be a boundaryless smooth m
-dimensional manifold in
denote the metric subspace of
of the M
-valued continuous functions on
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.
be a functional field over M
, assume that f
-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 3.1 The set X is 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
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 3.2 Let be such that is an accumulation point of nontrivial T-periodic pairs of (3.1). Then there exists such that , for any , and . Thus, any bifurcation point of (3.1) is a zero of w.
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).
, recalling 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 3.3 Let be a boundaryless smooth manifold
, be a functional field on M
-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 nontrivial T-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 3.5 There exist an open neighborhood U of in and a continuous map from U to
, with the following properties
is an extension of ψ;
is T-periodic on ;
Let now U
be an open subset of
as in the previous lemma and let f
be as in Theorem 3.3. Given
, 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 .
, 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 3.6 The fixed points of correspond to the T-periodic solutions of equation (3.1) in the following sense: ψ is a fixed point of if and only if it is the restriction to 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 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
is defined at least up to T
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.
be an open subset of
is compact and let
be such that
is contained in the domain D of P;
is relatively compact;
be an open subset of
as in Lemma 3.5. Given
, consider the initial value problem
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
, 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
is open in
because of the compactness of
. Moreover, observe that the slice
coincides with U
is contained in the domain D
of the operator P
defined above. Define
coincides with P
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 .
Then, for small values of
is defined and
To see this, let
be as in Step 4 and, given
. Clearly, k
is a locally compact map since it takes values in the locally compact space M
is actually compact since
is contained in
which is relatively compact by (b′) of Step 1. Now, observe that the composition
and that the set of fixed points of
is compact by (b′) of Step 1 and is contained in
by Step 4. Thus, the set of fixed points of
is compact so that, by applying the commutativity property of the fixed point index to the maps k
, we get
Consequently, since it is easy to verify that the composition
, 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
. By contradiction, suppose there exists a sequence
. Hence, there exists a sequence
. Because of (b′) of Step 1, we may assume that
so that, in particular,
. Now, by an argument similar to that used in the proof of Theorem 3.2, we get that
is constant and
. Moreover, since
, we also obtain that
, 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
and such that if
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
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 [1
Finally, notice that
is well defined since
is compact being homeomorphic to
. Also, observe that there are no zeros of w
. 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 ()
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 contains a connected set whose closure is noncompact and intersects K.
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 3.10 Let Y be 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.
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
To prove the assertion, suppose by contradiction that there exists a compact open neighborhood C
. Consequently, we can find an open subset W
. Therefore, denoted by
we have that
is a compact subset of
and is contained in the open slice
be an open subset of
. Since C
is compact and because of the local compactness of P
, we may suppose that
is relatively compact. Consequently, there exists
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
. Observe that V
is an isolating neighborhood of
. Thus, by formula (2.1), by the above equalities (3.5) and the assumption
, we get
is compact, by the generalized homotopy invariance property of the fixed point index, we get that
does not depend on
On the other hand, because of the compactness of C
, for some positive
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
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
. Observe that S
is actually contained in
. Taking into account Lemma 3.6, X
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
is locally compact, being open in
is locally compact and open in
. Denote by
the subsets of
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)
Let M and f be as in Theorem 3.3. Assume that M is closed in and that f sends bounded subsets of into bounded subsets of . Let V be an open subset of M such that , 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 , with , and is either unbounded or goes back to some , where .
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 3.13 Let M and f be as in Theorem 3.3. Assume that f sends bounded subsets of into bounded subsets of . In addition, suppose that M is compact with Euler-Poincaré characteristic . Then equation (3.1) has a connected unbounded set of nontrivial T-periodic pairs whose closure meets . Therefore, since is bounded, equation (3.1) has a T-periodic solution for any .
. 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)
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 into bounded subsets of
. Let V be a relatively compact open subset of M and assume that
along the boundary ∂V of V;
Then the equation
has a T-periodic orbit in V.
. Observe that
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 .
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. □