A Ducrot

M Marion

V Volpert

Spectrum of Some Integro-Differential Operators and Stability of Travelling Waves

Keywords: Spectral properties, nonlocal elliptic operator, travelling waves, stability. AMS subject classification: 34K08, 35K57

Spectral properties of some integro-differential operators on R 1 are studied. Characterization of the principal eigenvalue is obtained in terms of the positive eigenfunction. These results are used to prove local and global stability of travelling waves and to find their speed.

Introduction

In this paper we shall study some spectral properties of the linear integro-differential operator

Lu = u + cu + a(x)φ * u + b(x)u, ( 1.1)
acting from the Banach space E = C 2 0 (R) into the Banach space F = C 0 (R). Here C 0 (R) denotes the space of continuous functions from R into itself that tend to zero at x = ±∞ while E is the space of functions u of class C 2 such that u, u and u belong to F . The functions x → a(x) and x → b(x) are supposed to be bounded and continuous with the limits a ± and b ± when x → ±∞, respectively, while φ * u denotes the convolution product,

φ * u(x) = ∞ -∞ φ(x -y)u(y)dy x ∈ R,
where φ(x) is some integrable function.

The spectrum of the operator L consists of its essential part and of eigenvalues. Here and in the sequel, the essential spectrum is understood as the Fredholm spectrum, namely the set of complex λ for which the operator L -λ does not satisfy the Fredholm property.

On the one hand, the location of the essential spectrum is an important point when dealing with some nonlinear problems. Indeed, most of the tools of nonlinear analysis, such as the implicit function theorem, bifurcation analysis or topological degree, are based on the Fredholm property and on the related to it classical solvability conditions. Coming back to operator, the location of its essential spectrum is related to the so-called limiting operators

L ± u = u + cu + a ± φ * u + b ± u,
which are obtained from the operator L if we replace the coefficients a(x) and b(x) by their limits at infinity. The notion of limiting operators is well known in the context of elliptic operators. They were introduced for elliptic problems in general unbounded domains in [START_REF] Volpert | Normal solvability of general linear elliptic problems[END_REF] where the assumption that the coefficient have limits at infinity or that the domain has a special structure were not employed. In the particular situation considered in this work, the definition of the limiting operator is simple and rather explicit. The relationship between the limiting operators and the Fredholm spectrum was elucidated in [START_REF] Apreutesei | Travelling Waves for Integro-differential Equations[END_REF][START_REF] Apreutesei | Competition of species with intra-specific competition[END_REF]. It was shown that the operators L ± determine the set of λ ∈ C such that the operator L -λ is normally solvable, which is equivalent to the closeness of its image with a finite dimensional kernel. Under some additional conditions, the codimension of the image is also finite, leading to the Fredholm property. Hence we know the structure of the essential spectrum σ ess (L) of operator L.

On the other hand, similarly to elliptic operators in unbounded domains (see [START_REF] Volpert | Fredholm property of elliptic operators in unbounded domains[END_REF]), the spectrum of the operator L lies in some left-half plane. The Fredholm property ensures that the set of eigenvalues cannot have accumulation points in some finite part of the complex plane outside of the essential spectrum. Therefore, for any σ 0 > sup Re σ ess (L), there exists only a finite number of eigenvalues of the operator L in the half-plane Re λ ≥ σ 0 . Such a distribution of the eigenvalues of the operator L allows us to define the principal eigenvalue, that is the eigenvalue with the maximal real part. It plays an important role in the analysis of stability of solutions of nonlinear problems. In the case of scalar second order elliptic problems in bounded domains, the Krein-Rutman type theorems affirm that the principal eigenvalue is real, simple and the corresponding eigenfunction is positive. These results use the cone preserving property as well as the compactness of the inverse Laplace operator. The situation becomes more complex when dealing with unbounded domains because of the lack of compactness for the resolvent of the Laplace operator. However, this Perron-Frobenius like property of the principal eigenvalue remains valid (we refer to [START_REF] Volpert | Spectrum of elliptic operators and stability of travelling waves[END_REF]).

The aim of this work is to study the properties of the principal eigenvalue of the integrodifferential operator L when functions a(x) and φ(x) are non-negative. Then the corresponding evolution equations satisfy the maximum and comparison principles. They will allow us to prove that the principal eigenvalue of the operator is characterized by positiveness of the corresponding eigenfunction. This is the main result of the work (see Theorem 8). Note that such a property does not hold true without the conditions on the functions a and φ. Indeed, otherwise nonmonotone waves may exist and be numerically stable (see for instance [START_REF] Apreutesei | Spatial structures and generalized travelling waves for an integro-differential equation[END_REF][START_REF] Berestycki | The non-local Fisher-KPP equation: traveling waves and steady states[END_REF][START_REF] Genieys | Pattern and waves for a model in population dynamics with nonlocal consumption of resources[END_REF][START_REF] Gourley | Travelling front solutions of a nonlocal Fisher equation[END_REF]) signifying that the eigenfunction corresponding to the principal eigenvalue, which is the first derivative of the wave, is not positive.

These spectral properties have strong implications in nonlinear analysis and more particularly when looking at the stability of monotone travelling wave for some nonlocal parabolic problem of the form ∂u

(x, t) ∂t = ∂ 2 u(x, t) ∂x 2 + f (u(x, t), J(u)(x, t)) , t > 0, x ∈ R, (1.2)
wherein we have set

J(u)(x, t) = R φ(x -y)u(y, t)dy,
and where f = f (u, v) is some regular map. This kind of equations arises in many applications (see for instance [START_REF] Demin | Existence of waves for a nonlocal reaction-diffusion equation[END_REF][START_REF] Ma | Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem[END_REF][START_REF] Ma | Traveling waves for non-local delayed diffusion equations via auxiliary equations[END_REF] and the references cited therein).

Let us recall that a travelling wave for (1.2) is particular solution of the form u(x, t) = w(x-ct) where c ∈ R is the wave speed, so that w is a stationary solution of the problem

∂u(x, t) ∂t = ∂ 2 u(x, t) ∂x 2 + c ∂u(x, t) ∂x + f (u(x, t), J(u)(x, t)) , t > 0, x ∈ R.
The linear stability of w is related to the spectral properties of operator L with

a(x) = ∂f (w(x), J(w)(x)) ∂v , b(x) = ∂f (w(x), J(w)(x)) ∂u .
If we assume that the map f : R 2 → R is of the class C 1 and satisfies

∂f (u, v) ∂v ≥ 0, ∀(u, v) ∈ R 2 ,
we obtain, as a consequence of the spectral theory developed in this work, that the wave solution w is globally stability with shift in a scale of weighted spaces depending on the asymptotic behaviour at x = ±∞ of the initial data as well as on the location of the essential spectrum of operator L. This kind of result is well known for reaction-diffusion equations (see [START_REF] Volpert | Travelling wave solutions of parabolic systems[END_REF] and the references therein).

A result concerning the stability of waves for (1.2) was proved by Chen in [START_REF] Chen | Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations[END_REF] by using upper and lower-solution method, also well known for the scalar reaction-diffusion equations (see [START_REF] Fife | The approach of solutions of nonlinear diffusion equations to traveling wave solutions[END_REF]). The advantage of the spectral method is that it provides a more precise stability result depending on the properties of the initial data. It also provides a minimax representation of the wave speed. One may also notice that this method applies in the multidimensional situation and has rather simple extension to systems of equations. This point will be the aim of a forthcoming work. Let us mention that these results may also apply to reaction-diffusion equations with delay. The paper is organized as follow. In Section 2 we prove some comparison results. Section 3 investigates the location of the spectrum of operator L. Section 4 derives some consequences for the local stability of waves for problem (1.2) while Section 5 provides global stability result and the minimax representation of the wave speed. Finally Section 6 gives some examples of application in the context of population dynamics and chemical kinetics.

Positiveness and comparison theorems 2.1. Existence of solutions

Consider the equation involving a scalar function u = u(x, t), x ∈ R, t ≥ 0

∂u ∂t = ∂ 2 u ∂x 2 + f (u, J(u)) , ( 2.1)
together with the initial condition

u(x, 0) = u 0 (x), x ∈ R. (2.2)
Here the function

f is of class C 1 from R 2 into R with |f (u, v)| ≤ κ(1 + |u| + |v|), ∀(u, v) ∈ R 2 , ( 2.3)
where herefater κ denotes some positive constant. The term J(v) acts on a bounded function v = v(y), y ∈ R and is defined by the convolution product

J(v)(x) = φ * v(x) = ∞ -∞ φ(x -y)v(y)dy, ∀x ∈ R, (2.4)
where φ(x) is a given integrable function.

We first investigate the existence of solutions of this problem.

(R). For v ∈ L ∞ (R) we set w(x) = f v(x), ∞ -∞ φ(x -y)v(y)dy .
Clearly w ∈ L ∞ (R) and let F denote the mapping v → w.

Lemma 2. The mapping F is Lipschitz continuous on the bounded sets of L ∞ (R). Furthermore there exists a constant κ such that

F (v) ∞ ≤ κ v ∞ for all v ∈ L ∞ (R).
Next let S(t) denote the semi-group associated to the heat equation. It is well known that the following estimate holds :

S(t)u 0 ∞ ≤ u 0 ∞ , ∀t > 0 and u 0 ∈ L ∞ (R).
For u = u(x, t), let u(t) be the function x → u(x, t). Then the solution u of (2.1), (2.2) reads

u(t) = S(t)u 0 + t 0 S(t -s)F (u(s))ds. (2.5)
By looking for fixed points of the operator given by the right hand-side of (2.5) in the space L ∞ (R × [0, T ]) with T > 0, we easily conclude that such a fixed point exists first for sufficiently small T and then for all T > 0. This provides the existence and uniqueness result in Theorem 1.

Corollary 3. Under the assumptions of Theorem 1, assume furthermore that the initial data possesses limits at infinity

u 0 ± = lim x→±∞ u 0 (x).
Then the following limits

u ± (t) = lim x→±∞ u(x, t)
also exist and satisfy the equation

du ± (t) dt = f u ± (t) , R φ u ± (t) . (2.6)
Proof. It consists in taking the limits x → ±∞ in the iterative procedure in the proof of Theorem 1. For example we have

S(t)u 0 → u 0 ± , t 0 S(t -s)F (S(s)u 0 )ds → t 0 f u 0 ± , ( ∞ -∞ φ)u 0 ± ds.
The details are left to the reader.

Maximum and comparison principle

This section is devoted to some comparison results for (2.1)-(2.2) as well as for some corresponding linear equations. Let us introduce the operator

L[u] = ∂ 2 u ∂x 2 + c ∂u ∂x + f (u, J(u)) - ∂u ∂t , ( 2.7)
where c ∈ R is given. In addition to (2.3) we will assume that

supp φ ⊂ [-N, N ], for some given N > 0, φ(x) ≥ 0, (2.8
)

∂f (u, v) ∂v ≥ 0, ∀(u, v) ∈ R 2 .
(2.9)

Then the following comparison result holds :

Theorem 4. Under assumptions (2.3), (2.8) and (2.9), let T > 0 be given and let u and v be two

functions in C 0 (R × [0, T ]) C 2,1 (R × (0, T ]) L ∞ (R × [0, T ]) satisfying u(x, 0) ≤ v(x, 0), ∀x ∈ R, L[u](x, t) ≥ L[v](x, t) ∀t ∈ (0, T ), ∀x ∈ R.
(2.10)

Then we have u(x, t) ≤ v(x, t) for any x ∈ R and t ∈ [0, T ].

In order to prove this comparison result, we first investigate some associated linear equation. For that purpose we are given T > 0 and three functions a ≡ a(x, t), b ≡ b(x, t) and c ≡ c(x, t) defined on R × [0, T ]. We consider the operator

L[u] = ∂ 2 u ∂x 2 + c(x, t) ∂u ∂x + a(x, t)J(u) + b(x, t)u - ∂u ∂t .
(2.11)

We will assume that

The functions a, b and c are continuous and bounded on R × [0, T ], a(x, t) ≥ 0, ∀t ∈ (0, T ), ∀x ∈ R.

(2.12)

Then the following positiveness result holds:

Theorem 5. Let assumption (2.12) be satisfied. Let u ∈ C 0 (R × [0, T ]) C 2,1 (R × (0, T ]) satis- fying L[u](x, t) ≤ 0, ∀t ∈ (0, T ], ∀x ∈ R, u(x, 0) ≥ 0, ∀x ∈ R, u(x, t) ≥ -B, ∀t ∈ [0, T ], ∀x ∈ R, (2.13)
for some constant B > 0. Then we have u(x, t) ≥ 0 for any (x, t) ∈ R × [0, T ]. Furthermore the following alternative holds : either u(x, t) ≡ 0 for any

(x, t) ∈ R × [0, T ] or u(x, t) > 0 for any (x, t) ∈ R × (0, T ]. Proof. Let us consider the function v(x, t) = u(x, t) exp(- √ 1 + x 2 -νt) (2.14)
defined for x ∈ R and 0 ≤ t ≤ T where ν is some positive constant to be specified later. Obviously, we have :

v(x, t) ≥ -B exp(- √ 1 + x 2 ).
Therefore the following inequality holds

lim inf |x|→+∞ v(x, t) ≥ 0
uniformly with respect to t ∈ [0, T ] and ν ≥ 0. For convenience let us set θ(x) = √ 1 + x 2 and h(x, t) = exp(θ(x) + νt).

It follows easily from (2.13) and (2.14) that the function v satisfies the differential inequality

L[v] = ∂ 2 v ∂x 2 + c ∂v ∂x + a h J(vh) + bv - ∂v ∂t ≤ 0 (2.15) where c(x, t) = c(x, t) + 2θ (x), b(x, t) = b(x, t) + θ (x) + θ (x) 2 + θ (x)c(x, t) -ν. (2.16)
We claim that we can choose ν sufficiently large so that

L[1] = a h J(h) + b < 0 on R × [0, T ], (2.17)
Indeed, in view of the estimate

a(x, t) h(x, t) J(h)(x, t) ≤ a(x, t) φ ∞ N -N exp (θ(x -y) -θ(x)) dy the function a h J(h) is bounded on R × [0, T ].
Since a, b, c, θ and θ are also bounded the above property easily follows.

Next, for > 0, the function

w = v + satisfies lim inf |x|→+∞ w(x, t) ≥
uniformly with respect to t ∈ [0, T ]. Therefore there exists R > 0 such that w(x, t) > 0, for |x| ≥ R and 0 ≤ t ≤ T.

(2.18)

We aim to derive the positivity of w on [-R, R] × [0, T ]. Note that, thanks to (2.15) and (2.17), w satisfies the differential inequality

L[w] = L[v] + L[1] < 0 on R × (0, T ]. (2.19) Since w(x, 0) = u(x, 0) exp(- √ 1 + x 2 ) + > 0 we necessarily have w(x, t) > 0 on [-R, R] × [0, t * ]
for some t * > 0. Let t 0 denotes the supremum of the t * > 0 having this property. If t 0 < T , the function w(., t 0 ) is positive and vanishes at some point x 0 . Due to (2.18) we have

x 0 ∈] -R, R[ and L[w](x 0 , t 0 ) = ∂ 2 w ∂x 2 (x 0 , t 0 ) + a(x 0 , t 0 ) h(x 0 , t 0 ) J(wh)(x 0 , t 0 ) - ∂w ∂t (x 0 , t 0 ) ≥ 0.

This inequality contradicts (2.19). Hence t

0 = T and w ≥ 0 on [-R, R] × [0, T ].
Thanks to (2.18), we conclude that w ≥ 0 on R × [0, T ]. Since > 0 is arbitrary the positivity of w yields the one of v by taking the limit → 0. By (2.14), the positivity of u follows readily.

Finally note that u satisfies the differential inequality

∂ 2 u ∂x 2 + c(x, t) ∂u ∂x + b(x, t)u - ∂u ∂t ≤ -a(x, t)J(u) ≤ 0,
together with u(x, 0) ≥ 0. Hence classical properties of parabolic operators yield the alternative in Theorem 5 whose proof is now complete.

We now aim to complete the proof of Theorem 4.

Proof of Theorem 4. The function w = v -u satisfies

∂ 2 w ∂x 2 + c ∂w ∂x + f (v, J(v)) -f (u, J(u)) - ∂w ∂t ≤ 0, (2.20
)

together with w(x, 0) ≥ 0. Then f (v, J(v)) -f (u, J(u)) = b(x, t)w + a(x, t)J(w)
for some bounded functions a and b and a(x, t) ≥ 0 due to (2.9). Therefore Theorem 5 applies and provides the positivity of w.

Next let us investigate the maximum principle for some corresponding elliptic operator. In order to state our result, let us introduce two functions a ≡ a(x), b ≡ b(x), a real number c ∈ R and the one-dimensional elliptic operator

Lu = u + cu + a(x)φ * u + b(x)u.
(2.21) Besides (2.8) let us also assume that the functions a and b are continuous and bounded on R,

a(x) ≥ 0 ∀x ∈ R, R φ(t)dt a(x) + b(x) < 0 ∀|x| > N. (2.22)
Then we have the following elliptic comparison result:

Lemma 6. Let conditions (2.8) and (2.22) be satisfied. Let u ∈ C 2 (R) L ∞ (R) be a given function such that u ≥ 0 on [-N, N ], Lu ≤ 0 on R and lim inf |x|→∞ u(x) ≥ 0. ( 2

.23)

Then we have u ≥ 0 on R.

Proof. Let us assume that there exists x 0 ∈ R with |x 0 | > N such that u(x 0 ) < 0. In view of (2.23), one may assume that the function u realizes its minimum value at x 0 . Then, for any y ∈ R, we have u(x 0 -y) ≥ u(x 0 ), so that φ * u(x 0 ) -R φ(t)dt u(x 0 ) ≥ 0. On the other hand we have u (x 0 ) = 0, u (x 0 ) ≥ 0. Thus we get

u (x 0 ) + b(x 0 ) + R φ(t)dt a(x 0 ) u(x 0 ) + a(x 0 ) φ * u - R φ(t)dt u (x 0 ) > 0.
This contradicts Lu(x 0 ) ≤ 0. Thus u(x) ≥ 0 for any x ∈ R and the result follows.

Lemma 7. Let conditions (2.8) and (2.22) be satisfied. Let u ∈ C 2 (R) L ∞ (R) satisfy u ≥ 0 and Lu ≤ 0. Then either u ≡ 0 or u > 0 on R.
Proof. Let us first note that since u ≥ 0 and a ≥ 0 one has a(x

)φ * u(x) ≥ 0, ∀x ∈ R. Hence the inequality Lu ≤ 0 implies that u + cu + b(x)u ≤ 0,
and the result follows from classical properties of elliptic operators.

Spectral properties

The aim of this section is to study some spectral properties of a linear elliptic operator with a nonlocal term. The operator reads

Lu = u + cu + a(x)φ * u + b(x)u, ( 3.1)
and it is considered as acting from the Banach space E = C 2 0 (R) into the Banach space F = C 0 (R). Here C 0 (R) denotes the space of continuous functions from R into itself that tend to zero at x = ±∞ while E is the space of functions u of class C 2 such that u, u and u belong to F .

Here c ∈ R is some given constant while a and b are functions from R into itself satisfying a and b are continuous and bounded on R, the following limits

a ± = lim x→±∞ a(x), b ± = lim x→±∞ b(x) exist, a ± + b ± < 0, a(x) ≥ 0 ∀x ∈ R. (3.2) Also φ(x) ≥ 0, ∀x ∈ R, Supp φ ⊂ [-N, N ] for some given N > 0, ∞ -∞ φ(x)dx = 1. (3.3)
In this section we shall assume that the operator L has the zero eigenvalue associated with a positive eigenvector u 0 and we shall show a Perron-Frobenius like result for L. The precise result is the following: Theorem 8. Let assumptions (3.2)-(3.3) be satisfied. Furthermore suppose that there exists some u 0 ∈ E with u 0 > 0 solution of the equation

Lu 0 = 0.
Then zero is a simple eigenvalue of the operator L while all the other eigenvalues lie in the left half-plane {z ∈ C : z < 0}.

Remark 9. A similar result also holds true when the operator L is considered as acting on some Sobolev spaces. The proof in that case can be obtained by a slight modification of the one given below.

We start by deriving some asymptotic properties of the eigenvector u 0 (x) as x → ±∞. For that purpose let us introduce the so-called limiting operators defined by

L ± u = u + cu + a ± φ * u + b ± u,
where the coefficients a(x) and b(x) have been replaced by their limits at ±∞. The maps x → exp(-α ± x) are solutions of the equations L ± u = 0 provided that

α +2 -cα + + a + ∞ -∞ φ(z)e α + z dz + b + = 0 α -2 + cα -+ a - ∞ -∞ φ(z)e -α -z dz + b -= 0. (3.4)
The existence of such solutions will be guaranteed by the following lemma : Lemma 10. Under assumption (3.3), let δ and γ be two given real numbers with δ ≥ 0, δ + γ < 0.

Then the equations

α 2 ± cα + δ ∞ -∞ φ(z)e ∓αz dz + γ = 0, (3.5)
have respectively a unique solution α ± > 0. Moreover we have

0 < α ± ≤ c + c 2 + 4|γ| 2 . (3.6)
Proof. We shall only consider the equation

α 2 -cα + δ ∞ -∞ φ(z)e αz dz + γ = 0. (3.7) Let h : [0, ∞) → R be defined by h(α) = α 2 -cα + δ ∞ -∞ φ(z)e αz dz + γ.
In view of (3.3), h is of class C ∞ with h(0) = δ + γ < 0 and h(α) ≥ γ -cα + α 2 . These properties guarantee the existence of α + > 0 such that h(α + ) = 0 and provide the upper bound (3.6). Finally, due to

h (α) = 2 + δ ∞ -∞ z 2 φ(z)e αz dz > 0,
the function h is convex. Consequently, since h(0) < 0, the equation h(α) = 0 has a unique positive solution. This completes the proof of the lemma.

Choosing δ = a ± and γ = b ± , Lemma 10 applies thanks to assumption (3.2) and provides the existence of the two unique numbers α ± > 0 satisfying (3.4).

We can now state the following result on u 0 .

Lemma 11. Under assumptions (3.2)-(3.3), let ε 0 > 0 such that (a ± -ε 0 ) + + b ± -ε 0 < 0, with (a ± -ε 0 ) + = max(a ± -ε 0 , 0).
Then for each ε ∈ (0, ε 0 ) there exists a unique pair

(α + ε , α - ε ) ∈ (0, ∞) 2 satisfying the equations α +2 ε -cα + ε + (a + -ε) + ∞ -∞ φ(z)e α + ε z dz + b + -ε = 0 α -2 ε + cα - ε + (a --ε) + ∞ -∞ φ(z)e -α -ε z dz + b --ε = 0. (3.8)
Moreover we have

lim ε→0 + (α + ε , α - ε ) = (α + , α -), (3.9)
and for each ε ∈ (0, ε 0 ) there exists m ε > 0 such that

lim inf x→∞ e α + ε x u 0 (x) ≥ m ε , lim inf x→-∞ e -α -ε x u 0 (x) ≥ m ε . (3.10)
Proof. Note that the existence of ε 0 > 0 follows from (3.2). For ε ∈ (0, ε 0 ), we have (a

± -ε) + + b ± -ε < 0 and there exists N ε > N such that a(x) ≥ (a + -ε) + , b(x) ≥ b + -ε ∀x ≥ N ε .
Let L ε be the operator defined by

L ε u = u + cu + (a + -ε) + φ * u + (b + -ε)u, u ∈ E.
Since u 0 > 0, we have

L ε u 0 ≤ Lu 0 = 0, ∀x ≥ N ε .
Next, Lemma 10 provides the existence of a unique number α + ε > 0 such that

α + ε 2 -cα + ε + (a + -ε) + ∞ -∞ φ(z)e α + ε z dz + b + -ε = 0. (3.11) The function v ε (x) = u 0 (N ε )e -α + ε (x-Nε) satisfies the equality L ε v ε = 0 together with v ε (N ε ) = u 0 (N ε ). Now since u 0 (x) is a positive function, there exists ν > 0 such that νv ε (x) ≤ u 0 (x), N ε -N ≤ x ≤ N ε .
We aim to show that

νv ε (x) ≤ u 0 (x), ∀x ≥ N ε . (3.12)
Let us consider the function w

(x) = u 0 (x) -νv ε (x). It satisfies L ε w ≤ 0, ∀x ≥ N ε together with w(x) ≥ 0, ∀x ∈ [N ε -N, N ε ].
Suppose that w(x) < 0 for some x > N ε . Since lim x→∞ w(x) = 0, the function w attains its minimum on [N ε , +∞[ at some x 0 > N ε and w(x 0 ) < 0. Setting ŵ(x) = w(x) -w(x 0 ), we have

L ε w = L ε ŵ + L ε w(x 0 ) = L ε ŵ + (a + -ε) + + b + -ε w(x 0 ) ≤ 0, ∀x > N ε .
Let us check that the left hand-side of this inequality is strictly positive at x = x 0 . This contradiction will prove (3.12). Indeed, the last expression in the left-hand side is strictly positive. It remains to verify that L ε ŵ(x 0 ) ≥ 0. This follows from

ŵ (x 0 ) ≥ 0, ŵ (x 0 ) = 0, ŵ(x 0 ) = 0, φ * ŵ(x 0 ) = N -N φ(y) ŵ(x 0 -y)dy ≥ 0.
Here the last inequality holds true since φ ≥ 0 and ŵ(x) ≥ 0 for x > N ε -N . Thus, (3.12) is proved and yields the first inequality in (3.10). The second one can be proved similarly.

There remains to check that α + ε → α + when ε → 0. In view of (3.6), the family α + ε is bounded. Let β ≥ 0 be the limit of some converging subsequence α + ε n . Taking the limit n → ∞ in (3.11) we conclude that β ≥ 0 satisfies the equation

β 2 -cβ + a + ∞ -∞ φ(z)e βz dz + b + = 0.
Therefore due to the uniqueness result in Lemma 10 we conclude that β = α + so that α + ε → α + when ε → 0. The same proof holds for α - ε . This completes the proof of the lemma.

Next we aim to investigate the location as well as some geometrical properties of the essential spectrum of L under suitable transformation in some exponential weighted spaces. For that purpose, let us recall that some bounded linear operator A : E → F is said to be normally solvable (or a semi-Fredholm operator) if it has a finite dimensional kernel and a closed range. A linear operator A satisfies the Fredholm property if it is normally solvable and has a finite codimensional space. Then we can introduce the two following sets, the semi-Fredholm spectrum and the Fredholm spectrum:

σ N S (A) = {λ ∈ C : (A -λ) is not normally solvable}, σ ess (A) = {λ ∈ C : A -λ is not Fredholm}.
Coming back to the operator L, by using some results proved by Apreutesei et al. in [START_REF] Apreutesei | Travelling Waves for Integro-differential Equations[END_REF][START_REF] Apreutesei | Competition of species with intra-specific competition[END_REF], the location of the semi-Fredholm spectrum can be characterized in terms of the limiting operators L ± : Lemma 12. For each λ ∈ C, the operator L-λ has a closed range and a finite dimensional kernel if and only if the equations (L ± -λ)u = 0 have no bounded and nontrivial solution.

From this lemma we can derive the following useful characterization : Lemma 13. The semi-Fredholm spectrum of operator L : E → F is given by

σ N S (L) = {-ξ 2 + iξc + b ± + a ± ∞ -∞ φ(z)e -iξz dz; ξ ∈ R}.
(3.13)

Then we have the following link between the semi-Fredholm spectrum and the Fredholm spectrum: Lemma 14. The following holds:

(i) σ N S (L) ⊂ σ ess (L). (ii) Let λ ∈ C be given. If for any ρ ≥ 0 the operator L -(λ + ρ) is normally sovable then (L -λ) satisfies the Fredholm property.
The proof of this lemma can be found in Theorem 3.2 [START_REF] Apreutesei | Travelling Waves for Integro-differential Equations[END_REF] and in [START_REF] Apreutesei | Competition of species with intra-specific competition[END_REF]. We will investigate the operator L as acting in some exponential weighted spaces. We consider two regular functions θ 1 , θ 2 : R → R such that

θ 1 (x) = x if x ≥ 1 0 if x ≤ 0 θ 2 (x) = x if x ≤ -1 0 if x ≥ 0 . (3.14)
and the weighted function, for each τ ∈ R :

ω τ (x) = e τ θ(x) , x ∈ R, with θ(x) = α + θ 1 (x) -α -θ 2 (x). (3.15)
where α + and α -are given by (3.4). It satisfies

w τ (x) = e τ α + x if x ≥ 1 e -τ α -x if x ≤ -1.
Next we introduce the operator L τ : E → F defined by

L τ v = ω τ L v ω τ . (3.16)
The explicit form of L τ reads

L τ v =v + (c -2τ θ (x))v + {b(x) -cτ θ (x) + τ 2 θ (x) 2 -τ θ (x)}v + a(x) ∞ -∞
φ(y)e τ (θ(x)-θ(x-y)) v(x -y)dy.

(3.17)

The essential spectrum of operator L τ can easily be computed.

Lemma 15. For each τ ∈ R, the essential spectrum of the operator L τ is given by

σ N S (L τ ) = {Φ + (ξ, τ α + ), ξ ∈ R} ∪ {Φ -(ξ, τ α -), ξ ∈ R},
wherein we have set

Φ + (ξ, τ α + ) = -ξ 2 + iξ(c -2τ α + ) + (τ α + ) 2 -cτ α + + a + ∞ -∞ φ(z)e τ α + z-iξz dz + b + Φ -(ξ, τ α -) = -ξ 2 + iξ(c + 2τ α -) + (τ α -) 2 + cτ α -+ a - ∞ -∞ φ(z)e -τ α -z-iξz dz + b -.
(3.18)

Moreover for each λ ∈ C if for each ρ ≥ 0 we have λ + ρ / ∈ σ N S (L τ ) then λ / ∈ σ ess (L τ ).
In particular recalling (3.4), we have

Φ ± (0, α ± ) = 0. (3.19)
so that 0 ∈ σ N S (L 1 ).

We shall now prove some suitable geometrical continuation property for the essential spectrum.

Proposition 16. Under assumptions (3.2)-(3.3), the following properties hold true:

(i) For each τ ∈ [0, 1),

σ ess (L τ ) ⊂ {z ∈ C : z < 0}. (ii) For each λ ∈ {z ∈ C : z ≥ 0, z = 0} and each τ ∈ [0, 1], 0 / ∈ σ ess (L τ -λ).
Remark 17. For each λ ∈ {z ∈ C : z ≥ 0} let the number τ (λ) ≥ 0 be defined by

τ (λ) = inf{τ ≥ 0 : 0 ∈ σ ess (L τ -λ)}. Then τ (0) = 1 while τ (λ) > 1 for each λ ∈ {z ∈ C : z ≥ 0, z = 0}. Indeed the map τ → L τ is continuous from C into L(E, F )

while the set of Fredholm operators is an open subset of L(E, F ).

From a geometrical point of view, (i) and (ii) in Proposition 16 (for τ = 0) means that the essential spectrum of the operator L is in the left half plane. When it is shifted according to the parameter τ , that corresponds to increase the weighted function at infinity, the first imaginary point of the essential spectrum that is reached is the origin and it is obtained for τ = 1. It follows from (3.19) that λ = 0 belongs to the essential spectrum when τ = 1.

In order to explain this result, let us consider the following example. If the function φ in the definition of the operator L is the Dirac δ-function, then the integral disappears and we obtain a second order ordinary differential operator. Its essential spectrum is given by the parabolas λ ± (ξ) = Φ ± (ξ, 0). Assume that they are in the left-half plane of the complex plane. We move these curves to the right while their vertices come to the origin. In this case we obtain the curves λ ± (ξ) = Φ ± (ξ, τ α ± ). Then, the above result follows from the convexity of the essential spectrum for "local" operators.

The proof of Proposition 16 follows from Lemma 15 together with the following result:

Lemma 18. Under assumptions (3.2)-( 3.3), the following properties hold true:

(i) For each τ ∈ [0, 1), σ N S (L τ ) ⊂ {z ∈ C : z < 0}. (ii) For each λ ∈ {z ∈ C : z ≥ 0, z = 0} and each τ ∈ [0, 1], 0 / ∈ σ N S (L τ -λ).
Proof. Coming back to relations (3.18), for any τ ≥ 0 and ξ ∈ R we have

Φ + (ξ, τ α + ) = h + (τ α + ) -ξ 2 + a + ∞ -∞ φ(z)e τ α + z (1 -cos(ξz))dz , Φ -(ξ, τ α -) = h -(τ α -) -ξ 2 + a - ∞ -∞ φ(z)e -τ α -z (1 -cos(ξz))dz , ( 3.20)
wherein the maps h ± are defined by

h + (α) = α 2 -cα + a + ∞ -∞ φ(z)e αz dz + b + h -(β) = β 2 + cβ + a - ∞ -∞ φ(z)e -βz dz + b -. (3.21)
According to the proof of Lemma 10, the maps h ± are convex while by (3.19) we see that

h ± (α ± ) = Φ + (0, α + ) = 0. Consequently the first term in (3.20) satisfies h ± (τ α ± ) < 0 when τ ∈ [0, 1) > 0 when τ > 1
Since the other term in (3.20) is negative, we conclude that

Φ ± (ξ, τ α ± ) < 0 ∀ξ ∈ R, ∀τ ∈ [0, 1), (3.22)
that completes the proof of (i).

Let us now derive (ii). Let λ ∈ {z ∈ C : z ≥ 0, z = 0} be given. By the above inequality we have

Φ ± (ξ, τ α ± ) -λ ≤ -λ, ∀ξ ∈ R, ∀τ ∈ [0, 1].
This yields (ii) if λ > 0.

Let us now assume that λ = iµ 0 for some given µ 0 ∈ R \ {0}. We claim that the equation Φ ± (ξ, τ α ± ) = iµ 0 with ξ ∈ R and τ ∈ [0, 1] has no solution. We will only investigate the equation with the + signs. The other one can be held similarly. The equation Φ + (ξ, τ α + ) = iµ 0 re-writes as

h + (τ α + ) -ξ 2 + a + ∞ -∞ φ(z)e τ α + z (1 -cos(ξz))dz = 0 ξ(c -2τ α + ) -a + ∞ -∞ φ(z)e τ α + z sin(ξz)dz = µ 0 .
Here since τ ∈ [0, 1] we have h + (τ α + ) ≤ 0. Therefore the first equation implies that ξ = 0. Then the second equation yields µ 0 = 0, whence a contradiction. This shows (ii) and concludes the proof of Lemma 18.

Lemma 19. Under assumptions (3.2)-(3.

3), let λ ∈ C with λ ≥ 0 and λ = 0 be given. Then there exists τ > 1 such that the number of linearly independent solutions of the equation

(L τ -λ)u = 0, (3.23
)

does not depend on τ ∈ [0, τ ]. Moreover if u is a solution of (L -λ)u = 0 then ω b τ u ∈ E and lim |x|→∞ ω b τ (x)u(x) = 0.
The proof of this result relies on a fundamental result for Fredholm operators [START_REF] Gohberg | The basic propositions on defect numbers and indices of linear operators[END_REF] : Proof. We first extend the definition of the operator L τ for τ ∈ C by (3.17). Note that for any τ ∈ C we have L τ = L + τ K(τ ) where K(τ ) : E → F is a bounded linear operator defined by

Lemma
K(τ )v = -2θ (x)v +{-cθ (x)+τ θ (x) 2 -θ (x)}v+a(x) ∞ -∞ φ(y) e τ (θ(x)-θ(x-y)) -1 τ v(x-y)dy.
Then this operator can be extended into series power (converging for the topology of L(E, F )) with respect to τ ∈ C by

K(τ ) = K 0 + ∞ n=1 τ n K n , with K 0 v = b(x) ∞ -∞ φ(y) (θ(x) -θ(x -y)) v(x -y)dy -2θ (x)v -{cθ (x) + θ (x)}v, K 1 v = θ (x) 2 v + b(x) ∞ -∞ φ(y) (θ(x) -θ(x -y)) 2 2 v(x -y)dy, K n v = b(x) ∞ -∞ φ(y) (θ(x) -θ(x -y)) n+1 (n + 1)! v(x -y)dy, n ≥ 2.
Thus the map

H : C → L(E, F ) defined by H(τ ) = L τ -λ is holomorphic on C.
In view of Proposition 16 (ii) and Remark 17, there exists τ > 1 such that [0, τ ] is a subset of some connected component G of {τ ∈ C : (L τ -λ) is a Fredholm operator}. Lemma 20 yields that the quantity dim ker L τ is constant on G except for some possible isolated points τ where it takes a greater value. Let us check that such points can not exist. Indeed arguing by contradiction assume that τ 0 ∈ G is such an isolated point with m = dim ker H(τ 0 ) ≥ 1. For δ > 0 small enough we have τ 0 -δ ∈ G and dim ker H(τ 0 -δ) < m.

(3.24)

Let v 1 ,..., v m in E be linearly independent solutions of the equation H(τ 0 )u = 0. Then the func-

tions v i = v i ω δ ∈ E are linearly independent. Furthermore since ω τ 0 -δ = ω τ 0 ω -δ they satisfy : H(τ 0 -δ) v i = ω τ 0 -δ ω τ 0 H(τ 0 )v i = 0.
This contradicts (3.24) and provides that m = dim ker(L τ -λ) does not depend on τ ∈ [0, τ ].

Actually this proof shows that for each τ 0 ∈ G, the map δ → dim ker H(τ 0 -δ) is decreasing on the set {δ > 0 : τ 0 -δ ∈ G}. Let v1 ,...,v m denote some basis of ker(L b τ -λ). Then for any i = 1, .., m the functions w i = vi /ω b τ ∈ E satisfy (L -λ)w i = 0 and constitute a basis of ker(L -λ). Therefore any u satisfying (L -λ)u = 0 is a linear combination of the w i 's and the result follows.

Lemma 21. Under assumptions (3.2)- (3.3), let λ ∈ C with λ ≥ 0 and λ = 0 be given and let u be a solution of (L -λ)u = 0. Then

|u(x)| = o(u 0 (x)) x → ±∞.
Proof. We will only prove that

lim x→∞ u(x) u 0 (x) = 0. (3.25)
The proof of the analog limit as x → -∞ is similar. Recall that Lemma 11 guarantees the existence of m ε > 0 such that lim inf

x→∞ u 0 (x)e α + ε x ≥ m ε ,
where α + ε → α + when ε → 0 + . Next Lemma 19 enables to say that |u(x)| = o e -b τ α + x when x → ∞ with τ > 1. For ε > 0 small enough so that α + ε < τ α + , by combining the above properties we conclude that

|u(x)| u 0 (x) = o e (α + ε -b τ α + )x when x → ∞.
We now complete the proof of Theorem 8.

Proof of Theorem 8. Let us argue by contradiction and assume that there exists an eigenvalue λ = κ + iµ = 0 of L with κ ≥ 0. Let u = u 1 + iu 2 be an associated eigenvector with u 1 , u 2 ∈ E. We may assume that µ ≥ 0 and u 1 is not identically equal to zero. Indeed if u 1 ≡ 0 then µ = 0 and u 2 = 0 is an eigenvector associated with κ > 0. More precisely without loss of generality we may assume that

∃x 0 ∈ R u 1 (x 0 ) < 0. (3.26) Let us introduce the function v : R × [0, ∞) → R defined by v(x, t) = e κt u 1 (x) cos µt -u 2 (x) sin µt , ( 3.27)
that satisfies the parabolic equation

∂v ∂t = Lv, t > 0, x ∈ R, v(x, 0) = u 1 (x), ∀x ∈ R.
Then we introduce the set

A = {γ ≥ 0; γu 1 (x) + u 0 (x) ≥ 0, ∀x ∈ R}.
This set is not empty since 0 ∈ A. Furthermore it is bounded since due to (3.26) for sufficiently large γ, we have γu 1 (x 0 ) + u 0 (x 0 ) < 0, so that γ / ∈ A. Let us consider

γ 0 = sup A ∈ [0, ∞).
we can deduce from the definition of γ 0 that there exists x 1 ∈ R such that

γ 0 u 1 (x 1 ) + u 0 (x 1 ) = 0. (3.28)
Indeed in view of Lemma 21, we have |u 1 (x)| = o(u 0 (x)) as x → ∞ so that γu 1 (x) + u 0 (x) > 0 for large enough x and γ ∈ [γ 0 , γ 0 + 1]. Thus a straightforward contradiction argument readily yields (3.28).

Thus Lz = -u 0 < 0 on R while z ≥ 0 on [-N, N ]. Therefore, thanks to Lemmas 6 and 7, we obtain that z ≥ 0 and either z > 0 on R or z ≡ 0. Due to the definition of x ∈ [-N, N ], we have z ≡ 0 so that u = -γu 0 . Hence Lu = 0, that contradicts Lu = -u 0 < 0. This completes the proof of the theorem.

We conclude this section by giving a corollary of Theorem 8. For any τ ≥ 0 we consider the Banach spaces

E τ = {v ∈ E : ω τ v ∈ E}, F τ = {v ∈ F : ω τ v ∈ F },
and the operator L τ v = Lv, where L is defined in (3.1). The operator L is acting from E τ into F τ .

Then we have the following result:

Corollary 22. Let assumptions (3.2)-(3.
3) hold. Let us assume that there exists τ ∈ [0, 1) and

u 0 ∈ E τ such that u 0 > 0 and L τ u 0 = 0.
Then zero is a simple eigenvalue of the operator L τ : E τ → F τ while the remainder of its spectrum is strictly contained in the left half plane.

Proof. Let us first notice that L τ : E τ → F τ and L τ are related to each other through (3.16). Hence the resolvent sets of these two operators coincide as well as the complementary of their essential spectrum. Due to Proposition 16, since τ ∈ [0, 1), the essential spectrum of L τ : E τ → F τ is strictly contained in the left half plane. Moreover it is easily checked that any eigenvalue of the operator L τ is also an eigenvalue of the operator L. Therefore Theorem 8 applies and this completes the proof of the result.

Linear stability of travelling wave solutions

The aim of this section is to study the local stability of monotonic wave solution of problem (2.1).

For that purpose we shall assume that (3.3) holds, that the function f is of class C 1 on R 2 and that there exist two real numbers w -< w + such that

f (w ± , w ± ) = 0. (4.1)
A travelling wave solution of problem (2.1) consists in a bounded function w 0 : R → R of class C 2 and a real c ∈ R the wave speed such that the function

u(x, t) = w 0 (x -ct), x ∈ R, t > 0,
is a solution of (2.1) together with lim

z→±∞ w 0 (z) = w ± .
It is easily seen that such a travelling wave solution (w 0 , c) satisfies

w 0 + cw 0 + f (w 0 , J(w 0 )) = 0 lim z→±∞ w 0 (z) = w ± . (4.2)
Our aim is to investigate the local stability with shift of such a wave solution. We do not address the problem of existence of such a solution. This problem has been investigated for instance in [START_REF] Apreutesei | Travelling Waves for Integro-differential Equations[END_REF][START_REF] Apreutesei | Competition of species with intra-specific competition[END_REF][START_REF] Demin | Existence of waves for a nonlocal reaction-diffusion equation[END_REF].

The stability result is related to some spectral properties of the linearized operator around the wave solution w 0 , that reads

Lw = w + cw + a(x)φ * w + b(x)w, ( 4.3)
wherein we have set

a(x) = ∂f (w 0 (x), J(w 0 )(x)) ∂v , b(x) = ∂f (w 0 (x), J(w 0 )(x)) ∂u , x ∈ R. (4.4)
Let us also set

a ± = ∂f (w ± , w ± ) ∂v , b ± = ∂f (w ± , w ± ) ∂u . ( 4.5)
Let us assume that the following assumptions hold:

The

maps f : R 2 → R is of the class C 1 , ∂f (u, v) ∂v ≥ 0, ∀(u, v) ∈ [w -, w + ] 2 , ∂f (w ± , w ± ) ∂u + ∂f (w ± , w ± ) ∂v < 0. (4.6)
In addition, let us assume that Problem (4.2) has a solution (w 0 , c) such that w 0 (x) > 0 ∀x ∈ R.

In view of definitions (4.4) and (4.5), one can notice that the last inequality in (4.6) re-writes as a ± + b ± < 0.

Before stating our main result, let us introduce some notations and definitions. Let us first recall that due to Assumption (4.6) operator L defined in (3.1) satisfies Assumption (3.2). Therefore one may define the quantities α ± > 0 by using (3.19) as well as the map ω τ in (3.15) for any τ ∈ C. Together with these notations, let us introduce the Banach space X = C b (R, R), the space of bounded and continuous map from R into R. Moreover, for any τ ∈ R we consider the weighted space C τ defined by

C τ = {u ∈ C b (R) : lim x→±∞ ω τ (x)u(x) = 0}. (4.8)
We shall also consider the space

C 2 τ = {u ∈ C τ ∩ C 2 (R, R) : u , u ∈ C τ }. (4.9)
The space X is endowed with the usual supremum norm denoted by ∞ while the spaces C τ and C 2 τ are endowed with the weighted norms

u τ = ω τ (.)u ∞ , ∀u ∈ C τ , u 2,τ = u τ + u τ + u τ , ∀u ∈ C 2 τ . (4.10)
Then we have the following definition (see [START_REF] Volpert | Travelling wave solutions of parabolic systems[END_REF]) Definition 23. (Asymptotic stability with shift) Let (Y, Y ) be a Banach space contained in X. Let (w 0 , c) ∈ X × R be a solution of (4.2). Then (w 0 , c) is said to be asymptotically stable with shift with respect to (Y, Y ) if there exists ε > 0 such that for any u 0 ∈ X such that

u 0 -w 0 ∈ Y and u 0 -w 0 Y < ε, then the solution of the evolution problem    ∂u ∂t = ∂ 2 u ∂x 2 + c ∂u ∂x + f (u, J(u)) , x ∈ R, t > 0, u(., 0) = u 0 ,
exists for all t > 0, is unique and satisfies that

u(., t) -w 0 ∈ Y ∀t > 0 u(., t) -w 0 (. + h) Y ≤ M e -bt ∀t > 0
where h ∈ R depends on u 0 , M > 0 and b > 0 are independent of t > 0, u 0 and h.

We are now able to state our main result:

Theorem 24. Let Assumptions (4.6) and (4.7) be satisfied. Let us assume that w 0 ∈ C τ for some τ ∈ [0, 1). Then (w 0 , c) is asymptotically stable with shift with respect to the space (C τ , τ ).

If we recall definition (4.8), we obtain that

C 0 = {u ∈ C b (R) : lim x→±∞ u(x) = 0}.
Therefore Theorem 24 re-writes as follows with τ = 0.

Corollary 25. Let Assumptions (4.6) and (4.7) be satisfied. Let us assume that

lim x→±∞ w 0 (x) = 0.
Then (w 0 , c) is asymptotically stable with shift with respect to the space (C 0 , 0 ).

The proof of this result relies on the application of a general result that will be exposed in the next subsection, while the proof of Theorem 24 will be given in Section 4.2.

A general stability result

In this section, for the sake of completeness, we recall a general result concerning stability of a one dimensional manifold of equilibria for an abstract Cauchy problem. For that purpose let us consider a Banach space (X, ), a linear operator A : D(A) ⊂ X → X and a continuous map h : X → X. Then we can consider the following Cauchy problem

du(t) dt = Au(t) + h(u(t)), t > 0 u(0) = u 0 ∈ X. (4.11)
Then let us assume that the corresponding stationary equation

Au + h(u) = 0, u ∈ D(A),
has a one-parameter family of solutions ϕ α ∈ D(A) for any α ∈ I where I is some bounded interval, I ⊂ R. The aim of this section is to recall some local stability result for these family of equilibria with respect to small perturbations in some Banach space. For that purpose we introduce (Y, ) a Banach space such that Y ⊂ X. Then let us introduce some assumptions upon the family of equilibria {ϕ α } α∈I as well as upon A and h in (4.11).

• Assumption (H1)

-(a) The map α ∈ I → ϕ α ∈ X is differentiable and ϕ α ∈ Y for any α ∈ I. -(b) The map α ∈ I → ϕ α ∈ Y is Lipschitz continuous from I into (Y, Y ).
D(A Y ) = {u ∈ D(A) ∩ Y : Au ∈ Y } A Y u = Au ∀u ∈ D(A Y ),
is the generator of an analytic semigroup on Y . It follows from Assumption (H2) (b) that the operator L :

D(A Y ) ⊂ Y → Y defined by Lu = A Y u + h (ϕ 0 )u, ∀u ∈ D(A Y ),
is also the generator of an analytic semigroup on Y .

-(b) The spectrum of operator L has the following structure: zero is a simple eigenvalue while all the remainder of the spectrum is contained in a closed angle lying in the left complex half plane. This condition re-writes as there exists a 1 > 0 and b

1 > 0 such that λ + a 1 | λ| + b 1 ≤ 0 ∀λ ∈ σ(L) \ {0}.
Then we have the following result:

Theorem 26. Let assumptions (H1)-(H3) be satisfied. Then there exists ε > 0, such that for any

u 0 ∈ X satisfying u 0 -ϕ 0 ∈ Y and u 0 -ϕ 0 Y < ε,
the abstract Cauchy problem (4.11) has a unique and globally defined solution

t → u(t) ∈ X that satisfies u(t) -ϕ α Y ≤ M e -bt ∀t > 0,
for some α ∈ I depending on u 0 , while the constant M > 0 and b > 0 are independent of u 0 , α and t > 0.

The proof of this result can be found in the monograph of Volpert et al [START_REF] Volpert | Travelling wave solutions of parabolic systems[END_REF].

Proof of Theorem 24

Let us consider the linear operator A : D(A) ⊂ X → X defined by

D(A) = {u ∈ X : u , u ∈ X}, Au = u + cu ,
as well as the map h : X → X defined by

h(u)(x) = f u(x), ∞ -∞ φ(x -y)u(y)dy .
We also denote ϕ α = w 0 (. + α) ∈ X for any α ∈ R. Let us introduce τ ∈ [0, 1) such that w 0 ∈ C τ and let us set Y = C τ endowed with the norm τ . Then since w 0 ∈ C τ and φ is compactly supported, we can easily check that α → ϕ α satisfies Assumption (H1) while the map h : X → X satisfies Assumption (H2). It remains to check the operator A satisfies Assumption (H3). To do that, let us first introduce the operator A Y defined by Proof. Let us first notice that C 2 τ is dense in (C τ , τ ). The proof of this result is similar to the one given in the proof of Lemma 2.1 Chapter 5 in [START_REF] Volpert | Travelling wave solutions of parabolic systems[END_REF]. To complete the proof we shall show that L is a sectorial operator on C τ (see for instance [START_REF] Lunardi | Analytic semigroups and optimal regularity in parabolic problems[END_REF][START_REF] Pazy | Semigroups of operator and application to partial differential equation[END_REF] for details about sectorial operators and their links together with analytic semigroups). To do that let us introduce the operator

D(A Y ) = C 2 τ A Y u = u + cu .
T : C τ → C 0 , T v = ω τ v.
Note that T is a bounded linear operator that is invertible and such that T maps C 2 τ onto C 2 0 . Moreover we have T

-1 : C 0 → C τ with T -1 u = u/ω τ . Next we introduce the operator A : C 2 0 → C 0 defined by A = T A Y T -1 . Moreover operator A reads as Av = v + (c -2τ θ (x))v + {τ 2 θ (x) 2 -cτ θ (x) -τ θ (x)}v. It is clear that ρ(A Y ) = ρ( A) and that (λ -A Y ) -1 = T -1 (λ -A) -1 T, ∀λ ∈ ρ(A Y ).
To complete the proof it remains to show that A is a sectorial operator on C 0 . To do so let us recall that the operator ∆ : C 2 0 → C 0 defined by ∆ϕ = ϕ is a sectorial operator on C 0 (see for instance [START_REF] Henry | Geometric Theory of Semilinear Parabolic Equations[END_REF]). Moreover the operator B :

D(B) ⊂ C 0 → C 0 defined by D(B) = {ϕ ∈ C 0 : ϕ ∈ C 0 } and Bϕ = (c -2τ θ (x))ϕ + {τ 2 θ (x) 2 -cτ θ (x) -τ θ (x)}ϕ,
is a closed linear operator while for any γ ∈ (1/2, 1) D(∆ γ ) ⊂ D(B). Thus by using Corollary 2.4 page 81 in Pazy [START_REF] Pazy | Semigroups of operator and application to partial differential equation[END_REF] we get that A = ∆ + B is a sectorial operator and the result follows.

It remains to give some information about the location of the spectrum of the operator L : C 2 τ → C τ defined in (4.3). Now since problem (4.2) is invariant with respect to translation, the map w 0 ∈ C τ ⊂ C 0 is an eigenvalue of operator L associated to the eigenvalue zero. Moreover since w 0 is supposed to be monotonic, we have w 0 > 0. Now due to Assumption (4.6), Corollary 22 applies and shows that zero corresponds to the principle eigenvalue of the operator L and it is simple. Moreover, as explained in the proof of Corollary 22, its essential spectrum coincides with the one of operator L τ (see definition (3.17).) Now due to Assumption (4.6), Proposition 16 applies. Since τ ∈ [0, 1) it provides that σ ess ( L) ⊂ {z ∈ C : z < 0}. Due to the parabolic asymptotic shape of the curves Φ ± (ξ, τ α ± ), one may include these curves into a suitable angle strictly contained in the left half plane. This means that there exist a 1 > 0 and b 1 such that

σ N S ( L) ⊂ A := {λ ∈ C : λ + a 1 | λ| + b 1 ≤ 0}.
In addition, due to Lemma 15 one can derive that σ ess (( L) ⊂ A. Indeed for each λ ∈ C such that

λ + a 1 + b 1 | λ| > 0 we have λ + ρ + a 1 | λ| + b 1 > 0, ∀ρ ≥ 0.
Therefore λ + ρ / ∈ σ N S ( L) and λ / ∈ σ ess ( L). In addition, outside of this angle there is only a finite number of eigenvalues. Thus there exists an angle strictly contained in the left half plane containing the whole spectrum of L except zero which is a simple eigenvalue. This prove that Assumption (H3) holds true and complete the proof of Theorem 24 by applying Theorem 26.

Additional results and minimax representation of the wave speed

The aim of this section is to give some additional results that are a consequence of Theorem 24 as well as the comparison principle given in Theorem 4. The details of the proofs will be omitted. They directly follow the same steps as the one of Theorem 6.1 p. 250 and Theorem 7.1 page 255 in [START_REF] Volpert | Travelling wave solutions of parabolic systems[END_REF]. The first result is concerned with the asymptotic behaviour of the Cauchy problem Then the solution u(x, t) of (5.1) is globally defined and converges (with shift) to w 0 in the C τ -norm, that is lim t→∞ u(., t) -w 0 (. + x 0 ) τ = 0, wherein x 0 ∈ R depends on u 0 .

Next one can derive some minimax formula for the wave speed. To do so let us introduce for each τ ∈ R, the set K τ defined by 6. Some applications in biology and chemical kinetics where ku 2 (1 -u) describes bi-sexual reproduction of a population, u is the density of the population, (1 -u) corresponds to available resources, µu is mortality.

In order to explain the appearance of the integral term in the equation, consider the example of sexual plant reproduction with male and female plants. We assume that they have the same density u due to the same proportion at birth and to the same diffusion. Since male plants distribute their pollen in some area around their location, then u 2 in the equation should be replaced by u (density of female plants) times the integral (density of pollen).

Similar models can describe evolution of cell populations where cells can produce mitosis factors diffusing in the extra-cellular matrix and stimulating proliferation of surrounding cells. The existence of travelling waves for such integro-differential equations is proved in [START_REF] Apreutesei | Travelling Waves for Integro-differential Equations[END_REF], [START_REF] Apreutesei | Competition of species with intra-specific competition[END_REF] in the case of functions φ with a small support and in [START_REF] Demin | Existence of waves for a nonlocal reaction-diffusion equation[END_REF] for an arbitrary support.

Nonlocal problems in chemical kinetics

Distribution of concentrations in a chemically reacting medium can be described by the reactiondiffusion system of equations

∂A ∂t = d ∂ 2 A ∂x 2 + W (A)
, where A = (A 1 , ..., A m ) is the vector of concentrations, W = (W 1 , ..., W m ) is the vector of reaction rates, d is the diffusion matrix. Suppose that some of the species have fast reaction and fast diffusion, that is instead of the previous systems we can write

∂A i ∂t = d i ∂ 2 A i ∂x 2 + W i (A), i = 1, ..., k, ∂A i ∂t = d i ∂ 2 A i ∂x 2 + W i (A), i = k + 1, ..., m,

•

  Assumption (H2) -(a) The map h : X → X is of the class C 1 on X. -(b) For any α ∈ I and any v ∈ Y the map h (ϕ α + v) belongs to L(Y ), the space of bounded linear operators from Y into itself. Moreover for each α ∈ I the map v ∈Y → h (ϕ α + v) ∈ L(Y ) is Lipschitz continuous on B Y (0, 1) = {v ∈ Y : v Y ≤ 1}. • Assumption (H3) -(a) The part of A in Y denoted by A Y and defined by

  Then we have the following lemma:Lemma 27. The linear operator A Y : C 2 τ ⊂ C τ → C τis the infinitesimal generator of an analytic semigroup.

  u, J(u)) , x ∈ R, t > 0, u(., 0) = u 0 , (5.1) with monotonic initial condition. Recalling definitions (3.15), (4.8) and (4.10), the following result holds: Theorem 28. Let Assumptions (4.6) and (4.7) be satisfied and assume that w 0 ∈ C τ for some τ ∈ [0, 1). Let us assume that u 0 ∈ C b (R) is some given monotonic function such that lim x→±∞ |u 0 (x) -w ± |ω τ (x) = 0.

K

  τ = {ρ ∈ C 2 b (R) : ρ > 0 and lim x→±∞ |ρ(x) -w ± |ω τ (x) = 0}.Moreover let us set the map Ψ :K τ → C(R, R) defined by Ψ(ρ)(x) = -ρ (x) + f (ρ(x), φ * ρ(x)) ρ (x) , ∀x ∈ R.Then the following holdsTheorem 29. Let Assumptions (4.6) and (4.7) be satisfied. Let us assume that w 0 ∈ C τ for some τ ∈ [0, 1). Then the wave speed c has the following representation:

6. 1 .∂ 2 u ∂x 2 +

 12 Population dynamics with nonlocal stimulation of reproductionAs an application of the results given in this paper we consider the equation∂u ∂t = ∂ 2 u ∂x 2 + f (u, J(u)) , t > 0, x ∈ R,(6.1)wherein f (u, v) = kuv(1 -u) -µu, J(u) = ∞ -∞ φ(x -y)u(y, t)dy. (6.2)When φ = δ is a dirac delta function, then we obtain the local reaction-diffusion equation∂u ∂t = ku 2 (1 -u) -µu, (6.3)

where is a small parameter. Taking a formal limit as → 0, we obtain the equation

We can express the last variables A i , i = k + 1, ..., m. This will give us integro-differential equations for the first variables A i , i = 1, ..., k.

One of possible examples is given by the following systems written for travelling waves:

We can express C from the second equation and substitute in the first equation.

We now consider the map w : R × [0, ∞) → R given by w(x, t) = γ 0 v(x, t) + u 0 (x), (3.29) that satisfies

Theorem 5 provides that w(x, t) ≥ 0 for all

Let us check that w cannot have these properties. We first consider the case µ > 0. Then in view of definitions (3.29) and (3.27) we have

This implies µ = 0, whence a contradiction.

We now assume µ = 0 and κ > 0. Then for any t > 0 we have w(x 1 , t) = γ 0 e κt u 1 (x 1 ) + u 0 (x 1 ) < γ 0 u 1 (x 1 ) + u 0 (x 1 ) = 0, that is impossible Therefore the operator L has no eigenvalue with non-negative real part except the zero eigenvalue.

It remains to prove that zero is a simple eigenvalue (here simple is understood in the sense of the dimension of its generalized eigenspace). We first show that ker L = span u 0 . Indeed let us consider the equation Lu = 0, u ∈ E.

Let us consider γ ∈ R, the infimum value of γ ∈ R such that u + γu 0 ≥ 0 on [-N, N ]. Then the function w = u + γu 0 satisfies the equation

Therefore, thanks to Lemmas 6 and 7, we obtain that w ≥ 0 on R and either w > 0 on R or w ≡ 0.


N Apreutesei

V Volpert
email: volpert@math.univ-lyon1.fr

Properness and Topological Degree for Nonlocal Reaction-Diffusion Operators

Keywords: nonlocal reaction-diffusion equations, Fredholm property, properness, topological degree, travelling waves AMS subject classification: 35K57, 45K05, 47A53, 47H11

The paper is devoted to integro-differential operators, which correspond to nonlocal reaction-diffusion equations considered on the whole axis. Their Fredholm property and properness will be proved. This will allow one to define the topological degree.

Introduction

Consider the semi-linear parabolic equation

∂u ∂t = ∂ 2 u ∂x 2 + F (u, J (u)) , ( 1.1)
where

J (u) = ∞ -∞
φ (x -y) u (y, t) dy.

Here φ : R → R is a bounded function, not necessarily continuous, φ ≥ 0 on R. The support of the function φ is supposed to be bounded, supp φ ⊆ [-N, N ]. We will also assume that ∞ -∞ φ (y) dy = 1. Conditions on the function F will be specified below.

Integro-differential equations of this type arise in population dynamics (see [START_REF] Genieys | Pattern and waves for a model in population dynamics with nonlocal consumption of resources[END_REF], [START_REF] Volpert | Reaction-diffusion waves in biology[END_REF] and references therein). They are referred to as nonlocal reaction-diffusion equations. A travelling wave solution of equation (1.1) is a solution of this equation of the particular form u (x, t) = w (x -ct). It satisfies the equation w + cw + F (w, J (w)) = 0.

(

The constant c ∈ R is the wave speed. It is unknown and should be found together with the function w(x). There are numerous works devoted to the existence [START_REF] Ai | Traveling wave fronts for generalized Fisher equations with spatio-temporal delays[END_REF]- [START_REF] Berestycki | The non-local Fisher-KPP equation: travelling waves and steady states[END_REF], stability and nonlinear dynamics [START_REF] Genieys | Pattern and waves for a model in population dynamics with nonlocal consumption of resources[END_REF], [START_REF] Volpert | Reaction-diffusion waves in biology[END_REF], [START_REF] Britton | Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model[END_REF] - [START_REF] Perthame | Concentration in the nonlocal Fisher equation: The Hamilton-Jacobi limit[END_REF] of travelling wave solutions of some particular cases of equation (1.1). Properties of travelling waves are determined by the properties of the integro-differential operator in the left-hand side of (1.2). In this paper we will study the Fredholm property of this operator and its properness. We will use them to define the topological degree and will discuss some applications. Let E = C 2+α (R) , E 0 = C α (R) , 0 < α < 1 the usual Holder spaces endowed with the norms

||u|| E 0 = sup x,y∈R |u (x) -u (y)| |x -y| α + sup x∈R |u(x)|, ||u|| E = ||u|| E 0 + ||u || E 0 + ||u || E 0 .
We are interested in the solutions of equation (1.2) with the limits w ± at ±∞, where the values w ± are such that F (w ± , w ± ) = 0. We are looking for the solutions w of (1.2) under the form w = u + ψ, where ψ ∈ C ∞ (R) , such that ψ (x) = w + for x ≥ 1 and ψ (x) = w - for x ≤ -1. Thus equation (1.2) becomes

(u + ψ) + c (u + ψ) + F (u + ψ, J (u + ψ)) = 0. (1.3)
Denote by A the operator in the left-hand side of (1.3), that is A : E → E 0 , Au = (u + ψ) + c (u + ψ) + F (u + ψ, J (u + ψ)) . (1.4) Suppose that F is differentiable with respect to both variables. The linearization of A about a function u 1 ∈ E is the operator L : E → E 0 , Lu ≡ A (u 1 ) u = u +cu + ∂F ∂u (u 1 + ψ, J (u 1 + ψ)) u+ ∂F ∂U (u 1 + ψ, J (u 1 + ψ)) J (u) , (1.5) where ∂F/∂u and ∂F/∂U are the derivatives of F (u, U ) with respect to the first and to the second variable, respectively. For the linearized operator L, we introduce the limiting operators. Since for w 1 = u 1 + ψ, there exist the limits lim x→±∞ w 1 (x) = w ± , it follows that J(w 1 ) = J (u 1 + ψ) → w ± as x → ±∞ and the limiting operators are given by

L ± u = u + cu + a ± u + b ± J (u) , ( 1.6)
where

a ± = ∂F ∂u w ± , w ± , b ± = ∂F ∂U w ± , w ± .
We will now recall the main definitions and results concerning the essential spectrum and Fredholm property for linear operators and the properness of nonlinear operators.

Essential spectrum and Fredholm property. Let us recall that a linear operator M : E 1 → E 2 acting from a Banach space E 1 into another Banach space E 2 is called a Fredholm operator if its kernel has a finite dimension, its image is closed, and the codimension of the image is also finite. The last two conditions are equivalent to the following solvability condition: the equation Lu = f is solvable if and only if φ i (f ) = 0 for a finite number of functionals φ i from the dual space E * 2 . Suppose that E 1 ⊂ E 2 . By definition, the essential spectrum of the operator L is the set of all complex λ for which the operator L -λI, where I is the identity operator, does not satisfy the Fredholm property. The essential spectrum of general elliptic boundary value problems in unbounded domains can be determined in terms of limiting operators [START_REF] Volpert | Fredholm property of elliptic operators in unbounded domains[END_REF]. For the integro-differential operators under consideration, since they have constant coefficients at infinity, the essential spectrum can be found explicitly. It is proved [START_REF] Apreutesei | Competition of species with intra-specific competition[END_REF], [START_REF] Apreutesei | Travelling waves for integro-differential equations in population dynamics[END_REF] that the operator L -λI is normally solvable † with a finite-dimensional kernel if and only if the equations L ± u = λu do not have nonzero bounded solutions. Applying the Fourier transform to the last equations, we obtain

λ ± (ξ) = -ξ 2 + ciξ + a ± + b ± φ(ξ), ξ ∈ R,
where φ(ξ) is the Fourier transform of the function φ(x). Thus, the operator L is normally solvable with a finite-dimensional kernel if and only if the curves λ ± (ξ) on the complex plane do not pass through the origin. Under some additional conditions, it can be also shown that † By definition, an operator A : E 1 → E 2 is normally solvable if the equation Au = f has a solution if and only if φ(f ) = 0 for all functionals φ from some linear subspace in the dual space E * 2 . This is equivalent to the condition that the range of the operator is closed.

the codimension of the operator is finite, that is it satisfies the Fredholm property, and its index can be found.

A nonlinear operator B : E 1 → E 2 is called Fredholm if the linearized operator B satisfies this property. In what follows we will use the Fredholm property in some weighted spaces (see below).

Properness and topological degree. An operator B : E 1 → E 2 is called proper on closed bounded sets if the intersection of the inverse image of a compact set K ⊂ E 2 with any closed bounded set in E 1 is compact. For the sake of brevity, we will call such operators proper. It is an important property because it implies that the set of solution of the operator equation B(u) = 0 is compact.

It appears that elliptic (or ordinary differential) operators are not generally proper when considered in Hölder or Sobolev spaces in unbounded domains. We illustrate this situation with a simple example. Consider the equation

w + H(w) = 0, x ∈ R, (1.7)
where H(w) = w(w -1). It can be verified that this equation has a positive solution w(x), which converges to zero at infinity. This convergence is exponential. So the solution belongs to Hölder and to Sobolev spaces. Along with the function w(x), any shifted function w(x + h), h ∈ R is also a solution. Hence there is a family of solutions, and the set of solutions is not compact. Similar examples can be constructed for the integro-differential equation.

In order to obtain proper operators, we introduce weighted spaces C k+α µ (R) with a growing at infinity polynomial weight function µ(x). The norm in this space is given by the equality

u C k+α µ (R) = µu C k+α (R) .
Let us return to the previous example. The family of functions w(x + h) is not uniformly bounded in the weighted space. If we take any bounded closed set in the function space, it can contain the solutions w(x + h) only for a compact set of the values of h. Therefore the set of solutions is compact in any bounded closed set. This example shows the role of weighted spaces for the properness of the operators.

Properness of general nonlinear elliptic problems in unbounded domains and in weighted spaces is proved in [START_REF] Volpert | Properness and topological degree for general elliptic operators[END_REF]. In this work, we will prove properness of the integro-differential operators. After that, using the construction of the topological degree for Fredholm and proper operators with the zero index [START_REF] Volpert | Properness and topological degree for general elliptic operators[END_REF], we will define the degree for the integro-differential operators. We will finish this paper with some applications of these methods to travelling waves solutions.

Properness in weighted spaces

In this section we study the properness of the semilinear operator A.

Definition 2.1. If X, Y are Banach spaces, an operator A : X → Y is called proper if for any compact set D ⊂ Y and any bounded closed set B ⊂ X, the intersection A -1 (D) ∩ B is a compact set in X. Remark 2.2. The operator A : E = C 2+α (R) → E 0 = C α (R)
may not be proper from E to E 0 (see the comments related to equation (1.7) from the introduction).

We will show in the sequel that A is proper in some weighted spaces associated to E and E 0 .

Let µ : R → R be the function given by µ We begin with the following estimate for the integral term J (u) .

(x) = 1 + x 2 , x ∈ R. Denote E = C 2+α (R) , E 0 = C α (R)
Lemma 2.3. Suppose that µ (x) = 1 + x 2 and φ : R → R is a function such that φ ≥ 0 on R, suppφ = [-N, N ] is bounded, ∞ -∞ φ (y) dy = 1 and φ ∈ E 0 . Then ||J (u) || 0µ ≤ K||µu|| C(R) , (∀) u ∈ E 0 , (2.1)
for some constant K > 0.

Proof.

If supp φ = [-N, N ] , then supp φ (x -•) = [x -N, x + N ] . First we write µ (x) J (u) (x) = x+N x-N µ (x) µ (y) φ (x -y) µ (y) u (y) dy. Since µ (x) /µ (y) is bounded for |x -y| ≤ N and ∞ -∞ φ (y) dy = 1, we have ||µJ (u) || C(R) ≤ K 1 ||µu|| C(R) , for some positive constant K 1 .
For every

x 1 , x 2 ∈ R, x 1 = x 2 , denote H (x 1 , x 2 ) = |µ (x 1 ) J (u) (x 1 ) -µ (x 2 ) J (u) (x 2 )| |x 1 -x 2 | α . If |x 1 -x 2 | ≥ 1, we have H (x 1 , x 2 ) ≤ 2K 1 ||µu|| C(R) . If |x 1 -x 2 | < 1, then H (x 1 , x 2 ) = 1 |x 1 -x 2 | α | x 1 +N x 1 -N [µ (x 1 ) φ (x 1 -y) -µ (x 2 ) φ (x 2 -y)] u (y) dy+ + x 1 +N x 1 -N µ (x 2 ) φ (x 2 -y) u (y) dy - x 2 +N x 2 -N µ (x 2 ) φ (x 2 -y) u (y) dy| ≤ ≤ | x 1 +N x 1 -N { [µ (x 1 ) -µ (x 2 )]φ (x 1 -y) µ (y) |x 1 -x 2 | α + µ (x 2 ) [φ (x 1 -y) -φ (x 2 -y)] µ (y) |x 1 -x 2 | α }µ (y) u (y) dy|+ + 1 |x 1 -x 2 | α x 2 -N x 1 -N µ (x 2 ) µ (y) φ (x 2 -y) µ (y) u (y) dy + + 1 |x 1 -x 2 | α x 1 +N x 2 +N µ (x 2 ) µ (y) φ (x 2 -y) µ (y) u (y) dy . Since |x 1 -x 2 | < 1 and |x 1 -y| ≤ N , then |x 2 -y| ≤ N + 1.
In this case, the boundedness of φ and of

µ (x 1 ) -µ (x 2 ) µ (y) |x 1 -x 2 | α , implies that H (x 1 , x 2 ) ≤ K 2 ||µu|| C(R) + K 3 ||φ|| E 0 ||µu|| C(R) + 2K 4 |x 1 -x 2 | 1-α ||µu|| C(R) ,
for some K 2 , K 3 , K 4 > 0. Thus the desired estimate holds and the lemma is proved.

We study the operator A acting from E µ into E 0 µ . In order to introduce a topological degree (in Section 4), we prove the properness of A in the more general case when the coefficient c and function F depend also on a parameter τ

∈ [0, 1] . Let A τ : E µ → E 0 µ , τ ∈ [0, 1] be the operator defined through A τ u = (u + ψ) + c (τ ) (u + ψ) + F τ (u + ψ, J (u + ψ)) .
(2.2)

We note that the linearization

L τ of A τ about a function u 1 ∈ E µ is L τ u = u + c (τ ) u + ∂F τ ∂u (u 1 + ψ, J (u 1 + ψ)) u + ∂F τ ∂U (u 1 + ψ, J (u 1 + ψ)) J (u) ,
while its limiting operators are given by

L ± τ u = u + c (τ ) u + ∂F τ ∂u w ± , w ± u + ∂F τ ∂U w ± , w ± J (u) .
Assume that the following hypotheses are satisfied: (H1) For any τ ∈ [0, 1] , the function F τ (u, U ) and its derivatives with respect to u and U satisfy the Lipschitz condition: there exists K > 0 such that

|F τ (u 1 , U 1 ) -F τ (u 2 , U 2 )| ≤ K (|u 1 -u 2 | + |U 1 -U 2 |) , for any (u 1 , U 1 ) , (u 2 , U 2 ) ∈ R 2 .
Similarly for ∂F τ /∂u and ∂F τ /∂U :

∂F τ (u 1 , U 1 ) ∂u - ∂F τ (u 2 , U 2 ) ∂u ≤ K (|u 1 -u 2 | + |U 1 -U 2 |) , ∂F τ (u 1 , U 1 ) ∂U - ∂F τ (u 2 , U 2 ) ∂U ≤ K (|u 1 -u 2 | + |U 1 -U 2 |) .
(H2) c (τ ), F τ (u, U ) and the derivatives of F τ (u, U ) are Lipschitz continuous in τ, i. e. there exists a constant c > 0 such that

|c (τ ) -c (τ 0 ) | ≤ c |τ -τ 0 | , |F τ (u, U ) -F τ 0 (u, U ) | ≤ c |τ -τ 0 | , ∂F τ (u, U ) ∂u - ∂F τ 0 (u, U ) ∂u ≤ c|τ -τ 0 |, ∂F τ (u, U ) ∂U - ∂F τ 0 (u, U ) ∂U ≤ c|τ -τ 0 |. (∀) τ, τ 0 ∈ [0, 1]
, for all (u, U ) from any bounded set in R 2 . (H3) (Condition N S) For any τ ∈ [0, 1] , the limiting equations

u + c (τ ) u + ∂F τ ∂u w ± , w ± u + ∂F τ ∂U w ± , w ± J (u) = 0 do not have nonzero solutions in E. Lemma 2.4. Suppose that conditions (H1) -(H2) hold. If τ n → τ 0 and µu n → µu 0 in C (R), then ||F τ n (u n + ψ, J(u n + ψ)) -F τ 0 (u 0 + ψ, J(u 0 + ψ))|| 0µ → 0.
Proof. We have the equality

F τ n (u n + ψ, J(u n + ψ)) -F τ 0 (u 0 + ψ, J(u 0 + ψ)) = F τn (u n +ψ, J(u n +ψ))-F τn (u 0 +ψ, J(u 0 +ψ))+F τn (u 0 +ψ, J(u 0 +ψ))-F τ 0 (u 0 +ψ, J(u 0 +ψ)).
Condition (H1) leads to the estimate of the first difference

|F τ n (u n + ψ, J(u n + ψ)) -F τ n (u 0 + ψ, J(u 0 + ψ))| ≤ K (|u n -u 0 | + |J(u n -u 0 )|) .
In view of hypothesis µu n → µu 0 in C (R) , the above inequality allows us to conclude that the weighted norm converges to zero.

In order to estimate the second difference, we begin with the following representation:

F τ n (u 0 + ψ, J(u 0 + ψ)) -F τ n (ψ, J(ψ)) = J(u 0 ) 1 0 ∂F τn (u 0 + ψ, J(ψ) + tJ(u 0 )) ∂U dt + u 0 1 0 ∂F τn (tu 0 + ψ, J(ψ)) ∂u dt.
Similarly,

F τ 0 (u 0 + ψ, J(u 0 + ψ)) -F τ 0 (ψ, J(ψ)) = J(u 0 ) 1 0 ∂F τ 0 (u 0 + ψ, J(ψ) + tJ(u 0 )) ∂U dt + u 0 1 0 ∂F τ 0 (tu 0 + ψ, J(ψ)) ∂u dt.
Therefore,

F τ n (u 0 + ψ, J(u 0 + ψ)) -F τ 0 (u 0 + ψ, J(u 0 + ψ)) = J(u 0 ) 1 0 ∂F τ n (u 0 + ψ, J(ψ) + tJ(u 0 )) ∂U dt - 1 0 ∂F τ 0 (u 0 + ψ, J(ψ) + tJ(u 0 )) ∂U dt + +u 0 1 0 ∂F τn (tu 0 + ψ, J(ψ)) ∂u dt - 1 0 ∂F τ 0 (tu 0 + ψ, J(ψ)) ∂u dt + +F τn (ψ, J(ψ)) -F τ 0 (ψ, J(ψ)) ≡ A + B + C
(A denotes the first line in the right-hand side, B -the second, C -the third). The expressions A and B converge to zero in the weighted norm of E µ = C α µ (R) , due to the Lipschitz condition with respect to τ of the derivatives of F τ (see (H2)). The expression C is a function with a finite support. It also converges to zero in the weighted norm as τ n → τ 0 . This concludes the proof.

We can now prove the properness of the τ -dependent operator A τ . Denote by

E loc = u ∈ C 2+α (I) , (∀) I = bounded interval , E µ,loc = µu ∈ C 2+α (I) , (∀) I = bounded interval
and similarly E 0 loc and E 0 µ,loc . Theorem 2.5. If φ ∈ E 0 , under assumptions (H1) -(H3), the operator A τ (u) :

E µ × [0, 1] → E 0 µ from (2.2) is proper with respect to (u, τ ) on E µ × [0, 1] . Proof. Consider a convergent sequence f n ∈ E 0 µ , say f n → f 0 in E 0 µ . Let (u n , τ n ) be a solution in E µ × [0, 1] of the equation A τn (u n ) = f n , such that ||u n || µ ≤ M, (∀)n ≥ 1.
(

We prove that one can choose a convergent in E µ subsequence of the sequence u n . Without loss of generality we may assume that τ n → τ 0 as n → ∞. Equation A τ n (u n ) = f n can be written as

(u n + ψ) + c (τ n ) (u n + ψ) + F τn (u n + ψ, J (u n + ψ)) = f n . (2.4)
Multiplying the equation by µ and denoting v n (x) = µ (x) u n (x) , g n (x) = µ (x) f n (x) , we derive that

v n + -2 µ µ + c (τ n ) v n + - µ µ + 2 µ µ 2 -c (τ n ) µ µ v n + +µF τn (u n + ψ, J (u n + ψ)) + µ (ψ + c (τ n ) ψ ) = g n . (2.5) Indeed, since µu n = (µu n ) -µ u n = v n -µ v n /µ and µu n = (µu n ) -µ u n -2µ u n = v n -µ v n /µ -2µ (v n /µ -µ v n /µ 2 )
, by (2.4) one easily obtains (2.5) .

The sequence v n = µu n is uniformly bounded in E :

||v n || E = ||µu n || E = ||u n || µ ≤ M, (∀) n ≥ 1. (2.6)
Then it is locally convergent on a subsequence. More exactly, for every bounded interval [-N, N ] of x, there is a subsequence (denoted again

v n ) converging in C 2+α [-N, N ] to a limiting function v 0 ∈ C 2+α [-N, N ] . By a diagonalization process we can prolong v 0 to R such that v 0 ∈ E. Since ||v n || E ≤ M, (∀)n ≥ 1, we can easily see that ||v 0 || E ≤ M.
Let u 0 be the limit that corresponds to u n . Then µu n → µu 0 in E loc and v 0 = µu 0 .

We now want to pass to the limit as n → ∞ in (2.4) and (2.5) . To this end observe that (H2) implies that

||F τn (u n + ψ, J (u n + ψ)) -F τ 0 (u 0 + ψ, J (u 0 + ψ)) || 0µ = = ||F τ n (u n + ψ, J (u n + ψ)) -F τ 0 (u n + ψ, J (u n + ψ)) || 0µ + +||F τ 0 (u n + ψ, J (u n + ψ)) -F τ 0 (u 0 + ψ, J (u 0 + ψ)) || 0µ ≤ ≤ c 1 |τ n -τ 0 | + ||F τ 0 (u n + ψ, J (u n + ψ)) -F τ 0 (u 0 + ψ, J (u 0 + ψ)) || 0µ . Since F τ 0 is continuous from E 0 µ ×E 0 µ to E 0 µ (see (H1)) and J (u n + ψ) → J (u 0 + ψ) in E 0 µ,loc , we derive that F τn (u n + ψ, J (u n + ψ)) → F τ 0 (u 0 + ψ, J (u 0 + ψ)) as n → ∞, in E 0 µ,loc . (2.7)
By the change of variable x n -y = -z, it follows that

J (u n + ψ) (x + x n ) = ∞ -∞ φ (x -z) u n (x n + z) dz + ∞ -∞ φ (x -z) ψ (x n + z) dz → w ± ,
uniformly on bounded intervals of x. Hypothesis (H1) shows that

T n 2 → ∂F τ 0 ∂u w ± , w ± V 0 , as n → ∞ in E 0 loc . (2.16)
On the other hand,

T n 3 = µ (J (u n + ψ) -J (u 0 + ψ)) × F τ 0 (u 0 + ψ, J (u n + ψ)) -F τ 0 (u 0 + ψ, J (u 0 + ψ)) J(u n + ψ) -J(u 0 + ψ) (x + x n ) = I n (x) • ∂F τ 0 ∂U (u 0 + ψ, sJ(u n + ψ) + (1 -s) J(u 0 + ψ)) (x + x n ) ,
for some s ∈ [0, 1] , where

I n (x) = µ (J (u n + ψ) -J (u 0 + ψ)) . For µ (x) = 1 + x 2 , x ∈ R,
with the aid of (2.11) , we arrive at

I n (x) = ∞ -∞ µ (x + x n ) µ (z + x n ) φ (x -z) V n (z) dz → ∞ -∞ φ (x -z) V 0 (z) dz = J V 0 , in E 0 loc . As above, since J (u n ) (x + x n ) → 0, J (u 0 ) (x + x n ) → 0, J (ψ) (x + x n ) → w ± uniformly on bounded intervals of x, we deduce that T n 3 → ∂F τ 0 ∂U w ± , w ± J( V 0 ), as n → ∞ in E 0 loc .
(2.17)

Now we may pass to the limit in (2.13) . With the aid of (2.14) -(2.17) and (H2) , one arrives at

V 0 + c (τ 0 ) V 0 + ∂F τ 0 ∂u w ± , w ± V 0 + ∂F τ 0 ∂U w ± , w ± J V 0 = 0,
which contradicts (H3) . Therefore we have proved that

V n → 0 in C (R) . Now we have to show that V n → 0 in E = C 2+α (R) .
To this end, we write equation (2.10) in the form S (V n ) = h n , where S (V n ) is the linear part from the left-hand side and

h n = (µf n -µf 0 ) -µ[F τ n (u n + ψ, J (u n + ψ)) - -F τ 0 (u 0 + ψ, J (u 0 + ψ))] -[c (τ n ) -c (τ 0 )] v 0 - µ µ v 0 + µψ .
Using Lemma 2.1 from [START_REF] Apreutesei | Travelling waves for integro-differential equations in population dynamics[END_REF] for the linear operator S, we can write

||V n || E ≤ C ||S (V n ) || E 0 + ||V n || C(R) .
We make use of Lemma 2.4, hypothesis (H 2 ) for c (τ n ) -c (τ 0 ) , and of the convergence

f n → f 0 in E 0 µ , to deduce that S (V n ) = h n → 0 in E 0 = C α (R) . Since V n → 0 in C (R) , we conclude that u n → u 0 in E µ . The theorem is proved.
Definitions. Recall in the beginning the definition of the topological degree. Consider two Banach spaces E 1 , E 2 , a class Φ of operators acting from E 1 to E 2 and a class of homotopies

H = {A τ (u) : E 1 × [0, 1] → E 2 , such that A τ (u) ∈ Φ, (∀) τ ∈ [0, 1]}.
Let D ⊂ E 1 be an open bounded set and A ∈ Φ such that A (u) = 0, u ∈ ∂D, where ∂D is the boundary of D. Suppose that for such a pair (D, A), there exists an integer γ (A, D) with the following properties:

(i) Homotopy invariance.

If A τ (u) ∈ H and A τ (u) = 0, for u ∈ ∂D, τ ∈ [0, 1] , then γ (A 0 , D) = γ (A 1 , D) . (ii) Additivity. If A ∈ Φ, D is the closure of D and D 1 , D 2 ⊂ D are open sets, such that D 1 ∩ D 2 = and A (u) = 0, for all u ∈ D\ (D 1 ∪ D 2 ) , then γ (A, D) = γ (A, D 1 ) + γ (A, D 2 ) .
(iii) Normalization. There exists a bounded linear operator J : E 1 → E 2 with a bounded inverse defined on all E 2 such that, for every bounded set

D ⊂ E 1 with 0 ∈ D, γ (J, D) = 1.
The integer γ (A, D) is called a topological degree.

Degree for Fredholm and proper operators. We now recall a general result concerning the existence of a topological degree which was proved in [START_REF] Volpert | Topological degree for elliptic operators in unbounded cylinders[END_REF], [START_REF] Volpert | Properness and topological degree for general elliptic operators[END_REF].

Let E 1 and E 2 be Banach spaces, E 1 ⊆ E 2 algebraically and topologically and let G ⊂ E 1 be an open bounded set.

Denote by I : E 1 → E 2 the imbedding operator, Iu = u, and by Φ a class of bounded linear operators L : E 1 → E 2 satisfying the following conditions:

(a) The operator L -λI : E 1 → E 2 is Fredholm for all λ ≥ 0, (b) For every operator L ∈ Φ, there is λ 0 = λ 0 (L) such that L -λI has a uniformly bounded inverse for all λ > λ 0 .

Denote by F the class

F = {B∈C 1 (G, E 2 ) , B proper, B (x) ∈ Φ, (∀) x ∈ G}, ( 4.1)
where B (x) is the Fréchet derivative of the operator B. Finally, one introduces the class H of homotopies given by Remark 4.2. Condition (b) can be weakened. Let E 1 and E 2 be two Banach spaces such that E i ⊂ E i , i=1,2 where the inclusion is understood in the algebraic and topological sense. In the case of the Hölder space C k+α (R), this can be the space C k (R) with an integer nonnegative k. We can also consider some integral spaces W k,p ∞ (R) [START_REF] Volpert | Fredholm property of elliptic operators in unbounded domains[END_REF]. Instead of (b) above we can impose the following condition [START_REF] Volpert | Elliptic partial differential equations[END_REF]:

H = {B (x, τ ) ∈ C 1 (G × [0, 1] , E 2 ) , B proper, B (., τ ) ∈ F, (∀) τ ∈ [0, 1]}. ( 4
(b ) For every operator L : E 1 → E 2 , there is λ 0 = λ 0 (L) such that L-λI has a uniformly bounded inverse for all λ > λ 0 .

Degree for the integro-differential operators. Now, let E 1 = E µ and E 2 = E 0 µ be the weighted spaces introduced in the previous section, with µ (x) = 1 + x 2 , x ∈ R. We will apply Theorem 4.1 for the integro-differential operator A of the form (1.4) , where function

ψ ∈ C ∞ (R) , ψ (x) = w + , for x ≥ 1, ψ (x) = w -for x ≤ -1 and
(H4) F (u, U ) and its derivatives with respect to u and U are Lipschitz continuous in (u, U ) ;

(H5) The limiting equations

u + cu + ∂F ∂u w ± , w ± u + ∂F ∂U w ± , w ± J (u) -λu = 0
do not have nonzero solutions in E, (∀) λ ≥ 0. Under these hypotheses, Theorem 2.5 assures that operator A is proper. Moreover, its Fréchet derivative is A = L from (1.5) and it is a Fredholm operator with the index zero (Theorem 3.

Applications to travelling waves

In this section we will discuss some applications of the Fredholm property, properness and topological degree to study travelling wave solutions of equation (1.1). Let us begin with the classification of the nonlinearities. Denote F 0 (w) = F (w, w).

We obtain this function from F (w, J(w)) if we formally replace the kernel φ(x) of the integral by the δ-function. The corresponding reaction-diffusion equation

∂u ∂t = ∂ 2 u ∂x 2 + F 0 (u) (5.1)
is called bistable if F 0 (w ± ) < 0, monostable if one of these derivatives is positive and another one negative and, finally, unstable if F 0 (w ± ) > 0. As it is well-known, it can have travelling wave solutions, that is solutions, which satisfy the problem

w + cw + F 0 (w) = 0, w(±∞) = w ± . (5.2)
Let w 0 (x) be a solution of (5.2) with some c = c 0 . The operator L 0 linearized about this solution,

L 0 u = u + c 0 u + F 0 (w 0 )u
has the essential spectrum given by two parabolas:

λ 0 ± (ξ) = -ξ 2 + c 0 iξ + F 0 (w ± ), ξ ∈ R.
Therefore the operator L 0 satisfies the Fredholm property if and only if F 0 (w ± ) = 0. If this condition is satisfied, then the index of the operator is well defined. In the bistable case it equals 0, in the monostable case 1, in the unstable case 0 [START_REF] Collet | Computation of the index of linear elliptic operators in unbounded cylinders[END_REF].

In the case of the integro-differential operator

Lu = u + c 0 u + F u (w, J(w))u + F U (w, J(w))J(u),
the essential spectrum is given by the curves

λ ± (ξ) = -ξ 2 + c 0 iξ + F u (w ± , w ± ) + F U (w ± , w ± ) φ(ξ), ξ ∈ R,
where φ(ξ) is the Fourier transform of the function φ(x). If we replace J(u) by u, that is φ(x) by the δ-function, then the spectrum of the integro-differential operator coincides with the spectrum of the reaction-diffusion operator. We note that

F u (w ± , w ± ) + F U (w ± , w ± ) φ(0) = F 0 (w ± ) and Re φ(ξ) = ∞ -∞ φ(x) cos(ξx)dx < ∞ -∞ φ(x)dx = 1.
Fredholm property. Bistable case. Let

F 0 (w ± ) = F u (w ± , w ± ) + F U (w ± , w ± ) < 0
(we recall that φ(0) = 1). Suppose that F u (w + , w + ) < 0 and F U (w + , w + ) > 0. Then Re λ + (ξ) < 0 for all ξ ∈ R since

F u (w ± , w ± ) + Re F U (w ± , w ± ) φ(ξ) ≤ F u (w ± , w ± ) + F U (w ± , w ± ) = F 0 (w ± ).
Hence the essential spectrum is completely in the left-half plane. This allows us to prove properness of the corresponding operators in weighted spaces and to define the topological degree.

Consider now the case where F u (w + , w + ) > 0 and F U (w + , w + ) < 0. The principal difference with the previous case is that the essential spectrum of the integro-differential operator may not be completely in the left-half plane (Figure 1) though this is the case for the reaction-diffusion operator. Depending on the parameters, the essential spectrum can cross the imaginary axis for some pure imaginary values. However the linear operator remains Fredholm since the essential spectrum does not cross the origin; the nonlinear operator remains proper in the corresponding weighted spaces.

Thus, the bistable case for the reaction-diffusion equation gives rise to two different cases for the integro-differential equation. We will call both of them bistable but will distinguish them when necessary.

Monostable case. Suppose that F 0 (w + ) > 0 and F 0 (w -) < 0. Then λ 0 -(ξ) is in the left-half plane for all ξ ∈ R; λ 0 + (ξ) is partially in the right-half plane, λ 0 + (0) > 0. The essential spectrum of the integro-differential operator L given by the curves λ ± (ξ) has a similar structure. It does not cross the origin, so that the operator satisfies the Fredholm property. The curve λ + (ξ) is partially in the right-half plane, λ + (0) = λ 0 + (0) > 0. The curve λ -(ξ) can be completely in the left-half plane or partially in the right-half plane (Figure 1). Similar to the bistable case, there are two subcases in the monostable case.

Index. In order to find the index of the operator L, we consider the operator L τ which depends on the parameter τ characterizing the width of the support of the function φ τ , supp φ τ = [-N τ , N τ ]. We recall that ∞ -∞ φ τ (x)dx = 1. Let L 1 = L, that is the value τ = 1 corresponds to the function φ in the operator L.

Since the essential spectrum of the operator L τ can be determined explicitly, then we can affirm that it converges to the essential spectrum of the operator L 0 as τ → 0. Moreover, L τ converges to L 0 in the operator norm. The essential spectrum of the operator L τ does not cross the origin. Therefore it is normally solvable with a finite dimensional kernel. Hence the index of the operator L equals the index of the operator L 0 [START_REF] Gohberg | The basic propositions on defect numbers and indices of linear operators[END_REF]. It is 0 in the bistable case and 1 in the monostable case (cf. [START_REF] Collet | Computation of the index of linear elliptic operators in unbounded cylinders[END_REF]).

Topological degree and existence of solutions. In the bistable case we can define the topological degree for the integro-differential operator and use the Leray-Schauder method to prove existence of solutions. In order to use this method we need to obtain a priori estimates of solutions. In [START_REF] Demin | Existence of waves for a nonlocal reaction-diffusion equation[END_REF], a priori estimates are obtained in the case where F (u, J(u)) = J(u)u(1 -u) -αu. Thus, we can now conclude about the existence of waves for this particular form of the nonlinearity. More general functions will be considered in the subsequent works.

Local bifurcations and branches of solutions. Other conventional applications of the degree are related to local bifurcations and global branches of solutions (see, e.g., [START_REF] Krasnoselskii | Geometrical methods of nonlinear analysis[END_REF]). We can now use the corresponding results for the integro-differential operator in the bistable case. Let us emphasize that these results apply in particular for the case where the essential spectrum of the linearized operator crosses the imaginary axis (see above). Therefore the wave persists in this case unless a priori estimates are lost.

. 2 )

 2 Here the properness of B is understood in both variables x ∈ G and τ ∈ [0, 1] . Theorem 4.1. ([20]) For every B ∈ H and every open set D, with D ⊂ G, there exists a topological degree γ (B, D) .

Theorem 4 . 3 .

 43 [START_REF] Apreutesei | Competition of species with intra-specific competition[END_REF]. Consider F the class of operators A defined through(1.4) , such that (H4) -(H5) are satisfied. Consider also the class H of homotopies A τ : E µ → E 0 µ , τ ∈ [0, 1] , of the form (2.2) , satisfying (H1) -(H2) and (H6) For every τ ∈ [0, 1] , the equationsu + c (τ ) u + ∂F τ ∂u w ± , w ± u + ∂F τ ∂U w ± , w ± J (u) -λu =0 do not have nonzero solutions in E, (∀) λ ≥ 0. By Theorem 2.5 and Theorem 3.3, we infer that operators A τ (u) are Fréchet differentiable, proper with respect to (u, τ ) and their Fréchet derivatives A τ = L τ verify condition (a) above. Condition (b ) follows from the lemma in the appendix. Hence H has the form (4.2) . Applying Theorem 4.1 for the class of operators F and the class of homotopies H, we are led to the following result. Suppose that functions F τ and c (τ ) satisfy conditions (H1) -(H2) and (H4) -(H6) . Then a topological degree exists for the class F of operators and the class H of homotopies.

Figure 1 :

 1 Figure 1: Schematic representation of a possible location of the essential spectrum of the operator L.

  endowed with the usual norms || • || E and || • || E 0 . We will work in the weighted Holder spaces E µ and E 0 µ , which are E and E 0 , respectively, with the norms ||u|| µ = ||µu|| E and ||u|| 0µ = ||µu|| E 0 .

Acknowledgements

This work was partially supported by the LEA Math Mode between CNRS France and Romanian Academy through the joint project "Existence of travelling waves for nonlocal reaction-diffusion equations".

Passing to the limit as n → ∞, uniformly on bounded intervals of x in (2.4) and (2.5) , one obtains that

)

+µF τ 0 (u 0 + ψ, J (u 0 + ψ)) + µ (ψ + c (τ 0 ) ψ ) = µf 0 .

(2.9) Subtracting (2.9) from (2.5) and denoting V n = v n -v 0 , one finds

(2.10)

Recall that V n = v n -v 0 → 0 as n → ∞ in E loc . We show that V n → 0 in C (R) . Suppose that it is not the case. Then, without any loss of generality, we can chose a sequence

Writing (2.10) in x + x n , one obtains

We will pass to the limit as n → ∞ in (2.13) . First we note that by (2.11) and (2.6) , there exists

) implies a similar estimate for v 0 , so v 0 (x + x n ) and v 0 (x + x n ) are bounded in E. We also have ψ (x + x n ) = 0 for x + x n > 1 and for x + x n < -1 and

(

Denote by T n 1 , T n 2 , T n 3 the three terms in the right-hand side. Hypothesis (H2) for

Next, (2.11) leads to

Fredholm property in weighted spaces

Consider the operator L τ :

and its limiting operators

Recall here Condition NS for L τ , i. e. hypothesis (H3) : For each τ ∈ [0, 1] , the limiting equations L ± τ u = 0 do not have nonzero solutions. We prove now the Fredholm property of L τ as an operator acting between the above weighted Holder spaces.

µ (acting between weighted spaces) is normally solvable with a finite dimensional kernel.

Proof. Like in Theorem 2.2 from [START_REF] Apreutesei | Travelling waves for integro-differential equations in population dynamics[END_REF], we can prove that L τ from E to E 0 is normally solvable with a finite dimensional kernel. To verify the property in the weighted spaces, we use Lemma 2.24 in [START_REF] Volpert | Properness and topological degree for general elliptic operators[END_REF]: if L τ : E → E 0 is normally solvable with a finite dimensional kernel and the operator K :

µ is normally solvable with a finite dimensional kernel.

Let {u i } be a sequence such that ||u i | | Eµ ≤ M. We prove the existence of a subsequence of {Ku i } which converges in

one can find a subsequence, denoted again {v i } , which converges locally in C 2 to a function v 0 , which can be prolonged to R by a diagonalization process. We have v 0 ∈ E, ||v 0 || E ≤ M and v i → v 0 in E loc (in C 2+α (I) , for every bounded interval I).

Let u 0 be such that v 0 = µu 0 . Then

Observe that ||z i || E ≤ 2M and z i → 0 in E loc . Now we can write

, where h (x) → 0 as x → ±∞, z i (x) is uniformly bounded and z i (x) → 0 as i → ∞ locally with respect to x, it follows that µJ (z i /µ) -J (z i ) → 0 as i → ∞, uniformly with respect to x on R. Similarly, z i , z i are uniformly bounded, z i → 0, z i → 0 as i → ∞ locally and µ /µ → 0, µ /µ → 0 as x → ±∞, so the first two terms from (3.3) tend to zero uniformly with respect to x ∈ R, as i → ∞. This implies that

. Therefore, with the aid of the local convergence

The theorem is proved.

We prove now the Fredholm property for L τ : E µ → E 0 µ , under an additional hypothesis. To this end, let I be the identity operator on E µ .

Condition NS(λ) . For each τ ∈ [0, 1] , the limiting equations L ± τ u -λu = 0 associated to the operator L τ -λI do not have nonzero solutions in E µ , for any λ ≥ 0.

We recall an auxiliary result from [START_REF] Apreutesei | Travelling waves for integro-differential equations in population dynamics[END_REF] which will be employed below.

Lemma 3.2. Consider the operators L

. Then there exists ρ ≥ 0 large enough such that the limiting equations (L s ) ± u = 0 do not have nonzero solutions for any s ∈ [0, 1] . Theorem 3.3. If Condition NS (λ) is satisfied, then L τ , regarded as an operator from E µ to E 0 µ , has the Fredholm property and its index is zero. Proof. We put L 0 u = L τ u -λu, L 1 u = u -λu and L s :

. Condition NS(λ) for L τ implies Condition NS for L 0 = L τ -λI. Then, Theorem 3.1 ensures that L 0 = L τ -λI, regarded from E µ to E 0 µ , is normally solvable with a finite dimensional kernel. For operator L 1 , we have ker L 1 = {0} , ImL 1 = E 0 µ , hence L 1 is a Fredholm operator and its index is indL 1 = 0. By Lemma 3.2 applied for L s , there exists λ ≥ 0 large enough such that Condition NS holds for all L s , s ∈ [0, 1] . In view of Theorem 3.1, it follows that the operators L s are normally solvable with a finite dimensional kernel. In other words, the homotopy L s gives a continuous deformation from the operator L 0 to the operator L 1 , in the class of the normally solvable operators with finite dimensional kernels. Such deformation preserves the Fredholm property and the index. Since the index of L 1 is zero, we derive that the index of all L s is zero. In particular, for s = 0 and λ = 0, one arrives at the conclusion that L τ has the Fredholm property and its index is zero. This completes the proof.

Topological degree

In this section we apply the topological degree construction for Fredholm and proper operators with the zero index constructed in [START_REF] Volpert | Properness and topological degree for general elliptic operators[END_REF] to the integro-differential operators.

Appendix

Sectorial property of an operator implies certain location of its essential spectrum and an estimate of the resolvent. For general elliptic problems in unbounded domains it is proved in [START_REF] Volpert | Elliptic partial differential equations[END_REF]. A simple particular case of second-order operators on the axis is considered in [START_REF] Volpert | Traveling wave solutions of parabolic systems[END_REF]. In the lemma below we prove an estimate of the resolvent using this last result.

where the coefficients of this operator are sufficiently smooth bounded functions. Then the operator M λ u = M 0 u -λu, considered as acting in the same spaces, has a bounded inverse with the norm independent of λ for λ ≥ λ 0 > 0, where λ 0 is sufficiently large.

Proof. Consider the equation

We need to obtain the estimate u C 2 (R) ≤ K f C(R) (5.4) of this equation where K is independent of λ for all λ sufficiently large. Here and below we denote by K the constants independent of u, f and λ.

We first prove the estimate

(5.5)

Since the operator

is sectorial [START_REF] Volpert | Traveling wave solutions of parabolic systems[END_REF], then

Estimate (5.5) follows from the last one for λ sufficiently large. We can write equation (5.3) in the form

We can choose σ > 0 such that the operator in the left-hand side is invertible. Hence

This estimate and (5.5) give (5.4). The lemma is proved.

This lemma remains valid for the operators acting in the weighted spaces.