Let

*M* be a boundaryless smooth

*m*-dimensional manifold in

${\mathbb{R}}^{k}$. Given

$T>0$, let

$\stackrel{\u02c6}{M}:=C([-T,0],M)$

denote the metric subspace of

$C([-T,0],{\mathbb{R}}^{k})$ of the

*M*-valued continuous functions on

$[-T,0]$ and set

${\stackrel{\u02c6}{M}}_{\ast}:=\{\psi \in \stackrel{\u02c6}{M}:\psi (-T)=\psi (0)\}.$

Moreover, denote by ${C}_{T}({\mathbb{R}}^{k})$ the Banach space of the continuous *T*-periodic maps $x:\mathbb{R}\to {\mathbb{R}}^{k}$ (with the standard supremum norm) and by ${C}_{T}(M)$ the metric subspace of ${C}_{T}({\mathbb{R}}^{k})$ of the *M*-valued maps. Observe that, since *M* is locally compact, then $\stackrel{\u02c6}{M}$ and ${C}_{T}(M)$ (but not $\tilde{M}$) are locally complete. Moreover, they are complete if and only if *M* is closed.

Let

$f:\mathbb{R}\times \tilde{M}\to {\mathbb{R}}^{k}$ be a functional field over

*M*. Given

$T>0$, assume that

*f* is

*T*-periodic in the first variable. Consider the following RFDE depending on a parameter

$\lambda \ge 0$:

${x}^{\prime}(t)=\lambda f(t,{x}_{t}).$

(3.1)

As in the introduction, we call $(\lambda ,x)\in [0,+\mathrm{\infty})\times {C}_{T}(M)$ a *T-periodic pair* (of (3.1)) if the function $x:\mathbb{R}\to M$ is a (*T*-periodic) solution of (3.1) corresponding to *λ*. Let us denote by *X* the set of all *T*-periodic pairs of (3.1). Lemma 3.1 below states some properties of *X* that will be used in the sequel.

**Lemma 3.1** *The set* *X* *is closed in* $[0,+\mathrm{\infty})\times {C}_{T}(M)$ *and locally compact*.

*Proof* Let $\{({\lambda}^{n},{x}^{n})\}$ be a sequence of *T*-periodic pairs of (3.1) converging to $({\lambda}^{0},{x}^{0})$ in $[0,+\mathrm{\infty})\times {C}_{T}(M)$. Because of Lemma 2.4, $f(t,{x}_{t}^{n})$ converges uniformly to $f(t,{x}_{t}^{0})$ for $t\in \mathbb{R}$. Thus, ${({x}^{n})}^{\prime}(t)={\lambda}^{n}f(t,{x}_{t}^{n})\to {\lambda}^{0}f(t,{x}_{t}^{0})$ uniformly and, therefore, ${({x}^{0})}^{\prime}(t)={\lambda}^{0}f(t,{x}_{t}^{0})$, that is, $({\lambda}^{0},{x}^{0})$ belongs to *X*. This proves that *X* is closed in $[0,+\mathrm{\infty})\times {C}_{T}(M)$.

Now, as observed above, ${C}_{T}(M)$ is locally complete. Consequently, *X* is locally complete as well, as a closed subset of a locally complete space. Moreover, by using Ascoli’s theorem, we get that it is actually a locally compact space. □

We recall that, given $p\in M$, with the notation ${p}^{-}$ we mean the constant *p*-valued function defined on some real interval that will be clear from the context. Moreover, a *T*-periodic pair of the type $(0,{p}^{-})$ is said to be *trivial*, and an element $p\in M$ is a *bifurcation point* of equation (3.1) if any neighborhood of $(0,{p}^{-})$ in $[0,+\mathrm{\infty})\times {C}_{T}(M)$ contains a nontrivial *T*-periodic pair (*i.e.*, a *T*-periodic pair $(\lambda ,x)$ with $\lambda >0$). In some sense, *p* is a bifurcation point if, for $\lambda >0$ sufficiently small, there are *T*-periodic orbits of (3.1) arbitrarily close to *p*.

In the sequel, we are interested in the existence of branches of nontrivial

*T*-periodic pairs that, roughly speaking, emanate from a trivial pair

$(0,{p}^{-})$, with

*p* a bifurcation point of (3.1). To this end, we introduce the mean value tangent vector field

$w:M\to {\mathbb{R}}^{k}$ given by

$w(p)=\frac{1}{T}{\int}_{0}^{T}f(t,{p}^{-})\phantom{\rule{0.2em}{0ex}}dt.$

(3.2)

Throughout the paper, *w* will play a crucial role in obtaining our continuation results for (3.1). First, in Theorem 3.2 below, we provide a necessary condition for $p\in M$ to be a bifurcation point.

**Theorem 3.2** *Let* $x\in {C}_{T}(M)$ *be such that* $(0,x)$ *is an accumulation point of nontrivial* *T*-*periodic pairs of* (3.1). *Then there exists* $p\in M$ *such that* $x(t)=p$, *for any* $t\in \mathbb{R}$, *and* $w(p)=0$. *Thus*, *any bifurcation point of* (3.1) *is a zero of* *w*.

*Proof* By assumption there exists a sequence $\{({\lambda}^{n},{x}^{n})\}$ of *T*-periodic pairs of (3.1) such that ${\lambda}^{n}>0$, ${\lambda}^{n}\to 0$, and ${x}^{n}(t)\to x(t)$ uniformly on ℝ. As proved in Lemma 3.1, the set *X* of the *T*-periodic pairs is closed in $[0,+\mathrm{\infty})\times {C}_{T}(M)$. Thus, the pair $(0,x)$ belongs to *X* and, consequently, the function *x* must be constant, say $x={p}^{-}$ for some $p\in M$. Clearly, the point *p* is a bifurcation point of (3.1).

Now, given

$n\in \mathbb{N}$, recalling that

${x}^{n}(T)={x}^{n}(0)$ and that

${\lambda}^{n}\ne 0$, we get

${\int}_{0}^{T}f(t,{x}_{t}^{n})\phantom{\rule{0.2em}{0ex}}dt=0.$

Observe that the sequence of curves $t\mapsto (t,{x}_{t}^{n})\in \mathbb{R}\times \tilde{M}$ converges uniformly to $t\mapsto (t,{p}^{-})$ for $t\in [0,T]$. Hence, because of Lemma 2.4, $f(t,{x}_{t}^{n})\to f(t,{p}^{-})$ uniformly for $t\in [0,T]$ and the assertion follows passing to the limit in the above integral. □

Let now Ω be an open subset of $[0,+\mathrm{\infty})\times {C}_{T}(M)$. Our main result (Theorem 3.3 below) provides a sufficient condition for the existence of a bifurcation point *p* in *M* with $(0,{p}^{-})\in \mathrm{\Omega}$. More precisely, we give conditions which ensure the existence of a connected subset of Ω of nontrivial *T*-periodic pairs of equation (3.1) (a *global bifurcating branch* for short), whose closure in Ω is noncompact and intersects the set of trivial *T*-periodic pairs contained in Ω.

**Theorem 3.3** *Let* $M\subseteq {\mathbb{R}}^{k}$ *be a boundaryless smooth manifold*,

$f:\mathbb{R}\times \tilde{M}\to {\mathbb{R}}^{k}$ *be a functional field on* *M*,

*T*-

*periodic in the first variable and locally Lipschitz in the second one*,

*and* $w:M\to {\mathbb{R}}^{k}$ *be the autonomous tangent vector field* $w(p)=\frac{1}{T}{\int}_{0}^{T}f(t,{p}^{-})\phantom{\rule{0.2em}{0ex}}dt.$

*Let* Ω *be an open subset of* $[0,+\mathrm{\infty})\times {C}_{T}(M)$ *and let* $j:M\to [0,+\mathrm{\infty})\times {C}_{T}(M)$ *be the map* $p\mapsto (0,{p}^{-})$. *Assume that* $deg(w,{j}^{-1}(\mathrm{\Omega}))$ *is defined and nonzero*. *Then there exists a connected subset of* Ω *of nontrivial* *T*-*periodic pairs of equation* (3.1) *whose closure in* Ω *is noncompact and intersects* $\{0\}\times {C}_{T}(M)$ *in a* (*nonempty*) *subset of* $\{(0,{p}^{-})\in \mathrm{\Omega}:w(p)=0\}$.

**Remark 3.4** (On the meaning of global bifurcating branch)

In addition to the hypotheses of Theorem 3.3, assume that *f* sends bounded subsets of $\mathbb{R}\times \tilde{M}$ into bounded subsets of ${\mathbb{R}}^{k}$, and that *M* is closed in ${\mathbb{R}}^{k}$ (or, more generally, that the closure $\overline{\mathrm{\Omega}}$ of Ω in $[0,+\mathrm{\infty})\times {C}_{T}(M)$ is complete).

*Then a connected subset* Γ *of* Ω *as in Theorem 3.3 is either unbounded or, if bounded, its closure* $\overline{\mathrm{\Gamma}}$ *in* $\overline{\mathrm{\Omega}}$ *reaches the boundary* *∂* Ω *of* Ω.

To see this, assume that $\overline{\mathrm{\Gamma}}$ is bounded. Then, being $f(\overline{\mathrm{\Gamma}})$ bounded, because of Ascoli’s theorem, Γ is actually totally bounded. Thus, $\overline{\mathrm{\Gamma}}$ is compact, being totally bounded and, additionally, complete since $\overline{\mathrm{\Gamma}}$ is contained in $\overline{\mathrm{\Omega}}$. On the other hand, according to Theorem 3.3, the closure ${\overline{\mathrm{\Gamma}}}_{\mathrm{\Omega}}$ of Γ in Ω is noncompact. Consequently, the set $\overline{\mathrm{\Gamma}}\setminus {\overline{\mathrm{\Gamma}}}_{\mathrm{\Omega}}$ is nonempty, and this means that $\overline{\mathrm{\Gamma}}$ reaches the boundary of Ω.

The proof of Theorem 3.3 requires some preliminary steps. In the first one, we define a parametrized Poincaré-type *T*-translation operator whose fixed points are the restrictions to the interval $[-T,0]$ of the *T*-periodic solutions of (3.1). For this purpose, we need to introduce a suitable backward extension of the elements of $\stackrel{\u02c6}{M}$. The properties of such an extension are contained in Lemma 3.5 below, obtained in [33]. In what follows, by a *T*-periodic map on an interval *J*, we mean the restriction to *J* of a *T*-periodic map defined on ℝ.

**Lemma 3.5** *There exist an open neighborhood* *U* *of* ${\stackrel{\u02c6}{M}}_{\ast}$ *in* $\stackrel{\u02c6}{M}$ *and a continuous map from* *U* *to* $\tilde{M}$,

$\psi \mapsto \tilde{\psi}$,

*with the following properties*:

- 1.
$\tilde{\psi}$ *is an extension of* *ψ*;

- 2.
$\tilde{\psi}$ *is* *T*-*periodic on* $(-\mathrm{\infty},-T]$;

- 3.
$\tilde{\psi}$ *is* *T*-*periodic on* $(-\mathrm{\infty},0]$, *whenever* $\psi \in {\stackrel{\u02c6}{M}}_{\ast}$.

Let now

*U* be an open subset of

$\stackrel{\u02c6}{M}$ as in the previous lemma and let

*f* be as in Theorem 3.3. Given

$\lambda \ge 0$ and

$\psi \in U$, consider the initial value problem

$\{\begin{array}{c}{x}^{\prime}(t)=\lambda f(t,{x}_{t}),\hfill \\ {x}_{0}=\tilde{\psi},\hfill \end{array}$

(3.3)

where $\tilde{\psi}$ is the extension of *ψ* as in Lemma 3.5.

Let

$D=\{(\lambda ,\psi )\in [0,+\mathrm{\infty})\times U:\text{the maximal solution of (3.3) is defined up to}T\}.$

The set *D* is nonempty since it contains $\{0\}\times U$ (notice that for $\lambda =0$, the solution of problem (3.3) is constant for $t>0$). Moreover, it follows by Corollary 2.2 that *D* is open in $[0,+\mathrm{\infty})\times \stackrel{\u02c6}{M}$.

Given

$(\lambda ,\psi )\in D$, denote by

${x}^{(\lambda ,\tilde{\psi})}$ the maximal solution of problem (3.3) and define

$P:D\to \stackrel{\u02c6}{M}$

by

$P(\lambda ,\psi )(\theta )={x}^{(\lambda ,\tilde{\psi})}(\theta +T),\phantom{\rule{1em}{0ex}}\theta \in [-T,0].$

Observe that $P(\lambda ,\psi )$ is the restriction of ${x}_{T}^{(\lambda ,\tilde{\psi})}\in \tilde{M}$ to the interval $[-T,0]$.

The following lemmas regard crucial properties of the operator *P*. The proof of the first one is standard and will be omitted.

**Lemma 3.6** *The fixed points of* $P(\lambda ,\cdot )$ *correspond to the* *T*-*periodic solutions of equation* (3.1) *in the following sense*: *ψ* *is a fixed point of* $P(\lambda ,\cdot )$ *if and only if it is the restriction to* $[-T,0]$ *of a* *T*-*periodic solution*.

**Lemma 3.7** *The operator* *P* *is continuous and locally compact*.

*Proof* The continuity of *P* follows immediately from the continuous dependence on data stated in Corollary 2.2 and by the continuity of the map $\psi \mapsto \tilde{\psi}$ of Lemma 3.5 and of the map that associates to any $\phi \in \tilde{M}$ its restriction to the interval $[-T,0]$.

Let us prove that

*P* is locally compact. Take

$({\lambda}^{0},{\psi}^{0})\in D$ and denote, for simplicity, by

${x}^{0}$ the maximal solution

${x}^{({\lambda}^{0},\tilde{{\psi}^{0}})}$ of (3.3) corresponding to

$({\lambda}^{0},\tilde{{\psi}^{0}})$. Clearly,

${x}^{0}$ is defined at least up to

*T* and

$P({\lambda}^{0},{\psi}^{0})(\theta )={x}^{0}(\theta +T)$ for any

$\theta \in [-T,0]$. Set

$K=\{(t,{x}_{t}^{0})\in \mathbb{R}\times \tilde{M}:t\in [0,T]\}.$

Observe that *K* is compact, being the image of $[0,T]$ under the (continuous) curve $t\mapsto (t,{x}_{t}^{0})$. Let *O* be an open neighborhood of *K* in $\mathbb{R}\times \tilde{M}$ and $c>0$ such that $|f(t,\phi )|\le c$ for all $(t,\phi )\in \overline{O}$. Let us show that there exists an open neighborhood *W* of $({\lambda}^{0},{\psi}^{0})$ in *D* such that if $(\lambda ,\psi )\in W$, then $(t,{x}_{t}^{(\lambda ,\tilde{\psi})})\in O$ for $t\in [0,T]$, where ${x}^{(\lambda ,\tilde{\psi})}$ is the maximal solution of (3.3) corresponding to $(\lambda ,\tilde{\psi})$. By contradiction, for any $n\in \mathbb{N}$ suppose there exist $({\lambda}^{n},{\psi}^{n})\in D$ and ${t}^{n}\in [0,T]$ such that $({\lambda}^{n},{\psi}^{n})\to ({\lambda}^{0},{\psi}^{0})$ and $({t}^{n},{x}_{{t}^{n}}^{n})\notin O$, where ${x}^{n}$ denotes the maximal solution ${x}^{({\lambda}^{n},\tilde{{\psi}^{n}})}$ of (3.3) corresponding to $({\lambda}^{n},\tilde{{\psi}^{n}})$. We may assume ${t}^{n}\to \tau \in [0,T]$. Now, from the fact that in $\tilde{M}$ the convergence is uniform, we get the equicontinuity of the sequence $\{{x}_{T}^{n}\}$. This easily implies that $({t}^{n},{x}_{{t}^{n}}^{n})\to (\tau ,{x}_{\tau}^{0})$. A contradiction, since *O* is open and $(\tau ,{x}_{\tau}^{0})$ belongs to $K\subseteq O$. Thus, the existence of the required *W* is proved. Consequently, for any $(\lambda ,\psi )\in W$, the maximal solution ${x}^{(\lambda ,\tilde{\psi})}$ of (3.3) corresponding to $(\lambda ,\tilde{\psi})$ is such that $|{({x}^{(\lambda ,\tilde{\psi})})}^{\prime}(t)|=|\lambda f(t,{x}_{t}^{(\lambda ,\tilde{\psi})})|\le |\lambda |c$ for all $t\in [0,T]$.

Therefore, by Ascoli’s theorem and taking into account the local completeness of $\stackrel{\u02c6}{M}$, we get that *P* maps *W* into a compact subset of $\stackrel{\u02c6}{M}$. This proves that *P* is locally compact. □

The following result establishes the relationship between the fixed point index of the Poincaré-type operator $P(\lambda ,\cdot )$ and the degree of the mean value vector field *w*. It will be crucial in the proof of Lemma 3.10.

**Lemma 3.8**
*Let*
$\mathcal{V}$
*be an open subset of*
$\stackrel{\u02c6}{M}$
*such that*
$\mathcal{V}\cap \{{p}^{-}\in \stackrel{\u02c6}{M}:w(p)=0\}$
*is compact and let*
$\epsilon >0$
*be such that*
- (a)
$[0,\epsilon ]\times \overline{\mathcal{V}}$ *is contained in the domain* *D* *of* *P*;

- (b)
$P([0,\epsilon ]\times \mathcal{V})$ *is relatively compact*;

- (c)
$P(\lambda ,\psi )\ne \psi $ *for* $0<\lambda \le \epsilon $ *and* *ψ* *in the boundary* $\partial \mathcal{V}$ *of* $\mathcal{V}$.

*Consider the open set* $V=\{p\in M:{p}^{-}\in \mathcal{V}\}$.

*Then* $deg(-w,V)$ *is well defined and* ${ind}_{\stackrel{\u02c6}{M}}(P(\lambda ,\cdot ),\mathcal{V})=deg(-w,V),\phantom{\rule{1em}{0ex}}0<\lambda \le \epsilon .$

*Proof* Let

*U* be an open subset of

$\stackrel{\u02c6}{M}$ as in Lemma 3.5. Given

$\lambda \ge 0$,

$\mu \in [0,1]$ and

$\psi \in U$, consider the initial value problem

$\{\begin{array}{c}{x}^{\prime}(t)=\lambda ((1-\mu )f(t,{x}_{t})+\mu w(x(t))),\hfill \\ {x}_{0}=\tilde{\psi},\hfill \end{array}$

(3.4)

where

$\tilde{\psi}$ is associated to

*ψ* as in Lemma 3.5. Since

*f* is locally Lipschitz in the second variable, then it is easy to see that

*w* is locally Lipschitz as well. Hence, for any

$\lambda \in [0,+\mathrm{\infty})$ and

$\mu \in [0,1]$, the uniqueness of the solution of problem (3.4) is ensured (recall Remark 2.3). Denote by

${x}^{(\lambda ,\tilde{\psi},\mu )}$ the maximal solution of problem (3.4), and put

$E=\{(\lambda ,\psi ,\mu )\in [0,+\mathrm{\infty})\times U\times [0,1]:{x}^{(\lambda ,\tilde{\psi},\mu )}\text{is defined up to}T\}$

and

${D}^{\prime}=\{(\lambda ,\psi )\in [0,+\mathrm{\infty})\times U:(\lambda ,\psi ,\mu )\in E\text{for all}\mu \in [0,1]\}.$

Corollary 2.2 implies that

*E* is open in

$[0,+\mathrm{\infty})\times U\times [0,1]$. Therefore,

${D}^{\prime}$ is open in

$[0,+\mathrm{\infty})\times \stackrel{\u02c6}{M}$ because of the compactness of

$[0,1]$. Moreover, observe that the slice

${D}_{0}^{\prime}$ of

${D}^{\prime}$ at

$\lambda =0$ coincides with

*U* and that

${D}^{\prime}$ is contained in the domain

*D* of the operator

*P* defined above. Define

$H:{D}^{\prime}\times [0,1]\to \stackrel{\u02c6}{M}$ by

$H(\lambda ,\psi ,\mu )(\theta )={x}^{(\lambda ,\tilde{\psi},\mu )}(\theta +T),\phantom{\rule{1em}{0ex}}\theta \in [-T,0].$

Clearly,

$H(\cdot ,\cdot ,0)$ coincides with

*P* on

${D}^{\prime}$, while

$H(\cdot ,\cdot ,1)$ is the (infinite dimensional) operator associated to the undelayed problem

$\{\begin{array}{c}{x}^{\prime}(t)=\lambda w(x(t)),\hfill \\ {x}_{0}=\tilde{\psi}.\hfill \end{array}$

As in Lemmas 3.6 and 3.7, one can show that the fixed points of

$H(\lambda ,\cdot ,\mu )$ correspond to the

*T*-periodic solutions of the equation

${x}^{\prime}(t)=\lambda ((1-\mu )f(t,{x}_{t})+\mu w(x(t))),$

and that *H* is continuous and locally compact.

The assertion now will follow by proving some intermediate results on the homotopy

*H*. These results will be carried out in several steps. In what follows set

and, according to our notation,

${Z}^{-}=\{{p}^{-}\in \stackrel{\u02c6}{M}:p\in Z\}.$

Step 1. *There exist* $\sigma >0$ *and an open subset* ${\mathcal{V}}^{\prime}$ *of* $\stackrel{\u02c6}{M}$*, containing* $\mathcal{V}\cap {Z}^{-}$*, with* $\overline{{\mathcal{V}}^{\prime}}\subseteq \mathcal{V}$*, and such that*

(a′) $[0,\sigma ]\times \overline{{\mathcal{V}}^{\prime}}\subseteq {D}^{\prime}$ *(i.e., for* $0\le \lambda \le \sigma $*,* $H(\lambda ,\cdot ,\cdot )$ *is defined in* $\overline{{\mathcal{V}}^{\prime}}\times [0,1]$);

(b′) $H([0,\sigma ]\times {\mathcal{V}}^{\prime}\times [0,1])$ *is relatively compact*.

To prove Step 1, observe that $\{0\}\times (\mathcal{V}\cap {Z}^{-})\times [0,1]$ is compact and contained in ${D}^{\prime}\times [0,1]$, which is open in $[0,+\mathrm{\infty})\times \stackrel{\u02c6}{M}\times [0,1]$, and recall that *H* is locally compact.

Step 2. *For small values of* $\lambda >0$*,* $H(\lambda ,\psi ,\mu )\ne \psi $ *for any* $\psi \in \partial {\mathcal{V}}^{\prime}$ *and* $\mu \in [0,1]$*.*

By contradiction, suppose there exists a sequence $\{({\lambda}^{n},{\psi}^{n},{\mu}^{n})\}$ in ${D}^{\prime}\times [0,1]$ such that ${\lambda}^{n}>0$, ${\lambda}^{n}\to 0$, ${\psi}^{n}\in \partial {\mathcal{V}}^{\prime}$ and $H({\lambda}^{n},{\psi}^{n},{\mu}^{n})={\psi}^{n}$. Without loss of generality, taking into account (b′), we may assume that ${\psi}^{n}\to {\psi}^{0}$ and also that ${\mu}^{n}\to {\mu}^{0}$. Denote by ${x}^{n}$ the *T*-periodic solution ${x}^{({\lambda}^{n},\tilde{{\psi}^{n}},{\mu}^{n})}$ of (3.4) corresponding to $({\lambda}^{n},\tilde{{\psi}^{n}},{\mu}^{n})$. Since ${\psi}^{n}$ is the restriction of ${x}^{n}$ to $[-T,0]$, then $\{{x}^{n}(t)\}$ converges uniformly on ℝ to ${x}^{0}(t)$, where ${x}^{0}$ is the solution of (3.4) corresponding to the fixed point ${\psi}^{0}$ of $H(0,\cdot ,{\mu}^{0})$. Therefore, there exists $p\in M$ such that ${x}^{0}(t)=p$ for any $t\in \mathbb{R}$ and, as in the proof of Theorem 3.2, we can show that $w(p)=0$. Thus, ${\psi}^{0}={p}^{-}$ belongs to $\partial {\mathcal{V}}^{\prime}\cap {Z}^{-}$, contradicting the choice of ${\mathcal{V}}^{\prime}$. This proves Step 2.

Step 3. *For small values of* $\lambda >0$*,* $H(\lambda ,\psi ,0)\ne \psi $ *for any* $\psi \in \overline{\mathcal{V}}\mathrm{\setminus}{\mathcal{V}}^{\prime}$*.*

The proof is analogous to that of Step 2, noting that $H(\lambda ,\psi ,0)=P(\lambda ,\psi )$ for $(\lambda ,\psi )\in {D}^{\prime}$ and taking into account assumption b) and the fact that $\overline{\mathcal{V}}\mathrm{\setminus}{\mathcal{V}}^{\prime}$ is closed in $\stackrel{\u02c6}{M}$.

Step 4. *Let* $k:{\mathcal{V}}^{\prime}\to M$ *be defined by* $k(\psi )=\psi (0)$ *and consider the open set* ${V}^{\prime}=\{p\in M:{p}^{-}\in {\mathcal{V}}^{\prime}\}$*. Then there exists* ${\sigma}^{\prime}\in (0,\sigma ]$ *such that* $H(\lambda ,\psi ,1)\ne \psi $ *for any* $(\lambda ,\psi )\in (0,{\sigma}^{\prime}]\times (\overline{{\mathcal{V}}^{\prime}}\mathrm{\setminus}{k}^{-1}({V}^{\prime}))$*.*

By contradiction, suppose there exists a sequence $\{({\lambda}^{n},{\psi}^{n})\}$ in ${D}^{\prime}$ such that ${\lambda}^{n}>0$, ${\lambda}^{n}\to 0$, ${\psi}^{n}\in \overline{{\mathcal{V}}^{\prime}}\mathrm{\setminus}{k}^{-1}({V}^{\prime})$ and $H({\lambda}^{n},{\psi}^{n},1)={\psi}^{n}$. Without loss of generality, taking into account (b′), we may assume that ${\psi}^{n}\to {\psi}^{0}$. Therefore, by the continuity of *H*, we get $H(0,{\psi}^{0},1)={\psi}^{0}$ so that ${\psi}^{0}$ is a constant function of $\overline{{\mathcal{V}}^{\prime}}\mathrm{\setminus}{k}^{-1}({V}^{\prime})$. This is impossible, since any constant function of ${\mathcal{V}}^{\prime}$ is contained in ${k}^{-1}({V}^{\prime})$.

Step 5.

*Let* ${V}^{\prime}$ *and* ${\sigma}^{\prime}$ *be as in Step 4 and let* $Q:[0,{\sigma}^{\prime}]\times \overline{{V}^{\prime}}\to M$ *be the* *T-translation operator* $Q(\lambda ,p)={x}^{(\lambda ,{p}^{-},1)}(T)$*, where* ${x}^{(\lambda ,{p}^{-},1)}$ *is the maximal solution of the undelayed problem* $\{\begin{array}{c}{x}^{\prime}(t)=\lambda w(x(t)),\hfill \\ {x}_{0}={p}^{-}.\hfill \end{array}$

*Then, for small values of*
*λ,*
${ind}_{M}(Q(\lambda ,\cdot ),{V}^{\prime})$
*is defined and*
${ind}_{\stackrel{\u02c6}{M}}(H(\lambda ,\cdot ,1),{\mathcal{V}}^{\prime})={ind}_{M}(Q(\lambda ,\cdot ),{V}^{\prime}).$

To see this, let

$k:{\mathcal{V}}^{\prime}\to M$ be as in Step 4 and, given

$\lambda \in (0,{\sigma}^{\prime}]$, define

${h}_{\lambda}:{V}^{\prime}\to \stackrel{\u02c6}{M}$ by

${h}_{\lambda}(p)(\theta )={x}^{(\lambda ,{p}^{-},1)}(\theta +T)$,

$\theta \in [-T,0]$. Clearly,

*k* is a locally compact map since it takes values in the locally compact space

*M*. Moreover,

${h}_{\lambda}$ is actually compact since

${h}_{\lambda}({V}^{\prime})$ is contained in

$H([0,\sigma ]\times {\mathcal{V}}^{\prime}\times [0,1])$ which is relatively compact by (b′) of Step 1. Now, observe that the composition

${h}_{\lambda}k$ coincides with

$H(\lambda ,\cdot ,1)$ in

${k}^{-1}({V}^{\prime})$ and that the set of fixed points of

$H(\lambda ,\cdot ,1)$ in

$\overline{{\mathcal{V}}^{\prime}}$ is compact by (b′) of Step 1 and is contained in

${k}^{-1}({V}^{\prime})$ by Step 4. Thus, the set of fixed points of

${h}_{\lambda}k$ in

${k}^{-1}({V}^{\prime})$ is compact so that, by applying the commutativity property of the fixed point index to the maps

*k* and

${h}_{\lambda}$, we get

${ind}_{\stackrel{\u02c6}{M}}({h}_{\lambda}k,{k}^{-1}\left({V}^{\prime}\right))={ind}_{M}(k{h}_{\lambda},{h}_{\lambda}^{-1}\left({\mathcal{V}}^{\prime}\right)).$

Consequently, since it is easy to verify that the composition

$k{h}_{\lambda}$ coincides with

$Q(\lambda ,\cdot )$ in

${h}_{\lambda}^{-1}({\mathcal{V}}^{\prime})$, we obtain

${ind}_{\stackrel{\u02c6}{M}}(H(\lambda ,\cdot ,1),{k}^{-1}\left({V}^{\prime}\right))={ind}_{M}(Q(\lambda ,\cdot ),{h}_{\lambda}^{-1}\left({\mathcal{V}}^{\prime}\right)),$

and, because of Step 4, by the excision property of the index,

${ind}_{\stackrel{\u02c6}{M}}(H(\lambda ,\cdot ,1),{\mathcal{V}}^{\prime})={ind}_{\stackrel{\u02c6}{M}}(H(\lambda ,\cdot ,1),{k}^{-1}\left({V}^{\prime}\right)).$

To complete the proof of Step 5, let us show that for

*λ* sufficiently small,

$Q(\lambda ,p)\ne p$ for

$p\in \overline{{V}^{\prime}}\mathrm{\setminus}{h}_{\lambda}^{-1}({\mathcal{V}}^{\prime})$. By contradiction, suppose there exists a sequence

$\{({\lambda}^{n},{p}^{n})\}$ in

$[0,{\sigma}^{\prime}]\times \overline{{V}^{\prime}}$ such that

${\lambda}^{n}>0$,

${\lambda}^{n}\to 0$,

${p}^{n}\in \overline{{V}^{\prime}}\mathrm{\setminus}{h}_{{\lambda}_{n}}^{-1}({\mathcal{V}}^{\prime})$ and

$Q({\lambda}^{n},{p}^{n})={p}^{n}$. Hence, there exists a sequence

$\{{\psi}^{n}\}$ in

${\mathcal{V}}^{\prime}$ such that

${\psi}^{n}(0)={p}^{n}$ and

$H({\lambda}^{n},{\psi}^{n},1)={\psi}^{n}$. Because of (b′) of Step 1, we may assume that

${\psi}^{n}\to {\psi}^{0}$ so that, in particular,

${p}^{n}\to {p}^{0}$, where

${p}^{0}={\psi}^{0}(0)$. Now, by an argument similar to that used in the proof of Theorem 3.2, we get that

${\psi}^{0}$ is constant and

$w({p}^{0})=0$. Thus,

${p}^{0}\in Z$. Moreover, since

${\bigcap}_{\lambda >0}(\overline{{V}^{\prime}}\mathrm{\setminus}{h}_{\lambda}^{-1}({\mathcal{V}}^{\prime}))=\partial {V}^{\prime}$, we also obtain that

${p}^{0}$ belongs to

$\partial {V}^{\prime}$, contradicting the choice of

${V}^{\prime}$. Finally, again by excision, we get

${ind}_{M}(Q(\lambda ,\cdot ),{h}_{\lambda}^{-1}\left({\mathcal{V}}^{\prime}\right))={ind}_{M}(Q(\lambda ,\cdot ),{V}^{\prime}),$

and thus Step 5 is proved.

Let us now go back to the proof of our lemma. Step 1 and Step 2 above imply that there exist

${\epsilon}^{\prime}>0$ and an open subset

${\mathcal{V}}^{\prime}$ of

$\stackrel{\u02c6}{M}$, containing

$\mathcal{V}\cap {Z}^{-}$, with

$\overline{{\mathcal{V}}^{\prime}}\subseteq \mathcal{V}$ and such that if

$0<\lambda \le {\epsilon}^{\prime}$, then

${ind}_{\stackrel{\u02c6}{M}}(H(\lambda ,\cdot ,\mu ),{\mathcal{V}}^{\prime})$ is defined and is independent of

$\mu \in [0,1]$. Moreover, reducing

${\epsilon}^{\prime}$ if necessary, by Step 3 and by assumption (b), it follows that for

$\lambda \in (0,{\epsilon}^{\prime}]$, the fixed points of

$H(\lambda ,\cdot ,0)=P(\lambda ,\cdot )$ in

$\mathcal{V}$ are a compact subset of

${\mathcal{V}}^{\prime}$. Therefore, by the excision property and the homotopy invariance of the index, we get

${ind}_{\stackrel{\u02c6}{M}}(P(\lambda ,\cdot ),\mathcal{V})={ind}_{\stackrel{\u02c6}{M}}(P(\lambda ,\cdot ),{\mathcal{V}}^{\prime})={ind}_{\stackrel{\u02c6}{M}}(H(\lambda ,\cdot ,0),{\mathcal{V}}^{\prime})={ind}_{\stackrel{\u02c6}{M}}(H(\lambda ,\cdot ,1),{\mathcal{V}}^{\prime}).$

On the other hand, by Step 5, if

$\lambda >0$ is sufficiently small, we have

${ind}_{\stackrel{\u02c6}{M}}(H(\lambda ,\cdot ,1),{\mathcal{V}}^{\prime})={ind}_{M}(Q(\lambda ,\cdot ),{V}^{\prime}).$

Moreover, as shown in [

1],

${ind}_{M}(Q(\lambda ,\cdot ),{V}^{\prime})=deg(-w,{V}^{\prime}).$

Finally, notice that

$deg(-w,V)$ is well defined since

$V\cap Z$ is compact being homeomorphic to

$\mathcal{V}\cap {Z}^{-}$. Also, observe that there are no zeros of

*w* in

$V\mathrm{\setminus}\overline{{V}^{\prime}}$. Thus, by the excision property of the degree, we obtain

$deg(-w,{V}^{\prime})=deg(-w,V).$

This shows that for small values of $\lambda >0$, ${ind}_{\stackrel{\u02c6}{M}}(P(\lambda ,\cdot ),\mathcal{V})=deg(-w,V)$. The assertion of the lemma now follows by applying the homotopy invariance of the fixed point index to $P(\lambda ,\cdot )$ on $\mathcal{V}$. □

Lemma 3.10 below, whose proof makes use of the following Wyburn-type topological lemma, is another important step in the construction of the proof of Theorem 3.3.

**Lemma 3.9** ([31])

*Let* *K* *be a compact subset of a locally compact metric space* *Y*. *Assume that any compact subset of* *Y* *containing* *K* *has nonempty boundary*. *Then* $Y\mathrm{\setminus}K$ *contains a connected set whose closure is noncompact and intersects* *K*.

Before presenting Lemma 3.10, we introduce the sets

$S=\{(\lambda ,\psi )\in D:P(\lambda ,\psi )=\psi \}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{S}_{+}=\{(\lambda ,\psi )\in S:\lambda >0\},$

and we recall that $Z\subseteq M$ denotes the set of zeros of the tangent vector field *w*.

**Lemma 3.10** *Let* *Y* *be a locally compact open subset of* $(\{0\}\times {Z}^{-})\cup {S}_{+}$. *Assume that* $K:=Y\cap (\{0\}\times {Z}^{-})$ *is compact and that* $deg(w,V)\ne 0$, *where* $V\subseteq M$, *is an isolating neighborhood of* $\{p\in M:(0,{p}^{-})\in K\}$. *Then the pair* $(Y,K)$ *verifies the assumptions of Lemma * 3.9.

*Proof* First of all, observe that by Lemma 3.7,

*S* is closed in

*D* and locally compact. In addition,

*K* is clearly nonempty being

$deg(w,V)\ne 0$. Now, let

*G* be an open subset of

*D* such that

$G\cap ((\{0\}\times {Z}^{-})\cup {S}_{+})=Y.$

To prove the assertion, suppose by contradiction that there exists a compact open neighborhood

*C* of

*K* in

*Y*. Consequently, we can find an open subset

*W* of

*G* such that

$\overline{W}\subseteq G$ and

$C=W\cap Y=\overline{W}\cap Y$. Therefore, denoted by

${G}_{0}$ the slice

${G}_{0}=\{\psi \in \stackrel{\u02c6}{M}:(0,\psi )\in G\},$

we have that

${G}_{0}\cap {Z}^{-}$ is a compact subset of

$\stackrel{\u02c6}{M}$ and is contained in the open slice

${W}_{0}\subseteq {\overline{W}}_{0}\subseteq {G}_{0}$ of

*W* at

$\lambda =0$. Let

$\mathcal{V}$ be an open subset of

${W}_{0}$ such that

$\mathcal{V}\subseteq \overline{\mathcal{V}}\subseteq {W}_{0}$ and

$\mathcal{V}\cap {Z}^{-}={W}_{0}\cap {Z}^{-}$. Since

*C* is compact and because of the local compactness of

*P*, we may suppose that

$P(W)$ is relatively compact. Consequently, there exists

$\epsilon >0$ such that

- 1.
$[0,\epsilon ]\times \overline{\mathcal{V}}\subseteq W$;

- 2.
$P(\lambda ,\psi )\ne \psi $ for $\psi \in {\overline{W}}_{\lambda}\mathrm{\setminus}\mathcal{V}$ and $0<\lambda \le \epsilon $ (here, as usual, ${W}_{\lambda}$ denotes the slice $\{\psi \in \stackrel{\u02c6}{M}:(\lambda ,\psi )\in W\}$).

Notice that $P([0,\epsilon ]\times \mathcal{V})$ is relatively compact. This follows easily from the above condition 1 and the relative compactness of $P(W)$.

We can now apply Lemma 3.8 and the excision properties of the fixed point index and of the degree obtaining, for any

$0<\lambda \le \epsilon $,

${ind}_{\stackrel{\u02c6}{M}}(P(\lambda ,\cdot ),{W}_{\lambda})={ind}_{\stackrel{\u02c6}{M}}(P(\lambda ,\cdot ),\mathcal{V})=deg(-w,V),$

(3.5)

where

$V=\{p\in M:{p}^{-}\in \mathcal{V}\}$. Observe that

*V* is an isolating neighborhood of

$\{p\in M:(0,{p}^{-})\in K\}$. Thus, by formula (2.1), by the above equalities (3.5) and the assumption

$deg(w,V)\ne 0$, we get

${ind}_{\stackrel{\u02c6}{M}}(P(\lambda ,\cdot ),{W}_{\lambda})\ne 0,\phantom{\rule{1em}{0ex}}0<\lambda \le \epsilon .$

Since

*C* is compact, by the generalized homotopy invariance property of the fixed point index, we get that

${ind}_{\stackrel{\u02c6}{M}}(P(\lambda ,\cdot ),{W}_{\lambda})$ does not depend on

$\lambda >0$. Hence,

${ind}_{\stackrel{\u02c6}{M}}(P(\lambda ,\cdot ),{W}_{\lambda})\ne 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}\lambda >0.$

On the other hand, because of the compactness of

*C*, for some positive

$\overline{\lambda}$ the slice

${C}_{\overline{\lambda}}=\{\psi \in {W}_{\overline{\lambda}}:P(\overline{\lambda},\psi )=\psi \}$ is empty. Thus,

${ind}_{\stackrel{\u02c6}{M}}(P(\overline{\lambda},\cdot ),{W}_{\overline{\lambda}})=0,$

and we have a contradiction. Therefore, $(Y,K)$ verifies the assumptions of Lemma 3.9 and the proof is complete. □

*Proof of Theorem 3.3* Let

$\rho :[0,+\mathrm{\infty})\times {C}_{T}(M)\to [0,+\mathrm{\infty})\times {\stackrel{\u02c6}{M}}_{\ast}$ be the isometry given by

$\rho (\lambda ,x)=(\lambda ,\psi )$, where

*ψ* is the restriction of

*x* to the interval

$[-T,0]$. As previously, let

$X\subseteq [0,+\mathrm{\infty})\times {C}_{T}(M)$ denote the set of the

*T*-periodic pairs of (3.1) and, as in Lemma 3.10, let

*S* be the set of the pairs

$(\lambda ,\psi )$ such that

$P(\lambda ,\psi )=\psi $. Observe that

*S* is actually contained in

$[0,+\mathrm{\infty})\times {\stackrel{\u02c6}{M}}_{\ast}$. Taking into account Lemma 3.6,

*X* and

*S* correspond under

*ρ*. Analogously to the definition of

${S}_{+}$, let us denote

${X}_{+}=\{(\lambda ,x)\in X:\lambda >0\}.$

In addition, consider

${Z}^{T}=\{{p}^{-}\in {C}_{T}(M):w(p)=0\}.$

Theorem 3.2 implies that

$(\{0\}\times {Z}^{T})\cup {X}_{+}$ is a closed subset of

*X*. Therefore, it is locally compact since so is

*X* according to Lemma 3.1. Now, consider

${Y}^{T}=\mathrm{\Omega}\cap ((\{0\}\times {Z}^{T})\cup {X}_{+}).$

Observe that

${Y}^{T}$ is locally compact, being open in

$(\{0\}\times {Z}^{T})\cup {X}_{+}$. Then

$Y:=\rho \left({Y}^{T}\right)$

is locally compact and open in

$(\{0\}\times {Z}^{-})\cup {S}_{+}$. Denote by

${K}^{T}$ and

*K* the subsets of

${Y}^{T}$ and

*Y* defined as

${K}^{T}=\{(\lambda ,x)\in {Y}^{T}:\lambda =0\}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}K=\rho \left({K}^{T}\right).$

Now, observe that

${j}^{-1}(\mathrm{\Omega})$ is an isolating neighborhood of

$\{p\in M:(0,{p}^{-})\in K\}.$

Since $deg(w,{j}^{-1}(\mathrm{\Omega}))\ne 0$, we can apply Lemma 3.10 concluding that $(Y,K)$ verifies the assumptions of Lemma 3.9. Therefore, also $({Y}^{T},{K}^{T})$ verifies the same assumptions since the pairs $(Y,K)$ and $({Y}^{T},{K}^{T})$ correspond under the isometry *ρ*. Therefore, Lemma 3.9 implies that ${Y}^{T}\mathrm{\setminus}{K}^{T}$ contains a connected set Γ whose closure (in ${Y}^{T}$) is noncompact and intersects ${K}^{T}$. Now, observe that according to Theorem 3.2, ${Y}^{T}$ is closed in Ω. Thus, the closures of Γ in ${Y}^{T}$ and in Ω coincide. This concludes the proof. □

We give now some consequences of Theorem 3.3. The first one is in the spirit of a celebrated result due to Rabinowitz [5].

**Corollary 3.11** (Rabinowitz-type global bifurcation result)

*Let* *M* *and* *f* *be as in Theorem * 3.3. *Assume that* *M* *is closed in* ${\mathbb{R}}^{k}$ *and that* *f* *sends bounded subsets of* $\mathbb{R}\times \tilde{M}$ *into bounded subsets of* ${\mathbb{R}}^{k}$. *Let* *V* *be an open subset of* *M* *such that* $deg(w,V)\ne 0$, *where* *w* *is the mean value tangent vector field defined in formula* (3.2). *Then equation* (3.1) *has a connected subset of nontrivial* *T*-*periodic pairs whose closure contains some* $(0,{p}^{-})$, *with* $p\in V$, *and is either unbounded or goes back to some* $(0,{q}^{-})$, *where* $q\notin V$.

*Proof* Let Ω be the open set obtained by removing from

$[0,+\mathrm{\infty})\times {C}_{T}(M)$ the closed set

$\{(0,{q}^{-}):q\notin V\}$. In other words,

$\mathrm{\Omega}=([0,+\mathrm{\infty})\times {C}_{T}(M))\setminus (\{0\}\times {(M\setminus V)}^{-}).$

Observe that $\overline{\mathrm{\Omega}}$ is complete due to the closedness of *M*. Consider, by Theorem 3.3, a connected set $\mathrm{\Gamma}\subseteq \mathrm{\Omega}$ of nontrivial *T*-periodic pairs with noncompact closure (in Ω) and intersecting $\{0\}\times {C}_{T}(M)$ in a subset of $\{(0,{p}^{-})\in \mathrm{\Omega}:w(p)=0\}$. Suppose that Γ is bounded. From Remark 3.4 it follows that $\overline{\mathrm{\Gamma}}\setminus {\overline{\mathrm{\Gamma}}}_{\mathrm{\Omega}}$, where ${\overline{\mathrm{\Gamma}}}_{\mathrm{\Omega}}$ denotes the closure of Γ in Ω, is nonempty and hence contains a point $(0,{q}^{-})$ which does not belong to Ω, that is, such that $q\notin V$. □

**Remark 3.12** The assumption of Corollary 3.11 above on the existence of an open subset *V* of *M* such that $deg(w,V)\ne 0$ is clearly satisfied in the case when *w* has an isolated zero with nonzero index. For example, if $w(p)=0$ and *w* is ${C}^{1}$ with injective derivative ${w}^{\prime}(p):{T}_{p}M\to {\mathbb{R}}^{k}$, then *p* is an isolated zero of *w* and its index is either 1 or −1. In fact, in this case, ${w}^{\prime}(p)$ sends ${T}_{p}M$ into itself and, consequently, its determinant is well defined and nonzero. The index of *p* is just the sign of this determinant (see, *e.g.*, [29]).

The next consequence of Theorem 3.3 provides an existence result for *T*-periodic solutions already obtained in [6]. Moreover, it improves an analogous result in [3], in which the map *f* is continuous on $\mathbb{R}\times C((-\mathrm{\infty},0],M)$, with the compact-open topology in $C((-\mathrm{\infty},0],M)$. In fact, such a coarse topology makes the assumption of the continuity of *f* a more restrictive condition than the one we require here.

**Corollary 3.13** *Let* *M* *and* *f* *be as in Theorem * 3.3. *Assume that* *f* *sends bounded subsets of* $\mathbb{R}\times \tilde{M}$ *into bounded subsets of* ${\mathbb{R}}^{k}$. *In addition*, *suppose that* *M* *is compact with Euler*-*Poincaré characteristic* $\chi (M)\ne 0$. *Then equation* (3.1) *has a connected unbounded set of nontrivial* *T*-*periodic pairs whose closure meets* $\{0\}\times {C}_{T}(M)$. *Therefore*, *since* ${C}_{T}(M)$ *is bounded*, *equation* (3.1) *has a* *T*-*periodic solution for any* $\lambda \ge 0$.

*Proof* Choose

$V=M$. By the Poincaré-Hopf theorem, we have

$deg(w,M)=\chi (M)\ne 0,$

where *w* is the mean value tangent vector field defined in formula (3.2). The assertion follows from Corollary 3.11. □

Corollary 3.14 below is a kind of continuation principle in the spirit of a well-known result due to Jean Mawhin for ODEs in ${\mathbb{R}}^{k}$ [7, 8] and extends an analogous one for ODEs on differentiable manifolds [31]. In what follows, by a *T-periodic orbit* of ${x}^{\prime}(t)=\lambda f(t,{x}_{t})$, we mean the image of a *T*-periodic solution of this equation.

**Corollary 3.14** (Mawhin-type continuation principle)

*Let* *M* *and* *f* *be as in Theorem * 3.3

*and let* *w* *be the mean value tangent vector field defined in formula* (3.2).

*Assume that* *f* *sends bounded subsets of* $\mathbb{R}\times \tilde{M}$ *into bounded subsets of* ${\mathbb{R}}^{k}$.

*Let* *V* *be a relatively compact open subset of* *M* *and assume that* - 1.
$w(p)\ne 0$ *along the boundary* *∂V* *of* *V*;

- 2.

- 3.
*for any* $\lambda \in (0,1]$, *the* *T*-*periodic orbits of* ${x}^{\prime}(t)=\lambda f(t,{x}_{t})$ *lying in* $\overline{V}$ *do not meet* *∂V*.

*Then the equation*
${x}^{\prime}(t)=f(t,{x}_{t})$

*has a* *T*-*periodic orbit in V*.

*Proof* Define

$\mathrm{\Omega}=[0,1)\times {C}_{T}(V)$. Observe that

${C}_{T}(\overline{V})=\overline{{C}_{T}(V)}$. Therefore,

$\partial \mathrm{\Omega}=(\{1\}\times {C}_{T}(\overline{V}))\cup ([0,1)\times {C}_{T}(\overline{V})\setminus {C}_{T}(V)).$

According to Theorem 3.3, call Γ a connected subset of Ω of nontrivial *T*-periodic pairs of the equation ${x}^{\prime}(t)=\lambda f(t,{x}_{t})$, whose closure in Ω is noncompact and intersects $\{0\}\times {C}_{T}(M)$ in a subset of $\{(0,{p}^{-})\in \mathrm{\Omega}:w(p)=0\}$.

As

*V* has compact closure in

*M*, then the closure of Ω in

$[0,+\mathrm{\infty})\times {C}_{T}(M)$ is complete, being

$\overline{\mathrm{\Omega}}=[0,1]\times {C}_{T}(\overline{V}).$

Since *f* sends bounded subsets of $\mathbb{R}\times \tilde{M}$ into bounded subsets of ${\mathbb{R}}^{k}$, recalling Remark 3.4, one has that the closure $\overline{\mathrm{\Gamma}}$ of Γ in the whole space (which coincides with the closure in $\overline{\mathrm{\Omega}}$) must intersect *∂* Ω.

Now, because of the above condition 3, $\overline{\mathrm{\Gamma}}$ cannot contain elements of $(0,1)\times {C}_{T}(\overline{V})\setminus {C}_{T}(V)$. In addition, condition 1 and Theorem 3.2 imply that $\overline{\mathrm{\Gamma}}$ does not contain elements of $\{0\}\times ({C}_{T}(\overline{V})\setminus {C}_{T}(V))$. Therefore, the nonempty set $\overline{\mathrm{\Gamma}}\cap \partial \mathrm{\Omega}$ is composed of pairs of the form $(1,x)$, where *x* is a *T*-periodic solution of ${x}^{\prime}(t)=f(t,{x}_{t})$ whose image is contained in *V*. □