Global continuation of periodic solutions for retarded functional differential equations on manifolds
 Pierluigi Benevieri^{1, 2},
 Alessandro Calamai^{3},
 Massimo Furi^{1} and
 Maria Patrizia Pera^{1}Email author
DOI: 10.1186/16872770201321
© Benevieri et al.; licensee Springer. 2013
Received: 19 October 2012
Accepted: 18 January 2013
Published: 11 February 2013
Abstract
We consider Tperiodic parametrized retarded functional differential equations, with infinite delay, on (possibly) noncompact manifolds. Using a topological approach, based on the notions of degree of a tangent vector field and of the fixed point index, we prove a global continuation result for Tperiodic solutions of such equations.
Our main theorem is a generalization to the case of retarded equations of a global continuation result obtained by the last two authors for ordinary differential equations on manifolds. As corollaries we obtain a Rabinowitztype global bifurcation result and a continuation principle of Mawhin type.
MSC:34K13, 34C40, 37C25, 70K42.
Keywords
retarded functional differential equations global bifurcation fixed point index degree of a vector field1 Introduction
 1.
$f(t,\phi )=f(t+T,\phi )$, $\mathrm{\forall}(t,\phi )\in \mathbb{R}\times \tilde{M}$;
 2.
$f(t,\phi )\in {T}_{\phi (0)}M$, $\mathrm{\forall}(t,\phi )\in \mathbb{R}\times \tilde{M}$;
 3.
f is locally Lipschitz in the second variable.
A solution of (1.1) is a function x with values in the ambient manifold M, defined on an open real interval J with $infJ=\mathrm{\infty}$, bounded and uniformly continuous on any closed halfline $(\mathrm{\infty},b]\subset J$ such that the equality ${x}^{\prime}(t)=\lambda f(t,{x}_{t})$ is eventually verified. We use here the standard notation in functional equations: whenever it makes sense, ${x}_{t}\in \tilde{M}$ denotes the function $\theta \mapsto x(t+\theta )$.
To proceed with the exposition of our problem, we need some further notation. Given $p\in M$, ${p}^{}$ denotes the constant pvalued function defined on ℝ or on any convenient subinterval of ℝ. The actual domain of ${p}^{}$ will be clear from the context. Moreover, given any $A\subseteq M$, ${A}^{}$ stands for the set $\{{p}^{}:p\in A\}$. All the functions of ${A}^{}$ will be considered defined on the same interval, suggested by the context. By ${C}_{T}(M)$ we mean the set of all continuous Tperiodic maps $x:\mathbb{R}\to M$. This set, which contains ${M}^{}$, is a metric subspace of the Banach space ${C}_{T}({\mathbb{R}}^{k})$ with the standard supremum norm. We call $(\lambda ,x)\in [0,+\mathrm{\infty})\times {C}_{T}(M)$ a Tperiodic pair of equation (1.1) if $x:\mathbb{R}\to M$ is a solution of (1.1) corresponding to λ. Among these pairs, we distinguish the trivial ones, that is, the elements of the set $\{0\}\times {M}^{}$, which can be isometrically identified with M. Notice that any Tperiodic pair of the type $(0,x)$ is trivial since the function x turns out to be necessarily constant. An element $p\in M$ will be called a bifurcation point of (1.1) if any neighborhood of $(0,{p}^{})$ in $[0,+\mathrm{\infty})\times {C}_{T}(M)$ contains nontrivial Tperiodic pairs. Roughly speaking, $p\in M$ is a bifurcation point if any of its neighborhoods in M contains Tperiodic orbits corresponding to arbitrarily small values of $\lambda >0$.
and the fixed point index of a sort of Poincaré Ttranslation operator acting inside the Banach space $C([T,0],{\mathbb{R}}^{k})$.
The prelude of our approach can be found in some papers of the last two authors (see, for instance, [1]), where the notions of degree of a tangent vector field and of fixed point index of a suitable Poincaré Ttranslation operator are related in order to get continuation results for ODEs on differentiable manifolds.
Theorem 3.3 extends and unifies two results recently obtained by the authors in [2] and [3]. In [2] the ambient manifold M is not necessarily compact, but our investigation regards delay differential equations with finite time lag. On the other hand, in [3] we consider RFDEs with infinite delay; nevertheless, in this case M is compact and the map f is defined on $\mathbb{R}\times C((\mathrm{\infty},0],M)$ with a topology which is too weak, making the continuity assumption on f a too heavy condition.
We point out that, in order to obtain our continuation result for RFDEs with infinite delay without assuming the compactness of the ambient manifold M, we had to tackle strong technical difficulties. Therefore, we were forced to undertake a thorough preliminary investigation on the general properties of RFDEs with infinite delay on (possibly) noncompact manifolds. This was the purpose of our recent paper [4].
In our opinion the existence of a global bifurcating branch ensured by Theorem 3.3 should hold also without the assumption that f is locally Lipschitz in the second variable. However, we are not able to prove or disprove this conjecture because of some difficulties arising in this case. One is that the uniqueness of the initial value problem for equation (1.1) is not ensured and, consequently, a Poincaré Ttranslation operator is not defined as a single valued map. A classical tool to overcome this obstacle, usually applied in analogous problems, consists in considering a sequence of ${C}^{1}$ maps approximating f. In our situation, however, because of the peculiar domain of f, we do not know how to realize this approach, and this is another difficulty.
We conclude the paper with some consequences of Theorem 3.3. One is a Rabinowitztype global bifurcation result [5] obtained by assuming that the degree of the above tangent vector field w is nonzero on an open subset of M. Another corollary is deduced when M is compact: we get an existence result already proved in [6], and we extend an analogous one obtained in [3] in which the continuity assumption on f is too heavy. A third interesting case occurs when the degree of w is nonzero on a relatively compact open subset of M and suitable a priori bounds hold for the Tperiodic orbits of equation (1.1): in this case, we obtain a continuation principle à la Mawhin [7, 8].
The different and related cases of RFDEs with finite delay in Euclidean spaces have been investigated by many authors. For general reference, we suggest the monograph by Hale and Verduyn Lunel [9]. We refer also to the works of Gaines and Mawhin [10], Nussbaum [11, 12] and MalletParet, Nussbaum and Paraskevopoulos [13]. For RFDEs with infinite delay in Euclidean spaces, we recommend the article of Hale and Kato [14], the book by Hino, Murakami and Naito [15], and the more recent paper of Oliva and Rocha [16]. For RFDEs with finite delay on manifolds, we suggest the papers of Oliva [17, 18]. Finally, for RFDEs with infinite delay on manifolds we cite [4].
2 Preliminaries
2.1 Fixed point index
We recall that a metrizable space $\mathcal{X}$ is an absolute neighborhood retract (ANR) if, whenever it is homeomorphically embedded as a closed subset C of a metric space $\mathcal{Y}$, there exist an open neighborhood V of C in $\mathcal{Y}$ and a retraction $r:V\to C$ (see, e.g., [19, 20]). Polyhedra and differentiable manifolds are examples of ANRs. Let us also recall that a continuous map between topological spaces is called locally compact if each point in its domain has a neighborhood whose image is contained in a compact set.
Let $\mathcal{X}$ be a metric ANR and consider a locally compact (continuous) $\mathcal{X}$valued map k defined on a subset $\mathcal{D}(k)$ of $\mathcal{X}$. Given an open subset U of $\mathcal{X}$ contained in $\mathcal{D}(k)$, if the set of fixed points of k in U is compact, the pair $(k,U)$ is called admissible. We point out that such a condition is clearly satisfied if $\overline{U}\subseteq \mathcal{D}(k)$, $\overline{k(U)}$ is compact and $k(p)\ne p$ for all p in the boundary of U. To any admissible pair $(k,U)$, one can associate an integer ${ind}_{\mathcal{X}}(k,U)$  the fixed point index of k in U  which satisfies properties analogous to those of the classical LeraySchauder degree [21]. The reader can see, for instance, [12, 22–24] for a comprehensive presentation of the index theory for ANRs. As regards the connection with the homology theory, we refer to standard algebraic topology textbooks (e.g., [25, 26]).
We summarize below the main properties of the fixed point index.

(Existence) If${ind}_{\mathcal{X}}(k,U)\ne 0$, then k admits at least one fixed point in U.

(Normalization) If$\mathcal{X}$is compact, then${ind}_{\mathcal{X}}(k,\mathcal{X})=\mathrm{\Lambda}(k)$, where$\mathrm{\Lambda}(k)$denotes the Lefschetz number of k.

(Additivity) Given two disjoint open subsets${U}_{1}$,${U}_{2}$of U, if any fixed point of k in U is contained in${U}_{1}\cup {U}_{2}$, then${ind}_{\mathcal{X}}(k,U)={ind}_{\mathcal{X}}(k,{U}_{1})+{ind}_{\mathcal{X}}(k,{U}_{2})$.

(Excision) Given an open subset${U}_{1}$of U, if k has no fixed points in$U\mathrm{\setminus}{U}_{1}$, then${ind}_{\mathcal{X}}(k,U)={ind}_{\mathcal{X}}(k,{U}_{1})$.

(Commutativity) Let$\mathcal{X}$and$\mathcal{Y}$be metric ANRs. Suppose that U and V are open subsets of$\mathcal{X}$and$\mathcal{Y}$respectively and that$k:U\to \mathcal{Y}$and$h:V\to \mathcal{X}$are locally compact maps. Assume that the set of fixed points of either hk in${k}^{1}(V)$or kh in${h}^{1}(U)$is compact. Then the other set is compact as well and${ind}_{\mathcal{X}}(hk,{k}^{1}(V))={ind}_{\mathcal{Y}}(kh,{h}^{1}(U))$.

(Generalized homotopy invariance) Let I be a compact real interval and W be an open subset of$\mathcal{X}\times I$. For any$\lambda \in I$, denote${W}_{\lambda}=\{x\in \mathcal{X}:(x,\lambda )\in W\}$. Let$H:W\to \mathcal{X}$be a locally compact map such that the set$\{(x,\lambda )\in W:H(x,\lambda )=x\}$is compact. Then${ind}_{\mathcal{X}}(H(\cdot ,\lambda ),{W}_{\lambda})$is independent of λ.
2.2 Degree of a vector field
Let us recall some basic notions on degree theory for tangent vector fields on differentiable manifolds. Let $v:M\to {\mathbb{R}}^{k}$ be a continuous (autonomous) tangent vector field on a smooth manifold M, and let U be an open subset of M. We say that the pair $(v,U)$ is admissible (or, equivalently, that v is admissible in U) if ${v}^{1}(0)\cap U$ is compact. In this case, one can assign to the pair $(v,U)$ an integer, $deg(v,U)$, called the degree (or Euler characteristic, or rotation) of the tangent vector field v in U which, roughly speaking, counts algebraically the number of zeros of v in U (for general references, see, e.g., [27–30]). Notice that the condition for ${v}^{1}(0)\cap U$ to be compact is clearly satisfied if U is a relatively compact open subset of M and $v(p)\ne 0$ for all p in the boundary of U.
As a consequence of the PoincaréHopf theorem, when M is compact, $deg(v,M)$ equals $\chi (M)$, the EulerPoincaré characteristic of M.
In the particular case when U is an open subset of ${\mathbb{R}}^{k}$, $deg(v,U)$ is just the classical Brouwer degree of v in U when the map v is regarded as a vector field; namely, the degree $deg(v,U,0)$ of v in U with target value $0\in {\mathbb{R}}^{k}$. All the standard properties of the Brouwer degree in the flat case, such as homotopy invariance, excision, additivity, existence, still hold in the more general context of differentiable manifolds. To see this, one can use an equivalent definition of degree of a tangent vector field based on the fixed point index theory as presented in [1] and [31].
Let us stress that, actually, in [1] and [31] the definition of degree of a tangent vector field on M is given in terms of the fixed point index of a Poincarétype translation operator associated to a suitable ODE on M. Such a definition provides a formula that will play a central role in Lemma 3.8 below, and this will be a crucial step in the proof of our main result.
We point out that no orientability of M is required for $deg(v,U)$ to be defined. This highlights the fact that the extension of the Brouwer degree for tangent vector fields in the nonflat case does not coincide with the one regarding maps between oriented manifolds with a given target value (as illustrated, for example, in [28, 29]). This dichotomy of the notion of degree in the nonflat situation is not evident in ${\mathbb{R}}^{k}$: it is masked by the fact that an equation of the type $f(x)=y$ can be written as $f(x)y=0$. Anyhow, in the context of RFDEs (ODEs included), it is the degree of a vector field that plays a significative role.
where m denotes the dimension of M. Moreover, if v has an isolated zero p and U is an isolating (open) neighborhood of p, then $deg(v,U)$ is called the index of v at p. The excision property ensures that this is a welldefined integer.
2.3 Retarded functional differential equations
Notice that ${\tilde{\mathbb{R}}}^{k}$ is a Banach space, being closed in the space $BC((\mathrm{\infty},0],{\mathbb{R}}^{k})$ of the bounded and continuous functions from $(\mathrm{\infty},0]$ into ${\mathbb{R}}^{k}$ (endowed with the standard supremum norm).
Throughout the paper, the norm in ${\mathbb{R}}^{k}$ will be denoted by $\cdot $ and the norm in the infinite dimensional space ${\tilde{\mathbb{R}}}^{k}$ by $\parallel \cdot \parallel $. Thus, the distance between two elements ϕ and ψ of $\tilde{A}$ will be denoted $\parallel \varphi \psi \parallel $, even when $\varphi \psi $ does not belong to $\tilde{A}$. We observe that $\tilde{A}$, as a metric space, is complete if and only if A is closed in ${\mathbb{R}}^{k}$.
is said to be a retarded functional tangent vector field over M if $g(t,\phi )\in {T}_{\phi (0)}M$ for all $(t,\phi )\in \mathbb{R}\times \tilde{M}$. In the sequel, any map with this property will be briefly called a functional field (over M).
where $g:\mathbb{R}\times \tilde{M}\to {\mathbb{R}}^{k}$ is a functional field over M. Here, as usual and whenever it makes sense, given $t\in \mathbb{R}$, by ${x}_{t}\in \tilde{M}$ we mean the function $\theta \mapsto x(t+\theta )$.
A solution of (2.2) is a function $x:J\to M$, defined on an open real interval J with $infJ=\mathrm{\infty}$, bounded and uniformly continuous on any closed halfline $(\mathrm{\infty},b]\subset J$, which verifies eventually the equality ${x}^{\prime}(t)=g(t,{x}_{t})$. That is, $x:J\to M$ is a solution of (2.2) if ${x}_{t}\in \tilde{M}$ for all $t\in J$ and there exists $\tau \in J$ such that x is ${C}^{1}$ on the interval $(\tau ,supJ)$ and ${x}^{\prime}(t)=g(t,{x}_{t})$ for all $t\in (\tau ,supJ)$. Observe that the derivative of a solution x may not exist at $t=\tau $. However, the right derivative ${D}_{+}x(\tau )$ of x at τ always exists and is equal to $g(\tau ,{x}_{\tau})$. Also, notice that $t\mapsto {x}_{t}$ is a continuous curve in $\tilde{M}$ since x is uniformly continuous on any closed halfline $(\mathrm{\infty},b]$ of J.
A solution of (2.2) is said to be maximal if it is not a proper restriction of another solution. As in the case of ODEs, Zorn’s lemma implies that any solution is the restriction of a maximal solution.
A solution of (2.3) is a solution $x:J\to M$ of (2.2) such that $supJ>0$, ${x}^{\prime}(t)=g(t,{x}_{t})$ for $t>0$ and ${x}_{0}=\eta $.
The continuous dependence of the solutions on initial data is stated in Theorem 2.1 below and is a straightforward consequence of Theorem 4.4 of [4].
is open and the map $\eta \in \mathcal{D}\mapsto {x}_{T}^{\eta}\in \tilde{M}$, where ${x}^{\eta}(\cdot )$ is the unique maximal solution of problem (2.3), is continuous.
More generally, we will need to consider initial value problems depending on a parameter such as equation (1.1) with the initial condition ${x}_{0}=\eta $. For these problems the continuous dependence is ensured by the following consequence of Theorem 2.1.
Corollary 2.2 (Continuous dependence)
is open and the map $(\alpha ,\eta )\in {\mathcal{D}}^{\prime}\mapsto {x}_{T}^{(\alpha ,\eta )}\in \tilde{M}$, where ${x}^{(\alpha ,\eta )}(\cdot )$ is the unique maximal solution of problem (2.4), is continuous.
Proof
that can be regarded as an initial value problem of a RFDE on the ambient manifold ${\mathbb{R}}^{s}\times M\subseteq {\mathbb{R}}^{s+k}$. □
In Theorem 2.1 and in Corollary 2.2 above, the hypothesis of the uniqueness of the maximal solution of problems (2.3) and (2.4) is essential in order to make their statements meaningful. Sufficient conditions for the uniqueness are presented in Remark 2.3 below.
for all $(t,\phi ),(t,\psi )\in Q$. Moreover, we will say that g is locally cLipschitz if for any $(\tau ,\eta )\in \mathbb{R}\times \tilde{M}$ there exists an open neighborhood of $(\tau ,\eta )$ in which g is cLipschitz. In spite of the fact that a locally Lipschitz map is not necessarily (globally) Lipschitz, one could actually show that if g is locally cLipschitz, then it is also (globally) cLipschitz. As a consequence, if g is locally Lipschitz in the second variable, then it is cLipschitz as well. In [4] we proved that if g is a cLipschitz functional field, then problem (2.3) has a unique maximal solution for any $\eta \in \tilde{M}$. For a characterization of compact subsets of $\tilde{M}$ see, e.g., [[32], Part 1, IV.6.5].
We close this section with the following lemma whose elementary proof is given for the sake of completeness.
Lemma 2.4 Let $F:\mathcal{X}\to \mathcal{Y}$ be a continuous map between metric spaces and let $\{{\gamma}_{n}\}$ be a sequence of continuous functions from a compact interval $[a,b]$ (or, more generally, from a compact space) into $\mathcal{X}$. If $\{{\gamma}_{n}(s)\}$ converges to $\gamma (s)$ uniformly for $s\in [a,b]$, then also $F({\gamma}_{n}(s))\to F(\gamma (s))$ uniformly for $s\in [a,b]$.
Proof Notice that if K is a compact subset of $\mathcal{X}$, then for any $\epsilon >0$ there exists $\delta >0$ such that $x\in \mathcal{X}$, $k\in K$, ${dist}_{\mathcal{X}}(x,k)<\delta $ imply ${dist}_{\mathcal{Y}}(F(x),F(k))<\epsilon $. Now, our assertion follows immediately by taking the compact K to be the image of the limit function $\gamma :[a,b]\to \mathcal{X}$. □
3 Branches of periodic solutions
Moreover, denote by ${C}_{T}({\mathbb{R}}^{k})$ the Banach space of the continuous Tperiodic 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 Mvalued 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.
As in the introduction, we call $(\lambda ,x)\in [0,+\mathrm{\infty})\times {C}_{T}(M)$ a Tperiodic pair (of (3.1)) if the function $x:\mathbb{R}\to M$ is a (Tperiodic) solution of (3.1) corresponding to λ. Let us denote by X the set of all Tperiodic 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 Tperiodic 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 pvalued function defined on some real interval that will be clear from the context. Moreover, a Tperiodic 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 Tperiodic pair (i.e., a Tperiodic pair $(\lambda ,x)$ with $\lambda >0$). In some sense, p is a bifurcation point if, for $\lambda >0$ sufficiently small, there are Tperiodic orbits of (3.1) arbitrarily close to p.
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 Tperiodic 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 Tperiodic 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 Tperiodic 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).
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 Tperiodic pairs of equation (3.1) (a global bifurcating branch for short), whose closure in Ω is noncompact and intersects the set of trivial Tperiodic pairs contained in Ω.
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 Tperiodic 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 Poincarétype Ttranslation operator whose fixed points are the restrictions to the interval $[T,0]$ of the Tperiodic 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 Tperiodic map on an interval J, we mean the restriction to J of a Tperiodic map defined on ℝ.
 1.
$\tilde{\psi}$ is an extension of ψ;
 2.
$\tilde{\psi}$ is Tperiodic on $(\mathrm{\infty},T]$;
 3.
$\tilde{\psi}$ is Tperiodic on $(\mathrm{\infty},0]$, whenever $\psi \in {\stackrel{\u02c6}{M}}_{\ast}$.
where $\tilde{\psi}$ is the extension of ψ as in Lemma 3.5.
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 $[0,+\mathrm{\infty})\times \stackrel{\u02c6}{M}$.
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 Tperiodic 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 Tperiodic 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 $[T,0]$.
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 Poincarétype 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.
 (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 $\mathcal{V}$.
and that H is continuous and locally compact.
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 Tperiodic 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 ${k}^{1}({V}^{\prime})$.
and thus Step 5 is proved.
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 Wyburntype 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.
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.
 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 $P(W)$.
and we have a contradiction. Therefore, $(Y,K)$ verifies the assumptions of Lemma 3.9 and the proof is complete. □
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 (Rabinowitztype 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 Tperiodic pairs whose closure contains some $(0,{p}^{})$, with $p\in V$, and is either unbounded or goes back to some $(0,{q}^{})$, where $q\notin V$.
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 Tperiodic 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 Tperiodic 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 compactopen 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 EulerPoincaré characteristic $\chi (M)\ne 0$. Then equation (3.1) has a connected unbounded set of nontrivial Tperiodic pairs whose closure meets $\{0\}\times {C}_{T}(M)$. Therefore, since ${C}_{T}(M)$ is bounded, equation (3.1) has a Tperiodic solution for any $\lambda \ge 0$.
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 wellknown 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 Tperiodic orbit of ${x}^{\prime}(t)=\lambda f(t,{x}_{t})$, we mean the image of a Tperiodic solution of this equation.
Corollary 3.14 (Mawhintype continuation principle)
 1.
$w(p)\ne 0$ along the boundary ∂V of V;
 2.
$deg(w,V)\ne 0$;
 3.
for any $\lambda \in (0,1]$, the Tperiodic orbits of ${x}^{\prime}(t)=\lambda f(t,{x}_{t})$ lying in $\overline{V}$ do not meet ∂V.
has a Tperiodic orbit in V.
According to Theorem 3.3, call Γ a connected subset of Ω of nontrivial Tperiodic 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 $\{(0,{p}^{})\in \mathrm{\Omega}:w(p)=0\}$.
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 Tperiodic solution of ${x}^{\prime}(t)=f(t,{x}_{t})$ whose image is contained in V. □
Declarations
Acknowledgements
Dedicated to our friend and outstanding mathematician Jean Mawhin.
Pierluigi Benevieri is partially sponsored by Fapesp, Grant n. 2010/207274.
Authors’ Affiliations
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