# Properness and Topological Degree for Nonlocal Reaction-Diffusion Operators

Authors: N. Apreutesei; V. Volpert
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

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


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## Body

## 1. Introduction

Consider the semilinear parabolic equation(1.1)∂u∂t=∂2u∂x2+F(u,J(u)),
where (1.2)J(u)=∫-∞∞ϕ(x-y)u(y,t)dy.
Here ϕ:ℝ→ℝ is a bounded function, not necessarily continuous, ϕ≥0 on ℝ. The support of the function ϕ is supposed to be bounded, suppϕ⊆[-N,N]. We will also assume that ∫-∞∞ϕ(y)dy=1. Conditions on the function F will be specified below.Integro-differential equations of this type arise in population dynamics (see [1, 2] and references therein). They are referred to as nonlocal reaction-diffusion equations. A travelling wave solution of (1.1) is a solution of this equation of the particular form u(x,t)=w(x-ct). It satisfies the equation(1.3)w′′+cw′+F(w,J(w))=0.
The constant c∈ℝ is the wave speed. It is unknown and should be found together with the function w(x). There are numerous works devoted to the existence [3–7], stability and nonlinear dynamics [1, 1, 2, 8–16] of travelling wave solutions of some particular cases of (1.1). Properties of travelling waves are determined by the properties of the integro-differential operator in the left-hand side of (1.3). In this paper we will study the Fredholm property of this operator and its properness. We will use them to define the topological degree and will discuss some applications.LetE=C2+α(ℝ), E0=Cα(ℝ), 0<α<1 the usual Holder spaces endowed with the norms (1.4)‖u‖E0=sup⁡x,y∈R|u(x)-u(y)||x-y|α+sup⁡x∈R|u(x)|,‖u‖E=‖u‖E0+‖u′‖E0+‖u′′‖E0.
We are interested in the solutions of equation (1.3) with the limits w± at ±∞, where the values w± are such that F(w±,w±)=0. We are looking for the solutions w of (1.3) under the form w=u+ψ, where ψ∈C∞(ℝ), such that ψ(x)=w+ for x≥1 and ψ(x)=w- for x≤-1. Thus (1.3) becomes (1.5)(u+ψ)′′+c(u+ψ)′+F(u+ψ,J(u+ψ))=0.
Denote by A the operator in the left-hand side of (1.5), that is A:E→E0,(1.6)Au=(u+ψ)′′+c(u+ψ)′+F(u+ψ,J(u+ψ)).Suppose thatF is differentiable with respect to both variables. The linearization of A about a function u1∈E is the operator L:E→E0,(1.7)Lu≡A′(u1)u=u′′+cu′+∂F∂u(u1+ψ,J(u1+ψ))u+∂F∂U(u1+ψ,J(u1+ψ))J(u),
where ∂F/∂u and ∂F/∂U are the derivatives of F(u,U) with respect to the first and to the second variable, respectively.For the linearized operatorL, we introduce the limiting operators. Since for w1=u1+ψ, there exist the limits limx→±∞w1(x)=w±, it follows that J(w1)=J(u1+ψ)→w± as x→±∞ and the limiting operators are given by (1.8)L±u=u′′+cu′+a±u+b±J(u),
where
(1.9)a±=∂F∂u(w±,w±),b±=∂F∂U(w±,w±).We will now recall the main definitions and results concerning the essential spectrum and Fredholm property for linear operators and the properness of nonlinear operators.


### 1.1. Essential Spectrum and Fredholm Property

Let us recall that a linear operatorM:E1→E2 acting from a Banach space E1 into another Banach space E2 is called a Fredholm operator if its kernel has a finite dimension, its image is closed, and the codimension of the image is also finite. The last two conditions are equivalent to the following solvability condition: the equation Lu=f is solvable if and only if ϕi(f)=0 for a finite number of functionals ϕi from the dual space E2*.Suppose thatE1⊂E2. By definition, the essential spectrum of the operator L is the set of all complex λ for which the operator L-λI, where I is the identity operator, does not satisfy the Fredholm property. The essential spectrum of general elliptic boundary value problems in unbounded domains can be determined in terms of limiting operators [17]. For the integro-differential operators under consideration, since they have constant coefficients at infinity, the essential spectrum can be found explicitly. It is proved [5, 6] that the operator L-λI is normally solvable with a finite-dimensional kernel if and only if the equations L±u=λu do not have nonzero bounded solutions. Applying the Fourier transform to the last equations, we obtain
(1.10)λ±(ξ)=-ξ2+ciξ+a±+b±ϕ̃(ξ),ξ∈R,
where ϕ̃(ξ) is the Fourier transform of the function ϕ(x). Thus, the operator L is normally solvable with a finite-dimensional kernel if and only if the curves λ±(ξ) on the complex plane do not pass through the origin. Under some additional conditions, it can be also shown that the codimension of the operator is finite, that is, it satisfies the Fredholm property, and its index can be found.A nonlinear operatorB:E1→E2 is called Fredholm if the linearized operator B′ satisfies this property. In what follows we will use the Fredholm property in some weighted spaces (see below).


### 1.2. Properness and Topological Degree

An operatorB:E1→E2 is called proper on closed bounded sets if the intersection of the inverse image of a compact set K⊂E2 with any closed bounded set in E1 is compact. For the sake of brevity, we will call such operators proper. It is an important property because it implies that the set of solution of the operator equation B(u)=0 is compact.It appears that elliptic (or ordinary differential) operators are not generally proper when considered in Hölder or Sobolev spaces in unbounded domains. We illustrate this situation with a simple example. Consider the equation(1.11)w′′+H(w)=0,x∈R,
where H(w)=w(w-1). It can be verified that this equation has a positive solution w(x), which converges to zero at infinity. This convergence is exponential. So the solution belongs to Hölder and to Sobolev spaces. Along with the function w(x), any shifted function w(x+h), h∈ℝ is also a solution. Hence there is a family of solutions, and the set of solutions is not compact. Similar examples can be constructed for the integro-differential equation.In order to obtain proper operators, we introduce weighted spacesCμk+α(ℝ) with a growing at infinity polynomial weight function μ(x). The norm in this space is given by the equality(1.12)‖u‖Cμk+α(R)=‖μu‖Ck+α(R).
Let us return to the previous example. The family of functions w(x+h) is not uniformly bounded in the weighted space. If we take any bounded closed set in the function space, it can contain the solutions w(x+h) only for a compact set of the values of h. Therefore the set of solutions is compact in any bounded closed set. This example shows the role of weighted spaces for the properness of the operators.Properness of general nonlinear elliptic problems in unbounded domains and in weighted spaces is proved in [18]. In this work, we will prove properness of the integro-differential operators. After that, using the construction of the topological degree for Fredholm and proper operators with the zero index [18], we will define the degree for the integro-differential operators. We will finish this paper with some applications of these methods to travelling waves solutions.


## 1.1. Essential Spectrum and Fredholm Property

Let us recall that a linear operatorM:E1→E2 acting from a Banach space E1 into another Banach space E2 is called a Fredholm operator if its kernel has a finite dimension, its image is closed, and the codimension of the image is also finite. The last two conditions are equivalent to the following solvability condition: the equation Lu=f is solvable if and only if ϕi(f)=0 for a finite number of functionals ϕi from the dual space E2*.Suppose thatE1⊂E2. By definition, the essential spectrum of the operator L is the set of all complex λ for which the operator L-λI, where I is the identity operator, does not satisfy the Fredholm property. The essential spectrum of general elliptic boundary value problems in unbounded domains can be determined in terms of limiting operators [17]. For the integro-differential operators under consideration, since they have constant coefficients at infinity, the essential spectrum can be found explicitly. It is proved [5, 6] that the operator L-λI is normally solvable with a finite-dimensional kernel if and only if the equations L±u=λu do not have nonzero bounded solutions. Applying the Fourier transform to the last equations, we obtain
(1.10)λ±(ξ)=-ξ2+ciξ+a±+b±ϕ̃(ξ),ξ∈R,
where ϕ̃(ξ) is the Fourier transform of the function ϕ(x). Thus, the operator L is normally solvable with a finite-dimensional kernel if and only if the curves λ±(ξ) on the complex plane do not pass through the origin. Under some additional conditions, it can be also shown that the codimension of the operator is finite, that is, it satisfies the Fredholm property, and its index can be found.A nonlinear operatorB:E1→E2 is called Fredholm if the linearized operator B′ satisfies this property. In what follows we will use the Fredholm property in some weighted spaces (see below).


## 1.2. Properness and Topological Degree

An operatorB:E1→E2 is called proper on closed bounded sets if the intersection of the inverse image of a compact set K⊂E2 with any closed bounded set in E1 is compact. For the sake of brevity, we will call such operators proper. It is an important property because it implies that the set of solution of the operator equation B(u)=0 is compact.It appears that elliptic (or ordinary differential) operators are not generally proper when considered in Hölder or Sobolev spaces in unbounded domains. We illustrate this situation with a simple example. Consider the equation(1.11)w′′+H(w)=0,x∈R,
where H(w)=w(w-1). It can be verified that this equation has a positive solution w(x), which converges to zero at infinity. This convergence is exponential. So the solution belongs to Hölder and to Sobolev spaces. Along with the function w(x), any shifted function w(x+h), h∈ℝ is also a solution. Hence there is a family of solutions, and the set of solutions is not compact. Similar examples can be constructed for the integro-differential equation.In order to obtain proper operators, we introduce weighted spacesCμk+α(ℝ) with a growing at infinity polynomial weight function μ(x). The norm in this space is given by the equality(1.12)‖u‖Cμk+α(R)=‖μu‖Ck+α(R).
Let us return to the previous example. The family of functions w(x+h) is not uniformly bounded in the weighted space. If we take any bounded closed set in the function space, it can contain the solutions w(x+h) only for a compact set of the values of h. Therefore the set of solutions is compact in any bounded closed set. This example shows the role of weighted spaces for the properness of the operators.Properness of general nonlinear elliptic problems in unbounded domains and in weighted spaces is proved in [18]. In this work, we will prove properness of the integro-differential operators. After that, using the construction of the topological degree for Fredholm and proper operators with the zero index [18], we will define the degree for the integro-differential operators. We will finish this paper with some applications of these methods to travelling waves solutions.


## 2. Properness in Weighted Spaces

In this section we study the properness of the semilinear operatorA.Definition 2.1.
IfX, Y are Banach spaces, an operator A:X→Y is called proper if for any compact set D⊂Y and any bounded closed set B⊂X, the intersection A-1(D)∩B is a compact set in X.Remark 2.2.
The operatorA:E=C2+α(ℝ)→E0=Cα(ℝ) may not be proper from E to E0 (see the comments related to (1.11) from the introduction).We will show in the sequel thatA is proper in some weighted spaces associated to E and E0.Letμ:ℝ→ℝ be the function given by μ(x)=1+x2,x∈ℝ. Denote E=C2+α(ℝ), E0=Cα(ℝ) endowed with the usual norms ∥·∥E and ∥·∥E0. We will work in the weighted Holder spaces Eμ and Eμ0, which are E and E0, respectively, with the norms ∥u∥μ=∥μu∥E and ∥u∥0μ=∥μu∥E0.We begin with the following estimate for the integral termJ(u).Lemma 2.3.
Suppose thatμ(x)=1+x2  and ϕ:ℝ→ℝ is a function such that ϕ≥0 on ℝ, suppϕ=[-N,N] is bounded,∫-∞∞ϕ(y)dy=1 and ϕ∈E0. Then
(2.1)‖J(u)‖0μ≤K‖μu‖C(R),(∀)u∈E0,
for some constant K>0.Proof.
Ifsuppϕ=[-N,N], then suppϕ(x-·)=[x-N,x+N]. First we write
(2.2)μ(x)J(u)(x)=∫x-Nx+Nμ(x)μ(y)ϕ(x-y)μ(y)u(y)dy.
Since μ(x)/μ(y) is bounded for |x-y|≤N and ∫-∞∞ϕ(y)dy=1, we have ∥μJ(u)∥C(ℝ)≤K1∥μu∥C(ℝ), for some positive constant K1.
For everyx1,x2∈ℝ, x1≠x2, denote(2.3)H(x1,x2)=|μ(x1)J(u)(x1)-μ(x2)J(u)(x2)||x1-x2|α.
If|x1-x2|≥1, we have H(x1,x2)≤2K1∥μu∥C(ℝ). If |x1-x2|<1, then(2.4)H(x1,x2)=1|x1-x2|α|∫x1-Nx1+N[μ(x1)ϕ(x1-y)-μ(x2)ϕ(x2-y)]u(y)dy+∫x1-Nx1+Nμ(x2)ϕ(x2-y)u(y)dy-∫x2-Nx2+Nμ(x2)ϕ(x2-y)u(y)dy|≤|∫x1-Nx1+N{[μ(x1)-μ(x2)]ϕ(x1-y)μ(y)|x1-x2|α+μ(x2)[ϕ(x1-y)-ϕ(x2-y)]μ(y)|x1-x2|α}×μ(y)u(y)dy∫x1-Nx1+N|+1|x1-x2|α|∫x1-Nx2-Nμ(x2)μ(y)ϕ(x2-y)μ(y)u(y)dy|+1|x1-x2|α|∫x2+Nx1+Nμ(x2)μ(y)ϕ(x2-y)μ(y)u(y)dy|.
Since|x1-x2|<1 and |x1-y|≤N, then |x2-y|≤N+1. In this case, the boundedness of ϕ and of (μ(x1)-μ(x2))/μ(y)|x1-x2|α, implies that(2.5)H(x1,x2)≤K2‖μu‖C(R)+K3‖ϕ‖E0‖μu‖C(R)+2K4|x1-x2|1-α‖μu‖C(R),
for some K2,K3,K4>0. Thus the desired estimate holds and the lemma is proved.We study the operatorA acting from Eμ into Eμ0. In order to introduce a topological degree (in Section 4), we prove the properness of A in the more general case when the coefficient c and function F depend also on a parameter τ∈[0,1]. Let Aτ:Eμ→Eμ0, τ∈[0,1] be the operator defined through (2.6)Aτu=(u+ψ)′′+c(τ)(u+ψ)′+Fτ(u+ψ,J(u+ψ)).We note that the linearizationLτ of Aτ about a function u1∈Eμ is (2.7)Lτu=u′′+c(τ)u′+∂Fτ∂u(u1+ψ,J(u1+ψ))u+∂Fτ∂U(u1+ψ,J(u1+ψ))J(u),
while its limiting operators are given by (2.8)Lτ±u=u′′+c(τ)u′+∂Fτ∂u(w±,w±)u+∂Fτ∂U(w±,w±)J(u).Assume that the following hypotheses are satisfied(H1)
For anyτ∈[0,1], the function Fτ(u,U) and its derivatives with respect to u and U satisfy the Lipschitz condition: there exists K>0 such that
(2.9)|Fτ(u1,U1)-Fτ(u2,U2)|≤K(|u1-u2|+|U1-U2|),
for any (u1,U1),(u2,U2)∈ℝ2. Similarly for ∂Fτ/∂u and ∂Fτ/∂U:
(2.10)|∂Fτ(u1,U1)∂u-∂Fτ(u2,U2)∂u|≤K(|u1-u2|+|U1-U2|),|∂Fτ(u1,U1)∂U-∂Fτ(u2,U2)∂U|≤K(|u1-u2|+|U1-U2|).(H2)
c(τ), Fτ(u,U) and the derivatives of Fτ(u,U) are Lipschitz continuous in τ, that is, there exists a constant c>0 such that
(2.11)|c(τ)-c(τ0)|≤c|τ-τ0|,|Fτ(u,U)-Fτ0(u,U)|≤c|τ-τ0|,|∂Fτ(u,U)∂u-∂Fτ0(u,U)∂u|≤c|τ-τ0|,|∂Fτ(u,U)∂U-∂Fτ0(u,U)∂U|≤c|τ-τ0|,(∀)τ,τ0∈[0,1], for all (u,U) from any bounded set in ℝ2.(H3)
(Condition NS) For anyτ∈[0,1], the limiting equations
(2.12)u′′+c(τ)u′+∂Fτ∂u(w±,w±)u+∂Fτ∂U(w±,w±)J(u)=0
do not have nonzero solutions in E.Lemma 2.4.
Suppose that conditions(H1)-(H2) hold. If τn→τ0  and μun→μu0 in C(ℝ), then
(2.13)‖Fτn(un+ψ,J(un+ψ))-Fτ0(u0+ψ,J(u0+ψ))‖0μ⟶0.Proof.
We have the equality(2.14)Fτn(un+ψ,J(un+ψ))-Fτ0(u0+ψ,J(u0+ψ))=Fτn(un+ψ,J(un+ψ))-Fτn(u0+ψ,J(u0+ψ))+Fτn(u0+ψ,J(u0+ψ))-Fτ0(u0+ψ,J(u0+ψ)).
Condition (H1) leads to the estimate of the first difference
(2.15)|Fτn(un+ψ,J(un+ψ))-Fτn(u0+ψ,J(u0+ψ))|≤K(|un-u0|+|J(un-u0)|).
In view of hypothesis μun→μu0 in C(ℝ), the above inequality allows us to conclude that the weighted norm converges to zero.
In order to estimate the second difference, we begin with the following representation:(2.16)Fτn(u0+ψ,J(u0+ψ))-Fτn(ψ,J(ψ))=J(u0)∫01∂Fτn(u0+ψ,J(ψ)+tJ(u0))∂Udt+u0∫01∂Fτn(tu0+ψ,J(ψ))∂udt.
Similarly,
(2.17)Fτ0(u0+ψ,J(u0+ψ))-Fτ0(ψ,J(ψ))=J(u0)∫01∂Fτ0(u0+ψ,J(ψ)+tJ(u0))∂Udt+u0∫01∂Fτ0(tu0+ψ,J(ψ))∂udt.
Therefore,
(2.18)Fτn(u0+ψ,J(u0+ψ))-Fτ0(u0+ψ,J(u0+ψ))=J(u0)(∫01∂Fτn(u0+ψ,J(ψ)+tJ(u0))∂Udt-∫01∂Fτ0(u0+ψ,J(ψ)+tJ(u0))∂Udt)+u0(∫01∂Fτn(tu0+ψ,J(ψ))∂udt-∫01∂Fτ0(tu0+ψ,J(ψ))∂udt)+Fτn(ψ,J(ψ))-Fτ0(ψ,J(ψ))≡A+B+C
(A denotes the first line in the right-hand side, B: the second, C: the third). The expressions A and B converge to zero in the weighted norm of Eμ=Cμα(ℝ), due to the Lipschitz condition with respect to τ of the derivatives of Fτ (see (H2)). The expression C is a function with a finite support. It also converges to zero in the weighted norm as τn→τ0. This concludes the proof.We can now prove the properness of theτ-dependent operator Aτ. Denote by
(2.19)Eloc⁡={u∈C2+α(I),(∀)I=boundedinterval},Eμ,loc⁡={μu∈C2+α(I),(∀)I=boundedinterval}
and similarly Eloc0 and Eμ,loc0.Theorem 2.5.
Ifϕ∈E0, under assumptions (H1)–(H3), the operator Aτ(u):Eμ×[0,1]→Eμ0 from (2.6) is proper with respect to (u,τ) on Eμ×[0,1].Proof.
Consider a convergent sequencefn∈Eμ0, say fn→f0 in Eμ0. Let (un,τn) be a solution in Eμ×[0,1] of the equation Aτn(un)=fn, such that
(2.20)‖un‖μ≤M,(∀)n≥1.
We prove that one can choose a convergent in Eμ subsequence of the sequence un. Without loss of generality we may assume that τn→τ0 as n→∞. Equation Aτn(un)=fn can be written as
(2.21)(un+ψ)′′+c(τn)(un+ψ)′+Fτn(un+ψ,J(un+ψ))=fn.
Multiplying the equation by μ and denoting vn(x)=μ(x)un(x), gn(x)=μ(x)fn(x), we derive that
(2.22)vn′′+[-2μ′μ+c(τn)]vn′+[-μ′′μ+2(μ′μ)2-c(τn)μ′μ]vn+μFτn(un+ψ,J(un+ψ))+μ(ψ′′+c(τn)ψ′)=gn.
Indeed, since μun′=(μun)′-μ′un=vn′-μ′vn/μ and μun′′=(μun)′′-μ′′un-2μ′un′=vn′′-μ′′vn/μ-2μ′(vn′/μ-μ′vn/μ2), by (2.21) one easily obtains (2.22).
The sequencevn=μun is uniformly bounded in E:(2.23)‖vn‖E=‖μun‖E=‖un‖μ≤M,(∀)n≥1.
Then it is locally convergent on a subsequence. More exactly, for every bounded interval [-N,N] of x, there is a subsequence (denoted again vn) converging in C2+α[-N,N] to a limiting function v0∈C2+α[-N,N]. By a diagonalization process we can prolong v0 to ℝ such that v0∈E. Since ∥vn∥E≤M, (∀)n≥1, we can easily see that ∥v0∥E≤M.
Letu0 be the limit that corresponds to un. Then μun→μu0 in Eloc and v0=μu0.
We now want to pass to the limit asn→∞ in (2.21) and (2.22). To this end observe that (H2) implies that(2.24)‖Fτn(un+ψ,J(un+ψ))-Fτ0(u0+ψ,J(u0+ψ))‖0μ=‖Fτn(un+ψ,J(un+ψ))-Fτ0(un+ψ,J(un+ψ))‖0μ+‖Fτ0(un+ψ,J(un+ψ))-Fτ0(u0+ψ,J(u0+ψ))‖0μ≤c1|τn-τ0|+‖Fτ0(un+ψ,J(un+ψ))-Fτ0(u0+ψ,J(u0+ψ))‖0μ.
Since Fτ0 is continuous from Eμ0×Eμ0 to Eμ0 (see (H1)) and J(un+ψ)→J(u0+ψ) in Eμ,loc0, we derive that
(2.25)Fτn(un+ψ,J(un+ψ))⟶Fτ0(u0+ψ,J(u0+ψ))asn⟶∞,inEμ,loc⁡0.
Passing to the limit as n→∞, uniformly on bounded intervals of x in (2.21) and (2.22), one obtains that(2.26)(u0+ψ)′′+c(τ0)(u0+ψ)′+Fτ0(u0+ψ,J(u0+ψ))=f0,(2.27)v0′′+[-2μ′μ+c(τ0)]v0′+[-μ′′μ+2(μ′μ)2-c(τ0)μ′μ]v0+μFτ0(u0+ψ,J(u0+ψ))+μ(ψ′′+c(τ0)ψ′)=μf0.
Subtracting (2.27) from (2.22) and denoting Vn=vn-v0, one finds(2.28)Vn′′+[-2μ′μ+c(τn)]Vn′+[-μ′′μ+2(μ′μ)2-c(τn)μ′μ]Vn+μ[Fτn(un+ψ,J(un+ψ))-Fτ0(u0+ψ,J(u0+ψ))]+[c(τn)-c(τ0)](v0′-μ′μv0+μψ′)=μfn-μf0.
Recall thatVn=vn-v0→0 as n→∞ in Eloc. We show that Vn→0 in C(ℝ). Suppose that it is not the case. Then, without any loss of generality, we can chose a sequence xn→∞ such that |Vn(xn)|≥ɛ>0. This means that |vn(xn)-v0(xn)|≥ɛ>0. Let(2.29)Ṽn(x)=Vn(x+xn)=vn(x+xn)-v0(x+xn)=μ(x+xn)[un(x+xn)-u0(x+xn)].
Then,
(2.30)|Ṽn(0)|=|Vn(xn)|≥ɛ>0.
Writing (2.28) in x+xn, one obtains
(2.31)Ṽn′′(x)+[-2μ′(x+xn)μ(x+xn)+c(τn)]Ṽn′(x)+[-μ′′(x+xn)μ(x+xn)+2(μ′(x+xn)μ(x+xn))2-c(τn)μ′(x+xn)μ(x+xn)]Ṽn(x)+μ(x+xn)[Fτn(un+ψ,J(un+ψ))-Fτ0(u0+ψ,J(u0+ψ))](x+xn)+[c(τn)-c(τ0)][v0′-μ′μv0+μψ′](x+xn)=(μfn-μf0)(x+xn).
We will pass to the limit as n→∞ in (2.31). First we note that by (2.29) and (2.23), there exists Ṽ0∈E such that Ṽn→Ṽ0 as n→∞ in Eloc. Next, it is obvious that
(2.32)1μ(x+xn)⟶0,μ′(x+xn)μ(x+xn)⟶0,μ′′(x+xn)μ(x+xn)⟶0,n⟶∞,
while condition fn→f0 in Eμ0 leads to (μfn-μf0)(x+xn)→0. Inequality (2.23) implies a similar estimate for v0, so v0(x+xn) and v0′(x+xn) are bounded in E. We also have ψ′(x+xn)=0 for x+xn>1 and for x+xn<-1 and
(2.33)μ(x+xn)[Fτn(un+ψ,J(un+ψ))-Fτ0(u0+ψ,J((u0+ψ)))](x+xn)=μ(x+xn)[Fτn(un+ψ,J(un+ψ))-Fτ0(un+ψ,J(un+ψ))](x+xn)+μ(x+xn)[Fτ0(un+ψ,J(un+ψ))-Fτ0(u0+ψ,J(un+ψ))](x+xn)+μ(x+xn)[Fτ0(u0+ψ,J(un+ψ))-Fτ0(u0+ψ,J(u0+ψ))](x+xn).
Denote by T1n, T2n, and T3n the three terms in the right-hand side. Hypothesis (H2) for Fτ0 infers that
(2.34)T1n⟶0,n⟶∞inEloc⁡0.
Next, (2.29) leads to (2.35)T2n=Ṽn(x)[Fτ0(un+ψ,J(un+ψ))-Fτ0(u0+ψ,J(un+ψ))](x+xn)un(x+xn)-u0(x+xn)=Ṽn(x)∂Fτ0∂u(s(un+ψ)+(1-s)(u0+ψ),J(un+ψ))(x+xn)=Ṽn(x)∂Fτ0∂u(sun+(1-s)u0+ψ,J(un+ψ))(x+xn),
for some s∈[0,1]. By (2.20) we obtain |un(x+xn)|≤M/μ(x+xn), |u0(x+xn)|≤M/μ(x+xn), hence
(2.36)(sun+(1-s)u0+ψ)(x+xn)⟶w±,J(un+ψ)(x+xn)=∫-∞∞ϕ(x+xn-y)un(y)dy+∫-∞∞ϕ(x+xn-y)ψ(y)dy.
By the change of variable xn-y=-z, it follows that
(2.37)J(un+ψ)(x+xn)=∫-∞∞ϕ(x-z)un(xn+z)dz+∫-∞∞ϕ(x-z)ψ(xn+z)dz⟶w±,
uniformly on bounded intervals of x. Hypothesis (H1) shows that
(2.38)T2n⟶∂Fτ0∂u(w±,w±)Ṽ0,asn⟶∞inEloc⁡0.
On the other hand,(2.39)T3n=μ(J(un+ψ)-J(u0+ψ))×Fτ0(u0+ψ,J(un+ψ))-Fτ0(u0+ψ,J(u0+ψ))J(un+ψ)-J(u0+ψ)(x+xn)=In(x)⋅∂Fτ0∂U(u0+ψ,sJ(un+ψ)+(1-s)J(u0+ψ))(x+xn),
for some s∈[0,1], where In(x)=μ(J(un+ψ)-J(u0+ψ)). For μ(x)=1+x2, x∈ℝ, with the aid of (2.29), we arrive at
(2.40)In(x)=∫-∞∞μ(x+xn)μ(z+xn)ϕ(x-z)Ṽn(z)dz⟶∫-∞∞ϕ(x-z)Ṽ0(z)dz=J(Ṽ0),
in Eloc0. As above, since J(un)(x+xn)→0, J(u0)(x+xn)→0, J(ψ)(x+xn)→w± uniformly on bounded intervals of x, we deduce that
(2.41)T3n⟶∂Fτ0∂U(w±,w±)J(Ṽ0),asn⟶∞inEloc⁡0.
Now we may pass to the limit in (2.31). With the aid of (2.33)–(2.41) and (H2), one arrives at(2.42)Ṽ0′′+c(τ0)Ṽ0′+∂Fτ0∂u(w±,w±)Ṽ0+∂Fτ0∂U(w±,w±)J(Ṽ0)=0,
which contradicts (H3). Therefore we have proved that Vn→0 in C(ℝ).
Now we have to show thatVn→0 in E=C2+α(ℝ). To this end, we write (2.28) in the form S(Vn)=hn, where S(Vn) is the linear part from the left-hand side and(2.43)hn=(μfn-μf0)-μ[Fτn(un+ψ,J(un+ψ))-Fτ0(u0+ψ,J(u0+ψ))]-[c(τn)-c(τ0)](v0′-μ′μv0+μψ′).
Using Lemma 2.1 from [6] for the linear operator S, we can write
(2.44)‖Vn‖E≤C(‖S(Vn)‖E0+‖Vn‖C(R)).
We make use of Lemma 2.4, hypothesis (H2) for c(τn)-c(τ0), and of the convergence fn→f0 in Eμ0, to deduce that S(Vn)=hn→0 in E0=Cα(ℝ). Since Vn→0 in C(ℝ), we conclude that un→u0 in Eμ. The theorem is proved.


## 3. Fredholm Property in Weighted Spaces

Consider the operatorLτ:Eμ=Cμ2+α→Eμ0=Cμα,(3.1)Lτu=u′′+c(τ)u′+∂Fτ∂u(u1+ψ,J(u1+ψ))u+∂Fτ∂U(u1+ψ,J(u1+ψ))J(u),
and its limiting operators(3.2)Lτ±u=u′′+c(τ)u′+∂Fτ∂u(w±,w±)u+∂Fτ∂U(w±,w±)J(u).Recall here condition NS forLτ, that is, hypothesis (H3): for eachτ∈[0,1],the limiting equationsLτ±u=0do not have nonzero solutions.We prove now the Fredholm property ofLτ as an operator acting between the above weighted Holder spaces.Theorem 3.1.
If condition NS is satisfied, then the operatorLτ:Eμ=Cμ2+α→Eμ0=Cμα (acting between weighted spaces) is normally solvable with a finite-dimensional kernel.Proof.
Like in Theorem 2.2 from [6], we can prove that Lτ from E to E0 is normally solvable with a finite-dimensional kernel. To verify the property in the weighted spaces, we use Lemma 2.24 in [18]: if Lτ:E→E0 is normally solvable with a finite-dimensional kernel and the operator K:Eμ→E0, Ku=μLτu-Lτ(μu) is compact, then Lτ:Eμ→Eμ0 is normally solvable with a finite-dimensional kernel.
Let{ui} be a sequence such that ||ui||Eμ≤M. We prove the existence of a subsequence of {Kui} which converges in E0=Cα(ℝ). Consider the sequence vi=μui. Since ∥vi∥=∥ui∥Eμ≤M, one can find a subsequence, denoted again {vi}, which converges locally in C2 to a function v0, which can be prolonged to ℝ by a diagonalization process. We have v0∈E, ∥v0∥E≤M and vi→v0 in Eloc (in C2+α(I), for every bounded interval I).
Letu0 be such that v0=μu0. Then(3.3)‖Kui-Ku0‖E0=‖K(ziμ)‖E0,zi=vi-v0=μ(ui-u0).
Observe that ∥zi∥E≤2M and zi→0 in Eloc. Now we can write
(3.4)K(ziμ)=μLτ(ziμ)-Lτ(zi)=(-μ′′μ+2(μ′μ)2-c(τ)μ′μ)zi-2μ′μzi′+∂Fτ∂U(u1+ψ,J(u1+ψ))[μJ(ziμ)-J(zi)].
But
(3.5)μJ(ziμ)-J(zi)=∫-∞∞ϕ(x-y)zi(y)[μ(x)μ(y)-1]dy=∫-∞∞ϕ(ξ)zi(x-ξ)[μ(x)μ(x-ξ)-1]dξ.
Since μ(x)/μ(x-ξ)-1≤h(x), where h(x)→0 as x→±∞, zi(x) is uniformly bounded and zi(x)→0 as i→∞ locally with respect to x, it follows that μJ(zi/μ)-J(zi)→0 as i→∞, uniformly with respect to x on ℝ. Similarly, zi, zi′ are uniformly bounded, zi→0, zi′→0 as i→∞ locally and μ′′/μ→0, μ′/μ→0 as x→±∞, so the first two terms from (3.4) tend to zero uniformly with respect to x∈ℝ, as i→∞. This implies that K(zi/μ)→0 as i→∞ in C(ℝ). Therefore, with the aid of the local convergence zi→0 in C2, we conclude that Kui→Ku0 as i→∞ in E0=Cα(ℝ). The theorem is proved.We prove now the Fredholm property forLτ:Eμ→Eμ0, under an additional hypothesis. To this end, let I be the identity operator on Eμ.Condition NS[λ]
For eachτ∈[0,1], the limiting equations Lτ±u-λu=0 associated to the operator Lτ-λI  do not have nonzero solutions in Eμ, for any λ≥0.We recall an auxiliary result from [6] which will be employed below.Lemma 3.2.
Consider the operatorsL0,L1:C2+α(ℝ)→Cα(ℝ)  defined by L0u=Lτu-ρu, L1u=u′′-ρu(ρ≥0) and the homotopy Ls:C2+α(ℝ)→Cα(ℝ), Ls=(1-s)L0+sL1, s∈[0,1]. Then there exists ρ≥0 large enough such that the limiting equations (Ls)±u=0 do not have nonzero solutions for any s∈[0,1].Theorem 3.3.
If Condition NS(λ)  is satisfied, then Lτ, regarded as an operator from Eμ to Eμ0, has the Fredholm property and its index is zero.Proof.
We putL0u=Lτu-λu, L1u=u′′-λu and Ls:Eμ=Cμ2+α(ℝ)→Eμ0=Cμα(ℝ), Ls=(1-s)L0+sL1, s∈[0,1]. Condition NS(λ) for Lτ implies Condition NS for L0=Lτ-λI. Then, Theorem 3.1 ensures that L0=Lτ-λI, regarded from Eμ to Eμ0, is normally solvable with a finite-dimensional kernel. For operator L1, we have kerL1={0}, ImL1=Eμ0, hence L1 is a Fredholm operator and its index is indL1=0.
By Lemma3.2 applied for Ls, there exists λ≥0 large enough such that Condition NS holds for all Ls, s∈[0,1]. In view of Theorem 3.1, it follows that the operators Ls are normally solvable with a finite-dimensional kernel. In other words, the homotopy Ls gives a continuous deformation from the operator L0 to the operator L1, in the class of the normally solvable operators with finite-dimensional kernels. Such deformation preserves the Fredholm property and the index. Since the index of L1 is zero, we derive that the index of all Ls is zero. In particular, for s=0 and λ=0, one arrives at the conclusion that Lτ has the Fredholm property and its index is zero. This completes the proof.


## 4. Topological Degree

In this section we apply the topological degree construction for Fredholm and proper operators with the zero index constructed in [18] to the integro-differential operators.Definitions 4.
Recall in the beginning the definition of the topological degree. Consider two Banach spacesE1, E2, a class Φ of operators acting from E1 to E2 and a class of homotopies
(4.1)H={Aτ(u):E1×[0,1]⟶E2,suchthatAτ(u)∈Φ,(∀)τ∈[0,1]}.
LetD⊂E1 be an open bounded set and A∈Φ such that A(u)≠0, u∈∂D, where ∂D is the boundary of D. Suppose that for such a pair (D,A), there exists an integer γ(A,D) with the following properties.
(i)
Homotopy invariance.  If Aτ(u)∈H and Aτ(u)≠0, for u∈∂D, τ∈[0,1], then
(4.2)γ(A0,D)=γ(A1,D).(ii)
Additivity. If A∈Φ, D¯ is the closure of D and D1,D2⊂D are open sets, such that D1∩D2=⊘ and A(u)≠0,for all u∈D¯∖(D1∪D2), then
(4.3)γ(A,D)=γ(A,D1)+γ(A,D2).(iii)
Normalization. There exists a bounded linear operator J:E1→E2 with a bounded inverse defined on all E2 such that, for every bounded set D⊂E1 with 0∈D,
(4.4)γ(J,D)=1.
The integerγ(A,D) is called a topological degree.


### 4.1. Degree for Fredholm and Proper Operators

We now recall a general result concerning the existence of a topological degree which was proved in [18, 19].LetE1 and E2 be Banach spaces, E1⊆E2 algebraically and topologically and let G⊂E1 be an open bounded set.Denote byI:E1→E2 the imbedding operator, Iu=u, and by Φ a class of bounded linear operators L:E1→E2 satisfying the following conditions:(a)
the operatorL-λI:E1→E2 is Fredholm for all λ≥0,(b)
for every operatorL∈Φ, there is λ0=λ0(L) such that L-λI has a uniformly bounded inverse for all λ>λ0.Denote byℱ the class(4.5)F={B∈C1(G,E2),Bproper,B′(x)∈Φ,(∀)x∈G},
where B′(x) is the Fréchet derivative of the operator B.Finally, one introduces the classℋ of homotopies given by(4.6)H={B(x,τ)∈C1(G×[0,1],E2),Bproper,B(⋅,τ)∈F,(∀)τ∈[0,1]}.
Here the properness of B is understood in both variables x∈G and τ∈[0,1].Theorem 4.1 (see [18]).
For everyB∈ℋ  and every open set D, with D¯⊂G,there exists a topological degree γ(B,D).Remark 4.2.
Condition (b) can be weakened. LetE1′ and E2′ be two Banach spaces such that Ei⊂Ei′, i=1,2 where the inclusion is understood in the algebraic and topological sense. In the case of the Hölder space Ck+α(ℝ), this can be the space Ck(ℝ) with an integer nonnegative k. We can also consider some integral spaces W∞k,p(ℝ) [17]. Instead of (b) above we can impose the following condition [20]:(b′)
for every operatorL:E1′→E2′, there is λ0=λ0(L) such that L-λI has a uniformly bounded inverse for all λ>λ0.


### 4.2. Degree for the Integrodifferential Operators

Now, letE1=Eμ and E2=Eμ0 be the weighted spaces introduced in the previous section, with μ(x)=1+x2, x∈ℝ. We will apply Theorem 4.1 for the integro-differential operator A of the form (1.6), where function ψ∈C∞(ℝ), ψ(x)=w+, for x≥1, ψ(x)=w- for x≤-1 and(H4)
F(u,U) and its derivatives with respect to u and U are Lipschitz continuous in (u,U);(H5)
the limiting equations(4.7)u′′+cu′+∂F∂u(w±,w±)u+∂F∂U(w±,w±)J(u)-λu=0
do not have nonzero solutions in E, (∀)λ≥0.Under these hypotheses, Theorem2.5 assures that operator A is proper. Moreover, its Fréchet derivative is A′=L from (1.7) and it is a Fredholm operator with the index zero (Theorem 3.3).Considerℱ the class of operators A defined through (1.6), such that (H4)-(H5) are satisfied. Consider also the class ℋ of homotopies Aτ:Eμ→Eμ0, τ∈[0,1], of the form (2.6), satisfying (H1)-(H2) and(H6)
for everyτ∈[0,1], the equations
(4.8)u′′+c(τ)u′+∂Fτ∂u(w±,w±)u+∂Fτ∂U(w±,w±)J(u)-λu=0
do not have nonzero solutions in E, (∀)λ≥0. By Theorem 2.5 and Theorem 3.3, we infer that operators Aτ(u) are Fréchet differentiable, proper with respect to (u,τ) and their Fréchet derivatives Aτ′=Lτ verify condition (a) above. Condition (b′) follows from the lemma in the appendix. Hence ℋ has the form (4.6). Applying Theorem 4.1 for the class of operators ℱ and the class of homotopies ℋ, we are led to the following result.Theorem 4.3.
Suppose that functionsFτ and c(τ) satisfy conditions (H1)-(H2) and (H4)–(H6).  Then a topological degree exists for the class ℱ of operators and the class ℋ of homotopies.


## 4.1. Degree for Fredholm and Proper Operators

We now recall a general result concerning the existence of a topological degree which was proved in [18, 19].LetE1 and E2 be Banach spaces, E1⊆E2 algebraically and topologically and let G⊂E1 be an open bounded set.Denote byI:E1→E2 the imbedding operator, Iu=u, and by Φ a class of bounded linear operators L:E1→E2 satisfying the following conditions:(a)
the operatorL-λI:E1→E2 is Fredholm for all λ≥0,(b)
for every operatorL∈Φ, there is λ0=λ0(L) such that L-λI has a uniformly bounded inverse for all λ>λ0.Denote byℱ the class(4.5)F={B∈C1(G,E2),Bproper,B′(x)∈Φ,(∀)x∈G},
where B′(x) is the Fréchet derivative of the operator B.Finally, one introduces the classℋ of homotopies given by(4.6)H={B(x,τ)∈C1(G×[0,1],E2),Bproper,B(⋅,τ)∈F,(∀)τ∈[0,1]}.
Here the properness of B is understood in both variables x∈G and τ∈[0,1].Theorem 4.1 (see [18]).
For everyB∈ℋ  and every open set D, with D¯⊂G,there exists a topological degree γ(B,D).Remark 4.2.
Condition (b) can be weakened. LetE1′ and E2′ be two Banach spaces such that Ei⊂Ei′, i=1,2 where the inclusion is understood in the algebraic and topological sense. In the case of the Hölder space Ck+α(ℝ), this can be the space Ck(ℝ) with an integer nonnegative k. We can also consider some integral spaces W∞k,p(ℝ) [17]. Instead of (b) above we can impose the following condition [20]:(b′)
for every operatorL:E1′→E2′, there is λ0=λ0(L) such that L-λI has a uniformly bounded inverse for all λ>λ0.


## 4.2. Degree for the Integrodifferential Operators

Now, letE1=Eμ and E2=Eμ0 be the weighted spaces introduced in the previous section, with μ(x)=1+x2, x∈ℝ. We will apply Theorem 4.1 for the integro-differential operator A of the form (1.6), where function ψ∈C∞(ℝ), ψ(x)=w+, for x≥1, ψ(x)=w- for x≤-1 and(H4)
F(u,U) and its derivatives with respect to u and U are Lipschitz continuous in (u,U);(H5)
the limiting equations(4.7)u′′+cu′+∂F∂u(w±,w±)u+∂F∂U(w±,w±)J(u)-λu=0
do not have nonzero solutions in E, (∀)λ≥0.Under these hypotheses, Theorem2.5 assures that operator A is proper. Moreover, its Fréchet derivative is A′=L from (1.7) and it is a Fredholm operator with the index zero (Theorem 3.3).Considerℱ the class of operators A defined through (1.6), such that (H4)-(H5) are satisfied. Consider also the class ℋ of homotopies Aτ:Eμ→Eμ0, τ∈[0,1], of the form (2.6), satisfying (H1)-(H2) and(H6)
for everyτ∈[0,1], the equations
(4.8)u′′+c(τ)u′+∂Fτ∂u(w±,w±)u+∂Fτ∂U(w±,w±)J(u)-λu=0
do not have nonzero solutions in E, (∀)λ≥0. By Theorem 2.5 and Theorem 3.3, we infer that operators Aτ(u) are Fréchet differentiable, proper with respect to (u,τ) and their Fréchet derivatives Aτ′=Lτ verify condition (a) above. Condition (b′) follows from the lemma in the appendix. Hence ℋ has the form (4.6). Applying Theorem 4.1 for the class of operators ℱ and the class of homotopies ℋ, we are led to the following result.Theorem 4.3.
Suppose that functionsFτ and c(τ) satisfy conditions (H1)-(H2) and (H4)–(H6).  Then a topological degree exists for the class ℱ of operators and the class ℋ of homotopies.


## 5. Applications to Travelling Waves

In this section we will discuss some applications of the Fredholm property, properness and topological degree to study travelling wave solutions of (1.1). Let us begin with the classification of the nonlinearities. Denote(5.1)F0(w)=F(w,w).
We obtain this function from F(w,J(w)) if we formally replace the kernel ϕ(x) of the integral by the δ-function. The corresponding reaction-diffusion equation(5.2)∂u∂t=∂2u∂x2+F0(u)
is called bistable if F0′(w±)<0, monostable if one of these derivatives is positive and another one negative and, finally, unstable if F0′(w±)>0. As it is well-known, it can have travelling wave solutions, that is solutions, which satisfy the problem(5.3)w′′+cw′+F0(w)=0,w(±∞)=w±.
Let w0(x) be a solution of (5.3) with some c=c0. The operator L0 linearized about this solution,(5.4)L0u=u′′+c0u′+F0′(w0)u
has the essential spectrum given by two parabolas:(5.5)λ±0(ξ)=-ξ2+c0iξ+F0′(w±),ξ∈R.
Therefore the operator L0 satisfies the Fredholm property if and only if F0′(w±)≠0. If this condition is satisfied, then the index of the operator is well defined. In the bistable case it equals 0, in the monostable case 1, in the unstable case 0 [9].In the case of the integro-differential operator(5.6)Lu=u′′+c0u′+Fu′(w,J(w))u+FU′(w,J(w))J(u),
the essential spectrum is given by the curves(5.7)λ±(ξ)=-ξ2+c0iξ+Fu′(w±,w±)+FU′(w±,w±)ϕ̃(ξ),ξ∈R,
where ϕ̃(ξ) is the Fourier transform of the function ϕ(x). If we replace J(u) by u, that is, ϕ(x) by the δ -function, then the spectrum of the integro-differential operator coincides with the spectrum of the reaction-diffusion operator.We note that(5.8)Fu′(w±,w±)+FU′(w±,w±)ϕ̃(0)=F0′(w±),Reϕ̃(ξ)=∫-∞∞ϕ(x)cos⁡(ξx)dx<∫-∞∞ϕ(x)dx=1.


### 5.1. Fredholm Property

Bistable Case
Let(5.9)F0′(w±)=Fu′(w±,w±)+FU′(w±,w±)<0
(we recall that ϕ̃(0)=1). Suppose that Fu′(w+,w+)<0 and FU′(w+,w+)>0. Then Reλ+(ξ)<0 for all ξ∈ℝ since
(5.10)Fu′(w±,w±)+ReFU′(w±,w±)ϕ̃(ξ)≤Fu′(w±,w±)+FU′(w±,w±)=F0′(w±).
Hence the essential spectrum is completely in the left-half plane. This allows us to prove properness of the corresponding operators in weighted spaces and to define the topological degree.Consider now the case whereFu′(w+,w+)>0 and FU′(w+,w+)<0. The principal difference with the previous case is that the essential spectrum of the integro-differential operator may not be completely in the left-half plane (Figure 1) though this is the case for the reaction-diffusion operator. Depending on the parameters, the essential spectrum can cross the imaginary axis for some pure imaginary values. However the linear operator remains Fredholm since the essential spectrum does not cross the origin; the nonlinear operator remains proper in the corresponding weighted spaces.Figure 1
Schematic representation of a possible location of the essential spectrum of the operatorL.Thus, the bistable case for the reaction-diffusion equation gives rise to two different cases for the integro-differential equation. We will call both of them bistable but will distinguish them when necessary.Monostable Case
Suppose thatF0′(w+)>0 and F0′(w-)<0. Then λ-0(ξ) is in the left-half plane for all ξ∈ℝ; λ+0(ξ) is partially in the right-half plane, λ+0(0)>0. The essential spectrum of the integro-differential operator L given by the curves λ±(ξ) has a similar structure. It does not cross the origin, so that the operator satisfies the Fredholm property. The curve λ+(ξ) is partially in the right-half plane, λ+(0)=λ+0(0)>0. The curve λ-(ξ) can be completely in the left-half plane or partially in the right-half plane (Figure 1). Similar to the bistable case, there are two subcases in the monostable case.Index
In order to find the index of the operatorL, we consider the operator Lτ which depends on the parameter τ characterizing the width of the support of the function ϕτ, supp ϕτ=[-Nτ,Nτ]. We recall that ∫-∞∞ϕτ(x)dx=1. Let L1=L, that is the value τ=1 corresponds to the function ϕ in the operator L.Since the essential spectrum of the operatorLτ can be determined explicitly, then we can affirm that it converges to the essential spectrum of the operator L0 as τ→0. Moreover, Lτ converges to L0 in the operator norm. The essential spectrum of the operator Lτ does not cross the origin. Therefore it is normally solvable with a finite-dimensional kernel. Hence the index of the operator L equals the index of the operator L0 [12]. It is 0 in the bistable case and 1 in the monostable case (cf. [9]).


### 5.2. Topological Degree and Existence of Solutions

In the bistable case we can define the topological degree for the integro-differential operator and use the Leray-Schauder method to prove existence of solutions. In order to use this method we need to obtain a priori estimates of solutions. In [10], a priori estimates are obtained in the case where(5.11)F(u,J(u))=J(u)u(1-u)-αu.
Thus, we can now conclude about the existence of waves for this particular form of the nonlinearity. More general functions will be considered in the subsequent works.


### 5.3. Local Bifurcations and Branches of Solutions

Other conventional applications of the degree are related to local bifurcations and global branches of solutions (see, e.g., [14]). We can now use the corresponding results for the integro-differential operator in the bistable case. Let us emphasize that these results apply in particular for the case where the essential spectrum of the linearized operator crosses the imaginary axis (see above). Therefore the wave persists in this case unless a priori estimates are lost.


## 5.1. Fredholm Property

Bistable Case
Let(5.9)F0′(w±)=Fu′(w±,w±)+FU′(w±,w±)<0
(we recall that ϕ̃(0)=1). Suppose that Fu′(w+,w+)<0 and FU′(w+,w+)>0. Then Reλ+(ξ)<0 for all ξ∈ℝ since
(5.10)Fu′(w±,w±)+ReFU′(w±,w±)ϕ̃(ξ)≤Fu′(w±,w±)+FU′(w±,w±)=F0′(w±).
Hence the essential spectrum is completely in the left-half plane. This allows us to prove properness of the corresponding operators in weighted spaces and to define the topological degree.Consider now the case whereFu′(w+,w+)>0 and FU′(w+,w+)<0. The principal difference with the previous case is that the essential spectrum of the integro-differential operator may not be completely in the left-half plane (Figure 1) though this is the case for the reaction-diffusion operator. Depending on the parameters, the essential spectrum can cross the imaginary axis for some pure imaginary values. However the linear operator remains Fredholm since the essential spectrum does not cross the origin; the nonlinear operator remains proper in the corresponding weighted spaces.Figure 1
Schematic representation of a possible location of the essential spectrum of the operatorL.Thus, the bistable case for the reaction-diffusion equation gives rise to two different cases for the integro-differential equation. We will call both of them bistable but will distinguish them when necessary.Monostable Case
Suppose thatF0′(w+)>0 and F0′(w-)<0. Then λ-0(ξ) is in the left-half plane for all ξ∈ℝ; λ+0(ξ) is partially in the right-half plane, λ+0(0)>0. The essential spectrum of the integro-differential operator L given by the curves λ±(ξ) has a similar structure. It does not cross the origin, so that the operator satisfies the Fredholm property. The curve λ+(ξ) is partially in the right-half plane, λ+(0)=λ+0(0)>0. The curve λ-(ξ) can be completely in the left-half plane or partially in the right-half plane (Figure 1). Similar to the bistable case, there are two subcases in the monostable case.Index
In order to find the index of the operatorL, we consider the operator Lτ which depends on the parameter τ characterizing the width of the support of the function ϕτ, supp ϕτ=[-Nτ,Nτ]. We recall that ∫-∞∞ϕτ(x)dx=1. Let L1=L, that is the value τ=1 corresponds to the function ϕ in the operator L.Since the essential spectrum of the operatorLτ can be determined explicitly, then we can affirm that it converges to the essential spectrum of the operator L0 as τ→0. Moreover, Lτ converges to L0 in the operator norm. The essential spectrum of the operator Lτ does not cross the origin. Therefore it is normally solvable with a finite-dimensional kernel. Hence the index of the operator L equals the index of the operator L0 [12]. It is 0 in the bistable case and 1 in the monostable case (cf. [9]).


## 5.2. Topological Degree and Existence of Solutions

In the bistable case we can define the topological degree for the integro-differential operator and use the Leray-Schauder method to prove existence of solutions. In order to use this method we need to obtain a priori estimates of solutions. In [10], a priori estimates are obtained in the case where(5.11)F(u,J(u))=J(u)u(1-u)-αu.
Thus, we can now conclude about the existence of waves for this particular form of the nonlinearity. More general functions will be considered in the subsequent works.


## 5.3. Local Bifurcations and Branches of Solutions

Other conventional applications of the degree are related to local bifurcations and global branches of solutions (see, e.g., [14]). We can now use the corresponding results for the integro-differential operator in the bistable case. Let us emphasize that these results apply in particular for the case where the essential spectrum of the linearized operator crosses the imaginary axis (see above). Therefore the wave persists in this case unless a priori estimates are lost.



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*Source: 629692-2011-05-02.xml*