Open Access

Global continuation of periodic solutions for retarded functional differential equations on manifolds

  • Pierluigi Benevieri1, 2,
  • Alessandro Calamai3,
  • Massimo Furi1 and
  • Maria Patrizia Pera1Email author
Boundary Value Problems20132013:21

DOI: 10.1186/1687-2770-2013-21

Received: 19 October 2012

Accepted: 18 January 2013

Published: 11 February 2013

Abstract

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

1 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 λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq1_HTML.gif:
x ( t ) = λ f ( t , x t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equ1_HTML.gif
(1.1)
Let us present the setting of the problem. Consider a boundaryless smooth m-dimensional manifold M R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq2_HTML.gif and, given any p M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq3_HTML.gif, let T p M R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq4_HTML.gif stand for the tangent space of M at p. Denote by M ˜ : = B U ( ( , 0 ] , M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq5_HTML.gif the set of bounded and uniformly continuous maps from ( , 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq6_HTML.gif into M, and observe that this is a metric space as a subset of the Banach space R ˜ k : = B U ( ( , 0 ] , R k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq7_HTML.gif with the usual supremum norm. Given T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq8_HTML.gif, let f : R × M ˜ R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq9_HTML.gif be a continuous function verifying the following conditions:
  1. 1.

    f ( t , φ ) = f ( t + T , φ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq10_HTML.gif, ( t , φ ) R × M ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq11_HTML.gif;

     
  2. 2.

    f ( t , φ ) T φ ( 0 ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq12_HTML.gif, ( t , φ ) R × M ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq13_HTML.gif;

     
  3. 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 inf J = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq14_HTML.gif, bounded and uniformly continuous on any closed half-line ( , b ] J https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq15_HTML.gif such that the equality x ( t ) = λ f ( t , x t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq16_HTML.gif is eventually verified. We use here the standard notation in functional equations: whenever it makes sense, x t M ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq17_HTML.gif denotes the function θ x ( t + θ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq18_HTML.gif.

To proceed with the exposition of our problem, we need some further notation. Given p M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq3_HTML.gif, p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq19_HTML.gif denotes the constant p-valued function defined on or on any convenient subinterval of . The actual domain of p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq19_HTML.gif will be clear from the context. Moreover, given any A M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq20_HTML.gif, A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq21_HTML.gif stands for the set { p : p A } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq22_HTML.gif. All the functions of A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq21_HTML.gif will be considered defined on the same interval, suggested by the context. By C T ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq23_HTML.gif we mean the set of all continuous T-periodic maps x : R M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq24_HTML.gif. This set, which contains M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq25_HTML.gif, is a metric subspace of the Banach space C T ( R k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq26_HTML.gif with the standard supremum norm. We call ( λ , x ) [ 0 , + ) × C T ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq27_HTML.gif a T-periodic pair of equation (1.1) if x : R M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq24_HTML.gif 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 } × M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq28_HTML.gif, which can be isometrically identified with M. Notice that any T-periodic pair of the type ( 0 , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq29_HTML.gif is trivial since the function x turns out to be necessarily constant. An element p M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq3_HTML.gif will be called a bifurcation point of (1.1) if any neighborhood of ( 0 , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq30_HTML.gif in [ 0 , + ) × C T ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq31_HTML.gif contains nontrivial T-periodic pairs. Roughly speaking, p M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq3_HTML.gif is a bifurcation point if any of its neighborhoods in M contains T-periodic orbits corresponding to arbitrarily small values of λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq32_HTML.gif.

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 [ 0 , + ) × C T ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq33_HTML.gif, 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 nontrivial T-periodic pairs whose closure in Ω is noncompact and intersects the set of trivial T-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
w ( p ) = 1 T 0 T f ( t , p ) d t , p M , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equa_HTML.gif

and the fixed point index of a sort of Poincaré T-translation operator acting inside the Banach space C ( [ T , 0 ] , R k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq34_HTML.gif.

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 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 [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 R × C ( ( , 0 ] , M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq35_HTML.gif 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 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 C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq36_HTML.gif 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 [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 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 [9]. We refer also to the works of Gaines and Mawhin [10], Nussbaum [11, 12] and Mallet-Paret, 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 X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq37_HTML.gif is an absolute neighborhood retract (ANR) if, whenever it is homeomorphically embedded as a closed subset C of a metric space Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq38_HTML.gif, there exist an open neighborhood V of C in Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq38_HTML.gif and a retraction r : V C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq39_HTML.gif (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 X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq37_HTML.gif be a metric ANR and consider a locally compact (continuous) X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq37_HTML.gif-valued map k defined on a subset D ( k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq40_HTML.gif of X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq37_HTML.gif. Given an open subset U of X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq37_HTML.gif contained in D ( k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq40_HTML.gif, if the set of fixed points of k in U is compact, the pair ( k , U ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq41_HTML.gif is called admissible. We point out that such a condition is clearly satisfied if U ¯ D ( k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq42_HTML.gif, k ( U ) ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq43_HTML.gif is compact and k ( p ) p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq44_HTML.gif for all p in the boundary of U. To any admissible pair ( k , U ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq41_HTML.gif, one can associate an integer ind X ( k , U ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq45_HTML.gif - the fixed point index of k in U - which satisfies properties analogous to those of the classical Leray-Schauder degree [21]. The reader can see, for instance, [12, 2224] 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 X ( k , U ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq46_HTML.gif, then k admits at least one fixed point in U.

  • (Normalization) If X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq37_HTML.gifis compact, then ind X ( k , X ) = Λ ( k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq47_HTML.gif, where Λ ( k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq48_HTML.gifdenotes the Lefschetz number of k.

  • (Additivity) Given two disjoint open subsets U 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq49_HTML.gif, U 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq50_HTML.gifof U, if any fixed point of k in U is contained in U 1 U 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq51_HTML.gif, then ind X ( k , U ) = ind X ( k , U 1 ) + ind X ( k , U 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq52_HTML.gif.

  • (Excision) Given an open subset U 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq49_HTML.gifof U, if k has no fixed points in U U 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq53_HTML.gif, then ind X ( k , U ) = ind X ( k , U 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq54_HTML.gif.

  • (Commutativity) Let X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq37_HTML.gifand Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq38_HTML.gifbe metric ANRs. Suppose that U and V are open subsets of X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq37_HTML.gifand Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq38_HTML.gifrespectively and that k : U Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq55_HTML.gifand h : V X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq56_HTML.gifare locally compact maps. Assume that the set of fixed points of either hk in k 1 ( V ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq57_HTML.gifor kh in h 1 ( U ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq58_HTML.gifis compact. Then the other set is compact as well and ind X ( h k , k 1 ( V ) ) = ind Y ( k h , h 1 ( U ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq59_HTML.gif.

  • (Generalized homotopy invariance) Let I be a compact real interval and W be an open subset of X × I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq60_HTML.gif. For any λ I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq61_HTML.gif, denote W λ = { x X : ( x , λ ) W } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq62_HTML.gif. Let H : W X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq63_HTML.gifbe a locally compact map such that the set { ( x , λ ) W : H ( x , λ ) = x } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq64_HTML.gifis compact. Then ind X ( H ( , λ ) , W λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq65_HTML.gifis 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 R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq66_HTML.gif 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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq67_HTML.gif is admissible (or, equivalently, that v is admissible in U) if v 1 ( 0 ) U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq68_HTML.gif is compact. In this case, one can assign to the pair ( v , U ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq67_HTML.gif an integer, deg ( v , U ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq69_HTML.gif, 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., [2730]). Notice that the condition for v 1 ( 0 ) U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq68_HTML.gif to be compact is clearly satisfied if U is a relatively compact open subset of M and v ( p ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq70_HTML.gif for all p in the boundary of U.

As a consequence of the Poincaré-Hopf theorem, when M is compact, deg ( v , M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq71_HTML.gif equals χ ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq72_HTML.gif, the Euler-Poincaré characteristic of M.

In the particular case when U is an open subset of R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq73_HTML.gif, deg ( v , U ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq74_HTML.gif 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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq75_HTML.gif of v in U with target value 0 R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq76_HTML.gif. 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 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 deg ( v , U ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq69_HTML.gif 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 R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq73_HTML.gif: it is masked by the fact that an equation of the type f ( x ) = y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq77_HTML.gif can be written as f ( x ) y = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq78_HTML.gif. 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 ( v , U ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq67_HTML.gif is admissible, then
deg ( v , U ) = ( 1 ) m deg ( v , U ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equ2_HTML.gif
(2.1)

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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq69_HTML.gif 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 R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq73_HTML.gif, we denote by B U ( ( , 0 ] , A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq79_HTML.gif the set of bounded and uniformly continuous maps from ( , 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq80_HTML.gif into A. For brevity, we will use the notation
A ˜ : = B U ( ( , 0 ] , A ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equb_HTML.gif

Notice that R ˜ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq81_HTML.gif is a Banach space, being closed in the space B C ( ( , 0 ] , R k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq82_HTML.gif of the bounded and continuous functions from ( , 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq80_HTML.gif into R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq73_HTML.gif (endowed with the standard supremum norm).

Throughout the paper, the norm in R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq73_HTML.gif will be denoted by | | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq83_HTML.gif and the norm in the infinite dimensional space R ˜ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq81_HTML.gif by https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq84_HTML.gif. Thus, the distance between two elements ϕ and ψ of A ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq85_HTML.gif will be denoted ϕ ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq86_HTML.gif, even when ϕ ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq87_HTML.gif does not belong to A ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq85_HTML.gif. We observe that A ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq85_HTML.gif, as a metric space, is complete if and only if A is closed in R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq73_HTML.gif.

Let M be a boundaryless smooth manifold in R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq73_HTML.gif. A continuous map
g : R × M ˜ R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equc_HTML.gif

is said to be a retarded functional tangent vector field over M if g ( t , φ ) T φ ( 0 ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq88_HTML.gif for all ( t , φ ) R × M ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq89_HTML.gif. In the sequel, any map with this property will be briefly called a functional field (over M).

Let us consider a retarded functional differential equation (RFDE) of the type
x ( t ) = g ( t , x t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equ3_HTML.gif
(2.2)

where g : R × M ˜ R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq90_HTML.gif is a functional field over M. Here, as usual and whenever it makes sense, given t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq91_HTML.gif, by x t M ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq92_HTML.gif we mean the function θ x ( t + θ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq18_HTML.gif.

A solution of (2.2) is a function x : J M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq93_HTML.gif, defined on an open real interval J with inf J = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq94_HTML.gif, bounded and uniformly continuous on any closed half-line ( , b ] J https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq95_HTML.gif, which verifies eventually the equality x ( t ) = g ( t , x t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq96_HTML.gif. That is, x : J M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq93_HTML.gif is a solution of (2.2) if x t M ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq97_HTML.gif for all t J https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq98_HTML.gif and there exists τ J https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq99_HTML.gif such that x is C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq36_HTML.gif on the interval ( τ , sup J ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq100_HTML.gif and x ( t ) = g ( t , x t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq96_HTML.gif for all t ( τ , sup J ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq101_HTML.gif. Observe that the derivative of a solution x may not exist at t = τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq102_HTML.gif. However, the right derivative D + x ( τ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq103_HTML.gif of x at τ always exists and is equal to g ( τ , x τ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq104_HTML.gif. Also, notice that t x t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq105_HTML.gif is a continuous curve in M ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq106_HTML.gif since x is uniformly continuous on any closed half-line ( , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq107_HTML.gif 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 η M ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq108_HTML.gif, let us associate to equation (2.2) the initial value problem
{ x ( t ) = g ( t , x t ) , x 0 = η . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equ4_HTML.gif
(2.3)

A solution of (2.3) is a solution x : J M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq93_HTML.gif of (2.2) such that sup J > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq109_HTML.gif, x ( t ) = g ( t , x t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq96_HTML.gif for t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq110_HTML.gif and x 0 = η https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq111_HTML.gif.

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 2.1 Let M be a boundaryless smooth manifold and g : R × M ˜ R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq112_HTML.gif be a functional field. Assume, for any η M ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq113_HTML.gif, the uniqueness of the maximal solution of problem (2.3). Then, given T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq114_HTML.gif, the set
D = { η M ˜ : the maximal solution of (2.3) is defined up to T } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equd_HTML.gif

is open and the map η D x T η M ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq115_HTML.gif, where x η ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq116_HTML.gif 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 = η https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq111_HTML.gif. For these problems the continuous dependence is ensured by the following consequence of Theorem 2.1.

Corollary 2.2 (Continuous dependence)

Let M be a boundaryless smooth manifold and h : R s × R × M ˜ R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq117_HTML.gif a parametrized functional field. For any α R s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq118_HTML.gif and η M ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq108_HTML.gif, assume the uniqueness of the maximal solution of the problem
{ x ( t ) = h ( α , t , x t ) , x 0 = η . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equ5_HTML.gif
(2.4)
Then, given T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq114_HTML.gif, the set
D = { ( α , η ) R s × M ˜ : the maximal solution of (2.4) is defined up to T } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Eque_HTML.gif

is open and the map ( α , η ) D x T ( α , η ) M ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq119_HTML.gif, where x ( α , η ) ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq120_HTML.gif is the unique maximal solution of problem (2.4), is continuous.

Proof

Apply Theorem 2.1 to the problem
{ ( β ( t ) , x ( t ) ) = ( 0 , h ( β ( t ) , t , x t ) ) , ( β ( 0 ) , x 0 ) = ( α , η ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equf_HTML.gif

that can be regarded as an initial value problem of a RFDE on the ambient manifold R s × M R s + k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq121_HTML.gif. □

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 g : R × M ˜ R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq90_HTML.gif is said to be compactly Lipschitz (for short, c-Lipschitz) if, given any compact subset Q of R × M ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq122_HTML.gif, there exists L 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq123_HTML.gif such that
| g ( t , φ ) g ( t , ψ ) | L φ ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equg_HTML.gif

for all ( t , φ ) , ( t , ψ ) Q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq124_HTML.gif. Moreover, we will say that g is locally c-Lipschitz if for any ( τ , η ) R × M ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq125_HTML.gif there exists an open neighborhood of ( τ , η ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq126_HTML.gif 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 [4] we proved that if g is a c-Lipschitz functional field, then problem (2.3) has a unique maximal solution for any η M ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq113_HTML.gif. For a characterization of compact subsets of M ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq106_HTML.gif 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 : X Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq127_HTML.gif be a continuous map between metric spaces and let { γ n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq128_HTML.gif be a sequence of continuous functions from a compact interval [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq129_HTML.gif (or, more generally, from a compact space) into X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq37_HTML.gif. If { γ n ( s ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq130_HTML.gif converges to γ ( s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq131_HTML.gif uniformly for s [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq132_HTML.gif, then also F ( γ n ( s ) ) F ( γ ( s ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq133_HTML.gif uniformly for s [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq132_HTML.gif.

Proof Notice that if K is a compact subset of X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq37_HTML.gif, then for any ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq134_HTML.gif there exists δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq135_HTML.gif such that x X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq136_HTML.gif, k K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq137_HTML.gif, dist X ( x , k ) < δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq138_HTML.gif imply dist Y ( F ( x ) , F ( k ) ) < ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq139_HTML.gif. Now, our assertion follows immediately by taking the compact K to be the image of the limit function γ : [ a , b ] X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq140_HTML.gif. □

3 Branches of periodic solutions

Let M be a boundaryless smooth m-dimensional manifold in R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq73_HTML.gif. Given T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq114_HTML.gif, let
M ˆ : = C ( [ T , 0 ] , M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equh_HTML.gif
denote the metric subspace of C ( [ T , 0 ] , R k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq34_HTML.gif of the M-valued continuous functions on [ T , 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq141_HTML.gif and set
M ˆ : = { ψ M ˆ : ψ ( T ) = ψ ( 0 ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equi_HTML.gif

Moreover, denote by C T ( R k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq26_HTML.gif the Banach space of the continuous T-periodic maps x : R R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq142_HTML.gif (with the standard supremum norm) and by C T ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq23_HTML.gif the metric subspace of C T ( R k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq26_HTML.gif of the M-valued maps. Observe that, since M is locally compact, then M ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq143_HTML.gif and C T ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq23_HTML.gif (but not M ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq106_HTML.gif) are locally complete. Moreover, they are complete if and only if M is closed.

Let f : R × M ˜ R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq144_HTML.gif be a functional field over M. Given T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq114_HTML.gif, assume that f is T-periodic in the first variable. Consider the following RFDE depending on a parameter λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq1_HTML.gif:
x ( t ) = λ f ( t , x t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equ6_HTML.gif
(3.1)

As in the introduction, we call ( λ , x ) [ 0 , + ) × C T ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq27_HTML.gif a T-periodic pair (of (3.1)) if the function x : R M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq24_HTML.gif is a (T-periodic) solution of (3.1) corresponding to λ. Let us denote by X the set of all T-periodic pairs of (3.1). Lemma 3.1 below states some properties of X that will be used in the sequel.

Lemma 3.1 The set X is closed in [ 0 , + ) × C T ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq145_HTML.gif and locally compact.

Proof Let { ( λ n , x n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq146_HTML.gif be a sequence of T-periodic pairs of (3.1) converging to ( λ 0 , x 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq147_HTML.gif in [ 0 , + ) × C T ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq145_HTML.gif. Because of Lemma 2.4, f ( t , x t n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq148_HTML.gif converges uniformly to f ( t , x t 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq149_HTML.gif for t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq150_HTML.gif. Thus, ( x n ) ( t ) = λ n f ( t , x t n ) λ 0 f ( t , x t 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq151_HTML.gif uniformly and, therefore, ( x 0 ) ( t ) = λ 0 f ( t , x t 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq152_HTML.gif, that is, ( λ 0 , x 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq153_HTML.gif belongs to X. This proves that X is closed in [ 0 , + ) × C T ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq145_HTML.gif.

Now, as observed above, C T ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq23_HTML.gif 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 M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq3_HTML.gif, with the notation p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq19_HTML.gif we mean the constant p-valued function defined on some real interval that will be clear from the context. Moreover, a T-periodic pair of the type ( 0 , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq154_HTML.gif is said to be trivial, and an element p M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq3_HTML.gif is a bifurcation point of equation (3.1) if any neighborhood of ( 0 , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq154_HTML.gif in [ 0 , + ) × C T ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq155_HTML.gif contains a nontrivial T-periodic pair (i.e., a T-periodic pair ( λ , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq156_HTML.gif with λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq157_HTML.gif). In some sense, p is a bifurcation point if, for λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq157_HTML.gif sufficiently small, there are T-periodic orbits of (3.1) arbitrarily close to p.

In the sequel, we are interested in the existence of branches of nontrivial T-periodic pairs that, roughly speaking, emanate from a trivial pair ( 0 , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq158_HTML.gif, with p a bifurcation point of (3.1). To this end, we introduce the mean value tangent vector field w : M R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq159_HTML.gif given by
w ( p ) = 1 T 0 T f ( t , p ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equ7_HTML.gif
(3.2)

Throughout the paper, w will play a crucial role in obtaining our continuation results for (3.1). First, in Theorem 3.2 below, we provide a necessary condition for p M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq3_HTML.gif to be a bifurcation point.

Theorem 3.2 Let x C T ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq160_HTML.gif be such that ( 0 , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq29_HTML.gif is an accumulation point of nontrivial T-periodic pairs of (3.1). Then there exists p M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq161_HTML.gif such that x ( t ) = p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq162_HTML.gif, for any t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq150_HTML.gif, and w ( p ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq163_HTML.gif. Thus, any bifurcation point of (3.1) is a zero of w.

Proof By assumption there exists a sequence { ( λ n , x n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq164_HTML.gif of T-periodic pairs of (3.1) such that λ n > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq165_HTML.gif, λ n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq166_HTML.gif, and x n ( t ) x ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq167_HTML.gif uniformly on . As proved in Lemma 3.1, the set X of the T-periodic pairs is closed in [ 0 , + ) × C T ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq145_HTML.gif. Thus, the pair ( 0 , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq29_HTML.gif belongs to X and, consequently, the function x must be constant, say x = p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq168_HTML.gif for some p M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq169_HTML.gif. Clearly, the point p is a bifurcation point of (3.1).

Now, given n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq170_HTML.gif, recalling that x n ( T ) = x n ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq171_HTML.gif and that λ n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq172_HTML.gif, we get
0 T f ( t , x t n ) d t = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equj_HTML.gif

Observe that the sequence of curves t ( t , x t n ) R × M ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq173_HTML.gif converges uniformly to t ( t , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq174_HTML.gif for t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq175_HTML.gif. Hence, because of Lemma 2.4, f ( t , x t n ) f ( t , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq176_HTML.gif uniformly for t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq175_HTML.gif and the assertion follows passing to the limit in the above integral. □

Let now Ω be an open subset of [ 0 , + ) × C T ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq155_HTML.gif. Our main result (Theorem 3.3 below) provides a sufficient condition for the existence of a bifurcation point p in M with ( 0 , p ) Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq177_HTML.gif. More precisely, we give conditions which ensure the existence of a connected subset of Ω of nontrivial T-periodic pairs of equation (3.1) (a global bifurcating branch for short), whose closure in Ω is noncompact and intersects the set of trivial T-periodic pairs contained in Ω.

Theorem 3.3 Let M R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq178_HTML.gif be a boundaryless smooth manifold, f : R × M ˜ R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq144_HTML.gif be a functional field on M, T-periodic in the first variable and locally Lipschitz in the second one, and w : M R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq179_HTML.gif be the autonomous tangent vector field
w ( p ) = 1 T 0 T f ( t , p ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equk_HTML.gif

Let Ω be an open subset of [ 0 , + ) × C T ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq145_HTML.gif and let j : M [ 0 , + ) × C T ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq180_HTML.gif be the map p ( 0 , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq181_HTML.gif. Assume that deg ( w , j 1 ( Ω ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq182_HTML.gif is defined and nonzero. Then there exists a connected subset of Ω of nontrivial T-periodic pairs of equation (3.1) whose closure in Ω is noncompact and intersects { 0 } × C T ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq183_HTML.gif in a (nonempty) subset of { ( 0 , p ) Ω : w ( p ) = 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq184_HTML.gif.

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 R × M ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq122_HTML.gif into bounded subsets of R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq73_HTML.gif, and that M is closed in R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq73_HTML.gif (or, more generally, that the closure Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq185_HTML.gif of Ω in [ 0 , + ) × C T ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq145_HTML.gif is complete).

Then a connected subset Γ of Ω as in Theorem 3.3 is either unbounded or, if bounded, its closure Γ ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq186_HTML.gif in Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq185_HTML.gif reaches the boundary Ω of Ω.

To see this, assume that Γ ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq186_HTML.gif is bounded. Then, being f ( Γ ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq187_HTML.gif bounded, because of Ascoli’s theorem, Γ is actually totally bounded. Thus, Γ ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq186_HTML.gif is compact, being totally bounded and, additionally, complete since Γ ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq188_HTML.gif is contained in Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq189_HTML.gif. On the other hand, according to Theorem 3.3, the closure Γ ¯ Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq190_HTML.gif of Γ in Ω is noncompact. Consequently, the set Γ ¯ Γ ¯ Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq191_HTML.gif is nonempty, and this means that Γ ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq186_HTML.gif reaches the boundary of Ω.

The proof of Theorem 3.3 requires some preliminary steps. In the first one, we define a parametrized Poincaré-type T-translation operator whose fixed points are the restrictions to the interval [ T , 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq141_HTML.gif of the T-periodic solutions of (3.1). For this purpose, we need to introduce a suitable backward extension of the elements of M ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq143_HTML.gif. The properties of such an extension are contained in Lemma 3.5 below, obtained in [33]. In what follows, by a T-periodic map on an interval J, we mean the restriction to J of a T-periodic map defined on .

Lemma 3.5 There exist an open neighborhood U of M ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq192_HTML.gif in M ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq143_HTML.gif and a continuous map from U to M ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq106_HTML.gif, ψ ψ ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq193_HTML.gif, with the following properties:
  1. 1.

    ψ ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq194_HTML.gif is an extension of ψ;

     
  2. 2.

    ψ ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq194_HTML.gif is T-periodic on ( , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq195_HTML.gif;

     
  3. 3.

    ψ ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq194_HTML.gif is T-periodic on ( , 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq80_HTML.gif, whenever ψ M ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq196_HTML.gif.

     
Let now U be an open subset of M ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq143_HTML.gif as in the previous lemma and let f be as in Theorem 3.3. Given λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq197_HTML.gif and ψ U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq198_HTML.gif, consider the initial value problem
{ x ( t ) = λ f ( t , x t ) , x 0 = ψ ˜ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equ8_HTML.gif
(3.3)

where ψ ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq194_HTML.gif is the extension of ψ as in Lemma 3.5.

Let
D = { ( λ , ψ ) [ 0 , + ) × U :  the maximal solution of (3.3) is defined up to  T } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equl_HTML.gif

The set D is nonempty since it contains { 0 } × U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq199_HTML.gif (notice that for λ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq200_HTML.gif, the solution of problem (3.3) is constant for t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq110_HTML.gif). Moreover, it follows by Corollary 2.2 that D is open in [ 0 , + ) × M ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq201_HTML.gif.

Given ( λ , ψ ) D https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq202_HTML.gif, denote by x ( λ , ψ ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq203_HTML.gif the maximal solution of problem (3.3) and define
P : D M ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equm_HTML.gif
by
P ( λ , ψ ) ( θ ) = x ( λ , ψ ˜ ) ( θ + T ) , θ [ T , 0 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equn_HTML.gif

Observe that P ( λ , ψ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq204_HTML.gif is the restriction of x T ( λ , ψ ˜ ) M ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq205_HTML.gif to the interval [ T , 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq141_HTML.gif.

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 ( λ , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq206_HTML.gif correspond to the T-periodic solutions of equation (3.1) in the following sense: ψ is a fixed point of P ( λ , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq206_HTML.gif if and only if it is the restriction to [ T , 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq141_HTML.gif 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 ψ ψ ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq193_HTML.gif of Lemma 3.5 and of the map that associates to any φ M ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq207_HTML.gif its restriction to the interval [ T , 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq141_HTML.gif.

Let us prove that P is locally compact. Take ( λ 0 , ψ 0 ) D https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq208_HTML.gif and denote, for simplicity, by x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq209_HTML.gif the maximal solution x ( λ 0 , ψ 0 ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq210_HTML.gif of (3.3) corresponding to ( λ 0 , ψ 0 ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq211_HTML.gif. Clearly, x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq209_HTML.gif is defined at least up to T and P ( λ 0 , ψ 0 ) ( θ ) = x 0 ( θ + T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq212_HTML.gif for any θ [ T , 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq213_HTML.gif. Set
K = { ( t , x t 0 ) R × M ˜ : t [ 0 , T ] } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equo_HTML.gif

Observe that K is compact, being the image of [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq214_HTML.gif under the (continuous) curve t ( t , x t 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq215_HTML.gif. Let O be an open neighborhood of K in R × M ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq122_HTML.gif and c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq216_HTML.gif such that | f ( t , φ ) | c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq217_HTML.gif for all ( t , φ ) O ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq218_HTML.gif. Let us show that there exists an open neighborhood W of ( λ 0 , ψ 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq219_HTML.gif in D such that if ( λ , ψ ) W https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq220_HTML.gif, then ( t , x t ( λ , ψ ˜ ) ) O https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq221_HTML.gif for t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq175_HTML.gif, where x ( λ , ψ ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq203_HTML.gif is the maximal solution of (3.3) corresponding to ( λ , ψ ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq222_HTML.gif. By contradiction, for any n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq223_HTML.gif suppose there exist ( λ n , ψ n ) D https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq224_HTML.gif and t n [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq225_HTML.gif such that ( λ n , ψ n ) ( λ 0 , ψ 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq226_HTML.gif and ( t n , x t n n ) O https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq227_HTML.gif, where x n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq228_HTML.gif denotes the maximal solution x ( λ n , ψ n ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq229_HTML.gif of (3.3) corresponding to ( λ n , ψ n ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq230_HTML.gif. We may assume t n τ [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq231_HTML.gif. Now, from the fact that in M ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq106_HTML.gif the convergence is uniform, we get the equicontinuity of the sequence { x T n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq232_HTML.gif. This easily implies that ( t n , x t n n ) ( τ , x τ 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq233_HTML.gif. A contradiction, since O is open and ( τ , x τ 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq234_HTML.gif belongs to K O https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq235_HTML.gif. Thus, the existence of the required W is proved. Consequently, for any ( λ , ψ ) W https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq236_HTML.gif, the maximal solution x ( λ , ψ ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq203_HTML.gif of (3.3) corresponding to ( λ , ψ ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq222_HTML.gif is such that | ( x ( λ , ψ ˜ ) ) ( t ) | = | λ f ( t , x t ( λ , ψ ˜ ) ) | | λ | c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq237_HTML.gif for all t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq175_HTML.gif.

Therefore, by Ascoli’s theorem and taking into account the local completeness of M ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq143_HTML.gif, we get that P maps W into a compact subset of M ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq143_HTML.gif. This proves that P is locally compact. □

The following result establishes the relationship between the fixed point index of the Poincaré-type operator P ( λ , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq206_HTML.gif and the degree of the mean value vector field w. It will be crucial in the proof of Lemma 3.10.

Lemma 3.8 Let V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq238_HTML.gif be an open subset of M ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq143_HTML.gif such that V { p M ˆ : w ( p ) = 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq239_HTML.gif is compact and let ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq240_HTML.gif be such that
  1. (a)

    [ 0 , ε ] × V ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq241_HTML.gif is contained in the domain D of P;

     
  2. (b)

    P ( [ 0 , ε ] × V ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq242_HTML.gif is relatively compact;

     
  3. (c)

    P ( λ , ψ ) ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq243_HTML.gif for 0 < λ ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq244_HTML.gif and ψ in the boundary V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq245_HTML.gif of V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq238_HTML.gif.

     
Consider the open set V = { p M : p V } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq246_HTML.gif. Then deg ( w , V ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq247_HTML.gif is well defined and
ind M ˆ ( P ( λ , ) , V ) = deg ( w , V ) , 0 < λ ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equp_HTML.gif
Proof Let U be an open subset of M ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq143_HTML.gif as in Lemma 3.5. Given λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq248_HTML.gif, μ [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq249_HTML.gif and ψ U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq198_HTML.gif, consider the initial value problem
{ x ( t ) = λ ( ( 1 μ ) f ( t , x t ) + μ w ( x ( t ) ) ) , x 0 = ψ ˜ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equ9_HTML.gif
(3.4)
where ψ ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq194_HTML.gif 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 λ [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq250_HTML.gif and μ [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq249_HTML.gif, the uniqueness of the solution of problem (3.4) is ensured (recall Remark 2.3). Denote by x ( λ , ψ ˜ , μ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq251_HTML.gif the maximal solution of problem (3.4), and put
E = { ( λ , ψ , μ ) [ 0 , + ) × U × [ 0 , 1 ] : x ( λ , ψ ˜ , μ )  is defined up to  T } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equq_HTML.gif
and
D = { ( λ , ψ ) [ 0 , + ) × U : ( λ , ψ , μ ) E  for all  μ [ 0 , 1 ] } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equr_HTML.gif
Corollary 2.2 implies that E is open in [ 0 , + ) × U × [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq252_HTML.gif. Therefore, D https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq253_HTML.gif is open in [ 0 , + ) × M ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq254_HTML.gif because of the compactness of [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq255_HTML.gif. Moreover, observe that the slice D 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq256_HTML.gif of D https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq253_HTML.gif at λ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq200_HTML.gif coincides with U and that D https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq253_HTML.gif is contained in the domain D of the operator P defined above. Define H : D × [ 0 , 1 ] M ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq257_HTML.gif by
H ( λ , ψ , μ ) ( θ ) = x ( λ , ψ ˜ , μ ) ( θ + T ) , θ [ T , 0 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equs_HTML.gif
Clearly, H ( , , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq258_HTML.gif coincides with P on D https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq253_HTML.gif, while H ( , , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq259_HTML.gif is the (infinite dimensional) operator associated to the undelayed problem
{ x ( t ) = λ w ( x ( t ) ) , x 0 = ψ ˜ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equt_HTML.gif
As in Lemmas 3.6 and 3.7, one can show that the fixed points of H ( λ , , μ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq260_HTML.gif correspond to the T-periodic solutions of the equation
x ( t ) = λ ( ( 1 μ ) f ( t , x t ) + μ w ( x ( t ) ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equu_HTML.gif

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
Z = { p M : w ( p ) = 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equv_HTML.gif
and, according to our notation,
Z = { p M ˆ : p Z } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equw_HTML.gif

Step 1. There exist σ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq261_HTML.gif and an open subset V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq262_HTML.gif of M ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq143_HTML.gif, containing V Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq263_HTML.gif, with V ¯ V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq264_HTML.gif, and such that

(a′) [ 0 , σ ] × V ¯ D https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq265_HTML.gif (i.e., for 0 λ σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq266_HTML.gif, H ( λ , , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq267_HTML.gif is defined in V ¯ × [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq268_HTML.gif);

(b′) H ( [ 0 , σ ] × V × [ 0 , 1 ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq269_HTML.gif is relatively compact.

To prove Step 1, observe that { 0 } × ( V Z ) × [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq270_HTML.gif is compact and contained in D × [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq271_HTML.gif, which is open in [ 0 , + ) × M ˆ × [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq272_HTML.gif, and recall that H is locally compact.

Step 2. For small values of λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq157_HTML.gif, H ( λ , ψ , μ ) ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq273_HTML.gif for any ψ V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq274_HTML.gif and μ [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq249_HTML.gif.

By contradiction, suppose there exists a sequence { ( λ n , ψ n , μ n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq275_HTML.gif in D × [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq271_HTML.gif such that λ n > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq276_HTML.gif, λ n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq166_HTML.gif, ψ n V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq277_HTML.gif and H ( λ n , ψ n , μ n ) = ψ n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq278_HTML.gif. Without loss of generality, taking into account (b′), we may assume that ψ n ψ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq279_HTML.gif and also that μ n μ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq280_HTML.gif. Denote by x n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq281_HTML.gif the T-periodic solution x ( λ n , ψ n ˜ , μ n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq282_HTML.gif of (3.4) corresponding to ( λ n , ψ n ˜ , μ n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq283_HTML.gif. Since ψ n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq284_HTML.gif is the restriction of x n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq228_HTML.gif to [ T , 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq141_HTML.gif, then { x n ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq285_HTML.gif converges uniformly on to x 0 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq286_HTML.gif, where x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq209_HTML.gif is the solution of (3.4) corresponding to the fixed point ψ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq287_HTML.gif of H ( 0 , , μ 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq288_HTML.gif. Therefore, there exists p M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq161_HTML.gif such that x 0 ( t ) = p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq289_HTML.gif for any t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq150_HTML.gif and, as in the proof of Theorem 3.2, we can show that w ( p ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq163_HTML.gif. Thus, ψ 0 = p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq290_HTML.gif belongs to V Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq291_HTML.gif, contradicting the choice of V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq262_HTML.gif. This proves Step 2.

Step 3. For small values of λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq157_HTML.gif, H ( λ , ψ , 0 ) ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq292_HTML.gif for any ψ V ¯ V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq293_HTML.gif.

The proof is analogous to that of Step 2, noting that H ( λ , ψ , 0 ) = P ( λ , ψ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq294_HTML.gif for ( λ , ψ ) D https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq295_HTML.gif and taking into account assumption b) and the fact that V ¯ V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq296_HTML.gif is closed in M ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq143_HTML.gif.

Step 4. Let k : V M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq297_HTML.gif be defined by k ( ψ ) = ψ ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq298_HTML.gif and consider the open set V = { p M : p V } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq299_HTML.gif. Then there exists σ ( 0 , σ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq300_HTML.gif such that H ( λ , ψ , 1 ) ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq301_HTML.gif for any ( λ , ψ ) ( 0 , σ ] × ( V ¯ k 1 ( V ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq302_HTML.gif.

By contradiction, suppose there exists a sequence { ( λ n , ψ n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq303_HTML.gif in D https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq253_HTML.gif such that λ n > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq276_HTML.gif, λ n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq166_HTML.gif, ψ n V ¯ k 1 ( V ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq304_HTML.gif and H ( λ n , ψ n , 1 ) = ψ n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq305_HTML.gif. Without loss of generality, taking into account (b′), we may assume that ψ n ψ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq279_HTML.gif. Therefore, by the continuity of H, we get H ( 0 , ψ 0 , 1 ) = ψ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq306_HTML.gif so that ψ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq287_HTML.gif is a constant function of V ¯ k 1 ( V ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq307_HTML.gif. This is impossible, since any constant function of V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq262_HTML.gif is contained in k 1 ( V ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq308_HTML.gif.

Step 5. Let V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq309_HTML.gif and σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq310_HTML.gif be as in Step 4 and let Q : [ 0 , σ ] × V ¯ M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq311_HTML.gif be the T-translation operator Q ( λ , p ) = x ( λ , p , 1 ) ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq312_HTML.gif, where x ( λ , p , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq313_HTML.gif is the maximal solution of the undelayed problem
{ x ( t ) = λ w ( x ( t ) ) , x 0 = p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equx_HTML.gif
Then, for small values of λ, ind M ( Q ( λ , ) , V ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq314_HTML.gif is defined and
ind M ˆ ( H ( λ , , 1 ) , V ) = ind M ( Q ( λ , ) , V ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equy_HTML.gif
To see this, let k : V M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq297_HTML.gif be as in Step 4 and, given λ ( 0 , σ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq315_HTML.gif, define h λ : V M ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq316_HTML.gif by h λ ( p ) ( θ ) = x ( λ , p , 1 ) ( θ + T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq317_HTML.gif, θ [ T , 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq318_HTML.gif. Clearly, k is a locally compact map since it takes values in the locally compact space M. Moreover, h λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq319_HTML.gif is actually compact since h λ ( V ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq320_HTML.gif is contained in H ( [ 0 , σ ] × V × [ 0 , 1 ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq269_HTML.gif which is relatively compact by (b′) of Step 1. Now, observe that the composition h λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq321_HTML.gif coincides with H ( λ , , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq322_HTML.gif in k 1 ( V ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq308_HTML.gif and that the set of fixed points of H ( λ , , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq322_HTML.gif in V ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq323_HTML.gif is compact by (b′) of Step 1 and is contained in k 1 ( V ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq308_HTML.gif by Step 4. Thus, the set of fixed points of h λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq321_HTML.gif in k 1 ( V ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq308_HTML.gif is compact so that, by applying the commutativity property of the fixed point index to the maps k and h λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq319_HTML.gif, we get
ind M ˆ ( h λ k , k 1 ( V ) ) = ind M ( k h λ , h λ 1 ( V ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equz_HTML.gif
Consequently, since it is easy to verify that the composition k h λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq324_HTML.gif coincides with Q ( λ , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq325_HTML.gif in h λ 1 ( V ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq326_HTML.gif, we obtain
ind M ˆ ( H ( λ , , 1 ) , k 1 ( V ) ) = ind M ( Q ( λ , ) , h λ 1 ( V ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equaa_HTML.gif
and, because of Step 4, by the excision property of the index,
ind M ˆ ( H ( λ , , 1 ) , V ) = ind M ˆ ( H ( λ , , 1 ) , k 1 ( V ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equab_HTML.gif
To complete the proof of Step 5, let us show that for λ sufficiently small, Q ( λ , p ) p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq327_HTML.gif for p V ¯ h λ 1 ( V ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq328_HTML.gif. By contradiction, suppose there exists a sequence { ( λ n , p n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq329_HTML.gif in [ 0 , σ ] × V ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq330_HTML.gif such that λ n > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq276_HTML.gif, λ n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq331_HTML.gif, p n V ¯ h λ n 1 ( V ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq332_HTML.gif and Q ( λ n , p n ) = p n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq333_HTML.gif. Hence, there exists a sequence { ψ n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq334_HTML.gif in V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq335_HTML.gif such that ψ n ( 0 ) = p n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq336_HTML.gif and H ( λ n , ψ n , 1 ) = ψ n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq337_HTML.gif. Because of (b′) of Step 1, we may assume that ψ n ψ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq279_HTML.gif so that, in particular, p n p 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq338_HTML.gif, where p 0 = ψ 0 ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq339_HTML.gif. Now, by an argument similar to that used in the proof of Theorem 3.2, we get that ψ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq340_HTML.gif is constant and w ( p 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq341_HTML.gif. Thus, p 0 Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq342_HTML.gif. Moreover, since λ > 0 ( V ¯ h λ 1 ( V ) ) = V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq343_HTML.gif, we also obtain that p 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq344_HTML.gif belongs to V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq345_HTML.gif, contradicting the choice of V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq309_HTML.gif. Finally, again by excision, we get
ind M ( Q ( λ , ) , h λ 1 ( V ) ) = ind M ( Q ( λ , ) , V ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equac_HTML.gif

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 ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq346_HTML.gif and an open subset V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq262_HTML.gif of M ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq143_HTML.gif, containing V Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq263_HTML.gif, with V ¯ V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq264_HTML.gif and such that if 0 < λ ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq347_HTML.gif, then ind M ˆ ( H ( λ , , μ ) , V ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq348_HTML.gif is defined and is independent of μ [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq249_HTML.gif. Moreover, reducing ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq349_HTML.gif if necessary, by Step 3 and by assumption (b), it follows that for λ ( 0 , ε ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq350_HTML.gif, the fixed points of H ( λ , , 0 ) = P ( λ , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq351_HTML.gif in V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq238_HTML.gif are a compact subset of V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq262_HTML.gif. Therefore, by the excision property and the homotopy invariance of the index, we get
ind M ˆ ( P ( λ , ) , V ) = ind M ˆ ( P ( λ , ) , V ) = ind M ˆ ( H ( λ , , 0 ) , V ) = ind M ˆ ( H ( λ , , 1 ) , V ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equad_HTML.gif
On the other hand, by Step 5, if λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq157_HTML.gif is sufficiently small, we have
ind M ˆ ( H ( λ , , 1 ) , V ) = ind M ( Q ( λ , ) , V ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equae_HTML.gif
Moreover, as shown in [1],
ind M ( Q ( λ , ) , V ) = deg ( w , V ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equaf_HTML.gif
Finally, notice that deg ( w , V ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq247_HTML.gif is well defined since V Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq352_HTML.gif is compact being homeomorphic to V Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq353_HTML.gif. Also, observe that there are no zeros of w in V V ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq354_HTML.gif. Thus, by the excision property of the degree, we obtain
deg ( w , V ) = deg ( w , V ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equag_HTML.gif

This shows that for small values of λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq157_HTML.gif, ind M ˆ ( P ( λ , ) , V ) = deg ( w , V ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq355_HTML.gif. The assertion of the lemma now follows by applying the homotopy invariance of the fixed point index to P ( λ , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq356_HTML.gif on V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq238_HTML.gif. □

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

Lemma 3.9 ([31])

Let K be a compact subset of a locally compact metric space Y. Assume that any compact subset of Y containing K has nonempty boundary. Then Y K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq357_HTML.gif contains a connected set whose closure is noncompact and intersects K.

Before presenting Lemma 3.10, we introduce the sets
S = { ( λ , ψ ) D : P ( λ , ψ ) = ψ } and S + = { ( λ , ψ ) S : λ > 0 } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equah_HTML.gif

and we recall that Z M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq358_HTML.gif denotes the set of zeros of the tangent vector field w.

Lemma 3.10 Let Y be a locally compact open subset of ( { 0 } × Z ) S + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq359_HTML.gif. Assume that K : = Y ( { 0 } × Z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq360_HTML.gif is compact and that deg ( w , V ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq361_HTML.gif, where V M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq362_HTML.gif, is an isolating neighborhood of { p M : ( 0 , p ) K } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq363_HTML.gif. Then the pair ( Y , K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq364_HTML.gif verifies the assumptions of Lemma  3.9.

Proof First of all, observe that by Lemma 3.7, S is closed in D and locally compact. In addition, K is clearly nonempty being deg ( w , V ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq365_HTML.gif. Now, let G be an open subset of D such that
G ( ( { 0 } × Z ) S + ) = Y . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equai_HTML.gif
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 W ¯ G https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq366_HTML.gif and C = W Y = W ¯ Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq367_HTML.gif. Therefore, denoted by G 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq368_HTML.gif the slice
G 0 = { ψ M ˆ : ( 0 , ψ ) G } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equaj_HTML.gif
we have that G 0 Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq369_HTML.gif is a compact subset of M ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq143_HTML.gif and is contained in the open slice W 0 W ¯ 0 G 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq370_HTML.gif of W at λ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq200_HTML.gif. Let V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq238_HTML.gif be an open subset of W 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq371_HTML.gif such that V V ¯ W 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq372_HTML.gif and V Z = W 0 Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq373_HTML.gif. Since C is compact and because of the local compactness of P, we may suppose that P ( W ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq374_HTML.gif is relatively compact. Consequently, there exists ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq240_HTML.gif such that
  1. 1.

    [ 0 , ε ] × V ¯ W https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq375_HTML.gif;

     
  2. 2.

    P ( λ , ψ ) ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq376_HTML.gif for ψ W ¯ λ V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq377_HTML.gif and 0 < λ ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq378_HTML.gif (here, as usual, W λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq379_HTML.gif denotes the slice { ψ M ˆ : ( λ , ψ ) W } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq380_HTML.gif).

     

Notice that P ( [ 0 , ε ] × V ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq381_HTML.gif is relatively compact. This follows easily from the above condition 1 and the relative compactness of P ( W ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq374_HTML.gif.

We can now apply Lemma 3.8 and the excision properties of the fixed point index and of the degree obtaining, for any 0 < λ ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq382_HTML.gif,
ind M ˆ ( P ( λ , ) , W λ ) = ind M ˆ ( P ( λ , ) , V ) = deg ( w , V ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equ10_HTML.gif
(3.5)
where V = { p M : p V } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq246_HTML.gif. Observe that V is an isolating neighborhood of { p M : ( 0 , p ) K } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq363_HTML.gif. Thus, by formula (2.1), by the above equalities (3.5) and the assumption deg ( w , V ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq365_HTML.gif, we get
ind M ˆ ( P ( λ , ) , W λ ) 0 , 0 < λ ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equak_HTML.gif
Since C is compact, by the generalized homotopy invariance property of the fixed point index, we get that ind M ˆ ( P ( λ , ) , W λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq383_HTML.gif does not depend on λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq157_HTML.gif. Hence,
ind M ˆ ( P ( λ , ) , W λ ) 0 , λ > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equal_HTML.gif
On the other hand, because of the compactness of C, for some positive λ ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq384_HTML.gif the slice C λ ¯ = { ψ W λ ¯ : P ( λ ¯ , ψ ) = ψ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq385_HTML.gif is empty. Thus,
ind M ˆ ( P ( λ ¯ , ) , W λ ¯ ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equam_HTML.gif

and we have a contradiction. Therefore, ( Y , K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq364_HTML.gif verifies the assumptions of Lemma 3.9 and the proof is complete. □

Proof of Theorem 3.3 Let ρ : [ 0 , + ) × C T ( M ) [ 0 , + ) × M ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq386_HTML.gif be the isometry given by ρ ( λ , x ) = ( λ , ψ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq387_HTML.gif, where ψ is the restriction of x to the interval [ T , 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq141_HTML.gif. As previously, let X [ 0 , + ) × C T ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq388_HTML.gif 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 ( λ , ψ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq389_HTML.gif such that P ( λ , ψ ) = ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq390_HTML.gif. Observe that S is actually contained in [ 0 , + ) × M ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq391_HTML.gif. Taking into account Lemma 3.6, X and S correspond under ρ. Analogously to the definition of S + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq392_HTML.gif, let us denote
X + = { ( λ , x ) X : λ > 0 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equan_HTML.gif
In addition, consider
Z T = { p C T ( M ) : w ( p ) = 0 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equao_HTML.gif
Theorem 3.2 implies that ( { 0 } × Z T ) X + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq393_HTML.gif is a closed subset of X. Therefore, it is locally compact since so is X according to Lemma 3.1. Now, consider
Y T = Ω ( ( { 0 } × Z T ) X + ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equap_HTML.gif
Observe that Y T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq394_HTML.gif is locally compact, being open in ( { 0 } × Z T ) X + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq395_HTML.gif. Then
Y : = ρ ( Y T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equaq_HTML.gif
is locally compact and open in ( { 0 } × Z ) S + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq359_HTML.gif. Denote by K T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq396_HTML.gif and K the subsets of Y T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq394_HTML.gif and Y defined as
K T = { ( λ , x ) Y T : λ = 0 } and K = ρ ( K T ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equar_HTML.gif
Now, observe that j 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq397_HTML.gif is an isolating neighborhood of
{ p M : ( 0 , p ) K } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equas_HTML.gif

Since deg ( w , j 1 ( Ω ) ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq398_HTML.gif, we can apply Lemma 3.10 concluding that ( Y , K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq364_HTML.gif verifies the assumptions of Lemma 3.9. Therefore, also ( Y T , K T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq399_HTML.gif verifies the same assumptions since the pairs ( Y , K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq364_HTML.gif and ( Y T , K T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq399_HTML.gif correspond under the isometry ρ. Therefore, Lemma 3.9 implies that Y T K T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq400_HTML.gif contains a connected set Γ whose closure (in Y T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq394_HTML.gif) is noncompact and intersects K T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq396_HTML.gif. Now, observe that according to Theorem 3.2, Y T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq394_HTML.gif is closed in Ω. Thus, the closures of Γ in Y T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq394_HTML.gif and in Ω coincide. This concludes the proof. □

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

Corollary 3.11 (Rabinowitz-type global bifurcation result)

Let M and f be as in Theorem  3.3. Assume that M is closed in R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq73_HTML.gif and that f sends bounded subsets of R × M ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq122_HTML.gif into bounded subsets of R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq73_HTML.gif. Let V be an open subset of M such that deg ( w , V ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq365_HTML.gif, where w is the mean value tangent vector field defined in formula (3.2). Then equation (3.1) has a connected subset of nontrivial T-periodic pairs whose closure contains some ( 0 , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq154_HTML.gif, with p V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq401_HTML.gif, and is either unbounded or goes back to some ( 0 , q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq402_HTML.gif, where q V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq403_HTML.gif.

Proof Let Ω be the open set obtained by removing from [ 0 , + ) × C T ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq404_HTML.gif the closed set { ( 0 , q ) : q V } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq405_HTML.gif. In other words,
Ω = ( [ 0 , + ) × C T ( M ) ) ( { 0 } × ( M V ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equat_HTML.gif

Observe that Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq185_HTML.gif is complete due to the closedness of M. Consider, by Theorem 3.3, a connected set Γ Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq406_HTML.gif of nontrivial T-periodic pairs with noncompact closure (in Ω) and intersecting { 0 } × C T ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq183_HTML.gif in a subset of { ( 0 , p ) Ω : w ( p ) = 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq407_HTML.gif. Suppose that Γ is bounded. From Remark 3.4 it follows that Γ ¯ Γ ¯ Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq191_HTML.gif, where Γ ¯ Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq408_HTML.gif denotes the closure of Γ in Ω, is nonempty and hence contains a point ( 0 , q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq402_HTML.gif which does not belong to Ω, that is, such that q V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq403_HTML.gif. □

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 ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq409_HTML.gif is clearly satisfied in the case when w has an isolated zero with nonzero index. For example, if w ( p ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq410_HTML.gif and w is C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq36_HTML.gif with injective derivative w ( p ) : T p M R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq411_HTML.gif, then p is an isolated zero of w and its index is either 1 or −1. In fact, in this case, w ( p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq412_HTML.gif sends T p M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq413_HTML.gif into itself and, consequently, its determinant is well defined and nonzero. The index of p is just the sign of this determinant (see, e.g., [29]).

The next consequence of Theorem 3.3 provides an existence result for T-periodic solutions already obtained in [6]. Moreover, it improves an analogous result in [3], in which the map f is continuous on R × C ( ( , 0 ] , M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq414_HTML.gif, with the compact-open topology in C ( ( , 0 ] , M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq415_HTML.gif. 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 R × M ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq122_HTML.gif into bounded subsets of R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq73_HTML.gif. In addition, suppose that M is compact with Euler-Poincaré characteristic χ ( M ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq416_HTML.gif. Then equation (3.1) has a connected unbounded set of nontrivial T-periodic pairs whose closure meets { 0 } × C T ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq417_HTML.gif. Therefore, since C T ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq23_HTML.gif is bounded, equation (3.1) has a T-periodic solution for any λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq248_HTML.gif.

Proof Choose V = M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq418_HTML.gif. By the Poincaré-Hopf theorem, we have
deg ( w , M ) = χ ( M ) 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equau_HTML.gif

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 R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq73_HTML.gif [7, 8] and extends an analogous one for ODEs on differentiable manifolds [31]. In what follows, by a T-periodic orbit of x ( t ) = λ f ( t , x t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq419_HTML.gif, 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 R × M ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq122_HTML.gif into bounded subsets of R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq73_HTML.gif. Let V be a relatively compact open subset of M and assume that
  1. 1.

    w ( p ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq420_HTML.gif along the boundary ∂V of V;

     
  2. 2.

    deg ( w , V ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq409_HTML.gif;

     
  3. 3.

    for any λ ( 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq421_HTML.gif, the T-periodic orbits of x ( t ) = λ f ( t , x t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq422_HTML.gif lying in V ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq423_HTML.gif do not meet ∂V.

     
Then the equation
x ( t ) = f ( t , x t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equav_HTML.gif

has a T-periodic orbit in V.

Proof Define Ω = [ 0 , 1 ) × C T ( V ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq424_HTML.gif. Observe that C T ( V ¯ ) = C T ( V ) ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq425_HTML.gif. Therefore,
Ω = ( { 1 } × C T ( V ¯ ) ) ( [ 0 , 1 ) × C T ( V ¯ ) C T ( V ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equaw_HTML.gif

According to Theorem 3.3, call Γ a connected subset of Ω of nontrivial T-periodic pairs of the equation x ( t ) = λ f ( t , x t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq16_HTML.gif, whose closure in Ω is noncompact and intersects { 0 } × C T ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq417_HTML.gif in a subset of { ( 0 , p ) Ω : w ( p ) = 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq407_HTML.gif.

As V has compact closure in M, then the closure of Ω in [ 0 , + ) × C T ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq426_HTML.gif is complete, being
Ω ¯ = [ 0 , 1 ] × C T ( V ¯ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_Equax_HTML.gif

Since f sends bounded subsets of R × M ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq122_HTML.gif into bounded subsets of R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq73_HTML.gif, recalling Remark 3.4, one has that the closure Γ ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq186_HTML.gif of Γ in the whole space (which coincides with the closure in Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq185_HTML.gif) must intersect Ω.

Now, because of the above condition 3, Γ ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq186_HTML.gif cannot contain elements of ( 0 , 1 ) × C T ( V ¯ ) C T ( V ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq427_HTML.gif. In addition, condition 1 and Theorem 3.2 imply that Γ ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq186_HTML.gif does not contain elements of { 0 } × ( C T ( V ¯ ) C T ( V ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq428_HTML.gif. Therefore, the nonempty set Γ ¯ Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq429_HTML.gif is composed of pairs of the form ( 1 , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq430_HTML.gif, where x is a T-periodic solution of x ( t ) = f ( t , x t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-21/MediaObjects/13661_2012_Article_278_IEq431_HTML.gif 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. 2010/20727-4.

Authors’ Affiliations

(1)
Dipartimento di Matematica e Informatica, Università degli Studi di Firenze
(2)
Instituto de Matemática e Estatística, Universidade de São Paulo
(3)
Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche

References

  1. Furi M, Pera MP: A continuation principle for forced oscillations on differentiable manifolds. Pac. J. Math. 1986, 121: 321-338. 10.2140/pjm.1986.121.321MathSciNetView Article
  2. Benevieri P, Calamai A, Furi M, Pera MP: Delay differential equations on manifolds and applications to motion problems for forced constrained systems. Z. Anal. Anwend. 2009, 28: 451-474.MathSciNetView Article
  3. Benevieri, P, Calamai, A, Furi, M, Pera, MP: A continuation result for forced oscillations of constrained motion problems with infinite delay. Adv. Nonlinear Stud. (to appear)
  4. Benevieri P, Calamai A, Furi M, Pera MP: On general properties of retarded functional differential equations on manifolds. Discrete Contin. Dyn. Syst. 2013, 33(1):27-46. doi:10.3934/dcds.2013.33.27MathSciNetView Article
  5. Rabinowitz PH: Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 1971, 7: 487-513. 10.1016/0022-1236(71)90030-9MathSciNetView Article
  6. Benevieri P, Calamai A, Furi M, Pera MP: On the existence of forced oscillations of retarded functional motion equations on a class of topologically nontrivial manifolds. Rend. Ist. Mat. Univ. Trieste 2012, 44: 5-17.MathSciNet
  7. Mawhin J: Équations intégrales et solutions périodiques des systèmes différentiels non linéaires. Bull. Cl. Sci., Acad. R. Belg. 1969, 55: 934-947.MathSciNet
  8. Mawhin J: Periodic solutions of nonlinear functional differential equations. J. Differ. Equ. 1971, 10: 240-261. 10.1016/0022-0396(71)90049-0MathSciNetView Article
  9. Hale JK, Verduyn Lunel SM: Introduction to Functional Differential Equations. Springer, New York; 1993.View Article
  10. Gaines R, Mawhin J Lecture Notes in Math. 568. In Coincidence Degree and Nonlinear Differential Equations. Springer, Berlin; 1977.
  11. Nussbaum RD: Periodic solutions of some nonlinear autonomous functional differential equations. Ann. Mat. Pura Appl. 1974, 101: 263-306. 10.1007/BF02417109MathSciNetView Article
  12. Nussbaum RD: The fixed point index and fixed point theorems. Lecture Notes in Math. 1537. In Topological Methods for Ordinary Differential Equations (Montecatini Terme, 1991). Springer, Berlin; 1993:143-205.View Article
  13. Mallet-Paret J, Nussbaum RD, Paraskevopoulos P: Periodic solutions for functional-differential equations with multiple state-dependent time lags. Topol. Methods Nonlinear Anal. 1994, 3: 101-162.MathSciNet
  14. Hale JK, Kato J: Phase space for retarded equations with infinite delay. Funkc. Ekvacioj 1978, 21: 11-41.MathSciNet
  15. Hino Y, Murakami S, Naito T Lecture Notes in Math. 1473. In Functional-Differential Equations with Infinite Delay. Springer, Berlin; 1991.
  16. Oliva WM, Rocha C: Reducible Volterra and Levin-Nohel retarded equations with infinite delay. J. Dyn. Differ. Equ. 2010, 22: 509-532. 10.1007/s10884-010-9177-yMathSciNetView Article
  17. Oliva WM: Functional differential equations on compact manifolds and an approximation theorem. J. Differ. Equ. 1969, 5: 483-496. 10.1016/0022-0396(69)90089-8MathSciNetView Article
  18. Oliva WM: Functional differential equations-generic theory. In Dynamical Systems (Proc. Internat. Sympos., Brown Univ., Providence, R.I., 1974), Vol. I. Academic Press, New York; 1976:195-209.
  19. Borsuk K: Theory of Retracts. Polish Sci., Warsaw; 1967.
  20. Granas A, Dugundji J: Fixed Point Theory. Springer, New York; 2003.View Article
  21. Leray J, Schauder J: Topologie et équations fonctionnelles. Ann. Sci. Éc. Norm. Super. 1934, 51: 45-78.MathSciNet
  22. Brown RF: The Lefschetz Fixed Point Theorem. Scott, Foresman, Glenview; 1971.
  23. Granas A: The Leray-Schauder index and the fixed point theory for arbitrary ANR’s. Bull. Soc. Math. Fr. 1972, 100: 209-228.MathSciNet
  24. Nussbaum RD: The fixed point index for local condensing maps. Ann. Mat. Pura Appl. 1971, 89: 217-258. 10.1007/BF02414948MathSciNetView Article
  25. Dold A: Lectures on Algebraic Topology. Springer, Berlin; 1972.View Article
  26. Spanier E: Algebraic Topology. McGraw-Hill, New York; 1966.
  27. Guillemin V, Pollack A: Differential Topology. Prentice-Hall, Englewood Cliffs; 1974.
  28. Hirsch MW Graduate Texts in Math. 33. In Differential Topology. Springer, Berlin; 1976.View Article
  29. Milnor JM: Topology from the Differentiable Viewpoint. University of Virginia Press, Charlottesville; 1965.
  30. Tromba AJ: The Euler characteristic of vector fields on Banach manifolds and a globalization of Leray-Schauder degree. Adv. Math. 1978, 28: 148-173. 10.1016/0001-8708(78)90061-0MathSciNetView Article
  31. Furi M, Pera MP: A continuation principle for periodic solutions of forced motion equations on manifolds and applications to bifurcation theory. Pac. J. Math. 1993, 160: 219-244. 10.2140/pjm.1993.160.219MathSciNetView Article
  32. Dunford N, Schwartz JT: Linear Operators. Wiley, New York; 1957.
  33. Furi M, Pera MP, Spadini M: Periodic solutions of functional differential perturbations of autonomous differential equations. Commun. Appl. Anal. 2011, 15: 381-394.MathSciNet

Copyright

© Benevieri et al.; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.