# Periodic perturbations of a class of scalar second order functional
differential equations

Alessandro Calamai  Alessandro Calamai, Dipartimento di Ingegneria Civile,
Edile e Architettura, Università Politecnica delle Marche Via Brecce Bianche
I-60131 Ancona, Italy<EMAIL_ADDRESS>,  Maria Patrizia Pera  Maria
Patrizia Pera, Dipartimento di Matematica e Informatica “Ulisse Dini”,
Università degli Studi di Firenze, Via S. Marta 3, I-50139 Florence, Italy
<EMAIL_ADDRESS>and  Marco Spadini  Marco Spadini, Dipartimento di
Matematica e Informatica “Ulisse Dini”, Università degli Studi di Firenze, Via
S. Marta 3, I-50139 Florence, Italy<EMAIL_ADDRESS>
###### Abstract.

We study, by means of a topological approach, the forced oscillations of
second order functional retarded differential equations subject to periodic
perturbations. We consider a delay-type functional dependence involving a
gamma probability distribution. By a linear chain trick we obtain a first
order system of ODE’s whose $T$-periodic solutions correspond to those of the
functional equation.

###### Key words and phrases:

Functional differential equations, Branches of periodic solutions, Linear
chain trick, Brouwer degree.

###### 2010 Mathematics Subject Classification:

34K13, 34C25

The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la
Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta
Matematica (INdAM)

##  1\. Introduction and setting of the problem

In this paper we investigate the forced oscillations of a class of second
order functional retarded differential equations subject to periodic
perturbations. Namely, we study perturbations of _scalar, second order
functional differential equations_ of the following form:

(1)
$\ddot{x}(t)=g\left(x(t),\dot{x}(t),\int_{-\infty}^{t}\gamma_{a}^{b}(t-s)\varphi(x(s),\dot{x}(s))\,\mathrm{d}s\right),$

where $g\colon\mathbb{R}^{3}\to\mathbb{R}$ is a continuous map and
$\varphi\colon\mathbb{R}^{2}\to\mathbb{R}$ is locally Lipschitz. The integral
kernel $\gamma_{a}^{b}$, for $a>0$ and $b\in\mathbb{N}\setminus\\{0\\}$ is the
gamma probability distribution

(2)
$\gamma_{a}^{b}(s)=\frac{a^{b}s^{b-1}\mathrm{e}^{-as}}{(b-1)!}\;\;\text{for
$s\geq 0$},\qquad\gamma_{a}^{b}(s)=0\;\;\text{for $s<0$,}$

with mean $b/a$ and variance $b/a^{2}$.

The particular dependence on the past of the solution that is considered here
is one that naturally arises in many contexts (not necessarily for second
order equations) see, e.g., [5, 8, 17, 18, 28, 29]. In the equation we
examine, the delay is spread along the whole history but more concentrated at
a given point in the past. We can interpret this, from a probabilistic point
of view, as a delay that follows a gamma-type distribution with a given mean
(and variance). This approach seems to be reasonable in contexts when the
delay cannot be measured with precision, but its mean value and variance are
known, or it is genuinely spread in time as, e.g., in [14, 15]. Notice, in
particular, that letting $a$ and $b$ tend to infinity in such a way that the
quotient $r:=a/b$ remains constant (for instance, put $a=nr$ and $b=n$ and let
$n\to\infty$), one concentrates the memory effect close to the delay $r$.
Indeed, at least in the sense of distributions, (1) approximates the second
order delay differential equation

$\ddot{x}(t)=g\left(x(t),\dot{x}(t),\varphi\big{(}x(t-r),\dot{x}(t-r)\big{)}\right).$

Observe that the function $\gamma_{a}^{b}$ defined in (2) is continuous for
$b=2,3,\ldots$ but not for $b=1$. However, also in the latter case, assuming
$x$ in $C^{1}$, the function
$t\mapsto\int_{-\infty}^{t}\gamma_{a}^{1}(t-s)\varphi(x(s),\dot{x}(s))\,\mathrm{d}s$
is continuous. Then the right hand side of (1) is continuous as well;
consequently, any $C^{1}$ solution of (1) is actually of class $C^{2}$. It is
also worth noticing (and we will use this fact later) that, since in the
integral one has $t-s\geq 0$, then

$\int_{-\infty}^{t}\gamma_{a}^{1}(t-s)\varphi(x(s),\dot{x}(s))\,\mathrm{d}s=\int_{-\infty}^{t}a\mathrm{e}^{-a(t-s)}\varphi(x(s),\dot{x}(s))\,\mathrm{d}s,$

so that the function
$t\mapsto\int_{-\infty}^{t}\gamma_{a}^{1}(t-s)\varphi(x(s),\dot{x}(s))\,\mathrm{d}s$
is actually in $C^{1}$.

Given $T>0$, we consider the following $T$-periodic perturbation of (1):

(3)
$\ddot{x}(t)=g\left(x(t),\dot{x}(t),\int_{-\infty}^{t}\gamma_{a}^{b}(t-s)\varphi(x(s),\dot{x}(s))\,\mathrm{d}s\right)+\lambda
f\big{(}t,x(t),\dot{x}(t)\big{)},$

in which we assume that the map $f\colon\mathbb{R}^{3}\to\mathbb{R}$ is
continuous and $T$-periodic in the first variable, and $\lambda$ is a
nonnegative real parameter. Our main purpose is to investigate the set of
$T$-periodic solutions of (3). Here, given $\lambda\geq 0$, by a _$T$
-periodic solution_ (on $\mathbb{R}$) of (3) we mean a $C^{2}$ function
$x\colon\mathbb{R}\to\mathbb{R}$ of period $T$ that satisfies identically
equality (3).

Roughly speaking, we explicitly construct a scalar function $\Phi$ whose
change of sign implies the existence of a connected set, called a _“branch”_ ,
of nontrivial pairs $(\lambda,x)$ – with $x$ a $T$-periodic solution of (3)
corresponding to $\lambda\in[0,\infty)$ – that emanates out of the set of
zeros of $\Phi$ and whose closure is not compact.

We also give sufficient conditions yielding the multiplicity of $T$-periodic
solutions of (3) for $\lambda>0$ small. Such conditions are essentially based
upon the notion of _ejecting set_ (see, e.g., [22]).

The methods that we employ are topological in nature and are based on the
Brouwer degree. A familiarity with this notion, however, is not necessary to
use our main results since its application is restricted only to the proofs.
Nevertheless, we provide a brief summary.

Our results are so-to-speak dual to those of [10], where periodic
perturbations containing delay terms are applied to scalar, second order
ODE’s. The study, by means of topological methods, of the branching and
multiplicity of periodic solutions of periodically perturbed equations, is now
a well-investigated subject in the case of ODE’s both in Euclidean spaces and
on manifolds (see, e.g., [21, 24]). The case of ODE’s perturbed with delayed
forcing terms is also studied in the literature, although not so broadly (see,
e.g., [10, 11]). However, the presence of delay terms in the _unperturbed_
equation (1) is peculiar of the problem addressed here. In spite of the
apparent similarities, a different approach is called upon in order to manage
the unperturbed equation.

In the undelayed case, that is for periodically forced second-order scalar
autonomous ODE’s, both the problems of existence and multiplicity of periodic
solutions are quite classical. However, this is still the subject of active
research by the mathematical community. There are many approaches that have
been successfully pursued to get multiplicity results: among the others let us
mention here, e.g., the recent contributions [3, 4, 9, 19], the survey papers
[25, 26] as well as the monograph [20] and the references therein.

The strategy adopted in this paper is inspired to [30] and can be summarized
as follows. First we construct a system of $b+2$, $T$-periodically perturbed
first-order ordinary differential equations whose $T$-periodic solutions
correspond to that of (3). We then use known results about these perturbed
systems to ensure the existence of a branch as sought when the topological
degree of the unperturbed field $G$ has nonzero degree and finally, we show,
by homotopy techniques, that the change of sign of $\Phi$ imply that $G$ has
nonzero degree.

It should be noted that the result just described does not guarantee the
existence of forced oscillations even for very small values of $\lambda$. For
this reason, we also provide a nonlocal condition, based on an inequality of
J. A. Yorke [31], implying that the branch projects nontrivially onto
$[0,\infty)$.

In order to develop a better understanding of the nature of the branch of
$T$-periodic solutions and its relation with multiplicity results we discuss,
following [2] and [30], a method to visualize, in finite dimension, a
homemorphic set that retains all the relevant properties. We illustrate the
procedure with a numeric example.

##  2\. The linear chain trick

As pointed out in the introduction, a crucial remark is that the $T$-periodic
solutions of (3) and those of the following system of $b+2$ ordinary
differential equations correspond in some sense:

(4) $\dot{\xi}=G(\xi)+\lambda F(t,\xi),$

where $\lambda\geq 0$, $\xi=(u,v_{0},v_{1},\ldots,v_{b})\in\mathbb{R}^{b+2}$,
and the maps $G\colon\mathbb{R}^{b+2}\to\mathbb{R}^{b+2}$ and
$F\colon\mathbb{R}\times\mathbb{R}^{b+2}\to\mathbb{R}^{b+2}$ are respectively
defined as

(5) $\begin{split}G(u,v_{0},v_{1},\ldots,v_{b})=\>&\\\
\Big{(}v_{0},g(u,v_{0},v_{b})&,a\big{(}\varphi(u,v_{0})-v_{1}\big{)},a(v_{1}-v_{2})\ldots,a(v_{b-1}-v_{b})\Big{)}\end{split}$

and

$F(t,u,v_{0},v_{1},\ldots,v_{b})=\Big{(}0,f(t,u,v_{0}),\underbrace{0,\ldots,0}_{\text{$b$
times}}\Big{)},$

with $g$, $\varphi$ and $f$ as in equation (3). Clearly, $G$ and $F$ are
continuous maps. By a $T$-periodic solution (on $\mathbb{R}$) of system (4) we
mean a $C^{1}$ function $\xi\colon\mathbb{R}\to\mathbb{R}^{b+2}$ of period $T$
that satisfies (4) identically.

In our main results we will be concerned with the Brouwer degree of the map
$G$ (see Section 4 for details). Our first task will be to derive a formula
for the computation of the degree of $G$ in terms of the real-valued function
$\Phi\colon\mathbb{R}\to\mathbb{R}$, given by

(6) $\Phi(u)=g(u,0,\varphi(u,0)).$

######  Remark 2.1.

Observe that if $(\bar{u},v_{0},v_{1},\ldots,v_{b})\in G^{-1}(0)$ then
$\Phi(\bar{u})=0$, $v_{0}=0$ and
$v_{1}=v_{2}=\ldots=v_{b}=\varphi(\bar{u},0)$. Conversely, for any
$\bar{u}\in\Phi^{-1}(0)$, then
$G(\bar{u},0,\varphi(\bar{u},0)\ldots,\varphi(\bar{u},0))=0$.

Given a bounded open interval $(\alpha,\beta)\subseteq\mathbb{R}$, define the
open subset $W^{*}=(\alpha,\beta)\times\mathbb{R}^{b+1}$ of
$\mathbb{R}^{b+2}$. We have:

######  Theorem 2.2.

Suppose that $\Phi(\alpha)\cdot\Phi(\beta)<0$. Then $G$ is admissible for the
degree in $W^{*}$ and we have $\deg(G,W^{*})\neq 0$.

The proof of Theorem 2.2 is postponed to Section 4.

Let us show now how a “linear chain trick” (see, e.g., [29, 30]) can be used
to prove the correspondence between $T$-periodic solutions of the second order
equation (3) and of the first order system (4).

Let us introduce some notation. By $C_{T}^{n}(\mathbb{R}^{s})$, $n=0,1,2$, we
will denote the Banach space of the $T$-periodic $C^{n}$ maps
$x\colon\mathbb{R}\to\mathbb{R}^{s}$ with the the standard norm

$\|x\|_{C^{n}}=\sum_{i=0}^{n}\max_{t\in\mathbb{R}}|x^{(i)}(t)|.$

Here, $x^{(i)}$ denotes the $i$-th derivative of $x$, in particular $x^{(0)}$
coincides with $x$.

######  Theorem 2.3.

Suppose $x_{0}$ is a $T$-periodic solution of (3), and let

$\left\\{\begin{array}[]{l}y_{0}(t):=\dot{x}_{0}(t),\\\
y_{i}(t):=\int_{-\infty}^{t}\gamma_{a}^{i}(t-s)\varphi(x_{0}(s),y_{0}(s))\,\mathrm{d}s,\quad
i=1,\ldots,b\end{array}\right.$

for $t\in\mathbb{R}$. Then, $(x_{0},y_{0},y_{1}\ldots,y_{b})$ is a
$T$-periodic solution of (4).

###### Proof.

First notice that, since $x_{0}\in C_{T}^{2}(\mathbb{R})$ and
$\varphi\colon\mathbb{R}^{2}\to\mathbb{R}$ is continuous (we are actually
assuming it to be locally Lipschitz), it is not difficult to prove that the
functions $y_{i}$, $i=0,\ldots,b$, are in $C_{T}^{1}(\mathbb{R})$.

Moreover, we claim that, for any $t\in\mathbb{R}$,

$\left\\{\begin{array}[]{l}\dot{x}_{0}(t)=y_{0}(t),\\\
\dot{y}_{0}(t)=g\big{(}x_{0}(t),y_{0}(t),y_{b}(t)\big{)}+\lambda
f\big{(}t,x_{0}(t),y_{0}(t)\big{)},\quad\lambda\geq 0,\\\
\dot{y}_{1}(t)=a\big{(}\varphi(x_{0}(t),y_{0}(t))-y_{1}(t)),\\\
\dot{y}_{i}(t)=a\big{(}y_{i-1}(t)-y_{i}(t)\big{)},\quad
i=2,\ldots,b.\end{array}\right.$

Indeed, the first equality follows by definition. Since $x_{0}$ is a solution
of (3), for $\lambda\geq 0$ we have

$\displaystyle\dot{y}_{0}(t)$
$\displaystyle=\ddot{x}_{0}(t)=g\left(x_{0}(t),\dot{x}_{0}(t),\int_{-\infty}^{t}\gamma_{a}^{b}(t-s)\varphi(x_{0}(s),\dot{x}_{0}(s))\,\mathrm{d}s\right)+\lambda
f\big{(}t,x_{0}(t),\dot{x}_{0}(t)\big{)}$
$\displaystyle=g\big{(}x_{0}(t),y_{0}(t),y_{b}(t)\big{)}+\lambda
f\big{(}t,x_{0}(t)y_{0}(t)\big{)},\quad\forall t\in\mathbb{R}.$

Now observe that, when $i=1$, we have for any $t\in\mathbb{R}$,

$y_{1}(t):=\int_{-\infty}^{t}\gamma_{a}^{1}(t-s)\varphi(x_{0}(s),y_{0}(s))\,\mathrm{d}s=\int_{-\infty}^{t}a\mathrm{e}^{a(t-s)}\varphi(x_{0}(s),y_{0}(s))\,\mathrm{d}s.$

Thus, taking the derivative under the integral sign,

$\frac{\,\mathrm{d}\phantom{t}}{\,\mathrm{d}t}\,y_{1}(t)=\varphi(x_{0}(t),y_{0}(t))-a\int_{-\infty}^{t}\mathrm{e}^{a(t-s)}\,\varphi(x_{0}(s),y_{0}(s))\,\mathrm{d}s$

so that

$\dot{y}_{1}(t)=a\big{(}\varphi(x_{0}(t),y_{0}(t))-y_{1}(t)\big{)}.$

Also, for $i=2,\ldots,b$, we get for any $t\in\mathbb{R}$,

$\displaystyle\frac{\,\mathrm{d}\phantom{t}}{\,\mathrm{d}t}\,y_{i}(t)$
$\displaystyle=\gamma_{a}^{i}(0)\,\varphi(x_{0}(t),y_{0}(t))+\int_{-\infty}^{t}\frac{\,\mathrm{d}\phantom{t}}{\,\mathrm{d}t}\,\gamma_{a}^{i}(t-s)\,\varphi(x_{0}(s),y_{0}(s))\,\mathrm{d}s$
$\displaystyle=\int_{-\infty}^{t}a\left(\gamma_{a}^{i-1}(t-s)-\gamma_{a}^{i}(t-s)\right)\,\varphi(x_{0}(s),y_{0}(s))\,\mathrm{d}s$
$\displaystyle=a\left(\int_{-\infty}^{t}\gamma_{a}^{i-1}(t-s)\,\varphi(x_{0}(s),y_{0}(s))\,\mathrm{d}s-\int_{-\infty}^{t}\gamma_{a}^{i}(t-s)\,\varphi(x_{0}(s),y_{0}(s))\,\mathrm{d}s\right)$
$\displaystyle=a\left(y_{i-1}(t)-y_{i}(t)\right)$

proving our claim.

Finally, to see that $(x_{0},y_{0},y_{1}\ldots,y_{b})$ is $T$-periodic,
observe that so are $x_{0}$ and $y_{0}$, and with the change of variable
$\sigma=s-T$ we get

$\displaystyle y_{i}(t+T)=$
$\displaystyle\int_{-\infty}^{t+T}\gamma_{a}^{i}(T+t-s)\varphi(x_{0}(s),y_{0}(s))\,\mathrm{d}s=$
$\displaystyle\int_{-\infty}^{t}\gamma_{a}^{i}(t-\sigma)\varphi(x_{0}(\sigma+T),y_{0}(\sigma+T))\,\mathrm{d}\sigma=$
$\displaystyle\int_{-\infty}^{t}\gamma_{a}^{i}(t-\sigma)\varphi(x_{0}(\sigma),y_{0}(\sigma))\,\mathrm{d}\sigma=y_{i}(t),$

for $i=1,\ldots,b$ and any $t\in\mathbb{R}$, and this completes the proof. ∎

Conversely, we have the following:

######  Theorem 2.4.

Suppose that $(x_{0},y_{0},y_{1}\ldots,y_{b})$ is a $T$-periodic solution of
(4), then $x_{0}$ is a $T$-periodic solution of (3).

The proof of Theorem 2.4 is based on the following technical lemma on linear
systems of ODEs, see [30, Lemma 3.3], cf. also [29, Prop. 7.3]. The proof is
omitted.

######  Lemma 2.5 ([30], Lemma 3.3).

Given any continuous and bounded function $z_{0}\colon\mathbb{R}\to\mathbb{R}$
and any nonzero number $a$, there exists a unique $C^{1}$ solution
$z=(z_{1},\dots,z_{b}),\,z_{i}\colon\mathbb{R}\to\mathbb{R},\,i=1,\ldots,b,$
of the system in $\mathbb{R}^{b}$

$\dot{z}_{i}(t)=a\big{(}z_{i-1}(t)-z_{i}(t)\big{)},$

which is bounded in the $C^{1}$ norm. This solution is given by

$z_{i}(t)=\int_{-\infty}^{t}\gamma_{a}^{i}(t-s)z_{0}(s)\,\mathrm{d}s,\quad
i=1,\ldots,b.$

######  Remark 2.6.

Observe in particular that, for $i=1$ in the previous lemma, one has

$z_{1}(t)=\int_{-\infty}^{t}\gamma_{a}^{1}(t-s)z_{0}(s)\,\mathrm{d}s=a\int_{-\infty}^{t}\mathrm{e}^{a(t-s)}z_{0}(s)\,\mathrm{d}s,$

so that $z_{1}$ is actually a $C^{1}$ function, and thus so are all the
$z_{i}$’s for $i=2,\ldots,b$.

###### Proof of Theorem 2.4.

Let $(x_{0},y_{0},y_{1}\ldots,y_{b})$ be a $T$-periodic solution of (4), and
define $z_{0}(t)=\varphi(x_{0}(t),y_{0}(t))$ for $t\in\mathbb{R}$. Observe
that $z_{0}$ is bounded and (at least) continuous. Thus, by Lemma 2.5,

$y_{i}(t)=\int_{-\infty}^{t}\gamma_{a}^{i}(t-s)\varphi(x_{0}(s),y_{0}(s))\,\mathrm{d}s\quad
i=1,\ldots,b,$

is the unique solution of class $C^{1}$ of

$\left\\{\begin{array}[]{l}\dot{y}_{1}(t)=a\big{(}\varphi(x_{0}(t),y_{0}(t))-y_{1}(t)),\\\
\dot{y}_{i}(t)=a\big{(}y_{i-1}(t)-y_{i}(t)\big{)},\quad
i=2,\ldots,b.\end{array}\right.$

In particular, we have

$y_{b}(t)=\int_{-\infty}^{t}\gamma_{a}^{b}(t-s)\varphi(x_{0}(s),y_{0}(s))\,\mathrm{d}s.$

Thus, from (4),

$\ddot{x}_{0}(t)=\dot{y}_{0}(t)=g\big{(}x_{0}(t),y_{0}(t),y_{b}(t)\big{)}+\lambda
f\big{(}t,x_{0}(t)y_{0}(t)\big{)}=\\\
=g\left(x_{0}(t),\dot{x}_{0}(t),\int_{-\infty}^{t}\gamma_{a}^{b}(t-s)\varphi(x_{0}(s),\dot{x}_{0}(s))\,\mathrm{d}s\right)+\lambda
f\big{(}t,x_{0}(t),\dot{x}_{0}(t)\big{)},\quad\lambda\geq 0,$

for all $t\in\mathbb{R}$, whence the assertion. ∎

##  3\. Branches of $T$-pairs

This section investigates the structure of the set of $T$-periodic solutions
of (3). We begin by recalling some notation and basic facts.

Consider the following first-order parameterized ODE on $\mathbb{R}^{k}$:

(7)
$\dot{x}(t)=\mathfrak{g}\big{(}x(t)\big{)}+\lambda\mathfrak{f}\big{(}t,x(t)\big{)},$

where $\lambda\geq 0$, the maps
$\mathfrak{g}\colon\mathbb{R}^{k}\to\mathbb{R}^{k}$ and
$\mathfrak{f}\colon\mathbb{R}\times\mathbb{R}^{k}\to\mathbb{R}^{k}$ are
continuous and $\mathfrak{f}$ is $T$-periodic in the first variable.

We say that a pair $(\lambda,x)\in[0,\infty)\times C_{T}(\mathbb{R}^{k})$ is a
_$T$ -pair_ (for (7)) if $x$ is a $T$-periodic solution of (7) corresponding
to $\lambda$. If $\lambda=0$ and $x$ is constant then the $T$-pair is said
_trivial_. It is not hard to see that the trivial $T$-periodic pairs of (7)
correspond to the zeros of $\mathfrak{g}$. From now on, given any
$p\in\mathbb{R}^{k}$, we will denote by $\overline{p}$ the constant map
$t\mapsto p$, $t\in\mathbb{R}$. Given an open subset $\mathcal{O}$ of
$[0,\infty)\times C_{T}(\mathbb{R}^{k})$, we will denote by
$\widetilde{\mathcal{O}}\subseteq\mathbb{R}^{k}$ the open set
$\widetilde{\mathcal{O}}:=\big{\\{}p:(0,\overline{p})\in\mathcal{O}\big{\\}}$.

We have the following fact concerning the $T$-pairs of (7):

######  Theorem 3.1 ([23]).

Let $\mathcal{O}$ be open in $[0,\infty)\times C_{T}(\mathbb{R}^{k})$, and
assume that $\deg(\mathfrak{g},\widetilde{\mathcal{O}})$ is well defined and
nonzero. Then there exists a connected set $\Gamma$ of nontrivial $T$-pairs in
$[0,\infty)\times C_{T}(\mathbb{R}^{k})$ whose closure in $\mathcal{O}$ is not
compact and meets the set of trivial $T$-pairs contained in $\mathcal{O}$,
namely the set:
$\big{\\{}(0,\overline{p})\in\mathcal{O}:\mathfrak{g}(p)=0\big{\\}}$.

Let us now go back to the second-order equation (3). We need some further
notation. The pairs $(\lambda,x)\in[0,+\infty)\times C^{1}_{T}(\mathbb{R})$,
with $x\colon\mathbb{R}\to\mathbb{R}$ a $T$-periodic solution of (3)
corresponding to $\lambda$, will be called _$T$ -forced pairs_ (of (3)), and
we denote by $X$ the set of these pairs. Among the $T$-forced pairs we shall
consider as _trivial_ those of the type $(0,x)$ with $x$ constant. Given an
open subset $\Omega$ of $[0,\infty)\times C^{1}_{T}(\mathbb{R})$, we will
denote by
$\widetilde{\Omega}:=\big{\\{}u\in\mathbb{R}:(0,\overline{u})\in\Omega\big{\\}}$,
where by $\overline{u}$ we mean the constant map $t\mapsto u$,
$t\in\mathbb{R}$.

As pointed out in Section 2, there is a correspondence between $T$-periodic
solutions of the second order equation (3) and of the first order system in
(4). A similar correspondence holds between their “$T$-periodic pairs” in a
sense that we are going to specify. We point out that, since equation (4) can
be seen as a special case of (7), in accordance with the notation introduced
above, a pair $(\lambda,\xi)\in[0,\infty)\times C_{T}(\mathbb{R}^{b+2})$ is
called a _$T$ -pair_ if $\xi$ is a $T$-periodic solution of (4) corresponding
to $\lambda$. We also recall that the set of $T$-forced pairs of (3) is
regarded as a subset of $[0,+\infty)\times C^{1}_{T}(\mathbb{R})$.

As a first remark we relate the corresponding “trivial $T$-periodic pairs”.

######  Remark 3.2.

Observe that if $(0,\overline{q})$, with $q\in\mathbb{R}$, is a trivial
$T$-forced pair of (3), then $(0,\overline{\xi_{0}})$ is a $T$-pair for
equation (4), where $\xi_{0}\in\mathbb{R}^{b+2}$ is given by

$\xi_{0}:=(q,0,\varphi(q,0)\ldots,\varphi(q,0))$

that is, by Remark 2.1, $\xi_{0}\in G^{-1}(0)$. Conversely, any trivial
$T$-pair of (4) must be of the form $(0,\overline{\xi_{0}})$, where
$\xi_{0}\in\mathbb{R}^{b+2}$ is such that $G(\xi_{0})=0$. Thus, again by
Remark 2.1, we have $\xi_{0}=(q,0,\varphi(q,0)\ldots,\varphi(q,0))$ for some
$q\in\mathbb{R}$, and consequently $(0,\overline{q})$ is a trivial $T$-forced
pair for (4).

Let us now establish a general correspondence between the sets of $T$-forced
pairs and of $T$-pairs, that preserves the notion of triviality.

Let $\mathcal{J}\colon[0,+\infty)\times
C^{1}_{T}(\mathbb{R})\to[0,\infty)\times C_{T}(\mathbb{R}^{b+2})$ be the map
defined as follows:

(8) $\mathcal{J}:(\lambda,x_{0})\mapsto(\lambda,\xi)$

where $\xi:=(x_{0},y_{0},y_{1}\ldots,y_{b})$ is given by

$\left\\{\begin{array}[]{l}y_{0}(t):=\dot{x}_{0}(t),\\\
y_{i}(t):=\int_{-\infty}^{t}\gamma_{a}^{i}(t-s)\varphi(x_{0}(s),y_{0}(s))\,\mathrm{d}s,\quad
i=1,\ldots,b.\end{array}\right.$

As above, denote by $X$ the set of the $T$-forced pairs of (3), and let
$Y\subseteq[0,\infty)\times C_{T}(\mathbb{R}^{b+2})$ be the set of the
$T$-pairs of (4).

######  Lemma 3.3.

Let $\left.\mathcal{J}\right|_{X}\colon X\to Y$ be the restriction to $X$ of
$\mathcal{J}$. Then $\left.\mathcal{J}\right|_{X}$ is a homeomorphism of $X$
onto $Y$, which establishes a bijective correspondence between the trivial
$T$-forced pairs in $X$ and the trivial $T$-pairs in $Y$.

###### Proof.

Since $\varphi$ is locally Lipschitz continuous, the continuity of
$\mathcal{J}$ is obtained by construction. Whence we get the continuity of the
restriction $\left.\mathcal{J}\right|_{X}$. Injectivity and surjectivity of
$\left.\mathcal{J}\right|_{X}\colon X\to Y$ follow from theorems 2.3 and 2.4.

The inverse map $\left(\left.\mathcal{J}\right|_{X}\right)^{-1}$ can be seen
merely as the projection onto the first two components, so it is continuous.

Finally the last part of the assertion follows by Remark 3.2. ∎

We are now in a position to state and prove our main result concerning the set
of $T$-forced pairs of (3).

######  Theorem 3.4.

Denote by $X\subseteq[0,\infty)\times C^{1}_{T}(\mathbb{R})$ the set of
$T$-forced pairs of (3). Let $\Phi$ be the real-valued function defined in
(6), and suppose that $(\alpha,\beta)\subseteq\mathbb{R}$ is such that
$\Phi(\alpha)\cdot\Phi(\beta)<0$. Let $\Omega\subseteq[0,\infty)\times
C^{1}_{T}(\mathbb{R})$ be open and such that
$\widetilde{\Omega}=(\alpha,\beta)$. Then, in $X\cap\Omega$ there is a
connected subset $\Gamma$ of nontrivial $T$-forced pairs whose closure
relative to $\Omega$ is not compact and intersects the set

(9)
$\big{\\{}(0,\overline{u})\in\Omega:u\in(\alpha,\beta)\cap\Phi^{-1}(0)\big{\\}}.$

###### Proof.

Let $\mathcal{O}=\Omega\times C_{T}(\mathbb{R}\times\mathbb{R}^{b})$. Clearly,
$\widetilde{\mathcal{O}}=\widetilde{\Omega}\times\mathbb{R}\times\mathbb{R}^{b}=(\alpha,\beta)\times\mathbb{R}\times\mathbb{R}^{b}$.
Thus, Theorem 2.2 implies that $\deg(G,\widetilde{\mathcal{O}})$ is well-
defined and nonzero. So, there exists a connected set $\Upsilon$ of nontrivial
$T$-pairs for (4) as in Theorem 3.1. Finally, by Lemma 3.3, one sees that
$\Gamma:=\mathcal{J}^{-1}(\Upsilon)$ is a connected set as in the assertion. ∎

We remark that, as in the well-known case of the resonant harmonic oscillator

$\ddot{x}=-x+\lambda\sin t,$

this unbounded branch is possibly contained in the slice $\\{0\\}\times
C^{1}_{T}(\mathbb{R})$. We will discuss some conditions preventing this
“pathological” situation in Section 5.

##  4\. Computation of the degree

###  4.1. Brouwer degree in Euclidean spaces

We will make use of the Brouwer degree in $\mathbb{R}^{k}$ in a slightly
extended version (see e.g. [1, 16, 27])

Let $U$ be an open subset of $\mathbb{R}^{k}$, $f$ a continuous
$\mathbb{R}^{k}$-valued map whose domain contains the closure $\overline{U}$
of $U$, and $q\in\mathbb{R}^{k}$. We say that the triple $(f,U,q)$ is
_admissible (for the Brouwer degree)_ if $f^{-1}(q)\cap U$ is compact.

The Brouwer degree is a function that to any admissible triple $(f,U,q)$
assigns an integer, denoted by $\deg(f,U,q)$ and called the _Brouwer degree of
$f$ in $U$ with target $q$_. Roughly speaking, $\deg(f,U,q)$ is an algebraic
count of the solutions in $U$ of the equation $f(p)=q$. In fact, one of the
properties of this integer-valued function is given by the following
_computation formula._

Recall that, if $f\colon U\to\mathcal{\mathbb{R}}^{k}$ is a $C^{1}$ map, an
element $p\in U$ is said to be a _regular point_ (of $f$) if the differential
of $f$ at $p$, $df_{p}$, is surjective. Non-regular points are called
_critical (points)_. The _critical values_ of $f$ are those points of
$\mathbb{R}^{k}$ which lie in the image $f(C)$ of the set $C$ of critical
points. Any $q\in\mathcal{\mathbb{R}}^{k}$ which is not in $f(C)$ is a
_regular value_. Therefore, in particular, any element of $\mathbb{R}^{k}$
which is not in the image of $f$ is a regular value.

Computation formula. If $(f,U,q)$ is admissible, $f$ is smooth, and $q$ is a
regular value for $f$ in $U$, then

(10) $\deg(f,U,q)\;=\sum_{p\in f^{-1}(q)\cap
U}\operatorname{\mathrm{sign}}(df_{p}).$

This formula is actually the basic definition of the Brouwer degree, and the
integer associated to any admissible triple $(g,U,r)$ is defined by

(11) $\deg(g,U,r):=\deg(f,U,q),$

where $f$ and $q$ satisfy the assumptions of the Computation Formula and are,
respectively, “sufficiently close” to $g$ and $r$. It is known that this is a
well-posed definition.

The more classical and well-known definition of Brouwer degree is usually
given in the subclass of triples $(f,U,q)$ such that
$f\colon\overline{U}\to\mathbb{R}^{k}$ is continuous, $U$ is bounded and
$q\notin f(\partial U)$. However, all the standard properties of the degree,
such as homotopy invariance, excision, additivity, existence, are still valid
in this more general context.

Since in this paper the target point $q$ will always be the origin, for the
sake of simplicity, we will simply write $\deg(f,U)$ instead of $\deg(f,U,0)$.
In this context, we will say that an element $p\in f^{-1}(0)$ is a
_nondegenerate zero_ (of $f$) if $\det(df_{p})\neq 0$; this means,
equivalently, that $p$ is a regular point. Observe that $\deg(f,U)$ can be
regarded also as the degree (or characteristic, or rotation) of the map $f$
seen as a tangent vector field on $\mathbb{R}^{k}$.

###  4.2. The degree of the map $G$

The last part of this section is devoted to the proof of Theorem 2.2. Roughly
speaking we will relate the degree of the map $G$, defined in (5) by

$G(u,v_{0},v_{1},\ldots,v_{b})=\left(v_{0},g(u,v_{0},v_{b}),a(\varphi(u,v_{0})-v_{1}),a(v_{1}-v_{2})\ldots,a(v_{b-1}-v_{b})\right),$

with that of the function $\Phi\colon\mathbb{R}\to\mathbb{R}$ in (6), given by

$\Phi(u)=g(u,0,\varphi(u,0)).$

To simplify the computation of the degree, it is convenient to introduce the
following map on $\mathbb{R}^{b+2}$:

$\mathcal{G}(u,v_{0},v_{1},\ldots,v_{b}):=\left(v_{0},g(u,v_{0},v_{b}),a(\varphi(u,v_{0})-v_{b}),a(v_{1}-v_{2})\ldots,a(v_{b-1}-v_{b})\right)$

######  Lemma 4.1.

Let $G$ and $\mathcal{G}$ be as above, and let $V\subseteq\mathbb{R}^{b+2}$ be
open. Suppose that one of the maps $G$ or $\mathcal{G}$ is admissible for the
Brouwer degree in $V$; then, so is the other and they are admissibly homotopic
in $V$.

###### Proof.

For $(\lambda,u,v_{0},v_{1},\ldots,v_{b})\in[0,1]\times V$, consider the map

$\mathcal{H}(\lambda,u,v_{0},v_{1},\ldots,v_{b})=\\\
=\left(v_{0},g(u,v_{0},v_{b}),a(\varphi(u,v_{0})-[\lambda
v_{b}+(1-\lambda)v_{1}]),a(v_{1}-v_{2})\ldots,a(v_{b-1}-v_{b})\right),$

and observe that $\mathcal{H}(\lambda,u,v_{0},v_{1},\ldots,v_{b})=0$ if and
only if $(u,v_{0},v_{1},\ldots,v_{b})\in V\cap
G^{-1}(0)=V\cap\mathcal{G}^{-1}(0)$. Hence $\mathcal{H}$ is an admissible
homotopy. ∎

Now we prove that if the zeros of $\Phi$ are nondegenerate, so are those of
$\mathcal{G}$.

######  Lemma 4.2.

All zeros of $\Phi$ are nondegenerate if and only if all zeros of
$\mathcal{G}$ are nondegenerate.

###### Proof.

First observe that, as in Remark 2.1, the zeros of $\mathcal{G}$ and those of
$\Phi$ correspond. In fact, if
$\mathcal{G}(\bar{u},v_{0},v_{1},\ldots,v_{b})=0$ then $\Phi(\bar{u})=0$,
$v_{0}=0$ and $v_{1}=v_{2}=\ldots=v_{b}=\varphi(\bar{u},0)=:\bar{w}$.
Conversely, for any $\bar{u}\in\Phi^{-1}(0)$, then
$\mathcal{G}(\bar{u},0,\bar{w}\ldots,\bar{w})=0$.

Assume now that $\bar{\xi}:=(\bar{u},0,\bar{w}\ldots,\bar{w})$ is a
nondegenerate zero of $\mathcal{G}$. We compute the Jacobian matrix of
$\mathcal{G}$ at $\bar{\xi}$ as follows:

$J\mathcal{G}_{\bar{\xi}}=\left(\begin{array}[]{cc|ccccc|c}0&1&0&0&\ldots&\ldots&0&0\\\
\partial_{1}g(\bar{u},0,\bar{w})&\partial_{2}g(\bar{u},0,\bar{w})&0&0&\ldots&\ldots&0&\partial_{3}g(\bar{u},0,\bar{w})\\\
a\,\partial_{1}\varphi(\bar{u},0)&a\,\partial_{2}\varphi(\bar{u},0)&0&0&\ldots&\ldots&0&-a\\\
\hline\cr 0&0&a&-a&0&\ldots&0&0\\\ 0&0&0&a&-a&\ddots&0&0\\\
\vdots&\vdots&\ddots&\ddots&\ddots&\ddots&\vdots&\vdots\\\
\vdots&\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots\\\
0&0&0&0&\ldots&0&a&-a\end{array}\right)\;.$

By direct computation based on Laplace expansion we obtain

$\det J\mathcal{G}_{\bar{\xi}}=(-a)^{b-1}\,\det\begin{pmatrix}0&1&0\\\
\partial_{1}g(\bar{u},0,\bar{w})&\partial_{2}g(\bar{u},0,\bar{w})&\partial_{3}g(\bar{u},0,\bar{w})\\\
a\partial_{1}\varphi(\bar{u},0)&a\partial_{2}\varphi(\bar{u},0)&-a\end{pmatrix}=$
$=-(-a)^{b-1}\,\det\begin{pmatrix}\partial_{1}g(\bar{u},0,\bar{w})&\partial_{3}g(\bar{u},0,\bar{w})\\\
a\partial_{1}\varphi(\bar{u},0)&-a\end{pmatrix}=$
$=(-1)^{b-1}a^{b}\cdot\left[\partial_{1}g(\bar{u},0,\bar{w})+\partial_{3}g(\bar{u},0,\bar{w})\,\partial_{1}\varphi(\bar{u},0)\right]=(-1)^{b-1}a^{b}\cdot\Phi^{\prime}(\bar{u}).$

Thus $\bar{u}$ is a nondegenerate zero of the function $\Phi$. The proof of
the converse implication is analogous. ∎

Let us now compute the degree of the map $\mathcal{G}$.

######  Lemma 4.3.

Assume $\Phi$ is admissible on an open set $U\subseteq\mathbb{R}$ and all its
zeros are nondegenerate, then $\mathcal{G}$ is admissible in
$U^{*}=U\times\mathbb{R}^{b+1}$, and

$\deg(\mathcal{G},U^{*})=(-1)^{b-1}\deg(\Phi,U).$

###### Proof.

As in the proof of Lemma 4.2 we have that all zeros of $\mathcal{G}$ are of
the form $\bar{\xi}:=(\bar{u},0,\bar{w}\ldots,\bar{w})$ with
$\Phi(\bar{u})=0$. Moreover, they are all nondegenerate. In particular, if
$\Phi$ is admissible in $U$ then $\Phi^{-1}(0)\cap U$ is compact and so is
$\mathcal{G}^{-1}(0)\cap U^{*}$, whence the admissibility of $\mathcal{G}$ in
$U^{*}$. Let now $\bar{u}\in\Phi^{-1}(0)\cap U$ be a nondegenerate zero of
$\Phi$. As in the proof of Lemma 4.2 we have,

$\det J\mathcal{G}_{\bar{\xi}}=(-1)^{b-1}a^{b}\cdot\Phi^{\prime}(\bar{u})$

and, consequently,

$\operatorname{\mathrm{sign}}\det\left(d\mathcal{G}_{\bar{\xi}}\right)=(-1)^{b-1}\operatorname{\mathrm{sign}}\det\Phi^{\prime}(\bar{u})$

Thus, by formula (10),

$\deg(\mathcal{G},U^{*})=\sum_{\bar{\xi}\in\mathcal{G}^{-1}(0)\cap
U^{*}}\operatorname{\mathrm{sign}}\det\left(d\mathcal{G}_{\bar{\xi}}\right)=\\\
=\sum_{\bar{u}\in\Phi^{-1}(0)\cap
U}(-1)^{b-1}\operatorname{\mathrm{sign}}\det\Phi^{\prime}(\bar{u})=(-1)^{b-1}\deg(\Phi,U),$

whence the assertion. ∎

The above Lemmas imply now the assertion of Theorem 2.2.

###### Proof of Theorem 2.2.

Let $(\alpha,\beta)\subseteq\mathbb{R}$ be as in the assertion of Theorem 2.2.
As in equality (11), we can assume that all zeros of $\Phi$ in
$(\alpha,\beta)$ are nondegenerate. The assertion now follows from Lemma 4.3
and Lemma 4.1. ∎

##  5\. Ejecting points and small perturbations

As we have seen by the simple example at the end of Section 3, it may happen
that the branch $\Gamma$ of $T$-forced pairs of equation (3), as in the
assertion of Theorem 3.4, is completely contained in the slice $\\{0\\}\times
C_{T}^{1}(\mathbb{R})$. In this section we show simple conditions ensuring
that this is not the case. Such conditions are based on the key notion of
_ejecting set_ (see, e.g., [22]).

Let us first introduce the following notation: let $\Upsilon$ be a subset of
$[0,\infty)\times C_{T}^{1}(\mathbb{R})$. Given $\lambda\geq 0$, let
$\Upsilon_{\lambda}$ be the slice $\big{\\{}x\in
C_{T}^{1}(\mathbb{R}):(\lambda,x)\in\Upsilon\big{\\}}$. Below, we adapt to our
context the definition of ejecting set of [22]:

######  Definition 5.1.

Let $X\subseteq[0,\infty)\times C_{T}^{1}(\mathbb{R})$ be the set of
$T$-forced pairs of (3), and let $X_{0}$ be the slice of $X$ at $\lambda=0$.
We say that $A\subset X_{0}$ is an _ejecting set_ (for $X$) if it is
relatively open in $X_{0}$ and there exists a connected subset of $X$ which
meets $A$ and is not contained in $X_{0}$. In particular, when $A=\\{p_{0}\\}$
is a singleton we say that $p_{0}$ is an _ejecting point_.

We now discuss a sufficient condition for an isolated point of $\Phi$ to be
ejecting for the set $X$ of $T$-forced pairs of (3). This condition is based
on a result by J. Yorke ([31], see also [6]) concerning the period of
solutions of an autonomous ODE with Lipschitz continuous right-hand side.

We point out that an analysis of (4), for $\lambda=0$, linearized at its zeros
leads to a different, not entirely comparable, approach. A discussion of the
latter technique, which is based on the notion of $T$-resonance and requires
the knowledge of the spectrum of the linearized equation is outside the scope
of the present paper (see, e.g., [13, Ch. 7] and [12, Ch. 2], a similar idea
can be traced back to Poincaré see, e.g., [26]; see also [2] for an
application of this idea to multiplicity results).

The aforementioned result of Yorke is the following:

######  Theorem 5.2 ([31]).

Let $\xi$ be a nonconstant $\tau$-periodic solution of

$\dot{x}=\mathcal{F}(x)$

where $\mathcal{F}\colon W\subseteq\mathbb{R}^{k}\to\mathbb{R}^{k}$ is
Lipschitz of constant $L$. Then the period $\tau$ of $\xi$ satisfies $\tau\geq
2\pi/L$.

Inequalities on the period like that of Theorem 5.2 have been object of study
in different contexts, see e.g., the introduction of [7] for an interesting
discussion.

Consider the map $G$ defined in (5), and suppose that $g$ and $\varphi$ are
Lipschitz continuous. Then, so is $G$. Let $L_{G}$ be the Lipschitz constant
of $G$.

From Theorem 5.2 we immediately deduce the following facts concerning the
branch of $T$-forced pairs of (3), given by Theorem 3.4:

######  Corollary 5.3.

Let $I$, $\Omega$ and $\Phi$ be as in Theorem 3.4. If the period $T$ of the
forcing term $f$ satisfies $T<2\pi/L_{G}$, then the set $\Gamma$ of Theorem
3.4 cannot intersect the slice $\\{0\\}\times C_{T}^{1}(\mathbb{R})$.

###### Proof.

Assume by contradiction that there exists a nontrivial $T$-forced pair
$(0,x_{0})\in\Gamma$. Then $x_{0}$ corresponds (in the sense of Theorem 2.3)
to a $T$-periodic solution of $\dot{\xi}=G(\xi)$ with $T<2\pi/L_{G}$,
violating Theorem 5.2. ∎

A similar argument yields the following result:

######  Corollary 5.4.

Let $I$, $\Omega$ and $\Phi$ be as in Theorem 3.4. Let $p\in I$ be an isolated
zero of $\Phi$ and assume that $G$ is locally Lipschitz in a neighborhood $W$
of the corresponding zero
$P=\big{(}p,0,\varphi(p,0),\ldots,\varphi(p,0)\big{)}$ of $G$. Suppose that a
set $\Upsilon$ of $T$-pairs for (4) contains $\left(0,\overline{P}\right)$.
Let $\Lambda=\mathcal{J}^{-1}(\Upsilon)$ be the corresponding set of
$T$-forced pairs of (3). As above, call $L_{G}$ the Lipschitz constant of $G$
in $W$. Then, the trivial $T$-forced pair $(0,\bar{p})$ is isolated in the
slice $\Lambda_{0}$. In particular, if $T<2\pi/L_{G}$, the set $\Lambda$
cannot be contained in the slice $\\{0\\}\times C_{T}^{1}(\mathbb{R})$.

Observe that, unlike $\Gamma$ in Corollary 5.3, the set $\Upsilon$ in
Corollary 5.4 does not necessarily consist of nontrivial $T$-pairs.
Consequently, $\Lambda$ does not necessarily consist of nontrivial $T$-forced
pairs as well. Notice also that since $p$ is an isolated zero of $\Phi$ then
$\left(0,\overline{P}\right)$, is isolated in the set of trivial $T$-pairs of
(4) and, similarly, $(0,\bar{p})$ is isolated in the set of trivial $T$-forced
pairs of (3).

###### Proof of Corollary 5.4.

By construction of the map $\mathcal{J}$, the $T$-forced pairs of (3) that lie
in the slice $\\{0\\}\times C_{T}^{1}(\mathbb{R})$ corresponds bijectively to
the $T$-pairs of (4) contained in $\\{0\\}\times C_{T}(\mathbb{R}^{b+2})$.

Theorem 5.2 implies that there are not nonconstant $T$-periodic solutions of
(4) contained in $W$. Thus $\Upsilon$ cannot be contained in $\\{0\\}\times
C_{T}(\mathbb{R}^{b+2})$. Hence, the same is true for
$\Gamma=\mathcal{J}^{-1}(\Upsilon)$. ∎

######  Remark 5.5.

When $G$ is $C^{1}$ in a neghborhood of a nondegenerate zero, then it is
locally Lipschitz in this neighborhood. Then by Corollary 5.4 we have that _if
the frequency of the forcing term is sufficiently high, then a nondegenerate
zero of the unperturbed vector field is necessarily an ejecting point of
nontrivial $T$-forced pairs of (3)._

##  6\. Ejecting sets and multiplicity

This section is devoted to the illustration of sufficient conditions on $f$,
$g$, and $\varphi$ yielding the multiplicity of $T$-periodic solutions of (3)
for $\lambda>0$ small.

The following lemma of [22] about the notion of ejecting set (or point) is
crucial to get our multiplicity result. As in the previous section, we
reformulate the lemma in our setting. As above, let $G$ be as in (5), and let
$X\subseteq[0,\infty)\times C_{T}^{1}(\mathbb{R})$ be the set of nontrivial
$T$-forced pairs of (3).

######  Lemma 6.1 ([22, Lemma 3.1]).

Let $K$ be a relatively open compact subset of the slice $X_{0}$ of $X$ at
$\lambda=0$. Then, for any sufficiently small open neighborhood $U$ of $K$ in
$\\{0\\}\times C_{T}^{1}(\mathbb{R})$, there exists a positive number $\delta$
such that

$X\cap\big{(}[0,\delta]\times\partial U\big{)}=\emptyset\;.$

Corollary 5.4, Theorem 3.4 and Lemma 6.1 yield the following result about
multiple periodic solutions of (3).

######  Theorem 6.2.

Let $f$, $g$, and $\varphi$ be as in (3). Let $I\subseteq\mathbb{R}$ be an
open interval and assume that $\Phi$ changes sign at the isolated zeros
$p_{1},\ldots,p_{n}\in I$. For $i=1,\ldots,n$, put
$P_{i}=\big{(}p_{i},0,\varphi(p_{i},0)\ldots,\varphi(p_{i},0)\big{)}$. Assume
that $g$ and $\varphi$ are such that $G$ is Lipschitz with constant $L$ on a
neighborhood of $\\{P_{1},\ldots,P_{n}\\}$ and the period $T$ of $f$ satisfies
$T<2\pi/L$. Then:

  1. (1)

For $i=1,\ldots,n$, there exist connected sets $\Gamma_{i}$ of nontrivial
$T$-pairs of (3) emanating from $(0,\bar{p}_{i})$;

  2. (2)

There exists $\lambda_{*}>0$ such that the projection of each $\Gamma_{i}$ on
the first component contains $[0,\lambda_{*}]$. Consequently, (3) has at least
$n$ solutions $x_{1}^{\lambda},\ldots,x_{n}^{\lambda}$ of period $T$ for
$\lambda\in[0,\lambda_{*})$;

  3. (3)

The images of $x_{1}^{\lambda},\ldots,x_{n}^{\lambda}$ are pairwise disjoint.

###### Proof.

Assertion (1) follows immediately from Theorem 3.4 applied to pairwise
disjoint open subintervals $I_{i}$, $i=1,\ldots,n$, isolating the zero
$p_{i}$.

To prove assertion (2) let, for $i=1,\ldots,n$, $U_{i}=C_{T}^{1}(I_{i})$ and
$K_{i}=(0,\bar{p}_{i})$. By Corollary 5.4, $K_{i}$ is isolated, hence
relatively open and compact (being a singleton). Then, Lemma 6.1 yields
positive $\delta_{i}$’s such that$X\cap\big{(}[0,\delta]\times\partial
U_{i}\big{)}=\emptyset$.

Recall now that $X$ is locally complete. Since the closure of $\Gamma_{i}$ in
$X$ meets $(0,\bar{p}_{i})$ and is not contained in any compact subset of $X$,
it is not difficult using Ascoli-Arzelà theorem, to prove that that the
projection of $\Gamma_{i}$ on its first component is $[0,\delta_{i}]$. The
proof is completed by taking
$\lambda_{*}=\min\\{\delta_{1},\ldots,\delta_{n}\\}$.

To prove the last assertion, observe that by (2) there exists $\lambda_{*}>0$
such that each $\\{\lambda\\}\times U_{i}$, contains at least one $T$-forced
pair, say $(\lambda,x_{i}^{\lambda})$ for any $\lambda\in[0,\lambda_{*})$.
Hence, for $j,k=1,\ldots,n$ and $j\neq k$, the images of $x_{j}^{\lambda}$ and
$x_{k}^{\lambda}$ are confined to the disjoint sets $I_{i}$ and $I_{k}$. ∎

Restricting to a neighborhood of the set of zeros of $G$ corresponding to
$p_{1},\ldots,p_{n}$, we can give a somewhat less technical and perhaps more
elegant formulation of our multiplicity result.

######  Corollary 6.3.

Let $f$, $g$, and $\varphi$ be as in (3), with $g$ and $\varphi$ locally
Lipschitz. Assume that $\Phi$ changes sign at the isolated zeros
$p_{1},\ldots,p_{n}\in I$. Then, for sufficiently high frequency of the
perturbing term $f$ and sufficiently small $\lambda>0$, equation (3) has at
least $n$ solutions of period $T$ whose images are pairwise disjoint.

##  7\. Visual representation of branches

We briefly discuss a method allowing to represent graphically the infinite
dimensional set of $T$-forced pairs of (3). In other words, as in [2], we
create a homeomorphic finite dimensional image of the set $\Gamma$ yielded by
Theorem 3.4 and show a graph of some relevant functions of the point of
$\Gamma$ as, for instance, the sup-norm or the diameter of the orbit of the
solution $x$ in any $T$-forced pair $(\lambda,x)$.

In this section we assume $g$ and $\varphi$ as well as the perturbing term $f$
to be at least Lipschitz continuous, so that continuous dependence on initial
data of (4) holds.

Let us consider the set

$S=\left\\{(\lambda,q,p_{0},\ldots,p_{b})\in[0,\infty)\times\mathbb{R}^{b+2}\;\;\left|\;\parbox{182.1196pt}{\small$(q,p_{0},\ldots,p_{b})$
is an initial condition at $t=0$ for a $T$-periodic solution of
\eqref{eq:expanded}}\right.\right\\}$

The elements of $S$ are called _starting points_ for (4). A starting point
$(\lambda,q,p_{0},\ldots,p_{b})$ is _trivial_ when $\lambda=0$ and the
solution of (4) starting a time $t=0$ from $(q,p_{0},\ldots,p_{b})$ is
constant.

By uniqueness and continuous dependence on initial data the map
$\mathfrak{p}\colon Y\to S$ given by

$(\lambda,x_{0},y_{0},\ldots.y_{b})\mapsto\big{(}\lambda,x_{0}(0),y_{0}(0),\ldots.y_{b}(0)\big{)}$

is a homeomorphism that establishes a correspondence between trivial $T$-pairs
and trivial starting points. Thus, the composition
$\mathfrak{h}=\mathfrak{p}\circ\mathcal{J}^{-1}\colon X\to S$ is as well a
homeomorphism that establishes a correspondence between trivial $T$-forced
pairs and trivial starting points. In other words,
$\Sigma:=\mathfrak{h}(\Gamma)\subseteq[0,\infty)\times\mathbb{R}^{b+2}$ is the
desired homemorphic image of $\Gamma$.

|
---|---

(a) sup-norm of $x$ for all $(\lambda,x)\in X$ (b) diameter of the orbits of
$x$ for all $(\lambda,x)\in X$ (i.e.,
$\max_{t\in[0,T]}x(t)-\min_{t\in[0,T]}x(t)$)

Figure 1. Sup-norm and diameter of points in $\Gamma$

######  Example 7.1.

Consider the following equation:

(12)
$\ddot{x}(t)=-x(t)\left(1+\int_{-\infty}^{t}\gamma_{2}^{2}(t-s)\big{(}\dot{x}(s)-x(s)\big{)}\,\mathrm{d}s\right)+\lambda\big{(}1+x(t)\sin(2\pi
t)\big{)},$

with $\lambda\geq 0$. Here, $T=1$, $g(\xi,\eta,\zeta)=-\xi(1+\zeta)$,
$\varphi(p,q)=q-p$, so that

$\Phi(u)=g\big{(}u,0,\varphi(u,0)\big{)}=u(1-u),$

and

$G(u,v_{0},v_{1},v_{2})=\big{(}v_{0},-u(1+v_{2}),2(v_{0}-u-v_{1}),2(v_{1}-v_{2})\big{)}.$

Clearly, $\Phi$ changes sign at the zeros $0$ and $1$, the corresponding zeros
of $G$ being $P_{0}:=(0,0,0,0)$ and $P_{1}:=(1,0,-1,-1)$. Observe that $G$, in
suitably small neighborhoods of $P_{0}$ and $P_{1}$, is Lipschitz with
constant smaller than $2$. Thus, by Corollary 5.4, $(0,\overline{0})$ and
$(0,\overline{1})$ are ejecting for the set $X$ of $T$-forced pairs of (12).
Let

$\Omega_{0}=[0,\infty)\times
C_{T}^{1}(\mathbb{R})\setminus\\{(0,\overline{1})\\},\qquad\Omega_{1}=[0,\infty)\times
C_{T}^{1}(\mathbb{R})\setminus\\{(0,\overline{0})\\}.$

Theorem 3.4 implies the existence of connected sets $\Gamma_{0}$ and
$\Gamma_{1}$ of nontrivial $T$-forced pairs of (12) whose closures,
respectively, emanate from $(0,\overline{0})$ and $(0,\overline{1})$ and are
not contained in any compact subset of $\Omega_{0}$ and $\Omega_{1}$. Figure 1
shows sup-norm and diameter of the solutions of the $T$-forced pairs of (12).
Figure 2, instead shows the projections of $\Sigma$ on the plane $(\lambda,q)$
and on the 3-dimensional space $(q,p_{0},\lambda)$. Indeed, Figures 1 and 2
suggest that $\Gamma_{1}=\Gamma_{0}$. Observe also that, by Theorem 6.2 there
are two periodic solutions of (12) for small $\lambda>0$. The figures suggest
that the value of $\lambda_{*}$ in Theorem 6.2 is about $0.25$.

|
---|---

(a) projection of $X$ on the plane $(\lambda,q)$ (b) projection of $X$ on the
space $(q,p_{0},\lambda)$

Figure 2. Inital values and speed of $(\lambda,x)\in\Gamma$

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