$P(r(t);\lambda)=\sum_{k=0}^{n-1}\lambda^{k}r_{k}(t)=r_{0}(t)+\lambda
r_{1}(t)+\lambda^{2}r_{2}(t)+\cdots+\lambda^{n-1}r_{n-1}(t),$ (2.9)

the differential linear part with variable coefficients is

$\mathcal{L}(t,z):=\sum_{k=1}^{n-1}\frac{1}{k!}\frac{\partial^{k}}{\partial
x^{k}}[P(r;\lambda)]z^{(k-1)}(t).$ (2.10)

and the nonlinear term is given by

$\mathcal{F}(t,z,z^{\prime},\dots,z^{(n-2)})=\sum_{i=2}^{n}\sum_{j=0}^{n-i}{i+j\choose
j}(a_{i+j}+r_{i+j}(t))\lambda^{j}f_{i-1}(z,z^{\prime},\dots,z^{(i-1)})\,.$
(2.11)

Note that, the previous formulae above hold for $\lambda=0$ by assuming
$\lambda^{0}=1$ .

In order to study Eq. (2.7), it will be useful to understand operator
$\mathcal{D}$ so that we denote $P_{\mathcal{D}}(a;x)$ the characteristic
polynomial associated to the linear differential operator $\mathcal{D}$, i.e.,

$P_{\mathcal{D}}(a;x):=\sum_{k=1}^{n}\frac{1}{k!}\frac{\partial^{k}}{\partial
x^{k}}P(a;\lambda)x^{k-1}\,$ (2.12)

and we will use following result of [4].

######  Lemma 2.11.

Suppose that $\lambda_{l}$ , for $l=1,\dots n-1$ are the others different
roots of the polynomial $P(a;x)$. Then the roots of $P_{\mathcal{D}}(a;x)$ are
$\lambda_{l}-\lambda$ , for $l=1,\dots n-1$ . Furthermore, if $P(a,x)$ has
roots with different real parts, then the polynomial $P_{\mathcal{D}}(a,x)$
has roots with different and non zero real parts.

###  2.4. A more general non-linear equation

As in [4], we will study, in general way, equations of the form

$\mathcal{D}z(t)=b(t)+R(t,z(t),\dots,z^{(n-2)}(t))\,\quad t\in\mathbb{R},$
(2.13)

where $\mathcal{D}$ is a differential operator of degree $n-1$ with constant
coefficient and, the characteristic polynomial, associated to the differential
operator $\mathcal{D}$, has $n-1$ roots with real part not zero. Precisely,
let us denote by $\\{\gamma_{m}\\}_{m=1}^{n-1}$ the $n-1$ different roots of
the characteristic polynomial of $\mathcal{D}$ . We also denote
$\alpha_{m}:=\Re(\gamma_{m})$ , where $\Re$ denotes the real part. Also, we
shall assume that $R:\mathbb{R}\times\mathbb{C}^{n-1}:\to\mathbb{C}$. We will
impose some restrictions on $R$ later.

######  Remark 2.12.

Note that the reduction of the Eq (1.1) was rewritten in Eq. (2.7). This
equation is of type Eq. (2.13) denoting $b(t)=-P(r(t);\lambda)$ and
$R(t,z,\dots,z^{(n-2)})=-\mathcal{L}(t,z(t))-\mathcal{F}(t,z,\dots,z^{(n-2)})\,.$

We will denote by $Z:=(z_{0},\dots,z_{n-2})$ for a $(n-1)$-tuple, where
$z_{i}\in\mathcal{AP}(\mathbb{R})$ for $i=0,1,\dots,n-2$ . Arguing as in [23,
Lemma 7] we have the following fact. We ommit its proof.

######  Lemma 2.13.

If $R\in\mathcal{AP}(\mathbb{R}\times\mathbb{C}^{n-1})$ (resp.
$\mathcal{AAP}(\mathbb{R}\times\mathbb{C}^{n-1})$ or
$\mathcal{AAP}_{0}(\mathbb{R}\times\mathbb{C}^{n-1})$) and
$z_{0},z_{1},\dots,z_{n-2}\in\mathcal{AP}(\mathbb{R})$ (resp.
$\mathcal{AAP}(\mathbb{R})$ or (resp. $\mathcal{AAP}_{0}(\mathbb{R})$)), then
the map $t\mapsto R(t,z_{0}(t),\dots,z_{n-2}(t))\in\mathcal{AP}(\mathbb{R})$
(resp. $\mathcal{AAP}(\mathbb{R})$ or (resp.
$\mathcal{AAP}_{0}(\mathbb{R})$)).

The method of the variation of constants lead us to define the Green function
of the homogeneous equation $\mathcal{D}z=0$. Thus, we define the Green
function $\mathcal{G}(t,s)$ by

$\mathcal{G}(t,s)=\sum_{i=1}^{n-1}\frac{1}{\Gamma_{j}}g_{\gamma_{i}}(t,s)\quad\text{with}\quad\Gamma_{i}:=\prod_{k=1,\,k\not=i}^{n-1}(\gamma_{i}-\gamma_{k}),$
(2.14)

where $g_{\gamma}$ is given by (1.7). Therefore, our Green operator for
equation $\mathcal{D}z=f$ is given by

$G\big{[}f\big{]}(t):=\int_{-\infty}^{\infty}\mathcal{G}(s,t)f(s)\,ds=\sum_{j=1}^{n-1}\frac{1}{\Gamma_{j}}G_{\gamma_{j}}\big{[}f\big{]}(t)\,.$
(2.15)

Notice that for any $i=1,\dots,n-2$ we have that

$G\big{[}f\big{]}^{(i)}(t)=\sum_{j=1}^{n-1}\frac{\gamma_{j}^{i}}{\Gamma_{j}}G_{\gamma_{j}}\big{[}f\big{]}(t)$
(2.16)

and

$G\big{[}f\big{]}^{(n-1)}(t)=\sum_{j=1}^{n-1}\frac{\gamma_{j}^{n-1}}{\Gamma_{j}}G_{\gamma_{j}}\big{[}f\big{]}(t)+f(t),$
(2.17)

so that $\mathcal{D}(G[f])=f,$ since

$\sum_{j=1}^{n-1}\frac{\gamma_{j}^{i}}{\Gamma_{j}}=\begin{cases}0&i=0,\dots,n-2\\\
1&i=n-1\end{cases}.$

Hence, a direct consequence of Lemma 2.8, (2.16) and (2.17) is the following
result.

######  Lemma 2.14.

Suppose that $f\in\mathcal{AP}(\mathbb{R})$ , then the map
$G[f]^{(i)}\in\mathcal{AP}(\mathbb{R})$ for $i=0,\dots,n-1$ .

######  Remark 2.15.

Note that from (2.16) one can to estimate for a function
$f\in\mathcal{AP}(\mathbb{R})$ the value of $|G\big{[}f\big{]}^{(i)}(t)|$ in
the following way

$\begin{split}|G\big{[}f\big{]}^{(i)}(t)|&\leq\sum_{j=1}^{n-1}\left|{\gamma_{j}^{i}\over\Gamma_{j}}G_{\gamma_{j}}[f](t)\right|\leq\sum_{j=1}^{n-1}\frac{|\gamma_{j}|^{i}}{|\Gamma_{j}|}I_{\Re\gamma_{j}}[f](t)\leq\|f\|_{\infty}\sum_{j=1}^{n-1}\frac{|\gamma_{j}|^{i}}{|\Gamma_{j}\Re\gamma_{j}|}\,,\end{split}$

in view of $I_{\Re\gamma_{j}}[1](t)=\frac{1}{|\Re\gamma_{j}|}$ for all
$t\in\mathbb{R}$.

Given $Z\in\mathbb{C}$, with $Z=(z_{0},z_{1},\dots,z_{n-2})$ we consider the
norm $\|Z\|=\sum_{i=0}^{n-2}|z_{i}|$ . Returning to the study of Eq. (2.13),
we shall assume that there exists a constant $M>0$ and a function
$\xi_{M}:\mathbb{R}\to[0,\infty)$ such that for all $t\in\mathbb{R}$ and
$Z_{i}\in\mathbb{C}^{n-1}$ with $\|Z\|\leq M$ , for $i=1,2$

$|R(t,Z_{1})-R(t,Z_{2})|\leq\xi_{M}(t)\|Z_{1}-Z_{2}\|\,.$ (2.18)

Now, given a function $z:\mathbb{R}\to\mathbb{C}$, for simplicity we shall
denote $Z=Z(t)$ the $(n-1)$-tuple $Z=(z,z^{\prime},\dots,z^{(n-2)})$. Let us
introduce $\mathcal{AP}^{n-2}(\mathbb{R})$ , as the space of functions with
$n-2$ derivatives in $\mathcal{AP}(\mathbb{R})$, that is, the function space
$\mathcal{AP}^{n-2}(\mathbb{R})$ is defined by all functions
$z:\mathbb{R}\to\mathbb{C}$ such that $z^{(i)}\in\mathcal{AP}(\mathbb{R})$,
$i=0,1,\dots,n-2$. With a slight abuse of notation we denote
$\|Z(t)\|=\sum_{i=0}^{n-2}|z^{(i)}(t)|$ . Observe that
$\mathcal{AP}^{n-2}(\mathbb{R})$ is a Banach space with norm
$\|z\|_{\mathcal{AP}^{n-2}}=\sup_{t\in\mathbb{R}}\|Z(t)\|$.

For the next result, we denote
$\tilde{\alpha}_{j}=\sum_{i=0}^{n-2}|\gamma_{j}^{i}|>0$ for $j=1,\dots,n-1$
and $\tilde{\gamma}=\sum_{k=1}^{n-1}\frac{\tilde{\alpha}}{|\Gamma_{k}|}$.
Recall that $\alpha_{j}=\Re\gamma_{j}$.

######  Proposition 2.16.

Suppose that $b\in\mathcal{AP}(\mathbb{R})$, and
$R\in\mathcal{AP}(\mathbb{R}\times\mathbb{C}^{n-1})$. Moreover, assume that
$R(\cdot,0)=0$ on $\mathbb{R}$, and $R$ satisfies (2.18) for some constant
$M>0$. If there exists a positive constant $0<\epsilon_{0}<1$ such that

$\Big{\|}\sum_{j=1}^{n-1}\frac{\tilde{\alpha}_{j}}{\left|\Gamma_{j}\right|}I_{\alpha_{j}}\big{[}\xi_{M}\big{]}\Big{\|}_{\infty}\leq\epsilon_{0}\,,\quad\
\,\left\|\sum_{i=0}^{n-2}\left|G[b]^{(i)}\right|\right\|_{\infty}\leq(1-\epsilon_{0})M\,,$
(2.19)

where $\Gamma_{j}$ is defined in Eq. (2.14), then equation (2.13) has unique
solution $z$ on $\mathbb{R}$ such that $z^{(i)}\in\mathcal{AP}(\mathbb{R})$
for all $i=0,1,\dots,n-2$ and satisfies the integral equation

$z=G\big{[}b+R(\cdot,z,z^{\prime},\dots,z^{(n-2)})\big{]}.$ (2.20)

If, in addition, there exists $0<\beta<\min\\{|\alpha_{j}|\mid 1\leq j\leq
n-1\\}$ such that

$\Big{\|}\sum_{j=1}^{n-1}\frac{\tilde{\alpha}_{j}}{\left|\Gamma_{j}\right|}I_{\alpha_{j}-{\rm
sgn}(\alpha_{j})\beta}\big{[}\xi_{M}\big{]}\Big{\|}_{\infty}<\frac{1}{2},$
(2.21)

then

$z^{(i)}=O\big{(}(I_{\beta}+I_{-\beta})[b]\big{)}\qquad\text{for all
$i=0,1,\dots,n-2$}\,.$ (2.22)

###### Proof.

First, for functions $z\in\mathcal{AP}^{n-2}(\mathbb{R})$ define the operator

$Tz(t)=\int_{-\infty}^{+\infty}\mathcal{G}(t,s)\big{[}b(s)+R(s,Z(s))\big{]}\,ds=G\big{[}b+R(\cdot,Z)\big{]}(t)\,,$

where $Z=(z,z^{\prime},\dots,z^{(n-2)})$. Notice that from Lemmata 2.14 it
follows that $G[b]^{(i)}\in\mathcal{AP}$ for $i=0,\dots,n-2$. Furthermore,
from the definition of Green’s operator $G$ and its derivatives (2.16) and
Lemmata 2.13 and 2.14 it follows that $G[R(\cdot,Z)]^{(i)}\in\mathcal{AP}$ for
$i=0,\dots,n-2$, since $R(\cdot,Z)\in\mathcal{AP}$. Thus, we conclude that
$(Tz)^{(i)}\in\mathcal{AP}$ for $i=0,\dots,n-2$. Set
$B_{M}=\big{\\{}z\in\mathcal{AP}^{n-2}\,:\,\|z\|_{\mathcal{AP}^{n-2}}\leq
M\,\big{\\}}$. Note that $B_{M}$ is a closed subset of $\mathcal{AP}^{(n-2)}$.
Also, if $z\in B_{M}$ , we have that $\|z^{(i)}\|_{\infty}\leq M$ for all
$i=0,\dots,n-2$. Hence, arguing as in [4, Proposition 2.9] it follows that
$T(B_{M})\subseteq B_{M}$ and $T$ is a contractive operator. Therefore, there
exists a unique fixed point $z\in B_{M}$ .

Now, consider the sequence $\\{z_{k}\\}_{k}\subseteq\mathcal{AP}^{n-2}$ given
by $z_{0}=0$ and for $k\geq 0$ $z_{k+1}(t)=Tz_{k}(t)$. Again arguing as in [4,
Proposition 2.9] it follows estimate (2.22), by using Lemma 2.6, (2.21) and
that $z_{k}\to z$ as $k\to+\infty$ in $\mathcal{AP}^{n-2}$, since $T$ is a
contractive operator. This concludes the proof. ∎

###  2.5. Other spaces of solutions

Considering the notation of the Proposition 2.16, let $V$ be a closed subspace
of

$BC^{n-2}(\mathbb{R},\mathbb{C})=\\{f:\mathbb{R}\to\mathbb{C}\mid f^{(i)}\in
BC(\mathbb{R},\mathbb{C}),\ i=0,1,\dots,n-2\\},$ $\text{with
norm}\quad\|f\|_{BC^{n-2}}=\sup_{t\in\mathbb{R}}\sum_{i=0}^{n-2}\|f^{(i)}(t)\|.$

which is invariant under the operator $T$. Then a similar result to
Proposition 2.16 hold if $b\in V$, under assumptions (2.19) and (2.21). Thus,
there is a solution $z$ of (2.13) and (2.20) such that $z\in B\cap V$, where
$B=\\{g\in BC^{n-2}\mid\|g\|_{BC^{n-2}}\leq M\\}$. For instance,
$\mathcal{AAP}^{n-2}=\mathcal{AP}^{n-2}\oplus BC_{0}^{n-2}$ is a closed
subspace of $BC^{n-2}(\mathbb{R},C)$ invariant under $T$, where

$\mathcal{AAP}^{n-2}=\mathcal{AP}^{n-2}\oplus
BC_{0}^{n-2}=\\{f:\mathbb{R}\to\mathbb{C}\mid
f^{(i)}\in\mathcal{AAP}(\mathbb{R},\mathbb{C}),\ i=0,1,\dots,n-2\\}.$

However, notice that

$W_{1}=\\{f:\mathbb{R}\to\mathbb{C}\mid f^{(i)}\in L_{0}^{p}\cap
BC(\mathbb{R},\mathbb{C}),\ i=0,1,\dots,n-2\\}\qquad\text{and}$
$W_{2}=\\{f:\mathbb{R}\to\mathbb{C}\mid f^{(i)}\in L^{p}(\mathbb{R})\cap
BC(\mathbb{R},\mathbb{C}),\ i=0,1,\dots,n-2\\},$

are invariant under $T$, but they are not closed subspaces of
$BC^{n-2}(\mathbb{R},\mathbb{C})$, so that, we cannot obtain directly a
version of Proposition 2.16 in the subspaces either
$\mathcal{AP}^{n-2}(\mathbb{R},\mathbb{C},p)$ or
$\mathcal{AP}^{n-2}_{0}(\mathbb{R},\mathbb{C},p)$, where

$\mathcal{AP}^{n-2}(\mathbb{R},\mathbb{C},p)=\\{f:\mathbb{R}\to\mathbb{C}\mid
f^{(i)}\in\mathcal{AP}(\mathbb{R},\mathbb{C},p),\
i=0,1,\dots,n-2\\}\qquad\text{and}$
$\mathcal{AP}^{n-2}_{0}(\mathbb{R},\mathbb{C},p)=\\{f:\mathbb{R}\to\mathbb{C}\mid
f^{(i)}\in\mathcal{AP}_{0}(\mathbb{R},\mathbb{C},p),\ i=0,1,\dots,n-2\\}.$

Despite of this loss of completeness, we shall find such solutions in
$\mathcal{AP}^{n-2}(\mathbb{R},\mathbb{C},p)$ or
$\mathcal{AP}^{n-2}_{0}(\mathbb{R},\mathbb{C},p)$ by exploiting a
decomposition property.

Arguing as in [23, Corollary 1], we have that the decomposition in a sum of
$b$ and $R$ induces the direct sum of the solution $z$ and the equation (3.10)
in the direct sum (2.23), in view of
$\mathcal{AAP}^{n-2}=\mathcal{AP}^{n-2}\oplus BC_{0}^{n-2}$. Thus, we conclude
the following

######  Corollary 2.17.

Suppose that $b=\phi+g\in\mathcal{AAP}(\mathbb{R},\mathbb{C})$ with
$\phi\in\mathcal{AP}(\mathbb{R},\mathbb{C})$, $g\in
BC_{0}(\mathbb{R},\mathbb{C})$ and
$R=\varphi+h\in\mathcal{AAP}(\mathbb{R}\times\mathbb{C}^{n-1},\mathbb{C})$
with $\varphi\in\mathcal{AP}(\mathbb{R}\times\mathbb{C}^{n-1},\mathbb{C})$ and
$h(\cdot,x)\in BC_{0}(\mathbb{R},\mathbb{C})$ uniformly for $x$ in compact
subsets of $\mathbb{C}^{n-1}$. If $R$ satisfies (2.18) and (2.19) for some
$M>0$, then there is a solution $z$ of (2.13) such that
$z^{(i)}\in\mathcal{AAP}(\mathbb{R},\mathbb{C})$ (resp.
$z^{(i)}\in\mathcal{AAP}_{0}(\mathbb{R},\mathbb{C})$), $i=0,\dots,n-2$, is a
solution of the integral equation (2.20) and $z=\theta+\psi$ with
$\theta\in\mathcal{AP}^{n-2}$ and $\psi\in
BC_{0}^{n-2}(\mathbb{R},\mathbb{C})$ (resp. $\psi\in
C_{00}^{n-2}(\mathbb{R},\mathbb{C}))$), where $\theta$ and $\psi$ satisfy

$\theta=G[\phi+\varphi(\cdot,\Theta)]\qquad\text{and}\qquad\psi=G[g+h(\cdot,\theta)+R(\cdot,\Theta+\Psi)-R(\cdot,\Theta)]$
(2.23)

with $\Theta=(\theta,\theta^{\prime},\dots,\theta^{(n-2)})$and
$\Psi=(\psi,\psi^{\prime},\dots,\psi^{(n-2)})$. Furthermore, if it holds
(2.21) for some $0<\beta<\min\\{|\alpha_{j}|\ \mid 1\leq j\leq n-1\\}$, then
$z^{(i)}=O\big{(}(I_{\beta}+I_{-\beta})[b]\big{)}$ for all $i=0,\dots,n-2$.

Now, we present a result which will be useful to study the problem (2.13) when
$b$ belongs to this new class of almost periodic functions
$\mathcal{AP}(\mathbb{R},\mathbb{C},p)$ (resp.
$\mathcal{AP}_{0}(\mathbb{R},\mathbb{C},p)$) given in Definition 2 and $R$ is
a suitable function that allows us to find solutions to (2.13) in this class
of functions. See [23, Lemma 9], for more details.

######  Lemma 2.18.

Assume that $R\in\mathcal{AP}(\mathbb{R}\times\mathbb{C}^{n-1},\mathbb{C})$
and satisfies (2.18). If $z\in\mathcal{AP}(\mathbb{R},\mathbb{C},p)$ (resp.
$z\in\mathcal{AP}_{0}(\mathbb{R},\mathbb{C},p)$) then
$R(\cdot,Z(\cdot))\in\mathcal{AP}(\mathbb{R},\mathbb{C},p)$ (resp.
$R(\cdot,Z(\cdot))\in\mathcal{AP}_{0}(\mathbb{R},\mathbb{C},p)$). Furthermore,
if $R=\varphi+h\in\mathcal{AP}(\mathbb{R}\times\mathbb{C}^{n-1},\mathbb{C},p)$
(resp. $R\in\mathcal{AP}_{0}(\mathbb{R}\times\mathbb{C}^{n-1},\mathbb{C},p)$)
with $\varphi\in\mathcal{AP}(\mathbb{R}\times\mathbb{C}^{n-1},\mathbb{C})$
satisfying (2.18) as above and $h(\cdot,x)\in L^{p}_{0}$ (resp.
$L^{p}(\mathbb{R})$) uniformly for $x$ in compact subsets of
$\mathbb{C}^{n-1}$ and $z\in\mathcal{AP}(\mathbb{R},\mathbb{C},p)$ (resp.
$\mathcal{AP}_{0}(\mathbb{R},\mathbb{C},p)$), then
$R(\cdot,Z(\cdot))\in\mathcal{AP}(\mathbb{R},\mathbb{C},p)$ (resp.
$\mathcal{AP}_{0}(\mathbb{R},\mathbb{C},p)$).

In order to state the next result, let us denote for a suitable given function
$\theta$,

$\tilde{R}_{\theta}(t,W)=R(t,\Theta+W)-R(t,\Theta),$ (2.24)

where $W=(w,w^{\prime},\dots,w^{(n-2)})$. Moreover, assume that
$\tilde{R}_{\theta}$ satisfies (2.18) with $\zeta_{M}=\zeta_{M}(t)$ instead of
$\xi_{M}$ for some $M>0$. Applying Proposition 2.16 to equation for $\psi$ in
(2.23), namely, $T$ has a fixed point in $B=\\{g\in
BC^{n-2}\mid\|g\|_{BC^{n-2}}\leq M\\}$; we deduce the following fact. We omit
its proof.

######  Lemma 2.19.

Suppose that $b=\phi+g\in\mathcal{AP}(\mathbb{R},\mathbb{C},p)$ (resp.
$\mathcal{AP}_{0}(\mathbb{R},\mathbb{C},p)$) with
$\phi\in\mathcal{AP}(\mathbb{R},\mathbb{C})$ and $g\in L^{p}_{0}$ (resp.
$L^{p}(\mathbb{R})$), $R(\cdot,0)=0$ and
$R=\varphi+h\in\mathcal{AP}(\mathbb{R}\times\mathbb{C}^{n-1},\mathbb{C},p)$
(resp. $\mathcal{AP}_{0}(\mathbb{R}\times\mathbb{C}^{n-1},\mathbb{C},p)$) with
$\varphi\in\mathcal{AP}(\mathbb{R}\times\mathbb{C}^{n-1},\mathbb{C})$ and
$h(\cdot,x)\in L^{p}_{0}$ (resp. $L^{p}(\mathbb{R})$) uniformly for $x$ in
compact subsets of $\mathbb{C}^{n-1}$, so that, there is a solution
$\theta\in\mathcal{AP}^{n-2}$ of the integral equation

$\theta=G[\phi+\varphi(\cdot,\Theta)].$ (2.25)

If

$\Big{\|}\sum_{j=1}^{n-1}\frac{\tilde{\alpha}_{j}}{\left|\Gamma_{j}\right|}I_{\alpha_{j}}\big{[}\zeta_{M}\big{]}\Big{\|}_{\infty}\leq\epsilon_{0}\,,\quad\
\,\bigg{\|}\sum_{i=0}^{n-2}\left|G\big{[}g+h(\cdot,\Theta)\big{]}^{(i)}\right|\bigg{\|}_{\infty}\leq(1-\epsilon_{0})M\,,$

then there is a solution $w$ of the integral equation

$w=G[g+h(\cdot,\Theta)+\tilde{R}_{\theta}(\cdot,W)]$ (2.26)

where $\tilde{R}_{\theta}$ is given by (2.24) and $\theta$ satisfies (2.25).
If, in addition, there exists $0<\beta<\min\\{|\alpha_{j}|\mid 1\leq j\leq
n-1\\}$ such that

$\Big{\|}\sum_{j=1}^{n-1}\frac{\tilde{\alpha}_{j}}{\left|\Gamma_{j}\right|}I_{\alpha_{j}-{\rm
sgn}(\alpha_{j})\beta}\big{[}\zeta_{M}\big{]}\Big{\|}_{\infty}<\frac{1}{2},$
(2.27)

then for all $i=0,1,\dots,n-2$

$w^{(i)}=O\left((I_{\beta}+I_{-\beta})[g+h(\cdot,\theta)]\right)\qquad\text{and}\qquad
w^{(i)}\in L^{p}_{0}\quad\text{(resp. $L^{p}(\mathbb{R})$)}.$ (2.28)

Conversely, if $\theta$ and $w$ are such that
$\theta^{(i)}\in\mathcal{AP}(\mathbb{R},\mathbb{C})$ and $w^{(i)}\in
L^{p}_{0}$ (resp. $L^{p}(\mathbb{R})$) satisfy (2.25) and (2.26) respectively,
then $z=\theta+w\in\mathcal{AP}^{n-2}(\mathbb{R},\mathbb{C},p)$ (resp.
$\mathcal{AP}_{0}^{n-2}(\mathbb{R},\mathbb{C},p)$) is a solution to (2.13) and
(3.10) with $b=\phi+g$ and $R=\varphi+h$.

######  Remark 2.20.

Let us stress that the direct sum of $b$ and $R$ and the existence of the
“almost periodic part” $\theta$ induce the direct sum of the solution $z$ and
the equation (2.20) in the direct sum (2.25) with (2.26), in these new classes
of functions $\mathcal{AP}(\mathbb{R},\mathbb{C},p)$ or
$\mathcal{AP}_{0}(\mathbb{R},\mathbb{C},p)$. Furthermore, notice that we are
not using that $g$ and $h(\cdot,\theta)$ are bounded continuous functions, but
we need $G[g+h(\cdot,\theta)]\in BC(\mathbb{R},\mathbb{C})$. Notice that an
analogous version of Lemma 2.19 could be also obtained for
$b\in\mathcal{AAP}(\mathbb{R},\mathbb{C})$ or
$\mathcal{AAP}_{0}(\mathbb{R},\mathbb{C})$ and
$R\in\mathcal{AAP}(\mathbb{R}\times\mathbb{C}^{n-1},\mathbb{C})$ or
$\mathcal{AAP}_{0}(\mathbb{R}\times\mathbb{C}^{n-1},\mathbb{C})$. In other
words, we can study equation (2.25) in $\mathcal{AP}(\mathbb{R},\mathbb{C})$
and then equation (2.26) in either $BC_{0}(\mathbb{R},\mathbb{C})$ or
$C_{00}(\mathbb{R})$.

##  3\. Existence of solutions for the Poincaré-Perron Problem

We return to study Eq. (2.7), where $\mathcal{D}$ is defined (2.8),
$P(r;\lambda)$ and $\mathcal{L}$ are defined by (2.9) and (2.10),
respectively, and $\mathcal{F}$ is the non linear part of the equation given
by (2.11).

Note that given $\omega\in\mathbb{C}$, $\omega\neq 0$ it holds

$\int_{0}^{t}G_{\omega}[f](s)\,ds=-{1\over\omega}\int_{0}^{t}f(s)\,ds+{1\over\omega}\left\\{G_{\omega}[f](t)-G_{\omega}[f](0)\right\\}.$

Hence, it follows that

$\begin{split}\int_{0}^{t}G[f](s)\,ds=&\,(-1)^{n+1}\prod_{j=1}^{n-1}\gamma^{-1}_{j}\int_{0}^{t}f(s)\,ds+\sum_{j=1}^{n-1}{1\over\Gamma_{j}\gamma_{j}}\left[G_{\gamma_{j}}[f](t)-G_{\gamma_{j}}[f](0)\right],\end{split}$

in view of
$\displaystyle\sum_{j=1}^{n-1}{1\over\Gamma_{j}\gamma_{j}}=(-1)^{n}\prod_{j=1}^{n-1}\gamma^{-1}_{j}$.

Taking into account Lemma 2.11, let $\gamma_{j}=\lambda_{j}-\lambda$,
$j=1,\dots,n-1$ be the different roots of the characteristic polynomial
$P_{\mathcal{D}}$ associated to the differential operator $\mathcal{D}$ given
by (2.8) and denote by $\alpha_{j}=\Re\gamma_{j}\not=0$ and
$\tilde{\alpha}_{j}=\sum_{i=0}^{n-2}|\gamma_{j}|^{i}$ , for $j=1,\dots,n-1$.
Note that in this situation $\Gamma_{i}=\prod_{k=1,k\neq
i}^{n-1}(\lambda_{i}-\lambda_{k})$ in view of (2.14). Recall that $\lambda$ is
a fixed root of $P(a;x)$ and
$\tilde{\gamma}=\sum_{k=1}^{n-1}\frac{\tilde{\alpha}_{k}}{\left|\Gamma_{k}\right|}$.

###  3.1. General results.

For the next result we shall estimate $\mathcal{F}$ and $f_{i}$’s given by
(2.11) and (2.6), respectively. Denoting
$X=(x_{1},\dots,x_{i})\in\mathbb{R}^{i}$ and
$Y=(y_{1},\dots,y_{i})\in\mathbb{R}^{i}$ , one has that

$\dfrac{\big{|}f_{i}(X)-f_{i}(Y)\big{|}}{\|X-Y\|}<\infty\,,$

for $X\not=Y$ in $\overline{B(0;\delta)}$ , the closed ball of radius
$\delta>0$ with centre in $0$ in $\mathbb{R}^{i}$. It follows that the map

$m_{i}:\delta\mapsto\sup\left\\{\dfrac{\big{|}f_{i}(X)-f_{i}(Y)\big{|}}{\|X-Y\|}\mid
X\not=Y\,;\,X,Y\in B(0;\delta)\right\\}$ (3.1)

is also continuous and non negative. Indeed, one has that there is a constant
$C>0$ such that $m_{i}(\delta)\leq C\sup\\{\|\nabla f_{i}(X)\|\mid X\in
B(0;\delta)\\}$. Moreover, since $f_{i}$ is a polynomial with terms of degree
greater than 1, one obtains that $m_{i}(\delta)\to 0$ for $\delta\to 0$ and
$m_{i}(\delta)\to\infty$ for $\delta\to\infty$. Thus, we have the following
result.

######  Lemma 3.1.

Define $m(\delta)=\max_{i=2,\dots,n}\\{m_{i}(\delta)\\}\,$, where $m_{i}$ are
defined by equation (3.1). Then, $m$ extends to a continuous map defined on
$[0,\infty)$, such that $m(0)=0$, which satisfies $m(\delta)>0$ for all
$\delta>0$ , $m(\delta)\to+\infty$ as $\delta\to+\infty$ and for all
$i=1,\dots,n-1$

$\big{|}f_{i}(X)-f_{i}(Y)\big{|}\leq m(\delta)\|X-Y\|\,,\quad\text{ for all
}\,X,\,Y\in\overline{B(0;\delta)}\,.$ (3.2)

Then, for $\delta>0$ and $z_{1},\,z_{2}\in\mathcal{AP}^{n-2}(\mathbb{R})$ such
that $\|z_{i}\|_{\mathcal{AP}^{n-2}}<\delta$ , for $i=1,2$ , we have that

$\begin{split}\big{|}\mathcal{F}\big{(}t,Z_{1}(t)\big{)}-\mathcal{F}\big{(}t,Z_{2}(t)\big{)}\big{|}&\leq
m(\delta)\|Z_{1}(t)-Z_{2}(t)\|\sum_{i=2}^{n}\sum_{j=0}^{n-i}{i+j\choose
j}\big{(}|a_{i+j}|+|r_{i+j}(t)|\big{)}|\lambda|^{j}\end{split}$ (3.3)

Now, denote for $0\leq\beta<\min\\{|\alpha_{j}|\mid 0\leq j\leq n-1\\}$

$\begin{split}L_{\beta}&:=\sum_{j=1}^{n-1}\sum_{k=1}^{n-1}\left|\frac{\tilde{\alpha}_{j}}{\Gamma_{j}k!}\right|\left\|I_{\alpha_{j}-\operatorname{{sgn}}(\alpha_{j})\beta}\Big{[}\frac{\partial^{k}}{\partial
x^{k}}P(r(\cdot);\lambda)\Big{]}\\!\right\|_{\infty}\,,\end{split}$ (3.4)

and

$\begin{split}Q_{\beta}&:=\sum_{j=1}^{n-1}\sum_{i=2}^{n}\sum_{k=0}^{i-2}\frac{\tilde{\alpha}_{j}}{\left|\Gamma_{j}\right|}{i\choose
k}|\lambda|^{k}\Big{(}{|a_{i}|\over|\alpha_{j}-{\rm
sgn}(\alpha_{j})\beta|}+\left\|I_{\alpha_{j}-\operatorname{{sgn}}(\alpha_{j})\beta}\big{[}|r_{i}|\big{]}\right\|_{\infty}\Big{)}\,.\end{split}$
(3.5)

Hence, assuming $L_{0}<1$, let $\delta_{\lambda}>0$ be such that

$0<m(\delta)<\dfrac{1-L_{0}}{Q_{0}}\quad\text{ for all
$\delta\in(0,\delta_{\lambda})$ and
}m(\delta_{\lambda})=\dfrac{1-L_{0}}{Q_{0}}.$ (3.6)

Our main result reads as follows.

######  Theorem 3.2.

Consider the equation (2.7) with $r_{i}\in\mathcal{AP}(\mathbb{R})$ , for
$i=1,\dots,n-1$. Let $\gamma_{j}$ be the different roots of the differential
operator $\mathcal{D}$ given by (2.8) and denote by
$\alpha_{j}=\Re\gamma_{j}\not=0$ , for $j=1,\dots,n-1$. Let us also denote by
$\tilde{\alpha}_{j}=\sum_{i=0}^{n-2}|\gamma_{j}|^{i}$ . Assume that

$L_{0}<1\quad\text{ and }\quad H\leq\max\\{[1-L_{0}-m(\delta)Q_{0}]\delta\
\mid\ \delta\in(0,\delta_{\lambda})\\}\,,$ (3.7)

where $L_{0}$ and $Q_{0}$ are given by (3.4) and (3.5) with $\beta=0$,
respectively, and

$H:=\left\|\sum_{i=0}^{n-2}\left|G[P(r;\lambda)]^{(i)}\right|\right\|_{\infty}\,.$
(3.8)

Then, there exists a solution $y$ for the equation (1.1) such that

$\begin{split}y_{\lambda}(t)=&\,e^{\lambda
t}\exp\bigg{(}(-1)^{n}\prod_{j=1}^{n-1}\gamma^{-1}_{j}\int_{0}^{t}\big{[}P(r;\lambda)(s)+\mathcal{L}(s,z(s))+\mathcal{F}(s,Z(s))\big{]}\,ds\bigg{)}\\\
&\times\exp\bigg{(}-\sum_{j=1}^{n-1}{1\over\Gamma_{j}\gamma_{j}}G_{\gamma_{j}}\big{[}P(r;\lambda)+\mathcal{L}(\cdot,z)+\mathcal{F}(\cdot,Z)\big{]}(t)\bigg{)},\end{split}$
(3.9)

where $z\in\mathcal{AP}^{n-2}(\mathbb{R})$ satisfy the differential equation
(2.7) and integral equation

$z=-G[P(r;\lambda)+\mathcal{L}(\cdot,z)+\mathcal{F}(\cdot,Z)].$ (3.10)

In addition, if there exists a constant $0<\beta<\min\\{|\alpha_{j}|\mid 1\leq
j\leq n-1\\}$ such that

$L_{\beta}<\frac{1}{2},$ (3.11)

then $z$ satisfies
$z^{(i)}=O\big{(}(I_{\beta}+I_{-\beta})[P(r;\lambda)]\big{)}$ for
$i=0,1,\dots,n-2$ . Moreover

$\frac{y_{\lambda}^{(i)}(t)}{y_{\lambda}(t)}=\lambda^{i}+O\big{(}(I_{\beta}+I_{-\beta})[P(r;\lambda)]\big{)}$
(3.12)

###### Proof.

We shall apply Proposition 2.16 to equation (2.7) by taking
$b(t)=-P(r(t);\lambda)$ and
$R(t,z,z^{\prime},\dots,z^{(n-2)})=-\mathcal{L}(t,z)-\mathcal{F}(t,z,z^{\prime},\dots,z^{(n-2)})$,
so that, $b\in\mathcal{AP}$ and
$R(\cdot,Z)\in\mathcal{AP}(\mathbb{R}\times\mathbb{C}^{n-1})$, see (2.10) and
(2.11).

Arguing as in the proof of [4, Theorem 3.1], it follows that
$R=-\mathcal{L}-\mathcal{F}$ satisfy the Lipschitz condition (2.18) with the
choice

$\begin{split}\xi_{\delta}(t)&=\sum_{k=1}^{n-1}\frac{1}{k!}\bigg{|}\frac{\partial^{k}}{\partial
x^{k}}P(r(\cdot);\lambda)\bigg{|}+m(\delta)\sum_{i=2}^{n}\sum_{j=0}^{i-2}{i\choose
j}\big{(}|a_{i}|+|r_{i}(t)|\big{)}|\lambda|^{j}\end{split}$ (3.13)

and

$\sum_{j=0}^{n-1}\left|\frac{\tilde{\alpha}_{j}}{\Gamma_{j}}\right|I_{\alpha_{j}-{\rm
sgn}(\alpha_{j})\beta}\big{[}\xi_{\delta}\big{]}(t)\leq
m(\delta)Q_{\beta}+L_{\beta}\,.$

Note that for any function $f$ it holds $I_{\alpha_{j}}[f](t)\leq
I_{\alpha_{j}-\operatorname{{sgn}}(\alpha_{j})\beta}[f](t),$ so that,
$L_{0}\leq L_{\beta}$ and $Q_{0}\leq Q_{\beta}$. Hence, by definition of
$\delta_{\lambda}>0$ and (3.6) it follows that function
$g_{\lambda}(\delta)=[1-L_{0}-m_{\lambda}(\delta)Q_{0}]\delta$ satisfies
$g_{\lambda}(0)=0=g_{\lambda}(\delta_{\lambda})$ and $g_{\lambda}(\delta)>0$
for all $\delta\in(0,\delta_{\lambda})$. Then, we choose
$M\in(0,\delta_{\lambda})$ satisfying

$g_{\lambda}(M)=[1-L_{0}-m(M)Q_{0}]M=\max\\{[1-L_{0}-m(\delta)Q_{0}]\delta\
\mid\ \delta\in(0,\delta_{\lambda})\\}.$ (3.14)

Therefore, we have that $m(M)<\frac{1}{Q_{0}}(1-L_{0})$ and we choose
$\epsilon_{0}=m(M)Q_{0}+L_{0}<1$. Now, by the choice of $M$ in (3.14), it
follows that $H$ satisfies

$H\leq[1-L_{0}-m_{\lambda}(M)Q_{0}]M=[1-(m_{\lambda}(M)Q_{0}+L_{0})]M=(1-\epsilon_{0})M$

Hence, the conditions of Proposition 2.16 are satisfied, thus there exists a
solution $z\in\mathcal{AP}^{n-2}$ of the integral equation (3.10) which
implies that $z$ satisfies Eq. (2.13) that corresponds in this case to (2.7).
Hence, $y_{\lambda}$ given by (2.2) is a solution to equation (1.1). Moreover,
integrating (3.10) we find that

$\begin{split}\int_{0}^{t}z(s)\,ds=&\,(-1)^{n}\prod_{j=1}^{n-1}\gamma^{-1}_{j}\int_{0}^{t}\big{[}P(r;\lambda)(s)+\mathcal{L}(s,z(s))+\mathcal{F}(s,Z(s))\big{]}\,ds\\\
&\,-\sum_{j=1}^{n-1}{1\over\Gamma_{j}\gamma_{j}}\left\\{G_{\gamma_{j}}\big{[}P(r;\lambda)+\mathcal{L}(\cdot,z)+\mathcal{F}(\cdot,Z)\big{]}(t)+\text{const.}\right\\}\end{split}$

Thus, we conclude (3.9). Finally, by using (3.11) and again arguing as in the
proof of [4, Theorem 3.1] there exists $M$ small enough such that
$m(M)<\frac{1}{Q_{\beta}}({1\over 2}-L_{\beta})$ and condition (2.21) in
Proposition 2.16 is satisfied. Therefore, the solution $z$ of equation (2.7)
and also satisfies
$z^{(i)}=O\big{(}(I_{\beta}+I_{-\beta})\big{[}P(r;\lambda)]\big{)}$ for
$i=0,\dots,n-2$ . Moreover, from (2.3) we get immediately (3.12). This
finishes the proof. ∎

######  Remark 3.3.

Let us stress that inequality (3.7) is an explicit sufficient condition in
terms of $H$, $L_{0}$ and $Q_{0}$. The choice of $m(\delta)$ lead us to obtain
$M=M(\delta_{\lambda})$ satisfying
$g_{\lambda}(M)=\max\\{g_{\lambda}(\delta)\,:\,\delta\in(0,\delta_{\lambda})\\}$.
Notice that we can replace $m(\delta)$ in (3.2), (3.3) and (3.7) for any
increasing function, say $\tilde{m}(\delta)$, satisfying $\tilde{m}(0)=0$ and
$m(\delta)\leq\tilde{m}(\delta)$. However, assumption on $H$ becomes stronger
since

$\max\\{[1-L_{0}-\tilde{m}(\delta)Q_{0}]\delta\mid\delta\in(0,\tilde{\delta}_{\lambda})\\}\leq\max\\{[1-L_{0}-m(\delta)Q_{0}]\delta\mid\delta\in(0,\delta_{\lambda})\\}$

with $\tilde{\delta}_{\lambda}\leq\delta_{\lambda}$ and
$\tilde{m}(\tilde{\delta}_{\lambda})=\dfrac{1-L_{0}}{Q_{0}}$.

Note that $L_{0}$, $Q_{0}$ and $H$ in the hypothesis of Theorem 3.2 depend of
the election of $\lambda$. If we want to put in evidence this dependency we
rewrite $L_{0}^{\lambda}$, $Q_{0}^{\lambda}$ and $H_{\lambda}$.

Now, we shall prove the existence of fundamental system of solutions
$\\{y_{i}\\}_{i}$, $i=1,\dots,n$, assuming conditions of Theorem 3.2 for each
$\lambda_{i}$, $i=1,\dots,n$, the roots of $P(a;x)$. For simplicity, we will
only consider assumption (3.11). In this way, we recover Perron’s result [38].
To this purpose, we shall denote $P(r;\lambda_{i})$, $\mathcal{D}_{i}$,
$P_{\mathcal{D}_{i}}$ and $\mathcal{F}_{i}$ by (2.9), (2.8), (2.12) and (2.11)
with $\lambda=\lambda_{i}$ respectively. We shall assume that
$\Re(\lambda_{i})\neq\Re(\lambda_{j})$ for all $i\neq j$. Also, consider
$\mathcal{N}(i):=\\{1,\dots,n\\}\setminus\\{i\\}$ and the bijection
$\sigma_{i}:\mathcal{N}(i)\to\\{1,\dots,n-1\\}$ defined by

$\sigma_{i}(j)=\begin{cases}j&\text{if }j<i\\\ j-1&\text{if
}i<j\end{cases}\qquad\text{and}\qquad\sigma_{i}^{-1}(j)=\begin{cases}j&\text{if
}j<i\\\ j+1&\text{if }i<j\end{cases}.$

Thus, fixing $i\in\\{1,\dots,n\\}$ we have that
$\gamma_{ij}=\lambda_{\sigma_{i}^{-1}(j)}-\lambda_{i}$, $j=1,\dots,n-1$ are
the roots of polynomial $P_{\mathcal{D}_{i}}$ as shown in Lemma 2.11.
Furthermore, Green’s operator for equation $\mathcal{D}_{i}z=f$ is given by

$G^{\lambda_{i}}\big{[}f\big{]}(t):=\int_{-\infty}^{\infty}\mathcal{G}_{i}(t,s)f(s)\,ds=\sum_{j=1}^{n-1}\frac{1}{\Gamma_{ij}}G_{\gamma_{ij}}\big{[}f\big{]}(t)\,,$

where

$\Gamma_{ij}:=\prod_{k=1,\,k\not=j}^{n-1}(\gamma_{ij}-\gamma_{ik})\quad\text{and}\quad\mathcal{G}_{i}(t,s)=\sum_{j=1}^{n-1}\frac{1}{\Gamma_{ij}}g_{\gamma_{ij}}(t,s),$

with $g_{\gamma}$ is given by (1.7). Denote $\alpha_{ij}=\Re(\gamma_{ij})$ and
$\displaystyle\tilde{\alpha}_{ij}=\sum_{k=1}^{n-2}|\gamma_{ij}^{k}|$. For
$k\in\\{1,\dots,n\\}$, let $\delta_{\lambda_{k}}>0$ be such that

$0<m(\delta)<\dfrac{1-L_{0}^{\lambda_{k}}}{Q_{0}^{\lambda_{k}}}\quad\text{for
all $\delta\in(0,\delta_{\lambda_{k}})$ and
}m(\delta_{\lambda_{k}})=\dfrac{1-L_{0}^{\lambda_{k}}}{Q_{0}^{\lambda_{k}}}.$
(3.15)

For the next result and some consequences, we consider the submultiplicative
norms for a matrix $B=(b_{ik})$ defined by

$\|B\|_{1}=\max_{k}\sum_{i}^{n}|b_{ik}|\qquad\text{and}\qquad\|B\|_{\infty}=\max_{i}\sum_{k}^{n}|b_{ik}|.$

######  Theorem 3.4.

Consider the equation (2.7) with $r_{i}\in\mathcal{AP}(\mathbb{R})$ , for
$i=1,\dots,n-1$. Let $\lambda_{1},\dots,\lambda_{n}$ be the $n$ roots of the
polynomial $P(a,x)=0$ and $V$ the Vandermonde matrix of
$\lambda_{1},\dots,\lambda_{n}$. Assume that for every $k=1,\dots,n$ there is
a constant $M_{k}\in(0,\min\limits_{k}\delta_{\lambda_{k}})$ such that

$L_{0}^{\lambda_{k}}<1\,,\quad\text{ and }\quad
H_{\lambda_{k}}\leq\big{[}1-L_{0}^{\lambda_{k}}-m(M_{k})Q^{\lambda_{k}}_{0}\big{]}M_{k}$

and

$\max_{k}\left(M_{k}\big{(}m(M_{k})+1\big{)}\sum_{i=2}^{n}\big{[}(|\lambda_{k}|+1)^{i-1}-|\lambda_{k}|^{i-1}\big{]}\right)\\!\|V^{-1}\|_{1}<1.$
(3.16)

Then there exists a fundamental system of solutions $y_{k}$, $k=1,\dots,n$ to
equation (1.1) such that

$\begin{split}y_{k}(t)=&\,e^{\lambda_{k}t}\exp\bigg{(}-\sum_{j=1}^{n-1}{1\over\Gamma_{kj}\gamma_{kj}}G_{\gamma_{kj}}\big{[}P(r;\lambda_{k})+\mathcal{L}_{k}(\cdot,z_{k})+\mathcal{F}_{k}(\cdot,Z_{k})\big{]}(t)\bigg{)}\\\
&\times\exp\bigg{(}(-1)^{n}\prod_{j\in\mathcal{N}(k)}(\lambda_{j}-\lambda_{k})^{-1}\int_{0}^{t}\big{[}P(r;\lambda_{k})(s)+\mathcal{L}_{k}(s,z_{k}(s))+\mathcal{F}_{k}(s,Z_{k}(s))\big{]}\,ds\bigg{)},\end{split}$

where $\mathcal{L}_{i}(\cdot,z_{i})$ and $\mathcal{F}_{i}(\cdot,Z_{i})$ are
given by (2.10) and (2.11) with $\lambda=\lambda_{i}$, respectively, $z_{i}$
satisfies differential equation

$\mathcal{D}_{i}z+P(r;\lambda_{i})+\mathcal{L}_{i}(t,z)+\mathcal{F}_{i}\big{(}t,z,\dots,z^{(n-2)}\big{)}=0,$

the integral equation

$z=-G^{\lambda_{i}}[P(r;\lambda_{i})+\mathcal{L}_{i}(\cdot,z)+\mathcal{F}_{i}(\cdot,Z)]$

and $z_{i}\in\mathcal{AP}^{n-2}(\mathbb{R})$ for $i=1,\dots,n$.

###### Proof.

First, by using the choice of $M_{k}$ and the definition of
$\delta_{\lambda_{k}}$, one has that

$0<\big{[}1-L_{0}^{\lambda_{k}}-m(M_{k})Q_{0}^{\lambda_{k}}\big{]}M_{k}\leq\max\\{[1-L_{0}^{\lambda_{k}}-m(\delta)Q_{0}^{\lambda_{k}}]\delta\
\mid\ \delta\in(0,\delta_{\lambda_{k}})\\}\,,$

for each $k$, where $\delta_{\lambda_{k}}$ satisfies (3.15). It follows that
the hypothesis of Theorem 3.2 holds, then there exists a solution $y_{k}$ of
equation (1.1) for $\lambda=\lambda_{k}$, $k=1,\dots,n$.

It remains to prove that, $\\{y_{1},\dots,y_{n}\\}$ are linearly independent.
We shall show that the Wronskian of those functions, denoted by $W$, does not
vanish. By equation (2.3), one obtains for $i\geq 1$ that

$\displaystyle\frac{y_{k}^{(i)}(t)}{y_{k}(t)}$
$\displaystyle=\lambda_{k}^{i}+\sum_{j=1}^{i}{i\choose
j}\lambda_{k}^{i-j}\Big{(}f_{j-1}\big{(}z_{k}(t),z_{k}^{\prime}(t),\dots,z_{k}^{(j-2)}(t)\big{)}+z_{k}^{(j-1)}\Big{)}\,.$

where the polynomials $f_{j}$ are defined in (2.6). Then, denoting by
$A=(a_{ik})$ the matrix with entries

$a_{ik}=\begin{cases}\sum\limits_{j=1}^{i-1}{i-1\choose
j}\lambda_{k}^{i-1-j}\Big{(}f_{j-1}\big{(}z_{k}(t),z^{\prime}_{k}(t),\dots,z_{k}^{(j-2)}(t)\big{)}+z_{k}^{(j-1)}\Big{)}&i\geq
2\\\ 0&i=1\end{cases}\,,$

$W$ can be rewritten by

$W=(y_{1}y_{2}\cdots y_{n})\det\left[V(I+A)\right]=(y_{1}y_{2}\cdots
y_{n})\det[V]\det[I+V^{-1}A]\,,$

where $V$ is the Vandermonde matrix of $\lambda_{1},\dots,\lambda_{k}$. Thus,
it is enough to show that $I+V^{-1}A$ is an invertible matrix. It is well
known that if $\|A\|_{1}\|V^{-1}\|_{1}<1$ then the matrix $I+V^{-1}A$ is
invertible.

We now estimate the $\|A\|_{1}$. Since $f_{j}(0,\dots,0)=0$ for all $j$, and
using inequality (3.2), we obtain that

$\displaystyle|a_{ik}|$ $\displaystyle\leq\sum_{j=1}^{i-1}{i-1\choose
j}|\lambda_{k}|^{i-1-j}\big{(}m(M_{k})\|Z_{k}\|+\|Z_{k}\|\big{)}$
$\displaystyle\leq M_{k}(m(M_{k})+1)\sum_{j=1}^{i-1}{i-1\choose
j}|\lambda_{k}|^{i-1-j}$ $\displaystyle\leq
M_{k}(m(M_{k})+1)\big{[}(|\lambda_{k}|+1)^{i-1}-|\lambda_{k}|^{i-1}\big{]}$

for $\|Z_{k}\|\leq M_{k}$ and $i>1$ with
$Z_{k}=(z_{k},z_{k}^{\prime}\dots,z_{k}^{(n-2)})$. Notice that, if
$\lambda_{k}=0$ then $|a_{ik}|\leq M_{k}(m(M_{k})+1)$ for all $i$, so in
general one has

$\sum\limits_{i=1}^{n}|a_{ik}|=\sum\limits_{i=2}^{n}|a_{ik}|\leq
M_{k}(m(M_{k})+1)\sum_{i=2}^{n}\big{[}(|\lambda_{k}|+1)^{i-1}-|\lambda_{k}|^{i-1}\big{]}$

Thus, we obtain that

$\displaystyle\|A\|_{1}$
$\displaystyle\leq\,\max_{k}\left(M_{k}(m(M_{k})+1)\sum_{i=2}^{n}\big{[}(|\lambda_{k}|+1)^{i-1}-|\lambda_{k}|^{i-1}\big{]}\right).$

Therefore, by hypothesis, one obtains that $\|A\|_{1}\|V^{-1}\|_{1}<1$ ,
finishing the proof. ∎

In [26] it was shown that

$\|V^{-1}\|_{\infty}\leq\max_{k}\prod_{l=1,\,l\not=k}^{n}\frac{1+|\lambda_{l}|}{|\lambda_{k}-\lambda_{l}|}$

where equality holds if (but not only if) all complex numbers $\lambda_{k}$,
are on the same ray through the origin. It is well known that $\|B\|_{1}\leq
n\|B\|_{\infty}$. Hence, a different sufficient condition to obtain also the
previous result, but avoiding the computation of $V^{-1}$, is

$n\,\max_{k}\left(M_{k}\big{(}m(M_{k})+1\big{)}\sum_{i=2}^{n}\big{[}(|\lambda_{k}|+1)^{i-1}-|\lambda_{k}|^{i-1}\big{]}\right)\\!\max\left(\prod_{l=1,l\not=k}^{n}\frac{1+|\lambda_{l}|}{|\lambda_{k}-\lambda_{l}|}\right)<1\,.$

Another kind of perturbations are $r_{i}\in\mathcal{AAP}(\mathbb{R})$ for
$i=0,\dots,n-1$. Let us recall that
$\mathcal{AAP}^{n-2}(\mathbb{R},\mathbb{C})$ is a closed subspace of
$BC^{n-2}(\mathbb{R},\mathbb{C})$. From the ideas used in Theorem 3.2 and
Corollary 2.17 it readily follows the next result. In particular, the
decomposition of the coefficients $r_{i}$ , $i=0,\dots,n-1$ induces the direct
sum of the solution $z$. Direct computations from the definition of
$P(r;\lambda)$, $\mathcal{L}(t,z)$ and $\mathcal{F}(t,Z)$ in (2.9), (2.10) and
(2.11), respectively, binomial property of Bell’s polynomials (2.5) and
assuming that $r_{i}=\mu_{i}+\nu_{i}$ , $i=0,\dots,n-1$ where
$\mu_{i}\in\mathcal{AP}(\mathbb{R})$ and $\nu_{i}\in BC_{0}(\mathbb{R})$
(resp. $C_{00}(\mathbb{R})$) lead us to obtain that

$P(r;\lambda)=P(\mu;\lambda)+P(\nu;\lambda),$

with $\mu=(\mu_{0},\dots,\mu_{n})$, $\mu_{n}\equiv 0$,
$\nu=(\nu_{0},\dots,\nu_{n}),\quad\nu_{n}\equiv 0$,

$\mathcal{L}(\cdot,\theta+\psi)=\mathcal{L}_{\mu}(\cdot,\theta)+\mathcal{L}_{\nu}(\cdot,\theta)+\mathcal{L}(\cdot,\psi)$

and

$\mathcal{F}(\cdot,\Theta+\Psi)=\mathcal{F}_{\mu}(\cdot,\Theta)+\tilde{\mathcal{F}}_{\nu}(\cdot,\Theta)+\tilde{\mathcal{L}}_{\theta}(\cdot,\psi)+\tilde{\mathcal{F}}_{\theta}(\cdot,\Psi),$
(3.17)

where

$\mathcal{L}_{\mu}(t,\theta)=\sum_{k=1}^{n-1}\frac{1}{k!}\frac{\partial^{k}}{\partial
x^{k}}[P(\mu;\lambda)]\theta^{(k-1)}(t),\quad\mathcal{L}_{\nu}(t,\theta)=\sum_{k=1}^{n-1}\frac{1}{k!}\frac{\partial^{k}}{\partial
x^{k}}[P(\nu;\lambda)]\theta^{(k-1)}(t),$ (3.18)
$\mathcal{F}_{\mu}(t,\Theta)=\sum_{i=2}^{n}\sum_{j=0}^{n-i}{i+j\choose
j}(a_{i+j}+\mu_{i+j}(t))\lambda^{j}\big{(}B_{i}(\theta,\theta^{\prime},\dots,\theta^{(i-1)})-\theta^{(i-1)}\big{)}\,,$
(3.19)
$\tilde{\mathcal{F}}_{\nu}(t,\Theta)=\sum_{i=2}^{n}\sum_{j=0}^{n-i}{i+j\choose
j}\nu_{i+j}(t)\lambda^{j}\big{[}B_{i}(\theta,\theta^{\prime},\dots,\theta^{(i-1)})-\theta^{(i-1)}\big{]}\,,$
(3.20)
$\tilde{\mathcal{L}}_{\theta}(t,\psi)=\sum_{i=2}^{n}\sum_{j=0}^{n-i}{i+j\choose
j}(a_{i+j}+r_{i+j}(t))\lambda^{j}\sum_{k=1}^{i-1}{i\choose
k}B_{i-k}(\theta,\theta^{\prime},\dots,\theta^{(i-k-1)})\psi^{(k-1)}\,$ (3.21)

and

$\begin{split}\tilde{\mathcal{F}}_{\theta}(t,\psi)=&\,\sum_{i=2}^{n}\sum_{j=0}^{n-i}{i+j\choose
j}(a_{i+j}+r_{i+j}(t))\lambda^{j}\\\ &\times\sum_{k=2}^{i}{i\choose
k}B_{i-k}(\theta,\dots,\theta^{(i-k-1)})\big{[}B_{k}(\psi,\dots,\psi^{k-1})-\psi^{(k-1)}\big{]}\,.\end{split}$
(3.22)

Notice that $P(\mu;\lambda),\frac{\partial^{k}}{\partial
x^{k}}[P(\mu;\lambda)]\in\mathcal{AP}$ for all $k=1,\dots,n-1$ and
$\mathcal{L}_{\mu}(\cdot,\theta),\mathcal{F}_{\mu}(\cdot,\Theta)\in\mathcal{AP}$
if $\theta\in\mathcal{AP}^{n-2}$, in view of
$B_{i}(\theta,\theta^{\prime},\dots,\theta^{(i-1)})\in\mathcal{AP}$ for all
$i=1,\dots,n$. Furthermore, for any $z\in BC^{n-2}$,
$\mathcal{L}(\cdot,z)=\mathcal{L}_{\mu}(\cdot,z)+\mathcal{L}_{\nu}(\cdot,z)$
and
$\mathcal{F}(\cdot,z)=\mathcal{F}_{\mu}(\cdot,z)+\tilde{\mathcal{F}}_{\nu}(\cdot,z)$.

######  Theorem 3.5.

Consider the equation (1.1) with $r_{i}\in\mathcal{AAP}(\mathbb{R})$ (resp.
$\mathcal{AAP}_{0}(\mathbb{R})$), $i=0,\dots,n-1$. Assume that (3.7) holds.
Then, there exists a solution $y$ to equation (1.1) satisfying (3.9), where
$z$ is a solution to differential equation (2.7) and $z^{(i)}\in\mathcal{AAP}$
(resp. $\mathcal{AAP}_{0}(\mathbb{R})$) for $i=0,1,\dots,n-2$. Moreover, if
$r_{i}=\mu_{i}+\nu_{i}$ , $i=0,\dots,n-1$ where
$\mu_{i}\in\mathcal{AP}(\mathbb{R})$ and $\nu_{i}\in BC_{0}(\mathbb{R})$
(resp. $C_{00}(\mathbb{R})$), then $z=\theta+\psi$, where
$\theta\in\mathcal{AP}^{n-2}(\mathbb{R})$ and $\psi\in
BC_{0}^{n-2}(\mathbb{R})$ (resp. $C_{00}^{n-2}(\mathbb{R})$) satisfy

$\theta=-G[P(\mu;\lambda)+\mathcal{L}_{\mu}(\cdot,\theta)+\mathcal{F}_{\mu}(\cdot,\Theta)]$
(3.23)

and

$\psi=-G[P(\nu;\lambda)+\mathcal{L}_{\nu}(\cdot,\theta)+\tilde{\mathcal{F}}_{\nu}(\cdot,\Theta)+\mathcal{L}(\cdot,\psi)+\tilde{\mathcal{L}}_{\theta}(\cdot,\Psi)+\tilde{\mathcal{F}}_{\theta}(\cdot,\Psi)]$
(3.24)

Conversely, if $\theta\in\mathcal{AP}(\mathbb{R})$ and $\psi\in
BC_{0}(\mathbb{R})$ (resp. $C_{00}(\mathbb{R})$) satisfy equations (3.23) and
(3.24), respectively, then $z=\theta+\psi$ is solution of equation (1.1) in
$\mathcal{AAP}(\mathbb{R})$ (resp.$\mathcal{AAP}_{0}(\mathbb{R})$) .

###### Proof.

It is enough to prove the decomposition of the equation (3.10) in these two
equations (3.23) and (3.24). Note that, $P(r,\lambda)$ and ${\mathcal{L}}$ are
linear in the variable $r$ .

On the other hand, from property (2.5) it follows that

$\begin{split}B_{i}(\theta+\psi,\dots,\theta^{(i-1)}+\psi^{(i-1)})=&\,B_{i}(\theta,\dots,\theta^{(i-1)})\\\
&+\sum_{k=1}^{i}{i\choose
k}B_{i-k}\big{(}\theta,\dots,\theta^{(i-1)}\big{)}B_{k}(\psi,\dots,\psi^{(k-1)})\end{split}$

so that

$\begin{split}&B_{i}(\theta+\psi,\dots,\theta^{(i-1)}+\psi^{(i-1)})-[\theta^{(i-1)}+\psi^{(i-1)}]\\\
&=B_{i}(\theta,\dots,\theta^{(i-1)})-\theta^{(i-1)}+\sum_{k=1}^{i-1}{i\choose
k}B_{i-k}(\theta,\dots,\theta^{(i-1)})\psi^{(k-1)}\\\
&\quad+\sum_{k=1}^{i}{i\choose
k}B_{i-k}(\theta,\dots,\theta^{(i-1)})\big{[}B_{k}(\psi,\dots,\psi^{(k-1)})-\psi^{(k-1)}\big{]}.\end{split}$
(3.25)

Hence, we deduce that

$\begin{split}&(a_{i+j}+\mu_{i+j}+\nu_{i+j})\Big{(}B_{i}(\theta+\psi,\dots,\theta^{(i-1)}+\psi^{(i-1)})-[\theta^{(i-1)}+\psi^{(i-1)}]\Big{)}\\\
&=(a_{i+j}+\mu_{i+j})\big{[}B_{i}(\theta,\dots,\theta^{(i-1)})-\theta^{(i-1)}\big{]}+\nu_{i+j}\big{[}B_{i}(\theta,\dots,\theta^{(i-1)})-\theta^{(i-1)}\big{]}\\\
&\quad+(a_{i+j}+\mu_{i+j}+\nu_{i+j})\sum_{k=1}^{i-1}{i\choose
k}B_{i-k}(\theta,\dots,\theta^{(i-1)})\psi^{(k-1)}\\\
&\quad+(a_{i+j}+\mu_{i+j}+\nu_{i+j})\sum_{k=1}^{i}{i\choose
k}B_{i-k}(\theta,\dots,\theta^{(i-1)})\big{[}B_{k}(\psi,\dots,\psi^{(k-1)})-\psi^{(k-1)}\big{]}.\end{split}$
(3.26)

Therefore, we conclude (3.17). This finishes the proof. ∎

######  Remark 3.6.

In the same spirit of [23, Theorems 3 and 4], we could perturb
$r_{i}\in\mathcal{AP}(\mathbb{R},\mathbb{C})$, $i=0,\dots,n-1$ with
$L^{p}$-functions in equation (1.1), see [4, 13, 20, 22]. In other words,
equation (1.1) with $r_{i}\in\mathcal{AP}(\mathbb{R},\mathbb{C},p)$,
$i=0,\dots,n-1$. Let us stress that $L_{0}^{p}\cap BC(\mathbb{R},\mathbb{C})$
and $L^{p}(\mathbb{R})\cap BC(\mathbb{R},\mathbb{C})$ are not closed subspaces
of $BC(\mathbb{R},\mathbb{C})$, so that, we cannot obtain readily a version of
Theorem 3.5 in the subspaces $\mathcal{AP}(\mathbb{R},\mathbb{C},p)$ and
$\mathcal{AP}_{0}(\mathbb{R},\mathbb{C},p)$. However, we can find such a
solution $z$ in $\mathcal{AP}(\mathbb{R},\mathbb{C},p)$ or
$\mathcal{AP}_{0}(\mathbb{R},\mathbb{C},p)$ by exploiting the decomposition
property of coefficients $r_{i}=\mu_{i}+\nu_{i}$
($\mu_{i}\in\mathcal{AP}(\mathbb{R})$ and either $\nu_{i}\in
BC_{0}(\mathbb{R})$ or $C_{00}(\mathbb{R})$ or $L_{0}^{p}$ or
$L^{p}(\mathbb{R})$), $i=0,\dots,n-1$, studying equations (3.23) and (3.24),
and applying Proposition 2.16 to (3.23) and then Lemma 2.19. For simplicity,
we present a result only in case $n=3$ and next subsection. We omit its proof.

###  3.2. The case $n=3$

In this subsection we study Theorem 3.2 and present some consequences of
previous result (Theorem 3.5) in case $n=3$, namely,

$y^{\prime\prime\prime}+(a_{2}+r_{2}(t))y^{\prime\prime}+(a_{1}+r_{1}(t))y^{\prime}+(a_{0}+r_{0}(t))y=0.$
(3.27)

We assume that $\lambda$, $\lambda_{1}$ and $\lambda_{2}$ are the roots of
$P(a;x)=x^{3}+a_{2}x^{2}+a_{1}x+a_{0}$, so that
$\gamma_{i}=\lambda_{i}-\lambda$, $\alpha_{i}=\Re\gamma_{i}$, $i=1,2$,
$\tilde{\alpha}_{j}=1+|\gamma_{j}|$ and
$\Gamma_{i}=(-1)^{i}(\lambda_{2}-\lambda_{1})$. Notice that $P(r;\lambda)$,
$\mathcal{L}(\cdot,z)$ and $\mathcal{F}(\cdot,z,z^{\prime})$ are given by
(1.3)-(1.5), respectively. It is readily checked that $f_{1}(x_{1})=x_{1}^{2}$
and $f_{2}(x_{1},x_{2})=x_{1}^{3}+3x_{1}x_{2}$. Hence, it follows that for any
$\delta>0$

  * •

$|x|\leq\delta$ and $|y|\leq\delta$ implies $|f_{1}(x)-f_{1}(y)|\leq
2\delta|x-y|$;

  * •

$|x_{1}|+|x_{2}|\leq\delta$ and $|y_{1}|+|y_{2}|\leq\delta$ implies

$|f_{2}(x_{1},x_{2})-f_{2}(y_{1},y_{2})|\leq(3\delta^{2}+3\delta)\big{[}|x_{1}-y_{1}|+x_{2}-y_{2}|\big{]}.$

Thus, according to Remark 3.3 we can choose
$m(\delta)=\max\\{2\delta,3\delta^{2}+3\delta\\}=3\delta^{2}+3\delta$ to apply
Theorem 3.2. Furthermore, we obtain that for
$0<\beta<\min\\{|\alpha_{1}|,|\alpha_{2}|\\}$

$L_{\beta}=\sum_{i=1}^{2}\frac{1+|\gamma_{i}|}{|\lambda_{1}-\lambda_{2}|}\Big{[}\big{\|}I_{\alpha_{i}-{\rm
sgn}(\alpha_{i})\beta}[2\lambda
r_{2}+r_{1}]\big{\|}_{\infty}+\big{\|}I_{\alpha_{i}-{\rm
sgn}(\alpha_{i})\beta}[r_{2}]\big{\|}_{\infty}\Big{]}$

and

$Q_{\beta}=\sum_{i=1}^{2}\frac{1+|\gamma_{i}|}{|\lambda_{1}-\lambda_{2}|}\bigg{[}\frac{1+3|\lambda|+|a_{2}|}{|\alpha_{i}-{\rm
sgn}(\alpha_{i})\beta|}+\big{\|}I_{\alpha_{i}-{\rm
sgn}(\alpha_{i})\beta}[r_{2}]\big{\|}_{\infty}\bigg{]}.$

Recall that $a_{3}=1$ and $r_{3}\equiv 0$. Now, assume that $L_{0}<1$ and
consider the function $g(\delta)=[1-L_{0}-(3\delta^{2}+3\delta)Q_{0}]\delta$.
It is readily checked that
$\delta_{\lambda}=\frac{1}{2}\Big{(}\sqrt{1+\frac{4(1-L_{0})}{3Q_{0}}}-1\Big{)}>0$
satisfies $3Q_{0}\delta^{2}+3Q_{0}\delta+L_{0}-1=0$, namely,
$3\delta^{2}+3\delta=\frac{1-L_{0}}{Q_{0}}$ and that $g$ attains its maximum
in $(0,\delta_{\lambda})$ at
$M=\frac{1}{3}\Big{(}\sqrt{1+\frac{1-L_{0}}{Q_{0}}}-1\Big{)}$, namely,

$g(M)=M\Big{[}\frac{2}{3}(1-L_{0})-Q_{0}M\Big{]}=\max\\{g(\delta):\delta\in(0,\delta_{\lambda})\\}.$

Hence, if

$\|G[P(r;\lambda)]\|_{\infty}+\|G[P(r;\lambda)]^{\prime}\|_{\infty}\leq
M\Big{[}\frac{2}{3}(1-L_{0})-Q_{0}M\Big{]},$

then there exists a solution $y_{\lambda}$ to (1.1) with $n=3$ satisfying
(3.9) with $z$ solution to

$z=-G\big{[}\lambda^{2}r_{2}+\lambda r_{1}+r_{0}+(2\lambda
r_{2}+r_{1})z+r_{2}z^{\prime}+(3\lambda+a_{2}+r_{2})z^{2}+3zz^{\prime}+z^{3}\big{]}.$
(3.28)

In addition, if there exists a constant
$0<\beta<\min\\{|\alpha_{1}|,|\alpha_{2}|\\}$ such that
$L_{\beta}<\frac{1}{2}$ then $z$ satisfies
$z,z^{\prime}=O\big{(}(I_{\beta}+I_{-\beta})[P(r;\lambda)]\big{)}$.

On the other hand, assuming $r_{i}=\mu_{i}+\nu_{i}$, $i=0,1,2$ with
$\mu_{i}\in\mathcal{AP}(\mathbb{R})$ and $\nu_{i}\in BC_{0}$, $i=0,1,2$,
direct computations lead us to obtain that

$\theta=-G\big{[}\lambda^{2}\mu_{2}+\lambda\mu_{1}+\mu_{0}+(2\lambda\mu_{2}+\mu_{1})\theta+\mu_{2}\theta^{\prime}+(3\lambda+a_{2}+\mu_{2})\theta^{2}+3\theta\theta^{\prime}+\theta^{3}\big{]}$
(3.29)

and

$\begin{split}\psi=-G\Big{[}&P_{\nu}(\theta;\lambda)+r_{\theta}\psi+(r_{2}+\theta)\psi^{\prime}+(3\lambda+3\theta+a_{2}+r_{2})\psi^{2}+3\psi\psi^{\prime}+\psi^{3}\Big{]},\end{split}$
(3.30)

according to equations (3.23) and (3.24) in Theorem 3.5, where

$P_{\nu}(\theta;\lambda)=\lambda^{2}\nu_{2}+\lambda\nu_{1}+\nu_{0}+(2\lambda\nu_{2}+\nu_{1})\theta+\nu_{2}\theta^{\prime}+\nu_{2}\theta^{2}$
$\text{and}\qquad r_{\theta}=2\lambda
r_{2}+r_{1}+2(3\lambda+a_{2}+r_{2})\theta+3\theta^{2}+3\theta^{\prime}.$

Hence, for simplicity, we denote

$L_{\beta,\theta}=\sum_{i=1}^{2}\frac{1+|\gamma_{i}|}{|\lambda_{1}-\lambda_{2}|}\Big{[}\big{\|}I_{\alpha_{i}-{\rm
sgn}(\alpha_{i})\beta}[r_{\theta}]\big{\|}_{\infty}+\big{\|}I_{\alpha_{i}-{\rm
sgn}(\alpha_{i})\beta}[r_{2}+\theta]\big{\|}_{\infty}\Big{]}$

and

$Q_{\beta,\theta}=\sum_{i=1}^{2}\frac{1+|\gamma_{i}|}{|\lambda_{1}-\lambda_{2}|}\bigg{[}\frac{1+3|\lambda|+|a_{2}|}{|\alpha_{i}-{\rm
sgn}(\alpha_{i})\beta|}+\big{\|}I_{\alpha_{i}-{\rm
sgn}(\alpha_{i})\beta}[r_{2}+3\theta]\big{\|}_{\infty}\bigg{]}.$

Now, assume that $L_{0,\theta}<1$ and consider the function
$g_{\theta}(\delta)=[1-L_{0,\theta}-(3\delta^{2}+3\delta)Q_{0,\theta}]\delta$.
It is readily checked that
$\delta_{\lambda,\theta}=\frac{1}{2}\Big{(}\sqrt{1+\frac{4(1-L_{0,\theta})}{3Q_{0,\theta}}}-1\Big{)}>0$
satisfies $3Q_{0,\theta}\delta^{2}+3Q_{0,\theta}\delta+L_{0,\theta}-1=0$,
namely, $3\delta^{2}+3\delta=\frac{1-L_{0,\theta}}{Q_{0,\theta}}$ and that
$g_{\theta}$ attains its maximum in $(0,\delta_{\lambda,\theta})$ at
$M_{\theta}=\frac{1}{3}\Big{(}\sqrt{1+\frac{1-L_{0,\theta}}{Q_{0,\theta}}}-1\Big{)}$,
namely,

$g_{\theta}(M_{\theta})=M_{\theta}\Big{[}\frac{2}{3}(1-L_{0,\theta})-Q_{0,\theta}M_{\theta}\Big{]}=\max\\{g_{\theta}(\delta):\delta\in(0,\delta_{\lambda,\theta})\\}.$

######  Theorem 3.7.

Consider the equation (3.27) with
$r_{i}=\mu_{i}+\nu_{i}\in\mathcal{AAP}(\mathbb{R})$ (resp.
$\mathcal{AAP}_{0}(\mathbb{R})$, $\mathcal{AP}(\mathbb{R},\mathbb{C},p)$ or
$\mathcal{AP}_{0}(\mathbb{R},\mathbb{C},p)$), with
$\mu_{i}\in\mathcal{AP}(\mathbb{R})$ and $\nu_{i}\in BC_{0}(\mathbb{R})$
(resp. $C_{00}(\mathbb{R})$, $L_{0}^{p}$ or $L^{p}(\mathbb{R})$), $i=0,1,2$.
Assume that there is a solution $\theta\in\mathcal{AP}(\mathbb{R})$ to
equation (3.29), so that $L_{0,\theta}<1$ and

$\|G[P_{\nu}(\theta;\lambda)]\|_{\infty}+\|G[P_{\nu}(\theta;\lambda)]^{\prime}\|_{\infty}\leq
M_{\theta}\Big{[}\frac{2}{3}(1-L_{0,\theta})-Q_{0,\theta}M_{\theta}\Big{]}.$

Then, there exists a solution $y$ to equation (3.27) satisfying (1.2), where
$z=\theta+\psi$ is a solution to integral equation (3.28),
$z,z^{\prime}\in\mathcal{AAP}$ (resp. $\mathcal{AAP}_{0}(\mathbb{R})$,
$\mathcal{AP}(\mathbb{R},\mathbb{C},p)$ or
$\mathcal{AP}_{0}(\mathbb{R},\mathbb{C},p)$) and $\psi\in
BC_{0}^{1}(\mathbb{R})$ (resp. $C_{00}^{1}(\mathbb{R})$, $L_{0}^{p}$ or
$L^{p}(\mathbb{R})$) satisfies integral equation (3.30). In addition, if there
exists a constant $0<\beta<\min\\{|\alpha_{1}|,|\alpha_{2}|\\}$ such that
$L_{\beta,\theta}<\frac{1}{2}$ then $\psi$ satisfies
$\psi,\psi^{\prime}=O\big{(}(I_{\beta}+I_{-\beta})[P_{\nu}(\theta;\lambda)]\big{)}$.

##  4\. Comments and examples

Consider the equation for $n=3$

$y^{\prime\prime\prime}+r_{2}(t)y^{\prime\prime}-(1-r_{1}(t))y^{\prime}+r_{0}(t)y=0.$
(4.1)

We have that $\lambda_{1}=0$, $\lambda_{2}=1$ and $\lambda_{3}=-1$ and we
shall study the case $r_{2}=r_{1}\equiv 0$. Then, we obtain that
$\gamma_{11}=\alpha_{11}=1$, $\gamma_{12}=\alpha_{12}=-1$ and
$\Gamma_{1i}=(-1)^{i+1}2$. Also, it is readily checked that
$P(r;\lambda_{i})=r_{0}(t)$, $L_{\beta}^{\lambda_{i}}=0$, for all
$0<\beta<\min\\{|\alpha_{i1}|,|\alpha_{i2}|\\}=1$ and
$Q_{\beta}^{\lambda_{1}}=\frac{2}{1-\beta}$, $i=1,2,3$, since
$\gamma_{21}=\alpha_{21}=-1$, $\gamma_{22}=\alpha_{22}=-2$,
$\gamma_{31}=\alpha_{31}=1$ and $\gamma_{32}=\alpha_{32}=2$. Hence,
$M_{\lambda_{1}}=\frac{1}{3}\Big{(}\sqrt{1+\frac{1-L_{0}^{\lambda_{1}}}{Q_{0}^{\lambda_{1}}}}-1\Big{)}=\frac{\sqrt{6}-2}{6}\approx
0.0749$ and

$g_{\lambda_{1}}(M_{\lambda_{1}})=M_{\lambda_{1}}\Big{[}\frac{2}{3}(1-L_{0}^{\lambda_{1}})-Q_{0}^{\lambda_{1}}M_{\lambda_{1}}\Big{]}=\frac{3\sqrt{6}-7}{9}\approx
0.0387.$

If
$\|G^{\lambda_{1}}[r_{0}]\|_{\infty}+\|G^{\lambda_{1}}[r_{0}]^{\prime}\|_{\infty}\leq\frac{3\sqrt{6}-7}{9},$
then Theorem 3.2 applies, so that there exists a solution $y_{1}$ to (4.1)
satisfying

$\begin{split}y_{1}(t)=&\exp\bigg{(}\int_{0}^{t}\big{[}r_{0}(s)+3z_{1}(s)z_{1}^{\prime}(s)+z^{3}(s)\big{]}\,ds\bigg{)}\\\
&\times\exp\bigg{(}-\frac{1}{2}\sum_{j=1}^{2}G_{\gamma_{1j}}\big{[}r_{0}+3z_{1}z^{\prime}_{1}+z_{1}^{3}\big{]}(t)\bigg{)},\end{split}$

where $z_{1}$ satisfies the integral equation
$z=-G^{\lambda_{1}}\big{[}r_{0}+3zz^{\prime}+z^{3}\big{]}$, with
$G^{\lambda_{1}}=\frac{1}{2}G_{1}-\frac{1}{2}G_{-1}$. Furthermore,
$z_{1},z_{1}^{\prime}=O([I_{\beta}+I_{-\beta}][r_{0}])$ with $0<\beta<1$.

On the other hand, $\Gamma_{2i}=(-1)^{i+1}$ and
$Q_{\beta}^{\lambda_{2}}=\frac{8}{1-\beta}+\frac{12}{2-\beta}$. Hence, we
choose
$M_{\lambda_{2}}=\frac{1}{3}\Big{(}\sqrt{1+\frac{1-L_{0}^{\lambda_{2}}}{Q_{0}^{\lambda_{2}}}}-1\Big{)}=\frac{\sqrt{105}-10}{30}\approx
0.0082$ and

$g_{\lambda_{2}}(M_{\lambda_{2}})=M_{\lambda_{2}}\Big{[}\frac{2}{3}(1-L_{0}^{\lambda_{2}})-Q_{0}^{\lambda_{2}}M_{\lambda_{2}}\Big{]}=\frac{21\sqrt{105}-215}{45}\approx
0.0041.$

If
$\|G^{\lambda_{2}}[r_{0}]\|_{\infty}+\|G^{\lambda_{2}}[r_{0}]^{\prime}\|_{\infty}\leq\frac{21\sqrt{105}-215}{45},$
then Theorem 3.2 applies, so that there exists a solution $y_{2}$ to (4.1)
satisfying

$\begin{split}y_{2}(t)=&\,e^{t}\exp\bigg{(}-\frac{1}{2}\int_{0}^{t}\big{[}r_{0}(s)+3z_{2}^{2}(s)+3z_{2}(s)z_{2}^{\prime}(s)+z_{2}^{3}(s)\big{]}\,ds\bigg{)}\\\
&\times\exp\bigg{(}-\sum_{j=1}^{2}\frac{1}{\Gamma_{2j}\gamma_{2j}}G_{\gamma_{2j}}\big{[}r_{0}+3z_{2}^{2}+3z_{2}z^{\prime}_{2}+z_{2}^{3}\big{]}(t)\bigg{)},\end{split}$

where $z_{2}$ satisfies the integral equation
$z=-G^{\lambda_{2}}\big{[}r_{0}+3z^{2}+3zz^{\prime}+z^{3}\big{]}$, with
$G^{\lambda_{2}}=G_{-1}-G_{-2}$. Furthermore,
$z_{2},z_{2}^{\prime}=O([I_{\beta}+I_{-\beta}][r_{0}])$ with $0<\beta<1$.

For $k=3$, we obtain that $\Gamma_{3i}=(-1)^{i}$ and
$Q_{\beta}^{\lambda_{3}}=\frac{8}{1-\beta}+\frac{12}{2-\beta}$. Hence, we
choose
$M_{\lambda_{3}}=\frac{1}{3}\Big{(}\sqrt{1+\frac{1-L_{0}^{\lambda_{3}}}{Q_{0}^{\lambda_{3}}}}-1\Big{)}=\frac{\sqrt{105}-10}{30}\approx
0.0082$ and

$g_{\lambda_{2}}(M_{\lambda_{3}})=M_{\lambda_{3}}\Big{[}\frac{2}{3}(1-L_{0}^{\lambda_{3}})-Q_{0}^{\lambda_{3}}M_{\lambda_{3}}\Big{]}=\frac{21\sqrt{105}-215}{45}\approx
0.0041.$

If
$\|G^{\lambda_{3}}[r_{0}]\|_{\infty}+\|G^{\lambda_{3}}[r_{0}]^{\prime}\|_{\infty}\leq\frac{21\sqrt{105}-215}{45},$
then Theorem 3.2 applies, so that there exists a solution $y_{3}$ to (4.1)
satisfying

$\begin{split}y_{3}(t)=&\,e^{t}\exp\bigg{(}-\frac{1}{2}\int_{0}^{t}\big{[}r_{0}(s)-3z_{3}^{2}(s)+3z_{3}(s)z_{3}^{\prime}(s)+z_{3}^{3}(s)\big{]}\,ds\bigg{)}\\\
&\times\exp\bigg{(}-\sum_{j=1}^{2}\frac{1}{\Gamma_{3j}\gamma_{3j}}G_{\gamma_{3j}}\big{[}r_{0}-3z_{3}^{2}+3z_{3}z^{\prime}_{3}+z_{3}^{3}\big{]}(t)\bigg{)},\end{split}$

where $z_{3}$ satisfies the integral equation
$z=-G^{\lambda_{3}}\big{[}r_{0}-3z^{2}+3zz^{\prime}+z^{3}\big{]}$, with
$G^{\lambda_{3}}=-G_{1}+G_{2}$. Furthermore,
$z_{3},z_{3}^{\prime}=O([I_{\beta}+I_{-\beta}][r_{0}])$ with $0<\beta<1$.

On the other hand, it is readily checked that the Vandermonde matrix is

$V=\left(\begin{array}[]{crr}1&1&1\\\ 0&1&-1\\\
0&1&1\end{array}\right)\qquad\text{and}\qquad
V^{-1}=\left(\begin{array}[]{ccc}1&0&-1\\\ 0&1/2&1/2\\\
0&-1/2&1/2\end{array}\right)$

its inverse, so that $\|V^{-1}\|_{1}=\|V^{-1}\|_{\infty}=2$. Furthermore, it
is readily checked that

$\sum_{i=2}^{3}\big{[}(|\lambda_{k}|+1)^{i-1}-|\lambda_{k}|^{i-1}\big{]}=\begin{cases}2&\text{if
}k=1\\\ 4&\text{if }k=2,3\end{cases}$
$M_{k}\big{(}m(M_{k})+1\big{)}=\begin{cases}\dfrac{6\sqrt{6}-8}{9}&\text{if
}k=1\\\\[14.22636pt] \dfrac{21\sqrt{105}-200}{225}&\text{if
}k=2,3\end{cases},$ $\dfrac{6\sqrt{6}-8}{9}\approx
0.093\qquad\text{and}\qquad\dfrac{21\sqrt{105}-200}{225}\approx 0.0084.$

Thus, assumption (3.16) is fulfilled and we conclude that
$\\{y_{1},y_{2},y_{3}\\}$ are linearly independent.

Now, we shall study equations (3.29) and (3.30) for $\lambda=\lambda_{1}=0$.
Assume $r_{0}=\mu_{0}+\nu_{0}$ with $\mu_{0}\in\mathcal{AP}(\mathbb{R})$ and
$\nu_{0}\in BC_{0}$. For simplicity we omit the dependency of $\lambda_{1}$.
Direct computations lead us to obtain that

$\theta=-G\big{[}\mu_{0}+3\theta\theta^{\prime}+\theta^{3}\big{]}$ (4.2)

and

$\psi=-G\Big{[}\nu_{0}+(3\theta^{2}+3\theta^{\prime})\psi+\theta\psi^{\prime}+3\theta\psi^{2}+3\psi\psi^{\prime}+\psi^{3}\Big{]}.$
(4.3)

According to previous computations, if
$\|G[\mu_{0}]\|_{\infty}+\|G[\mu_{0}]^{\prime}\|_{\infty}\leq\frac{3\sqrt{6}-7}{9}=g_{\lambda_{1}}(M_{\lambda_{1}}),$
then there exists a solution $\theta\in\mathcal{AP}^{1}$ to (4.2), with
$\|\theta\|_{\infty}+\|\theta^{\prime}\|_{\infty}\leq\frac{\sqrt{6}-2}{6}=M_{\lambda_{1}}$
and $\theta=O([I_{\beta}+I_{-\beta}][\mu_{0}])$ with $0<\beta<1$. Hence,
$P_{\nu}(\theta;\lambda)=\nu_{0}$, $r_{\theta}=3\theta^{2}+3\theta^{\prime}$,

$L_{\beta,\theta}=\big{\|}I_{1-\beta}[3\theta^{2}+3\theta^{\prime}]\big{\|}_{\infty}+\big{\|}I_{1-\beta}[\theta]\big{\|}_{\infty}+\big{\|}I_{-1+\beta}[3\theta^{2}+3\theta^{\prime}]\big{\|}_{\infty}+\big{\|}I_{-1+\beta}[\theta]\big{\|}_{\infty}$

and

$Q_{\beta,\theta}=\frac{1}{|1-\beta|}+\big{\|}I_{1-\beta}[3\theta]\big{\|}_{\infty}+\frac{1}{|-1+\beta|}+\big{\|}I_{-1+\beta}[3\theta]\big{\|}_{\infty}.$

Notice that according to Remark 2.15 we deduce that

$L_{0,\theta}\leq
6\|\theta\|_{\infty}^{2}+6\|\theta^{\prime}\|_{\infty}+2\|\theta\|_{\infty}\leq
6M^{2}_{\lambda_{1}}+6M_{\lambda_{1}}=\frac{\sqrt{6}-1}{3}<\frac{1}{2}<1$

so that,
$M_{\theta}=\frac{1}{3}\Big{(}\sqrt{1+\frac{1-L_{0,\theta}}{Q_{0,\theta}}}-1\Big{)}$
is well defined and

$g_{\theta}(M_{\theta})=M_{\theta}\Big{[}\frac{2}{3}(1-L_{0,\theta})-Q_{0,\theta}M_{\theta}\Big{]}=\max\\{g_{\theta}(\delta):\delta\in(0,\delta_{\lambda,\theta})\\},$

where
$\delta_{\lambda,\theta}=\frac{1}{2}\Big{(}\sqrt{1+\frac{4(1-L_{0,\theta})}{3Q_{0,\theta}}}-1\Big{)}>0$
and it satisfies $3\delta^{2}+3\delta=\frac{1-L_{0,\theta}}{Q_{0,\theta}}$
Furthermore, we get that

$2\leq
Q_{0,\theta}=2+\|I_{1}[3\theta]\|_{\infty}+\|I_{-1}[3\theta]\|_{\infty}\leq
2+6\|\theta\|_{\infty}\leq\sqrt{6}$

and $1-L_{0,\theta}\geq\frac{4-\sqrt{6}}{3}$. This implies that

$\frac{1}{2}\geq\frac{1}{Q_{0,\theta}}\geq\frac{1-L_{0,\theta}}{Q_{0,\theta}}\geq\frac{\frac{4-\sqrt{6}}{3}}{\sqrt{6}}=\frac{2\sqrt{6}-3}{9}$

and

$\frac{\sqrt{6}-2}{6}=\frac{1}{3}\bigg{(}\sqrt{1+\frac{1}{2}}-1\bigg{)}\geq
M_{\theta}\geq\frac{1}{3}\bigg{(}\sqrt{1+\frac{2\sqrt{6}-3}{9}}-1\bigg{)}=\frac{\sqrt{6+2\sqrt{6}}}{9}-\frac{1}{3}.$

Taking into account previous inequalities we obtain that

$\frac{2}{9}\Big{(}\sqrt{6+2\sqrt{6}}-3\Big{)}\leq
M_{\theta}Q_{0,\theta}\leq\sqrt{6}\cdot\frac{\sqrt{6}-2}{6}$

and

$g_{\theta}(M_{\theta})\geq\frac{\sqrt{6+2\sqrt{6}}}{9}-\frac{1}{3}\bigg{[}\frac{2}{3}\cdot\frac{4-\sqrt{6}}{3}-\frac{6-2\sqrt{6}}{6}\bigg{]}=\frac{\sqrt{6}-1}{81}\Big{(}\sqrt{6+2\sqrt{6}}-3\Big{)}.$

If
$\|G[\nu_{0}]\|_{\infty}+\|G[\nu_{0}]^{\prime}\|_{\infty}\leq\frac{\sqrt{6}-1}{81}\Big{(}\sqrt{6+2\sqrt{6}}-3\Big{)},$
then Theorem 3.7 applies, namely, there exists a solution $\psi\in BC_{0}$ to
(4.3), with $\|\psi\|_{\infty}+\|\psi^{\prime}\|_{\infty}\leq M_{\theta}$ and
$\psi=O([I_{\beta}+I_{-\beta}][\nu_{0}])$ with $0<\beta<1$. Thus, there exists
a solution $y$ to equation (1.1) satisfying (3.9), where $z=\theta+\psi$.

Suppose that $r_{0}=\mu_{0}+\nu_{0}$ with $\mu_{0}(t)=\eta_{1}\left[2+\cos
t+\cos(\sqrt{2}t)\right]$ and $\nu_{0}(t)={\eta_{2}\over 1+t^{2}}$, where
$\eta_{1},\eta_{2}\geq 0$. It is clear that $\mu_{0}$ and $\nu_{0}$ are non-
negative functions, $\mu_{0}\in\mathcal{AP}(\mathbb{R},\mathbb{C})$,
$\nu_{0}\in C_{00}(\mathbb{R},\mathbb{C})\cap L^{1}(\mathbb{R})$. Direct
computations show that

$G^{\lambda_{1}}[\mu_{0}](t)=\frac{1}{2}G_{1}[\mu_{0}](t)-\frac{1}{2}G_{-1}[\mu_{0}](t)=--2\eta_{1}-\frac{\eta_{1}}{2}\cos
t-\frac{\eta_{1}}{3}\cos(\sqrt{2}t),$

since for any $\lambda\in\mathbb{R}$ and integration by parts

$\int_{t}^{\mp\infty}e^{\mp(t-s)}\cos(\lambda s)\,ds=\mp\cos(\lambda
t)\pm\lambda\int_{t}^{\mp\infty}e^{\mp(t-s)}\sin(\lambda
s)\,ds={\mp\cos(\lambda t)-\lambda\sin(\lambda t)\over 1+\lambda^{2}}.$

Furthermore, we get that

$G^{\lambda_{1}}[\mu_{0}]^{\prime}(t)=\frac{\eta_{1}}{2}\sin
t+\frac{\eta_{1}}{3}\sqrt{2}\sin(\sqrt{2}t),$

and

$\|G^{\lambda_{1}}[\mu_{0}]\|_{\infty}+\|G^{\lambda_{1}}[\mu_{0}]^{\prime}\|_{\infty}\leq\frac{17}{6}\eta_{1}+\eta_{1}\Big{(}\frac{1}{2}+\frac{\sqrt{2}}{3}\Big{)}\leq\frac{3\sqrt{6}-7}{9}$

Theorem 3.2 applies. $\|G_{\pm 1}[\nu_{0}]\|_{\infty}\leq\|\nu_{0}\|_{\infty}$
and $\|\nu_{0}\|_{\infty}=\eta_{2}$, so that
$\|G^{\lambda_{1}}[\nu_{0}]\|_{\infty}\leq\|\nu_{0}\|_{\infty}$ and
$\|G^{\lambda_{1}}[\nu_{0}]^{\prime}\|_{\infty}\leq\|\nu_{0}\|_{\infty}$.
Hence, it is enough

$2\eta_{2}\leq\frac{\sqrt{6}-1}{81}\Big{(}\sqrt{6+2\sqrt{6}}-3\Big{)}$

to apply Theorem 3.7. Similar computations can be perform for $\lambda_{2}$
and $\lambda_{3}$.

Acknowledgements

The second author has been partially supported by grant Fondecyt Regular N∘
1201884\. The third author has been supported by grant Fondecyt Regular N∘
1170466.

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