Properness and Topological Degree for Nonlocal Reaction-Diffusion Operators

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.

1. Introduction

Consider the semilinear parabolic equation 𝜕𝑢𝜕𝑡=𝜕²𝑢𝜕𝑥²+𝐹(𝑢,𝐽(𝑢)),(1.1) where 𝐽(𝑢)=^∞−∞𝜙(𝑥−𝑦)𝑢(𝑦,𝑡)𝑑𝑦.(1.2) 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 equation𝑤^+𝑐𝑤^+𝐹(𝑤,𝐽(𝑤))=0.(1.3) 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.

Let 𝐸=𝐶^2+𝛼(ℝ), 𝐸⁰=𝐶^𝛼(ℝ), 0<𝛼<1 the usual Holder spaces endowed with the norms ‖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.(1.4) 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 (𝑢+𝜓)^+𝑐(𝑢+𝜓)^+𝐹(𝑢+𝜓,𝐽(𝑢+𝜓))=0.(1.5) Denote by 𝐴 the operator in the left-hand side of (1.5), that is 𝐴∶𝐸→𝐸⁰,𝐴𝑢=(𝑢+𝜓)^+𝑐(𝑢+𝜓)^+𝐹(𝑢+𝜓,𝐽(𝑢+𝜓)).(1.6)

Suppose that 𝐹 is differentiable with respect to both variables. The linearization of 𝐴 about a function 𝑢₁∈𝐸 is the operator 𝐿∶𝐸→𝐸⁰,𝐿𝑢≡𝐴^𝑢₁𝑢=𝑢^+𝑐𝑢^+𝜕𝐹𝜕𝑢𝑢¹+𝜓,𝐽𝑢₁+𝜓𝑢+𝜕𝐹𝜕𝑈𝑢¹+𝜓,𝐽𝑢₁+𝜓𝐽(𝑢),(1.7) where 𝜕𝐹/𝜕𝑢 and 𝜕𝐹/𝜕𝑈 are the derivatives of 𝐹(𝑢,𝑈) with respect to the first and to the second variable, respectively.

For the linearized operator 𝐿, we introduce the limiting operators. Since for 𝑤₁=𝑢₁+𝜓, there exist the limits lim_𝑥→±∞𝑤₁(𝑥)=𝑤^±, it follows that 𝐽(𝑤₁)=𝐽(𝑢₁+𝜓)→𝑤^± as 𝑥→±∞ and the limiting operators are given by 𝐿^±𝑢=𝑢^+𝑐𝑢^+𝑎^±𝑢+𝑏^±𝐽(𝑢),(1.8) where 𝑎^±=𝜕𝐹𝜕𝑢𝑤^±,𝑤^±,𝑏^±=𝜕𝐹𝜕𝑈𝑤^±,𝑤^±.(1.9)

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 operator 𝑀∶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 𝜙_𝑖(𝑓)=0 for a finite number of functionals 𝜙_𝑖 from the dual space 𝐸^∗2.

Suppose that 𝐸₁⊂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 𝜆_±(𝜉)=−𝜉²+𝑐𝑖𝜉+𝑎^±+𝑏^±𝜙(𝜉),𝜉∈ℝ,(1.10) where 𝜙(𝜉) is the Fourier transform of the function 𝜙(𝑥). Thus, the operator 𝐿 is normally solvable with a finite-dimensional kernel if and only if the curves 𝜆_±(𝜉) on the complex plane do not pass through the origin. Under some additional conditions, it can be also shown that the codimension of the operator is finite, that is, it satisfies the Fredholm property, and its index can be found.

A nonlinear operator 𝐵∶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 operator 𝐵∶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 𝑤^+𝐻(𝑤)=0,𝑥∈ℝ,(1.11) where 𝐻(𝑤)=𝑤(𝑤−1). It can be verified that this equation has a positive solution 𝑤(𝑥), which converges to zero at infinity. This convergence is exponential. So the solution belongs to Hölder and to Sobolev spaces. Along with the function 𝑤(𝑥), any shifted function 𝑤(𝑥+ℎ), ℎ∈ℝ is also a solution. Hence there is a family of solutions, and the set of solutions is not compact. Similar examples can be constructed for the integro-differential equation.

In order to obtain proper operators, we introduce weighted spaces 𝐶^{𝜇𝑘+𝛼}(ℝ) with a growing at infinity polynomial weight function 𝜇(𝑥). The norm in this space is given by the equality‖𝑢‖_𝐶^{𝜇𝑘+𝛼}_(ℝ)=‖𝜇𝑢‖_𝐶^𝑘+𝛼_(ℝ).(1.12) 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 operator 𝐴.

Definition 2.1. If 𝑋, 𝑌 are Banach spaces, an operator 𝐴∶𝑋→𝑌 is called proper if for any compact set 𝐷⊂𝑌 and any bounded closed set 𝐵⊂𝑋, the intersection 𝐴⁻¹(𝐷)∩𝐵 is a compact set in 𝑋.

Remark 2.2. The operator 𝐴∶𝐸=𝐶^2+𝛼(ℝ)→𝐸⁰=𝐶^𝛼(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 that 𝐴 is proper in some weighted spaces associated to 𝐸 and 𝐸⁰.

Let 𝜇∶ℝ→ℝ be the function given by 𝜇(𝑥)=1+𝑥²,𝑥∈ℝ. Denote 𝐸=𝐶^2+𝛼(ℝ), 𝐸⁰=𝐶^𝛼(ℝ) endowed with the usual norms ‖⋅‖_𝐸 and ‖⋅‖_𝐸⁰. We will work in the weighted Holder spaces 𝐸_𝜇 and 𝐸^0𝜇, 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 term 𝐽(𝑢).

Lemma 2.3. Suppose that 𝜇(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 ‖𝐽(𝑢)‖_0𝜇≤𝐾‖𝜇𝑢‖_𝐶(ℝ),(∀)𝑢∈𝐸⁰,(2.1) for some constant 𝐾>0.

Proof. If supp𝜙=[−𝑁,𝑁], then supp𝜙(𝑥−⋅)=[𝑥−𝑁,𝑥+𝑁]. First we write 𝜇(𝑥)𝐽(𝑢)(𝑥)=^{𝑥+𝑁𝑥−𝑁}𝜇(𝑥)𝜇(𝑦)𝜙(𝑥−𝑦)𝜇(𝑦)𝑢(𝑦)𝑑𝑦.(2.2) 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 every 𝑥₁,𝑥₂∈ℝ, 𝑥₁≠𝑥₂, denote𝐻𝑥₁,𝑥₂=||𝜇𝑥¹𝐽(𝑢)𝑥₁−𝜇𝑥₂𝐽(𝑢)𝑥₂||||𝑥₁−𝑥₂||^𝛼.(2.3)
If |𝑥₁−𝑥₂|≥1, we have 𝐻(𝑥₁,𝑥₂)≤2𝐾₁‖𝜇𝑢‖_𝐶(ℝ). If |𝑥₁−𝑥₂|<1, then𝐻𝑥₁,𝑥₂=1||𝑥₁−𝑥₂||^𝛼||||^𝑥₁^𝑥+𝑁_1−𝑁+𝜇𝑥₁𝜙𝑥₁−𝑦−𝜇𝑥₂𝜙𝑥₂−𝑦𝑢(𝑦)𝑑𝑦_{𝑥1𝑥+𝑁1−𝑁}≤|||||𝜇𝑥₂𝜙𝑥₂−𝑦𝑢(𝑦)𝑑𝑦−^{𝑥2𝑥+𝑁}_2−𝑁𝜇𝑥₂𝜙𝑥₂−𝑦𝑢(𝑦)𝑑𝑦_{𝑥1𝑥+𝑁1−𝑁}||||𝜇𝑥¹−𝜇𝑥₂𝜙𝑥₁−𝑦𝜇(𝑦)||𝑥₁−𝑥₂||^𝛼+𝜇𝑥²𝜙𝑥₁−𝑦−𝜙𝑥₂−𝑦+×𝜇(𝑦1)𝑢(𝑦)𝑑𝑦||||𝜇(𝑦)||𝑥₁−𝑥₂||^𝛼||𝑥₁−𝑥₂||^𝛼||||^𝑥₂^𝑥−𝑁_1−𝑁𝜇𝑥₂𝜇(𝑦)𝜙𝑥²+1−𝑦𝜇(𝑦)𝑢(𝑦)𝑑𝑦||||||𝑥₁−𝑥₂||^𝛼||||^𝑥₁^𝑥+𝑁_2+𝑁𝜇𝑥₂𝜇(𝑦)𝜙𝑥²−𝑦𝜇(𝑦)𝑢(𝑦)𝑑𝑦||||.(2.4)
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 that𝐻𝑥₁,𝑥₂≤𝐾₂‖𝜇𝑢‖_𝐶(ℝ)+𝐾₃‖𝜙‖_𝐸⁰‖𝜇𝑢‖_𝐶(ℝ)+2𝐾₄||𝑥₁−𝑥₂||^1−𝛼‖𝜇𝑢‖_𝐶(ℝ),(2.5) for some 𝐾₂,𝐾₃,𝐾₄>0. Thus the desired estimate holds and the lemma is proved.

We study the operator 𝐴 acting from 𝐸_𝜇 into 𝐸^0𝜇. In order to introduce a topological degree (in Section 4), we prove the properness of 𝐴 in the more general case when the coefficient 𝑐 and function 𝐹 depend also on a parameter 𝜏∈[0,1]. Let 𝐴_𝜏∶𝐸_𝜇→𝐸^0𝜇, 𝜏∈[0,1] be the operator defined through 𝐴_𝜏𝑢=(𝑢+𝜓)^+𝑐(𝜏)(𝑢+𝜓)^+𝐹_𝜏(𝑢+𝜓,𝐽(𝑢+𝜓)).(2.6)

We note that the linearization 𝐿_𝜏 of 𝐴_𝜏 about a function 𝑢₁∈𝐸_𝜇 is 𝐿_𝜏𝑢=𝑢^+𝑐(𝜏)𝑢^+𝜕𝐹^𝜏𝜕𝑢𝑢¹+𝜓,𝐽𝑢₁+𝜓𝑢+𝜕𝐹^𝜏𝜕𝑈𝑢¹+𝜓,𝐽𝑢₁+𝜓𝐽(𝑢),(2.7) while its limiting operators are given by 𝐿^±𝜏𝑢=𝑢^+𝑐(𝜏)𝑢^+𝜕𝐹^𝜏𝜕𝑢𝑤^±,𝑤^±𝑢+𝜕𝐹^𝜏𝜕𝑈𝑤^±,𝑤^±𝐽(𝑢).(2.8)

Assume that the following hypotheses are satisfied(𝐻1) For any 𝜏∈[0,1], the function 𝐹_𝜏(𝑢,𝑈) and its derivatives with respect to 𝑢 and 𝑈 satisfy the Lipschitz condition: there exists 𝐾>0 such that ||𝐹_𝜏𝑢₁,𝑈₁−𝐹_𝜏𝑢₂,𝑈₂||≤𝐾||𝑢₁−𝑢₂||+||𝑈₁−𝑈₂||,(2.9) for any (𝑢₁,𝑈₁),(𝑢₂,𝑈₂)∈ℝ². Similarly for 𝜕𝐹_𝜏/𝜕𝑢 and 𝜕𝐹_𝜏/𝜕𝑈: ||||𝜕𝐹_𝜏𝑢₁,𝑈₁𝜕𝑢−𝜕𝐹^𝜏𝑢₂,𝑈₂𝜕𝑢||||≤𝐾||𝑢¹||||𝜕𝐹_𝜏−𝑢₂||+||𝑈₁−𝑈₂||,𝑢₁,𝑈₁𝜕𝑈−𝜕𝐹^𝜏𝑢₂,𝑈₂𝜕𝑈||||≤𝐾||𝑢¹−𝑢₂||+||𝑈₁−𝑈₂||.(2.10)(𝐻2)𝑐(𝜏), 𝐹_𝜏(𝑢,𝑈) and the derivatives of 𝐹_𝜏(𝑢,𝑈) are Lipschitz continuous in 𝜏, that is, there exists a constant 𝑐>0 such that ||||𝜕𝐹_𝜏||𝑐(𝜏)−𝑐𝜏₀||≤𝑐||𝜏−𝜏₀||,||𝐹_𝜏(𝑢,𝑈)−𝐹_𝜏0(𝑢,𝑈)||≤𝑐||𝜏−𝜏₀||,(𝑢,𝑈)𝜕𝑢−𝜕𝐹^𝜏0(𝑢,𝑈)𝜕𝑢||||≤𝑐||𝜏−𝜏⁰||,||||𝜕𝐹^𝜏(𝑢,𝑈)𝜕𝑈−𝜕𝐹^𝜏0(𝑢,𝑈)𝜕𝑈||||≤𝑐||𝜏−𝜏⁰||,(2.11)(∀)𝜏,𝜏₀∈[0,1], for all (𝑢,𝑈) from any bounded set in ℝ².(𝐻3) (Condition NS) For any 𝜏∈[0,1], the limiting equations 𝑢^+𝑐(𝜏)𝑢^+𝜕𝐹^𝜏𝜕𝑢𝑤^±,𝑤^±𝑢+𝜕𝐹^𝜏𝜕𝑈𝑤^±,𝑤^±𝐽(𝑢)=0(2.12) do not have nonzero solutions in 𝐸.

Lemma 2.4. Suppose that conditions (𝐻1)-(𝐻2) hold. If 𝜏_𝑛→𝜏₀ and 𝜇𝑢_𝑛→𝜇𝑢₀ in 𝐶(ℝ), then ‖‖𝐹_𝜏𝑛𝑢_𝑛+𝜓,𝐽𝑢_𝑛+𝜓−𝐹_𝜏0𝑢₀+𝜓,𝐽𝑢₀+𝜓‖‖_0𝜇⟶0.(2.13)

Proof. We have the equality 𝐹_𝜏𝑛=𝑢_𝑛𝐹+_𝜏𝑛𝜓,𝐽𝑢_𝑛+𝜓−𝐹_𝜏0𝑢₀+𝜓,𝐽𝑢₀+𝜓+𝐹_𝜏𝑛𝑢_𝑛+𝜓,𝐽𝑢_𝑛+𝜓−𝐹_𝜏𝑛𝑢₀+𝜓,𝐽𝑢₀+𝜓𝑢₀+𝜓,𝐽𝑢₀+𝜓−𝐹_𝜏0𝑢₀+𝜓,𝐽𝑢₀+𝜓.(2.14) Condition (𝐻1) leads to the estimate of the first difference ||𝐹_𝜏𝑛𝑢_𝑛+𝜓,𝐽𝑢_𝑛+𝜓−𝐹_𝜏𝑛𝑢₀+𝜓,𝐽𝑢₀+𝜓||≤𝐾||𝑢_𝑛−𝑢₀||+||𝐽𝑢_𝑛−𝑢₀||.(2.15) In view of hypothesis 𝜇𝑢_𝑛→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:𝐹_𝜏𝑛=𝑢₀𝐽+𝜓𝑢₀,𝐽𝑢₀+𝜓−𝐹_𝜏𝑛(𝜓,𝐽(𝜓))¹⁰𝜕𝐹_𝜏𝑛𝑢₀+𝜓,𝐽(𝜓)+𝑡𝐽𝑢₀𝜕𝑈𝑑𝑡+𝑢⁰¹⁰𝜕𝐹_𝜏𝑛𝑡𝑢₀+𝜓,𝐽(𝜓)𝜕𝑢𝑑𝑡.(2.16) Similarly, 𝐹_𝜏0=𝑢₀+𝐽𝜓𝑢,𝐽𝑢₀+𝜓−𝐹_𝜏0(𝜓,𝐽(𝜓))₀¹⁰𝜕𝐹_𝜏0𝑢₀+𝜓,𝐽(𝜓)+𝑡𝐽𝑢₀𝜕𝑈𝑑𝑡+𝑢⁰¹⁰𝜕𝐹_𝜏0𝑡𝑢₀+𝜓,𝐽(𝜓)𝜕𝑢𝑑𝑡.(2.17) Therefore, 𝐹_𝜏𝑛=𝑢₀𝐽+𝜓𝑢₀,𝐽𝑢₀+𝜓−𝐹_𝜏0𝑢₀+𝜓,𝐽𝑢₀+𝜓¹⁰𝜕𝐹_𝜏𝑛𝑢₀+𝜓,𝐽(𝜓)+𝑡𝐽𝑢₀𝜕𝑈𝑑𝑡−¹⁰𝜕𝐹_𝜏0𝑢₀+𝜓,𝐽(𝜓)+𝑡𝐽𝑢₀+𝑢₀𝜕𝑈𝑑𝑡¹⁰𝜕𝐹_𝜏𝑛𝑡𝑢₀+𝜓,𝐽(𝜓)𝜕𝑢𝑑𝑡−¹⁰𝜕𝐹_𝜏0𝑡𝑢₀+𝜓,𝐽(𝜓)+𝐹_𝜏𝑛𝜕𝑢𝑑𝑡(𝜓,𝐽(𝜓))−𝐹_𝜏0(𝜓,𝐽(𝜓))≡𝐴+𝐵+𝐶(2.18) (𝐴 denotes the first line in the right-hand side, 𝐵: the second, 𝐶: the third). The expressions 𝐴 and 𝐵 converge to zero in the weighted norm of 𝐸_𝜇=𝐶^𝛼𝜇(ℝ), due to the Lipschitz condition with respect to 𝜏 of the derivatives of 𝐹_𝜏 (see (𝐻2)). The expression 𝐶 is a function with a finite support. It also converges to zero in the weighted norm as 𝜏_𝑛→𝜏₀. This concludes the proof.

We can now prove the properness of the 𝜏-dependent operator 𝐴_𝜏. Denote by 𝐸𝐸_loc=𝑢∈𝐶^2+𝛼(𝐼),(∀)𝐼=boundedinterval_𝜇,loc,=𝜇𝑢∈𝐶^2+𝛼(𝐼),(∀)𝐼=boundedinterval(2.19) and similarly 𝐸^0loc and 𝐸^0𝜇,loc.

Theorem 2.5. If 𝜙∈𝐸⁰, under assumptions (𝐻1)−(𝐻3), the operator 𝐴_𝜏(𝑢)∶𝐸_𝜇×[0,1]→𝐸^0𝜇 from (2.6) is proper with respect to (𝑢,𝜏) on 𝐸_𝜇×[0,1].

Proof. Consider a convergent sequence 𝑓_𝑛∈𝐸^0𝜇, say 𝑓_𝑛→𝑓₀ in 𝐸^0𝜇. Let (𝑢_𝑛,𝜏_𝑛) be a solution in 𝐸_𝜇×[0,1] of the equation 𝐴_𝜏𝑛(𝑢_𝑛)=𝑓_𝑛, such that ‖‖𝑢_𝑛‖‖_𝜇≤𝑀,(∀)𝑛≥1.(2.20) We prove that one can choose a convergent in 𝐸_𝜇 subsequence of the sequence 𝑢_𝑛. Without loss of generality we may assume that 𝜏_𝑛→𝜏₀ as 𝑛→∞. Equation 𝐴_𝜏𝑛(𝑢_𝑛)=𝑓_𝑛 can be written as 𝑢_𝑛+𝜓^+𝑐𝜏_𝑛𝑢_𝑛+𝜓^+𝐹_𝜏𝑛𝑢_𝑛+𝜓,𝐽𝑢_𝑛+𝜓=𝑓_𝑛.(2.21) Multiplying the equation by 𝜇 and denoting 𝑣_𝑛(𝑥)=𝜇(𝑥)𝑢_𝑛(𝑥), 𝑔_𝑛(𝑥)=𝜇(𝑥)𝑓_𝑛(𝑥), we derive that 𝑣^𝑛+−2𝜇^𝜇+𝑐𝜏^𝑛𝑣^𝑛+−𝜇^𝜇+2𝜇𝜇^²−𝑐𝜏_𝑛𝜇^𝜇𝑣^𝑛+𝜇𝐹_𝜏𝑛𝑢_𝑛+𝜓,𝐽𝑢_𝑛+𝜓+𝜇𝜓^+𝑐𝜏_𝑛𝜓^=𝑔_𝑛.(2.22) Indeed, since 𝜇𝑢^𝑛=(𝜇𝑢_𝑛)^−𝜇^𝑢_𝑛=𝑣^𝑛−𝜇^𝑣_𝑛/𝜇 and 𝜇𝑢^𝑛=(𝜇𝑢_𝑛)^−𝜇^𝑢_𝑛−2𝜇^𝑢^𝑛=𝑣^𝑛−𝜇^𝑣_𝑛/𝜇−2𝜇^(𝑣^𝑛/𝜇−𝜇^𝑣_𝑛/𝜇²), by (2.21) one easily obtains (2.22).
The sequence 𝑣_𝑛=𝜇𝑢_𝑛 is uniformly bounded in 𝐸:‖‖𝑣_𝑛‖‖_𝐸=‖‖𝜇𝑢_𝑛‖‖_𝐸=‖‖𝑢_𝑛‖‖_𝜇≤𝑀,(∀)𝑛≥1.(2.23) Then it is locally convergent on a subsequence. More exactly, for every bounded interval [−𝑁,𝑁] of 𝑥, there is a subsequence (denoted again 𝑣_𝑛) converging in 𝐶^2+𝛼[−𝑁,𝑁] to a limiting function 𝑣₀∈𝐶^2+𝛼[−𝑁,𝑁]. By a diagonalization process we can prolong 𝑣₀ to ℝ such that 𝑣₀∈𝐸. Since ‖𝑣_𝑛‖_𝐸≤𝑀, (∀)𝑛≥1, we can easily see that ‖𝑣₀‖_𝐸≤𝑀.
Let 𝑢₀ be the limit that corresponds to 𝑢_𝑛. Then 𝜇𝑢_𝑛→𝜇𝑢₀ in 𝐸_loc and 𝑣₀=𝜇𝑢₀.
We now want to pass to the limit as 𝑛→∞ in (2.21) and (2.22). To this end observe that (𝐻2) implies that‖‖𝐹_𝜏𝑛𝑢_𝑛+𝜓,𝐽𝑢_𝑛+𝜓−𝐹_𝜏0𝑢₀+𝜓,𝐽𝑢₀+𝜓‖‖_0𝜇=‖‖𝐹_𝜏𝑛𝑢_𝑛+𝜓,𝐽𝑢_𝑛+𝜓−𝐹_𝜏0𝑢_𝑛+𝜓,𝐽𝑢_𝑛+𝜓‖‖_0𝜇+‖‖𝐹_𝜏0𝑢_𝑛+𝜓,𝐽𝑢_𝑛+𝜓−𝐹_𝜏0𝑢₀+𝜓,𝐽𝑢₀+𝜓‖‖_0𝜇≤𝑐₁||𝜏_𝑛−𝜏₀||+‖‖𝐹_𝜏0𝑢_𝑛+𝜓,𝐽𝑢_𝑛+𝜓−𝐹_𝜏0𝑢₀+𝜓,𝐽𝑢₀+𝜓‖‖_0𝜇.(2.24) Since 𝐹_𝜏0 is continuous from 𝐸^0𝜇×𝐸^0𝜇 to 𝐸^0𝜇 (see (𝐻1)) and 𝐽(𝑢_𝑛+𝜓)→𝐽(𝑢₀+𝜓) in 𝐸^0𝜇,loc, we derive that 𝐹_𝜏𝑛𝑢_𝑛+𝜓,𝐽𝑢_𝑛+𝜓⟶𝐹_𝜏0𝑢₀+𝜓,𝐽𝑢₀+𝜓as𝑛⟶∞,in𝐸^0𝜇,loc.(2.25) Passing to the limit as 𝑛→∞, uniformly on bounded intervals of 𝑥 in (2.21) and (2.22), one obtains that𝑢𝑣₀+𝜓^+𝑐𝜏₀𝑢₀+𝜓^+𝐹_𝜏0𝑢₀+𝜓,𝐽𝑢₀+𝜓=𝑓₀,(2.26)_0+−2𝜇^𝜇+𝑐𝜏⁰𝑣^0+−𝜇^𝜇+2𝜇𝜇^²−𝑐𝜏₀𝜇^𝜇𝑣⁰+𝜇𝐹_𝜏0𝑢₀+𝜓,𝐽𝑢₀+𝜓+𝜇𝜓^+𝑐𝜏₀𝜓^=𝜇𝑓₀.(2.27)
Subtracting (2.27) from (2.22) and denoting 𝑉_𝑛=𝑣_𝑛−𝑣₀, one finds𝑉^𝑛+−2𝜇^𝜇+𝑐𝜏^𝑛𝑉^𝑛+−𝜇^𝜇+2𝜇𝜇^²−𝑐𝜏_𝑛𝜇^𝜇𝑉^𝑛++𝜇𝑐𝐹𝜏_𝜏𝑛𝑢_𝑛+𝜓,𝐽𝑢_𝑛+𝜓−𝐹_𝜏0𝑢₀+𝜓,𝐽𝑢₀+𝜓_𝑛−𝑐𝜏₀𝑣^0−𝜇𝜇^𝑣⁰+𝜇𝜓^=𝜇𝑓^𝑛−𝜇𝑓₀.(2.28)
Recall that 𝑉_𝑛=𝑣_𝑛−𝑣₀→0 as 𝑛→∞ in 𝐸_loc. We show that 𝑉_𝑛→0 in 𝐶(ℝ). Suppose that it is not the case. Then, without any loss of generality, we can chose a sequence 𝑥_𝑛→∞ such that |𝑉_𝑛(𝑥_𝑛)|≥𝜀>0. This means that |𝑣_𝑛(𝑥_𝑛)−𝑣₀(𝑥_𝑛)|≥𝜀>0. Let𝑉_𝑛(𝑥)=𝑉_𝑛𝑥+𝑥_𝑛=𝑣_𝑛𝑥+𝑥_𝑛−𝑣₀𝑥+𝑥_𝑛=𝜇𝑥+𝑥_𝑛𝑢_𝑛𝑥+𝑥_𝑛−𝑢₀𝑥+𝑥_𝑛.(2.29) Then, ||𝑉_𝑛(0)||=||𝑉_𝑛𝑥_𝑛||≥𝜀>0.(2.30) Writing (2.28) in 𝑥+𝑥_𝑛, one obtains 𝑉^𝑛(𝑥)+−2𝜇^𝑥+𝑥_𝑛𝜇𝑥+𝑥_𝑛+𝑐𝜏^𝑛+⎡⎢⎢⎣−𝜇𝑉^𝑛(𝑥)_𝑥+𝑥_𝑛𝜇𝑥+𝑥_𝑛+2𝜇^𝑥+𝑥_𝑛𝜇𝑥+𝑥_𝑛²−𝑐𝜏_𝑛𝜇^𝑥+𝑥_𝑛𝜇𝑥+𝑥_𝑛⎤⎥⎥⎦𝑉^𝑛+𝜇𝑥+𝑥_𝑛(𝑥)+𝑐𝜏_𝑛𝐹_𝜏𝑛𝑢_𝑛+𝜓,𝐽𝑢_𝑛+𝜓−𝐹_𝜏0𝑢₀+𝜓,𝐽𝑢₀+𝜓𝑥+𝑥_𝑛−𝑐𝜏₀𝑣^0−𝜇𝜇^𝑣⁰+𝜇𝜓^𝑥+𝑥^𝑛=𝜇𝑓_𝑛−𝜇𝑓₀𝑥+𝑥_𝑛.(2.31) We will pass to the limit as 𝑛→∞ in (2.31). First we note that by (2.29) and (2.23), there exists 𝑉₀∈𝐸 such that 𝑉_𝑛→𝑉₀ as 𝑛→∞ in 𝐸_loc. Next, it is obvious that 1𝜇𝑥+𝑥_𝑛⟶0,𝜇^𝑥+𝑥_𝑛𝜇𝑥+𝑥_𝑛⟶0,𝜇^𝑥+𝑥_𝑛𝜇𝑥+𝑥_𝑛⟶0,𝑛⟶∞,(2.32) while condition 𝑓_𝑛→𝑓₀ in 𝐸^0𝜇 leads to (𝜇𝑓_𝑛−𝜇𝑓₀)(𝑥+𝑥_𝑛)→0. Inequality (2.23) implies a similar estimate for 𝑣₀, so 𝑣₀(𝑥+𝑥_𝑛) and 𝑣^0(𝑥+𝑥_𝑛) are bounded in 𝐸. We also have 𝜓^(𝑥+𝑥_𝑛)=0 for 𝑥+𝑥_𝑛>1 and for 𝑥+𝑥_𝑛<−1 and 𝜇𝑥=+𝜇𝑥_𝑛𝑥+𝐹𝑥_𝜏𝑛𝑛𝑢_𝑛+𝜓,𝐽𝑢_𝑛+𝜓−𝐹_𝜏0𝑢₀+𝜓,𝐽𝑢₀+𝜓𝑥+𝑥_𝑛+𝜇𝑥+𝑥_𝑛𝐹_𝜏𝑛𝑢_𝑛+𝜓,𝐽𝑢_𝑛+𝜓−𝐹_𝜏0𝑢_𝑛+𝜓,𝐽𝑢_𝑛+𝜓𝑥+𝑥_𝑛+𝜇𝑥+𝑥_𝑛𝐹_𝜏0𝑢_𝑛+𝜓,𝐽𝑢_𝑛+𝜓−𝐹_𝜏0𝑢₀+𝜓,𝐽𝑢_𝑛+𝜓𝑥+𝑥_𝑛𝐹_𝜏0𝑢₀+𝜓,𝐽𝑢_𝑛+𝜓−𝐹_𝜏0𝑢₀+𝜓,𝐽𝑢₀+𝜓𝑥+𝑥_𝑛.(2.33) Denote by 𝑇^𝑛1, 𝑇^𝑛2, and 𝑇^𝑛3 the three terms in the right-hand side. Hypothesis (𝐻2) for 𝐹_𝜏0 infers that 𝑇^𝑛1⟶0,𝑛⟶∞in𝐸^0loc.(2.34)
Next, (2.29) leads to 𝑇^𝑛2=𝑉_𝑛(𝑥)𝐹^𝜏0𝑢_𝑛+𝜓,𝐽𝑢_𝑛+𝜓−𝐹_𝜏0𝑢₀+𝜓,𝐽𝑢_𝑛+𝜓𝑥+𝑥_𝑛=𝑉_𝑛𝑢_𝑛𝑥+𝑥_𝑛−𝑢₀𝑥+𝑥_𝑛(𝑥)𝜕𝐹^𝜏0𝜕𝑢𝑠𝑢^𝑛=𝑉_𝑛+𝜓+(1−𝑠)𝑢₀+𝜓,𝐽𝑢_𝑛+𝜓𝑥+𝑥_𝑛(𝑥)𝜕𝐹^𝜏0𝜕𝑢𝑠𝑢^𝑛+(1−𝑠)𝑢₀+𝜓,𝐽𝑢_𝑛+𝜓𝑥+𝑥_𝑛,(2.35) for some 𝑠∈[0,1]. By (2.20) we obtain |𝑢_𝑛(𝑥+𝑥_𝑛)|≤𝑀/𝜇(𝑥+𝑥_𝑛), |𝑢₀(𝑥+𝑥_𝑛)|≤𝑀/𝜇(𝑥+𝑥_𝑛), hence 𝐽𝑢_𝑛𝑠𝑢_𝑛+(1−𝑠)𝑢₀+𝜓𝑥+𝑥_𝑛⟶𝑤^±,+𝜓𝑥+𝑥_𝑛=^∞−∞𝜙𝑥+𝑥_𝑛−𝑦𝑢_𝑛(𝑦)𝑑𝑦+^∞−∞𝜙𝑥+𝑥_𝑛−𝑦𝜓(𝑦)𝑑𝑦.(2.36) By the change of variable 𝑥_𝑛−𝑦=−𝑧, it follows that 𝐽𝑢_𝑛+𝜓𝑥+𝑥_𝑛=^∞−∞𝜙(𝑥−𝑧)𝑢_𝑛𝑥_𝑛+𝑧𝑑𝑧+^∞−∞𝜙(𝑥−𝑧)𝜓𝑥_𝑛+𝑧𝑑𝑧⟶𝑤^±,(2.37) uniformly on bounded intervals of 𝑥. Hypothesis (𝐻1) shows that 𝑇^𝑛2⟶𝜕𝐹^𝜏0𝜕𝑢𝑤^±,𝑤^±𝑉₀,as𝑛⟶∞in𝐸^0loc.(2.38)
On the other hand,𝑇^𝑛3=×𝜇𝐹𝐽𝑢_𝑛+𝜓−𝐽𝑢₀+𝜓_𝜏0𝑢₀+𝜓,𝐽𝑢_𝑛+𝜓−𝐹_𝜏0𝑢₀+𝜓,𝐽𝑢₀+𝜓𝐽𝑢_𝑛+𝜓−𝐽𝑢₀+𝜓𝑥+𝑥^𝑛=𝐼_𝑛(𝑥)⋅𝜕𝐹^𝜏0𝜕𝑈𝑢⁰+𝜓,𝑠𝐽𝑢_𝑛+𝜓+(1−𝑠)𝐽𝑢₀+𝜓𝑥+𝑥_𝑛,(2.39) for some 𝑠∈[0,1], where 𝐼_𝑛(𝑥)=𝜇(𝐽(𝑢_𝑛+𝜓)−𝐽(𝑢₀+𝜓)). For 𝜇(𝑥)=1+𝑥², 𝑥∈ℝ, with the aid of (2.29), we arrive at 𝐼_𝑛(𝑥)=^∞−∞𝜇𝑥+𝑥_𝑛𝜇𝑧+𝑥_𝑛𝜙(𝑥−𝑧)𝑉^𝑛(𝑧)𝑑𝑧⟶^∞−∞𝜙(𝑥−𝑧)𝑉₀(𝑧)𝑑𝑧=𝐽𝑉₀,(2.40) in 𝐸^0loc. As above, since 𝐽(𝑢_𝑛)(𝑥+𝑥_𝑛)→0, 𝐽(𝑢₀)(𝑥+𝑥_𝑛)→0, 𝐽(𝜓)(𝑥+𝑥_𝑛)→𝑤^± uniformly on bounded intervals of 𝑥, we deduce that 𝑇^𝑛3⟶𝜕𝐹^𝜏0𝜕𝑈𝑤^±,𝑤^±𝐽𝑉₀,as𝑛⟶∞in𝐸^0loc.(2.41)
Now we may pass to the limit in (2.31). With the aid of (2.33)–(2.41) and (𝐻2), one arrives at𝑉^0+𝑐𝜏₀𝑉^0+𝜕𝐹^𝜏0𝜕𝑢𝑤^±,𝑤^±𝑉₀+𝜕𝐹^𝜏0𝜕𝑈𝑤^±,𝑤^±𝐽𝑉₀=0,(2.42) which contradicts (𝐻3). Therefore we have proved that 𝑉_𝑛→0 in 𝐶(ℝ).
Now we have to show that 𝑉_𝑛→0 in 𝐸=𝐶^2+𝛼(ℝ). To this end, we write (2.28) in the form 𝑆(𝑉_𝑛)=ℎ_𝑛, where 𝑆(𝑉_𝑛) is the linear part from the left-hand side andℎ_𝑛=−𝜇𝑐𝑓_𝑛𝜏−𝜇𝑓₀−𝜇𝐹_𝜏𝑛𝑢_𝑛+𝜓,𝐽𝑢_𝑛+𝜓−𝐹_𝜏0𝑢₀+𝜓,𝐽𝑢₀+𝜓_𝑛−𝑐𝜏₀𝑣^0−𝜇𝜇^𝑣⁰+𝜇𝜓^.(2.43) Using Lemma 2.1 from [6] for the linear operator 𝑆, we can write ‖‖𝑉_𝑛‖‖_𝐸≤𝐶‖‖𝑆𝑉_𝑛‖‖_𝐸⁰+‖‖𝑉_𝑛‖‖_𝐶(ℝ).(2.44) We make use of Lemma 2.4, hypothesis (𝐻₂) for 𝑐(𝜏_𝑛)−𝑐(𝜏₀), and of the convergence 𝑓_𝑛→𝑓₀ in 𝐸^0𝜇, to deduce that 𝑆(𝑉_𝑛)=ℎ_𝑛→0 in 𝐸⁰=𝐶^𝛼(ℝ). Since 𝑉_𝑛→0 in 𝐶(ℝ), we conclude that 𝑢_𝑛→𝑢₀ in 𝐸_𝜇. The theorem is proved.

3. Fredholm Property in Weighted Spaces

Consider the operator 𝐿_𝜏∶𝐸_𝜇=𝐶^𝜇2+𝛼→𝐸^0𝜇=𝐶^𝛼𝜇,𝐿_𝜏𝑢=𝑢^+𝑐(𝜏)𝑢^+𝜕𝐹^𝜏𝜕𝑢𝑢¹+𝜓,𝐽𝑢₁+𝜓𝑢+𝜕𝐹^𝜏𝜕𝑈𝑢¹+𝜓,𝐽𝑢₁+𝜓𝐽(𝑢),(3.1) and its limiting operators𝐿^±𝜏𝑢=𝑢^+𝑐(𝜏)𝑢^+𝜕𝐹^𝜏𝜕𝑢𝑤^±,𝑤^±𝑢+𝜕𝐹^𝜏𝜕𝑈𝑤^±,𝑤^±𝐽(𝑢).(3.2)

Recall here condition NS for 𝐿_𝜏, that is, hypothesis (𝐻3): for each 𝜏∈[0,1], the limiting equations 𝐿^±𝜏𝑢=0 do not have nonzero solutions.

We prove now the Fredholm property of 𝐿_𝜏 as an operator acting between the above weighted Holder spaces.

Theorem 3.1. If condition NS is satisfied, then the operator 𝐿_𝜏∶𝐸_𝜇=𝐶^𝜇2+𝛼→𝐸^0𝜇=𝐶^𝛼𝜇 (acting between weighted spaces) is normally solvable with a finite-dimensional kernel.

Proof. Like in Theorem 2.2 from [6], we can prove that 𝐿_𝜏 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 𝐿_𝜏∶𝐸→𝐸⁰ is normally solvable with a finite-dimensional kernel and the operator 𝐾∶𝐸_𝜇→𝐸⁰, 𝐾𝑢=𝜇𝐿_𝜏𝑢−𝐿_𝜏(𝜇𝑢) is compact, then 𝐿_𝜏∶𝐸_𝜇→𝐸^0𝜇 is normally solvable with a finite-dimensional kernel.
Let {𝑢_𝑖} be a sequence such that ||𝑢_𝑖||_𝐸𝜇≤𝑀. We prove the existence of a subsequence of {𝐾𝑢_𝑖} which converges in 𝐸⁰=𝐶^𝛼(ℝ). Consider the sequence 𝑣_𝑖=𝜇𝑢_𝑖. Since ‖𝑣_𝑖‖=‖𝑢_𝑖‖_𝐸𝜇≤𝑀, one can find a subsequence, denoted again {𝑣_𝑖}, 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 𝑣_𝑖→𝑣₀ in 𝐸_loc (in 𝐶^2+𝛼(𝐼), for every bounded interval 𝐼).
Let 𝑢₀ be such that 𝑣₀=𝜇𝑢₀. Then‖‖𝐾𝑢_𝑖−𝐾𝑢₀‖‖_𝐸0=‖‖‖𝐾𝑧^𝑖𝜇‖‖‖_𝐸0,𝑧_𝑖=𝑣_𝑖−𝑣₀=𝜇𝑢_𝑖−𝑢₀.(3.3) Observe that ‖𝑧_𝑖‖_𝐸≤2𝑀 and 𝑧_𝑖→0 in 𝐸_loc. Now we can write 𝐾𝑧^𝑖𝜇=𝜇𝐿^𝜏𝑧𝜇^𝑖−𝐿^𝜏𝑧_𝑖=−𝜇𝜇^+2𝜇𝜇^²−𝑐(𝜏)𝜇^𝜇𝑧^𝑖−2𝜇^𝜇𝑧^𝑖+𝜕𝐹^𝜏𝜕𝑈𝑢¹+𝜓,𝐽𝑢₁+𝜓𝜇𝐽𝑧^𝑖𝜇−𝐽𝑧^𝑖.(3.4) But 𝜇𝐽𝑧^𝑖𝜇−𝐽𝑧^𝑖=^∞−∞𝜙(𝑥−𝑦)𝑧_𝑖(𝑦)𝜇(𝑥)=𝜇(𝑦)−1𝑑𝑦_∞−∞𝜙(𝜉)𝑧_𝑖(𝑥−𝜉)𝜇(𝑥)𝜇(𝑥−𝜉)−1𝑑𝜉.(3.5) Since 𝜇(𝑥)/𝜇(𝑥−𝜉)−1≤ℎ(𝑥), where ℎ(𝑥)→0 as 𝑥→±∞, 𝑧_𝑖(𝑥) is uniformly bounded and 𝑧_𝑖(𝑥)→0 as 𝑖→∞ locally with respect to 𝑥, it follows that 𝜇𝐽(𝑧_𝑖/𝜇)−𝐽(𝑧_𝑖)→0 as 𝑖→∞, uniformly with respect to 𝑥 on ℝ. Similarly, 𝑧_𝑖, 𝑧^𝑖 are uniformly bounded, 𝑧_𝑖→0, 𝑧^𝑖→0 as 𝑖→∞ locally and 𝜇^/𝜇→0, 𝜇^/𝜇→0 as 𝑥→±∞, so the first two terms from (3.4) tend to zero uniformly with respect to 𝑥∈ℝ, as 𝑖→∞. This implies that 𝐾(𝑧_𝑖/𝜇)→0 as 𝑖→∞ in 𝐶(ℝ). Therefore, with the aid of the local convergence 𝑧_𝑖→0 in 𝐶², we conclude that 𝐾𝑢_𝑖→𝐾𝑢₀ as 𝑖→∞ in 𝐸⁰=𝐶^𝛼(ℝ). The theorem is proved.

We prove now the Fredholm property for 𝐿_𝜏∶𝐸_𝜇→𝐸^0𝜇, under an additional hypothesis. To this end, let 𝐼 be the identity operator on 𝐸_𝜇.

Condition NS[𝜆]
For each 𝜏∈[0,1], the limiting equations 𝐿^±𝜏𝑢−𝜆𝑢=0 associated to the operator 𝐿_𝜏−𝜆𝐼 do not have nonzero solutions in 𝐸_𝜇, for any 𝜆≥0.

We recall an auxiliary result from [6] which will be employed below.

Lemma 3.2. Consider the operators 𝐿⁰,𝐿¹∶𝐶^2+𝛼(ℝ)→𝐶^𝛼(ℝ) defined by 𝐿⁰𝑢=𝐿_𝜏𝑢−𝜌𝑢, 𝐿¹𝑢=𝑢^−𝜌𝑢(𝜌≥0) and the homotopy 𝐿^𝑠∶𝐶^2+𝛼(ℝ)→𝐶^𝛼(ℝ), 𝐿^𝑠=(1−𝑠)𝐿⁰+𝑠𝐿¹, 𝑠∈[0,1]. Then there exists 𝜌≥0 large enough such that the limiting equations (𝐿^𝑠)^±𝑢=0 do not have nonzero solutions for any 𝑠∈[0,1].

Theorem 3.3. If Condition NS(𝜆) is satisfied, then 𝐿_𝜏, regarded as an operator from 𝐸_𝜇 to 𝐸^0𝜇, has the Fredholm property and its index is zero.

Proof. We put 𝐿⁰𝑢=𝐿_𝜏𝑢−𝜆𝑢, 𝐿¹𝑢=𝑢^−𝜆𝑢 and 𝐿^𝑠∶𝐸_𝜇=𝐶^𝜇2+𝛼(ℝ)→𝐸^0𝜇=𝐶^𝛼𝜇(ℝ), 𝐿^𝑠=(1−𝑠)𝐿⁰+𝑠𝐿¹, 𝑠∈[0,1]. Condition NS(𝜆) for 𝐿_𝜏 implies Condition NS for 𝐿⁰=𝐿_𝜏−𝜆𝐼. Then, Theorem 3.1 ensures that 𝐿⁰=𝐿_𝜏−𝜆𝐼, regarded from 𝐸_𝜇 to 𝐸^0𝜇, is normally solvable with a finite-dimensional kernel. For operator 𝐿¹, we have ker𝐿¹={0}, Im𝐿¹=𝐸^0𝜇, hence 𝐿¹ is a Fredholm operator and its index is ind𝐿¹=0.
By Lemma 3.2 applied for 𝐿^𝑠, there exists 𝜆≥0 large enough such that Condition NS holds for all 𝐿^𝑠, 𝑠∈[0,1]. In view of Theorem 3.1, it follows that the operators 𝐿^𝑠 are normally solvable with a finite-dimensional kernel. In other words, the homotopy 𝐿^𝑠 gives a continuous deformation from the operator 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 𝐿^𝑠 is zero. In particular, for 𝑠=0 and 𝜆=0, one arrives at the conclusion that 𝐿_𝜏 has the Fredholm property and its index is zero. This completes the proof.

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 spaces 𝐸₁, 𝐸₂, a class Φ of operators acting from 𝐸₁ to 𝐸₂ and a class of homotopies 𝐻=𝐴_𝜏(𝑢)∶𝐸₁×[0,1]⟶𝐸₂,suchthat𝐴_𝜏(𝑢)∈Φ,(∀)𝜏∈[0,1].(4.1)
Let 𝐷⊂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 𝐴_𝜏(𝑢)∈𝐻 and 𝐴_𝜏(𝑢)≠0, for 𝑢∈𝜕𝐷, 𝜏∈[0,1], then 𝛾𝐴₀,𝐷=𝛾𝐴₁,𝐷.(4.2)(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 𝛾(𝐴,𝐷)=𝛾𝐴,𝐷₁+𝛾𝐴,𝐷₂.(4.3)(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, 𝛾(𝐽,𝐷)=1.(4.4)
The integer 𝛾(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].

Let 𝐸₁ and 𝐸₂ be Banach spaces, 𝐸₁⊆𝐸₂ algebraically and topologically and let 𝐺⊂𝐸₁ be an open bounded set.

Denote by 𝐼∶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 operator 𝐿−𝜆𝐼∶𝐸₁→𝐸₂ is Fredholm for all 𝜆≥0,(b)for every operator 𝐿∈Φ, there is 𝜆₀=𝜆₀(𝐿) such that 𝐿−𝜆𝐼 has a uniformly bounded inverse for all 𝜆>𝜆₀.

Denote by ℱ the classℱ=𝐵∈𝐶¹𝐺,𝐸₂,𝐵proper,𝐵^(𝑥)∈Φ,(∀)𝑥∈𝐺,(4.5) where 𝐵^(𝑥) is the Fréchet derivative of the operator 𝐵.

Finally, one introduces the class ℋ of homotopies given byℋ=𝐵(𝑥,𝜏)∈𝐶¹𝐺×[0,1],𝐸₂,𝐵proper,𝐵(⋅,𝜏)∈ℱ,(∀)𝜏∈[0,1].(4.6) Here the properness of 𝐵 is understood in both variables 𝑥∈𝐺 and 𝜏∈[0,1].

Theorem 4.1 (see [18]). For every 𝐵∈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. Let 𝐸^1 and 𝐸^2 be two Banach spaces such that 𝐸_𝑖⊂𝐸^𝑖, 𝑖=1,2 where the inclusion is understood in the algebraic and topological sense. In the case of the Hölder space 𝐶^𝑘+𝛼(ℝ), this can be the space 𝐶^𝑘(ℝ) with an integer nonnegative 𝑘. We can also consider some integral spaces 𝑊^∞𝑘,𝑝(ℝ) [17]. Instead of (b) above we can impose the following condition [20]:(b^)for every operator 𝐿∶𝐸^1→𝐸^2, 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, let 𝐸₁=𝐸_𝜇 and 𝐸₂=𝐸^0𝜇 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 𝑢^+𝑐𝑢^+𝜕𝐹𝜕𝑢𝑤^±,𝑤^±𝑢+𝜕𝐹𝜕𝑈𝑤^±,𝑤^±𝐽(𝑢)−𝜆𝑢=0(4.7) do not have nonzero solutions in 𝐸, (∀)𝜆≥0.

Under these hypotheses, Theorem 2.5 assures that operator 𝐴 is proper. Moreover, its Fréchet derivative is 𝐴^=𝐿 from (1.7) and it is a Fredholm operator with the index zero (Theorem 3.3).

Consider ℱ the class of operators 𝐴 defined through (1.6), such that (𝐻4)-(𝐻5) are satisfied. Consider also the class ℋ of homotopies 𝐴_𝜏∶𝐸_𝜇→𝐸^0𝜇, 𝜏∈[0,1], of the form (2.6), satisfying (𝐻1)-(𝐻2) and(𝐻6)for every 𝜏∈[0,1], the equations 𝑢^+𝑐(𝜏)𝑢^+𝜕𝐹^𝜏𝜕𝑢𝑤^±,𝑤^±𝑢+𝜕𝐹^𝜏𝜕𝑈𝑤^±,𝑤^±𝐽(𝑢)−𝜆𝑢=0(4.8) do not have nonzero solutions in 𝐸, (∀)𝜆≥0. By Theorem 2.5 and Theorem 3.3, we infer that operators 𝐴_𝜏(𝑢) are Fréchet differentiable, proper with respect to (𝑢,𝜏) and their Fréchet derivatives 𝐴^𝜏=𝐿_𝜏 verify condition (a) above. Condition (b^) follows from the lemma in the appendix. Hence ℋ has the form (4.6). 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 functions 𝐹_𝜏 and 𝑐(𝜏) satisfy conditions (𝐻1)-(𝐻2) and (𝐻4)−(𝐻6). Then a topological degree exists for the class ℱ of operators and the class ℋ of homotopies.

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. Denote𝐹₀(𝑤)=𝐹(𝑤,𝑤).(5.1) We obtain this function from 𝐹(𝑤,𝐽(𝑤)) if we formally replace the kernel 𝜙(𝑥) of the integral by the 𝛿-function. The corresponding reaction-diffusion equation𝜕𝑢𝜕𝑡=𝜕²𝑢𝜕𝑥²+𝐹₀(𝑢)(5.2) is called bistable if 𝐹^0(𝑤^±)<0, monostable if one of these derivatives is positive and another one negative and, finally, unstable if 𝐹^0(𝑤^±)>0. As it is well-known, it can have travelling wave solutions, that is solutions, which satisfy the problem𝑤^+𝑐𝑤^+𝐹₀(𝑤)=0,𝑤(±∞)=𝑤^±.(5.3) Let 𝑤₀(𝑥) be a solution of (5.3) with some 𝑐=𝑐₀. The operator 𝐿₀ linearized about this solution,𝐿₀𝑢=𝑢^+𝑐₀𝑢^+𝐹^0𝑤₀𝑢(5.4) has the essential spectrum given by two parabolas:𝜆^0±(𝜉)=−𝜉²+𝑐₀𝑖𝜉+𝐹^0𝑤^±,𝜉∈ℝ.(5.5) Therefore the operator 𝐿₀ satisfies the Fredholm property if and only if 𝐹^0(𝑤^±)≠0. If this condition is satisfied, then the index of the operator is well defined. In the bistable case it equals 0, in the monostable case 1, in the unstable case 0 [9].

In the case of the integro-differential operator𝐿𝑢=𝑢^+𝑐₀𝑢^+𝐹^𝑢(𝑤,𝐽(𝑤))𝑢+𝐹^𝑈(𝑤,𝐽(𝑤))𝐽(𝑢),(5.6) the essential spectrum is given by the curves𝜆_±(𝜉)=−𝜉²+𝑐₀𝑖𝜉+𝐹^𝑢𝑤^±,𝑤^±+𝐹^𝑈𝑤^±,𝑤^±𝜙(𝜉),𝜉∈ℝ,(5.7) where 𝜙(𝜉) is the Fourier transform of the function 𝜙(𝑥). If we replace 𝐽(𝑢) by 𝑢, that is, 𝜙(𝑥) by the 𝛿 -function, then the spectrum of the integro-differential operator coincides with the spectrum of the reaction-diffusion operator.

We note thatRe𝐹𝜙^𝑢(𝜉)𝑤=^±,𝑤^±_∞−∞+𝐹^𝑈𝑤^±,𝑤^±𝜙(0)=𝐹^0𝑤^±,𝜙(𝑥)cos(𝜉𝑥)𝑑𝑥<^∞−∞𝜙(𝑥)𝑑𝑥=1.(5.8)

5.1. Fredholm Property

Bistable Case
Let 𝐹^0𝑤^±=𝐹^𝑢𝑤^±,𝑤^±+𝐹^𝑈𝑤^±,𝑤^±<0(5.9) (we recall that 𝜙(0)=1). Suppose that 𝐹^𝑢(𝑤₊,𝑤₊)<0 and 𝐹^𝑈(𝑤₊,𝑤₊)>0. Then Re𝜆₊(𝜉)<0 for all 𝜉∈ℝ since 𝐹^𝑢𝑤^±,𝑤^±+Re𝐹^𝑈𝑤^±,𝑤^±𝜙(𝜉)≤𝐹^𝑢𝑤^±,𝑤^±+𝐹^𝑈𝑤^±,𝑤^±=𝐹^0𝑤^±.(5.10) Hence the essential spectrum is completely in the left-half plane. This allows us to prove properness of the corresponding operators in weighted spaces and to define the topological degree.

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

Figure 1 

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

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

Monostable Case
Suppose that 𝐹^0(𝑤₊)>0 and 𝐹^0(𝑤₋)<0. Then 𝜆⁰⁻(𝜉) is in the left-half plane for all 𝜉∈ℝ; 𝜆⁰⁺(𝜉) is partially in the right-half plane, 𝜆⁰⁺(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)=𝜆⁰⁺(0)>0. The curve 𝜆₋(𝜉) can be completely in the left-half plane or partially in the right-half plane (Figure 1). Similar to the bistable case, there are two subcases in the monostable case.

Index
In order to find the index of the operator 𝐿, we consider the operator 𝐿_𝜏 which depends on the parameter 𝜏 characterizing the width of the support of the function 𝜙_𝜏, supp 𝜙_𝜏=[−𝑁_𝜏,𝑁_𝜏]. We recall that ∫^∞−∞𝜙_𝜏(𝑥)𝑑𝑥=1. Let 𝐿₁=𝐿, that is the value 𝜏=1 corresponds to the function 𝜙 in the operator 𝐿.

Since the essential spectrum of the operator 𝐿_𝜏 can be determined explicitly, then we can affirm that it converges to the essential spectrum of the operator 𝐿₀ as 𝜏→0. Moreover, 𝐿_𝜏 converges to 𝐿₀ in the operator norm. The essential spectrum of the operator 𝐿_𝜏 does not cross the origin. Therefore it is normally solvable with a finite-dimensional kernel. Hence the index of the operator 𝐿 equals the index of the operator 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 where𝐹(𝑢,𝐽(𝑢))=𝐽(𝑢)𝑢(1−𝑢)−𝛼𝑢.(5.11) 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.

Appendix

Sectorial property of an operator implies certain location of its essential spectrum and an estimate of the resolvent. For general elliptic problems in unbounded domains it is proved in [20]. A simple particular case of second-order operators on the axis is considered in [21]. In the lemma below we prove an estimate of the resolvent using this last result.

Lemma A.1. Let 𝑀₀∶𝐶²(𝑅)→𝐶(𝑅), 𝑀₀𝑢=𝑢^+𝑏(𝑥)𝑢^(𝑥)+𝑐(𝑥)𝑢+𝑑(𝑥)𝐽(𝑢),(A.1) where the coefficients of this operator are sufficiently smooth bounded functions. Then the operator 𝑀_𝜆𝑢=𝑀₀𝑢−𝜆𝑢, considered as acting in the same spaces, has a bounded inverse with the norm independent of 𝜆 for 𝜆≥𝜆₀>0, where 𝜆₀ is sufficiently large.

Proof. Consider the equation 𝑀_𝜆𝑢=𝑓.(A.2) We need to obtain the estimate ‖𝑢‖_𝐶²_(ℝ)≤𝐾‖𝑓‖_𝐶(ℝ)(A.3) of this equation where 𝐾 is independent of 𝜆 for all 𝜆 sufficiently large. Here and below we denote by 𝐾 the constants independent of 𝑢, 𝑓, and 𝜆.
We first prove the estimate‖𝑢‖_𝐶(ℝ)≤𝐾‖𝑓‖^𝐶(ℝ)𝜆.(A.4) Since the operator 𝑀𝑢=𝑢^+𝑏(𝑥)𝑢^(𝑥)+𝑐(𝑥)𝑢(A.5) is sectorial [21], then ‖𝑢‖_𝐶(ℝ)≤𝐾‖‖𝑀𝑢−𝜆𝑢‖‖^𝐶(ℝ)𝜆=𝐾‖𝑓−𝑑(𝑥)𝜆𝐽(𝑢)‖^𝐶(ℝ)≤𝐾𝜆‖𝑓‖^𝐶(ℝ)+‖𝑢‖_𝐶(ℝ).(A.6) Estimate (A.4) follows from the last one for 𝜆 sufficiently large.
We can write (A.2) in the form𝑀₀𝑢−𝜎𝑢=𝑓+𝜆𝑢−𝜎𝑢.(A.7) We can choose 𝜎>0 such that the operator in the left-hand side is invertible. Hence ‖𝑢‖_𝐶²_(ℝ)≤𝐾‖𝑓‖_𝐶(ℝ)+𝜆‖𝑢‖_𝐶(ℝ)+𝜎‖𝑢‖_𝐶(ℝ).(A.8) This estimate and (A.4) give (A.3). The lemma is proved.

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

Acknowledgments

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