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
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 mdimensional manifold
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
To proceed with the exposition of our problem, we need some further notation. Given
The main outcome of this paper, Theorem 3.3 below, is a global continuation result for Tperiodic solutions of equation (1.1). That is, given an open subset Ω of
and the fixed point index of a sort of Poincaré Ttranslation 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, [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
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
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
Let
We summarize below the main properties of the fixed point index.
• (Existence) If
• (Normalization) If
• (Additivity) Given two disjoint open subsets
• (Excision) Given an open subset
• (Commutativity) Let
• (Generalized homotopy invariance) LetIbe a compact real interval andWbe an open subset of
2.2 Degree of a vector field
Let us recall some basic notions on degree theory for tangent vector fields on differentiable
manifolds. Let
As a consequence of the PoincaréHopf theorem, when M is compact,
In the particular case when U is an open subset of
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
It is known that if
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
2.3 Retarded functional differential equations
Given an arbitrary subset A of
Notice that
Throughout the paper, the norm in
Let M be a boundaryless smooth manifold in
is said to be a retarded functional tangent vector field overM if
Let us consider a retarded functional differential equation (RFDE) of the type
where
A solution of (2.2) is a function
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
A solution of (2.3) is a solution
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].
Theorem 2.1LetMbe a boundaryless smooth manifold and
is open and the map
More generally, we will need to consider initial value problems depending on a parameter
such as equation (1.1) with the initial condition
Corollary 2.2 (Continuous dependence)
LetMbe a boundaryless smooth manifold and
Then, given
is open and the map
Proof
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
for all
We close this section with the following lemma whose elementary proof is given for the sake of completeness.
Lemma 2.4Let
Proof Notice that if K is a compact subset of
3 Branches of periodic solutions
Let M be a boundaryless smooth mdimensional manifold in
denote the metric subspace of
Moreover, denote by
Let
As in the introduction, we call
Lemma 3.1The setXis closed in
Proof Let
Now, as observed above,
We recall that, given
In the sequel, we are interested in the existence of branches of nontrivial Tperiodic pairs that, roughly speaking, emanate from a trivial pair
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
Theorem 3.2Let
Proof By assumption there exists a sequence
Now, given
Observe that the sequence of curves
Let now Ω be an open subset of
Theorem 3.3Let
Let Ω be an open 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
Then a connected subset Γ of Ω as in Theorem 3.3 is either unbounded or, if bounded, its closure
To see this, assume that
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
Lemma 3.5There exist an open neighborhoodUof
1.
2.
3.
Let now U be an open subset of
where
Let
The set D is nonempty since it contains
Given
by
Observe that
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
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
Let us prove that P is locally compact. Take
Observe that K is compact, being the image of
Therefore, by Ascoli’s theorem and taking into account the local completeness of
The following result establishes the relationship between the fixed point index of
the Poincarétype operator
Lemma 3.8Let
(a)
(b)
(c)
Consider the open set
Proof Let U be an open subset of
where
and
Corollary 2.2 implies that E is open in
Clearly,
As in Lemmas 3.6 and 3.7, one can show that the fixed points of
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
(a′)
(b′)
To prove Step 1, observe that
Step 2. For small values of
By contradiction, suppose there exists a sequence
Step 3. For small values of
The proof is analogous to that of Step 2, noting that
Step 4. Let
By contradiction, suppose there exists a sequence
Step 5. Let
Then, for small values ofλ,
To see this, let
Consequently, since it is easy to verify that the composition
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,
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
On the other hand, by Step 5, if
Moreover, as shown in [1],
Finally, notice that
This shows that for small values of
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])
LetKbe a compact subset of a locally compact metric spaceY. Assume that any compact subset ofYcontainingKhas nonempty boundary. Then
Before presenting Lemma 3.10, we introduce the sets
and we recall that
Lemma 3.10LetYbe a locally compact open subset of
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
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
we have that
1.
2.
Notice that
We can now apply Lemma 3.8 and the excision properties of the fixed point index and
of the degree obtaining, for any
where
Since C is compact, by the generalized homotopy invariance property of the fixed point index,
we get that
On the other hand, because of the compactness of C, for some positive
and we have a contradiction. Therefore,
Proof of Theorem 3.3 Let
In addition, consider
Theorem 3.2 implies that
Observe that
is locally compact and open in
Now, observe that
Since
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)
LetMandfbe as in Theorem 3.3. Assume thatMis closed in
Proof Let Ω be the open set obtained by removing from
Observe that
Remark 3.12 The assumption of Corollary 3.11 above on the existence of an open subset V of M such that
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
Corollary 3.13LetMandfbe as in Theorem 3.3. Assume thatfsends bounded subsets of
Proof Choose
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
Corollary 3.14 (Mawhintype 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
1.
2.
3. for any
Then the equation
has aTperiodic orbit in V.
Proof Define
According to Theorem 3.3, call Γ a connected subset of Ω of nontrivial Tperiodic pairs of the equation
As V has compact closure in M, then the closure of Ω in
Since f sends bounded subsets of
Now, because of the above condition 3,
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed to each part of the work equally. All authors read and approved the final version of the manuscript.
Acknowledgements
Dedicated to our friend and outstanding mathematician Jean Mawhin.
Pierluigi Benevieri is partially sponsored by Fapesp, Grant n. 2010/207274.
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