Cristina Leon

Irina Kutsenko
email: kutsenko-il@rudn.ru

S M Nikol'skii

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

S M Nikol

EXISTENCE OF SOLUTIONS FOR A NONLOCAL REACTION-DIFFUSION EQUATION IN BIOMEDICAL APPLICATIONS BY

The paper is devoted to a nonlocal semi-linear elliptic equation in R n arising in various biological and biomedical applications. The Fredholm property studied for the corresponding linear elliptic operators with discontinuous coefficients allows the application of the implicit function theorem to prove the persistence of solutions under a small perturbation of the problem. Furthermore, the existence of solutions is established by the Leray-Schauder method based on the topological degree for Fredholm and proper operators and on a priori estimates of solutions in some special weighted spaces.

Introduction

In this work, we study the existence of solutions of the equation (1.1) Δu + au(1 -I(u)) -F (u, x) = 0 considered in the whole space R n , n = 1, 2, 3, where

I(u) = R n u(x)dx.
Such equations arise in various biological and biomedical applications where u(x) corresponds to the density of some population (animals, cells, viruses) [START_REF] Bessonov | Mathematics of Darwin's Diagram[END_REF][START_REF] Bessonov | Genotype dependent virus distribution and competition of virus strains[END_REF].

The diffusion term describes random motion of the individual entities of the population, the second term characterizes their reproduction, and the last term their mortality. The reproduction term is proportional to the population density and to the available resources (1 -I(u)). Here K = 1 is dimensionless carrying capacity, and I(u) corresponds to consumed resources proportional to the total population. Thus, instead of the conventional logistic term u(1u) appropriate for the case of local consumption of resources, we consider the integral term for the global consumption [START_REF] Volpert | Spectrum of elliptic operators and stability of travelling waves[END_REF]. The specific form of the mortality term depends on the applications. In particular, in a model of viral infection development,

F (u, x) = uf (u) + σ(x)u,
where the first term in this function describes virus elimination due to the immune response, and the second term its natural mortality.

Let us note that in the models of population dynamics the space variable x can have two different meanings. The first one corresponds to the conventional physical space where the function u(x, t) describes the distribution of animals in some habitats or cell (viruses) in the tissue. In the second interpretation, the variable x characterizes the genotype or the phenotype of the population. The population density distribution as a function of its genotype describes the existence and the evolution of biological species, cell lineages and cell clones in cancer, or virus strains. In this case, a conventional mathematical question about the existence of solutions acquires a clear and important biological significance allowing the determination of the conditions of the existence of biological species (clones, strains) (see [START_REF] Bessonov | Mathematics of Darwin's Diagram[END_REF][START_REF] Bessonov | Genotype dependent virus distribution and competition of virus strains[END_REF]). Mathematical analysis of equation (1.1) has some specific features because it is considered in an unbounded domain, and also because of the presence of the integral term and possibly discontinuous coefficients. General elliptic problems in bounded domains with a sufficiently smooth boundary satisfy the Fredholm property if and only if the ellipticity condition, the condition of proper ellipticity and the Lopatinskii conditions are satisfied [START_REF] Agmon | Estimates near the boundary for solutions of elliptic partial diffrential equations satisfying general boundary conditions[END_REF][START_REF] Agmon | Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions[END_REF][START_REF] Volevich | Solvability of boundary problems for general elliptic systems[END_REF]. In the case of unbounded domains, these conditions may not be sufficient. Elliptic problems in unbounded domains are characterized by the presence of the essential spectrum determined by limiting operators [START_REF] Rabinovich | Limit Operators and their Applications in Operator Theory[END_REF][START_REF] Volpert | Elliptic Partial Differential Equations[END_REF]. Solvability conditions for linear problems, their index and Fredholm property determine the properties of nonlinear operators including their properness and topological degree.

The topological degree for elliptic operators was introduced by Leray and Schauder [START_REF] Leray | Topologie et équations fonctionnelles[END_REF] with the reduction to the operators I + K, where I is the identity operator and K is a compact operator. This construction is not applicable to elliptic problems in unbounded domains since, contrary to the case of bounded domains, the inverse to the Laplace operator is not compact. There are various degree constructions in an abstract setting [START_REF] Borisovich | Nonlinear Fredholm maps and the Leray-Schauder theory[END_REF][START_REF] Elworthy | Degree theory on Banach manifolds[END_REF][START_REF] Elworthy | Differential structures and Fredholm maps on Banach manifolds[END_REF][START_REF] Fenske | Analytische Theorie des Abbildungrades für Abbildungen in Banachräumen[END_REF], [START_REF] Fitzpatrick | The parity as an invariant for detecting bifurcaton of the zeroes of one parameter families of nonlinear Fredholm maps[END_REF]- [START_REF] Fitzpatrick | Orientability of Fredholm families and topological degree for orientable nonlinear Fredholm mappings[END_REF] and in the framework of elliptic problems [START_REF] Dancer | Boundary value problems for ordinary differential equations on infinite intervals[END_REF][START_REF] Volpert | Applications of the rotation theory of vector fields to the study of wave solutions of parabolic equations[END_REF]. We will use the degree construction for Fredholm and proper operators with the zero index [START_REF] Volpert | Properness and topological degree for general elliptic operators[END_REF]. Together with a priori estimates of solutions, the topological degree allows the application of the Leray-Schauder method to prove the existence of solutions.

We will apply these methods of linear and nonlinear analysis to study equation (1.1). The presence of the integral term implies the integrability of the solution and imposes some constraints on the function spaces. We will also be interested in equations with discontinuous coefficients because they can admit explicit analytical solutions. The latter can be used as a starting point for the continuation of solutions. However, they also require a special functional setting which will be discussed below. Altogether, this extended formulation of classical elliptic problems leads us to revisit the existing theory. Since these results can represent an independent interest for the mathematical theory, and in view of other possible applications, we will prove more general results and apply them to study equation (1.1). In the next section, we study linear operators with discontinuous coefficients. Section 3 is devoted to the application of the implicit function theorem in the case of discontinuous functions. Finally, in Section 4 we introduce the topological degree and study the existence of solutions by the Leray-Schauder method.

Linear operators with discontinuous coefficients

Fredholm property of general elliptic problems in unbounded domains is studied in [START_REF] Volpert | Elliptic Partial Differential Equations[END_REF] in the case where the coefficients of the problem belong to some Hölder spaces. In this section we will use these results in order to study a class of elliptic operators with discontinuous coefficients.

2.1. Operators and spaces. Consider a Banach space E with the norm • E , and the space E ∞ with the norm

u E∞ = sup y∈R n u(•)ω(• -y) E ,
where ω(x) is an infinitely differentiable function, 0 ≤ ω(x) ≤ 1 for x ∈ R n , ω(x) = 1 for |x| ≤ 1/2 and ω(x) = 0 for |x| ≥ 1 (see [START_REF] Volpert | Elliptic Partial Differential Equations[END_REF] for more details). In particular, we will consider the spaces

L 2 ∞ (R n ) and H 2 ∞ (R n ) for which E = L 2 (R n ) and E = H 2 (R n ), respectively, n = 1, 2, 3.
Consider the linear second-order elliptic operator

Lu = Δu + n i=1 a i (x) ∂u ∂x i + b(x)u,
where

a i (x) = a 0 i (x) + a 1 i (x), b(x) = b 0 (x) + b 1 (x), a 0 i , b 0 ∈ C α (R n ),
(R n ) into L 2 ∞ (R n
). We will also consider the operator

L 0 u = Δu + n i=1 a 0 i (x) ∂u ∂x i + b 0 (x)u.

2.2.

Limiting operators and a priori estimates. Consider an arbitrary sequence

x k ∈ R n such that |x k | → ∞ as k → ∞. Set a k i (x) = a 0 i (x + x k ), b k (x) = b 0 (x + x k ), k = 1, 2
, . . . . Since these functions belong to the Hölder space, they converge locally uniformly to some functions a * i (x), b * (x). These limiting functions can depend on the choice of the sequence x k . The operator L with the limiting coefficients

Lu = Δu + m i=1 a * i (x) ∂u ∂x i + b * (x)u
is called the limiting operator.

Condition NS. Any limiting equation Lu =0 has only zero solution in H 2 ∞ (R n ). The following lemma is a particular case of a more general result proved in [START_REF] Volpert | Elliptic Partial Differential Equations[END_REF] (Chapter 4, Theorem 2.10, p. 155).

Lemma 2.1: Let Condition NS be satisfied. Then there exist numbers M and R such that the following estimate holds:

(2.1) u H 2 ∞ (R n ) ≤ M ( L 0 u L 2 ∞ (R n ) + u L 2 (BR) ) for any u ∈ H 2 ∞ (R n ). Here B R = {x ∈ R n , |x| ≤ R}.
We will use this lemma to get a similar estimate for the operator L. Lemma 2.1 holds under the assumption that the coefficients of the operator L 0 belong to the Hölder space. This result cannot be directly applied to the operator L with discontinuous coefficients. We prove it in the following theorem. Let us note that limiting operators for the operator L 0 and L are the same. Theorem 2.2: Let Condition NS be satisfied. Then there exist numbers M and R such that the following estimate holds:

(2.2) u H 2 ∞ (R n ) ≤ M ( Lu L 2 ∞ (R n ) + u L 2 (BR) ) for any u ∈ H 2 ∞ (R n ). Here B R = {x ∈ R n , |x| ≤ R}. Proof. Set Bu = m i=1 (a i (x) -a 0 i (x)) ∂u ∂x i + (b(x) -b 0 (x))u.
Then L = L 0 + B, and estimate (2.1) can be written as follows:

(2.3) u H 2 ∞ (R n ) ≤ M ( (L -B)u L 2 ∞ (R n ) + u L 2 (BR) ) ≤ M ( Lu L 2 ∞ (R n ) + Bu L 2 ∞ (R n ) + u L 2 (BR) ).
We need to estimate the term Bu L 2 ∞ (R n ) . We have

Bu = m i=1 a 1 i (x) ∂u ∂x i + b 1 (x)u.
Since a 1 i (x) and b 1 (x) are bounded functions with bounded supports, then

|a 1 i (x), b 1 (x)| ≤ K
for some positive constant K, and R can be chosen sufficiently large such that their supports belong to the ball B R . Hence,

b 1 u L 2 ∞ (R n ) ≤ R n (b 1 (x)u(x)) 2 dx 1/2 ≤ K BR u 2 (x)dx 1/2 , a 1 i ∂u ∂x i L 2 ∞ (R n ) ≤ K BR | ∂u ∂x i | 2 dx 1/2 ≤ u H 2 (BR) + C u L 2 (BR) .
The last estimate holds for any positive and some constant C depending on .

For sufficiently small, these estimates and (2.3) provide (2.2). The theorem is proved.

2.3. Properness, normal solvability, the Fredholm property. Recall that a linear operator acting in Banach spaces is normally solvable if and only if its image is closed. The operator is called proper on closed bounded sets if the inverse image of any compact set is compact in any bounded closed set. If a linear operator is proper, then it is normally solvable with a finite-dimensional kernel (see [START_REF] Volpert | Elliptic Partial Differential Equations[END_REF], Theorem 2.13, p. 163). By definition, a linear operator satisfies the Fredholm property if it is normally solvable, the dimension of its kernel is finite and the codimension of its image is also finite. The latter is equivalent to a finite number of solvability conditions of the non-homogeneous equation.

Condition NS is a necessary and sufficient condition for general elliptic problems in unbounded domains to be normally solvable with a finite-dimensional kernel [START_REF] Volpert | Elliptic Partial Differential Equations[END_REF]. This result is proved in the case of Hölder continuous coefficients of the operator. The next theorem affirms the sufficiency of this condition for the operators with discontinuous coefficients. The necessity of Condition NS is proved in [START_REF] Volpert | Elliptic Partial Differential Equations[END_REF] by construction of non-compact sequences providing a nonzero solution of some limiting operator. A similar construction may be also applicable here since the limiting operators of the operators L 0 and L are the same.

Continuous deformation in the class of normally solvable operators with a finite-dimensional kernel does not change the index of the operator [START_REF] Gohberg | The basic propositions on defect numbers and indices of linear operators[END_REF]. Therefore, we can find the index of the operator reducing it to another operator with a known index by a continuous deformation, assuming that Condition NS is satisfied in the process of this deformation. This approach is used in Theorem 2.4 and Lemma 3.3 below. In particular, if the reduction can be applied to an invertible operator, then the index equals 0.

Theorem 2.3: Let Condition NS be satisfied. Then the operator L is normally solvable with a finite-dimensional kernel.

Proof. It is sufficient to prove that the operator L is proper. This property follows from Theorem 2.2. Indeed, let Lu n = f n , f n → f 0 , and the sequence u n be bounded in H 2 ∞ (R n ). We will verify that we can choose a convergent subsequence from this sequence. Since H 2 is locally compact in L 2 , then we can choose a subsequence u nk , which forms a fundamental sequence in L 2 (B R ). Applying Theorem 2.2 to the equation

L(u nk -u nj ) = f nk -f nj , we conclude that this sequence is also fundamental in H 2 ∞ (R n ). Therefore, it converges to some function u 0 ∈ H 2 ∞ (R n ).
The theorem is proved.

Theorem 2.4: The operator L satisfies the Fredholm property if the operator L 0 satisfies it. In this case, their indices are equal to each other.

Proof. Suppose that the operator L 0 satisfies the Fredholm property. Then Condition NS is satisfied for L 0 . Since the limiting operators for the operator L 0 and L are the same, then Condition NS is also satisfied for the operator L and for their linear combination L τ = τL 0 + (1τ )L. Hence, the operator L τ is normally solvable with a finite-dimensional kernel for all τ ∈ [0, 1], and its index does not depend on τ . The theorem is proved.

Example 2.5: Suppose that b 0 (x) → β as |x| → ∞, where β is a negative number. Then the essential spectrum of the operator L 0 lies in the sector of the complex plane which does not contain the positive half-axis [START_REF] Volpert | Elliptic Partial Differential Equations[END_REF], and its index equals 0. According to the previous theorem, the same properties hold for the operator L.

Solvability of the nonlocal equation.

Let us briefly recall some definitions and spectral properties used below (see [START_REF] Gohberg | The basic propositions on defect numbers and indices of linear operators[END_REF]). Let L be a linear operator acting in Banach spaces. The set of all complex numbers λ such that the operator L λ = Lλ does not satisfy the Fredholm property is called the essential spectrum of the operator L. The essential spectrum splits the complex plane into a finite or a countable number of connected components.

Inside each such connected component, the index κ = αβ of the operator L λ , where α is the dimension of the kernel and β the codimension of the image, has a constant value. The numbers α and β are also constants except, possibly, for some discrete set of λ. Next, suppose that the essential spectrum of the operator L belongs to a half-plane Re λ ≤ r of the complex plane, where r is a real number, and the index of the operator equals 0 in the complementing half-plane Re λ > r. This property is satisfied for general elliptic problems in unbounded domains [START_REF] Volpert | Elliptic Partial Differential Equations[END_REF]. For any > 0, the half-plane Re λ ≥ r + contains a finite number of points λ of the spectrum. The operator L λ satisfies the Fredholm property for such λ but it is not invertible, and α = β. These points λ are eigenvalues of finite multiplicity. The eigenvalue with the maximal real part is called the principal eigenvalue. Due to the Krein-Rutman theorem applicable for the scalar second-order elliptic problems in bounded domains, the principal eigenvalue is real, simple, and the corresponding eigenfunction is positive. Moreover, there are no other positive eigenfunctions. This result is generalized for unbounded domains [START_REF] Volpert | Applications of the rotation theory of vector fields to the study of wave solutions of parabolic equations[END_REF] (see also [START_REF] Berestycki | The principal eigenvalue and maximum principle for second-order elliptic operators in general domains[END_REF][START_REF] Volpert | Spectrum of elliptic operators and stability of travelling waves[END_REF]).

We will now use these spectral properties in order to study the solvability of the equation

(2.4) Δu + uf (I(u)) -σ(x)u = 0,
where

I(u)= R n u(x)dx, f (•) is a continuous function, σ(x) ∈ C α (R n ), σ(x) → σ 0 as |x| → ∞, σ 0 is a positive constant.
We will see below that the solution of equation (2.4) is an eigenfunction of a Fredholm operator. Therefore, it decays exponentially at infinity [START_REF] Volpert | Elliptic Partial Differential Equations[END_REF], and the integral I(u) is well defined. We will obtain the conditions for the existence of a positive integrable solution of this equation.

Theorem 2.6: Suppose that the principal eigenvalue λ 0 of the operator

L0 = Δu -σ(x)u satisfies the inequality λ 0 > -σ 0 . Then equation (2.

4) has a positive integrable solution if and only if the equation

(2.5) f (h) = -λ 0 has a positive solution h.
Proof. The essential spectrum of the operator L0 fills the half-axis λ ≤ -σ 0 . Since λ 0 > -σ 0 , then the principal eigenvalue λ 0 is real and simple, the corresponding eigenfunction u 0 (x) is positive, and there are no positive eigenfunctions corresponding to other eigenvalues [START_REF] Volpert | Applications of the rotation theory of vector fields to the study of wave solutions of parabolic equations[END_REF].

Suppose that equation (2.5) has a positive solution h = h 0 . Since λ 0 is the principal eigenvalue of the operator L0 with a positive eigenfunction u 0 (x), then u 0 (x) is a solution of the equation

(2.6) Δu -σ(x)u = -f (h 0 )u.
The eigenfunction u 0 (x) is exponentially decaying at infinity [START_REF] Volpert | Elliptic Partial Differential Equations[END_REF], and the integral I(u 0 ) is well defined. For any real constant c, the function u(x) = cu 0 (x) is also a solution of equation (2.6). We can choose c in such a way that

I(u) = h 0 .
Then u is a solution of equation (2.4).

Next, suppose that u(x) is a positive integrable solution of equation (2.4). Set h = I(u). Then u(x) is a solution of the equation

Δu -σ(x)u = -f (h)u.
Since a positive eigenfunction corresponds only to the principal eigenvalue [START_REF] Volpert | Spectrum of elliptic operators and stability of travelling waves[END_REF], then -f (h) = λ 0 . The theorem is proved.

Example 2.7: Set f (h) = 1h, and suppose that σ(x) ≥ 0 for all x ∈ R n , σ 0 > 1, -1 < λ 0 < 0. Then equation (2.5) has a positive solution h. According to the theorem, equation (2.4) has a positive integrable solution.

2.5. Spectral properties of operators with discontinuous coefficients. In the analysis of equation (2.4) we used the spectral properties of the operator L0 , namely, the location of its essential spectrum and the properties of the principal eigenvalue. In order to apply these properties, we assumed that the function σ(x) is Hölder continuous. Theorem 2.4 allows us to determine the essential spectrum for a wider class of functions including discontinuous functions if the discontinuities belong to a bounded domain on R n . Let us now discuss the properties of the principal eigenvalue in the case of discontinuous coefficients. We will verify that the corresponding eigenfunction is positive and that there are no positive eigenfunctions corresponding to other eigenvalues.

Consider the operator L0 u = Δuσ(x)u assuming that σ(x) = σ 0 (x) + σ 1 (x), where σ 0 (x) ∈ C α (R n ) and σ 1 (x) is a bounded function with a bounded support, σ(x) ≥ 0 for all x ∈ R n . Suppose that σ 0 (x) → σ 0 as |x| → ∞, where σ 0 is a positive constant, and the principal eigenvalue λ 0 of the operator L 0 satisfies the inequality λ 0 > -σ 0 .

Lemma 2.8: The eigenfunction u 0 (x) corresponding to the principle eigenvalue λ 0 is positive.

Proof. Consider a sequence of functions σ k (x) ∈ C α (R n ) which coincide with σ 0 (x) for |x| ≥ R for some R sufficiently large, and

σ k (x) → σ(x) in L 2 (R n ).
We suppose that the support of the function

σ 1 (x) is inside the ball B R of radius R. The corresponding operators Lk u = Δu -σ k (x)u
have the same limiting operators and essential spectrum as the operator L0 , and they satisfy the Fredholm property. Denote by λ k the principle eigenvalue of the operator Lk , and by u k (x) the corresponding eigenfunction.

Since

u C(R n ) ≤ u H 2 ∞ (R n ) for u ∈ H 2 ∞ (R n ) (n = 1, 2, 3), we have Lk u -L0 u L 2 ∞ (R n ) ≤ BR (σ k (x) -σ(x)) 2 u 2 (x)dx 1/2 ≤ u H 2 ∞ (R n ) BR (σ k (x) -σ(x)) 2 dx 1/2 → 0, k → ∞.
Therefore, the operators converge in the operator norm. By virtue of the properties of Fredholm operators, λ k → λ 0 , and u k → u 0 as k → ∞, where u 0 (x) is the eigenfunction of the operator L0 corresponding to the eigenvalue λ 0 [START_REF] Gohberg | The basic propositions on defect numbers and indices of linear operators[END_REF]. Since u k (x) > 0 for all x, then u 0 (x) ≥ 0. Let us show that this inequality is strict. Suppose that u 0 (x 0 ) = 0 for some x 0 . If σ(x) is continuous at x = x 0 , then we obtain a contradiction with the usual positiveness theorem. Therefore, u 0 (x) > 0 for |x| > R, where σ(x) is continuous. We can now apply the generalized maximum principle to affirm that u 0 (x) > 0 for |x| ≤ R. The lemma is proved.

Lemma 2.9: The eigenfunction u 0 (x) corresponding to the principle eigenvalue λ 0 is the only positive eigenfunction.

Proof. We will prove that a positive eigenfunction cannot exist for other eigenvalues and that a positive eigenfunction is unique for the eigenvalue λ 0 up to a constant factor.

Suppose that there exists a positive eigenfunction u 1 (x) for a real eigenvalue λ 1 < λ 0 . As in the proof of the previous lemma, consider a sequence of smoothed problems with the corresponding sequence of eigenvalues λ k 1 → λ 1 .

Since the corresponding eigenfunctions u k 1 (x) converge to u 1 (x) locally uniformly, then u k 1 (x) > 0 for |x| = N for N fixed and for k sufficiently large. If N is sufficiently large, then u k 1 (x) > 0 for |x| ≥ N (see [START_REF] Volpert | Location of spectrum and stability of solutions for monotone parabolic systems[END_REF], Lemma 3.2). Therefore, u k 1 (x k ) ≤ 0 for some x k , |x k |≤ N . Passing to the limit, we conclude that the function u 1 (x) is not positive. This contradiction proves the required assertion.

If a positive eigenfunction u 1 (x) corresponds to the principal eigenvalue λ 0 and it is linearly independent with the eigenfunction u 0 (x), then consider the function z(x) = τu 0 (x)u 1 (x). Let us take the minimal value of τ for which z(x) ≥ 0 for |x| ≤ N and for the same N as above. Then z(x) ≥ 0 for |x| ≥ N . Hence z(x) ≥ 0 for all x ∈ R n . If z(x) > 0 for all |x| ≤ N , then τ is not minimal, and we obtain a contradiction with the assumption above. If z(x) ≥ 0 and z(x 0 ) = 0 for some x 0 , then we obtain a contradiction with the maximum principle. The lemma is proved.

We proved in Lemmas 2.8 and 2.9 that the properties of the principal eigenfunction remain valid for the operators with discontinuous coefficients. These properties were used in Theorem 2.6. Therefore, this theorem is now applicable for the operators with discontinuous coefficients. Similar results can be obtained for more general operators.

Persistence of solutions

3.1. Operators and spaces. Along with the spaces L 2 ∞ (R n ) and H 2 ∞ (R n ) introduced in the previous section, we consider the corresponding weighted spaces

L 2 ∞,μ (R n ) and H 2 ∞,μ (R n ) with the norms u L 2 ∞,μ (R n ) = uμ L 2 ∞ (R n ) , u H 2 ∞,μ (R n ) = uμ H 2 ∞ (R n )
and the weight function μ(x) = exp(ν 1 + |x| 2 ). The value of constant ν > 0 will be specified below. Consider the operator

A(u, τ ) = Δu + F (u, I(u), x, τ)
depending on a parameter τ ∈ [0, 1] and acting from

E 1 = H 2 ∞,μ (R n ) into E 2 = L 2 ∞,μ (R n ). Here I(u) = R n u(x)dx, F (u, v, x, τ
) is a real function of the variables u, v, x and of parameter τ . Since we consider the space with an exponential weight, the integral I(u) is well defined. We will assume that the function F (u, v, x, τ ) satisfies the following conditions.

Condition 1:

The following inequalities hold:

|F (u 1 , v, x, τ) -F (u 2 , v, x, τ)| ≤ K|u 1 -u 2 |, (3.1) |F (u, v 1 , x, τ) -F (u, v 2 , x, τ)| ≤ K|v 1 -v 2 ||u|, (3.2) |F (u, v, x, τ 1 ) -F (u, v, x, τ 2 )| ≤ Kg(x, τ 1 , τ 2 )|u|, (3.3)
where K is a positive constant, and

g(x, τ 1 , τ 2 ) L 2 ∞,μ (R n ) → 0 as |τ 1 -τ 2 | → 0.
These inequalities hold for any

u 1 , u 2 , v 1 , v 2 such that |u i |, |v i | ≤ M , i=1,2,
where M is an arbitrary positive constant, and K depends on M , and it is independent of

x ∈ R n ; τ 1 , τ 2 ∈ [0, 1].

Condition 2:

The following inequalities hold:

|F u (u 1 , v, x, τ) -F u (u 2 , v, x, τ)|, |F v (u 1 , v, x, τ) -F v (u 2 ,v, x, τ)| ≤ K|u 1 -u 2 |, (3.4) |F u (u, v 1 , x, τ) -F u (u, v 2 , x, τ)|, |F v (u, v 1 , x, τ) -F v (u,v 2 , x, τ)| ≤ K|v 1 -v 2 ||u|, (3.5) |F u (u, v, x, τ 1 ) -F u (u, v, x, τ 2 )|, |F v (u, v, x, τ 1 ) -F v (u,v, x, τ 2 )| ≤ Kg(x, τ 1 , τ 2 ), (3.6)
where K is a positive constant, and

g(x, τ 1 , τ 2 ) L 2 ∞,μ (R n ) → 0 as |τ 1 -τ 2 | → 0.
These inequalities hold for any

u 1 , u 2 , v 1 , v 2 such that |u i |, |v i | ≤ M , i = 1, 2,
where M is an arbitrary positive constant, and K depends on M , and it is independent of

x ∈ R n ; τ 1 , τ 2 ∈ [0, 1].
Condition 3: There exists a positive constant N such that

F u (u, v, x, τ ), F v (u, v, x, τ ) ∈ C α (R N )
for any u, v, and τ ∈ [0, 1] fixed, where R N = {x ∈ R n , |x| ≥ N }, α ∈ (0, 1). Furthermore, F u (0, 0, x, τ), F u (0, 0, x, τ) <for all |x| ≥ N , τ ∈ [0, 1], and some > 0.

Lemma 3.1: If Condition 1 is satisfied, then the operator

A(u, τ ) : E 1 × R → E 2
is bounded and continuous.

Proof. Let u 1 , u 2 ∈ B ρ ⊂ E 1
, where B ρ is a ball with radius ρ. Then

u i C(R n ) ≤ M, i = 1, 2,
where the constant M depends on ρ. From (3.1)-( 3.3) we have the following estimates:

|A(u 1 , τ 1 ) -A(u 2 , τ 2 )| ≤ |Δ(u 1 -u 2 )| + K(|u 1 -u 2 | + |I(u 1 ) -I(u 2 )||u 2 | + g(x, τ 1 , τ 2 )|u 2 |).
Here and in what follows K denotes any constant which depends only on ρ. Then (3.7)

A(u 1 , τ 1 ) -A(u 2 , τ 2 ) L 2 ∞,μ (R n ) ≤K u 1 -u 2 H 2 ∞,μ (R n ) + K|I(u 1 ) -I(u 2 )| u 2 L 2 ∞,μ (R n ) + K u 2 C(R n ) g(x, τ 1 , τ 2 ) L 2 ∞,μ (R n ) . Since |I(u 1 ) -I(u 2 )| ≤ Rn 1 μ(x) |u 1 (x) -u 2 (x)|μ(x)dx ≤ K (u 1 -u 2 )μ C(R n ) ≤ u 1 -u 2 H 2 ∞,μ (R n ) , then it follows from (3.7) that A(u 1 , τ 1 ) -A(u 2 , τ 2 ) L 2 ∞,μ (R n ) → 0 as u 1 -u 2 H 2 ∞,μ (R n ) → 0, |τ 1 -τ 2 | → 0.
Hence the operator is continuous. Its boundedness obviously follows from (3.7).

The lemma is proved.

Next, consider the linearized operator L(u 0 , τ 0 )u = Δu + F u (u 0 , I(u 0 ), x, τ 0 )u + F v (u 0 , I(u 0 ), x, τ 0 )I(u).

Lemma 3.2: If Conditions 2 and 3 are satisfied, then the operator L(u 0 , τ 0 ) is continuous with respect to (u 0 , τ 0 ) in the operator norm.

Proof. We have

L(u 1 , τ 1 )u -L(u 0 , τ 0 )u =(F u (u 1 , I(u 1 ), x, τ 1 ) -F u (u 0 , I(u 0 ), x, τ 0 ))u + (F v (u 1 , I(u 1 ), x, τ 1 ) -F v (u 0 , I(u 0 ), x, τ 0 ))I(u).
It follows from Condition 2 that

(F u (u 1 , I(u 0 ), x, τ 0 ) -F u (u 0 , I(u 0 ), x,τ 0 ))u L 2 ∞,μ (R n ) ≤ K u 1 -u 2 C(R n ) u L 2 ∞,μ (R n ) , (F u (u 1 , I(u 1 ), x, τ 0 ) -F u (u 1 , I(u 0 ), x,τ 0 ))u L 2 ∞,μ (R n ) ≤ K|I(u 1 ) -I(u 2 )| u L 2 ∞,μ (R n ) , (F u (u 1 , I(u 1 ), x, τ 1 ) -F u (u 1 , I(u 1 ), x,τ 0 ))u L 2 ∞,μ (R n ) ≤ K g(x, τ 1 , τ 2 )u L 2 ∞,μ (R n ) ≤ K g(x, τ 1 , τ 2 ) L 2 ∞,μ (R n ) u C(R n ) ≤ K g(x, τ 1 , τ 2 ) L 2 ∞,μ (R n ) u H 2 ∞,μ (R n ) , and from Condition 3 that (F v (u 1 , I(u 0 ), x, τ 0 ) -F v (u 0 , I(u 0 ), x,τ 0 ))I(u) L 2 ∞,μ (R n ) ≤ K|I(u)| u 1 -u 2 L 2 ∞,μ (R n ) , (F v (u 1 , I(u 1 ), x, τ 0 ) -F v (u 1 , I(u 0 ), x,τ 0 ))I(u) L 2 ∞,μ (R n ) ≤ |I(u 1 ) -I(u 0 )||I(u)| u 1 L 2 ∞,μ (R n ) , (F v (u 1 , I(u 1 ), x, τ 1 ) -F v (u 1 , I(u 1 ), x,τ 0 ))I(u) L 2 ∞,μ (R n ) ≤ K|I(u)| g(x, τ 1 , τ 2 ) L 2 ∞,μ (R n ) . Taking into account that u 1 -u 2 C(R n ) , |I(u 1 ) -I(u 2 )| ≤ K u 1 -u 2 H 2 ∞,μ (R n )
, we conclude from the previous estimates that

L(u 1 ,τ 1 )u -L(u 0 , τ 0 )u L 2 ∞,μ (R n ) ≤K u 1 -u 2 H 2 ∞,μ (R n ) u H 2 ∞,μ (R n ) + g(x, τ 1 , τ 2 ) L 2 ∞,μ (R n ) u H 2 ∞,μ (R n ) . Therefore, L(u 1 , τ 1 )u -L(u 0 , τ 0 )u → 0 in the operator norm as u 1 -u 2 H 2 ∞,μ (R n ) → 0 and |τ 1 -τ 2 | → 0.
The lemma is proved. Proof. Consider, first, the operator

L 1 u = Δu + F u (u 0 , I(u 0 ), x, τ 0 )u.
By virtue of Conditions 2 and 3, we can represent the derivative as a sum of two functions,

F u (u 0 , I(u 0 ), x, τ 0 ) = a 0 (x) + a 1 (x),
where a 0 (x) ∈ C α (R n ) and a 1 (x) is a bounded function with a bounded support.

The operator L 0 1 u = Δu+a 0 (x)u is normally solvable with a finite-dimensional kernel. Indeed, all its limiting operators Lu = Δu + â(x)u are invertible since the limiting coefficients are such that â(x) ≤ -< 0 for all x ∈ R n [START_REF] Volpert | Elliptic Partial Differential Equations[END_REF]. Furthermore, it satisfies the Fredholm property, and it has the zero index since it can be reduced by a continuous deformation to an invertible operator in the class of normally solvable operators with a finite-dimensional kernel. From Theorem 2.4 it follows that the operator L 1 is Fredholm with the zero index. It remains to note that the operator L(u 0 , τ 0 ) differs from the operator L 1 by a finite-dimensional operator which does not change the Fredholm property and index. The lemma is proved.

Example 3.4: Consider the following function arising in various applications:

F (u, I(u), x, τ) = u(1 -I(u)) -σ τ (x)u.
Suppose that σ τ (x) = σ 0 (x) + σ 1,τ (x), where σ 0 (x) ∈ C α (R n ), and σ 1,τ (x) is a bounded function with a bounded support such that σ 1,τ1 (x) → σ 1,τ0 (x) in L 2 (R n ) as τ 1 → τ 0 . Then the function F satisfies Condition 1, and its derivatives satisfy Condition 2. Finally, if there exist limits σ 0 (±∞) < 0, then Condition 3 is also satisfied.

Implicit function theorem.

The results of the previous section allow us to use the implicit function theorem in order to study the persistence of solutions under small perturbations. Theorem 3.5: Suppose that Conditions 1-3 are satisfied, and A(u 0 , τ 0 ) = 0 for some u 0 ∈ H 2 μ,∞ (R n ) and τ 0 ∈ (0, 1). If the linearized operator A u (u 0 , τ 0 ) does not have a zero eigenvalue, then the equation A(u, τ ) = 0 has a solution u ∈ H 2 μ,∞ (R n ) for all τ sufficiently close to τ 0 .

The proof of the theorem directly follows from Lemmas 3.1-3.3 and the implicit function theorem.

Example 3.6: In the previous theorem, we supposed that the linearized operator does not have a zero eigenvalue. Consider an example where this condition can be explicitly verified. Suppose that the equation

(3.8) Δu + au(1 -I(u)) -σ(x)u = 0
has a positive solution u 0 (x) decaying at infinity. Here a > 0, and the function σ(x) satisfies the conditions of Example 3.4. Consider the nonlinear operator

A(u) = Δu + au(1 -I(u)) -σ(x)u
and the operator L linearized about u 0 :

Lv = A (u 0 )v = Δv + av(1 -I(u 0 )) -σ(x)v -au 0 (x)I(v).
Suppose that it has a zero eigenvalue. Then (3.9) Δv + av(1 -I(u 0 ))σ(x)v = au 0 (x)I(v).

The operator

Lv = Δv + av(1 -I(u 0 )) -σ(x)v
has a zero eigenvalue with the eigenfunction u 0 (x). Since it is positive, then λ = 0 is the principal eigenvalue of this operator, and it is simple (Lemmas 2.8 and 2.9). Multiplying equation (3.9) by u 0 (x) and integrating, we conclude that

I(v) ∞ -∞ u 2 0 (x)dx = 0.
Hence, I(v) = 0, and from (3.9) we obtain

(3.10) Δv + av(1 -I(u 0 )) -σ(x)v = 0.
Since v = u 0 (x) is the only solution of this equation, and it is positive, we obtain a contradiction with the equality I(v) = 0. This contradiction shows that the linearized operator does not have a zero eigenvalue, and the implicit function theorem is applicable.

Existence of solutions

4.1. Topological degree. In this section, we will prove the existence of solutions of equation (1.1) by the Leray-Schauder method using the topological degree for elliptic operators in unbounded domains [START_REF] Volpert | Elliptic Partial Differential Equations[END_REF]. The construction of the degree for Fredholm and proper operators is based on the notion of orientation o(L) for a linear Fredholm operator L. It is given by the formula (-1) ν , where ν is the sum of multiplicities of all its positive eigenvalues. The orientation of the operator is a homotopy invariant, it does not change under a continuous deformation in the class of Fredholm operators. Next, we consider the equation A(u) = a for a nonlinear Fredholm operator A. It is possible to choose sufficiently small a (in the norm of the space) such that this operator equation has a finite number of solutions u i , and the sum of the orientations

γ = i o(A (u i ))
does not depend on a. Here A (u i ) is the operator linearized about the solution u i . The integer value γ is the topological degree.

In order to define the degree, the operator A(u, τ ) : E 1 ×R → E 2 (Section 2.1) should be:

(a) proper with respect to both variables u and τ , (b) it is assumed that the linearized operator Lv = A u (u, τ )v satisfies the Fredholm property together with the operator L λ v = Lvλv for all real λ ≥ 0, and (c) the operator L λ has a uniformly bounded inverse for all λ ≥ λ 0 for some λ 0 depending on the operator A τ .

We will verify these conditions assuming that the function F (u, v, x, τ ) satisfies the following condition in addition to Conditions 1-3 formulated above. Proof. First, we prove that the operator A(u, τ ) is proper for a fixed τ . Consider a bounded sequence u k ∈ E 1 and suppose that it converges locally in C to some u 0 ∈ E 1 . It is proved in [START_REF] Volpert | Elliptic Partial Differential Equations[END_REF] (Lemma 2.1, Chapter 11) that the operator is proper if the convergence (4.1)

A(u 0 , τ) -A(u k , τ) -L(u 0 )(u 0 -u k ) E2 → 0
occurs for any sequence u k and its limiting function u 0 , and the operator L(u 0 ) has a closed range and a finite-dimensional kernel. Here L(u 0 ) = A u (u 0 , τ).

The required properties of the operator L(u 0 ) follow from Condition 3. Convergence (4.1) is proved for conventional elliptic operators without the integral term, that is, it holds for the operator

A 0 (u, τ ) = Δu + F (u, I(u 0 ), x, τ).
We have

A(u 0 , τ) -A(u k , τ) = A 0 (u 0 , τ) -A 0 (u k , τ) + A 0 (u k , τ) -A(u k , τ).
From convergence (4.1) for the operator A 0

A 0 (u 0 , τ) -A 0 (u k , τ) -L(u 0 )(u 0 -u k ) E2 → 0,
and from (3.2)

A 0 (u k , τ) -A(u k , τ) E2 = F (u k , I(u 0 ), x, τ) -F (u k , I(u k ), x, τ) E2 ≤ |I(u 0 ) -I(u k )| u k E2 . It remains to show that |I(u 0 ) -I(u k )| → 0. Indeed, I(u 0 ) -I(u k ) = R n 1 μ(x) ((u 0 (x) -u k (x))μ(x))dx.
Since the function 1/μ(x) is integrable, and the functions

z k (x) = (u 0 (x) -u k (x))μ(x)
are uniformly bounded and locally converge to 0, then the desired convergence takes place. Thus, convergence (4.1) is shown.

In order to prove the properness of the operator A(u, τ ) with respect to both variables, it is sufficient to verify that the operator is continuous with respect to τ in the operator norm (Section 2.3, Chapter 11, [START_REF] Volpert | Elliptic Partial Differential Equations[END_REF]). We have

A(u, τ ) -A(u, τ 0 ) E2 = F (u, I(u), x, τ) -F (u, I(u), x, τ 0 ) E2 ≤ K g(x, τ 1 , τ 2 )|u| E2 .
It remains to note that |u| is bounded and g(x, τ 1 , τ 2 )|u| E2 → 0. The lemma is proved.

L λ v = Δv + F u (u, I(u), x, τ)v + F v (u, I(u), x, τ)I(v) -λv,
satisfies Condition 3 for any λ ≥ 0. Therefore, this is a Fredholm operator with the zero index. The operator L 0,λ v = Δv + F u (u, I(u), x, τ)vλv is sectorial ( [START_REF] Volpert | Elliptic Partial Differential Equations[END_REF], Chapter 6). For all λ sufficiently large it satisfies the estimate

v E1 ≤ K L 0,λ v E2 |λ| .
Hence, by virtue of Condition 4,

v E1 ≤ K L λ v E2 |λ| + K F v (u, I(u), x, τ)I(v) E2 |λ| ≤ K L λ v E2 |λ| + K 2 u E1 v E1 |λ| .
For all λ sufficiently large, this inequality provides the estimate of the inverse to the operator L λ . The lemma is proved.

Lemmas 4.1 and 4.2 provide the existence and uniqueness of the topological degree for the considered class of operators. Therefore, we can use the Leray-Schauder method to prove the existence of solutions. In the next section, we obtain a priori estimates of solutions. Under these conditions, we will obtain a priori estimates of solutions given by the following theorem. The proof of the theorem will be preceded by some auxiliary results. Proof. The left inequality in the assertion of the lemma follows from the positiveness of the solution. If the right inequality does not hold, then we consider two cases, I(u 0 ) > 1 and I(u 0 ) = 1. In the first case, we obtain a contradiction in signs in equation ( 4.2) at the point x = x 0 where the function u 0 (x) attains its maximum. In the second case, if f (u 0 (x 0 ), x 0 ) > 0, we obtain the same contradiction in signs, as before. If f (u 0 (x 0 ), x 0 ) = 0, then u 1 (x) ≡ u 0 (x 0 ) is a solution of equation ( 4 Since z(x) ≥ 0 for all x ∈ R n , and z(x 0 ) = 0, we obtain a contradiction with the maximum principle.

u 0 (x) ≥ u 0 (x 0 ) 1 - k 1 2 (x -x 0 ) 2 .
Proof. Let us recall that x 0 is a global maximum of the function u 0 (x). This maximum exists since this function is positive and decaying at infinity. We consider an arbitrary straight line through the point x = x 0 , and we will estimate the function u 0 (x) from below along this line.

We set v(t) = u 0 (x 0 + x * t), where x * belongs to the unit sphere, |x * | = 1. Then, the function v(t) is positive, decaying at infinity, it has a global maximum at t = 0, v(0) = m, and, by virtue of Lemma 4.5, |v (t)| ≤ k 1 m. Hence,

|v (t)| ≤ t 0 |v (y)|dy ≤ k 1 mt, v(t) ≥ v(0) - t 0 |v (y)|dy ≥ v(0) - k 1 m 2 t 2 .
This estimate implies (4.7).

Lemma 4.7: In the conditions of Lemma 4.5,

u 0 C 2+α (R n ) ≤ k 1 .
Proof. From Lemmas 4.4 and 4.6, it follows that

1 > I(u 0 ) > S u 0 (x)dx > u 0 (x 0 ) S 1 - k 1 2 (x -x 0 ) 2 dx,
where S is the ball |xx 0 | ≤ 2 k1 . From this inequality, we get the estimate u 0 C ( R n ) ≤ k 1 . We recall that k 1 here is an arbitrary constant which depends only on a and K. The assertion of the lemma follows now from the Schauder estimate. Leray-Schauder method. We can now prove the existence of solutions by the Leray-Schauder method. Let us briefly recall the corresponding construction. Consider an operator A(u, τ ) : E 1 → E 2 acting in Banach spaces, and a bounded domain Ω ⊂ E 1 . Suppose that

A(u, τ ) = 0 for u ∈ ∂Ω, τ ∈ [0, 1].
Denote by γ(τ ) the value of the degree corresponding to the operator A(u, τ ) and to the domain Ω. If γ(0) = 0, then, by virtue of homotopy invariance of the degree, γ(1) = 0, and there exists a solution of the equation A(u, 1) = 0 inside the domain Ω. The operator A(u, 1) is the one for which we want to prove the existence of solutions, while A(u, 0) is a model operator for which we can prove that the degree is different from 0.

Thus, in order to prove the existence of solutions we need to construct a homotopy (continuous deformation) from some model operator A(u, 0) to the given operator A(u, 1) in such a way that the operator does not vanish at the boundary of the domain ∂Ω. Usually, Ω is taken to be a ball B R ⊂ E 1 of a sufficiently large radius R. Then a priori estimates of solutions, which affirm that the norm of the solution is less than R, insure that there are no solutions at the boundary of the domain. Since we want to prove the existence of only positive solutions, we will use below a more complex construction of domain Ω.

Model operator. Consider the operator

A 0 (u) = Δu + au(1 -I(u)) -f (0, x)u.
Suppose that the principle eigenvalue λ 0 of the operator

L 0 u = Δu -f (0, x)u
satisfies the inequality -a < λ 0 < 0. Then equation A 0 (u) = 0 has a single positive solution u 0 (x) (Theorem 2.6), and the linearized operator A (u 0 ) does not have zero eigenvalue (Example 3.6). The index of the stationary point ind(u 0 ), that is, the value of the degree with respect to a small ball containing the solution, equals (-1) ν , where ν is the number of positive eigenvalues of the linearized operator. It is well defined if the linearized operator does not have a zero eigenvalue. Depending on the value of ν, it can be ±1. In what follows, we will use that ind(u 0 ) = 0.

Existence theorem. The main existence result of this work is given by the following theorem. Theorem 4.9: Suppose that a function f (u, x) satisfies Conditions 5 and 6, and the derivatives f u 2 and f u 3 are bounded and continuous (cf. Condition 4). Furthermore, the principal eigenvalue λ 0 of the operator L 0 u = Δuf (0, x)u satisfies the inequality -a < λ 0 < 0. Then the equation Δu + au(1 -I(u))uf (u, x) = 0 has a positive solution in the space H 2 μ,∞ (R n ).

Proof. We consider the operator A(u, τ ) :

E 1 → E 2 , where E 1 = H 2 μ,∞ (R n ) and E 2 = L 2 μ,∞ (R n ),
A(u, τ ) = Δu + au(1 -I(u))u(τf (u, x) + (1τ )f (0, x)).

Due to the imposed conditions and the results of the previous sections, it is a Fredholm and proper operator with the zero index, and the topological degree can be defined. Since we want to prove the existence of positive solutions, and not to take into account the trivial solution u = 0, we will construct a domain Ω which contains all positive solutions and which does not contain non-positive solutions (including the trivial solution). Set

D = {u ∈ E 1 , δ < u E1 < R, u(x) > 0, x ∈ R n },
where R > M, and M is the constant in Theorem 4.3, δ is a positive number. The value of δ can be chosen sufficiently small such that any positive solution of the equation A(u, τ ) = 0, τ ∈ [0, 1] belongs to D. Indeed, suppose that this assertion does not hold, and there is a sequence of positive solutions u k for τ = τ k such that u E1 → 0 as k → ∞. Without loss of generality we can assume that τ k → τ 0 ∈ [0, 1]. Then the linearized operator A (0, τ 0 )v = Δv + avf (0, x)v has a zero eigenvalue with a positive eigenfunction. Hence, this is the principal eigenvalue. This contradicts the condition of the theorem.

It can be easily verified that if u 0 is a solution of the equation A(u, τ ) = 0 for some τ ∈ [0, 1] and u 0 ∈ D, then any other solution u 1 (possibly for a different τ ) sufficiently close to u 0 in the norm E 1 also belongs to D. Indeed, u 1 (x) is positive in any bounded domain since it is sufficiently close to a positive solution u 0 (x). Its positiveness at infinity follows from Condition 6 which ensures that the solution cannot have a negative minimum. Thus, a ball B ρ (u 0 ) with center u 0 and a sufficiently small radius ρ 0 contains only positive solutions (no trivial solution, negative or alternating sign solutions). It should be noted that B ρ (u 0 ) is not a subset of D because a function from this ball (not a solution) can become negative for large |x|.

Consider the set S ⊂ D of all positive solutions of the equation A(u, τ ) = 0, τ ∈ [0, 1] and its covering with balls B ρ (u), where ρ can depend on the solution. Since the operator A(u, τ ) is proper with respect to an ensemble of two variables, the set S is compact, and we can choose a finite sub-covering Ω of this covering. By virtue of the properties of the set D, domain Ω contains all positive solutions, and it does not contain non-positive solutions. We consider the topological degree with respect to the domain Ω.

The model problem A(u, 0) = 0 has a single positive solution (in Ω), and γ(0) = 0. Since A(u, τ ) = 0 for u ∈ ∂Ω, then γ(1) = 0, and there exists a solution of equation A(u, 1) = 0 in Ω. The theorem is proved.

Lemma 3 . 3 :

 33 If Conditions 2 and 3 are satisfied, then the operator L(u 0 , τ 0 ) satisfies the Fredholm property with the zero index.

Condition 4 :

 4 Function F (u, v, x, τ ) is bounded together with its first derivative with respect to x ∈ R n , and up to the third derivative with respect to u in any bounded interval. It also satisfies estimates (3.2) and (3.3) in Condition 1. Furthermore, |F v (u, v, x, τ )| ≤ K|u| for any |u|, |v| ≤ M , τ ∈ [0, 1], where K depends on M .

Lemma 4 . 1 :

 41 If Conditions 3 and 4 are satisfied, then the operator A(u, τ ) is proper with respect to both variables u and τ , that is, the inverse image of any compact set in E 2 is compact in any bounded closed set in E 1 × R.

Lemma 4 . 2 :

 42 If Conditions 3 and 4 are satisfied, then the operator L λ v = Lv-λv satisfies the Fredholm property for all real λ ≥ 0, and it has a uniformly bounded inverse for all λ ≥ λ 0 for some λ 0 depending on the operator A τ .Proof. The operator L λ ,

4. 2 .Condition 5 :Condition 6 :

 256 A priori estimates. We consider the equation(4.2) Δu + au(1 -I(u))uf (u, x) = 0, in R n , n = 1, 2, 3, where x ∈ R, a > 0, I(u) = R n u(y)dy. We suppose that u(x)is integrable, and the integral I(u) is well defined. Function f (u, x) is continuous together with its first partial derivatives, |f u (u, x)|, |f x (u, x)| ≤ K, and 0 ≤ f (u, x) ≤ K for all u ≥ 0, x ∈ R n . Furthermore, f (u, x)a ≥ > 0 for all u ≥ 0, |x| ≥ N and some positive N .

Theorem 4 . 3 :

 43 Suppose that the function f (u, x) satisfies Conditions 5 and 6. Then any positive solution u 0 (x) of equation (4.2) decaying at infinity admits the estimate u 0 H 2 μ,∞ (R n ) ≤ M with some positive constant M independent of the solution.

Lemma 4 . 4 :

 44 Suppose that Condition 5 is satisfied and equation (4.2) has a positive integrable solution u 0 (x) decaying at infinity, |u 0 (x)| → 0 as |x| → ∞. Then 0 < I(u 0 ) < 1.

  .2). The function z(x) = u 1 (x)u 0 (x) satisfies the corresponding linear equation Δz + b(x)z = 0, where b(x) = (F (u 1 (x), x) -F (u 0 (x), x))/(u 1 (x)u 0 (x)), F(u, x) = uf (u, x).

Lemma 4 . 6 :

 46 Under the conditions of Lemma 4.4, (4.7)

Remark 4 . 8 : 3 .

 483 A similar estimate remains valid for the more general equation Δu + F (u, I(u), x) = 0 if F (u, v, x) < 0 for v > v 0 with some v 0 > 0, and all u and x. Proof of Theorem 4.3. From Lemma 4.7 it follows that sup x∈R n |u 0 (x)| ≤ M . Here and below we denote by M positive constants independent of the solutions. Consider the equation Δuu = 0 in the exterior domain |x| > N with the boundary condition u(x) = M for |x| = N . Here N is determined in Condition 6. It can be easily verified that u 0 (x) ≤ u(x) for |x| ≥ N . Therefore, u 0 (x) ≤ M e νx for all x ∈ R n and some ν > 0, and u 0 L∞,μ(R n ) ≤ M . Set v 0 (x) = u 0 (x)μ(x). Then v 0 Lμ(R n ) ≤ M . Estimate (4.4) considered for the corresponding operator completes the proof of the theorem. 4.Existence of solutions.

Acknowledgement. This paper has been supported by the RUDN University Strategic Academic Leadership Program.

Let us write equation (4.2) in the following form:

where g(x) = -au 0 (x)(1 -I(u 0 )) + u 0 (x)f (u 0 (x), x), and u 0 (x) is a positive solution of (4.2). Then u 0 (x) is a solution of this equation, and by virtue of Theorem 3.2 (see Chapter 3, [START_REF] Volpert | Elliptic Partial Differential Equations[END_REF]), we have

Here and below we denote by k positive constants independent of the solution.

Lemma 4.5: Under the conditions of Lemma 4.4, the following inequality holds:

where 0 < α < 1, k is a positive constant independent of the solution.

Proof. Note that

, where m = sup x∈R n |u 0 (x)|. Due to Lemma 2.1, the function g(x) defined above admits the following estimate:

From the embedding theorems (n = 1, 2, 3) it follows that

Here and below k 1 is a positive constant which depends only on a and K.

Next, we can estimate the Hölder norm of the function g(x) in (4.3):

Finally, from the Schauder estimate applied for equation (4.3), we obtain estimate (4.5).


Arnaud Martin
email: arnaud.martin@irisa.fr

Belief functions on ordered frames of discernment

Keywords: ordinal variable, ordered frame of discernment, ordered and fuzzy elements, ordered power set, distance

Most questionnaires offer ordered responses whose order is poorly studied via belief functions. In this paper, we study the consequences of a frame of discernment consisting of ordered elements on belief functions. This leads us to redefine the power space and the union of ordered elements for the disjunctive combination. We also study distances on ordered elements and their use. In particular, from a membership function, we redefine the cardinality of the intersection of ordered elements, considering them fuzzy.

Introduction

The theory of belief functions is used in more and more applications such as machine learning, pattern recognition, clustering, etc. To apply the theory of belief functions, we consider the frame of discernment given by Ω = {ω 1 , ω 2 , . . . , ω n } and the basic belief assignments are defined as a mapping from the power set of Ω, noted 2 Ω , to [0, 1] [START_REF] Shafer | A mathematical theory of evidence[END_REF]. The elements ω i of Ω are considered exclusive and exhaustive. The assumption of the exclusivity can be lifted, considering the hyper power set D Ω [START_REF] Martin | Implementing general belief function framework with a practical codification for low complexity[END_REF].

However, we may be confronted with applications where a semantic or proximity link exists between the elements of the frame of discernment. For example, in questionnaires graduated or ordered answers can be proposed: What is the distance between Lannion and Paris?: 314km, 414km, 514km, 614km Other possible answers use the Likert scale such as:

Not happy, neutral, happy Strongly disagree; Disagree; Neither agree nor disagree; Agree; Strongly agree These answers can be represented by an ordinal variable. In these cases, there is no way to deal with this link between the elements of the frame of discernment. Forcing these elements to have a zero mass is not stable in the combination and decision processes. In this paper, we explore the possibilities of considering an order between the elements of the frame of discernment with the theory of belief functions. First in section 2 we introduce the ordered power set, then we show how we can combine mass functions defined on an ordered power set in section 3. In section 4, we propose the definition of a distance between ordered elements used to define a distance between belief functions, that can be used for conflict measure and decision in section 5. In section 6, we consider ordered fuzzy elements to redefine the cardinality of the intersection of elements before concluding.

Power set of ordered elements

Let us take again the example of the question about the distance between Paris and Lannion. The frame of discernment is given by Ω = {ω 1 , ω 2 , ω 3 , ω 4 }, with the answers ω 1 = 314km, ω 2 = 414km, ω 3 = 514km, ω 4 = 614km. The right answer is 514km. Here, the elements of the frame of discernment are obviously exclusive. However, if someone answers 614km, the error is smaller than if the answer given is 314km. If we consider imprecise answers as in [START_REF] Thierry | Modeling uncertainty and inaccuracy on data from crowdsourcing platforms: MONITOR[END_REF], {ω 3 , ω 4 } make sense while {ω 1 , ω 3 } does not.

We therefore consider that we should not take into account all the elements of the power set 2 Ω , but only those which have a meaning. Thus disjunctions that do not contain consecutive elements should not be considered. Let us consider an ordinal variable having values in a finite set Ω = {ω 1 , . . . , ω n } of ordered ω i , i = 1, . . . , n, i being an ordinal number.

Definition The ordered power set, noted oP S Ω , is a subset of the power set composed by the empty set and all the disjunctions of consecutive elements of Ω.

A disjunction of consecutive elements from endpoint elements is noted by:

{ω i , ω j } o = {ω i , ω i+1 , . . . , ω j-1 , ω j }, with 1 ≤ i ≤ j ≤ n (1)
Hence, the ordered power set is given by:

oP S Ω = {∅, {{ω i , ω j } o } i,j=1,...,n } (2)
The number of elements of 2 Ω is 2 n , but it is smaller for oP S Ω . Proposition The number of elements of oP S Ω , with: Ω = {ω 1 , . . . , ω n } is:

1 + n(n + 1) 2 (3)
Proof The set oP S Ω contains the empty set and ordered elements determined by the endpoint elements. The number of these elements is the number of pairs (i, j) with 1

≤ i ≤ j ≤ n. This number is n! 2!(n-2)! = n(n+1) 2 . So |oP S Ω | = 1 + n(n+1) 2 .
⊓ ⊔ Let Ω = {ω 1 , . . . , ω n } be a frame of discernment of ordered, exclusive and exhaustive elements ω i , i = 1, . . . , n. A mass function m is the mapping from elements of the ordered power set oP S Ω onto [0, 1] such that:

X∈oP S Ω m(X) = 1. (4)
A focal element X is an element of oP S Ω such that m(X) = 0. From this definition of mass function, all the definitions of special mass functions are available (simple mass functions, non dogmatic, consonant, etc.). A categorical mass function with m(X) = 1 is noted m X . Definitions of classical belief functions such as credibility, plausibility and pignistic probability are also available on the ordered power set oP S Ω . The credibility function is given for all X ∈ oP S Ω by: bel

(X) = Y ⊆X,Y =∅ m(Y ).
(

) 5
The plausibility function is given for all X ∈ oP S Ω by:

pl(X) = Y ∈oP S Ω ,Y ∩X =∅ m(Y ). (6)
However, it is not possible to compute this function as a dual of the credibility function, because an ordered power set is not invariant by the complement. Indeed, if we consider Ω = {ω 1 , ω 2 , ω 3 }, the complementary of ω 2 is {ω 1 , ω 3 } / ∈ oP S Ω . The pignistic probability [START_REF] Smets | Constructing the pignistic probability function in a context of uncertainty[END_REF] can be written for all ω ∈ Ω by:

BetP(ω) = X∈oP S Ω ,X =∅,ω∈X 1 |X| m(X) 1 -m(∅) . (7)
3 Combination of belief functions on ordered power set

When considering several mass functions on an ordered power set oP S Ω from several sources or persons, we need to be able to combine them. The combination operator must therefore produce a mass function in the same ordered power set oP S Ω . Therefore, an ordered power set must be invariant by the combination operator. There are a large number of combination operators [START_REF] Martin | Conflict management in information fusion with belief functions[END_REF] and not all of them verify this property. Proposition An ordered power set is invariant by the conjunctive combinations (normalized, not normalized, Yager [START_REF] Martin | Conflict management in information fusion with belief functions[END_REF]).

Proof Let two focal elements of an ordered power set oP S Ω : {ω i1 , ω j1 } o and {ω i2 , ω j2 } o . The intersection of these elements is empty or given by:

{max(ω i1 , ω i2 ), min(ω j1 , ω j2 )} o ∈ oP S Ω
Therefore, an ordered power set is invariant by all conjunctive operators. ⊓ ⊔ Proposition An ordered power set is not invariant by the disjunctive combination.

Proof Let two categorical mass functions m ω1 and m ω3 , with two non consecutive focal elements such as ω 1 and ω 3 . The disjunctive combination of these two mass functions is the categorical mass function with {ω 1 , ω 3 } such as unique focal element. As {ω 1 , ω 3 } / ∈ oP S Ω , the ordered power set is not invariant by the disjunctive combinations.

⊓ ⊔ Thus, an ordered power set is not invariant by all the mixed combination operators based on the disjunctive operator. Here we propose a new disjunctive combination operator that makes an ordered power set invariant. Let two elements Y i and Y j ∈ oP S Ω , we note Y i = {ω i1 , ω in i } o and Y j = {ω j1 , ω jn j } o . We define the union of these two ordered elements by:

Y i o ∪ Y j = {min(ω i1 , ω j1 ), max(ω in i , ω jn j )} o (8)
The union of s ordered elements is given by the extension of the previous equation or recursively by ((Y

1 o ∪ Y 2 ) o ∪ Y 3 ) . . . o ∪ Y s .
Let two mass functions defined on the ordered power set oP S Ω , for all X ∈ oP S Ω , the disjunctive combination is given by:

m oDis (X) = Yi o ∪ Yj =X m i (Y i )m j (Y j ). ( 9
)
The disjunction combination of s mass functions on the ordered power set oP S Ω , is given for all X ∈ oP S Ω by:

m oDis (X) = Y1 o ∪... o ∪ Ys=X S j=1 m j (Y j ). ( 10
)
where Y j ∈ oP S Ω is a focal element of the source S j , and m j (Y j ) the associated mass function. We can thus rewrite the mixed combination operators [START_REF] Martin | Conflict management in information fusion with belief functions[END_REF], such as the Dubois and Prade one [START_REF] Dubois | Representation and combination of uncertainty with belief functions and possibility measures[END_REF] given for all X ∈ oP S Ω , X = ∅ by:

m oDP (X) = Y1∩...∩Ys=X S j=1 m j (Y j ) + Y 1 o ∪ . . . o ∪ Y s = X Y 1 ∩ . . . ∩ Y s = ∅ S j=1
m j (Y j ). [START_REF] Martin | Conflict management in information fusion with belief functions[END_REF] Proposition An ordered power set is stable by the average combination. Proof The proof is obvious, because the set of focal elements obtained by the combination is the union of the sets focal elements from the mass functions to be combined.

⊓ ⊔

Distances on belief functions on ordered power set

In the context of belief functions, we often use a distance between mass functions, in order to measure similarity for clustering, to define some measures or to decide.

There are many distances that can be considered [START_REF] Jousselme | Distances in evidence theory: Comprehensive survey and generalizations[END_REF]. The most commonly used distance between the belief functions is the distance defined in [START_REF] Jousselme | A new distance between two bodies of evidence[END_REF]. This distance can obviously be defined for two mass functions m 1 and m 2 on oP S Ω by:

d J (m 1 , m 2 ) = 1 2 (m 1 -m 2 ) T D(m 1 -m 2 ), ( 12
)
where D is an (1 + n(n+1)

2

) × (1 + n(n+1)

2

) matrix based on Jaccard dissimilarity whose elements are:

D(A, B) =          1, if A = B = ∅, |A ∩ B| |A o ∪ B| , ∀A, B ∈ oP S Ω . ( 13
)
where |X| is the cardinality of X ∈ oP S Ω . Of course we have:

|A ∩ B| |A o ∪ B| = |A ∩ B| |A ∪ B| because if A ∩ B = ∅ then A o ∪ B = A ∪ B with A, B ∈ oP S Ω .
However, with this distance, we have without any distinction:

d J (m ω1 , m ω2 ) = d J (m ω1 , m ω3 ) = 1.
The value 1 is the maximum value of this distance.

Distance between ordered elements

Let us take the example of the question about the distance between Paris and Lannion. We have noticed that if someone answers 614km, the error is smaller than if the answer given is 314km. Thus, the distance between elements of an ordered frame of discernment Ω can be considered differently according to their order. On a Likert scale, we have the same argument. We can consider that the minimal distance between two elements of Ω is that between two consecutive elements. Inthe same way, the maximum distance between two elements of Ω is the distance between the first ω 1 and the last ω n element of Ω.

In order to have a distance with a normality property, we propose the following distance between two elements of Ω:

d o (ω i , ω j ) = |i -j| n -1 ( 14
)
where |x| is the absolute value of x = 1, . . . , n. This distance takes obviously its values in [0, 1]. The distance between an element of Ω and an element X = {ω 1x , ω nx } o of oP S Ω can be defined by one of the following equations:

d omin (ω i , X) = min(d o (ω i , ω 1x ), d o (ω i , ω nx )) ( 15
)
d omax (ω i , X) = max(d o (ω i , ω 1x ), d o (ω i , ω nx )) (16) d oav (ω i , X) = 1 [X| nx k=1x d o (ω i , ω k ) (17)
The distance between two elements X = {ω 1x , ω nx } o and Y = {ω 1y , ω ny } o of oP S Ω can be defined by one of the following equations:

d omin (X, Y ) = min ωy i ∈Y d omin (X, ω yi ) (18) d omax (X, Y ) = max ωy i ∈Y d omax (X, ω yi ) (19) d oav (X, Y ) = 1 |XY | nx kx=1x ny ky=1y d o (ω kx , ω ky ) (20)
These distances take their values in [0, 1]. The use of one of these distances rather than another may depend on the application. For the sake of simplicity, we will note d o in the following.

Distance between belief functions

As we have seen, the Jousselme distance does not fit the ordered elements of the frame of discernment. Indeed, with Jaccard dissimilarity, the dissimilarity is zero if the intersection is empty. However, on ordered and exclusive elements, the dissimilarity can be different depending on the order. Therefore, we modify the dissimilarity of Jaccard on empty intersection elements from the distance defined in the previous section. Since the minimum strictly positive value of the Jaccard dissimilarity is 1 n , the proposed dissimilarity takes its values on [0, 1 n ]. we define a modified Jaccard dissimilarity for ordered elements by:

D o (A, B) =          1, if A = B = ∅, |A ∩ B| |A o ∪ B| + (1 -Int(A, B)) 1 -d o (A, B) n , ∀A, B ∈ oP S Ω . ( 21
)
where Int is the intersection index defined such as in [START_REF] Jousselme | Distances in evidence theory: Comprehensive survey and generalizations[END_REF] by Int(A, B) = 1 if A ∩ B = ∅ and 0 otherwise.

According to [START_REF] Bouchard | A proof for the positive definiteness of the Jaccard index matrix[END_REF], D o define a matrix positive definite. The distance obtained is given for two mass functions m 1 and m 2 on oP S Ω by:

d Jo (m 1 , m 2 ) = 1 2 (m 1 -m 2 ) T D o (m 1 -m 2 ), ( 22
)
where D o is given by Equation ( 21).

If we consider Ω = {ω 1 , ω 2 , ω 3 } and the distance given by Equation (20), we obtain:

D o =            1 0 0 0 0 0 0 0 1 1 6 1 2 0 1 2 1 3 0 1 6 1 1 2 1 6 1 2 1 3 0 1 2 1 2 1 1 12 1 3 2 3 0 0 1 6 1 12 1 1 2 1 3 0 1 12 1 2 1 3 1 2 1 2 3 0 1 3 1 3 2 3 1 3 2 3 1            (23)
Hence:

d Jo (m ω1 , m ω2 ) = 5 6 ≃ 0.91 and d Jo (m ω1 , m ω3 ) = 1 (24)
In this way, we show that we can take into account the proximity of ordered elements in the distance.

The modification of the Jaccard dissimilarity, could induce an interpretation of the non-exclusivity of the elements. We will discuss this in section 6.

Decision and conflict on ordered elements

After combining the mass functions, we generally want to make a decision about the resulting mass function. This mass function with the operators seen in section 3 takes its values on oP S Ω . It is common to make the decision on Ω by maximum of credibility, plausibility or by compromise with the pignistic probability. We have seen in section 2, that we can calculate these belief functions in oP S Ω and if we denote f d one of these functions the decision is made by:

ω d = argmax ω∈Ω (f d (ω)) . (25)
However, these functions do not consider the difference of proximity of the elements of Ω and do not allow to decide on the disjunctions, i.e. on some elements of oP S Ω . In [START_REF] Essaid | A distance-based decision in the credal level[END_REF], we introduced another decision process based on a distance given by:

A = argmax X∈D (d Jo (m, m X )) , (26)
where m X is the categorical mass function m(X) = 1, m is the mass function coming from the combination rule and d Jo is the distance introduced in Equation (22). The subset D ⊆ oP S Ω is the set of elements on which we want to decide. This last decision process also allows to decide on imprecise elements of the ordered power set oP S Ω and to take into account with the distance the difference of proximity of ordered element of Ω. Let us take the example of the Paris-Lannion trip, if we get 3 categorical answers from 3 people such as: Person 1 said: 314km, person 2 said: 514km, and person 3 said: 614km. Person 1 disagrees more with person 2 than with person 3. The distance introduced in section 4 can measure this difference on the ordered elements of the frame of discernment. Therefore, we can extend here the conflict measure introduced in [START_REF] Martin | About conflict in the theory of belief functions, Belief[END_REF], between two mass functions m 1 and m 2 by:

Conf(m 1 , m 2 ) = (1 -δ inc (m 1 , m 2 ))d Jo (m 1 , m 2 ) ( 27
)
where d Jo is the distance defined by equation ( 22) and δ inc is a degree of inclusion measuring how m 1 and m 2 and included each other. For more details on possible degree of inclusion see [START_REF] Martin | About conflict in the theory of belief functions, Belief[END_REF]. To measure the conflict between more than two mass functions, the average of the conflicts two by two can be considered.

Belief functions on ordered fuzzy elements

Now assume that the answers to the question about the distance between Lannion and Paris are: about 300km, about 400km, about 500km, about 600km The possible answers are always ordered, and if they can always be considered exclusive, they are fuzzy answers.

The link between belief functions and fuzzy sets has already been discussed in different ways [START_REF] Denoeux | Modeling vague beliefs using fuzzy-valued belief structures[END_REF][START_REF] Denoeux | Belief functions induced by random fuzzy sets: A general framework for representing uncertain and fuzzy evidence, Fuzzy Set and Systems[END_REF]. Here we wish to study the representation and the consideration of ordered and fuzzy elements of the frame of discernment.

As we saw in section 4 by redefining Jaccard's dissimilarity, the exclusivity of the ordered elements can be questioned. The fact of having ordered fuzzy elements does not however call into question the exclusivity of the elements and thus the intersection of the elements. Indeed, there is no semantic sense in having about 400km and about 500km at the same time. On the other hand, considering fuzzy elements allows to redefine the cardinality of the intersection of the elements from the membership functions.

In [START_REF] Petković | Fuzzy Jaccard Index: A robust comparison of ordered lists[END_REF], the cardinality of the intersection of ordered elements is defined from a definition of the membership function, which allows them to propose another modified version of Jaccard dissimilarity. Inspired by this work, we define here the membership function of an element of ω ∈ Ω to a set of X ∈ oP S Ω by:

µ X (ω) =    1, if ω ∈ X, αe -γdo(ω,X) , otherwise, (28)
with 0 ≤ α ≤ 1, 0 ≤ γ ≤ 1 two thresholds to control the membership function and d o the distance defined by Equations ( 15), ( 16) or (17). Based on this membership function, we define the cardinality of the intersection of two elements X and Y ∈ oP S Ω by:

|X ∩ Y | o = ω∈X o ∪ Y,ω∈Ω min(µ X (ω), µ Y (ω)) (29) If α = 0 then |X ∩ Y | o = |X ∩ Y |.
Therefore, we define a modified Jaccard dissimilarity for ordered elements by: [START_REF] Petković | Fuzzy Jaccard Index: A robust comparison of ordered lists[END_REF]. A new distance d Jo between the ordered and fuzzy elements can thus be defined by Equation ( 22).

D o (A, B) =          1, if A = B = ∅, |A ∩ B| o |A o ∪ B| , ∀A, B ∈ oP S Ω . (30) We have 0 ≤ D o (A, B) ≤ 1 and D(A, B) ≤ D o (A, B)
The cardinality of the intersection defined by Equation (29) allows its use in the mixed rules [START_REF] Martin | Toward a combination rule to deal with partial conflict and specificity in belief functions theory[END_REF] to regulate the conjunctive/disjunctive behaviour by taking into account the partial combinations according to the cardinality of the elements. The mixed rule is given for m 1 and m 2 for all X ∈ 2 Ω by:

m Mix (X) = Y1 o ∪ Y2=X δ 1 m 1 (Y 1 )m 2 (Y 2 ) + Y1∩Y2=X δ 2 m 1 (Y 1 )m 2 (Y 2 ). ( 31
)
The choice of δ 1 = 1δ 2 can also be made from Jaccard dissimilarity: Questionnaires whose responses are in the form of a Likert scale can also be modeled by a frame of discernment of ordered and fuzzy elements and thus use the modified Jaccard dissimilarity defined by Equation (30).

δ 2 (Y 1 , Y 2 ) = |Y 1 ∩ Y 2 | o |Y 1 o ∪ Y 2 | . ( 32

Conclusion

In this paper, we explored the modeling and integration of an ordered element of frame of discernment by belief functions. The fact of considering ordered elements led us to redefine the power space of unions of ordered elements making sense, called ordered power space. Thus, to consider disjunctive combination rules leaving invariant the ordered power space, we defined the union of ordered elements. From a distance between ordered elements we redefined a distance between mass functions that can be used in a conflict measure and for decision making. Without redefining the intersection of ordered elements, which we still consider exclusive, we have redefined, from a membership function, the cardinality of the intersection of ordered elements that can be considered fuzzy.

A large number of applications can be addressed by modeling on ordered sets of elements. We are thinking in particular of the answers to questionnaires which very often use Likert scales. This type of questionnaire can, for example, be used to evaluate the knowledge and skills of students allowing to consider the closest answers in the sense of the order of the right answer, without considering them as totally wrong. Concrete applications will be considered in future work.

1 o∪ Y 2

 12 )Thus, if we have a partial conflict between Y 1 and Y 2 , Y 1 ∩ Y 2 = ∅, the rule transfers the mass on Y according to the difference in order of the elements of Ω constituting Y 1 and Y 2 .


Toufic Wehbe

Gilles Dessein
email: gilles.dessein@enit.fr

Lionel Arnaud
email: lionel.arnaud@enit.fr

EXPERIMENTAL STUDY OF THIN PART VIBRATION MODES IN MACHINING

Keywords: milling, chatter, flexible part, modal shape, displacement field

The machining of thin walls generally generates milling chatter, that damage surface roughness and manufacturing tools. Stability lobes which include natural frequencies are successful in case of tool chatter. When milling thin webs models are less adequate, because the interaction with the tool disrupts the behaviour of the work piece. The modal approach generally used for stability charts may be not adequate enough because of neglecting the tool and the work piece contact. This paper presents the experimental phase of a work aiming at analyse vibration modes of a thin web during machining. A finite element calculation shows the influence of a contact on natural frequencies of the part. For a better investigation, field displacements of the work piece are analysed. This work eventually aims at better knowledge of the contact between the tool and the part to improve the hardiness of models.

INTRODUCTION

In milling operations, the machining system is prone to vibratory phenomena that result from the dynamic interaction between the tool and the part. These vibrations called milling chatter generate a poor surface quality, the cutting edge chipping, and the tool and the spindle wear. Many works have been leaded for more than fifty years in order to study this complex phenomenon. They have shown that chatter control constitutes an industrial and scientific stake.

The source of these vibrations was first identified for turning operation by Tobias [START_REF] Tobias | A theory of regenerative chatter[END_REF] and led to the stability lobes theory. Afterwards, this theory was extended to milling [2,[START_REF] Budak | Analytical prediction of the chatter stability in milling -Part II: Application of the general formulation to common milling systems[END_REF], and its efficiency was demonstrated for tool vibrations.

However if stability lobes are transposed to flexible work piece, the theory is confronted with a more complex configuration. While vibrating, a thin wall has several modes of vibration, with nodes and antinodes which can be met by the tool. Thus the variable dynamic response of the part according to the tool localization was underlined by [START_REF] Lapoujoulade | Prédiction du comportement vibratoire du fraisage latéral de finition des pièces à parois minces (Vibratory behavior prediction of thin-walled parts during lateral finish milling)[END_REF]. In the same way [START_REF] Thevenot | Influence of material removal on dynamic behaviour of thin walled structure in peripheral milling[END_REF] showed that the contact with the tool as well as the progressive metal removal involve a variation of its dynamic characteristics. A three-dimensional chart of stability lobes including nodes and antinodes makes it possible to see the importance of the position of the tool on the part. Moreover the fast variation of surface quality show behavior changes during the operation, which is specific of thin walls milling. Thus [START_REF] Seguy | Surface roughness variation of thin wall milling, related to modal interactions[END_REF] simulates by temporal integration the machining chatter with a moving tool-part contact along the operation. By taking into account the ploughing effect, the mode coupling and the exit of the cut, the model leads to a good correlation with the tests.

Although various works include changes of the part characteristics, stability charts can still contain significant inaccuracy. Indeed the part natural frequencies are measured before the operation in order to plot stability charts. So the stability lobes theory neglects the tool-part interaction. The disturbance brought to natural modes by the tool presence remains ignored. A better comprehension of this effect requires knowing the real vibratory behavior of the part during its machining.

Experimental works were done to analyze the machining behavior by studying surface qualities, [START_REF] Corduan | Modelling and experimental approach of milling stability: Application to thin walled parts[END_REF] can be quoted. It can be noticed that the complexity is linked to the fact that when a mark of vibration is left by a tooth, the following tooth can erase this information. As a consequence the surface shape which is only a partial trace of the vibratory behavior of the part, does not allow reconstituting the complete history of the operation. To bypass that, we were brought to observe the displacement fields of the part during the operation.

Acoustics also gives only very partial information on machining vibrations [8,9]. Acceleration measurements usually are utilized [10] but sensors mounting significantly change the structure dynamical properties, according to the added mass proportion. Moreover, displacement obtained by signal integration doesn't give the static deflection of the motion, dependent on the average cutting dynamics. In fact, vibration frequencies obtained by Fast Fourier Transform are mainly utilized for process monitoring. In this application, fixing thresholds on frequencies rather than on measured accelerations is easier and more efficient to detect a change of cutting conditions [11].

As regards chatter control, accelerometer measurements on parts are used especially for the modal analysis before the operation, in order to plot stability lobes. In addition the displacement velocity can be measured by specific Laser velocimetry. This contactless recording has no influence on the operation dynamical behavior. Anyway it can be restricted by lubricant or chips which cross the beam. Velocity measurements were used by [START_REF] Seguy | Surface roughness variation of thin wall milling, related to modal interactions[END_REF] and displacement sensors by [START_REF] Corduan | Modelling and experimental approach of milling stability: Application to thin walled parts[END_REF]. Contactless or not measurements are efficient to study specific frequencies values, but do not allow to know the entire work piece behavior. For such a study, field measurement makes it possible to know the behavior of a structure in any point. This already has been used to identify modal properties of a thin wall [12], but not for machining application.

In order to know real modes of a thin part in machining, a space-time measurement protocol was set up. It is described in the second part. The third part details the machining test procedure. Specific frequencies and real modes shapes analysis are respectively given in fourth part.

EXPERIMENTAL SETUP

The measurement device illustrated by Fig. 1 was used during machining. On the left side of the part (on the illustration) four accelerometers are fixed in lower part of the tool. Their location will be justified in § 3.2.

On the right side, a Laser vibrometer measures the work piece velocity at its corner, because it is a antinode for all modes shapes. Low frequencies were first removed from the raw signal. Secondly it is integrated to provide the vibratory displacement.

This same side is entirely covered by black and white painting drops. The random resulting gray levels are necessary to measure displacement fields by images stereovision processing. This technique uses images taken at the same instant by the two cameras. The accuracy of stereovision depends on the angle and distance between cameras, sample gray levels, the lighting and an initial calibration phase. Lower accuracy will be reached in case of too weak angle between cameras, too important distance from the work piece, and non homogeneous grey levels or lighting. On top of that noise will be added to results if stereo vision processing parameters are not adapted enough.

Usual applications as tensile tests enable optimized measurement conditions. To measure dynamic phenomena in machining, a specific optimization work has been done. During the machining the two cameras simultaneously take photos at their maximal shooting frequency namely 5.7 Hertz. The cameras are 1 meter far from the work piece. To reduce noise, tests are launched in the dark and the stroboscope is trigged by cameras to visualize the instantaneous deformed part.

All signals are recorded in the same temporal reference mark. Thus accelerations or velocity give the exact starting instant of the machining. The stroboscope signal and the machining feed speed give the position of the tool for each displacement field.

MACHINING TESTS PROTOCOL

Validation of the measurement device capacity

Once the device was ready for machining, its capability was evaluated. Beyond a 0.02 mm static displacement perpendicular to the work piece plane, field measurement guarantees an error lower than 5%. For vibration, its capability was validated with a shaker by raising the same modal shapes from accelerations and field measurement [13].

When associated with laser and accelerometer measurements, field measurement is very interesting. In addition to showing the space behavior, it gives the sum of the static and dynamic deformations.

Taking into account the part instrumentation

The machined part is a rectangular aluminum alloy plate (2017 A). It is 50 mm height 2 mm thickness and its clamped length is 110 mm.

The location of the accelerometers was optimized by finite elements calculation to detect the first three modes. They are vertically put at middle height of the part to avoid interference with the tool. The mass of the only plate is 30.7 Grams. It becomes 58.3 grams after instrumentation. This modification was taken into account by hammer test to plot stability lobes (Table 1).

Machining conditions

The operation is a radial down milling. The tool is a cylindrical milling cutter with 4 teeth and 12 mm diameter. Its helix angle is 45°. The feed rate f is 0.05 mm/tooth and the radial depth of cut is 0.2 mm. The tangential and radial milling force coefficients (Kt and Kr) drawn from the literature [START_REF] Seguy | Surface roughness variation of thin wall milling, related to modal interactions[END_REF] are respectively 400 and 40 N/mm². The spindle speed that is worth 24 000 rev/min corresponds at 900 m/min cutting speed. The axial depth of cut Ap is of 4 mm. On stability lobes according to [2], the test is in unstable zone of lobes of 2 nd and 3 rd mode (Fig. 2).

This test takes 1.39 seconds and provides 9 instantaneous field measurements (image gust at 5.7 Hz). In order to statistically obtain images in many configurations (two directions of displacement, crossing at reference position…), the test was repeated 13 times with a different shift of the machining starting departure.

Before each test, part straightness defects are integrated in the tool path in order to guarantee a constant radial engagement. Thereafter this translation of the part is taken into consideration in the analysis. Moreover structural frequencies of each part are systematically measured before the test. This experiment revealed variations going up to 7% on Eigen frequencies, and 57% on damping ratios. These significant variations have many causes: embedding conditions, accelerometers and their cables location errors, and the temperature of the metal.

Plates slightly differ and the plate position related to the angular tool position is not known enough precisely. So conditions of the 1st contact between the tooth and the work piece are not repeatable and tests will not be strictly identical.

WORK PIECE SURFACE AND FREQUENCY ANALYSIS

Analysis of final generated surfaces

The non homogeneous surface quality according to the tool feed reveals the variable part behavior during the machining chronology. In spite of inevitable differences between tests, the most of them shows strong grooves at the beginning. These marks disappear at the center and reappear at exit. At this stage, surface qualities are gathered in three types (Fig. 3) corresponding to similar zones on several parts. In order to validate this procedure and enhance images number, displacements frequencies and amplitudes were compared for the three zones on various parts. This showed a strong agreement and validated the selection of the 3 types.

Frequency analysis of the three zones

For the first zone which is deeply scratched displacements FFT show four frequencies (Fig. 4).

The two of them being 573 Hz and 1600 Hz have low amplitudes and their impact is not visible on the part. As regards the other frequencies 741 Hz is not far from second mode and 800 Hz exactly corresponds to the halftooth frequency.

The proximity of these two frequencies induces a global undulation that is visible on temporal signals and on the work piece. This phenomenon frequency is (800-741 = 59) Hz and exactly corresponds to the grooves spacing on the part. Its total amplitude goes from 200 µm to 50 µm but has not been related to grooves height.

The second mode Eigen frequency is worth 688Hz. Its associated chatter frequency should be close to 850 Hz according to common models. From Flip Lobe models predict it between 700 Hz and 800 Hz. The calculations are not developed here. For the second zone a better surface quality is shown by Fig. 4. Now dominant frequencies are 3200 Hz (2nd harmonic of tooth crossing frequency), 1600Hz (tooth frequency), 1200 Hz (next to third natural frequency) and 800Hz (half-blow of tooth). Third type looks like the first one by symmetry. The part vibrates at 800 Hz and 758 Hz which are relatively close to the first case. Anyway the scratches of third type are a little bit deeper than those of first one.

Simplified tool-part contact modeling.

The tool stiffness is worth more than 10 times the parts one, so that it can be considered as undeformable.

By allowing only rotations at the cutting zone, the Eigen frequencies of the part were calculated by finite elements method. This calculation which integrates the metal removal shows that mass loss induces a 5 Hertz variation of the Eigen frequencies. Consequently it is negligible and makes it possible to consider the position of the connection until the medium of the part. Figure 5 gives comparative values of the first three Eigen frequencies with this connection to those of the free part.

This too simplistic modeling does not represent the interaction between the tool tooth and the part enough finely. In the machining the tool and the part are in contact only 50% of time. Nevertheless it is clearly observed that the additional connection significantly increases eigen frequencies. Moreover this effect varies differently according to the connection position and the considered mode. The second structural frequency is highly shifted with the utilized contact.

A less restrictive connection type may bring eigen frequency closer to 741 Hz or 800 Hz. So considering the tool as a constant support on the part seems to be wrong and sophisticated analysis and calculus with other parameters are needed to evaluate that precisely.

Synthesis of frequencies analysis

Behaviors of the first and third type are similar. In both cases the cutting zone is strongly vibrating. Grooves spacing may be linked to the plate behavior by main vibration periods but not by amplitudes. The recorded vibration frequencies 741 Hz, 758 Hz, 800Hz, 1600Hz and 3200 Hz do not give the contribution of each mode because real modes can't be given by specific recording. The following part details displacement field analysis.

SHAPE ANALYSIS DURING MACHINING

Problematic of instantaneous measured modes

On Fig. 6, the red line called H50 shows the total part displacement next to the tool.

The displacement caused by the only machining is obtained by subtracting the translation H0 of the machine table to correct straightness defects.

When the tool is machining around 20.4 mm from the part edge, several fields show that the deformation has an almost fixed point at 88 mm of the edge. The work piece seems to turn around this point.

As regards the cutting zone it is pushed 125 µm far from the tool and this displacement could correspond to the second modal shape. Anyway the measured node is not central as for the free second mode (symmetrical torsion of the plate), so an overall translation caused by the first mode could be implied. When the tool is at 25.1 mm an additional antinode appears against the tool showing that more complex phenomena can be brought into play. For the second zone the 40 µm amplitude of the temporal vibrations is hardly linkable to instantaneous displacements when the tool is at 54.9. Indeed in this case the image noise takes an important ratio of weak displacements. Anyway it can be said the part is nearly 30 µm pushed back because of milling dynamics.

At this stage real modes remain impossible to identify visually. That led us to develop a calculus method.

Modes identification method

In order to calculate free modes weights a specific procedure has been built. It aims at finding the weight of each mode so that their sum is superimposed on instantaneous H50 line shape.

First, normalized free modes are calculated by finite elements. Then they are separately interpolated by polynomials with the least square method (see Fig. 7). For 1 st 2 nd 3 rd and 4 th mode the degrees are respectively 0, 3, 4 and 5.

So each polynomial coefficient takes into account curvatures of the mode in question. Then each polynomial is entirely weighted by a factor (to be determined), that will represent its displacement amplitude contribution.

Secondly for each recording, point coordinates are extracted of the displacement line (H50-H0). The points are then interpolated with least square method by the mode polynomial sum described before (Fig. 8). This step enables to numerically determine modal weights.

Instantaneous free modes recognition

In this test modes higher than third mode don't appear on stability charts. Owing to that only the first four modes are included. Once the identification procedure was applied to the image contributions found are -0.051 for M1, 0.102 for M2, -0.006 for M3 and -0.003 for M4. It clearly appears that the second and first modes are dominant on this snap. The third and fourth one have very low amplitudes.

Quote that the snap is instantaneous while the plate is vibrating. At this instant they may be crossing their reference positions and it cannot be concluded that they have no participation. In other words an only one displacement field doesn't let understand the phenomenon during an entire period. Many snaps while milling must be analyzed to know the real behavior.

Modal contribution for the entire milling

Figure 9 displays the modal amplitudes found for the three milling zones of the operation. The second mode amplitude is dominant. The fact that its value can be positive or negative attests a vibratory component of the motion. The amplitude calculated for the first mode is always negative. That shows that the plate seems to vibrate around a static negative value. To verify this hypothesis, static translation induced by milling forces has been investigated. To do that a finite elements calculus led to a 4 Newton cutting force, corresponding to a 33 µm displacement of the cutting zone (averaged value from 13 measurements in 2 nd zone of the part). The same effort is found with 1 st and 3 rd zones, and also correlated by coarse calculus of chip section due to cutting conditions (see § 3.3). Thus for all images the plate deflexion was given by Finite Element method, by applying 4 Newton at the corresponding tool position. The deformed shape was interpolated and then added to modes in the procedure.

Figure 10 displays new modal calculated amplitudes. The second mode amplitude is still dominant. Contrary to the previous results the amplitudes of the first mode oscillate around zero. These more realistic results show the addition of static deflexion to natural modes. In addition interesting information is brought by the envelope of the second mode weights (dark triangles on the graphic). It may be in good agreement to the second modal torsional shape and confirms the higher response of the mode when the tool is exciting its antinodes.

Anyway note that we assumed the milling effort to be constant. That means the feed rate and the radial tooth engagement are also constant. This assumption is still not realistic enough because the measured temporal displacement can reach 100 µm ( § 4.1). Moreover in a minority of cases the modal sum does not superpose on measured points. So the modal calculated weights prove to be not valid for some photos. To be more realistic non constant plate deflexion must be tested. The identification procedure will give solutions but the amplitudes can mathematically be negative. In other words the work piece would be grabbed by the tool. So the identification procedure reaches its limit.

Synthesis of shape analysis

The test location on stability lobes suggests the part should mainly vibrate according to its second mode. Nevertheless field analysis showed that first and second free modes and a static deflexion simultaneously appear during machining. The modal identification procedure brings supplementary interesting information. Anyway care must be taken because the first modal shape is much closer to the static deflexion. So to enhance results robustness additional calculus will be done without first mode in the polynomial sum. The analysis of the surface quality and machining behavior led to two reports:

CONCLUSIONS

On the one hand measured frequencies are not closed to those predicted by common chatter models which integrate tool free modes. So the tool presence disturbs the natural modes and this effect must be analysed. On the other hand field measurements show that the real behavior of the part seems result from a static deflection due to the milling forces added to vibration.

[13] Wehbe, T., Seguy, S., Dessein, G., Arnaud, L., Fazzini. M.

(2007). Mesure de champs de déplacements en vibrations : Application à l'usinage de paroi mince (Vibration displacement field measurement of thin web machining), August 2007, 18ème Congrès Français de Mécanique, Grenoble, France.

As a consequence complementary tests with different cutting conditions must be done to complete this analysis. First they will enable more sophisticated analyses to quantify the contribution of each mode. From that, we will search how the tooth and work-piece interaction disturbs the natural modes. To validate it, the link between modal shapes and frequencies and generated surface will be investigated. Final objectives of this work are to understand the influence of the tool presence on structural modes. It will provide more specific hypothesis to improve current modeling.

Fig. 1 .

 1 Fig. 1. Measurement set up.

Fig. 2 .

 2 Fig. 2. Layout of the stability chart from [2].

Fig. 4 .

 4 Fig.4. First zone temporal displacement and frequency.

Fig. 3 .

 3 Fig. 3. Selected machining zones.

Fig. 5 .

 5 Fig. 5. Natural frequencies variation according to the location of the connection.

Fig. 6 .

 6 Fig. 6. Displacement fields versus tool position.

Fig. 7 .

 7 Fig. 7. Polynomials of third modal shape.

Fig. 8 .

 8 Fig. 8. Interpolation of measurement points.

Fig. 9 .

 9 Fig. 9. Modal contributions for the whole operations.

[ 8 ]

 8 Delio, T., Tlusty, J., Smith, S. (1992). Use of audio signal for chatter detection and control, Transaction of the ASME, Journal of Engineering for industry, Vol. 114, May 1992, pp. 27-34. [9] Prateepasen, A., Au, Y.H.J., Jones, B. (2001). Acoustic Emission and Vibration for Tool Wear Monitoring in Single-Point Machining Using Belief Network, IEEE Instrumentation and measurement, May 2001, Technology Conference, Budapest, Hungary. [10] Rehorn, A.G., Jiang, J., Orban, P.E. (2005). State-of-theart methods and results in tool condition monitoring : a review, Int J Adv Manuf Technol, Vol. 26, pp. 693-710.

[ 11 ]

 11 Orhan, S., Osman, A., Camuscu, N., Aslan, E. (2007). Tool wear evaluation by vibration analysis during end milling of AISID3 cold work tool steel with 35 HRC hardness, Independent Nondestructive Testing and Evaluation, Vol. 40, pp. 121-126.

[ 12 ]

 12 Girodeau, A., Guo, B. Pierron. F. (2006). Stiffness and Damping Identification from Full Field Measurements on Vibrating Plates, Experimental Mechanics, Vol. 46, pp. 777-787.

Table 1 Eigen frequencies of the instrumented part 1 st mode 2 nd mode 3 rd mode Natural frequency (Hz)

 1

		520	688	1160
	Damping ratio ξn (%)	1.65	3.6	3.92
	Mod. stiffness Kn (N/mm)	132	134	611

Authors:

Toufic WEHBE, PhD student, Laboratoire Génie de Production, Ecole Nationale d'Ingénieurs de Tarbes, France, E-mail: toufic.wehbe@enit.fr Gilles DESSEIN, Assistant professor, Laboratoire Génie de Production, Ecole Nationale d'Ingénieurs de Tarbes, France,