We consider T-periodic 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 T-periodic 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 Rabinowitz-type 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 field
In this paper we prove a global continuation result for periodic solutions of the following retarded functional differential equation (RFDE for short) on a manifold, depending on a parameter :
Let us present the setting of the problem. Consider a boundaryless smooth m-dimensional manifold and, given any , let stand for the tangent space of M at p. Denote by the set of bounded and uniformly continuous maps from into M, and observe that this is a metric space as a subset of the Banach space with the usual supremum norm. Given , let be a continuous function verifying the following conditions:
1. , ;
2. , ;
3. f is locally Lipschitz in the second variable.
A solution of (1.1) is a function x with values in the ambient manifoldM, defined on an open real interval J with , bounded and uniformly continuous on any closed half-line such that the equality is eventually verified. We use here the standard notation in functional equations: whenever it makes sense, denotes the function .
To proceed with the exposition of our problem, we need some further notation. Given , denotes the constant p-valued function defined on ℝ or on any convenient subinterval of ℝ. The actual domain of will be clear from the context. Moreover, given any , stands for the set . All the functions of will be considered defined on the same interval, suggested by the context. By we mean the set of all continuous T-periodic maps . This set, which contains , is a metric subspace of the Banach space with the standard supremum norm. We call a T-periodic pair of equation (1.1) if is a solution of (1.1) corresponding to λ. Among these pairs, we distinguish the trivial ones, that is, the elements of the set , which can be isometrically identified with M. Notice that any T-periodic pair of the type is trivial since the function x turns out to be necessarily constant. An element will be called a bifurcation point of (1.1) if any neighborhood of in contains nontrivial T-periodic pairs. Roughly speaking, is a bifurcation point if any of its neighborhoods in M contains T-periodic orbits corresponding to arbitrarily small values of .
The main outcome of this paper, Theorem 3.3 below, is a global continuation result for T-periodic solutions of equation (1.1). That is, given an open subset Ω of , it is a result which provides sufficient conditions for the existence of a global bifurcating branch in Ω, meaning a connected subset of Ω of nontrivialT-periodic pairs whose closure in Ω is noncompact and intersects the set of trivialT-periodic pairs. The proof of Theorem 3.3 is based on a relation, obtained in a technical result, Lemma 3.8 below, between the degree (in an open subset of M) of the tangent vector field
and the fixed point index of a sort of Poincaré T-translation operator acting inside the Banach space .
The prelude of our approach can be found in some papers of the last two authors (see, for instance, ), where the notions of degree of a tangent vector field and of fixed point index of a suitable Poincaré T-translation 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  and . In  the ambient manifold M is not necessarily compact, but our investigation regards delay differential equations with finite time lag. On the other hand, in  we consider RFDEs with infinite delay; nevertheless, in this case M is compact and the map f is defined on 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 .
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 T-translation 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 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 Rabinowitz-type global bifurcation result  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 , and we extend an analogous one obtained in  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 T-periodic 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 . We refer also to the works of Gaines and Mawhin , Nussbaum [11,12] and Mallet-Paret, Nussbaum and Paraskevopoulos . For RFDEs with infinite delay in Euclidean spaces, we recommend the article of Hale and Kato , the book by Hino, Murakami and Naito , and the more recent paper of Oliva and Rocha . 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 .
2.1 Fixed point index
We recall that a metrizable space is an absolute neighborhood retract (ANR) if, whenever it is homeomorphically embedded as a closed subset C of a metric space , there exist an open neighborhood V of C in and a retraction (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 be a metric ANR and consider a locally compact (continuous) -valued map k defined on a subset of . Given an open subset U of contained in , if the set of fixed points of k in U is compact, the pair is called admissible. We point out that such a condition is clearly satisfied if , is compact and for all p in the boundary of U. To any admissible pair , one can associate an integer - the fixed point index of k in U - which satisfies properties analogous to those of the classical Leray-Schauder degree . 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 , thenkadmits at least one fixed point inU.
• (Normalization) If is compact, then , where denotes the Lefschetz number ofk.
• (Additivity) Given two disjoint open subsets , ofU, if any fixed point ofkinUis contained in , then .
• (Excision) Given an open subset ofU, ifkhas no fixed points in , then .
• (Commutativity) Let and be metric ANRs. Suppose thatUandVare open subsets of and respectively and that and are locally compact maps. Assume that the set of fixed points of eitherhkin orkhin is compact. Then the other set is compact as well and .
• (Generalized homotopy invariance) LetIbe a compact real interval andWbe an open subset of . For any , denote . Let be a locally compact map such that the set is compact. Then is independent ofλ.
2.2 Degree of a vector field
Let us recall some basic notions on degree theory for tangent vector fields on differentiable manifolds. Let 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 is admissible (or, equivalently, that v is admissible in U) if is compact. In this case, one can assign to the pair an integer, , 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 to be compact is clearly satisfied if U is a relatively compact open subset of M and for all p in the boundary of U.
As a consequence of the Poincaré-Hopf theorem, when M is compact, equals , the Euler-Poincaré characteristic of M.
In the particular case when U is an open subset of , is just the classical Brouwer degree of v in U when the map v is regarded as a vector field; namely, the degree of v in U with target value . 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  and .
Let us stress that, actually, in  and  the definition of degree of a tangent vector field on M is given in terms of the fixed point index of a Poincaré-type 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 to be defined. This highlights the fact that the extension of the Brouwer degree for tangent vector fields in the non-flat 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 non-flat situation is not evident in : it is masked by the fact that an equation of the type can be written as . Anyhow, in the context of RFDEs (ODEs included), it is the degree of a vector field that plays a significative role.
It is known that if is admissible, then
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 is called the index of v at p. The excision property ensures that this is a well-defined integer.
2.3 Retarded functional differential equations
Given an arbitrary subset A of , we denote by the set of bounded and uniformly continuous maps from into A. For brevity, we will use the notation
Notice that is a Banach space, being closed in the space of the bounded and continuous functions from into (endowed with the standard supremum norm).
Throughout the paper, the norm in will be denoted by and the norm in the infinite dimensional space by . Thus, the distance between two elements ϕ and ψ of will be denoted , even when does not belong to . We observe that , as a metric space, is complete if and only if A is closed in .
Let M be a boundaryless smooth manifold in . A continuous map
is said to be a retarded functional tangent vector field overM if for all . In the sequel, any map with this property will be briefly called a functional field (overM).
Let us consider a retarded functional differential equation (RFDE) of the type
where is a functional field over M. Here, as usual and whenever it makes sense, given , by we mean the function .
A solution of (2.2) is a function , defined on an open real interval J with , bounded and uniformly continuous on any closed half-line , which verifies eventually the equality . That is, is a solution of (2.2) if for all and there exists such that x is on the interval and for all . Observe that the derivative of a solution x may not exist at . However, the right derivative of x at τ always exists and is equal to . Also, notice that is a continuous curve in since x is uniformly continuous on any closed half-line 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.
Given , let us associate to equation (2.2) the initial value problem
A solution of (2.3) is a solution of (2.2) such that , for and .
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 .
Theorem 2.1LetMbe a boundaryless smooth manifold and be a functional field. Assume, for any , the uniqueness of the maximal solution of problem (2.3). Then, given , the set
is open and the map , where 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 . For these problems the continuous dependence is ensured by the following consequence of Theorem 2.1.
Corollary 2.2 (Continuous dependence)
LetMbe a boundaryless smooth manifold and a parametrized functional field. For any and , assume the uniqueness of the maximal solution of the problem
Then, given , the set
is open and the map , where is the unique maximal solution of problem (2.4), is continuous.
Apply Theorem 2.1 to the problem
that can be regarded as an initial value problem of a RFDE on the ambient manifold . □
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.
Remark 2.3 A functional field is said to be compactly Lipschitz (for short, c-Lipschitz) if, given any compact subset Q of , there exists such that
for all . Moreover, we will say that g is locally c-Lipschitz if for any there exists an open neighborhood of in which g is c-Lipschitz. In spite of the fact that a locally Lipschitz map is not necessarily (globally) Lipschitz, one could actually show that if g is locally c-Lipschitz, then it is also (globally) c-Lipschitz. As a consequence, if g is locally Lipschitz in the second variable, then it is c-Lipschitz as well. In  we proved that if g is a c-Lipschitz functional field, then problem (2.3) has a unique maximal solution for any . For a characterization of compact subsets of see, e.g., [, Part 1, IV.6.5].
We close this section with the following lemma whose elementary proof is given for the sake of completeness.
Lemma 2.4Let be a continuous map between metric spaces and let be a sequence of continuous functions from a compact interval (or, more generally, from a compact space) into . If converges to uniformly for , then also uniformly for .
Proof Notice that if K is a compact subset of , then for any there exists such that , , imply . Now, our assertion follows immediately by taking the compact K to be the image of the limit function . □
3 Branches of periodic solutions
Let M be a boundaryless smooth m-dimensional manifold in . Given , let
denote the metric subspace of of the M-valued continuous functions on and set
Moreover, denote by the Banach space of the continuous T-periodic maps (with the standard supremum norm) and by the metric subspace of of the M-valued maps. Observe that, since M is locally compact, then and (but not ) are locally complete. Moreover, they are complete if and only if M is closed.
Let be a functional field over M. Given , assume that f is T-periodic in the first variable. Consider the following RFDE depending on a parameter :
As in the introduction, we call a T-periodic pair (of (3.1)) if the function 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 3.1The setXis closed in and locally compact.
Proof Let be a sequence of T-periodic pairs of (3.1) converging to in . Because of Lemma 2.4, converges uniformly to for . Thus, uniformly and, therefore, , that is, belongs to X. This proves that X is closed in .
Now, as observed above, 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 , with the notation 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 is said to be trivial, and an element is a bifurcation point of equation (3.1) if any neighborhood of in contains a nontrivial T-periodic pair (i.e., a T-periodic pair with ). In some sense, p is a bifurcation point if, for 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 , with p a bifurcation point of (3.1). To this end, we introduce the mean value tangent vector field given by
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 to be a bifurcation point.
Theorem 3.2Let be such that is an accumulation point of nontrivialT-periodic pairs of (3.1). Then there exists such that , for any , and . Thus, any bifurcation point of (3.1) is a zero ofw.
Proof By assumption there exists a sequence of T-periodic pairs of (3.1) such that , , and uniformly on ℝ. As proved in Lemma 3.1, the set X of the T-periodic pairs is closed in . Thus, the pair belongs to X and, consequently, the function x must be constant, say for some . Clearly, the point p is a bifurcation point of (3.1).
Now, given , recalling that and that , we get
Observe that the sequence of curves converges uniformly to for . Hence, because of Lemma 2.4, uniformly for and the assertion follows passing to the limit in the above integral. □
Let now Ω be an open subset of . Our main result (Theorem 3.3 below) provides a sufficient condition for the existence of a bifurcation point p in M with . 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 3.3Let be a boundaryless smooth manifold, be a functional field onM, T-periodic in the first variable and locally Lipschitz in the second one, and be the autonomous tangent vector field
Let Ω be an open subset of and let be the map . Assume that is defined and nonzero. Then there exists a connected subset of Ω of nontrivialT-periodic pairs of equation (3.1) whose closure in Ω is noncompact and intersects in a (nonempty) subset of .
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 into bounded subsets of , and that M is closed in (or, more generally, that the closure of Ω in is complete).
Then a connected subset Γ of Ω as in Theorem 3.3 is either unbounded or, if bounded, its closure in reaches the boundary∂Ω of Ω.
To see this, assume that is bounded. Then, being bounded, because of Ascoli’s theorem, Γ is actually totally bounded. Thus, is compact, being totally bounded and, additionally, complete since is contained in . On the other hand, according to Theorem 3.3, the closure of Γ in Ω is noncompact. Consequently, the set is nonempty, and this means that 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 of the T-periodic solutions of (3.1). For this purpose, we need to introduce a suitable backward extension of the elements of . The properties of such an extension are contained in Lemma 3.5 below, obtained in . 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 3.5There exist an open neighborhoodUof in and a continuous map fromUto , , with the following properties:
1. is an extension ofψ;
2. isT-periodic on ;
3. isT-periodic on , whenever .
Let now U be an open subset of as in the previous lemma and let f be as in Theorem 3.3. Given and , consider the initial value problem
where is the extension of ψ as in Lemma 3.5.
The set D is nonempty since it contains (notice that for , the solution of problem (3.3) is constant for ). Moreover, it follows by Corollary 2.2 that D is open in .
Given , denote by the maximal solution of problem (3.3) and define
Observe that is the restriction of to the interval .
The following lemmas regard crucial properties of the operator P. The proof of the first one is standard and will be omitted.
Lemma 3.6The fixed points of correspond to theT-periodic solutions of equation (3.1) in the following sense: ψis a fixed point of if and only if it is the restriction to of aT-periodic solution.
Lemma 3.7The operatorPis 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 of Lemma 3.5 and of the map that associates to any its restriction to the interval .
Let us prove that P is locally compact. Take and denote, for simplicity, by the maximal solution of (3.3) corresponding to . Clearly, is defined at least up to T and for any . Set
Observe that K is compact, being the image of under the (continuous) curve . Let O be an open neighborhood of K in and such that for all . Let us show that there exists an open neighborhood W of in D such that if , then for , where is the maximal solution of (3.3) corresponding to . By contradiction, for any suppose there exist and such that and , where denotes the maximal solution of (3.3) corresponding to . We may assume . Now, from the fact that in the convergence is uniform, we get the equicontinuity of the sequence . This easily implies that . A contradiction, since O is open and belongs to . Thus, the existence of the required W is proved. Consequently, for any , the maximal solution of (3.3) corresponding to is such that for all .
Therefore, by Ascoli’s theorem and taking into account the local completeness of , we get that P maps W into a compact subset of . This proves that P is locally compact. □
The following result establishes the relationship between the fixed point index of the Poincaré-type operator and the degree of the mean value vector field w. It will be crucial in the proof of Lemma 3.10.
Lemma 3.8Let be an open subset of such that is compact and let be such that
(a) is contained in the domainDofP;
(b) is relatively compact;
(c) for andψin the boundary of .
Consider the open set . Then is well defined and
Proof Let U be an open subset of as in Lemma 3.5. Given , and , consider the initial value problem
where 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 and , the uniqueness of the solution of problem (3.4) is ensured (recall Remark 2.3). Denote by the maximal solution of problem (3.4), and put
Corollary 2.2 implies that E is open in . Therefore, is open in because of the compactness of . Moreover, observe that the slice of at coincides with U and that is contained in the domain D of the operator P defined above. Define by
Clearly, coincides with P on , while is the (infinite dimensional) operator associated to the undelayed problem
As in Lemmas 3.6 and 3.7, one can show that the fixed points of correspond to the T-periodic solutions of the equation
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,
Step 1. There exist and an open subset of , containing , with , and such that
(a′) (i.e., for , is defined in );
(b′) is relatively compact.
To prove Step 1, observe that is compact and contained in , which is open in , and recall that H is locally compact.
Step 2. For small values of , for any and .
By contradiction, suppose there exists a sequence in such that , , and . Without loss of generality, taking into account (b′), we may assume that and also that . Denote by the T-periodic solution of (3.4) corresponding to . Since is the restriction of to , then converges uniformly on ℝ to , where is the solution of (3.4) corresponding to the fixed point of . Therefore, there exists such that for any and, as in the proof of Theorem 3.2, we can show that . Thus, belongs to , contradicting the choice of . This proves Step 2.
Step 3. For small values of , for any .
The proof is analogous to that of Step 2, noting that for and taking into account assumption b) and the fact that is closed in .
Step 4. Let be defined by and consider the open set . Then there exists such that for any .
By contradiction, suppose there exists a sequence in such that , , and . Without loss of generality, taking into account (b′), we may assume that . Therefore, by the continuity of H, we get so that is a constant function of . This is impossible, since any constant function of is contained in .
Step 5. Let and be as in Step 4 and let be theT-translation operator , where is the maximal solution of the undelayed problem
Then, for small values ofλ, is defined and
To see this, let be as in Step 4 and, given , define by , . Clearly, k is a locally compact map since it takes values in the locally compact space M. Moreover, is actually compact since is contained in which is relatively compact by (b′) of Step 1. Now, observe that the composition coincides with in and that the set of fixed points of in is compact by (b′) of Step 1 and is contained in by Step 4. Thus, the set of fixed points of in is compact so that, by applying the commutativity property of the fixed point index to the maps k and , we get
Consequently, since it is easy to verify that the composition coincides with in , we obtain
and, because of Step 4, by the excision property of the index,
To complete the proof of Step 5, let us show that for λ sufficiently small, for . By contradiction, suppose there exists a sequence in such that , , and . Hence, there exists a sequence in such that and . Because of (b′) of Step 1, we may assume that so that, in particular, , where . Now, by an argument similar to that used in the proof of Theorem 3.2, we get that is constant and . Thus, . Moreover, since , we also obtain that belongs to , contradicting the choice of . Finally, again by excision, we get
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 and an open subset of , containing , with and such that if , then is defined and is independent of . Moreover, reducing if necessary, by Step 3 and by assumption (b), it follows that for , the fixed points of in are a compact subset of . Therefore, by the excision property and the homotopy invariance of the index, we get
On the other hand, by Step 5, if is sufficiently small, we have
Moreover, as shown in ,
Finally, notice that is well defined since is compact being homeomorphic to . Also, observe that there are no zeros of w in . Thus, by the excision property of the degree, we obtain
This shows that for small values of , . The assertion of the lemma now follows by applying the homotopy invariance of the fixed point index to on . □
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 ()
LetKbe a compact subset of a locally compact metric spaceY. Assume that any compact subset ofYcontainingKhas nonempty boundary. Then contains a connected set whose closure is noncompact and intersectsK.
Before presenting Lemma 3.10, we introduce the sets
and we recall that denotes the set of zeros of the tangent vector field w.
Lemma 3.10LetYbe a locally compact open subset of . Assume that is compact and that , where , is an isolating neighborhood of . Then the pair 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 . Now, let G be an open subset of D such that
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 and . Therefore, denoted by the slice
we have that is a compact subset of and is contained in the open slice of W at . Let be an open subset of such that and . Since C is compact and because of the local compactness of P, we may suppose that is relatively compact. Consequently, there exists such that
2. for and (here, as usual, denotes the slice ).
Notice that is relatively compact. This follows easily from the above condition 1 and the relative compactness of .
We can now apply Lemma 3.8 and the excision properties of the fixed point index and of the degree obtaining, for any ,
where . Observe that V is an isolating neighborhood of . Thus, by formula (2.1), by the above equalities (3.5) and the assumption , we get
Since C is compact, by the generalized homotopy invariance property of the fixed point index, we get that does not depend on . Hence,
On the other hand, because of the compactness of C, for some positive the slice is empty. Thus,
and we have a contradiction. Therefore, verifies the assumptions of Lemma 3.9 and the proof is complete. □
Proof of Theorem 3.3 Let be the isometry given by , where ψ is the restriction of x to the interval . As previously, let 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 such that . Observe that S is actually contained in . Taking into account Lemma 3.6, X and S correspond under ρ. Analogously to the definition of , let us denote
In addition, consider
Theorem 3.2 implies that is a closed subset of X. Therefore, it is locally compact since so is X according to Lemma 3.1. Now, consider
Observe that is locally compact, being open in . Then
is locally compact and open in . Denote by and K the subsets of and Y defined as
Now, observe that is an isolating neighborhood of
Since , we can apply Lemma 3.10 concluding that verifies the assumptions of Lemma 3.9. Therefore, also verifies the same assumptions since the pairs and correspond under the isometry ρ. Therefore, Lemma 3.9 implies that contains a connected set Γ whose closure (in ) is noncompact and intersects . Now, observe that according to Theorem 3.2, is closed in Ω. Thus, the closures of Γ in 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 .
Corollary 3.11 (Rabinowitz-type global bifurcation result)
LetMandfbe as in Theorem 3.3. Assume thatMis closed in and thatfsends bounded subsets of into bounded subsets of . LetVbe an open subset ofMsuch that , wherewis the mean value tangent vector field defined in formula (3.2). Then equation (3.1) has a connected subset of nontrivialT-periodic pairs whose closure contains some , with , and is either unbounded or goes back to some , where .
Proof Let Ω be the open set obtained by removing from the closed set . In other words,
Observe that is complete due to the closedness of M. Consider, by Theorem 3.3, a connected set of nontrivial T-periodic pairs with noncompact closure (in Ω) and intersecting in a subset of . Suppose that Γ is bounded. From Remark 3.4 it follows that , where denotes the closure of Γ in Ω, is nonempty and hence contains a point which does not belong to Ω, that is, such that . □
Remark 3.12 The assumption of Corollary 3.11 above on the existence of an open subset V of M such that is clearly satisfied in the case when w has an isolated zero with nonzero index. For example, if and w is with injective derivative , then p is an isolated zero of w and its index is either 1 or −1. In fact, in this case, sends 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., ).
The next consequence of Theorem 3.3 provides an existence result for T-periodic solutions already obtained in . Moreover, it improves an analogous result in , in which the map f is continuous on , with the compact-open topology in . 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 3.13LetMandfbe as in Theorem 3.3. Assume thatfsends bounded subsets of into bounded subsets of . In addition, suppose thatMis compact with Euler-Poincaré characteristic . Then equation (3.1) has a connected unbounded set of nontrivialT-periodic pairs whose closure meets . Therefore, since is bounded, equation (3.1) has aT-periodic solution for any .
Proof Choose . By the Poincaré-Hopf theorem, we have
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 [7,8] and extends an analogous one for ODEs on differentiable manifolds . In what follows, by a T-periodic orbit of , we mean the image of a T-periodic solution of this equation.
Corollary 3.14 (Mawhin-type continuation principle)
LetMandfbe as in Theorem 3.3 and letwbe the mean value tangent vector field defined in formula (3.2). Assume thatfsends bounded subsets of into bounded subsets of . LetVbe a relatively compact open subset ofMand assume that
1. along the boundary∂VofV;
3. for any , theT-periodic orbits of lying in do not meet∂V.
Then the equation
has aT-periodic orbit in V.
Proof Define . Observe that . Therefore,
According to Theorem 3.3, call Γ a connected subset of Ω of nontrivial T-periodic pairs of the equation , whose closure in Ω is noncompact and intersects in a subset of .
As V has compact closure in M, then the closure of Ω in is complete, being
Since f sends bounded subsets of into bounded subsets of , recalling Remark 3.4, one has that the closure of Γ in the whole space (which coincides with the closure in ) must intersect ∂Ω.
Now, because of the above condition 3, cannot contain elements of . In addition, condition 1 and Theorem 3.2 imply that does not contain elements of . Therefore, the nonempty set is composed of pairs of the form , where x is a T-periodic solution of whose image is contained in V. □
The authors declare that they have no competing interests.
All authors contributed to each part of the work equally. All authors read and approved the final version of the manuscript.
Dedicated to our friend and outstanding mathematician Jean Mawhin.
Pierluigi Benevieri is partially sponsored by Fapesp, Grant n. 2010/20727-4.
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