The coincidence degree of some functional

differential operators in spaces of periodic

functions and related continuation theorems∗

A. Capietto J. Mawhin F. Zanolin

To Ken Cooke, who witnessed and encouraged the development of earliercontinuation theorems.

1 Introduction

In a recent paper [CMZ], we have shown that if Ω is an open bounded set of thespace Cω of continuous and ω-periodic functions with values in R

m such thatthe autonomous equation

x′(t) − f(x(t)) = 0, (1)

with f : Rm → R

m continuous, has no ω-periodic solution on ∂Ω, then thecoincidence degree of the operator in Cω associated to the left-hand member of(1) is equal to (−1)m times the Brouwer degree of f with respect to Ω ∩ R

m.Of course, we identify here R

m with the space of constant mappings from R toR

m.In other words, this coincidence degree (which is essentially a Leray-Schauder

degree of an equivalent fixed point operator, see [M1, M2] ), is “blind” to thenon-trivial ω-periodic solutions and depends only on the equilibria of (1). Thepositive aspect of this result is that this degree is rather easy to compute andprovides then effective continuation theorems for the existence of ω-periodicsolutions of nonautonomous differential equations of the form

x′(t) − f(x(t)) = e(t, x)

with e “small” or having restricted growth, via the homotopy

x′(t) − f(x(t)) = λe(t, x), λ ∈ [0, 1].

∗Delay differential equations and dynamical systems (Claremont, 1990), Busenberg andMartelli eds, Springer, Berlin, 1991n 76–87

1

The proof of the mentioned result depends upon the Kupka-Smale approxima-tion theorem [K, S] for the closed orbits of autonomous systems and on somedelicate degree computations. The aim of this paper consists in stating and

proving the corresponding result for the autonomous retarded functional differ-ential equation (RFDE)

x′(t) − f(xt) = 0, (2)

where f : Cr → Rm is continuous and takes bounded sets into bounded sets,

Cr = C ([−r, 0], Rm) and, for each t, xt is the element of Cr defined by xt(θ) =x(t + θ), θ ∈ [−r, 0] (see [H] for these notations and a thorough treatment ofRFDE). If Ω is like above, it is well known [M1, M3] that the coincidence degreeof the mapping in Cω defined by the left-hand member of (2) exists and we shallprove in Theorem 1 that it is again equal, up to a factor (−1)m, to the Brouwerdegree of the restriction of f to Ω ∩ R

m.A basic ingredient in the proof will be an extension to RFDE of the Kupka-Smale theorem due to Mallet-Paret [MP] and following earlier generic resultsfor fixed points of a RFDE defined on a compact manifold due to Oliva [O]. Seealso interesting remarks in [CM] and [HMO].Although our proof will follow the same main lines as the one given in [CMZ] forthe ordinary differential equation case, the different nature of (2) will require atvarious stages nontrivial variants of the arguments and even completely differentones due in particular to the fact that time-scaling involve modifications of thedelay in a RFDE. We then relate the coincidence degree associated to a RFDE

of the formx′(t) − h(t, xt) = 0, t ∈ R

with h ω-periodic in t and positively homogeneous of degree α 6= 1 in itssecond variable, to that associated to the autonomous RFDE

x′(t) − h(xt) = 0, t ∈ R

with

h(ϕ) = ω−1

∫ ω

0

h(s, ϕ)ds

and we give related existence theorems. We finally state and prove a continua-

tion thorem for the ω-periodic solutions of non-autonomous RFDE

x′(t) = F (t, xt), t ∈ R

based upon the previous degree calculations.

2

2 A formula for the coincidence

degree of an autonomous RFDE in the space

of ω-periodic functions

Let Cr = C ([−r, 0], Rm) , (r ≥ 0) and let f : Cr → Rm be continuous and such

that it takes bounded sets into bounded sets. We consider the

ω − periodic solutions ( ω > 0 fixed )

of the corresponding RFDEx′(t) = f(xt), (3)

i.e., the functions x : R → Rm of class C1 such that

x(t + ω) = x(t), t ∈ R

which satisfy (3) on R. We denote by Cω the Banach space of continuous ω-periodic functions x : R → R

m with the uniform norm ‖ x ‖= maxt∈R

|x(t)|.

If we define L : D(L) ⊂ Cω → Cω by D(L) = x ∈ Cω : x is of class C1 andLx = x′, and F : Cω → Cω by F (x)(t) = f(xt), t ∈ R (Nemitzky operator),then it is well known [M1, M3] that L is a Fredholm operator of index zero, Fis L-completely continuous on Cω and the existence of ω-periodic solutions of(3) is equivalent to the abstract equation

Lx = Fx, x ∈ D(L).

Moreover, if Ω ⊂ Cω is an open bounded set such that

Lx 6= Fx, x ∈ D(L) ∩ ∂Ω,

then the coincidence degree DL(L − F, Ω) is defined as the Leray-Schauder de-gree of an associated fixed point problem.We denote by dB the Brouwer degree of a mapping from R

m into Rm (see, e.g.

[M1, M2] for details).

Theorem 1 . Assume that Ω ⊂ Cω is an open bounded set such that there isno x ∈ ∂Ω such that x′(t) = f(xt), t ∈ R. Then

DL(L − F, Ω) = (−1)mdB (f |Rm , Ω ∩ Rm, 0) .

Proof. First of all, we observe that, as Ω is bounded, there is a constant R > 0such that ‖x‖ < R for every x ∈ clΩ. Furthermore, we point out that theassumption is equivalent to

Lx 6= Fx, (4)

for all x ∈ D(L) ∩ ∂Ω; therefore, the coincidence degree DL(L − F, Ω) is welldefined, and as we also have

f(c) 6= 0

3

for all c ∈ Rm ∩ ∂Ω, the Brouwer degree dB (f |Rm , Ω ∩ R

m, 0) is defined as well.The proof is performed by means of Mallet-Paret’s extension of the Kupka-Smale’s theorem [MP]; this result ensures the existence of a sequence of C1-functions (ϕk), ϕk : Cr → R

m, taking bounded sets into bounded sets, and suchthat

(a) (ϕk) → f0 uniformly on closed bounded sets

(b) for every closed bounded subset B of Cr and for all k ∈ N, the equation

x′ = ϕk(xt)

has finitely many singular orbits in Cr (i.e., rest points and closed orbits)with minimal period in [0, ω + 1] which are contained in B and they areall hyperbolic.

Let Nk,µ be the Nemitzky operator induced by the functions x 7−→ µf(x·) +(1−µ)ϕk(x·), µ ∈ [0, 1]. We claim that there is k0 > 0 such that, for all k ≥ k0

and for all µ ∈ [0, 1],

Lx 6= Nk,µx for all x ∈ D(L) ∩ ∂Ω. (5)

This fact will imply, in particular, that

ϕk(z) 6= 0 for all z ∈ ∂Ω ∩ Rm, k ≥ k0.

Then, a classical compactness argument ensures that, for any k ≥ k0, there isδ1 = δ1(k) such that

ϕk(y) 6= 0 for all y ∈ B(∂Ω ∩ Rm, δ1).

To obtain (5), it is sufficient to observe that the sequence of operators Nk,µ

converges, as k → +∞, to F in Cω uniformly on clΩ × [0, 1] and that, by (4),

inf ‖ (L − F )x ‖: x ∈ D(L) ∩ ∂Ω > 0.

Hence, the claim is proved and, using the homotopy property of the coincidencedegree (see [M1 , p. 15]), we can write

DL(L − F, Ω) = DL(L − Nk,1, Ω) = DL(L − Nk,0, Ω),

for every k ≥ k0 and, in particular,

dB(f |Rm , Ω ∩ Rm, 0) = dB(ϕk|Rm , Ω ∩ R

m, 0),

for every k ≥ k0 .Let us fix k∗ ≥ k0 . For brevity, we set

ϕ := ϕk∗ , Nϕ := Nk∗,0 , δ1 := δ1(k∗).

4

Consider the singular orbits (i.e. rest points and closed orbits) with minimalperiod in [0, ω + 1] of the equation

x′ = ϕ(xt). (6)

By the Kupka-Smale’s property, there exist finitely many such orbits which arecontained in B(0, R) ⊂ Cr and they are hyperbolic. Recal that for a rest point,this means that the spectrum of the infinitesimal generator of its linearizedequation contains no purely imaginary values and, for a nonconstant periodicsolution, this means that the characteristic multiplier µ = 1 of the linearizedequation is simple and no other characteristic multiplier satisfies |µ| = 1 (see[H]). We denote these orbits by S1, ..., Sn. They are mutually disjoint (two orbitsof (6) may cross in Cr because uniqueness of the Cauchy problem only holds inthe future, but this may no happen to closed orbits). Pick, for each i = 1, ..., na point (in Cr ) φi ∈ Si. Then, φi is a periodic point (possibly a rest point).We can assume that φi is a rest point ζi for 1 ≤ i ≤ p (p ≥ 0 an integer)and a periodic point for p + 1 ≤ i ≤ n. We denote its minimal period by Ti

(p + 1 ≤ i ≤ n). We can also assume that Ti ≤ ω for p + 1 ≤ i ≤ q andω < Ti ≤ ω + 1 for q + 1 ≤ i ≤ n. We denote by ki the largest integer such thatkiTi ≤ ω (p + 1 ≤ i ≤ q), so that (ki + 1)Ti > ω (p + 1 ≤ i ≤ q). We denoteby xi(·) the solution of (6) with xi

0 = φi (p + 1 ≤ i ≤ n).We claim that for each ω′ such that

ω < ω′ < min (kp+1 + 1)Tp+1, ..., (kq + 1)Tq,

Tq+1, ..., Tn, ω + 1 := τ ,

the problemx′ = ϕ(xt), x(t + ω′) = x(ω′) (7)

has no solution x(·), with xt ∈ B(0, R), and hence |x(t)| < R, for all t, otherthan the equilibria ζ1, ..., ζp. Indeed, if x satisfies (7) and xt ∈ B(0, R), for all

t, then S = xt : t ∈ R is a singular orbit of (6) contained in B(0, R). If it isnot a rest point, then S = Si for some p + 1 ≤ i ≤ n and hence there existsti ∈ R such that

xt = xit+ti

, t ∈ R

(indeed ti is such that x−ti= φi = xi

0 ).In particular,

xi(ω′ + ti) = xi(ti).

This is impossible for q+1 ≤ i ≤ n as then ω′ < Ti and Ti is the smallest period.This is impossible for p + 1 ≤ i ≤ q as in this case kiTi < ω′ < (ki + 1)Ti .Therefore the claim is proved. Now, the solutions of (7) correspond, by the

transformation

y(t) = x

(

ω′

ωt

)

, t ∈ R,

5

to the solutions of the problem

y′(t) =ω′

ωϕ

(

yt

( ω

ω′(·)

))

, y(t + ω) = y(t). (8)

Thus, problem (8) has, by construction, no nontrivial (i.e. non-equilibrium)solution on clΩ and, by assumption, no rest point on ∂Ω (as its rest points arethe same as those of (6) and all its possible solutions in B(0, R) are rest points).Defining

Φ : Cω × [ω, τ [→ Cω ,

by

Φ(y, ω′)(t) =ω′

ωϕ

(

yt

( ω

ω′(·)

))

, t ∈ R,

we shall show that Φ is continuous so that we can apply a homotopy argument.If ε > 0, y ∈ Cω , ω′ ∈ [ω, τ [ are given, then as

(

yt

(

ωω′

(·)))

:t ∈ [0, ω] is compact, there exists δ > 0 such that

∣

∣

∣ϕ(φ) − ϕ

(

yt

( ω

ω′(·)

))∣

∣

∣≤ ε

whenever t ∈ [0, ω] (and hence whenever t ∈ R ) and

∣

∣

∣φ − yt

( ω

ω′(·)

)∣

∣

∣

Cr

≤ δ.

Now, y being uniformly continuous, there exists δ′ > 0, such that

|y(t′) − y(t′′)| ≤δ

2

when |t′ − t′′| ≤ δ′ and hence if ω < ω′′ < τ and ω′′ ≥ ω′/2, we shall have

∣

∣

∣y

(

t +ω

ω′′(θ)

)

− y(

t +ω

ω′(θ)

)∣

∣

∣≤

δ

2

when∣

∣

∣

ω

ω′′−

ω

ω′

∣

∣

∣|θ| ≤ δ′

which will be the case if

|ω′′ − ω′| ≤δ′(ω′)2

2ωr.

Now, if z ∈ Cω is such that ‖ z − y ‖≤ δ/2, we have

∣

∣

∣z

(

t +ω

ω′′(θ)

)

− y(

t +ω

ω′(θ)

)∣

∣

∣

≤∣

∣

∣z

(

t +ω

ω′′(θ)

)

− y(

t +ω

ω′′(θ)

)∣

∣

∣

+∣

∣

∣y

(

t +ω

ω′′(θ)

)

− y(

t +ω

ω′(θ)

)∣

∣

∣

≤‖ z − y ‖ +δ

2≤ δ

6

for all t ∈ R and θ ∈ [−r, 0] and hence∣

∣

∣zt

( ω

ω′′(·)

)

− yt

( ω

ω′(·)

)∣

∣

∣

Cr

≤ δ.

Summarizing, if z ∈ Cω with ‖ z − y ‖≤ δ/2, and if ω′′ ∈ [ω, τ [ with

|ω′′ − ω′| ≤ min

ω′

2,δ′(ω′)2

2ωr

,

we shall have∣

∣

∣ϕ

(

zt

( ω

ω′′(·)

))

− ϕ(

yt

( ω

ω′(·)

))∣

∣

∣≤ ε.

for all t ∈ R and this easily implies the continuity of Φ on Cω × [ω, τ [.Now, as (6) has no solution on ∂Ω, the homotopy invariance of the coincidencedegree implies that

DL

(

L − Nϕ, Ω)

= DL

(

L − Φ(·, ω′), Ω)

(9)

for all ω ≤ ω′ < τ.Thus, by excision,

DL

(

L − Φ(·, ω′), Ω)

=∑

1≤j≤pζj∈Ω

DL

(

L − Φ(·, ω′), B(ζj , δ))

, (10)

where

δ = min

δ1, η/2

, η = min

d(Si, Sj) : 1 ≤ i 6= j ≤ n

.

But, by the choice of δ, L − Nϕ has no solution on ∂B(ζj , δ) and hence, byhomotopy invariance again we have, for all ω ≤ ω′ < τ,

DL

(

L − Φ(·, ω′), B(ζj , δ))

= DL

(

L − Nϕ, B(ζj , δ))

(11)

1 ≤ j ≤ p, ζj ∈ Ω.

As ϕ is of class C1, the same is true for Nϕ and hence, by the linearizationproperty of the degree (see e.g. [M1, Prop. VIII.3]),

DL

(

L − Nϕ, B(ζj , δ))

= DL

(

L − N ′ϕ(ζi), B(0, 1)

)

, (12)

1 ≤ j ≤ p, ζj ∈ Ω.

where N ′ϕ(ζi) has the form

N ′ϕ(ζi)φ = ϕ′(ζi)φ =

∫ 0

−r

φ(θ)dηj(θ)

for some function ηj whose elements are of bounded variation ([H]). Moreover,the hyperbolicity of ζj implies that the corresponding characteristic equation

det∆j(µ) = 0,

7

where

det∆j(µ) = µI −

∫ 0

−r

eµθdηj(θ),

has all its roots with nonzero real part (see e.g. [H] ).Consequently, the same is true for the characteristic equation of the equationsin the family

x′(t) = λϕ′(ζj)xt , λ ∈]0, 1]

which therefore only admit the trivial ω-periodic solution. A standard argument(see e.g. [M1, Th. IV.12] ) then shows that the same is true for the family

x′(t) = (1 − λ)ϕ′(ζj)x + λϕ′(ζj)xt , λ ∈ [0, 1]

where

x = ω−1

∫ ω

0

x(s)ds = Px.

Consequently, by the homotopy invariance, and a classical property of coinci-dence degree (see e.g. [M1, Prop. II.12] ),

DL

(

L − N ′ϕ(ζi), B(0, 1)

)

= DL

(

L − N ′ϕ(ζi)P, B(0, 1)

)

= (−1)mdB (ϕ′(ζi)|Rm , B(0, 1) ∩ Rm, 0)

= (−1)mdB (ϕ|Rm , B(ζi, δ) ∩ Rm, 0) , 1 ≤ j ≤ p, ζj ∈ Ω.

Therefore, combining (9), (10), (11), (12) and (13), we obtain

DL

(

L − Nϕ, Ω)

=∑

1≤j≤pζj∈Ω

(−1)mdB (ϕ|Rm , B(ζj , δ) ∩ Rm, 0)

= (−1)mdB (ϕ|Rm , Ω ∩ Rm, 0) ,

and the proof is complete.

3 A formula for the coincidence degree of non-

linear homogeneous nonautonomous RFDE in

the space of ω-periodic functions

We consider the RFDE

x′(t) = h(t, xt) + δ(α)p(t) (14)

where h: R×Cr → Rm, (t, ϕ) 7→ h(t, ϕ) is a continuous mapping, taking bounded

set into bounded sets, ω-periodic in t and positively homogeneous of order α 6= 1in ϕ, δ(α) = max(0, (1−α)/|1−α|) and p ∈ Cω. We define the averaged vectorfield h: Cr → R

m by

h(ϕ) = (1/ω)

∫ ω

0

h(s, ϕ)ds

8

and the corresponding Nemitzky mappings H : Cω → Cω and H : Cω → Cω by

H(x)(t) = h(t, xt), H(x)(t) = h(xt),

for all t ∈ R.

Theorem 2 Assume that h(z) 6= 0 for |z| = 1 in Rm. Then there exists r0 > 0

such that, if α < 1 and r ≥ r0 or α > 1 and 0 < r ≤ r0, (14) has no ω-periodicsolutions x with ‖x‖ = r and

DL(L − H − δ(α)p, B(0, r)) = DL(L − H, B(0, r)).

Proof. Let us define H: Cω × [0, 1] → Cω by

H(x, λ) = (1 − λ)H(x) + λ(H(x) + δ(α)p).

We have only to show that there is some r0 > 0 such that for each r ≥ r0 ifα < 1 or each 0 < r ≤ r0 if α > 1, and for each λ ∈ [0, 1], the equation

Lx = H(x; λ)

has no solution x with ‖x‖ = r.If this is not the case, there are sequences (rk) in R+, (xk) in Cω and (λk) in[0, 1] such that ‖xk‖ = rk, rk ≤ 1/k if α > 1, rk ≥ k if α < 1 and

x′k(t) = (1 − λk)h((xk)t) + λk[h(t, (xk)t) + δ(α)p(t)]

(k ∈ N). Letting uk = xk/‖xk‖ = r−1

k xk, we get

u′k(t) = rα−1

k [(1 − λk)h((uk)t) + λk[h(t, (uk)t)]

+r−1

k λkδ(α)p(t), (15)

so that|u′

k(t)| ≤ rα−1

k β + γ

for some β, γ > 0 and all t ∈ R. Consequently there are subsequences (λjk),

(ujk) and λ∗ ∈ [0, 1], v ∈ Cω, ‖v‖ = 1 such that (ujk

) → v uniformly on R and(λjk

) → λ∗.From

uk(t) − uk(0) = rα−1

k

∫ t

0

[(1 − λk)h(uk)s)+

+λkh(s, (uk)s)]ds + r−1

k

∫ t

0

δ(α)p(s)ds,

we getv(t) − v(0) = 0, t ∈ R,

so that v is constant and ‖v‖ = 1. From (15) we also get

0 =

∫ 0

0

[(1 − λk)h((uk)s) + λkh(s, (uk)s) + r−αk δ(α)λkp(s)]ds = 0

9

and hence, letting jk → ∞,0 = ωh(v),

a contradiction. Hence, for 0 < r ≤ r0 if α > 1 and r ≥ r0 if α < 1, we have

DL(L − H − δ(α)p, B(0, r)) = DL(L −H(·, 1), B(0, r)) =

= DL(L −H(·, 0), B(0, r)) = DL(L − H, B(0, r)),

and the proof is complete.

By using Theorem 1 and 2, the existence property of degree and its invariancefor sufficiently small perturbations of the nonlinear term we immediately deducethe following existence results.

Corollary 1 1 Assume that h satisfies the assumptions of Theorem 2 and that

dB(h|Rm , B(0, 1) ∩ Rm, 0) 6= 0.

Then, if α < 1, the RFDE

x′(t) = h(t, xt) + p(t) (16)

has at least one ω-periodic solution for each p ∈ Cω and, if α > 1, there existsε0 > 0 such that (16) has at least one ω-periodic solution for each p ∈ Cω with‖p‖ ≤ ε0.

4 Continuation theorems for ω-periodic soluti-

ons of nonautonomous RFDE.

Let F : R×Cr → Rm, (t, ϕ) 7→ F (t, ϕ) be a continuous mapping, taking bounded

sets into bounded sets and such that

F (t + ω, ϕ) = F (t, ϕ)

for some ω > 0 and all t ∈ R and ϕ ∈ Cr. We consider the existence of ω-periodicsolutions of the RFDE

x′(t) = F (t, xt), t ∈ R, (17)

i.e. of solutions x such that

x(t) = x(t + ω), t ∈ R. (18)

As it is the case in any continuation theorem, we introduce a mapping f : R ×Cr × [0, 1] → R

m which is continuous, takes bounded sets into bounded sets andis such that

f(t + ω, ϕ, λ) = f(t, ϕ, λ)

for all t ∈ R, ϕ ∈ Cr, λ ∈ [0, 1],

f(t, ϕ, 0) = f0(ϕ)

10

for all t ∈ R, ϕ ∈ Cr (i.e. f(·, ·, 0) is autonomous), and

f(t, ϕ, 1) = F (t, ϕ)

for all t ∈ R, ϕ ∈ Cr.

Theorem 3 Let Ω ⊂ Cω be an open bounded set such that the following condi-tions are satisfied:

(p1) there is no x ∈ ∂Ω such that

x′(t) = f(t, xt, λ), t ∈ R, λ ∈ [0, 1); (19)

(p2) dB(f0|Rm , Ω ∩ Rm, 0) 6= 0.

Then (17)-(18) have at least one solution x ∈ clΩ.

Proof. We use the framework of coincidence degree as in Theorem 1. Theclassical Leray-Schauder continuation theorem [LS] could be used instead bythe equivalence stated at the beginning of Section 2. Besides the spaces andoperators considered there, we further define M : = M(x; λ): Cω × [0, 1] → Cω:

M(x; λ)(t) = f(t, xt; λ).

Observe that M(·; 0) = M0 where

M0: Cω → Cω, x 7→ f0(x.).

According to [M1, Chapter I], M is L-compact on clΩ× [0, 1]. We remark that xis an ω-periodic solution of x′(t) = f(t, xt; λ), λ ∈ [0, 1], if and only if x ∈ D(L)is a solution of the coincidence equation Lx = M(x; λ), λ ∈ [0, 1]. In particular,(17)-(18) is equivalent to Lx = M(x; 1). Without loss of generality, we supposethat (p1) holds for λ ∈ [0, 1] in (19). Otherwise, the result is proved for x ∈ ∂Ω.Accordingly, by the definition of M(·, λ) and using (p1) we have:

Lx 6= M(x; λ), λ ∈ [0, 1]

for all x ∈ D(L) ∩ ∂Ω. Thus we can apply the homotopy property of thecoincidence degree and obtain:

DL(L − M0, Ω) = DL(L − M(·; 0), Ω) = DL(L − M(·; 1), Ω). (20)

Assumption (p1) (for λ = 0) ensures that Theorem 1 can be applied, so that(20) and (p2) imply:

|DL(L − M0, Ω)| = (dB(f0|Rm , Ω ∩ Rm, 0)| 6= 0.

Hence, by the existence property of the coincidence degree, there is x ∈ D(L)∩Ωsuch that Lx = M(x; 1); thus x(·) is a solution to (17)-(18), with x ∈ D(L)∩Ω.The proof is complete.

11

Theorem 3 is particularly suitable for the study of ω-periodic solutions of per-turbed autonomous RFDE of the form

x′(t) = f(xt) + e(t), t ∈ R,

with e ∈ Cω, through the homotopy

x′(t) = f(xt) + λe(t), t ∈ R, λ ∈ [0, 1].

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[M2] J. Mawhin, “Points fixes, points critiques et problemes aux limites”,Seminaire de Mathematiques Superieures, vol. 92, Les Presses del’Universite de Montreal, 1985.

[O] W.M. Oliva, Functional differential equations on compact manifolds andan approximation theorem, J. Differential Equations, 5 (1969), 483-496.

[S] S.Smale, Stable manifolds for differential equations and diffeomorphisms,Ann. Scuola Norm. Sup. Pisa, 17 (1963), 97-116.

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International School for Advanced Studies, Strada Costiera 11, 34014 Trieste, Italy

Universite de Louvain, Institut Mathematique, Chemin du Cyclotron 2, B-1348 Louvain-

la-Neuve, Belgium

Department of Mathematics and Computer Sciences, Via Zanon 6, 33100 Udine, Italy

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