The Exponential Dichotomy under Discretization on General Approximation Scheme

Abstract

This paper is devoted to the numerical analysis of abstract parabolic problem 𝑢^(𝑡)=𝐴𝑢(𝑡); 𝑢(0)=𝑢⁰, with hyperbolic generator 𝐴. We are developing a general approach to establish a discrete dichotomy in a very general setting in case of discrete approximation in space and time. It is a well-known fact that the phase space in the neighborhood of the hyperbolic equilibrium can be split in a such way that the original initial value problem is reduced to initial value problems with exponential decaying solutions in opposite time direction. We use the theory of compact approximation principle and collectively condensing approximation to show that such a decomposition of the flow persists under rather general approximation schemes. The main assumption of our results is naturally satisfied, in particular, for operators with compact resolvents and condensing semigroups and can be verified for finite element as well as finite difference methods.

1. Introduction

Many problems like approximation of attractors, traveling waves, shadowing e.c. involve the notion of dichotomy. In numerical analysis of such problems, it is very important to know if they keep some kind of exponential estimates uniformly in discretization parameter.

Let 𝐵(𝐸) denote the Banach algebra of all bounded linear operators on a complex Banach space 𝐸. The set of all linear-closed densely defined operators in 𝐸 will be denoted by 𝒞(𝐸). For 𝐵∈𝒞(𝐸), let 𝜎(𝐵) be its spectrum and 𝜌(𝐵), its resolvent set. In the following let 𝐴∶𝐷(𝐴)⊆𝐸→𝐸 be a closed linear operator, such that‖‖(𝜆𝐼−𝐴)⁻¹‖‖_𝐵(𝐸)≤𝑀1+||𝜆||foranyRe𝜆≥0.(1.1) Under condition (1.1), the spectrum of 𝐴 is on the left: sup{Re𝜆∶𝜆∈𝜎(𝐴)}<0, so the fractional power operators (−𝐴)^𝛼,𝛼∈ℝ⁺, (see [1, 2]) associated to 𝐴 and 𝐸^𝛼, can be constructed and the corresponding fractional power spaces too, that is, 𝐸^𝛼∶=𝐷((−𝐴)^𝛼) endowed with the graph norm ‖𝑥‖_𝐸^𝛼=‖(−𝐴)^𝛼𝑥‖_𝐸. Define the ball 𝒰_𝐸^𝛼(0;𝜌) with the center at 0 of radius 𝜌>0 in 𝐸^𝛼 space.

To show how the dichotomy problems appear in numerical analysis, we are considering for example the semilinear equation in Banach space 𝐸^𝛼𝑢𝑢^′((0𝑡))==𝐴𝑢₀𝑢(𝑡)+𝑓(𝑢(𝑡)),𝑡≥0,∈𝐸^𝛼,(1.2) where 𝑓(⋅)∶𝐸^𝛼⊆𝐸→𝐸,0≤𝛼<1, is assumed to be continuous, bounded, and continuously Fréchet differentiable function. More precisely, we assume that the following condition holds.(F1) For any 𝜖>0, there is 𝛿>0 such that ‖𝑓′(𝑤)−𝑓′(𝑧)‖_𝐵(𝐸^𝛼_,𝐸)≤𝜖 as ‖𝑤−𝑧‖_𝐸^𝛼≤𝛿 for all 𝑤,𝑧∈𝒰_𝐸^𝛼(𝑢^∗;𝜌), where 𝑢^∗ is a hyperbolic equilibrium point of (1.2).

By means of the change of variables 𝑣(⋅)=𝑢(⋅)−𝑢^∗ in the problem (1.2), where 𝑢^∗ is the hyperbolic equilibrium, we obtain the problem𝑣𝑣^((0𝑡))==𝑢𝐴₀+𝑓^𝑢^∗𝑣(𝑡)+𝑓𝑣(𝑡)+𝑢^∗−𝑓𝑢^∗−𝑓^𝑢^∗𝑣(𝑡),−𝑢^∗=𝑣⁰.(1.3) Such problem can be written in the form𝑣^(𝑡)=𝐴_𝑢∗𝑣(𝑡)+𝐹_𝑢∗(𝑣(𝑡)),𝑣(0)=𝑣⁰,𝑡≥0,(1.4) where 𝐴_𝑢∗=𝐴+𝑓^(𝑢^∗), 𝐹_𝑢∗(𝑣(𝑡))=𝑓(𝑣(𝑡)+𝑢^∗)−𝑓(𝑢^∗)−𝑓^(𝑢^∗)𝑣(𝑡). We note that from condition (F1) it follows that the function 𝐹_𝑢∗(𝑣(𝑡))=𝑓(𝑣(𝑡)+𝑢^∗)−𝑓(𝑢^∗)−𝑓^(𝑢^∗)𝑣(𝑡) for small ‖𝑣⁰‖_𝐸^𝛼 is of order 𝑜(‖𝑣(𝑡)‖_𝐸^𝛼). Since 𝑓′(𝑢^∗)∈𝐵(𝐸^𝛼,𝐸),0≤𝛼<1, the operator 𝐴_𝑢∗=𝐴+𝑓′(𝑢^∗) is the generator of an analytic 𝐶₀-semigroup [3]. It can happen that the spectrum of operator 𝐴_𝑢∗ can be split into two parts 𝜎⁺ and 𝜎⁻.

We assume that the part 𝜎⁺ of the spectrum of operator 𝐴+𝑓′(𝑢^∗), which is located strictly to the right of the imaginary axis, consists of a finite number of eigenvalues with finite multiplicity. This assumption is satisfied, for instance, if the resolvent of operator 𝐴 is compact. The conditions under which the operator 𝐴_𝑢∗ has the dichotomy property were studied say in [4–6]. In case of hyperbolic equilibrium point 𝑢^∗, there is no spectrum of 𝐴_𝑢∗ on 𝑖ℝ. Let 𝑈(𝜎⁺)⊂{𝜆∈𝐂∶Re𝜆>0} be an open connected neighborhood of 𝜎⁺ which has a closed rectifiable curve 𝜕𝑈(𝜎⁺) as a boundary. We decompose 𝐸^𝛼 using the Riesz projection𝑃𝜎⁺∶=𝑃𝜎⁺,𝐴_𝑢∗∶=12𝜋𝑖_{𝜕𝑈(𝜎+)}𝜁𝐼−𝐴_𝑢^∗⁻¹𝑑𝜁(1.5) defined by 𝜎⁺. Due to this definition and analyticity of the 𝐶₀-semigroup 𝑒^{𝑡𝐴𝑢∗},𝑡∈ℝ₊, we have positive constants 𝑀₁,𝛽>0, such that‖‖‖‖𝑒𝑒^{𝑡𝐴𝑢∗}𝑧‖‖_𝐸𝛼≤𝑀₁𝑒^−𝛽𝑡‖𝑧‖_𝐸^𝛼,𝑡≥0,_{𝑡𝐴𝑢∗}𝑣‖‖_𝐸𝛼≤𝑀₁𝑒^𝛽𝑡‖𝑣‖_𝐸^𝛼,𝑡≤0,(1.6) for all 𝑣∈𝑃(𝜎⁺)𝐸^𝛼 and 𝑧∈(𝐼−𝑃(𝜎⁺))𝐸^𝛼. Since 𝐹_𝑢∗(𝑣(𝑡))=𝑜(‖𝑣(𝑡)‖_𝐸^𝛼) for small 𝑣(⋅), the estimates (1.6) are crucial to describe the behavior of solution of the problem (1.2) at the vicinity of the hyperbolic equilibrium point 𝑢^∗.

If 𝑣⁰ is close to 0, that is, say 𝑣⁰∈𝒰_𝐸^𝛼(0;𝜌) with small 𝜌>0, then the mild solution 𝑣(𝑡;𝑣⁰) of (1.4) can stay in the ball 𝒰_𝐸^𝛼(0;𝜌), for some time. We denote now the maximal time of staying 𝑣(𝑡;𝑣⁰) in 𝒰_𝐸^𝛼(0;𝜌) by 𝑇=𝑇(𝑣⁰)=sup{𝑡≥0∶‖𝑣(𝑡;𝑣⁰)‖_𝐸^𝛼≤𝜌or𝑣(𝑡;𝑣⁰)∈𝒰_𝐸^𝛼(0;𝜌)}. Now coming back to solution of (1.4) for any two 𝑣⁰,𝑣^𝑇∈𝒰_𝐸^𝛼(0;𝜌) we consider the boundary value problem𝐼−𝑃𝜎₊𝑣^′(𝑡)=𝐴_𝑢∗𝑣(𝑡)+𝐹_𝑢∗(𝑣(𝑡)),0≤𝑡≤𝑇,𝑣(0)=𝐼−𝑃𝜎⁺𝑣⁰,𝑃𝜎⁺𝑣(𝑇)=𝑃𝜎⁺𝑣^𝑇.(1.7) A mild solution of problem (1.7) as was shown in [7] satisfies the integral equation𝑣(𝑡)=+𝑒^{(𝑡−𝑇)𝐴𝑢∗}𝑃𝜎⁺𝑣^𝑇+𝑒^{𝑡𝐴𝑢∗}𝐼−𝑃𝜎⁺𝑣⁰+^𝑡0𝑒^{(𝑡−𝑠)𝐴𝑢∗}𝐼−𝑃𝜎⁺𝐹_𝑢^∗(𝑣(𝑠))𝑑𝑠_𝑇𝑡𝑒^{(𝑡−𝑠)𝐴𝑢∗}𝑃𝜎⁺𝐹_𝑢^∗(𝑣(𝑠))𝑑𝑠,0≤𝑡≤𝑇.(1.8)

In case one would like to discretize the problem (1.4) in space and time variables it is very important to know what will happen to estimates like (1.6) for approximation solutions. If the estimates like (1.6) hold uniformly in parameter of discretization, then one can expect to get a similar behavior of approximated solutions of (1.8).

So, in this paper we are going to consider general approximation approach for keeping the dichotomy estimates (1.6) for approximations of trajectory 𝑢(⋅).

2. Preliminaries

Let 𝕋(𝑟)={𝜆∶𝜆∈𝐂,|𝜆|=𝐫},𝕋=𝕋(1).

Definition 2.1. A 𝐶₀-semigroup 𝑒^𝑡𝐴,𝑡≥0, defined on a Banach space 𝐸 is called hyperbolic if 𝜎(𝑒^𝑡𝐴)∩𝕋=∅ for all 𝑡>0. The generator 𝐴 is called hyperbolic if 𝜎(𝐴)∩𝑖ℝ=∅.

Let us denote by Υ(ℝ;𝐸) any of the spaces 𝐿^𝑝(ℝ;𝐸),1≤𝑝<∞,𝐶₀(ℝ;𝐸) or Stepanov’s space 𝑆^𝑝(ℝ;𝐸),1≤𝑝<∞. We consider in the Banach space Υ(ℝ;𝐸) (we call this space as Palmer’s space, see [8], where Fredholm property was mentioned for the first time) the linear differential operatorℒ=−𝑑𝑑𝑡+𝐴∶𝐷(ℒ)⊆Υ(ℝ;𝐸)⟶Υ(ℝ;𝐸),(2.1) where 𝐴 generates 𝐶₀-semigroup, domain of ℒ is assumed to consist of the functions 𝑢(⋅)∈Υ(ℝ;𝐸) such that for some function 𝑔(⋅)∈Υ(ℝ;𝐸) one has 𝑢(𝑡)=𝑒^{(𝑡−𝑠)𝐴}𝑢(𝑠)−∫^𝑡𝑠𝑒^{(𝑡−𝜂)𝐴}𝑔(𝜂)𝑑𝜂,𝑠≤𝑡,𝑡∈ℝ, and ℒ𝑢(⋅)=𝑔(⋅). Let us note [9] that the operator ℒ is the generator of 𝐶₀-semigroup 𝑒^𝑡ℒ on a Banach space Υ(ℝ;𝐸), which is defined for all 𝑣(⋅)∈Υ(ℝ;𝐸) by formula𝑒^𝑡ℒ𝑣(𝑠)=𝑒^𝑡𝐴𝑣(𝑠−𝑡)forany𝑠∈ℝ,𝑡≥0.(2.2)

Definition 2.2. A 𝐶₀-semigroup 𝑒^𝑡𝐴,𝑡≥0, has an exponential dichotomy on ℝ with exponential dichotomy data (𝑀≥1,𝛽>0) if there exists projector 𝑃∶𝐸→𝐸 such that(i)𝑒^𝑡𝐴𝑃=𝑃𝑒^𝑡𝐴 for all 𝑡≥0,(ii)the restriction 𝑒^𝑡𝐴|_ℛ(𝑃),𝑡≥0, is invertible on 𝑃(𝐸) and‖‖𝑒‖‖𝑒^−𝑡𝐴𝑃𝑥‖‖≤𝑀𝑒^−𝛽𝑡‖𝑃𝑥‖,𝑡≥0,𝑥∈𝐸,_𝑡𝐴(𝐼−𝑃)𝑥‖‖≤𝑀𝑒^−𝛽𝑡‖(𝐼−𝑃)𝑥‖,𝑡≥0,𝑥∈𝐸.(2.3)

Theorem 2.3 (see [10]). The operator ℒ in the Banach space Υ(ℝ;𝐸) is invertible if and only if the condition 𝜎𝑒^1𝐴∩𝕋=∅(2.4) holds. If condition (2.4) is satisfied, then ℒ⁻¹𝑓(𝑡)=^∞−∞𝐺(𝑡−𝑠)𝑓(𝑠)𝑑𝑠,𝑡∈ℝ,𝑓(⋅)∈Υ(ℝ;𝐸),(2.5) where the Green’s function 𝐺(𝜂)=−𝑒^𝜂𝐴𝑒𝑃₋,𝜂≥0,_𝜂𝐴‖𝐺(𝜂)‖≤𝑀𝑃₊₊,𝜂<0,(2.6)𝑀𝑒^−𝛾+𝜂,𝜂≥0,₋𝑒^𝛾−𝜂,𝜂<0,(2.7) where 𝑀₊=2𝑀(ℒ)1+12(ℒ)²,𝑀₋=2𝑀(ℒ)1−12(ℒ)²𝛾₊,=ln1+12(ℒ),𝛾⁻=−ln1−12(ℒ),(2.8) and æ (ℒ)=1+𝐶(Υ)(𝑀+𝑀²‖ℒ⁻¹‖).

We note that the constant 𝐶(Υ) is defined as 𝐶(Υ)=1 if Υ(ℝ;𝐸)=𝐿^∞(ℝ,𝐸) or Υ(ℝ;𝐸)=𝐶₀(ℝ,𝐸) and 𝐶(Υ)=2^1−1/𝑝 if Υ(ℝ;𝐸)=𝐿^𝑝(ℝ,𝐸) or Υ(ℝ;𝐸)=𝑆^𝑝(ℝ,𝐸),𝑝∈[1,∞).

Denote by Υ(ℤ;𝐸) the Banach space of 𝐸-valued sequences with corresponding discrete norm which is consistent with the norm of Υ(ℝ;𝐸). For any 𝑢(⋅)∈Υ(ℤ;𝐸), that is, {𝑢(𝑘)}_𝑘∈ℤ, and 𝐵=𝑒^1𝐴∈𝐵(𝐸), we define an operator ℬ∶𝐷(ℬ)⊆𝑙^𝑝(ℤ;𝐸)→𝑙^𝑝(ℤ;𝐸) by formula(ℬ𝑢)(𝑘)=𝐵𝑢(𝑘−1),𝑘∈ℤ,𝑢(⋅)∈𝑙^𝑝(ℤ;𝐸).(2.9) Now we define an operator 𝒟=𝐼−ℬ∶𝐷(𝒟)=𝐷(ℬ)⊆𝑙^p(ℤ;𝐸)→𝑙^𝑝(ℤ;𝐸) as(𝒟𝑢)(𝑘)=𝑢(𝑘)−𝐵𝑢(𝑘−1),𝑢(⋅)∈𝐷(ℬ),𝑘∈ℤ.(2.10)

Proposition 2.4 (see [10]). Let an operator ℒ=−𝑑/𝑑𝑡+𝐴∶𝐷(ℒ)⊆Υ(ℝ;𝐸)→Υ(ℝ;𝐸) be invertible. Then, an operator 𝒟∶𝐷(𝒟)⊆Υ(ℤ;𝐸)→Υ(ℤ;𝐸) is invertible too and ‖‖𝒟⁻¹‖‖≤1+𝐶(Υ)𝑀+𝑀²‖‖ℒ⁻¹‖‖.(2.11) Conversely, if 𝒟 is invertible, then ℒ∶𝐷(ℒ)⊆Υ(ℝ;𝐸)→Υ(ℝ;𝐸) is invertible and ‖‖ℒ⁻¹‖‖≤𝐶(Υ)𝑀+𝑀²‖‖𝒟⁻¹‖‖.(2.12)

Theorem 2.5 (see [11]). The difference operator (𝒟𝑢)(𝑘)=u(𝑘)−𝐵𝑢(𝑘−1),𝑘∈ℕ,𝑢(⋅)∈𝑙^𝑝(ℝ;𝐸),1≤𝑝≤∞,(2.13) is invertible if and only if 𝜎(𝐵)∩𝕋=∅.(2.14) If condition (2.14) is satisfied, then the inverse operator has the form 𝒟⁻¹𝑣(𝑘)=Σ_𝑚∈ℤΓ(𝑘−𝑚)𝑣(𝑚),(2.15) where 𝑣(⋅)∈𝑙^𝑝(ℤ;𝐸),1≤𝑝≤∞, and the function Γ(⋅)∶ℤ→𝐵(𝐸) is defined by Γ(𝑘)=𝐵^𝑘−𝐵(𝐼_0−𝑘−𝑃),𝑘≥0,𝑃,𝑘≤−1,(2.16) where 𝐵₀ is the restriction of 𝐵 on ℛ(𝑃).

Definition 2.6. The operator 𝐵∈𝐵(𝐸) has an exponential discrete dichotomy with data (𝑀,𝑟,𝑃) if 𝑃∈𝐵(𝐸) is a projector in 𝐸 and 𝑀,𝑟 are constants with 0≤𝑟<1 such that the following properties hold(i)𝐵^𝑘𝑃=𝑃𝐵^𝑘 for all 𝑘∈ℕ,(ii)‖𝐵^𝑘(𝐼−𝑃)‖≤𝑀𝑟^𝑘 for all 𝑘∈ℕ,(iii)𝐵∶=𝐵|_ℛ(𝑃)∶ℛ(𝑃)↦ℛ(𝑃) is a homeomorphism that satisfies ‖𝐵^−𝑘𝑃‖≤𝑀𝑟^𝑘,𝑘∈ℕ.

The following result relates the constants in an exponential discrete dichotomy to a bound on the resolvent (𝜆𝐼−𝐵)⁻¹ for 𝜆∈𝕋.

Theorem 2.7. For 𝐵∈𝐵(𝐸), the following conditions are equivalent:(i)𝜆∈𝜌(𝐵) for all 𝜆∈𝕋 and ‖(𝜆𝐼−𝐵)⁻¹‖≤𝛽<∞∀𝜆∈𝕋;(ii)𝐵 has an exponential dichotomy with data (𝑀,𝑟,𝑃). More precisely, one shows that (i) implies (ii) with 𝑀=2𝛽²/(𝛽−1) and 𝑟=1−1/2𝛽. Conversely, (ii) implies (i) with 𝛽=𝑀((1+𝑟)/(1−𝑟)).

Proof. First assume (i) and without loss of generality let 𝛽>1. For 𝑧∈𝐂,𝐳≠𝟎, we have ||||𝑧−𝑧|𝑧|||||=||1−|𝑧|||.(2.17) Hence, if |1−|𝑧||𝛽<1, the classical perturbation estimate shows that 𝑧∈𝜌(𝐵) and ‖‖(𝑧𝐼−𝐵)⁻¹‖‖≤𝛽1−𝛽||1−|𝑧|||.(2.18) We define 𝑃 as the Riesz projector defined by the formula 𝐼−𝑃=12𝜋𝑖_|𝑧|=1(𝑧𝐼−𝐵)⁻¹𝑑𝑧.(2.19) Since 𝐵 commutes with the resolvent, condition (i) of Definition 2.6 holds. Further, using (2.18) and Cauchy’s theorem, we can shift the contour 𝐼−𝑃=12𝜋𝑖_|𝑧|=𝑟(𝑧𝐼−𝐵)⁻¹𝑑𝑧,if||1−𝑟||<1𝛽.(2.20) Now we claim 𝐵^𝑘(𝐼−𝑃)=12𝜋𝑖_|𝑧|=𝑟𝑧^𝑘(𝑧𝐼−𝐵)⁻¹𝑑𝑧if||1−𝑟||<1𝛽.(2.21) For 𝑘=0 this follows from (2.20). If (2.21) holds for some 𝑘, then we obtain 𝐵^𝑘+1𝑃=12𝜋𝑖_|𝑧|=𝑟=1(𝐵−𝑧𝐼+𝑧𝐼)𝑧^𝑘(𝑧𝐼−𝐵)⁻¹𝑑𝑧2𝜋𝑖_|𝑧|=𝑟𝑧^𝑘+1(𝑧𝐼−𝐵)⁻¹𝑑𝑧−12𝜋𝑖_|𝑧|=𝑟𝑧^𝑘𝑑𝑧,(2.22) and thus the assertion holds for 𝑘+1. Equations (2.21) and (2.18) immediately lead to the first dichotomy estimate for 1−1/𝛽<𝑟≤1‖‖𝐵^𝑘𝑃‖‖≤12𝜋2𝜋𝑟𝑟^𝑘𝛽1−𝛽(1−𝑟)=1−𝛽𝛽𝑟(1^𝑘+1−𝑟)for𝑘≥0.(2.23) For the second dichotomy estimate, we use the resolvent equation (𝑧𝐼−𝐵)⁻¹=1z𝐼+𝑧⁻¹𝐵(𝑧𝐼−𝐵)⁻¹.(2.24) For |1−𝑟|<1/𝛽, (2.20) and (2.24) lead to 𝐼−𝑃=1=−2𝜋𝑖1_|𝑧|=11𝑧𝐼−(𝑧𝐼−𝐵)⁻¹𝑑𝑧2𝜋𝑖_|𝑧|=𝑟1𝑧𝐵(𝑧𝐼−𝐵)⁻¹𝑑𝑧.(2.25) This shows that the following equality holds for 𝑘=1𝐼−𝑃=−𝐵^𝑘12𝜋𝑖_|𝑧|=𝑟𝑧^−𝑘(𝑧𝐼−𝐵)⁻¹𝑑𝑧,for𝑘≥1.(2.26) If (2.26) is known for some 𝑘, then use (2.24) and find 𝐼−𝑃=−𝐵^𝑘12𝜋𝑖_|𝑧|=𝑟𝑧^−(𝑘+1)=−𝐵_𝑘+1+𝑧^−(𝑘+1)𝐵(𝑧𝐼−𝐵)⁻¹𝑑𝑧12𝜋𝑖_|𝑧|=𝑟𝑧^−(𝑘+1)(𝑧𝐼−𝐵)⁻¹𝑑𝑧.(2.27) We apply (2.26) to 𝑥∈𝐸, using that 𝐼−𝑃 commutes with 𝐵 as well as the estimate (2.18) ‖(𝐼−𝑃)𝑥‖=‖‖‖12𝜋𝑖_|𝑧|=𝑟≤𝑟𝑧^−𝑘(𝑧𝐼−𝐵)⁻¹𝑑𝑧𝐵^𝑘(𝐼−𝑃)𝑥_−𝑘‖‖‖2𝜋2𝜋𝑟1−𝛽𝛽(𝑟−1)‖‖𝐵^𝑘(𝐼−𝑃)𝑥‖‖.(2.28) Summarizing, we have shown for all 𝑘≥1,1≤𝑟<1+1/𝛽,𝑢∈𝐸‖(𝐼−𝑃)𝑢‖≤𝛽𝑟^−𝑘+11−𝛽(𝑟−1)‖‖𝐵^𝑘(𝐼−𝑃)𝑢‖‖.(2.29) For 𝑘=1, this estimate shows that 𝐵=𝐵_|𝒩(𝑃)∶𝒩(𝑃)↦𝒩(𝑃) is one-to-one with a bound for the inverse. To show that 𝐵 is onto, we take 𝑓∈𝒩(𝑃) and set 𝑣=−12𝜋𝑖_|𝑧|=1𝑧⁻¹(𝑧𝐼−𝐵)⁻¹𝑓𝑑𝑧.(2.30) From this equation, we have (𝐼−𝑃)𝑣=0, and using (2.24), we find 𝐵𝑣=12𝜋𝑖_|𝑧|=1𝑧⁻¹𝐼−(𝑧𝐼−𝐵)⁻¹𝑓𝑑𝑧=(𝐼−𝑃)𝑓=𝑓.(2.31) Therefore, 𝐵 is a linear homeomorphism on 𝒩(𝑃) satisfying ‖‖𝐵^−𝑘(𝐼−𝑃)𝑢‖‖≤𝛽𝑟^−𝑘+11−𝛽(𝑟−1)‖(𝐼−𝑃)𝑢‖for1≤𝑟<1+𝛽1.(2.32) This proves exponential dichotomy.
To arrive at the specific constants, we choose 𝑟=1−1/2𝛽 in (2.23) and obtain the bound (2𝛽−1)𝑟^𝑘. In order to have the same rate in the opposite direction, we apply (2.32), with 𝑟=(1−1/2𝛽)⁻¹<1+1/𝛽. In (2.32) we then find the upper bound 𝑀𝑟^−𝑘 with the constant 𝑀=2𝛽²/(𝛽−1). Since 𝑀>2𝛽−1, our assertion follows.
Now we assume exponential dichotomy and prove condition (i). For |𝜆|=1 the equation (𝜆𝐼−𝐵)𝑢=𝑓 is equivalent to the system𝜆𝐼−(𝐵𝜆𝐼−(𝐼𝐵−𝑃𝑃)𝑃)𝑢𝑢==(𝑃𝐼𝑓−,𝑃)𝑓,(2.33) which we can rewrite as 𝐼−𝜆𝐵𝐼₋₁−𝜆⁻¹𝐵𝑃𝑃𝑢=𝜆⁻¹𝑃𝑓,(𝐼−𝑃)𝑢=−𝐵⁻¹(𝐼−𝑃)𝑓.(2.34) Both equations have a unique solution given by a geometric series 𝑃𝑢=^∞_𝑘=0(𝐼−𝑃)𝑢=−_∞𝜆^−(𝑘+1)𝐵^𝑘𝑃𝑓,_𝑘=0𝐵^−(𝑘+1)𝜆^𝑘(𝐼−𝑃)𝑓.(2.35) The exponential dichotomy then implies the estimates ‖𝑃𝑢‖≤𝑀‖(𝐼−𝑃)𝑢‖≤1−𝑀𝑟‖𝑟𝑓‖,1−𝑟‖𝑓‖.(2.36) By the triangle inequality, we obtain condition (i) with 𝛽=𝑀((1+𝑟)/(1−𝑟)).

3. Discretization of Operators and Semigroups

In the papers [12–15], a general framework was developed that allows to analyze convergence properties of numerical discretizations in a unifying way. This approach is able to cover such seemingly different methods as (conforming and nonconforming) finite elements, finite differences, or collocation methods. It is the purpose of this paper to show that it is also possible to handle dichotomy properties with discretization in space and time on general approximation scheme. Moreover, we also consider the case when resolvent of operator 𝐴 is not necessarily compact.

3.1. General Approximation Scheme

Let 𝐸_𝑛 and 𝐸 be Banach spaces and {𝑝_𝑛} a sequence of linear bounded operators 𝑝_𝑛∶𝐸→𝐸_𝑛,𝑝_𝑛∈𝐵(𝐸,𝐸_𝑛),𝑛∈ℕ={1,2,…}, with the property:‖‖𝑝_𝑛𝑥‖‖_𝐸𝑛⟶‖𝑥‖_𝐸as𝑛→∞forany𝑥∈𝐸.(3.1)

Definition 3.1. The sequence of elements {𝑥_𝑛},𝑥_𝑛∈𝐸_𝑛,𝑛∈ℕ, is said to be 𝒫-convergent to 𝑥∈𝐸 if and only if ‖𝑥_𝑛−𝑝_𝑛𝑥‖_𝐸𝑛→0 as 𝑛→∞, and we write this 𝑥−−_𝑛𝒫→𝑥.

Definition 3.2. The sequence of bounded linear operators 𝐵_𝑛∈𝐵(𝐸_𝑛), 𝑛∈ℕ, is said to be 𝒫𝒫-convergent to the bounded linear operator 𝐵∈𝐵(𝐸) if for every 𝑥∈𝐸 and for every sequence {𝑥_𝑛},𝑥_𝑛∈𝐸_𝑛,𝑛∈ℕ, such that 𝑥−−_𝑛𝒫→𝑥 one has 𝐵_𝑛𝑥−−_𝑛𝒫→𝐵𝑥. We write then 𝐵−−−_𝑛𝒫𝒫−→𝐵.

In the case of unbounded operators as it occurs for general infinitesimal generators of PDE’s, the notion of compatibility turns out to be useful.

Definition 3.3. The sequence of closed linear operators {𝐴_𝑛},𝐴_𝑛∈𝒞(𝐸_𝑛),𝑛∈ℕ, is called compatible with a closed linear operator 𝐴∈𝒞(𝐸) if and only if for each 𝑥∈𝐷(𝐴) there is a sequence {𝑥_𝑛},𝑥_𝑛∈𝐷(𝐴_𝑛)⊆𝐸_𝑛,𝑛∈ℕ, such that 𝑥−−_𝑛𝒫→𝑥 and 𝐴_𝑛𝑥−−_𝑛𝒫→𝐴𝑥. We write that (𝐴_𝑛,𝐴) are compatible.

For analytic 𝐶₀-semigroups, the following ABC Theorem holds.

Theorem 3.4 (see [16]). Let operators 𝐴 and 𝐴_𝑛 generate analytic 𝐶₀-semigroups. The following conditions (𝐴) and (𝐵₁) are equivalent to condition (𝐶₁).(𝐴) Compatibility. There exists 𝜆∈𝜌(𝐴)∩∩_𝑛𝜌(𝐴_𝑛) such that the resolvents converge (𝜆𝐼−𝐴_𝑛)−−−^−1𝒫𝒫−→(𝜆𝐼−𝐴)⁻¹.(𝐵₁)Stability. There are some constants 𝑀₁≥1 and 𝜔₁∈ℝ such that‖‖𝜆𝐼_𝑛−𝐴_𝑛⁻¹‖‖≤𝑀¹||𝜆−𝜔₁||,Re𝜆>𝜔¹,𝑛∈ℕ.(3.2)(𝐶₁) Convergence. For any finite 𝜇>0 and some 0<𝜃<𝜋/2, one hasmax_{𝜂∈Σ(𝜃,𝜇)}‖‖𝑒^𝜂𝐴𝑛𝑢^0𝑛−𝑝_𝑛𝑒^𝜂𝐴𝑢⁰‖‖⟶0as𝑛⟶∞whenever𝑢−−^0𝑛𝒫→𝑢⁰.(3.3)Here one used the sector of angle 2𝜃 and radius 𝜌 given by Σ(𝜃,𝜇)={𝑧∈Σ(𝜃)∶|𝑧|≤𝜇}, and Σ(𝜃)={𝑧∈ℂ∶|arg𝑧|≤𝜃}.

It is natural to assume in semidiscretization that conditions like (A) and (𝐵₁) are satisfied.

Definition 3.5. The region of stability Δ_𝑠=Δ_𝑠({𝐴_𝑛}),𝐴_𝑛∈𝒞(𝐵_𝑛), is defined as the set of all 𝜆∈𝐂 such that 𝜆∈𝜌(𝐴_𝑛) for almost all 𝑛 and such that the sequence {‖(𝜆𝐼_𝑛−𝐴_𝑛)⁻¹‖}_𝑛∈ℕ is bounded. The region of convergence Δ_𝑐=Δ_𝑐({𝐴_𝑛}),𝐴_𝑛∈𝒞(𝐸_𝑛), is defined as the set of all 𝜆∈𝐂 such that 𝜆∈Δ_𝑠({𝐴_𝑛}) and such that the sequence of operators {(𝜆𝐼_𝑛−𝐴_𝑛)⁻¹}_𝑛∈ℕ is 𝒫𝒫-convergent to some operator 𝑆(𝜆)∈𝐵(𝐸).

Definition 3.6. A sequence of operators {𝐵_𝑛},𝐵_𝑛∈𝐵(𝐸_𝑛),𝑛∈ℕ, is said to be stably convergent to an operator 𝐵∈𝐵(𝐸) if and only if 𝐵−−−_𝑛𝒫𝒫−→𝐵 and ‖𝐵^𝑛−1‖_{𝐵(𝐸𝑛)}=𝑂(1), 𝑛→∞. We will write this as: 𝐵−−−_𝑛𝒫𝒫−→𝐵 stably.

Definition 3.7. A sequence of operators {𝐵_𝑛}, 𝐵_𝑛∈𝐵(𝐸_𝑛), is called regularly convergent to the operator 𝐵∈𝐵(𝐸) if and only if 𝐵−−−_𝑛𝒫𝒫−→𝐵 and the following implication holds ‖‖𝑥_𝑛‖‖_𝐸𝑛=𝑂(1),𝐵_𝑛𝑥_𝑛is𝒫-compact⟹𝑥_𝑛is𝒫-compact.(3.4) We write this as: 𝐵−−−_𝑛𝒫𝒫−→𝐵 regularly.

Theorem 3.8 (see [15]). Let 𝐶_𝑛,𝑄_𝑛∈𝐵(𝐸_𝑛), 𝐶,𝑄∈𝐵(𝐸) and ℛ(𝑄)=𝐸. Assume also that 𝐶−−−_𝑛𝒫𝒫−→𝐶 compactly and 𝑄−−−_𝑛𝒫𝒫−→𝑄 stably. Then, 𝑄_𝑛+𝐶−−−_𝑛𝒫𝒫−→𝑄+𝐶 converge regularly.

Theorem 3.9 (see [15]). For 𝑄_𝑛∈𝐵(𝐸_𝑛) and 𝑄∈𝐵(𝐸), the following conditions are equivalent:(i)𝑄−−−_𝑛𝒫𝒫−→𝑄 regularly, 𝑄_𝑛 are Fredholm operators of index 0 and 𝒩(𝑄)={0};(ii)𝑄−−−_𝑛𝒫𝒫−→𝑄 stably and ℛ(𝑄)=𝐸;(iii)𝑄−−−_𝑛𝒫𝒫−→𝑄 stably and regularly; (iv)if one of conditions (i)–(iii) holds, then there exist 𝑄^𝑛−1∈𝐵(𝐸_𝑛), 𝑄⁻¹∈𝐵(𝐸), and 𝑄−−−^{𝑛−1𝒫𝒫}−→𝑄⁻¹ regularly and stably.

Theorem 3.10. Let operators 𝜆𝐼_𝑛−𝐵_𝑛∈𝐵(𝐸_𝑛) be Fredholm operators with 𝑖𝑛𝑑(𝜆𝐼_𝑛−𝐵_𝑛)=0 for any 𝜆∈𝕋,𝑛∈ℕ. Assume also that 𝐵∈𝐵(𝐸) has the property 𝕋∩𝜎(𝐵)=∅ and 𝜆𝐼_𝑛−𝐵−−−_𝑛𝒫𝒫−→𝜆𝐼−𝐵 regularly for any 𝜆∈𝕋. Then 𝜆𝐼_𝑛−𝐵−−−_𝑛𝒫𝒫−→𝜆𝐼−𝐵 stably for any 𝜆∈𝕋 and sup_𝜆∈𝕋‖(𝜆𝐼_𝑛−𝐵_𝑛)⁻¹‖<∞.

Proof. Assume that there are some sequences {𝜆_𝑛},𝜆_𝑛∈𝕋, and {𝑥_𝑛},𝑥_𝑛∈𝐸_𝑛, such that ‖𝑥_𝑛‖=1 and (𝜆_𝑛𝐼_𝑛−𝐵_𝑛)𝑥−−_𝑛𝒫→0 as 𝑛→∞. Since 𝕋 is compact, one can find ℕ^′⊂ℕ such that 𝜆_𝑛→𝜆₀∈𝕋 as 𝑛∈ℕ^. In the meantime, 𝜆₀𝐼_𝑛−𝐵−−−_𝑛𝒫𝒫−→𝜆₀𝐼−𝐵 regularly for such 𝜆₀∈𝕋, and (𝜆₀𝐼_𝑛−𝐵_𝑛)𝑥_𝑛=(𝜆₀𝐼_𝑛−𝜆_𝑛𝐼_𝑛)𝑥_𝑛+(𝜆_𝑛𝐼_𝑛−𝐵_𝑛)𝑥−−_𝑛𝒫→0 for 𝑛∈ℕ^. Therefore, there is ℕ^⊂ℕ^ such that 𝑥−−_𝑛𝒫→𝑥₀≠0 as 𝑛∈ℕ^. But in such case (𝜆₀𝐼_𝑛−𝐵_𝑛)𝑥−−−_𝑛𝒫𝒫−→(𝜆₀𝐼−𝐵)𝑥₀=0 as 𝑛∈ℕ^, which contradicts our assumption 𝕋∩𝜎(𝐵)=∅.

Definition 3.11. The operator 𝐵_𝑛∈𝐵(𝐸_𝑛) has a uniform exponential discrete dichotomy with data (𝑀,𝑟,𝑃_𝑛) if 𝑃_𝑛∈𝐵(𝐸_𝑛) is a projector in 𝐸_𝑛 and 𝑀,𝑟 are constants with 0≤𝑟<1 such that the following properties hold(i)𝐵^𝑘𝑛𝑃_𝑛=𝑃_𝑛𝐵^𝑘𝑛 and ‖𝑃_𝑛‖≤𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 for all 𝑘,𝑛∈ℕ;(ii)‖𝐵^𝑘𝑛(𝐼_𝑛−𝑃_𝑛)‖≤𝑀𝑟^𝑘 for all 𝑘,𝑛∈ℕ;(iii)𝐵_𝑛∶=𝐵_𝑛|_{ℛ(𝑃𝑛)}∶ℛ(𝑃_𝑛)↦ℛ(𝑃_𝑛) is a homeomorphism that satisfies ‖‖𝐵_𝑛−𝑘𝑃_𝑛‖‖≤𝑀𝑟_𝑘,𝑘,𝑛∈ℕ.(3.5)

Theorem 3.12. The following conditions are equivalent:(i)𝜆𝐼_𝑛−𝐵−−−_𝑛𝒫𝒫−→𝜆𝐼−𝐵 stably and 𝜆∈𝜌(𝐵) for any 𝜆∈𝕋;(ii)operator 𝒟=𝐼−ℬ is invertible and 𝒟−−−_𝑛𝒫𝒫−→𝒟 stably, where (ℬ𝑢)(𝑘)=𝐵𝑢(𝑘−1),𝑘∈ℕ;(iii)𝐵−−−_𝑛𝒫𝒫−→𝐵 and 𝜆𝐼−𝐵 be invertible for any 𝜆∈𝕋 and the operators 𝐵_𝑛 have an exponential discrete dichotomy with data (𝑀,𝑟,𝑃_𝑛) uniformly in 𝑛∈ℕ.

Proof. The equivalence (i)⇔(iii) follows from Theorem 2.7. Indeed, by formula (2.19), one gets from (i) that 𝑃−−−_𝑛𝒫𝒫−→𝑃 and ‖𝑃_𝑛‖≤𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡. By Theorem 3.10, one has sup_𝜆∈𝕋‖(𝜆𝐼_𝑛−𝐵_𝑛)⁻¹‖<∞, and by Theorem 2.7 (ii) we have (iii). Conversely, from condition (iii), it follows by Theorem 2.7 (i) that 𝜆𝐼_𝑛−𝐵−−−_𝑛𝒫𝒫−→𝜆𝐼−𝐵 stably for all 𝜆∈𝕋.
To prove (ii)⇒(i), we note that (ii) means that for any 𝑢(⋅)∈𝑙^𝑝(ℤ;𝐸) one has Σ^{∞𝑘=−∞}‖𝑝_𝑛𝑢(𝑘)−𝐵_𝑛𝑝_𝑛𝑢(𝑘−1)−𝑝_𝑛𝑢(𝑘)+𝑝_𝑛𝐵𝑢(𝑘−1)‖^𝑝𝐸_𝑛→0 as 𝑛→∞, that is, 𝐵−−−_𝑛𝒫𝒫−→𝐵. Assume now that 𝐼_𝑛−𝐵_𝑛 is not uniformly invertible in 𝑙^∞(ℤ;𝐸_𝑛), that is, for some sequence ‖𝑥_𝑛‖=1, one has (𝜆₀𝐼_𝑛−𝐵_𝑛)𝑥−−_𝑛𝒫→0 as 𝑛→∞ for some 𝜆₀=1∈𝕋. This means that for stationary sequence 𝑢_𝑛(𝑘)=𝑥_𝑛,𝑘∈ℤ,𝑛∈𝕋, one has (𝒟_𝑛𝑢_𝑛)(𝑘)=𝑢_𝑛(𝑘)−𝐵_𝑛𝑢_𝑛(𝑘−1)=𝑥_𝑛−𝐵_𝑛𝑥−−_𝑛𝒫→0 for any 𝑘∈ℕ and 𝑛→∞. But, 𝒟−−−_𝑛𝒫𝒫−→𝒟 stably, that is, ‖𝒟_𝑛𝑢_𝑛‖_𝑙^∞_{(ℤ;𝐸𝑛)}≥𝛾‖𝑢_𝑛‖_𝑙^∞_{(ℤ;𝐸𝑛)} which contradicts (𝜆₀𝐼_𝑛−𝐵_𝑛)𝑥−−_𝑛𝒫→0 as 𝑛→∞. Now we show that ℛ(𝜆₀𝐼_𝑛−𝐵_𝑛)=𝐸_𝑛. For any 𝑦_𝑛∈𝐸_𝑛,‖𝑦_𝑛‖=1, consider 𝑣_𝑛(𝑘)=𝑦_𝑛,𝑘∈ℤ,𝑛∈ℕ. The solution of 𝒟_𝑛𝑢_𝑛=𝑣_𝑛 is a sequence 𝑢_𝑛(𝑘), which is stationary too, that is, (𝜆₀𝐼_𝑛−𝐵_𝑛)𝑥_𝑛=𝑦_𝑛, where 𝑥_𝑛=𝑢_𝑛(𝑘),𝑘∈ℤ,𝑛∈ℕ. To show (i)⇒(ii), we note that ‖𝒟_𝑛‖_𝐵(𝑙^𝑝_{(ℤ;𝐸𝑛))}≤𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡,𝑛∈ℕ. Now for any 𝑢(⋅)∈𝑙^𝑝(ℤ;𝐸) and for any 𝜖>0, one can find 𝐾∈ℕ such that (Σ^∞𝑘=𝐾+Σ^{−∞𝑘=−𝐾})‖𝑢(𝑘)‖^𝑝≤𝜖. In the meantime, Σ^𝐾𝑘=−_K‖𝑝_𝑛𝑢(𝑘)−𝐵_𝑛𝑝_𝑛𝑢(𝑘−1)−𝑝_𝑛𝑢(𝑘)+𝑝_𝑛𝐵𝑢(𝑘−1)‖^𝑝𝐸_𝑛→0 as 𝑛→∞, since 𝐵−−−_𝑛𝒫𝒫−→𝐵. So we have 𝒟−−−_𝑛𝒫𝒫−→𝒟. The convergence 𝒟−−−^{𝑛−1𝒫𝒫}−→𝒟⁻¹ follows from formula (2.15). The theorem is proved.

3.2. Dichotomy for Compact Resolvents in Semidiscretization

In the case of operators which have compact resolvent, it is natural to consider approximate operators that “preserve” the property of compactness.

Definition 3.13. A sequence of operators {𝐵_𝑛},𝐵_𝑛∶𝐸_𝑛→𝐸_𝑛,𝑛∈ℕ, converges compactly to an operator 𝐵∶𝐸→𝐸 if 𝐵−−−_𝑛𝒫𝒫−→𝐵 and the following compactness condition holds ‖‖𝑥_𝑛‖‖_𝐸𝑛=𝑂(1)⟹𝐵_𝑛𝑥_𝑛is𝒫-compact.(3.6)

Definition 3.14. The region of compact convergence of resolvents, Δ_𝑐𝑐=Δ_𝑐𝑐(𝐴_𝑛,𝐴), where 𝐴_𝑛∈𝒞(𝐸_𝑛) and 𝐴∈𝒞(𝐸) is defined as the set of all 𝜆∈Δ_𝑐∩𝜌(𝐴) such that (𝜆𝐼_𝑛−𝐴_𝑛)−−−^−1𝒫𝒫−→(𝜆𝐼−𝐴)⁻¹ compactly.

Proposition 3.15. Assume that operators 𝐵,𝐵_n are compact, 𝕋⊂𝜌(𝐵) and Δ_𝑐𝑐(𝐵_𝑛,𝐵)≠∅. Then, 𝜆𝐼_𝑛−𝐵−−−_𝑛𝒫𝒫−→𝜆𝐼−𝐵 stably for any 𝜆∈𝕋 and sup_𝜆∈𝕋‖(𝜆𝐼_𝑛−𝐵_𝑛)⁻¹‖<∞.

Proof. The proof follows from Theorems 3.8, 3.9, and 3.10.
Here we continue considering an example of discretization of problem (1.2). To this end we will also use the operators 𝑝^𝛼𝑛=(−𝐴_𝑛)^−𝛼𝑝_𝑛(−𝐴)^𝛼∈𝐵(𝐸^𝛼,𝐸^𝛼𝑛) which satisfy property (3.1), but for the spaces 𝐸^𝛼,𝐸^𝛼𝑛. The operators 𝐴_𝑛 and 𝐴 are supposed to be related by condition (1.1), and conditions (𝐴) and (𝐵₁) of Theorem 3.4 are assumed to hold. So we say that 𝑥−−−→_𝑛𝒫^𝛼𝑥 if and only if ‖𝑥_𝑛−𝑝^𝛼𝑛𝑥‖_𝐸^𝛼𝑛→0 as 𝑛→∞. One can see that ‖𝑥_𝑛−𝑝^𝛼𝑛𝑥‖_𝐸^𝛼𝑛=‖(−𝐴_𝑛)^𝛼𝑥_𝑛−𝑝_𝑛(−𝐴)^𝛼𝑥‖ and ‖𝑝^𝛼𝑛𝑥‖_𝐸^𝛼𝑛=‖𝑝_𝑛(−𝐴)^𝛼𝑥‖_𝐸𝑛→‖(−𝐴)^𝛼𝑥‖_𝐸=‖𝑥‖_𝐸^𝛼 for any 𝑥∈𝐷((−𝐴)^𝛼) and 𝑛→∞.
Consider in Banach spaces 𝐸^𝛼𝑛 the family of parabolic problems𝑢𝑢^′_𝑛(𝑡)=𝐴_𝑛𝑢_𝑛(𝑡)+𝑓_𝑛𝑢_𝑛(𝑡),𝑡≥0,_𝑛(0)=𝑢^0𝑛∈𝐸^𝛼𝑛,(3.7) where 𝑢−−−→^{0𝑛𝒫𝛼}𝑢⁰, operators (𝐴_𝑛,𝐴) are compatible, 𝑓_𝑛(⋅)∶𝐸^𝛼𝑛→𝐸_𝑛 are globally bounded and globally Lipschitz continuous both uniformly in 𝑛∈ℕ, and continuously Fréchet differentiable.
Under the above assumptions, the mild solution 𝑢_𝑛(⋅) of (3.7) is defined for all 𝑡≥0 (see [1, 17]), and we define it by 𝑢_𝑛(⋅)=𝑇_𝑛(⋅)𝑢^0𝑛∶ℝ⁺→𝐸_𝑛. The nonlinear semigroup 𝑇_𝑛(⋅) satisfies the variation of constants formula𝑇_𝑛(𝑡)𝑢^0𝑛=𝑒^𝑡𝐴𝑛𝑢^0𝑛+^𝑡0𝑒^{(𝑡−𝑠)𝐴𝑛}𝑓_𝑛𝑇_𝑛(𝑠)𝑢^0𝑛𝑑𝑠,𝑡≥0.(3.8)
Recall that a hyperbolic equilibrium point 𝑢^∗ is a solution of equation 𝐴𝑢+𝑓(𝑢)=0, or equivalently 𝑢^∗=−𝐴⁻¹𝑓(𝑢^∗). Since the operator 𝐴 has a compact resolvent, the operator 𝐴⁻¹𝑓(⋅) is compact. In case of Δ_𝑐𝑐≠∅, the operators 𝐴^𝑛−1𝑓_𝑛(⋅)−−−^𝒫𝒫−→𝐴⁻¹𝑓(⋅) compactly. From [15], it follows that equations 𝑢_𝑛=−𝐴^𝑛−1𝑓_𝑛(𝑢_𝑛) have solutions {𝑢^∗𝑛},𝐴_𝑛𝑢^∗𝑛+𝑓_𝑛(𝑢^∗𝑛)=0, such that 𝑢−−^∗𝑛𝒫→𝑢^∗.
Now, consider the problems (3.7) near hyperbolic equilibrium points 𝑢^∗𝑛. In this case one gets𝑣^𝑛(𝑡)=𝐴_𝑢^∗𝑛_,𝑛𝑣_𝑛(𝑡)+𝐹_𝑢^∗𝑛_,𝑛𝑣_𝑛(𝑡),𝑣_𝑛(0)=𝑣^0𝑛,𝑡≥0,(3.9) where 𝐴_𝑢^∗𝑛_,𝑛=𝐴_𝑛+𝑓^𝑛(𝑢^∗𝑛),𝐹_𝑢^∗𝑛_,𝑛(𝑣_𝑛(𝑡))=𝑓_𝑛(𝑣_𝑛(𝑡)+𝑢^∗𝑛)−𝑓_𝑛(𝑢^∗𝑛)−𝑓^𝑛(𝑢^∗𝑛)𝑣_𝑛(𝑡). From now on, we consider hyperbolic point 𝑢^∗ and hyperbolic points 𝑢−−−→^{∗𝑛𝒫𝛼}𝑢^∗.
We decompose 𝐸^𝛼𝑛 using the projection𝑃_𝑛𝜎^+𝑛∶=𝑃_𝑛𝜎^+𝑛,𝐴_𝑢^∗𝑛_,𝑛∶=12𝜋𝑖_{𝜕𝑈(𝜎}^+𝑛₎𝜁𝐼_𝑛−𝐴_𝑢^∗𝑛_,𝑛⁻¹𝑑𝜁(3.10) defined by the set 𝜎^+𝑛, which is enclosed in a contour consisting of a part of 𝑖ℝ and the contour from condition (𝐵₁) for operators 𝐴_𝑢^∗𝑛_,𝑛.
There are some positive 𝑀₂,𝛾>0, because of analyticity of 𝐶₀-semigroup 𝑒^{𝑡𝐴𝑢∗𝑛,𝑛} and condition Δ_𝑐𝑐≠∅, which is applied to operator 𝐴_𝑢^∗𝑛_,𝑛, such that [7, 18]‖‖‖‖𝑒𝑒^{𝑡𝐴𝑢∗𝑛,𝑛}𝑧_𝑛‖‖_𝐸𝛼𝑛≤𝑀₂𝑒^−𝛾𝑡‖‖𝑧_𝑛‖‖_𝐸𝛼𝑛,𝑡≥0,_{𝑡𝐴𝑢∗𝑛,𝑛}𝑣_𝑛‖‖_𝐸𝛼n≤𝑀₂𝑒^𝛾𝑡‖‖𝑣_𝑛‖‖_𝐸𝛼𝑛,𝑡≤0,(3.11) for all 𝑣_𝑛∈𝑃_𝑛(𝜎^+𝑛)𝐸^𝛼𝑛 and 𝑧_𝑛∈(𝐼_𝑛−𝑃_𝑛(𝜎^+𝑛))𝐸^𝛼𝑛. One has to note that the neighborhood of any part of 𝑖ℝ does not intersect 𝜎(𝐴_𝑢^∗𝑛_,𝑛). Moreover, the condition Δ_𝑐𝑐≠∅ implies that 𝑃_𝑛(𝜎^+𝑛)−−−^𝒫𝒫−→𝑃(𝜎⁺) compactly, and therefore, as was shown in [14], dim𝑃_𝑛(𝜎^+𝑛)=dim𝑃(𝜎⁺) for 𝑛≥𝑛₀. One can consider for the 𝑇=𝑇(𝑣^0𝑛)=sup{𝑡≥0∶𝑣_𝑛(𝑡,𝑣^0𝑛)∈𝒰_𝐸^𝛼𝑛(0;𝜌)} with 𝑣^0𝑛∈𝒰_𝐸^𝛼𝑛(0;𝜌) the problem 𝐼𝑣^′_𝑛(𝑡)=𝐴_𝑢^∗𝑛_,𝑛𝑣_𝑛(𝑡)+𝐹_𝑢^∗𝑛_,𝑛𝑣_𝑛(𝑡),0≤𝑡≤𝑇,_𝑛−𝑃_𝑛𝜎^+𝑛𝑣_𝑛(0)=𝐼_𝑛−𝑃_𝑛𝜎^+𝑛𝑣^0𝑛,𝑃_𝑛𝜎^+𝑛𝑣_𝑛(𝑇)=𝑃_𝑛𝜎^+𝑛𝑣^𝑇𝑛.(3.12) The mild solution of problem (3.12) is given by the formula (for 0≤𝑡≤𝑇) 𝑣_𝑛(𝑡)=𝑒^{(𝑡−𝑇)𝐴𝑢∗𝑛,𝑛}𝑃_𝑛𝜎^+𝑛𝑣^𝑇𝑛+𝑒^{𝑡𝐴𝑢∗𝑛,𝑛}𝐼_𝑛−𝑃_𝑛𝜎^+𝑛𝑣^0𝑛+^𝑡0𝑒^{(𝑡−𝑠)𝐴𝑢∗𝑛,𝑛}𝐼_𝑛−𝑃_𝑛𝜎^+𝑛𝐹_𝑢^∗𝑛_,𝑛𝑣_𝑛(𝑠)𝑑𝑠+^𝑇𝑡𝑒^{(𝑡−𝑠)𝐴𝑢∗𝑛,𝑛}𝑃_𝑛𝜎^+𝑛𝐹_𝑢^∗𝑛_,𝑛𝑣_𝑛(𝑠)𝑑𝑠.(3.13) Now we are in a position to state our main result on uniform in index 𝑛 estimates for the terms of discrete solutions (3.13).

Theorem 3.16. Let operators 𝐴_𝑛,𝐴 be generators of analytic 𝐶₀-semigroups and let condition (𝐵₁) be satisfied. Assume also that the semigroup 𝑒^{𝑡𝐴𝑢∗} is hyperbolic, 𝜎(𝐴_𝑢∗)∩{𝜆∶Re𝜆≥0}=𝑃𝜎(𝐴_𝑢∗),dim𝑃(𝜎+)<∞, and for 𝜌>0 such that {𝜆∶−𝜌≤Re𝜆≤𝜌}⊂𝜌(𝐴_𝑢∗), operators 𝜆𝐼_𝑛−𝐴_𝑢^∗𝑛_,𝑛 are Fredholm operators of ind 0, and operators 𝜆𝐼_𝑛−𝐴_𝑢^∗𝑛_,𝑛,𝜆𝐼−𝐴_𝑢∗ are regularly consistent for any Re𝜆≥−𝜌. Then, 𝑃_𝑛(𝜎^+𝑛)−−−^𝒫𝒫−→𝑃(𝜎⁺) compactly and ‖‖𝑒‖‖𝑒^{𝑡𝐴𝑢∗𝑛,𝑛}𝐼_𝑛−𝑃_𝑛𝜎^+𝑛‖‖_𝐸𝑛≤𝑀₂𝑒^−𝛾𝑡,𝑡≥0,_{𝑡𝐴𝑢∗𝑛,𝑛}𝑃_𝑛𝜎^+𝑛‖‖_𝐸𝑛≤𝑀₂𝑒^𝛾𝑡,𝑡≤0,(3.14) where 𝛾>0.

Proof. The condition (𝐵₁) implies that (𝜆𝐼_𝑛−𝐴_𝑢^∗𝑛_,𝑛)−−−^−1𝒫𝒫−→(𝜆𝐼−𝐴_𝑢∗)⁻¹ as −𝜌≤Re𝜆≤𝜌 for |𝜆| big enough. For the other −𝜌≤Re𝜆≤𝜌, the convergence (𝜆𝐼_𝑛−𝐴_𝑢^∗𝑛_,𝑛)−−−^−1𝒫𝒫−→(𝜆𝐼−𝐴_𝑢∗)⁻¹ follows from analogy of Theorem 3.9 for closed operators. Now the compact convergence 𝑃_𝑛(𝜎^+𝑛)−−−^𝒫𝒫−→𝑃(𝜎⁺) can be obtained in the same way as in [3] and the estimates (3.14) follow say as in [7, 18]. The theorem is proved.

Remark 3.17. Of course, Theorem 3.16 holds for the case of any operator 𝐴 which generates analytic hyperbolic 𝐶₀-semigroup with condition 𝜎(𝐴)∩{𝜆∶Re𝜆≥0}=𝑃𝜎(𝐴),dim𝑃(𝜎+)<∞ and corresponding conditions on approximation of operators. The structure of operator like 𝐴+𝑓^(𝑢^∗) is not necessary.

Theorem 3.18. Let the operators 𝐴_𝑛,𝐴 be generators of analytic 𝐶₀-semigroups and let the condition (𝐵₁) be satisfied. Assume also that the 𝐶₀-semigroup 𝑒^{𝑡𝐴𝑢∗} is hyperbolic, 𝜎(𝐴_𝑢∗)∩{𝜆∶Re𝜆≥0}=𝑃𝜎(𝐴_𝑢∗),dim𝑃(𝜎+)<∞. Assume also that Δ_𝑐𝑐(𝐴_𝑛,𝐴)≠∅ and resolvents of 𝐴_𝑛,𝐴 are compact operators. Then, (3.14) holds.

Proof. We set 𝐵_𝑛=𝑒^{1𝐴𝑢∗𝑛,𝑛} and 𝐵=𝑒^1𝐴𝑢∗ and apply Proposition 3.15. It is known [3] that Δ_𝑐𝑐≠∅ is equivalent to compact convergence 𝐵−−−_𝑛𝒫𝒫−→𝐵. Then, the condition (i) of Theorem 3.12 is satisfied and one gets discrete dichotomy for 𝐵_𝑛. From the other side, since we have exactly the form 𝐵_𝑛=𝑒^{1𝐴𝑢∗𝑛,𝑛}, this means by Theorem 2.7 and Proposition 2.4 that operators ℒ_𝑛=−𝑑/𝑑𝑡+𝐴_𝑢^∗𝑛_,𝑛 are invertible, and thus by Theorem 2.3 the estimates (3.14) are followed. The theorem is proved.

3.3. Dichotomy for Condensing Operators in Semidiscretization

Let Λ⊆𝐂 be some open connected set, and let 𝐵∈𝐵(𝐸). For an isolated point 𝜆∈𝜎(𝐵), the corresponding maximal invariant space (or generalized eigenspace) will be denoted by 𝒲(𝜆;𝐵)=𝑄(𝜆)𝐸, where 𝑄(𝜆)=(1/2𝜋𝑖)∫_{|𝜁−𝜆|=𝛿}(𝜁𝐼−𝐵)⁻¹𝑑𝜁 and 𝛿 is small enough so that there are no points of 𝜎(𝐵) in the disc {𝜁∶|𝜁−𝜆|≤𝛿} different from 𝜆. The isolated point 𝜆∈𝜎(𝐵) is a Riesz point of 𝐵 if 𝜆𝐼−𝐵 is a Fredholm operator of index zero and 𝑄(𝜆) is of finite rank. Denote 𝒲𝜆,𝛿;𝐵_𝑛=_{𝜆𝑛𝐵𝑛,𝑛∈𝜎|𝜆−𝜆|<𝛿}𝒲𝜆_𝑛,𝐵_𝑛.(3.15) It is clear that 𝒲(𝜆,𝛿;𝐵_𝑛)=𝑄_𝑛(𝜆)𝐸_𝑛, where𝑄_𝑛(𝜆)=12𝜋𝑖_{|𝜁−𝜆|=𝛿}𝜁𝐼_𝑛−𝐵_𝑛⁻¹𝑑𝜁.(3.16)

Definition 3.19. The function 𝜇(⋅) is said to be the measure of noncompactness if for any bounded sequence {𝑥_𝑛},𝑥_𝑛∈𝐸_𝑛, one has 𝜇𝑥_𝑛=inf𝜖>0∶∀ℕ^⊆ℕ,∃ℕ^⊆ℕ^,𝑥^∈𝐸suchthat‖‖𝑥_𝑛−𝑝_𝑛𝑥^′‖‖≤𝜖,𝑛∈ℕ^.(3.17)

Definition 3.20. We say that the operators 𝐵_𝑛∈𝐵(𝐸_𝑛) are jointly condensing with constant 𝑞>0 with respect to measure 𝜇(⋅) if for any bounded sequence {𝑥_𝑛},𝑥_𝑛∈𝐸_𝑛, one has 𝜇𝐵_𝑛𝑥_𝑛≤𝑞𝜇𝑥_𝑛.(3.18)

It is known [19, page 82], that outside a closed disc of radius 𝑞 centered at zero each operator 𝐵_𝑛 has only isolated points of spectrum, each of which can only be an eigenvalue of finite multiplicity.

Proposition 3.21. Let 𝐵−−−_𝑛𝒫𝒫−→𝐵 for 𝐵_𝑛∈𝐵(𝐸_𝑛),𝐵∈𝐵(𝐸) and 𝜇({𝐵_𝑛𝑥_𝑛})≤𝑞𝜇({𝑥_𝑛}), for any bounded sequence {𝑥_𝑛},𝑥_𝑛∈𝐸_𝑛. Assume that 𝜎(𝐵)∩Ψ=∅, where bounded closed set Ψ⊂𝐂⧵{𝜆∶|𝜆|≤𝑞} and 𝜎(𝐵)⧵{𝜆∶|𝜆|≤𝐪} consists only of discrete spectrum. Then, there is a constant 𝐶>0 such that ‖(𝜆𝐼_𝑛−𝐵_𝑛)⁻¹‖≤𝐶,𝜆∈Ψ,𝑛∈ℕ.

Proof. Any point 𝜆∈Ψ belongs to 𝑃𝜎(𝐵_𝑛)∪𝜌(𝐵_𝑛). This means that one gets for a sequence ‖𝑥_𝑛‖=1,𝑥_𝑛∈𝐸_𝑛, two cases: (𝜆𝐼_𝑛−𝐵_𝑛)𝑥_𝑛=0 or ‖(𝜆𝐼_𝑛−𝐵_𝑛)𝑥_𝑛‖≥𝛾_𝜆,𝑛‖𝑥_𝑛‖ with some 𝛾_𝜆,𝑛>0,𝜆∈Ψ. We are going to show that in reality we have ‖(𝜆𝐼_𝑛−𝐵_𝑛)𝑥_𝑛‖≥𝛾_Ψ‖𝑥_𝑛‖,𝜆∈Ψ.
Assume in contradiction that there are sequences {𝜆_𝑛},𝜆_𝑛∈Ψ,{𝑥_𝑛},‖𝑥_𝑛‖=1, such that𝜆_𝑛𝐼_𝑛−𝐵_𝑛𝑥−−_𝑛𝒫→0as𝑛∈ℕ.(3.19) Then, 𝜆_𝑛→𝜆₀∈Ψ,𝑛∈ℕ^⊆ℕ. One has for 𝑟̃=inf{|𝜉|∶𝜉∈Ψ}𝜇𝑥_𝑛≤||𝜆^𝑛||𝑟̃𝜇𝑥^𝑛≤𝜇𝐵^𝑛𝑥_𝑛𝑟̃≤𝑞𝑟̃𝜇𝑥^𝑛,(3.20) which means because of 𝑞/𝑟̃<1, that 𝜇({𝑥_𝑛})=0, that is, {𝑥_𝑛} is 𝒫-compact. Now 𝑥−−_𝑛𝒫→𝑥₀,𝑛∈ℕ^⊆ℕ^ and 𝐵_𝑛𝑥−−_𝑛𝒫→𝐵𝑥₀,𝜆_𝑛𝑥−−_𝑛𝒫→𝜆₀𝑥₀,𝑛∈ℕ^, that is, 𝜆₀𝑥₀=𝐵𝑥₀ with ‖𝑥₀‖=1, which contradicts our assumption 𝜎(𝐵)∩Ψ=∅. The proposition is proved.

Proposition 3.22. Let 𝐵−−−_𝑛𝒫𝒫−→𝐵 and 𝜇({𝐵_𝑛𝑥_𝑛})≤𝑞𝜇({𝑥_𝑛}) for any bounded sequence {𝑥_𝑛},𝑥_𝑛∈𝐸_𝑛. Assume that any 𝜆₀∈𝜎(𝐵),|𝜆₀|>𝑞, is an isolated eigenvalue with the finite dimensional projector 𝑄(𝜆₀). Then, there are sequence {𝜆_𝑛},𝜆_𝑛∈𝜎(𝐵_𝑛), and sequence of projectors 𝑄_𝑛(𝜆₀)∈𝐵(𝐸_𝑛) such that 𝜆_𝑛→𝜆₀ and 𝑄_𝑛(𝜆₀)−−−^𝒫𝒫−→𝑄(𝜆₀) converge compactly.

Proof. Note first that for Γ_𝑟={𝜆∶|𝜆−𝜆₀|=𝑟}⊂𝐂⧵{𝜆∶|𝜆|≤𝐪}, where 𝑟 can be taken small enough, we have by Proposition 3.21𝜆𝐼_𝑛−𝐵_𝑛−−−^−1𝒫𝒫−→(𝜆𝐼−𝐵)⁻¹as𝜆∈Γ_𝑟,𝑛∈ℕ.(3.21) Therefore, 𝑄_𝑛(𝜆₀)−−−^𝒫𝒫−→𝑄(𝜆₀). To show compact convergence of these projectors one can note that 𝜇𝜆₀𝐼_𝑛−𝐵_𝑛𝑥_𝑛≥||𝜆₀||𝜇𝑥_𝑛−𝜇𝐵_𝑛𝑥_𝑛≥||𝜆₀||𝜇𝑥_𝑛−𝑞𝜇𝑥_𝑛≥𝛾𝜇𝑥_𝑛,(3.22) where 𝛾=|𝜆₀|−𝑞>0. This means that 𝜇𝜆₀𝐼_𝑛−𝐵_𝑛^𝑘𝑥_𝑛≥𝛾^𝑘𝜇𝑥_𝑛forany𝑘∈ℕ.(3.23) By functional calculus 𝜆₀𝐼_𝑛−𝐵_𝑛^𝑘𝑄_𝑛𝜆₀𝑥_𝑛=21𝜋𝑖_Γ𝑟𝜆₀−𝜆^𝑘𝜆𝐼_𝑛−𝐵_𝑛⁻¹𝑥_𝑛𝑑𝜆.(3.24) From this representation using (3.23), one has 𝛾^𝑘𝜇𝑄_𝑛𝜆₀𝑥_𝑛≤‖‖𝜆₀𝐼_𝑛−𝐵_𝑛^𝑘𝑄_𝑛𝜆₀𝑥_𝑛‖‖≤𝐶2𝜋𝑟^𝑘‖‖𝑥_𝑛‖‖.(3.25) It is clear that from 𝑟/𝛾<1 it follows that (𝑟/𝛾)^𝑘→0 as 𝑘→∞. This means that 𝑄_𝑛(𝜆₀)−−−^𝒫𝒫−→𝑄(𝜆₀) compactly. The proposition is proved.

Theorem 3.23. Let the conditions (𝐴) and (𝐵₁) be satisfied, and let the analytic 𝐶₀-semigroup 𝑒^𝑡𝐴,𝑡∈ℝ₊, be hyperbolic such that the set 𝜎(𝐴)∩{𝜆∶Re𝜆≥0} consists of a finite number of points 𝑃𝜎(𝐴) and dim𝑃(𝜎+)<∞. Assume also that 𝜇({𝐵_𝑛𝑥_𝑛})≤𝑞𝜇({𝑥_𝑛}), for any bounded sequence {𝑥_𝑛},𝑥_𝑛∈𝐸_𝑛, with 𝑞<1, where 𝐵_𝑛=𝑒^1𝐴𝑛. Then, conclusions of Theorem 3.16, that is, (3.14), are hold and 𝑃_𝑛(𝜎_𝑛+)−−−^𝒫𝒫−→𝑃(𝜎+) converge compactly.

Proof. Because of the spectral mapping theorem, the spectrum of operator 𝐵=𝑒^1𝐴 which is located outside of unit disc 𝕋 consists of finite number of points of the set 𝑃𝜎(𝑒^1𝐴)={𝜁∶𝜁=𝑒^𝜆,𝜆∈𝑃𝜎(𝐴)∩{𝜉∶Re𝜉≥0}}. Moreover, since 𝑞<1, for any Ψ which contains 𝕋 and Ψ⊂𝜌(𝐵),𝐵=𝑒^1𝐴, one has ‖(𝜆𝐼_𝑛−𝐵_𝑛)⁻¹‖≤𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 as 𝜆∈Ψ by Proposition 3.22. Now from Theorem 3.12 it follows that 𝐵_𝑛 have discrete dichotomy. By Theorem 2.5, the operator 𝒟_𝑛 is invertible, and by Proposition 2.4 and by Theorem 2.3 we get that the semigroups 𝑒^𝑡𝐴𝑛,𝑡∈ℝ₊, have an exponential dichotomy uniformly in index 𝑛∈ℕ and (2.7), that is, (3.14) holds. By Theorem 3.9  𝑄_𝑛(𝜆₀)−−−^𝒫𝒫−→𝑄(𝜆₀) compactly, which implies that conditions of Theorem 3.16 are satisfied. The theorem is proved.

3.4. Discretisation in Time Variable and Δ_𝑐𝑐≠∅

Consider now the discretization of the problem (3.9) in time by the following scheme:𝑉_𝑛𝑡+𝜏_𝑛−𝑉_𝑛(𝑡)𝜏_𝑛=𝐴_𝑢^∗𝑛_,𝑛𝑉_𝑛𝑡+𝜏_𝑛+𝐹_𝑢^∗𝑛_,𝑛𝑉_𝑛(𝑡),𝑡=𝑘𝜏_𝑛,(3.26) with initial data 𝑉_𝑛(0)=𝑣^0𝑛. The solution of such problem is given by formula𝑉_𝑛𝑡+𝜏_𝑛==𝐼𝐼_𝑛−𝜏_𝑛𝐴_𝑢^∗𝑛_,𝑛⁻¹𝑉_𝑛(𝑡)+𝜏_𝑛𝐼_𝑛−𝜏_𝑛𝐴_𝑢^∗𝑛_,𝑛⁻¹𝐹_𝑢^∗𝑛_,𝑛𝑉_𝑛(𝑡)_𝑛−𝜏_𝑛𝐴_𝑢^∗𝑛_,𝑛^−𝑘𝑉_𝑛(0)+𝜏_𝑛𝑘_𝑗=0𝐼_𝑛−𝜏_𝑛𝐴_𝑢^∗𝑛_,𝑛^{−(𝑘−𝑗+1)}𝐹_𝑢^∗𝑛_,𝑛𝑉_𝑛𝑗𝜏_𝑛,𝑡=𝑘𝜏_𝑛,(3.27) where 𝑉_𝑛(0)=𝑣^0𝑛.

The problem (3.12) also can be discretized following (3.26) approach, so we have𝑉_𝑛𝑡+𝜏_𝑛−𝑉_𝑛(𝑡)𝜏_𝑛𝐼=𝐴_𝑢^∗𝑛_,𝑛𝑉_𝑛𝑡+𝜏_𝑛+𝐹_𝑢^∗𝑛_,𝑛𝑉_𝑛(𝑡),𝑡=𝑘𝜏_𝑛,(3.28)_𝑛−𝑃_𝑛𝑉_𝑛(0)=𝐼_𝑛−𝑃_𝑛𝑣^0𝑛,𝑃_𝑛𝑉_𝑛(𝑇)=𝑃_𝑛𝑣^𝑇𝑛.(3.29) The solution of problem (3.28) could be obtained by using the formulas𝐼_𝑛−𝑃_𝑛𝑉_𝑛𝑡+𝐼𝜏_𝑛=𝐼_𝑛−𝜏_𝑛𝐴_𝑢^∗𝑛_,𝑛⁻¹𝐼_𝑛−𝑃_𝑛𝑉_𝑛(𝑡)+𝜏_𝑛𝐼_𝑛−𝜏_𝑛𝐴_𝑢^∗𝑛_,𝑛⁻¹𝐼_𝑛−𝑃_𝑛𝐹_𝑢^∗𝑛_,𝑛𝑉_𝑛𝑗𝜏_𝑛_𝑛,−𝜏_𝑛𝐴_𝑢^∗𝑛_,𝑛𝑃_𝑛𝑉_𝑛𝑡+𝜏_𝑛=𝑃_𝑛𝑉_𝑛(𝑡)+𝜏_𝑛𝑃_𝑛𝐹_𝑢^∗𝑛_,𝑛𝑉_𝑛𝑘𝜏_𝑛,𝑡=𝑘𝜏_𝑛.(3.30) So one has a representation of solution of problem (3.28) as𝑉_𝑛𝑡;𝑣^0𝑛,𝑣^𝑇𝑛==𝐼𝐼_𝑛−𝑃_𝑛𝑉_𝑛(𝑡)+𝑃_𝑛𝑉_𝑛(𝑡)_𝑛−𝜏_𝑛𝐴_𝑢^∗𝑛_,𝑛^−𝑘+1𝐼_𝑛−𝑃_𝑛𝑣^0𝑛+𝜏_𝑛𝑘−1_𝑗=0𝐼_𝑛+𝐼−𝜏_𝑛𝐴_𝑢^∗𝑛_,𝑛^{−(𝑘−𝑗)}𝐼_𝑛−𝑃_𝑛𝐹_𝑢^∗𝑛_,𝑛𝑉_𝑛𝑗𝜏_𝑛_𝑛−𝜏_𝑛𝐴_𝑢^∗𝑛_,𝑛^𝐾−𝑘𝑃_𝑛𝑣^𝑇𝑛−𝜏_𝑛𝐾−1_𝑗=𝑘𝐼_𝑛−𝜏_𝑛A_𝑢^∗𝑛_,𝑛^{𝐾−1−𝑗}𝑃_𝑛𝐹_𝑢^∗𝑛_,𝑛𝑉_𝑛𝑗𝜏_𝑛,𝑡=𝑘𝜏_𝑛.(3.31) From (3.31), it is clear that corresponding estimates on powers of operators (𝐼_𝑛−𝜏_𝑛𝐴_𝑢^∗𝑛_,𝑛)^−𝑘+1(𝐼_𝑛−𝑃_𝑛),(𝐼_𝑛−𝜏_𝑛𝐴_𝑢^∗𝑛_,𝑛)^𝐾−𝑘𝑃_𝑛 play the main role in approximation of solutions of (1.4) in the vicinity of 𝑢^∗.

Theorem 3.24. Let operators 𝐴_𝑛,𝐴 be generators of analytic 𝐶₀-semigroups and let condition (𝐵₁) be satisfied. Assume also that the analytic 𝐶₀-semigroup 𝑒^{𝑡𝐴𝑢∗},𝑡∈ℝ₊, is hyperbolic and for 𝜌>0 such that {𝜆∶−𝜌≤Re𝜆≤𝜌}⊂𝜌(𝐴), operators 𝜆𝐼_𝑛−𝐴_𝑢^∗𝑛_,𝑛 are Fredholm operators of ind 0, and operators 𝜆𝐼_𝑛−𝐴_𝑢^∗𝑛_,𝑛,𝜆𝐼−𝐴_𝑢∗ are regularly consistent for any Re𝜆≥−𝜌. Then, 𝑃_𝑛(𝜎+)−−−^𝒫𝒫−→𝑃(𝜎+) compactly and ‖‖𝐼‖‖𝐼_𝑛−𝜏_𝑛𝐴_𝑢^∗𝑛_,𝑛^−𝑘𝑛𝑃_𝑛‖‖_𝐸𝑛≤𝑀₂𝑒^−𝛾𝑡,𝑡≥0,_𝑛−𝜏_𝑛𝐴_𝑢^∗𝑛_,𝑛^𝑘𝑛𝐼_𝑛−𝑃_𝑛‖‖_𝐸𝑛≤𝑀₂𝑒^𝛾𝑡,𝑡≤0,(3.32) for some 𝛾>0.

Proof. From Theorem 3.16 it follows that the analytic 𝐶₀-semigroups 𝑒^{𝑡𝐴𝑢∗𝑛,𝑛} have dichotomy uniformly in 𝑛∈ℕ. Now one can see as in [20] that ‖(𝐼_𝑛−𝜏_𝑛𝐴_𝑢^∗𝑛_,𝑛)^−𝑘𝑛−𝑒^{𝑡𝐴𝑢∗𝑛,𝑛}‖≤𝑀𝜏_𝑛𝑒^𝜔𝑡/𝑡, where 𝑡=𝑘_𝑛𝜏_𝑛=1. Using perturbation dichotomy theorem from [1, p. 254], one gets that (3.32) holds. The Theorem is proved.

Theorem 3.25. Let Δ_𝑐𝑐(𝐴_𝑛,𝐴)≠∅ and resolvents of 𝐴_𝑛,𝐴 be compact operators. Assume also that the analytic 𝐶₀-semigroup 𝑒^{𝑡𝐴𝑢∗},𝑡∈ℝ₊, is hyperbolic and condition (𝐵₁) is satisfied. Then, ‖‖𝐼‖‖𝐼_𝑛−𝜏_𝑛𝐴_𝑢^∗𝑛_,𝑛^−𝑘𝑛𝑃_𝑛‖‖_𝐸𝑛≤𝑀₂𝑟^[𝑡],𝑡=𝑘_𝑛𝜏_𝑛≥0,_𝑛−𝜏_𝑛𝐴_𝑢^∗𝑛_,𝑛^𝑘𝑛𝐼_𝑛−𝑃_𝑛‖‖_𝐸𝑛≤𝑀₂𝑟^−[𝑡],−𝑡=−𝑘_𝑛𝜏_𝑛≤0,(3.33) where 𝑟<1.

Proof. Compact convergence of resolvents (𝜆𝐼_𝑛−𝐴_𝑛)−−−^−1𝒫𝒫−→(𝜆𝐼−𝐴)⁻¹ implies [18] that 𝜆𝐼_𝑛−𝐴_𝑢^∗𝑛_,𝑛−−−^−1𝒫𝒫−→𝜆𝐼−𝐴_𝑢∗⁻¹compactly.(3.34) We set 𝐵_𝑛=(𝐼_𝑛−𝜏_𝑛𝐴_𝑢^∗𝑛_,𝑛)^−𝑘𝑛,𝜏_𝑛𝑘_𝑛=1, and 𝐵=𝑒^1𝐴𝑢∗. Then, 𝐵−−−_𝑛𝒫𝒫−→𝐵, since the operators 𝐴_𝑢^∗𝑛_,𝑛,𝐴_𝑢∗ are consistent. Note that by condition (𝐵₁) one has [21, 22] ‖𝜏_𝑛𝑘_𝑛𝐴_𝑢^∗𝑛_,𝑛(𝐼_𝑛−𝜏_𝑛𝐴_𝑢^∗𝑛_,𝑛)^−𝑘𝑛‖_{𝐵(𝐸𝑛)}≤𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, which implies 𝐵_𝑛=𝐴^𝑢−1_∗𝑛,𝑛𝜏_𝑛𝑘_𝑛𝐴_𝑢^∗𝑛_,𝑛(𝐼_𝑛−𝜏_𝑛𝐴_𝑢^∗𝑛_,𝑛)−−−^{−𝑘𝑛𝒫𝒫}−→𝐵 compactly, since 𝐴−−−^𝑢−1_{∗𝑛,𝑛𝒫𝒫}−→𝐴^𝑢−1_∗ compactly. Now by applying Proposition 3.15 one gets discrete dichotomy for 𝐵_𝑛 by Theorem 3.12. The theorem is proved.

3.5. Discretisation in Time Variable and Condensing Property

Theorem 3.26. Let condition (𝐴) and condition (𝐵₁) be satisfied. Assume that 𝜇({𝐵_𝑛𝑥_𝑛})≤𝑞𝜇({𝑥_𝑛}), for any bounded sequence {𝑥_𝑛},𝑥_𝑛∈𝐸_𝑛, with 𝑞<1 and 𝐵_𝑛=𝑒^{𝐴𝑢∗𝑛,𝑛}. Assume also that the analytic 𝐶₀-semigroup 𝑒^{𝑡𝐴𝑢∗},𝑡∈ℝ₊, is hyperbolic.Then, ‖‖𝐼‖‖𝐼_𝑛−𝜏_𝑛𝐴_𝑢^∗𝑛_,𝑛^−𝑘𝑛𝑃_𝑛‖‖_𝐸𝑛≤𝑀₂𝑟^[𝑡],𝑡=𝑘_𝑛𝜏_𝑛≥0,_𝑛−𝜏_𝑛𝐴_𝑢^∗𝑛_,𝑛^𝑘𝑛𝐼_𝑛−𝑃_𝑛‖‖_𝐸𝑛≤𝑀₂𝑟^−[𝑡],−𝑡=−𝑘_𝑛𝜏_𝑛≤0,(3.35) where 𝑟<1.

Proof. By Theorem 3.23 it follows that (3.11) holds. By perturbation theorem from [1], one gets (3.35) which follows in the same way as in the proof of Theorem 3.24. The theorem is proved.

4. Example

The condition 𝜇(𝐵_𝑛𝑥_𝑛)≤𝑞𝜇(𝑥_𝑛) with 𝑞<1 in Theorems 3.23 and 3.26 can be checked for instance in case of compact convergence of operators 𝐴^𝑛−1𝑓^′_𝑛(𝑢^∗𝑛)−−−^𝒫𝒫−→𝐴⁻¹𝑓′(𝑢^∗). We present here an example where the analogy of such condition is naturally satisfied.

Example 4.1. Consider in 𝐿²(ℝ) an operator (𝐴𝑣)(𝑥)=𝑣^(𝑥)+𝑎𝑣^(𝑥)+𝑏𝑣(𝑥),𝑥∈(−∞,∞).(4.1) Since we took 𝐸=𝐿²(𝑅), we can take (𝑝_𝑛𝑣)(𝑥)=(1/ℎ)∫^{ℎ/2−ℎ/2}𝑣(𝑥+𝑦)𝑑𝑦, and the main condition ‖𝑝_𝑛𝑣‖_𝐿^2ℎ_(𝑍)→‖𝑣‖_𝐿²_(𝑅) is satisfied [15].
As in Section 5.4 of [1], one can see that 𝜎_ess(−𝐴)⊂{𝜆∶Re𝜆−((𝐼𝑚𝜆)²/𝑎²≥−𝑏}. For the case of 𝑎=0, we have 𝜎(𝐴)∈(−∞,𝑏). So for 𝑏<0 the operator 𝐴 is a negative self-adjoint operator. Then, we have the same for some difference scheme, say for central difference scheme,𝐴_𝑛𝑣_𝑛(𝑥)=𝑣^𝑛(𝑥+ℎ)−2𝑣_𝑛(𝑥)+𝑣_𝑛(𝑥−ℎ)ℎ²+𝑏𝑣_𝑛(𝑥),(4.2) that is, 𝜔_ess(𝐴_𝑛)≤𝜔₁<0 uniformly in ℎ>0. Moreover, it is easy to see that ‖‖𝑒^𝑡𝐴𝑛‖‖≤𝑀𝑒^𝜔2𝑡,𝑡≥0,with𝜔₂<0,(4.3) that is, 𝜇𝑒^𝑡𝐴𝑛𝑥_𝑛≤𝛾𝜇𝑥_𝑛with𝛾<1forsome𝑡=𝑡₀>0.(4.4) Now to get to the range of Theorem 3.23, let us denote 𝐵=𝑒^𝑡0𝐴,𝐵_𝑛=𝑒^{𝑡0𝐴𝑛}. For analytic 𝐶₀-semigroups, the spectrums of operators 𝐴 and 𝐵 are strictly related, which is also concerned to point spectrum 𝑃𝜎(𝐵)=𝑒^{𝑡0𝑃𝜎(𝐴)}. This means that operators 𝐴_𝑛 have for almost all 𝑛 the spectrums 𝜎(𝐴_𝑛)∩{𝜆∶Re𝜆>0} which approximate the spectrum 𝜎(𝐴)∩{𝜆∶Re𝜆>0}.
Let us now consider in 𝐿²(ℝ) the case of perturbed operator with smooth function 𝑏(𝑥)𝐴𝑣(𝑥)=𝑣^′(𝑥)+𝑏(𝑥)𝑣(𝑥)(4.5) and its approximation, say like 𝐴^𝑛𝑣_𝑛(𝑥)=𝑣^𝑛(𝑥+ℎ)−2𝑣_𝑛(𝑥)+𝑣_𝑛(𝑥−ℎ)ℎ²+𝑏(𝑥)𝑣_𝑛(𝑥),(4.6) with condition 𝑏(𝑥)→𝑏 as 𝑥→±∞ for simplicity. The operator ((𝐴−𝐴)𝑣)(𝑥)=(𝑏(𝑥)−𝑏)𝑣(𝑥) is an additive perturbation. We assume that 𝐶₀-semigroup 𝑒^𝑡𝐴,𝑡∈ℝ₊, is hyperbolic. The perturbation 𝐴−𝐴 is a relatively compact perturbation like in [23]. The same must happen to 𝐴_𝑛 because of 𝐴_𝑛=𝐴_𝑛+(𝐴_𝑛−𝐴_𝑛) and 𝑒^𝑡0𝐴_𝑛=𝑒^{𝑡0𝐴𝑛}+^𝑡0₀𝐴^𝛼𝑛𝑒^{(𝑡0−𝑠)𝐴𝑛}𝐴^𝑛−𝛼𝐴^𝑛−𝐴_𝑛𝑒^𝑠𝐴_𝑛𝑑𝑠.(4.7) The crucial point is that such perturbation gives us from (4.4) the estimate 𝜇𝑒^𝑡𝐴_𝑛𝑥_𝑛≤𝛾𝜇𝑥^𝑛with𝛾<1forsomet=𝑡₀>0,(4.8) since the integral part in (4.7) could be estimated by any small 𝜖>0 as 𝜇(∫^{𝑡00−𝜖}+∫^𝑡0_𝑡0−𝜖)≤𝑐𝜖^1−𝛼.
Then any point of spectrum of 𝐴 which is located to the right of 𝑏 belongs to 𝑃𝜎(𝐴) and it is of finite dimensional generalized eigenspace. The same is true for 𝐴_𝑛 with 𝐵_𝑛=𝑒^𝑡𝐴_𝑛, since we have (4.8). Using property (4.8), we get from Theorems 3.8 and 3.9 the regular consistence of operators 𝜆𝐼_𝑛−𝐴_𝑛,𝜆𝐼−𝐴 for any 𝜆∈𝑖ℝ and any Re𝜆>𝑏.
If as before 𝑃 is a dichotomy projector, then one has dim𝑃<∞ and 𝑃𝐴=𝐴𝑃. We can also state that 𝑃_𝑛→𝑃 compactly by Proposition 3.22. This means that say from dichotomy of 𝐴 we get dichotomy for 𝐴_𝑛 uniformly in 𝑛 by Theorem 3.16.
The similar situation for concrete differential operator was considered in [24].

Acknowledgments

This research was supported by grants of Russian Foundation for Basic Research 10-01-00297_a, by Grant SFB 701 “Spectral Structures and Topological Methods in Mathematics,” Bielefeld University, by University Roma Tre, INdAM, Polish Government grant N 6081/B/H03/2011/40, Fudan University, and by Universitat de Valencia, Grant UV-AE-09-5915. The authors wish to thank the anonymous referee for many useful comments which improved presentation very much.