On the set of harmonic solutions of a class of perturbed coupled and nonautonomous differential equations on manifolds

We study the set of $T$-periodic solutions of a class of $T$-periodically perturbed coupled and nonautonomous differential equations on manifolds. By using degree-theoretic methods we obtain a global continuation result for the $T$-periodic solutions of the considered equations.


Introduction
This paper is concerned with topological properties of the set of harmonic solutions of a class of parametrized coupled periodic differential equations. Namely, we consider a perturbed periodic linear nonhomogeneous differential equation in R k coupled with a (tangent) periodic perturbation of the zero vector field on a smooth differentiable manifold. More precisely, let M ⊆ R s be a boundaryless smooth manifold, given T > 0, we consider T -periodic solutions to the following system of equations on R k × M (1.1) ẋ = A(t)x + c(t) + λf 1 (t, x, y, λ), y = λf 2 (t, x, y, λ), λ ≥ 0, where A : R → GL(R k ) ⊆ R k×k , is a continuous matrix-valued map, c : R → R k is a sufficiently regular vector-valued map, f 1 and f 2 are continuous with f 1 : R × R k × M × [0, ∞) → R k and f 2 : R × R k × M × [0, ∞) → R s and, in particular, f 2 is a tangent vector field to M , in the sense that for every (t, p, q, λ) ∈ R×R k ×M ×[0, ∞) it holds that f 2 (t, p, q, λ) ∈ T q M . Moreover, all of these maps are assumed to be T -periodic, T > 0 given, with respect to the t-variable. By applying fixed point index and degree-theoretic methods, in our main result (Theorem 2.2 below) we deduce a global continuation principle for the T -periodic solutions of (1.1). More precisely, assuming a non-T -resonance condition on A we provide a condition based on the degree of the vector field f 2 averaged along a solution of the first equation of (1.1) for λ = 0, that ensures the existence of a connected set of (nontrivial i.e. corresponding to λ > 0, see definition 2.1 below) T -periodic solutions of (1.1) whose closure is not compact emanating from a specific set Γ of T -periodic solutions of the unperturbed (1.1).
The arrangement of equations in (1.1) may look unusual at first sight but it is pretty natural in some contexts. Consider, for instance, a moving point in the plane or space whose radial distance from the origin is governed by a possibly perturbed linear equation (the perturbation itself depending on the position in the plane or space), while its "angular" position with respect to any given frame of reference is influenced only by a small forcing. The motion of such a point in the plane could be described by a curve in polar coordinates t → r(t), θ(t) ∈ R × S 1 (allowing negative r in the usual sense) that is determined by a system of equations as follows: ṙ = a(t)r + c(t) + λf 1 (t, r, θ, λ), r ∈ Ṙ θ = λf 2 (t, r, θ, λ), θ ∈ S 1 λ ≥ 0, where f 1 and f 2 , that represent perturbations, are T -periodic in t of the same period as the scalar functions a and c.
In fact the form of system (1.1) is rather flexible. In later sections we will show, throughout some examples, how it lends itself to quite distant applications.
To get a rough idea of the kind of result that we seek, consider the following equation in R × R: (1.2) ẋ = −x + 1 + 1 10 sin t − λ y − x , y = −λ 1 2 + y + 2x sin t . The numerical diagram in figure 1 shows a portion of the set of points (λ, x λ 0 , y λ 0 ) where (x λ 0 , y λ 0 ) is the initial data (for t = 0) of a T -periodic solutions of (1.2) corresponding to λ. Roughly speaking, Theorem 2.2 yields a connected set Γ of "nontrivial" Tperiodic solutions emanating from a subset of initial points as in the figure 1. Later on, we will analyze the behavior of this set in more detail (see Definiton 4.1 and Theorem 4.2 below).
On a different tack, observe that system (1.1) is a periodic perturbation of a nonautomous differential equation. In general, periodic problems of such equations are rather difficult to study, at least with topological methods. One reason being that it is not easy to compute the fixed point index of the Poincarè T -translation operator associated with a nonautonomous equation. The present study does this in a particular case. A different situation, but in the same direction was considered in [15] where perturbations of separated variables equations are tackled by a change of variable approach. Here, the difficulties arising from the fact that the unperturbed part of system (1.1) is nonautonomous are solved thanks to the simple observation that the first equation in of (1.1) admits a unique T -periodic solution when λ = 0 and some assumptions are made on A.
The technique used here could probably be generalized to the case when the unperturbed first equation of (1.1) is nonlinear but admits a finite number of "nondegenerate" periodic solutions. Moreover, our methods open the way to the analysis of coupled differential equations of type (1.1) involving delays in the perturbing term (see, e.g., [3,4,9] for some recent contributions about similar issues). These lines of research would take us too far from the core topic of the present paper, but will be investigated in future studies.
Finally, we point out that our arguments rely on the notions of degree (or characteristic) of a tangent vector field and of fixed point index of a map on a manifold. For an exposition, we refer to standard texts as, e.g., [11,13].

Preliminaries and main result
We begin this section by recalling some basic facts and definitions about the function spaces used throughout the paper.
Let I ⊆ R be an interval and let X ⊆ R n . The set of all X-valued functions defined on I is denoted by C(I, X). When I = R, we simply write C(X) instead of C(R, X). Let T > 0 be given, by C T (R n ) we mean the Banach space of all the continuous T -periodic functions ζ : R → R n whereas C T (X) denotes the metric subspace of C T (R n ) consisting of all those ζ ∈ C T (R n ) that take values in X. In fact, X can be regarded as a subset of C T (X) by identifying it with the image of the closed embedding that associates to any point q ∈ X the constant function There is a natural homeomorphism between C T (Y )× C T (W ) and C T (Y × W ). Then, for the reminder of the paper we will use C T (Y × W ) and C T (Y ) × C T (W ) interchangeably, and denote the elements of C T (Y × W ) as pairs (x, y) with x ∈ C T (Y ) and y ∈ C T (W ).
In the sequel, if S is a topological space, given A ⊆ [0, ∞) × S and λ ∈ [0, ∞), we will denote the λ-slice {x ∈ S : (λ, x) ∈ A} by the symbol A λ . Observe that Our main result concerns the properties of the set of the T -periodic solutions of (1.1). Let us recall the following definition: Observe that a trivial T -triple (0, x, y) has necessarily a specific form: y is clearly constant and x is a T -periodic solution of the ODE where A and c are as in (1.1).
Let Φ = Φ(t) ∈ R k×k be the principal fundamental matrix of equationẋ(t) = A(t)x(t). Namely, Φ : R → R k×k is such thaṫ Throughout this section we posit the following non-T -resonance assumption on A: When A(t) ≡ A is a constant matrix, we have that Φ(t) = e tA and the condition (2.2) can be written as det(I − e T A ) = 0 which holds true if and only if A has no eigenvalues of the form 2nπi/T , n ∈ Z, i being the imaginary unit. When the matrix-valued function A has the "non-T -resonance" property above (see also, e.g., [5]) something can be said about the T -periodic solutions of (2.1), as shown in the following well-known technical lemma whose standard proof is provided only for the sake of completeness: Then, equation (2.1) admits a unique T -periodic solution given by: Proof. The initial value problem admits a unique solution that can be written as We seek initial values x 0 that correspond to T -periodic solutions. Hence, we put and solve for x 0 . Since I − Φ(T ) is invertible we get which is the unique initial condition that yields a T -periodic solution of (2.4). Susbstituting in (2.5) we get the assertion.
It is also convenient to define the following average vector field: where f 2 andx are as in (1.1) and (2.3), respectively. Clearly, w is tangent to M .
We are now in a position to state our main result; its proof, that requires several preliminary steps, is postponed to a later section. We now illustrate the setting of Theorem 2.2 with an explicit example. Example 2.3. Consider the following system of constrained ODEs in R 3 : Where f : R × R 3 → R, continuous and 2π-periodic in t, is given. System (2.8) can be regarded as an ODE on R × S 1 (here S 1 lies in the plane y, z) because the timedependent vector field defined by the second and third equations in (2.8) is tangent to S 1 . Takingx and w as in Theorem 2.2 (with T = 2π), we getx(t) = − cos(t) and Clearly w has only two zeros on S 1 given by N := (0, 1) and S := (0, −1). Consider, for instance, U N := S 1 \ {S} and let It is not difficult to see that Hence, Theorem 2.2 yields a connected set Γ N of nontrivial T -triples for (2.8) whose is not contained in any compact subset of Ω N and meets the set The previous example opens the way to the study of a more significant situation concerning the coupled differential equations describing a simple mechanical system, and the possible connections to a particular class of Differential-Algebraic Equations (DAEs). Example 2.4. Consider a block of mass m constrained to a unitary circular rail lying on an horizontal plane. Put the origin of the plane in the center of the rail, and suppose the block can slide along the rail without friction and that it is acted upon by a continuous T -periodic force of the form φ(t, x, y) ( −y x ) (which is clearly tangent to the constraint). Let us denote by r(t) the modulus of the centripetal force (the vincular reaction of the rail) exerted, at time t, on the block. One customary way of writing the equations governing such a system is through the formalism of Differential-Algebraic Equations (DAEs) see, e.g. [12], as follows: (See also, e.g., [10, Example 4.7, §4].) Consider now a linear oscillator of (fixed) unit mass and elastic constant k subject to a T -periodic forcing c(t), and assume, finally, that the acceleration r(t)/m is used as imput by some nonspecified mechanism to influence this oscillator by the addition of a force f r(t)/m . Thus this oscillator is described by the following equation: The system under consideration is schematized in figure 2. We are interested in the set of T -periodic solutions of the coupled system (2.9) as m > 0 varies.
Setting λ 2 = 1/m, ζ =ż,ẋ = λu,ẏ = λv, and assuming that f is homogeneous of degree α, the equations governing (2.9) become: System (2.10) is actually a DAE with differentiation index 2 (see [12]). Differentiating with respect to t the "algebraic" constraint x(t) 2 + y(t) 2 = 1, we get 2ẋ(t)x(t) + 2ẏ(t)y(t) = 0. Using (2.10) and dividing by 2λ > 0, we get x(t)u(t) + y(t)v(t) = 0. Differentiationg again and using the other relations in (2.10) we get that any solution t → x(t), y(t), u(t), v(t), r(t), z(t), ζ(t) of (2.10) satisfies Thus, we have a differential equation for r: Pluggiung this differential equation in (2.10) we get an equivalent equation on R 2 × M ⊆ R 7 , given by: where M is the 2-dimensional submanifold of R 5 defined by the constraints obtained above, namely: here we use capital letters to denote the coordinates in R 5 in order to distinguish from the dependent variables in (2.10) and (2.11).
Equation (2.11) can be extended to the case λ ∈ [0, ∞), although its physical meaning is lost when λ = 0. However, Theorem 2.2 can be used to investigate the limiting behaviour of the system (2.9) and to gain insights into the topological structure of its T -periodic solutions.
Finally, note that (2.10), after index reduction, falls within the framework considered in [16,1] (see also [4,2] for the case of similar DAEs involving delay).

Fixed point index and Poincaré translation operator
Throughout this section we will assume that f 1 and f 2 of equation (1.1) are locally Lipschitz so that the initial value problem admits a unique solution for λ ≥ 0.
We denote by P λ τ (p, q) the value of the solution of (3.1) at time τ , when defined: That is, P λ τ is the so-called Poincaré τ -translation operator associated with (3.1). Well-known properties of differential equations show that the domain As the functions A, c, f 1 and f 2 are T -periodic in t, it is well-known that Tperiodic solutions of (1.1) (for any given λ ≥ 0) correspond bijectively to fixed points of the T -translation operator P λ T , in the sense that an initial condition (p, q) of (3.1) yields a T -periodic solution if and only if (p, q) is a fixed point of P λ T . Since our purpose is to study the T -periodic solutions of (1.1), it makes sense to investigate the fixed point index of the T -translation operator P λ T . The following theorem gives a useful formula; it is a major step towards Theorem 2.2 and the main result of this section.
The proof of this theorem requires some preliminary steps. Let us begin with a simple formula for the fixed point index of the T -translation operator associated to equation (2.1). More precisely, let F T (p) ∈ R k be the value, at time t = T , of the maximal solution of (2.1) that satisfies x(0) = p. Since A and c are bounded functions (they are assumed T -periodic), it is well known that all solutions of (2.1) are continuable in R; that is, the domain of the function F T is the whole R k .
The formula we need is the content of the following preliminary lemma: Lemma 3.2. Let F T be as above, and let U ⊆ R k be an open and such that x(0) / ∈ ∂U , then where, as in Theorem 3.1, 1 U denotes the characteristic function of U .
Proof. Let Φ be as in Lemma 2.1. Then, by (2.5), for all p ∈ R k . Taking the Fréchet derivative of F T at p 0 :=x(0), we get Hence, since Φ(T ) is nonsingular, by the properties of the fixed point index we have that The assertion follows.
We will need a slight extension of a result of [7] that concerns the translation operator associated to equations of the following form: (3.5)ẏ = λϕ(t, y, λ), λ ≥ 0, where ϕ : R × N × [0, ∞) → R n is a C 1 map, tangent to a manifold N ⊆ R n in the sense that ϕ(t, q, λ) ∈ T q N for all (t, q, λ) ∈ R × N × [0, ∞), which is T -periodic in the first variable.
We will denote by Q λ ϕ,T the T -translation operator associated with (3.5). Namely, Q λ ϕ,T (q), when defined, will be the value for t = T of the (unique) solution of (3.5) satisfying y(0) = q. As discussed above for the case of P λ T one can prove that the domainn of the map (λ, q) → Q λ ϕ,T (q) is open in [0, ∞) × N . We also define the average vector field which is clearly tangent to N .
3. Let f , w ϕ and N be as above. Let U be a relatively compact open subset of N and assume that (w ϕ , U ) is admissible for the degree. Then, there exists λ 0 > 0 such that, for 0 < λ ≤ λ 0 , the T -translation operator Q λ ϕ,T associated with (3.5) is defined on U , fixed point free on ∂U and ind Q λ ϕ,T , U = deg(−w ϕ , U ). Proof. The same argument of the proof of Theorem 3.11 in [7] applies to this more general assertion with only minimal adaptation.
The following fact is a direct consequence of the properties of the fixed point index of tangent vector fields on manifolds. Its simple proof is left to the reader. ind Taken together, Theorem 3.3 and Proposition 3.4 yield the following result about the T -translation operator of decoupled systems: Corollary 3.5. Let A, c and f 2 be as in (1.1). Consider the following system of equations wherex is the unique T -periodic solution of (2.1). Let Q λ T be the T -translation operator associated with (3.8), and let w be given by (2.7). Given an open U × V ⊆ R k × M such thatx(0) / ∈ ∂U and w −1 (0) ∩ ∂V = ∅, we have, for sufficiently small values of λ ≥ 0, . Proof. Let F T be as in Lemma 3.2. Letx be as in (2.3) and define, for any t ∈ R and q ∈ M the map ϕ : R × M → R s given by ϕ(t, q, λ) := f 2 t,x(t), q, λ .
Let Q λ ϕ,T be as in Theorem 3.3. Since equations (3.8) are completely decoupled (they are independent of each other) it follows that Observe that a fixed point of Q λ T is necessarily of the form x(0), q , with q a fixed point of the T -translation operator associated to equatioṅ y = λf 2 t,x(t), y, λ .
Thus, the set of fixed points of Q λ T coincides with that of the T -translation operator associated with the following weakly coupled system: This remark alone, however, is not enough to prove Theorem 3.1 because (1.1) are coupled when λ > 0. To overcome this difficulty, we will use a homotopy argument as shown below.
Proof of Theorem 3.1. Let us consider the following system of coupled equations depending on two parameters: where t → x λ,µ (p, q, t), y λ,µ (p, q, t) denotes the unique maximal solution of the initial-value problem for the system (3.10) supplemented by the initial conditions x(0) = p and y(0) = q, (p, q) ∈ R k × M . Well-known properties of differential equations imply that D H is an open set. Clearly, if we consider λ =λ and µ =μ to be fixed parameters, the map (p, q) → H(λ, p, q,μ) is the T -translation operator associated with (3.10) for λ =λ and µ =μ. As such, its domain is an open subset of R k × M . It is convenient to put H λ (µ, p, q) := H(λ, p, q, µ) whenever defined.
For compactness reasons we can assume, as n → +∞, that up to a subsequence µ n → µ 0 ∈ [0, 1] and (p n , q n ) → (p 0 , q 0 ) ∈ ∂U × ∂V . Let us denote by (x n , y n ) the T -periodic solution of (3.10) corresponding to (µ n , λ n ) and starting from (p n , q n ) at time t = 0. Due to the continuous dependence on initial data, it follows that x n →x and y n (t) → y 0 (t) ≡ q 0 uniformly on [0, T ]. In particular we have that x(0) = p 0 . Moreover, from the second equation in (3.10) it follows that So, dividing by λ n > 0, and taking n → +∞, we get This is a contradiction since we assumed that w(q) = 0 for every q ∈ ∂V . Claim 2. Identity (3.3) holds. As a consequence of Claim 1 we have that there exists λ * > 0 so small that H λ is admissible for all λ ∈ (0, λ * ]. Then, by the homotopy invariance property, we have that (3.12) ind T is the T -translation operator of Corollary 3.5. Using the chain of identities (3.12) along with Corollary 3.5, we get , and the assertion follows.

Proof of the main result
In order to prove Theorem 2.2, it is convenient to consider first what we can call its 'finite dimensional version'. A crucial notion is the following: In other words, a starting triple (λ, p, q) is such that (p, q) is an initial point for a T -periodic solution of (1.1) corresponding to λ.
In what follows, the set of all starting triples for (1.1) is denoted by S, and the subset of nontrivial ones is called N .
An immediate consequence of the continuous dependence on data is that the hence it is locally compact. Thus N , as a closed subset of D, is locally compact as well.
The technical result below characterizes the elements of the 0-slice (N ) 0 .
Similarly to what was done for Theorem 2. To help clarify the relations between the different projections introduced so far, consider the following commutative diagram: where the vertical arrows, from left to right, are defined as the closed embeddings (λ, p) → (λ, p), (λ, p, q) → (λ, p, q) and (λ, q) → (λ, q).
The main result concerning the starting triples for (1.1) is Theorem 4.2 below. The argument of the proof follows, with some adaptations, that of [15,Thm. 3], see also [8,Thm. 3.1].
Theorem 4.2. Let A, c, f 1 and f 2 be as in (1.1), and letx be as in (2.3). Assume, as in Theorem 3.1 that f 1 and f 2 are locally Lipschitz in x and y. Define w as in Theorem 2.2. Let U be a given open subset of D, and assume that deg w, pr 2 (U) 0 is well-defined and nonzero and thatx(0) ∈ pr 1 (U) 0 . Then, there exists a connected set G of nontrivial starting triples for (1.1) in U whose closure in D meets the set Z := (0,x(0), q) ∈ U : w(q) = 0 and is not contained in any compact subset of U.
The diagram shown in figure 1 illustrates the situation described in Theorem 4.2 in the case of equation (1.2). More specifically, one directly sees thatx(t) = 1 20 (sin t − cos t) + 1, so thatx(0) = 19 20 and (following (2.7)) that Hence, by Theorem 4.2, there is a connected set of nontrivial starting points emanating from (0, 19 20 , − 11 20 ) . In figure 3 we exhibit the projections of the portion of Γ considered in figure 1 on the planes xy and yλ.  Before providing the proof of Theorem 4.2 we recall the following well known global connection result (see [6]). Lemma 4.3. Let Y be a locally compact metric space and let Z be a compact subset of Y . Assume that any compact subset of Y containing Z has nonempty boundary. Then Y \ Z contains a connected set whose closure (in Y ) intersects Z and is not compact.
Proof of Theorem 4.2. Observe that sincex(0) ∈ pr 1 (U) 0 and deg w, pr 2 (U) 0 is well-defined and nonzero one has that the set Z is compact and nonempty.
Since N is locally compact, so is N ∩ U. It is enough to prove that the pair In particular, sincex(0) ∈ pr 1 (U) 0 , we havex(0) ∈ U . By Theorem 3.1 and the excision property of the degree, we get, for all 0 < λ ≤ ε, Now, C being compact, there necessarily exists δ > 0 such that C λ = ∅. That is, P δ T is fixed point free on A δ . Thus, from the generalized homotopy invariance property of the degree, we have Starting triples are important because, when some regularity is imposed on the differential equation, they are closely related to the T -triples. More precisely, there exists a homeomorphism between the relative sets that respects the notion of triviality, as shown by the following lemma: Lemma 4.4. Let A, c, f 1 , and f 2 be as in Theorem 4.2, and let X and S be, respectively, the sets of T -triples and of starting triples for (1.1). Let h : X → S be the map given by h(λ, x, y) = λ, x(0), y(0) . Then h is a homeomorphism that makes trivial T -triples correspond to trivial starting pairs and vice versa.
Proof. Obiously, h is continuous and surjective. Since f 1 , and f 2 are locally Lipschitz in x and y, it is also clearly injective. The continuity with respect to initial data implies that the inverse of h is continuous. Thus h is a homeomorphism.
Finally observe that by the definition of h, a T -triple whose λ-component is 0 can only correspond to a starting triple with the same property and vice versa. We are now ready to proceed with the proof of Theorem 2.2.
Proof of Theorem 2.2. Assume first that f 1 and f 2 are locally Lipschitz, so that the Cauchy problem (3.1) admits unique solution for any (p, q) ∈ R k × M and λ ≥ 0.
Consider the set S Ω := (λ, p, q) ∈ S : the solution of (3.1) is contained in Ω .  1). It is not difficult to see that h maps X ∩ Ω homeomorphically onto S Ω . It is not difficult to verify thatx(0) ∈ O 1 (Ω) if and only ifx(0) ∈ pr 1 (U Ω ) 0 . Also, by inspection of equation (1.1), one has that for any q ∈ M , 0,x(0), q ∈ S and that, conversely, the projection of S 0 onto the third component is the whole M .
In particular, for any q ∈ M we have 0,x(0), q ∈ Ω if and only if 0,x(0), q ∈ S Ω . Thus, by the definition of O 2 (Ω) Consequently, by the choice of U Ω , whenx(0) ∈ O 1 (Ω) we have that O 2 (Ω) coincides with pr 2 (U Ω ) 0 , so that Thus, by Theorem 4.2 there exists a connected set G of nontrivial starting triples for (1.1) in U Ω whose closure in D meets the set (0,x(0), q) ∈ U Ω : w(q) = 0 and is not contained in any compact subset of U Ω . It is not difficult to see that the set Γ := h −1 (G) has the properties required in the assertion.
In order to conclude the proof we need to remove the local Lipschitzianity assumptions on f 1 and f 2 . Denote by X the subset of X consisting of nontrivial T -triples. Let Z := (0,x, q) ∈ Ω : w(q) = 0 . As a consequence of Remark 4.5 we have that X ∪ Z ⊆ X coincides with the closure X of X in [0, ∞) × C T (R k × M ). As in the case of starting triples, it is not difficult to show that X is locally compact.
We only have to prove that the pair X ∩ Ω , Z satisfies the assumptions of Lemma 4.4. Assume by contradiction that there exists a relatively open compact subset C of X ∩ Ω that contains Z. Thus, there exists an open set W ⊆ Ω such that W ∩ X = C.
Since C is compact, it is not difficult to show that W can be chosen with the following properties: Clearly, the following subset is relatively compact.
By known approximation results (see, e.g., [11]), there exist sequences {f i 1 } i∈N and {f i 2 } i∈N of T -periodic smooth tangent vector fields uniformly approximating f 1 and f 2 . Put As a consequence of Remark 4.5 we see that and also that w i is nonzero on the boundary of O 2 (W) relative to M . Thus, for i ∈ N large enough, we get The last equality being a consequence of the excision property of the degree. Thus, Consider the system x, y, λ), and let X i be the set of nontrivial T -triples of (4.6).
For i large enough, by (4.4) and (4.5), the first part of the proof applied to equation (4.6) and the open set W yields a connected subset Γ i of X i ∩ W whose closure in [0, ∞) × C T (R k × M ) is nonempty (as, for any i, it intersects the set (0,x, q) ∈ W : w i (q) = 0 ) and is not contained in any compact subset of W. Let us prove that, for i large enough, Γ i ∩ ∂W = ∅. It is sufficient to show that X i ∩ W is compact. In fact, if (λ, x, y) ∈ X i ∩ W we have, for any t ∈ [0, T ], where K is as in (iii), and |·| k , |·| s and |·| k+s denote the usual norms in R k , R s and R k+s , respectively (recall that we are assuming M ⊆ R s ). Hence, by Ascoli-Arzelà theorem, X i ∩W is totally bounded and, consequently, compact since W is complete by (i). Thus, for i large enough, there exists a T -triple (λ i , x i , y i ) ∈ Γ i ∩ ∂W of (4.6). Again by Ascoli-Arzelà theorem, we may assume that (x i , y i ) → (x 0 , y 0 ) in C T (R k × M ), and λ i → λ 0 with (λ 0 , x 0 , y 0 ) ∈ ∂W. Therefore ẋ 0 (t) = A(t)x(t) + c(t) + λ 0 f 1 t, x 0 (t), y 0 (t), λ 0 ẏ 0 (t) = λ 0 f 2 t, x 0 (t), y 0 (t), λ 0 .
Hence (λ 0 , x 0 , y 0 ) is a T -triple in ∂W. This contradicts the choice of W, in particular the assumption (ii) that ∂W ∩ X = ∅.

Further examples
The following example shows how Theorem 2.2 can be used to get information about the set of T -periodic solutions of a parametrized differential equation containing a distributed continuous delay. The strategy consists in reducing (5.1) to a coupled system of equations (see, e.g., [14]).
Example 5.1. Let us consider the following equation with distributed continuous delay where φ : R × R n → R is continuous and T -periodic, T > 0 given, in t. Since we are interested in T -periodic (hence bounded) solutions it makes sense to set Differentiating this relation and integrating by parts, we geṫ Thus, given any T -periodic solution y of (5.1), we have that (x, y) is an obviously T -periodic solution of the coupled system (5.2) ẋ = x − λy, y = λ h(t, y) + φ(t, y)x , which is a system of type (1.1). Conversely, if (x, y) is a T -periodic solution of (5.2) then y is a T -periodic solution of (5.1). Observe that the unique T -periodic solution of the first equation in (5.2), corresponding to λ = 0, isx ≡ 0. Take Ω = [0, ∞) × C T (R n × R n ) and put w(q) := 1 TˆT 0 h t, q)dt.
Let π 2 : [0, ∞) × C T (R n × R n ) → [0, ∞) × C T (R n ) be as in (2.6). Then π 2 (Γ) is an unbounded connected set of pairs (λ, y), with y a T -periodic solution of (5.1) corresponding to λ, whose closure meets the set  Figure 4. The mechanical system of Example 5.2 P 2 confined to a linear rail, a fixed point O on the rail and two connecting springs S 1 and S 2 disposed as in figure 4. We assume that S 1 is a linear spring (it obeys Hooke's law) and S 2 is nonlinear (we assume that the elastic force is a strictly increasing odd function φ of the displacement). Moreover, P 1 is subject to friction and is attached to O through an actuator that displaces periodically the leftmost extremum of the spring S 1 by an amount δ(t). In (5.3), α > 0 is the friction coefficient and µ ≥ 0 is a parameter used to control the stiffness of S 2 .
Let us now see how Theorem 2.2 can be used to get information about the set of triples (µ, x, y) ∈ (0, ∞) × C 1 T (R 2 ), with (x, y) a solution of (5.3) corresponding to µ. Here by C 1 T (R 2 ) we mean the Banach space of all the C 1 and T -periodic functions ζ : R → R 2 endowed with the standard C 1 norm.