# A short proof for Hopf bifurcation in Gurtin-MacCamy’s population dynamics
model

Arnaud Ducrot(a) Hao Kang(a) and Pierre Magal(b)
(a)Normandie Univ, UNIHAVRE, LMAH, FR-CNRS-3335, ISCN, 76600 Le Havre, France,
(b)Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France.
CNRS, IMB, UMR 5251, F-33400 Talence, France. Corresponding author. e-mail:
<EMAIL_ADDRESS>
###### Abstract

In this paper, we provide a short proof for the Hopf bifurcation theorem in
the Gurtin-MacCamy’s population dynamics model. Here we use the Crandall and
Rabinowitz’s approach, based on the implicit function theorem. Compared with
previous methods, here we require the age-specific birth rate to be slightly
smoother (roughly of bounded variation), but we have a huge gain for the
length of the proof.

##  1 Age-Structured Models

We consider the existence of periodic solutions for the following equation,

$\begin{cases}(\partial_{t}+\partial_{a})u(t,a)=-m\,u(t,a),\quad
a\in(0,+\infty),\\\ u(t,0)=f\left(\nu,\int_{0}^{\infty}b(a)u(t,a)da\right),\\\
u(0,\cdot)=\psi\in L^{1}_{+}((0,\infty),\mathbb{R}),\end{cases}$ (1.1)

where $m>0$ and $b\in L^{\infty}_{+}(0,\infty)$ are mortality rate and
fertility rate of the population respectively, the nonlinear function $f\in
C^{2}\left(\mathbb{R}\times\mathbb{R},\mathbb{R}\right)$ describes the birth
limitations for the population while $\nu\in\mathbb{R}$ is regarded as a
bifurcation parameter. Here $u(t,a)$ denotes the density of a population at
time $t$ with age $a$. This equation is referred as the Gurtin-MacCamy’s age-
structured equation and was introduced in its nonlinear form by Gurtin and
MacCamy in [8] to study temporal evolution of biological populations.

The existence of nontrivial periodic solution induced by Hopf bifurcation has
been observed in various specific age-structured models (Cushing [3, 4], Prüss
[16], Swart [17], Kostava and Li [11], Bertoni [1], Magal and Ruan [14]). In
this paper we shall use the implicit function theorem to establish the Hopf
bifurcation theorem that is used to obtain the existence of nontrivial
periodic solutions of the age-structured model (1.1), that is, a nontrivial
periodic solution bifurcated from the equilibrium of (1.1) when the
bifurcation parameter $\nu$ takes some critical values.

For two dimensional ordinary differential equations Crandall and Rabinowitz
[2] requires less regularity ($C^{2}$-right hand side) than the standard
result by Hale and Kocack [9] (which requires $C^{3}$-right hand side) and
Hassard, Kazarinoff and Wan [10] (which requires $C^{4}$-right hand side).
Here we assume that the function $f$ is only $C^{2}$ which corresponds to the
regularity imposed by Crandall and Rabinowitz [2] for their result applied to
ordinary differential equations. Such a regularity assumption has been
mentioned already in Liu, Magal and Ruan [12, see Remark 2.5]. In the context
of partial differential equations Crandall and Rabinowitz [2] original theorem
only applies to parabolic PDE since their proof strongly uses the fact that
the semigroup is generated by a sectorial operator $A:D(A)\subset X\to X$ on a
Banach space $X$ and verifies that the map $t\to e^{At}x$ (with $x\in X$) is
of class $C^{1}$ on $(0,\infty)$. Here we are working with hyperbolic
operator, therefore such a property is not satisfied. Nevertheless by
exploiting the special structure of the system, and imposing some extra
regularity for the birth function $a\to b(a)$ (i.e. $b$ has bounded variation
on $[0,\infty)$) we are still able to apply Crandall and Rabinowitz’s ideas.

Note that the previous result by Magal and Ruan [14] and Liu, Magal and Ruan
[12] only assume that the birth function $b$ belongs to
$L^{\infty}(0,\infty)$. Here the fact that $a\to b(a)$ has bounded variation
on each $[0,\infty)$, means that $a\to b(a)$ has a finite number of
discontinuity point on each bounded interval on $[0,\infty)$ and that $b(a)$
is continuous in between two successive discontinuity points. Such an
assumption is sufficient for most practical examples. Finally this paper is
closely related to the work of Cushing [3] in which he considered an equation
with age and delay at birth. In [3], the function $a\to b(a)$ is assumed to be
of class $C^{1}$ which is stronger than our bounded variation assumption.

This paper is organized as follows. In Section 2, we give the well-posedness
result of (1.1). In Section 3, we provide the assumptions for our Hopf
bifurcation theorem, while Section 4 is devoted to state and prove the Hopf
bifurcation theorem.

##  2 Well-posedness

Set

$X=\mathbb{R}\times L^{1}((0,\infty),\mathbb{R})\text{ and
}X_{0}=\left\\{0_{\mathbb{R}}\right\\}\times L^{1}((0,\infty),\mathbb{R}).$

Assume that $X$ is endowed with the product norm

$\left\lVert
x\right\rVert=|\alpha|+\left\lVert\psi\right\rVert_{L^{1}((0,\infty),\mathbb{R})},\quad\forall
x=\begin{pmatrix}\alpha\\\ \psi\end{pmatrix}\in X.$

Consider the linear operator $A:D(A)\subset X\to X$ given by

$A\begin{pmatrix}0\\\ \psi\end{pmatrix}=\begin{pmatrix}-\psi(0)\\\
-\psi^{\prime}-m\psi\end{pmatrix}$

with

$D(A)=\\{0_{\mathbb{R}}\\}\times W^{1,1}((0,\infty),\mathbb{R}).$

Recall that $X_{0}$ is the closure of $D(A)$ in $X$. In addition, note that
$A_{0}$, the part of $A$ in $X_{0}$, generates a $C_{0}-$semigroup of bounded
linear operators, denoted by $\\{T_{A_{0}}(t)\\}_{t\geq 0}$ and explicitly
given by

$T_{A_{0}}(t)\begin{pmatrix}0\\\ \psi\end{pmatrix}=\begin{pmatrix}0\\\
\widehat{T}_{A_{0}}(t)\psi\end{pmatrix},$

wherein we have set, for all $t\geq 0$ and $\psi\in
L^{1}((0,\infty),\mathbb{R})$,

$\widehat{T}_{A_{0}}(t)(\psi)(a)=\begin{cases}e^{-m\,t}\psi(a-t),\quad&\text{if}\quad
a\geq t,\\\ 0,\quad&\text{if}\quad a\leq t.\end{cases}$

Moreover $A$ generates an integrated semigroup of $X$, denoted by
$\\{S_{A}(t)\\}_{t\geq 0}$ and defined, for $t\geq 0$ by

$S_{A}(t)\begin{pmatrix}\alpha\\\ \psi\end{pmatrix}=\begin{pmatrix}0\\\
L(t)\alpha+\int_{0}^{t}\widehat{T}_{A_{0}}(s)\psi
ds\end{pmatrix},\;\forall\begin{pmatrix}\alpha\\\ \psi\end{pmatrix}\in X,$

where

$L(t)(\alpha)(a)=\begin{cases}0,\quad&\text{if}\quad a\geq t,\\\
e^{-m\,a}\alpha,\quad&\text{if}\quad a\leq t.\end{cases}$

Define the map $H:\mathbb{R}\times X_{0}\to X$ by

$H\begin{pmatrix}\nu,\begin{pmatrix}0\\\
\psi\end{pmatrix}\end{pmatrix}=\begin{pmatrix}f\left(\nu,\int_{0}^{\infty}b(a)\psi(a)da\right)\\\
0\end{pmatrix}.$

Then by identifying $u(t)$ with $v(t)=\begin{pmatrix}0\\\ u(t)\end{pmatrix}$,
problem (1.1) can be rewritten as the following abstract Cauchy problem

$\frac{dv(t)}{dt}=Av(t)+H(\nu,v(t)),\quad t\geq 0,\quad v(0)=y\in X_{0}.$
(2.1)

Since $f$ is Lipschitz continuous on bounded sets, the general results proved
in Magal and Ruan [13, see section 5] or [15, Chapter 5] ensure that the
Cauchy problem (2.1) generated a maximal semiflow (with eventually some
blowup), denoted below by $\\{U_{\nu}(t)\\}_{t\geq 0}$.

##  3 Assumptions

Regularity of the birth function: Let us recall some definition and properties
of the so called bounded variation functions. Let $F:[0,\infty)\to\mathbb{R}$
be some function. For each $a>0$ define

$T_{F}(a):=\sup\big{\\{}\sum_{j=1}^{n}|F(a_{j})-F(a_{j-1})|:n\in\mathbb{N},0=a_{0}<\cdots<a_{n}=a\big{\\}}\in[0,\infty],$

where the supremum is taken over all finite strictly increasing sequences in
$[0,a]$.

###### Definition 3.1

A function $F:[0,\infty)\to\mathbb{R}$ is said to be of bounded variation on
$[0,\infty)$ if

$\sup\limits_{a>0}T_{F}(a)<\infty.$

In that case the function $a\to T_{F}(a)$ is bounded and increasing on
$[0,\infty)$.

Let $F:[0,\infty)\to\mathbb{R}$ be a right continuous function of bounded
variation on $[0,\infty)$, then $a\to T_{F}(a)$ is also right continuous and
according to Folland [7, Theorem 3.29], there exists a unique Borel measure
$\mu_{F}$ such that

$\mu_{F}((a,b])=F(b)-F(a),\text{ for all }a,b\in[0,\infty),\text{ with }a<b.$

Furthermore, its total variation $|\mu_{F}|$ is the positive and bounded Borel
measure associated to the right continuous and increasing function $a\to
T_{F}(a)$.

Next let us recall the integration by parts formula proved by Folland [7,
Theorem 3.36] as well. If $G:[0,\infty)\to\mathbb{R}$ is of class $C^{1}$ then
for all $0\leq a<b$, one has

$\int_{a}^{b}G(s)\mu_{F}(ds)=[G(b)F(b-)-G(a)F(a+)]-\int_{a}^{b}G^{\prime}(s)F(s)ds.$
(3.1)

We now make a set of assumptions on the fertility rate $b$.

###### Assumption 3.2

Assume that the function $\chi(a):=b(a)e^{-m\,a}$ satisfies the two following
properties

  * (i)

Assume that $\chi\in L^{1}(0,\infty)$ with

$\int_{0}^{\infty}\chi(a)da=1\Leftrightarrow\int_{0}^{\infty}b(a)e^{-m\,a}da=1.$

  * (ii)

We assume that the function $\tau:a\to a\chi(a)$ is right continuous and of
bounded variation on $[0,\infty)$. We let $\mu_{\tau}$ be the unique Borel
measure associated to $\tau$.

###### Remark 3.3

Note that when $b\in L^{\infty}(0,\infty)$, for each integer $n\in\mathbb{N}$
one has

$a\to a^{n}\chi(a)=a^{n}b(a)e^{-m\,a}\in L^{1}((0,\infty),\mathbb{R}).$

Equilibria:  Recall that $\begin{pmatrix}0\\\ \bar{u}\end{pmatrix}\in D(A)$ is
an equilibrium of the semiflow $\\{U_{\nu}(t)\\}_{t\geq 0}$ if and only if

$\begin{pmatrix}0\\\ \bar{u}\end{pmatrix}\in D(A)\quad\text{and}\quad
A\begin{pmatrix}0\\\
\bar{u}\end{pmatrix}+H\begin{pmatrix}\nu,\begin{pmatrix}0\\\
\bar{u}\end{pmatrix}\end{pmatrix}=0.$

As a consequence a positive equilibrium is given by

$\bar{u}(a)=\overline{w}_{\nu}e^{-m\,a},\;a\geq 0$

where $\overline{w}_{\nu}>0$ becomes a solution of the following equation

$\overline{w}_{\nu}=f\left(\nu,\overline{w}_{\nu}\right).$

Our next assumption is concerned with the existence of such equilibrium point
and its regularity with respect to the parameter $\nu$.

###### Assumption 3.4

We assume that there exists an open interval $I$ such that for each $\nu\in I$
there exists a constant solution $\overline{w}_{\nu}\in\mathbb{R}$ of the
equation

$\overline{w}_{\nu}=f(\nu,\overline{w}_{\nu}).$

We assume further that the map $\nu\to\overline{w}_{\nu}$ is continuously
differentiable on the interval $I$.
In the sequel we set

$\bar{v}_{\nu}=\begin{pmatrix}0\\\
\bar{u}_{\nu}\end{pmatrix}\quad\text{with}\quad\bar{u}_{\nu}(a)=\overline{w}_{\nu}e^{-m\,a},\quad\forall\nu\in
I.$

In the following we will use the notation $\mathcal{L}(Y,Z)$ to denote the
space of the linear bounded operators from $Y$ to $Z$ where $Y$ and $Z$ are
two Banach spaces. Define for $\nu\in I$ the linear operator
$B_{\nu}:D(B_{\nu})\subset X\to X$ as follows,

$D(B_{\nu})=D(A)\text{ and
}B_{\nu}x=Ax+\partial_{v}H(\nu,\bar{v}_{\nu})x,\;\forall x\in D(B_{\nu}),$

wherein $\partial_{v}$ corresponds to the partial derivative of $H(\nu,v)$
with respect to $v$. The bounded linear operator
$\partial_{v}H(\nu,\bar{v}_{\nu})\in\mathcal{L}(X_{0},X)$ is defined by

$\partial_{v}H(\nu,\bar{v}_{\nu})\begin{pmatrix}0_{\mathbb{R}}\\\
\varphi\end{pmatrix}=\begin{pmatrix}\partial_{w}f\left(\nu,\overline{w}_{\nu}\right)\int_{0}^{\infty}b(l)\varphi(l)dl\\\
0_{L^{1}}\end{pmatrix},\;\forall\begin{pmatrix}0_{\mathbb{R}}\\\
\varphi\end{pmatrix}\in X_{0}.$

Herein $\partial_{w}$ denotes the partial derivative of $f=f(\nu,w)$ with
respect to $w$.

By using the result of Ducrot, Liu and Magal [6], the essential growth rate of
the semigroup generated by $(B_{\nu})_{0}$, the part of $B_{\nu}$ in the
closure of its domain, satisfies

$\omega_{0,ess}((B_{\nu})_{0})\leq-\mu<0.$

The following result follows from [15, Theorem 4.3.27, Lemma 4.4.2, Theorem
4.4.3-(ii)] to which we refer the reader for a proof and more details.

###### Lemma 3.5

The spectrum of $B_{\nu}$ in the half plane

$\Omega:=\left\\{\lambda\in\mathbb{C}:{\rm Re}\,\lambda>-m\right\\}$

contains only isolated eigenvalues which are poles of the resolvent of
$B_{\nu}$.

Recall that the characteristic function, describing the spectrum of $B_{\nu}$
in $\Omega$, is obtained by computing the resolvent of $B_{\nu}$ as presented
in Liu, Magal and Ruan [12]. We define the characteristic function for $\nu\in
I$ and $\lambda\in\Omega$ as follows

$\Delta(\nu,\lambda):=1-\partial_{w}f\left(\nu,\overline{w}_{\nu}\right)\int_{0}^{\infty}b(l)e^{-\left(m+\lambda\right)l}dl.$

Recall that the resolvent set $\rho(A)$ of $A$ contains $\Omega$ and for each
$\lambda\in\Omega$ the resolvent of $A$ is defined by the following formula

$\begin{array}[]{l}\left(\lambda
I-A\right)^{-1}\left(\begin{array}[]{c}\alpha\\\
\psi\end{array}\right)=\left(\begin{array}[]{c}0\\\
\varphi\end{array}\right)\\\ \\\
\Leftrightarrow\varphi(a)=e^{-\left(\lambda+m\right)a}\alpha+\int_{0}^{a}e^{-\left(\lambda+m\right)\left(a-s\right)}\psi(s)ds.\end{array}$

We now recall some result already presented (in a more general framework in
the Section 5.2 in [12])

###### Lemma 3.6

For each $\nu\in I$ the resolvent set $\rho(B_{\nu})$ of $B_{\nu}$ satisfies

$\lambda\in\rho(B_{\nu})\cap\Omega\Leftrightarrow\Delta(\nu,\lambda)\neq 0,$

or equivalently the spectrum
$\sigma(B_{\nu}):=\mathbb{C}\setminus\rho(B_{\nu})$ of $B_{\nu}$ satisfies

$\lambda\in\sigma(B_{\nu})\cap\Omega\Leftrightarrow\Delta(\nu,\lambda)=0.$

Moreover one has

$\begin{array}[]{l}\left(\lambda
I-B_{\nu}\right)^{-1}\left(\begin{array}[]{c}\alpha\\\
\psi\end{array}\right)=\left(\begin{array}[]{c}0\\\
\varphi\end{array}\right)\\\ \\\
\Leftrightarrow\varphi(a)=e^{-\left(\lambda+m\right)a}\alpha_{1}+\int_{0}^{a}e^{-\left(\lambda+m\right)\left(a-s\right)}\psi(s)ds,\end{array}$

where

$\alpha_{1}:=\Delta(\nu,\lambda)^{-1}\left[\alpha-\partial_{w}f\left(\nu,\overline{w}_{\nu}\right)\int_{0}^{\infty}b(a)\int_{0}^{a}e^{-\left(\lambda+m\right)\left(a-s\right)}\psi(s)dsda\right].$

Proof. For each $\lambda\in\Omega$ the linear operator $\lambda I-B_{\nu}$ is
invertible if and only if the linear operator is invertible

$I-\partial_{v}H(\nu,\bar{v}_{\nu})\left(\lambda I-A\right)^{-1}.$

In that case we have

$(\lambda I-B_{\nu})^{-1}=\left(\lambda
I-A\right)^{-1}\left[I-\partial_{v}H(\nu,\bar{v}_{\nu})\left(\lambda
I-A\right)^{-1}\right]^{-1}.$

Computation of the inverse of $I-\partial_{v}H(\nu,\bar{v}_{\nu})\left(\lambda
I-A\right)^{-1}$ : We have

$\begin{array}[]{l}\left(\begin{array}[]{c}\widehat{\alpha}\\\
\widehat{\varphi}\end{array}\right)=\left[I-\partial_{v}H(\nu,\bar{v}_{\nu})\left(\lambda
I-A\right)^{-1}\right]\left(\begin{array}[]{c}\alpha\\\
\varphi\end{array}\right)\\\
\Leftrightarrow\left(\begin{array}[]{c}\widehat{\alpha}\\\
\widehat{\varphi}\end{array}\right)=\left(\begin{array}[]{c}\alpha-\partial_{w}f\left(\nu,\overline{w}_{\nu}\right)\int_{0}^{\infty}b(a)\left[e^{-\left(\lambda+m\right)a}\alpha+\int_{0}^{a}e^{-\left(\lambda+m\right)\left(a-s\right)}\
\varphi(s)ds\right]da\\\ \varphi\end{array}\right).\end{array}$

Therefore $I-\partial_{v}H(\nu,\bar{v}_{\nu})\left(\lambda I-A\right)^{-1}$ is
invertible (for $\lambda\in\Omega$) if and only if $\Delta(\nu,\lambda)\neq 0$
and one has

$\begin{array}[]{l}\left(\begin{array}[]{c}\alpha\\\
\varphi\end{array}\right)=\left[I-\partial_{v}H(\nu,\bar{v}_{\nu})\left(\lambda
I-A\right)^{-1}\right]^{-1}\left(\begin{array}[]{c}\widehat{\alpha}\\\
\widehat{\varphi}\end{array}\right)\\\
\Leftrightarrow\left(\begin{array}[]{c}\alpha\\\
\varphi\end{array}\right)=\left(\begin{array}[]{c}\Delta(\nu,\lambda)^{-1}\left[\widehat{\alpha}-\partial_{w}f\left(\nu,\overline{w}_{\nu}\right)\int_{0}^{\infty}b(a)\int_{0}^{a}e^{-\left(\lambda+m\right)\left(a-s\right)}\widehat{\varphi}(s)dsda\right]\\\
\widehat{\varphi}\end{array}\right).\end{array}$

The result follows.

###### Assumption 3.7

There exists $\nu_{0}\in I$ and $\omega_{0}>0$ such that the following
properties are satisfied

  * (i)

$\Delta(\nu_{0},\omega_{0}i)=0.$

  * (ii)

Crandall and Rabinowitz’s condition

$\Delta(\nu_{0},k\,\omega_{0}i)\neq 0,\forall k\in\mathbb{N}\text{ and }k\neq
1.$

  * (iii)

Simplicity of the eigenvalue $\omega_{0}i$

$\partial_{\lambda}\Delta(\nu_{0},\omega_{0}i)\neq 0,$

which is also equivalent to

$\partial_{w}f\left(\nu_{0},\overline{w}_{\nu_{0}}\right)\int_{0}^{\infty}b(l)\,l\,e^{-\left(m+\omega_{0}i\right)l}dl\neq
0.$

  * (iv)

Transversality condition

${\rm
Re}\,\left(\partial_{\nu}\Delta(\nu_{0},\omega_{0}i)\times\overline{\partial_{\lambda}\Delta(\nu_{0},\omega_{0}i)}\right)\neq
0.$

We can observe that by combining (i) and (iii) and by using the implicit
function theorem there exists a branch
$\lambda:(\nu_{0}-\eta,\nu_{0}+\eta)\subset I\to\mathbb{C}$ with some $\eta>0$
small enough such that for each $\nu\in(\nu_{0}-\eta,\nu_{0}+\eta)$,
$\lambda(\nu)=\alpha(\nu)+i\omega(\nu)$ and
$\overline{\lambda(\nu)}=\alpha(\nu)-i\omega(\nu)$ satisfying solution of

$\Delta(\nu,\lambda(\nu))=0$ (3.2)

and

$\lambda(\nu_{0})=i\omega_{0}.$

Moreover by combining (iii)-(iv) we deduce that the transversality condition
is satisfied. Namely, one has

${\rm Re}\,\frac{d\lambda(\nu_{0})}{d\nu}\neq 0.$

Moreover by using the property (iii) in Lemma 5.8 in [12] we also deduce that
$i\omega_{0}$ is a simple eigenvalue of $B_{\nu_{0}}$ since

$\lim_{\lambda\to\lambda_{0}}\dfrac{\Delta(\nu_{0},\lambda)}{\lambda-\lambda_{0}}\neq
0\Leftrightarrow\partial_{\lambda}\Delta(\nu_{0},\lambda_{0})\neq
0.\quad(\text{with }\lambda_{0}=\lambda(\nu_{0})).$

The condition (ii) avoid to assume that the purely imaginary spectrum is
reduced to the a single pair of purely imaginary eigenvalues. Such a condition
has been introduced in the Crandall and Rabinowitz’s proof [2].

##  4 Main Result

In this section we state the main theorem of this paper. The following result
is inspired by Crandall and Rabinowitz [2].

###### Theorem 4.1

Let Assumptions 3.2, 3.4 and 3.7 be satisfied. Then there exist a constant
$\delta>0$ and two $C^{1}$ maps, $s\to\nu(s)$ from $(-\delta,\delta)$ to
$\mathbb{R}$ and $s\to\omega(s)$ from $(-\delta,\delta)$ to $\mathbb{R}$ such
that for each $s\in(-\delta,\delta)$ there exists a $2\pi/\omega(s)-$periodic
solution $u(s)$ of class of $C^{1}$ which is a solution of (1.1) with the
parameter $\nu=\nu(s)$. Moreover, the branch of periodic orbit is bifurcating
from $\nu_{0}$ at $\nu=\nu_{0}$, that is to say that

$\nu(0)=\nu_{0},\quad\omega(0)=\omega_{0}.$

Proof. (Proof of Theorem 4.1) Up to time rescaling we can assume, without loss
of generality, that $\omega_{0}=1$. Observe that Assumption 3.7-(i) implies
that (1.1) linearized about $u=\bar{u}_{\nu_{0}}$ for $\nu=\nu_{0}$ has
nontrivial $2\pi-$periodic solutions. We now seek nontrivial
$2\pi/\omega-$periodic solutions of (1.1) with $\omega$ close to $1$ and
$(u,\nu)$ close to $(\bar{u}_{\nu_{0}},\nu_{0})$.

Solving (1.1) along the characteristic line $t-a=$constant, one obtains

$u(t,a)=u(t-a,0)e^{-m\,a},\;t\in\mathbb{R},a>0.$

Thus $v=v(t)$ given by

$v(t)=u(t,0),$

satisfies the renewal equation

$v(t)=f\left(\nu,\int_{0}^{\infty}b(a)v(t-a)e^{-m\,a}da\right),\;t\in\mathbb{R}.$

Setting

$w(t)=v\left(\frac{t}{\omega}\right),$

yields the following equation for the $2\pi-$periodic function $w=w(t)$

$w(t)=v\left(\frac{t}{\omega}\right)=f\left(\nu,\int_{0}^{\infty}b(a)v(t/\omega-a)e^{-m\,a}da\right)=f\left(\nu,\int_{0}^{\infty}b(a)w(t-\omega
a)e^{-m\,a}da\right),\;t\in\mathbb{R}.$

Recalling the definition of $\chi$ in Assumption 3.2 and using the change of
the variable $l=\omega a$ in the integral lead to the following equation for
$w=w(t)$

$w(t)=f\left(\nu,\int_{0}^{\infty}\frac{1}{\omega}\chi\left(\frac{l}{\omega}\right)w(t-l)dl\right),\;t\in\mathbb{R}.$
(4.1)

Now the existence of nontrivial $2\pi/\omega-$periodic solution of (1.1)
becomes equivalent to the one of nontrivial $2\pi-$ periodic solution of
(4.1). Next we shall apply the implicit function theorem to investigate the
existence of nontrivial $2\pi-$periodic solution of (4.1) for $\nu$ close to
$\nu_{0}$.

Let $C_{2\pi}(\mathbb{R})$ be the Banach space of the continuous
$2\pi-$periodic functions. Define the map $F:\mathbb{R}^{2}\times
C_{2\pi}(\mathbb{R})\to C_{2\pi}(\mathbb{R})$ by

$F(\omega,\nu,x)(t)=x(t)-f\left(\nu,\int_{0}^{\infty}\frac{1}{\omega}\chi\left(\frac{l}{\omega}\right)x(t-l)dl\right),\quad\forall(\omega,\nu,x)\in\mathbb{R}^{2}\times
C_{2\pi}(\mathbb{R}).$

We now aim at investigating the zeros of the equation

$F(\omega,\nu,x)=0,$

for $(\omega,\nu,x)$ close to $(1,\nu_{0},\overline{w}_{\nu_{0}})$ using the
implicit function theorem. To do so, we need first to verify the smoothness of

$\int_{0}^{\infty}\frac{1}{\omega}\chi\left(\frac{l}{\omega}\right)x(t-l)dl=\int_{0}^{\infty}\chi(l)x(t-\omega
l)dl,$

with respect to $\omega$.

The first main step of this proof is the following lemma.

###### Lemma 4.2

The map $G:\mathbb{R}\times C_{2\pi}(\mathbb{R})\to C_{2\pi}(\mathbb{R})$
defined by

$G(\omega,x)(t)=\int_{0}^{\infty}\chi(l)x(t-\omega l)dl,\;t\in\mathbb{R},$

is continuously differentiable with respect to $\omega\in\mathbb{R}$ and its
partial derivative with respect to $\omega$, denoted by $\partial_{\omega}G$,
is given

$\partial_{\omega}G(\omega,x)(\cdot)=\int_{0}^{\infty}x(\cdot-\omega
l)\mu_{\tau}(dl),\;\forall(\omega,x)\in\mathbb{R}\times C_{2\pi}(\mathbb{R}).$

Herein $\tau$ and $\mu_{\tau}$ are defined in Assumption 3.2-(ii).

Proof.

First observe that using Fubini theorem, for any $x\in C^{1}(\mathbb{R})\cap
C_{2\pi}(\mathbb{R})$ and $\omega\in\mathbb{R}$ we have, for any
$t\in\mathbb{R}$,

$\begin{array}[]{ll}\int_{0}^{\omega}\int_{0}^{\infty}-x^{\prime}(t-\sigma
l)\,l\,\chi(l)\,dl\,d\sigma&=\int_{0}^{\infty}\int_{0}^{\omega}-lx^{\prime}(t-\sigma
l)\,d\sigma\,\chi(l)\,dl\\\ &=\int_{0}^{\infty}\left[x(t-\sigma
l)\right]_{\sigma=0}^{\sigma=\omega}\,\chi(l)\,dl\\\
&=\int_{0}^{\infty}\chi(l)x(t-\omega
l)dl-x(t)\int_{0}^{\infty}\chi(l)dl,\end{array}$

so that, since $\int_{0}^{\infty}\chi(l)dl=1$ (see Assumption 3.2-(i)), we get

$\int_{0}^{\omega}\int_{0}^{\infty}-x^{\prime}(t-\sigma
l)l\chi(l)dld\sigma=\int_{0}^{\infty}\chi(l)x(t-\omega l)dl-x(t),$ (4.2)

that is for all $(\omega,x)\in\mathbb{R}\times\left(C^{1}(\mathbb{R})\cap
C_{2\pi}(\mathbb{R})\right)$ and $t\in\mathbb{R}$

$G(\omega,x)(t)=x(t)-\int_{0}^{\omega}\int_{0}^{\infty}x^{\prime}(t-\sigma
l)\tau(l)dld\sigma.$

We deduce that for all $x\in C^{1}(\mathbb{R})\cap C_{2\pi}(\mathbb{R})$ the
map $\omega\to G(\omega,x)=\int_{0}^{\infty}\chi(l)x(\cdot-\omega l)dl$ is of
class $C^{1}$ and

$\frac{d}{d\omega}\int_{0}^{\infty}\chi(l)x(t-\omega
l)dl=-\int_{0}^{\infty}x^{\prime}(t-\omega l)l\chi(l)dl.$ (4.3)

Moreover by using again the formula (4.2) we get

$\int_{\omega}^{\omega+\varepsilon}\int_{0}^{\infty}-x^{\prime}(t-\sigma
l)l\chi(l)dld\sigma=\int_{0}^{\infty}\chi(l)x(t-(\omega+\varepsilon)l)dl-\int_{0}^{\infty}\chi(l)x(t-\omega
l)dl.$

hence

$\int_{\omega}^{\omega+\varepsilon}\frac{d}{d\sigma}\int_{0}^{\infty}\chi(l)x(t-\sigma
l)dld\sigma=\int_{0}^{\infty}\chi(l)x(t-(\omega+\varepsilon)l)dl-\int_{0}^{\infty}\chi(l)x(t-\omega
l)dl.$ (4.4)

By using the integration by parts formula (3.1) and (4.3), we obtain for all
$(\omega,x)\in\mathbb{R}\times\left(C^{1}(\mathbb{R})\cap
C_{2\pi}(\mathbb{R})\right)$

$\partial_{\omega}G(\omega,x)(t)=\frac{d}{d\omega}\int_{0}^{\infty}\chi(l)x(t-\omega
l)dl=-\int_{0}^{\infty}x^{\prime}(t-\omega
l)l\chi(l)dl=\int_{0}^{\infty}x(t-\omega l)\mu_{\tau}(dl).$

Then using Assumption 3.2-(ii) we infer from the above equality that

$\left\lVert\partial_{\omega}G(\omega,x)\right\rVert_{C_{2\pi}(\mathbb{R})}\leq\left\lVert
x\right\rVert_{C_{2\pi}(\mathbb{R})}|\mu_{\tau}|((0,\infty))<\infty,\,\forall(\omega,x)\in\mathbb{R}\times
C^{1}(\mathbb{R})\cap C_{2\pi}(\mathbb{R}),$

wherein $|\mu_{\tau}|((0,\infty))$ is nothing but the variation of $\tau(a)$
on $(0,\infty)$. That is supremum over all the subdivision
$0=t_{0}<t_{1}<t_{2}<\ldots<t_{n}=M$ of

$|\mu_{\tau}|((0,M])=\sup_{0=t_{0}<t_{1}<t_{2}<\ldots<t_{n}=M}\sum_{i=0}^{n}|\tau(t_{i+1})-\tau(t_{i})|,$

and

$|\mu_{\tau}|((0,\infty))=\lim_{M\to+\infty}|\mu_{\tau}|((0,M]).$

We now define $L_{\omega}\in\mathcal{L}\left(C_{2\pi}(\mathbb{R})\right)$ by

$L_{\omega}(x)(t):=\int_{0}^{\infty}x(t-\omega l)\mu_{\tau}(dl).$

Now the result follows by using the (4.4) and the density of
$C^{1}(\mathbb{R})\cap C_{2\pi}(\mathbb{R})$ into $C_{2\pi}(\mathbb{R})$,
since it implies that

$\int_{\omega}^{\omega+\varepsilon}L_{\sigma}(x)(t)d\sigma=\int_{0}^{\infty}\chi(l)x(t-(\omega+\varepsilon)l)dl-\int_{0}^{\infty}\chi(l)x(t-\omega
l)dl.$

It remains to prove that the map
$(\omega,x)\to\partial_{\omega}G(\omega,x)=\int_{0}^{\infty}x(\cdot-\omega
l)\mu_{\tau}(dl)$ is continuous from $\mathbb{R}\times C_{2\pi}(\mathbb{R})$
into $C_{2\pi}(\mathbb{R})$. To see this fix $(\omega_{1},x_{1})$ in
$\mathbb{R}\times C_{2\pi}(\mathbb{R})$ and observe that for all
$(\omega,x)\in\mathbb{R}\times C_{2\pi}(\mathbb{R})$ one has:

$\begin{array}[]{l}\partial_{\omega}G(\omega,x)-\partial_{\omega}G(\omega_{1},x_{1})=\int_{0}^{\infty}\left[x(\cdot-\omega
l)-x_{1}(\cdot-\omega_{1}l)\right]\mu_{\tau}(dl)={\rm
J_{1}+J_{2}},\end{array}$

wherein we have set

${\rm J_{1}}:=\int_{0}^{\infty}\left[x(\cdot-\omega l)-x_{1}(\cdot-\omega
l)\right]\mu_{\tau}(dl),\;{\rm
J_{2}}:=\int_{0}^{\infty}\left[x_{1}(\cdot-\omega
l)-x_{1}(\cdot-\omega_{1}l)\right]\mu_{\tau}(dl).$

We first observe that

$\|{\rm
J_{1}}\|_{C_{2\pi}(\mathbb{R})}\leq\|x-x_{1}\|_{C_{2\pi}(\mathbb{R})}|\mu_{\tau}|((0,\infty))\to
0\text{ uniformy for $\omega\in\mathbb{R}$ as
$\|x-x_{1}\|_{C_{2\pi}(\mathbb{R})}\to 0$}.$

On the other hand fix $\varepsilon>0$ and since one has

$|\mu_{\tau}|((M,\infty)):=|\mu_{\tau}|((0,\infty))-|\mu_{\tau}|((0,M])\to
0\text{ as }M\to\infty,$

choose $M>0$ large enough so that
$2\|x_{1}\|_{C_{2\pi}(\mathbb{R})}|\mu_{\tau}|((M,\infty))\leq\varepsilon$.
With such a choice we split ${\rm J_{2}}$ as follows ${\rm J_{2}=I_{1}+I_{2}}$
with

${\rm I_{1}}:=\int_{0}^{M}\left[x_{1}(\cdot-\omega
l)-x_{1}(\cdot-\omega_{1}l)\right]\mu_{\tau}(dl),\;{\rm
I_{2}}:=\int_{M}^{\infty}\left[x_{1}(\cdot-\omega
l)-x_{1}(\cdot-\omega_{1}l)\right]\mu_{\tau}(dl).$

Hence we get

$\|{\rm I_{2}}\|_{C_{2\pi}(\mathbb{R})}\leq
2\|x_{1}\|_{C_{2\pi}(\mathbb{R})}|\mu_{\tau}|((M,\infty))\leq\varepsilon,$

and, since $x_{1}$ is continuous and $2\pi$-periodic, it is uniformly
continuous on $\mathbb{R}$. Thus for all $\omega$ is sufficiently close to
$\omega_{1}$, we have

$\|{\rm I_{1}}\|_{C_{2\pi}(\mathbb{R})}\leq\sup_{0\leq l\leq M}|x_{1}(t-\omega
l)-x_{1}(t-\omega_{1}l)||\mu_{\tau}|((0,M])\leq\varepsilon.$

As a consequence $\partial_{\omega}G$ is continuous and the proof is
completed.  Computation of the derivatives of $F$:  One can calculate the
following derivatives directly,

$\partial_{x}F(\omega,\nu,\overline{w}_{\nu})(x)(t)=x(t)-\partial_{w}f(\nu,\overline{w}_{\nu})\int_{0}^{\infty}\chi(l)x(t-\omega
l)dl,$ (4.5)
$\partial_{\nu}\partial_{x}F(\omega,\nu,\overline{w}_{\nu})(x)(t)=-\left[\partial_{\nu}\partial_{w}f(\nu,\overline{w}_{\nu})+\partial_{w}^{2}f(\nu,\overline{w}_{\nu})\partial_{\nu}\overline{w}_{\nu}\right]\int_{0}^{\infty}\chi(l)x(t-\omega
l)dl,$ (4.6)

and by Lemma 4.2

$\partial_{\omega}\partial_{x}F(\omega,\nu,\overline{w}_{\nu})(x)(t)=-\partial_{w}f(\nu,\overline{w}_{\nu})\int_{0}^{\infty}x(t-\omega
l)\mu_{\tau}(dl).$ (4.7)

State space decomposition: Note that by Assumption 3.7-(i) we have

$1=\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})\int_{0}^{\infty}\chi(l)e^{\pm
i\,l}dl,$

which re-writes as

$1=\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})\int_{0}^{\infty}\chi(l)\cos
ldl\text{ and
}0=\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})\int_{0}^{\infty}\chi(l)\sin
ldl,$ (4.8)

and thus the kernel $N(\partial_{x}F(1,\nu_{0},\overline{w}_{\nu_{0}}))$
contains the space $X_{1}$ given by

$X_{1}:=\text{span}\\{\cos,\sin\\}\subset C_{2\pi}(\mathbb{R}).$

Now define the closed space

$X_{2}:=\bigg{\\{}z\in
C_{2\pi}(\mathbb{R}):\int_{0}^{2\pi}z(t)e^{i\,t}dt=0\bigg{\\}}=\bigg{\\{}z\in
C_{2\pi}(\mathbb{R}):\int_{0}^{2\pi}z(t)\cos(t)dt=\int_{0}^{2\pi}z(t)\sin(t)dt=0\bigg{\\}}.$

This space turns out to be a complement of $X_{1}$ as stated in the next
lemma.

###### Lemma 4.3

We have the following state space decomposition

$C_{2\pi}(\mathbb{R})=X_{1}\oplus X_{2}.$

Proof. This property is directly inherited from the decomposition of
$L^{2}((0,2\pi),\mathbb{R})$ as

$L^{2}((0,2\pi),\mathbb{R})=X_{1}\oplus X_{1}^{\perp},$

with

$X_{1}^{\perp}=\bigg{\\{}z\in
L^{2}((0,2\pi),\mathbb{R}):\int_{0}^{2\pi}z(t)e^{i\,t}dt=0\bigg{\\}},$

Now if $z\in C_{2\pi}(\mathbb{R})$ then $z\in L^{2}((0,2\pi),\mathbb{R})$ and
the above $L^{2}((0,2\pi),\mathbb{R})-$decomposition ensures that there exist
unique $z_{1}\in X_{1}$ and $z_{2}\in X_{1}^{\perp}$ such that

$z=z_{1}+z_{2}.$

Now since $z_{1}=c_{1}\cos+c_{2}\sin$ (for some constants $c_{1}$ and $c_{2}$)
this ensures that $z_{2}=z-z_{1}$ is also continuous and $z_{2}\in X_{2}$. The
state space decomposition follows.

Now let us define the map $h:\mathbb{R}^{3}\times X_{2}\to
C_{2\pi}(\mathbb{R})$ by

$h(s,\omega,\nu,z)=\begin{cases}s^{-1}F(\omega,\nu,\overline{w}_{\nu}+s(u_{1}+z)),\quad&\text{if}\,s\neq
0,\\\
\partial_{x}F(\omega,\nu,\overline{w}_{\nu})(u_{1}+z),\quad&\text{if}\,s=0,\end{cases}$

where

$u_{1}(t)=\cos(t),\;\forall t\in\mathbb{R}.$

Now let us observe that since $f=f(\nu,x)$ is of class $C^{2}$, $h$ is of
class $C^{1}$. One also has $h(0,1,\nu_{0},0)=0$ while the derivative with
respect to $(\omega,\nu,z)$ is given, for all
$(\tilde{\omega},\tilde{\nu},\tilde{z})\in\mathbb{R}\times\mathbb{R}\times
C_{2\pi}(\mathbb{R})$, by

$\begin{array}[]{l}D_{(\omega,\nu,z)}h(0,1,\nu_{0},0)(\tilde{\omega},\tilde{\nu},\tilde{z})\par=\partial_{x}F(1,\nu_{0},\overline{w}_{\nu_{0}})\tilde{z}+\tilde{\nu}\partial_{\nu}\partial_{x}F(1,\nu_{0},\overline{w}_{\nu_{0}})u_{1}+\tilde{\omega}\partial_{\omega}\partial_{x}F(1,\nu_{0},\overline{w}_{\nu_{0}})u_{1},\par\par\end{array}$

hence by using (4.5)-(4.7) we obtain

$\begin{array}[]{ll}D_{(\omega,\nu,z)}h(0,1,\nu_{0},0)(\tilde{\omega},\tilde{\nu},\tilde{z})=&\tilde{z}(t)-\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})\int_{0}^{\infty}\chi(l)\tilde{z}(t-l)dl\vspace{0.2cm}\\\
&-\tilde{\nu}\left[\partial_{\nu}\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})+\partial_{w}^{2}f(\nu_{0},\overline{w}_{\nu_{0}})\partial_{\nu}\overline{w}_{\nu_{0}}\right]\int_{0}^{\infty}\chi(l)u_{1}(t-l)dl\vspace{0.2cm}\\\
&-\tilde{\omega}\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})\int_{0}^{\infty}u_{1}(t-l)\mu_{\tau}(dl).\end{array}$

The second main step of the proof is the following lemma.

###### Lemma 4.4

The bounded linear operator

$D_{(\omega,\nu,z)}h(0,1,\nu_{0},0)\in\mathcal{L}(\mathbb{R}^{2}\times
X_{2},C_{2\pi}(\mathbb{R}))$

is invertible.

Proof. Let us first define the linear bounded operator $K:X_{2}\to
C_{2\pi}(\mathbb{R})$ by

$K\phi:=\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})\int_{0}^{\infty}\chi(l)\phi(\cdot-l)dl.$

Step 1: Let us prove that $K(X_{2})\subset X_{2}$. Indeed, by using Fubini’s
theorem, for all $\phi\in C_{2\pi}(\mathbb{R})$ one has

$\int_{0}^{2\pi}\int_{0}^{\infty}\chi(l)\phi(t-l)dle^{i\,t}dt=\int_{0}^{\infty}\int_{0}^{2\pi}\phi(t-l)e^{i\,(t-l)}dte^{i\,l}\chi(l)dl$

and since $t\to\phi(t)e^{i\,t}$ is $2\pi-$periodic we deduce that

$\int_{0}^{2\pi}\phi(t-l)e^{i\,(t-l)}dt=0,\forall l\geq 0.$

This completes the first step.

Step 2: Let us now prove that $N(I-K)=\\{0\\}$ whenever
$I-K\in\mathcal{L}(X_{2})$. In order to compute the kernel of $I-K$ in
$X_{2}$, consider $g\in N(I-K)$, that is $g\in X_{2}$ such that

$g(t)-\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})\int_{0}^{\infty}\chi(l)g(t-l)dl=0,\;\forall
t\in\mathbb{R}.$

Multiplying the above equality by $e^{-int}$, for some $n\in\mathbb{Z}$ and
integrating between $0$ and $2\pi$ we obtain

$\left[1-\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})\widehat{\chi}(n)\right]\widehat{g}(n)=0,$

wherein we have set

$\widehat{g}(n):=\int_{0}^{2\pi}g(l)e^{-inl}dl,$

and

$\widehat{\chi}(n):=\int_{0}^{\infty}\chi(l)e^{-inl}dl.$

Now for $n\neq\pm 1$ we deduce by Assumption 3.7-(ii) that

$\widehat{g}(n)=\left[1-\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})\widehat{\chi}(n)\right]^{-1}0=0.$

Since $g\in X_{2}$, it follows that $g=0$ and $N(I-K)=\\{0\\}$.

Step 3: Let us prove that $I-K\in\mathcal{L}(X_{2})$ is invertible. Next note
that it follows from the continuity of the translation in $L^{1}$ that $K$ is
a compact operator. Thus one has $R(I-K)=X_{2}$ by Fredholm Alternative (see
[15, Lemma 4.3.17]), where $R(I-K)$ denotes the range of $I-K$. Hence we have
that $I-K$ is invertible and that the inverse is continuous by bounded inverse
theorem.

Step 4: Let us prove that $D_{(\omega,\nu,z)}h(0,1,\nu_{0},0)$ is invertible.
To prove this, let $y\in C_{2\pi}(\mathbb{R})$ be given. Set
$L:=D_{(\omega,\nu,z)}h(0,1,\nu_{0},0)$ and let us consider the equation

$\left(\tilde{\omega},\tilde{\nu},\tilde{z}\right)\in\mathbb{R}\times\mathbb{R}\times
X_{2},\;\;L(\tilde{\omega},\tilde{\nu},\tilde{z})=y.$ (4.9)

Define the projectors $P_{1}:C_{2\pi}(\mathbb{R})\to X_{1}$ and
$P_{2}:C_{2\pi}(\mathbb{R})\to X_{2}$ associated to the state space
decomposition of Lemma 4.3 and set $y_{1}:=P_{1}y$ and $y_{2}:=P_{2}y$. Next
projecting (4.9) along $P_{1}$ and $P_{2}$ yields the following system, for
all $t\in\mathbb{R}$,

$\begin{array}[]{ll}y_{1}(t)=&-\tilde{\nu}\left[\partial_{\nu}\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})+\partial_{w}^{2}f(\nu_{0},\overline{w}_{\nu_{0}})\partial_{\nu}\overline{w}_{\nu_{0}}\right]\int_{0}^{\infty}\chi(l)u_{1}(t-l)dl\vspace{0.1cm}\\\
&-\tilde{\omega}\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})\int_{0}^{\infty}u_{1}(t-l)\mu_{\tau}(dl),\end{array}$
(4.10)

and

$\begin{array}[]{ll}y_{2}(t)=&\tilde{z}_{2}(t)-\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})\int_{0}^{\infty}\chi(l)\tilde{z}_{2}(t-l)dl.\end{array}$

Observe that $I-K$ is invertible in $X_{2}$, thus $\tilde{z}_{2}$ can be
solved by

$\tilde{z}_{2}=(I-K)^{-1}y_{2}.$

Let us focus on the resolution of (4.10). To that aim, recall that
$u_{1}=\cos(\cdot)$ and (4.8) so that (4.10) rewrites

$\begin{array}[]{ll}y_{1}(t)=&-\tilde{\nu}\left[\partial_{\nu}\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})+\partial_{w}^{2}f(\nu_{0},\overline{w}_{\nu_{0}})\partial_{\nu}\overline{w}_{\nu_{0}}\right]\int_{0}^{\infty}\chi(l)\cos
ldl\cos t\vspace{0.1cm}\\\
&-\tilde{\omega}\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})\left[\int_{0}^{\infty}\cos
l\mu_{\tau}(dl)\cos t+\int_{0}^{\infty}\sin l\mu_{\tau}(dl)\sin
t\right],\end{array}$

Furthermore by applying the integration by parts formula (3.1), we obtain

$\begin{array}[]{l}\int_{0}^{\infty}\cos
l\mu_{\tau}(dl)=\lim_{M\to\infty}\int_{0}^{M}\cos l\mu_{\tau}(dl)\\\
=\lim_{M\to\infty}\left\\{\left[\cos M\tau(M)-\cos
0\tau(0)\right]+\int_{0}^{M}\tau(l)\sin ldl\right\\},\end{array}$

hence

$\int_{0}^{\infty}\cos(l)\,\mu_{\tau}(dl)=\int_{0}^{\infty}l\chi(l)\sin(l)dl.$

Similarly, one can obtain

$\int_{0}^{\infty}\sin l\mu_{\tau}(dl)=-\int_{0}^{\infty}l\chi(l)\cos ldl.$

On the other hand, since $y_{1}\in X_{1}$, there exist two constants
$c_{1},c_{2}\in\mathbb{R}$ such that
$y_{1}=c_{1}\cos(\cdot)+c_{2}\sin(\cdot)$, while the $y_{1}$-equation can be
rewritten as the following system, for all $t\in\mathbb{R}$,

$\begin{array}[]{ll}&c_{1}\cos t+c_{2}\sin t\vspace{0.1cm}\\\
=&-\tilde{\nu}\left[\partial_{\nu}\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})+\partial_{w}^{2}f(\nu_{0},\overline{w}_{\nu_{0}})\partial_{\nu}\overline{w}_{\nu_{0}}\right]\int_{0}^{\infty}\chi(l)\cos
ldl\cos t\vspace{0.1cm}\\\
&-\tilde{\omega}\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})\left[\int_{0}^{\infty}l\chi(l)\sin
ldl\cos t-\int_{0}^{\infty}l\chi(l)\cos ldl\sin t\right],\end{array}$

and identifying the coefficients of $\cos(\cdot)$ and $\sin(\cdot)$ we end-up
with the resolution of the following two-dimensional linear system

$\begin{pmatrix}-\left[\partial_{\nu}\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})+\partial_{w}^{2}f(\nu_{0},\overline{w}_{\nu_{0}})\partial_{\nu}\overline{w}_{\nu_{0}}\right]\int_{0}^{\infty}\chi(l)\cos
ldl&-\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})\int_{0}^{\infty}l\chi(l)\sin
ldl\vspace{0.1cm}\\\
0&\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})\int_{0}^{\infty}l\chi(l)\cos
ldl\end{pmatrix}\begin{pmatrix}\tilde{\nu}\\\
\tilde{\omega}\end{pmatrix}=\begin{pmatrix}c_{1}\\\ c_{2}\end{pmatrix},$

To solve this linear system, let us show that our assumptions for the
characteristic equation , namely Assumption 3.7, ensures that the determinant
of the above matrix is non-zero, that reads as

$-\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})\left[\partial_{\nu}\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})+\partial_{w}^{2}f(\nu_{0},\overline{w}_{\nu_{0}})\partial_{\nu}\overline{w}_{\nu_{0}}\right]\int_{0}^{\infty}\chi(l)\cos
ldl\int_{0}^{\infty}l\chi(l)\cos ldl\neq 0.$ (4.11)

To check this property, recalling (4.8), it is sufficient to check that

$\left[\partial_{\nu}\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})+\partial_{w}^{2}f(\nu_{0},\overline{w}_{\nu_{0}})\partial_{\nu}\overline{w}_{\nu_{0}}\right]\int_{0}^{\infty}l\chi(l)\cos(l)dl\neq
0.$

Next set for $\theta\in\mathbb{R}$,

$\widehat{\chi}(\theta)=\int_{0}^{\infty}\chi(l)e^{-i\theta l}dl,$

and observe that the above condition (or equivalently (4.11)) is equivalent to

$\left[\partial_{\nu}\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})+\partial_{w}^{2}f(\nu_{0},\overline{w}_{\nu_{0}})\partial_{\nu}\overline{w}_{\nu_{0}}\right]{\rm
Im}\,\widehat{\chi}^{\prime}(1)\neq 0$ (4.12)

On the other hand, by differentiating the characteristic equation (3.2) with
respect to $\nu$ at $\nu=\nu_{0}$ (and recalling that $\omega_{0}=1$), we have

$\frac{d\lambda(\nu_{0})}{d\nu}\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})\int_{0}^{\infty}l\chi(l)e^{-i\,l}dl=\left[\partial_{\nu}\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})+\partial_{w}^{2}f(\nu_{0},\overline{w}_{\nu_{0}})\partial_{\nu}\overline{w}_{\nu_{0}}\right]\widehat{\chi}(1),$

which implies

$\frac{d\lambda(\nu_{0})}{d\nu}=\frac{\left[\partial_{\nu}\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})+\partial_{w}^{2}f(\nu_{0},\overline{w}_{\nu_{0}})\partial_{\nu}\overline{w}_{\nu_{0}}\right]\int_{0}^{\infty}\chi(l)e^{-il}dl}{\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})\int_{0}^{\infty}l\chi(l)e^{-il}dl}.$

Thus the transversality condition, that is Assumption 3.7-(iv), becomes

$\begin{array}[]{l}{\rm Re}\dfrac{d\lambda(\nu_{0})}{d\nu}={\rm
Re}\left\\{\dfrac{\left[\partial_{\nu}\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})+\partial_{w}^{2}f(\nu_{0},\overline{w}_{\nu_{0}})\partial_{\nu}\overline{w}_{\nu_{0}}\right]\widehat{\chi}(1)}{i\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})\widehat{\chi}^{\prime}(1)}\right\\}\vspace{0.2cm}\\\
={\rm
Re}\left\\{\left[\partial_{\nu}\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})+\partial_{w}^{2}f(\nu_{0},\overline{w}_{\nu_{0}})\partial_{\nu}\overline{w}_{\nu_{0}}\right]\underbrace{\widehat{\chi}(1)\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})}_{=1}\overline{i\widehat{\chi}^{\prime}(1)}\right\\}\vspace{0.2cm}\\\
=\left[\partial_{\nu}\partial_{w}f(\nu_{0},\overline{w}_{\nu_{0}})+\partial_{w}^{2}f(\nu_{0},\overline{w}_{\nu_{0}})\partial_{\nu}\overline{w}_{\nu_{0}}\right]{\rm
Im}\,\left(\widehat{\chi}^{\prime}(1)\right)\neq 0,\end{array}$

therefore (4.12) holds true and we can find a unique
$\left(\tilde{\nu},\tilde{\omega}\right)\in\mathbb{R}^{2}$ solving the above
two-dimensional linear system. This completes the proof of the lemma.

Last part of the proof of Theorem 4.1:  To conclude the proof of the Hopf
bifurcation Theorem 4.1, we apply the implicit function theorem (see Deimling
[5, Theorem 15.2]) to the function $h:\mathbb{R}^{3}\times X_{2}\rightarrow
C_{2\pi}(\mathbb{R})$ and we deduce that there exists a $C^{1}-$mapping
$(\omega,\nu,z):(-\delta,\delta)\rightarrow\mathbb{R}^{2}\times X_{2}$, for
some $\delta>0$ small enough, such that

$h(s,\omega(s),\nu(s),z(s))=0,\quad\forall s\in(-\delta,\delta).$

By the definition of $h$, this is equivalent to say that

$F(\omega(s),\nu(s),\overline{w}_{\nu(s)}+s(u_{1}+z(s)))=0,$

when $s\neq 0$ with
$(\omega(0),\nu(0),z(0))=(1,\nu_{0},\overline{w}_{\nu_{0}})$. We see that
$(\omega(s),\nu(s),\overline{w}_{\nu(s)}+s(u_{1}+z(s)))$ is the desired curve
of solutions of $F=0$. Thus the proof is complete.

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