# The Exponential Dichotomy under Discretization on General Approximation Scheme

Authors: Javier Pastor; Sergey Piskarev
Journal: Advances in Numerical Analysis
 (2011)

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## Abstract

This paper is devoted to the numerical analysis of abstract parabolic problemu′(t)=Au(t);
u0=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.


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## Body

## 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.LetB(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(1.1)‖(λI-A)-1‖B(E)≤M1+|λ|foranyReλ≥0.
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 Eα, can be constructed and the corresponding fractional power spaces too, that is, Eα:=D((-A)α) endowed with the graph norm ∥x∥Eα=∥(-A)αx∥E. Define the ball 𝒰Eα(0;ρ) with the center at 0 of radius ρ>0 in Eα space.To show how the dichotomy problems appear in numerical analysis, we are considering for example the semilinear equation in Banach spaceEα(1.2)u′(t)=Au(t)+f(u(t)),t≥0,u(0)=u0∈Eα,
where f(·):Eα⊆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 anyϵ>0, there is δ>0 such that ∥f′(w)-f′(z)∥B(Eα,E)≤ϵ as ∥w-z∥Eα≤δ for all w,z∈𝒰Eα(u*;ρ), where u* is a hyperbolic equilibrium point of (1.2).By means of the change of variablesv(·)=u(·)-u* in the problem (1.2), where u* is the hyperbolic equilibrium, we obtain the problem(1.3)v′(t)=(A+f′(u*))v(t)+f(v(t)+u*)-f(u*)-f′(u*)v(t),v(0)=u0-u*=v0.
Such problem can be written in the form(1.4)v′(t)=Au*v(t)+Fu*(v(t)),v(0)=v0,t≥0,
where Au*=A+f′(u*), Fu*(v(t))=f(v(t)+u*)-f(u*)-f′(u*)v(t). We note that from condition (F1) it follows that the function Fu*(v(t))=f(v(t)+u*)-f(u*)-f′(u*)v(t) for small ∥v0∥Eα is of order o(∥v(t)∥Eα). Since f′(u*)∈B(Eα,E),0≤α<1, the operator Au*=A+f′(u*) is the generator of an analytic C0-semigroup [3]. It can happen that the spectrum of operator Au* can be split into two parts σ+ and σ-.We assume that the partσ+ 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 Au* has the dichotomy property were studied say in [4–6]. In case of hyperbolic equilibrium point u*, there is no spectrum of Au* 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 Eα using the Riesz projection(1.5)P(σ+)∶=P(σ+,Au*)∶=12πi∫∂U(σ+)(ζI-Au*)-1dζ
defined by σ+. Due to this definition and analyticity of the C0-semigroup etAu*,t∈ℝ+, we have positive constants M1,β>0, such that(1.6)‖etAu*z‖Eα≤M1e-βt‖z‖Eα,t≥0,‖etAu*v‖Eα≤M1eβt‖v‖Eα,t≤0,
for all v∈P(σ+)Eα and z∈(I-P(σ+))Eα. Since Fu*(v(t))=o(∥v(t)∥Eα) 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*.Ifv0 is close to 0, that is, say v0∈𝒰Eα(0;ρ) with small ρ>0, then the mild solution v(t;v0) of (1.4) can stay in the ball 𝒰Eα(0;ρ), for some time. We denote now the maximal time of staying v(t;v0) in 𝒰Eα(0;ρ) by T=T(v0)=sup{t≥0:∥v(t;v0)∥Eα≤ρorv(t;v0)∈𝒰Eα(0;ρ)}. Now coming back to solution of (1.4) for any two v0,vT∈𝒰Eα(0;ρ) we consider the boundary value problem(1.7)v′(t)=Au*v(t)+Fu*(v(t)),0≤t≤T,(I-P(σ+))v(0)=(I-P(σ+))v0,P(σ+)v(T)=P(σ+)vT.
A mild solution of problem (1.7) as was shown in [7] satisfies the integral equation(1.8)v(t)=e(t-T)Au*P(σ+)vT+etAu*(I-P(σ+))v0+∫0te(t-s)Au*(I-P(σ+))Fu*(v(s))ds+∫tTe(t-s)Au*P(σ+)Fu*(v(s))ds,0≤t≤T.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

Let𝕋(r)={λ:λ∈C,|λ|=r},𝕋=𝕋(1).Definition 2.1.
AC0-semigroup etA,t≥0, defined on a Banach space E is called hyperbolic if σ(etA)∩𝕋=∅ for all t>0. The generator A is called hyperbolic if σ(A)∩iℝ=∅.Let us denote byΥ(ℝ;E) any of the spaces Lp(ℝ;E),1≤p<∞,C0(ℝ;E) or Stepanov’s space Sp(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 operator(2.1)L=-ddt+A:D(L)⊆Υ(R;E)⟶Υ(R;E),
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)Au(s)-∫ste(t-η)Ag(η)dη,s≤t,t∈ℝ, and ℒu(·)=g(·). Let us note [9] that the operator ℒ is the generator of C0-semigroup etℒ on a Banach space Υ(ℝ;E), which is defined for all v(·)∈Υ(ℝ;E) by formula(2.2)(etLv)(s)=etAv(s-t)foranys∈R,t≥0.Definition 2.2.
AC0-semigroup etA,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)
etAP=PetA for all t≥0,(ii)
the restrictionetA|ℛ(P),t≥0, is invertible on P(E) and(2.3)‖e-tAPx‖≤Me-βt‖Px‖,t≥0,x∈E,‖etA(I-P)x‖≤Me-βt‖(I-P)x‖,t≥0,x∈E.Theorem 2.3 (see [10]).
The operatorℒ in the Banach space Υ(ℝ;E) is invertible if and only if the condition
(2.4)σ(e1A)∩T=∅
holds. If condition (2.4) is satisfied, then
(2.5)(L-1f)(t)=∫-∞∞G(t-s)f(s)ds,t∈R,f(⋅)∈Υ(R;E),
where the Green’s function
(2.6)G(η)={-eηAP-,η≥0,eηAP+,η<0,(2.7)‖G(η)‖≤{M+e-γ+η,η≥0,M-eγ-η,η<0,
where
(2.8)M+=2Mæ(L)(1+12æ(L))2,M-=2Mæ(L)(1-12æ(L))2,γ+=ln⁡(1+12æ(L)),γ-=-ln⁡(1-12æ(L)),
and æ (ℒ)=1+C(Υ)(M+M2∥ℒ-1∥).We note that the constantC(Υ) is defined as C(Υ)=1 if Υ(ℝ;E)=L∞(ℝ,E) or Υ(ℝ;E)=C0(ℝ,E) and C(Υ)=21-1/p if Υ(ℝ;E)=Lp(ℝ,E) or Υ(ℝ;E)=Sp(ℝ,E),p∈[1,∞).Denote byΥ(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=e1A∈B(E), we define an operator ℬ:D(ℬ)⊆lp(ℤ;E)→lp(ℤ;E) by formula(2.9)(Bu)(k)=Bu(k-1),k∈Z,u(⋅)∈lp(Z;E).
Now we define an operator 𝒟=I-ℬ:D(𝒟)=D(ℬ)⊆lp(ℤ;E)→lp(ℤ;E) as(2.10)(Du)(k)=u(k)-Bu(k-1),u(⋅)∈D(B),k∈Z.Proposition 2.4 (see [10]).
Let an operatorℒ=-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
(2.11)‖D-1‖≤1+C(Υ)(M+M2‖L-1‖).
Conversely, if 𝒟 is invertible, then ℒ:D(ℒ)⊆Υ(ℝ;E)→Υ(ℝ;E) is invertible and
(2.12)‖L-1‖≤C(Υ)(M+M2‖D-1‖).Theorem 2.5 (see [11]).
The difference operator(2.13)(Du)(k)=u(k)-Bu(k-1),k∈N,u(⋅)∈lp(R;E),1≤p≤∞,
is invertible if and only if
(2.14)σ(B)∩T=∅.
If condition (2.14) is satisfied, then the inverse operator has the form
(2.15)(D-1v)(k)=Σm∈ZΓ(k-m)v(m),
where v(·)∈lp(ℤ;E),1≤p≤∞, and the function Γ(·):ℤ→B(E) is defined by
(2.16)Γ(k)={Bk(I-P),k≥0,-B0-kP,k≤-1,
where B0 is the restriction of B on ℛ(P).Definition 2.6.
The operatorB∈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)
BkP=PBk for all k∈ℕ,(ii)
∥Bk(I-P)∥≤Mrk for all k∈ℕ,(iii)
B̂:=B|ℛ(P):ℛ(P)↦ℛ(P) is a homeomorphism that satisfies ∥B̂-kP∥≤Mrk,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.
ForB∈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 letβ>1. For z∈C,z≠0, we have
(2.17)|z-z|z||=|1-|z||.
Hence, if |1-|z||β<1, the classical perturbation estimate shows that z∈ρ(B) and
(2.18)‖(zI-B)-1‖≤β1-β|1-|z||.
We define P as the Riesz projector defined by the formula
(2.19)I-P=12πi∫|z|=1(zI-B)-1dz.
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
(2.20)I-P=12πi∫|z|=r(zI-B)-1dz,if|1-r|<1β.
Now we claim
(2.21)Bk(I-P)=12πi∫|z|=rzk(zI-B)-1dzif|1-r|<1β.
For k=0 this follows from (2.20). If (2.21) holds for some k, then we obtain
(2.22)Bk+1P=12πi∫|z|=r(B-zI+zI)zk(zI-B)-1dz=12πi∫|z|=rzk+1(zI-B)-1dz-12πi∫|z|=rzkdz,
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(2.23)‖BkP‖≤12π2πrrkβ1-β(1-r)=βrk+11-β(1-r)fork≥0.
For the second dichotomy estimate, we use the resolvent equation
(2.24)(zI-B)-1=1zI+z-1B(zI-B)-1.
For |1-r|<1/β, (2.20) and (2.24) lead to
(2.25)I-P=12πi∫|z|=1(1zI-(zI-B)-1)dz=-12πi∫|z|=r1zB(zI-B)-1dz.
This shows that the following equality holds for k=1(2.26)I-P=-Bk12πi∫|z|=rz-k(zI-B)-1dz,fork≥1.
If (2.26) is known for some k, then use (2.24) and find
(2.27)I-P=-Bk12πi∫|z|=r(z-(k+1)+z-(k+1)B(zI-B)-1)dz=-Bk+112πi∫|z|=rz-(k+1)(zI-B)-1dz.
We apply (2.26) to x∈E, using that I-P commutes with B as well as the estimate (2.18)
(2.28)‖(I-P)x‖=‖12πi∫|z|=rz-k(zI-B)-1dzBk(I-P)x‖≤r-k2π2πrβ1-β(r-1)‖Bk(I-P)x‖.
Summarizing, we have shown for all k≥1,1≤r<1+1/β,u∈E(2.29)‖(I-P)u‖≤βr-k+11-β(r-1)‖Bk(I-P)u‖.
For k=1, this estimate shows that B̂=B|𝒩(P):𝒩(P)↦𝒩(P) is one-to-one with a bound for the inverse. To show that B̂ is onto, we take f∈𝒩(P) and set
(2.30)v=-12πi∫|z|=1z-1(zI-B)-1fdz.
From this equation, we have (I-P)v=0, and using (2.24), we find
(2.31)Bv=12πi∫|z|=1(z-1I-(zI-B)-1)fdz=(I-P)f=f.
Therefore, B̂ is a linear homeomorphism on 𝒩(P) satisfying
(2.32)‖B̂-k(I-P)u‖≤βr-k+11-β(r-1)‖(I-P)u‖for1≤r<1+1β.
This proves exponential dichotomy.
To arrive at the specific constants, we chooser=1-1/2β in (2.23) and obtain the bound (2β-1)rk. 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(2.33)(λI-BP)Pu=Pf,(λI-B̂)(I-P)u=(I-P)f,
which we can rewrite as
(2.34)(I-λ-1BP)Pu=λ-1Pf,(I-λB̂-1)(I-P)u=-B̂-1(I-P)f.
Both equations have a unique solution given by a geometric series
(2.35)Pu=∑k=0∞λ-(k+1)BkPf,(I-P)u=-∑k=0∞B̂-(k+1)λk(I-P)f.
The exponential dichotomy then implies the estimates
(2.36)‖Pu‖≤M1-r‖f‖,‖(I-P)u‖≤Mr1-r‖f‖.
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

LetEn and E be Banach spaces and {pn} a sequence of linear bounded operators pn:E→En,pn∈B(E,En),n∈ℕ={1,2,…}, with the property:(3.1)‖pnx‖En⟶‖x‖Easn→∞foranyx∈E.Definition 3.1.
The sequence of elements{xn},xn∈En,n∈ℕ, is said to be 𝒫-convergent to x∈E if and only if ∥xn-pnx∥En→0 as n→∞, and we write this xn→𝒫x.Definition 3.2.
The sequence of bounded linear operatorsBn∈B(En), 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 {xn},xn∈En,n∈ℕ, such that xn→𝒫x one has Bnxn→𝒫Bx. We write then Bn→𝒫𝒫B.In the case of unbounded operators as it occurs for general infinitesimal generators of PDE’s, the notion ofcompatibility turns out to be useful.Definition 3.3.
The sequence of closed linear operators{An},An∈𝒞(En),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 {xn},xn∈D(An)⊆En,n∈ℕ, such that xn→𝒫x and Anxn→𝒫Ax. We write that (An,A) are compatible.For analyticC0-semigroups, the following ABC Theorem holds.Theorem 3.4 (see [16]).
Let operatorsA and An generate analytic C0-semigroups. The following conditions (A) and (B1) are equivalent to condition (C1).(A)
Compatibility. There existsλ∈ρ(A)∩∩nρ(An) such that the resolvents converge (λI-An)-1→𝒫𝒫(λI-A)-1.(B1)
Stability. There are some constantsM1≥1 and ω1∈ℝ such that(3.2)‖(λIn-An)-1‖≤M1|λ-ω1|,Reλ>ω1,n∈N.(C1)
Convergence. For any finiteμ>0 and some 0<θ<π/2, one has(3.3)max⁡η∈Σ(θ,μ)‖eηAnun0-pneηAu0‖⟶0asn⟶∞wheneverun0→Pu0.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 stabilityΔs=Δs({An}),An∈𝒞(Bn), is defined as the set of all λ∈C such that λ∈ρ(An) for almost all n and such that the sequence {∥(λIn-An)-1∥}n∈ℕ is bounded. The region of convergence Δc=Δc({An}),An∈𝒞(En), is defined as the set of all λ∈C such that λ∈Δs({An}) and such that the sequence of operators {(λIn-An)-1}n∈ℕ is 𝒫𝒫-convergent to some operator S(λ)∈B(E).Definition 3.6.
A sequence of operators{Bn},Bn∈B(En),n∈ℕ, is said to be stably convergent to an operator B∈B(E) if and only if Bn→𝒫𝒫B and ∥Bn-1∥B(En)=O(1), n→∞. We will write this as: Bn→𝒫𝒫B stably.Definition 3.7.
A sequence of operators{Bn}, Bn∈B(En), is called regularly convergent to the operator B∈B(E) if and only if Bn→𝒫𝒫B and the following implication holds
(3.4)‖xn‖En=O(1),{Bnxn}isP-compact⟹{xn}isP-compact.
We write this as: Bn→𝒫𝒫B regularly.Theorem 3.8 (see [15]).
LetCn,Qn∈B(En), C,Q∈B(E) and ℛ(Q)=E. Assume also that Cn→𝒫𝒫C compactly and Qn→𝒫𝒫Q stably. Then, Qn+Cn→𝒫𝒫Q+C converge regularly.Theorem 3.9 (see [15]).
ForQn∈B(En) and Q∈B(E), the following conditions are equivalent:(i)
Qn→𝒫𝒫Q regularly, Qn are Fredholm operators of index 0 and 𝒩(Q)={0};(ii)
Qn→𝒫𝒫Q stably and ℛ(Q)=E;(iii)
Qn→𝒫𝒫Q stably and regularly;(iv)
if one of conditions (i)–(iii) holds, then there existQn-1∈B(En), Q-1∈B(E), and Qn-1→𝒫𝒫Q-1 regularly and stably.Theorem 3.10.
Let operatorsλIn-Bn∈B(En) be Fredholm operators with ind(λIn-Bn)=0 for any λ∈𝕋,n∈ℕ. Assume also that B∈B(E) has the property 𝕋∩σ(B)=∅ and λIn-Bn→𝒫𝒫λI-B regularly for any λ∈𝕋. Then λIn-Bn→𝒫𝒫λI-B stably for any λ∈𝕋 and supλ∈𝕋∥(λIn-Bn)-1∥<∞.Proof.
Assume that there are some sequences{λn},λn∈𝕋, and {xn},xn∈En, such that ∥xn∥=1 and (λnIn-Bn)xn→𝒫0 as n→∞. Since 𝕋 is compact, one can find ℕ′⊂ℕ such that λn→λ0∈𝕋 as n∈ℕ′. In the meantime, λ0In-Bn→𝒫𝒫λ0I-B regularly for such λ0∈𝕋, and (λ0In-Bn)xn=(λ0In-λnIn)xn+(λnIn-Bn)xn→𝒫0 for n∈ℕ′. Therefore, there is ℕ′′⊂ℕ′ such that xn→𝒫x0≠0 as n∈ℕ′′. But in such case (λ0In-Bn)xn→𝒫𝒫(λ0I-B)x0=0 as n∈ℕ′′, which contradicts our assumption 𝕋∩σ(B)=∅.Definition 3.11.
The operatorBn∈B(En) has a uniform exponential discrete dichotomy with data (M,r,Pn) if Pn∈B(En) is a projector in En and M,r are constants with 0≤r<1 such that the following properties hold(i)
BnkPn=PnBnk and ∥Pn∥≤constant for all k,n∈ℕ;(ii)
∥Bnk(In-Pn)∥≤Mrk for all k,n∈ℕ;(iii)
B̂n:=Bn|ℛ(Pn):ℛ(Pn)↦ℛ(Pn) is a homeomorphism that satisfies
(3.5)‖Bn̂-kPn‖≤Mrk,k,n∈N.Theorem 3.12.
The following conditions are equivalent:(i)
λIn-Bn→𝒫𝒫λI-B stably and λ∈ρ(B) for any λ∈𝕋;(ii)
operator𝒟=I-ℬ is invertible and 𝒟n→𝒫𝒫𝒟 stably, where (ℬu)(k)=Bu(k-1),k∈ℕ;(iii)
Bn→𝒫𝒫B and λI-B be invertible for any λ∈𝕋 and the operators Bn have an exponential discrete dichotomy with data (M,r,Pn) uniformly in n∈ℕ.Proof.
The equivalence (i)⇔(iii) follows from Theorem 2.7. Indeed, by formula (2.19), one gets from (i) that Pn→𝒫𝒫P and ∥Pn∥≤constant. By Theorem 3.10, one has supλ∈𝕋∥(λIn-Bn)-1∥<∞, and by Theorem 2.7 (ii) we have (iii). Conversely, from condition (iii), it follows by Theorem 2.7 (i) that λIn-Bn→𝒫𝒫λI-B stably for all λ∈𝕋.
To prove (ii)⇒(i), we note that (ii) means that for any u(·)∈lp(ℤ;E) one has Σk=-∞∞∥pnu(k)-Bnpnu(k-1)-pnu(k)+pnBu(k-1)∥Enp→0 as n→∞, that is, Bn→𝒫𝒫B. Assume now that In-Bn is not uniformly invertible in l∞(ℤ;En), that is, for some sequence ∥xn∥=1, one has (λ0In-Bn)xn→𝒫0 as n→∞ for some λ0=1∈𝕋. This means that for stationary sequence un(k)=xn,k∈ℤ,n∈𝕋, one has (𝒟nun)(k)=un(k)-Bnun(k-1)=xn-Bnxn→𝒫0 for any k∈ℕ and n→∞. But, 𝒟n→𝒫𝒫𝒟 stably, that is, ∥𝒟nun∥l∞(ℤ;En)≥γ∥un∥l∞(ℤ;En) which contradicts (λ0In-Bn)xn→𝒫0 as n→∞. Now we show that ℛ(λ0In-Bn)=En. For any yn∈En,∥yn∥=1, consider vn(k)=yn,k∈ℤ,n∈ℕ. The solution of 𝒟nun=vn is a sequence un(k), which is stationary too, that is, (λ0In-Bn)xn=yn, where xn=un(k),k∈ℤ,n∈ℕ. To show (i)⇒(ii), we note that ∥𝒟n∥B(lp(ℤ;En))≤constant,n∈ℕ. Now for any u(·)∈lp(ℤ;E) and for any ϵ>0, one can find K∈ℕ such that (Σk=K∞+Σk=-K-∞)∥u(k)∥p≤ϵ. In the meantime, Σk=-KK∥pnu(k)-Bnpnu(k-1)-pnu(k)+pnBu(k-1)∥Enp→0 as n→∞, since Bn→𝒫𝒫B. So we have 𝒟n→𝒫𝒫𝒟. The convergence 𝒟n-1→𝒫𝒫𝒟-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{Bn},Bn:En→En,n∈ℕ, converges compactly to an operator B:E→E if Bn→𝒫𝒫B and the following compactness condition holds
(3.6)‖xn‖En=O(1)⟹{Bnxn}isP-compact.Definition 3.14.
The region of compact convergence of resolvents,Δcc=Δcc(An,A), where An∈𝒞(En) and A∈𝒞(E) is defined as the set of all λ∈Δc∩ρ(A) such that (λIn-An)-1→𝒫𝒫(λI-A)-1 compactly.Proposition 3.15.
Assume that operatorsB,Bn are compact, 𝕋⊂ρ(B) and Δcc(Bn,B)≠∅. Then, λIn-Bn→𝒫𝒫λI-B stably for any λ∈𝕋 and supλ∈𝕋∥(λIn-Bn)-1∥<∞.Proof.
The proof follows from Theorems3.8, 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 pnα=(-An)-αpn(-A)α∈B(Eα,Enα) which satisfy property (3.1), but for the spaces Eα,Enα. The operators An 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 xn→𝒫αx if and only if ∥xn-pnαx∥Enα→0 as n→∞. One can see that ∥xn-pnαx∥Enα=∥(-An)αxn-pn(-A)αx∥ and ∥pnαx∥Enα=∥pn(-A)αx∥En→∥(-A)αx∥E=∥x∥Eα for any x∈D((-A)α) and n→∞.
Consider in Banach spacesEnα the family of parabolic problems(3.7)un′(t)=Anun(t)+fn(un(t)),t≥0,un(0)=un0∈Enα,
where un0→𝒫αu0, operators (An,A) are compatible, fn(·):Enα→En are globally bounded and globally Lipschitz continuous both uniformly in n∈ℕ, and continuously Fréchet differentiable.
Under the above assumptions, the mild solutionun(·) of (3.7) is defined for all t≥0 (see [1, 17]), and we define it by un(·)=Tn(·)un0:ℝ+→En. The nonlinear semigroup Tn(·) satisfies the variation of constants formula(3.8)Tn(t)un0=etAnun0+∫0te(t-s)Anfn(Tn(s)un0)ds,t≥0.
Recall that a hyperbolic equilibrium pointu* 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 Δcc≠∅, the operators An-1fn(·)→𝒫𝒫A-1f(·) compactly. From [15], it follows that equations un=-An-1fn(un) have solutions {un*},Anun*+fn(un*)=0, such that un*→𝒫u*.
Now, consider the problems (3.7) near hyperbolic equilibrium points un*. In this case one gets(3.9)vn′(t)=Aun*,nvn(t)+Fun*,n(vn(t)),vn(0)=vn0,t≥0,
where Aun*,n=An+fn′(un*),Fun*,n(vn(t))=fn(vn(t)+un*)-fn(un*)-fn′(un*)vn(t). From now on, we consider hyperbolic point u* and hyperbolic points un*→𝒫αu*.
We decomposeEnα using the projection(3.10)Pn(σn+)∶=Pn(σn+,Aun*,n)∶=12πi∫∂U(σn+)(ζIn-Aun*,n)-1dζ
defined by the set σn+, which is enclosed in a contour consisting of a part of iℝ and the contour from condition (B1) for operators Aun*,n.
There are some positiveM2,γ>0, because of analyticity of C0-semigroup etAun*,n and condition Δcc≠∅, which is applied to operator Aun*,n, such that [7, 18](3.11)‖etAun*,nzn‖Enα≤M2e-γt‖zn‖Enα,t≥0,‖etAun*,nvn‖Enα≤M2eγt‖vn‖Enα,t≤0,
for all vn∈Pn(σn+)Enα and zn∈(In-Pn(σn+))Enα. One has to note that the neighborhood of any part of iℝ does not intersect σ(Aun*,n). Moreover, the condition Δcc≠∅ implies that Pn(σn+)→𝒫𝒫P(σ+) compactly, and therefore, as was shown in [14], dimPn(σn+)=dimP(σ+) for n≥n0. One can consider for the T=T(vn0)=sup{t≥0:vn(t,vn0)∈𝒰Enα(0;ρ)} with vn0∈𝒰Enα(0;ρ) the problem
(3.12)vn′(t)=Aun*,nvn(t)+Fun*,n(vn(t)),0≤t≤T,(In-Pn(σn+))vn(0)=(In-Pn(σn+))vn0,Pn(σn+)vn(T)=Pn(σn+)vnT.
The mild solution of problem (3.12) is given by the formula (for 0≤t≤T)
(3.13)vn(t)=e(t-T)Aun*,nPn(σn+)vnT+etAun*,n(In-Pn(σn+))vn0+∫0te(t-s)Aun*,n(In-Pn(σn+))Fun*,n(vn(s))ds+∫tTe(t-s)Aun*,nPn(σn+)Fun*,n(vn(s))ds.
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 operatorsAn,A be generators of analytic C0-semigroups and let condition (B1) be satisfied. Assume also that the semigroup etAu* is hyperbolic, σ(Au*)∩{λ:Reλ≥0}=Pσ(Au*),dimP(σ+)<∞, and for ρ>0 such that {λ:-ρ≤Reλ≤ρ}⊂ρ(Au*), operators λIn-Aun*,n are Fredholm operators of ind 0, and operators λIn-Aun*,n,λI-Au* are regularly consistent for any Reλ≥-ρ. Then, Pn(σn+)→𝒫𝒫P(σ+) compactly and
(3.14)‖etAun*,n(In-Pn(σn+))‖En≤M2e-γt,t≥0,‖etAun*,nPn(σn+)‖En≤M2eγt,t≤0,
where γ>0.Proof.
The condition(B1) implies that (λIn-Aun*,n)-1→𝒫𝒫(λI-Au*)-1 as -ρ≤Reλ≤ρ for |λ| big enough. For the other -ρ≤Reλ≤ρ, the convergence (λIn-Aun*,n)-1→𝒫𝒫(λI-Au*)-1 follows from analogy of Theorem 3.9 for closed operators. Now the compact convergence Pn(σn+)→𝒫𝒫P(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, Theorem3.16 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 operatorsAn,A be generators of analytic C0-semigroups and let the condition (B1) be satisfied. Assume also that the C0-semigroup etAu* is hyperbolic, σ(Au*)∩{λ:Reλ≥0}=Pσ(Au*),dimP(σ+)<∞. Assume also that Δcc(An,A)≠∅ and resolvents of An,A are compact operators. Then, (3.14) holds.Proof.
We setBn=e1Aun*,n and B=e1Au* and apply Proposition 3.15. It is known [3] that Δcc≠∅ is equivalent to compact convergence Bn→𝒫𝒫B. Then, the condition (i) of Theorem 3.12 is satisfied and one gets discrete dichotomy for Bn. From the other side, since we have exactly the form Bn=e1Aun*,n, this means by Theorem 2.7 and Proposition 2.4 that operators ℒn=-d/dt+Aun*,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

LetΛ⊆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 pointof B if λI-B is a Fredholm operator of index zero and Q(λ) is of finite rank. Denote (3.15)W(λ,δ;Bn)=⋃|λn-λ|<δλn∈σ(Bn),W(λn,Bn).
It is clear that 𝒲(λ,δ;Bn)=Qn(λ)En, where(3.16)Qn(λ)=12πi∫|ζ-λ|=δ(ζIn-Bn)-1dζ.Definition 3.19.
The functionμ(·) is said to be the measure of noncompactness if for any bounded sequence {xn},xn∈En, one has
(3.17)μ({xn})=inf⁡{ϵ>0:∀N′⊆N,∃N′′⊆N′,x′∈Esuchthat‖xn-pnx′‖≤ϵ,n∈N′′}.Definition 3.20.
We say that the operatorsBn∈B(En) are jointly condensing with constant q>0 with respect to measure μ(·) if for any bounded sequence {xn},xn∈En, one has
(3.18)μ({Bnxn})≤qμ({xn}).It is known [19, page 82], that outside a closed disc of radius q centered at zero each operator Bn has only isolated points of spectrum, each of which can only be an eigenvalue of finite multiplicity.Proposition 3.21.
LetBn→𝒫𝒫B for Bn∈B(En),B∈B(E) and μ({Bnxn})≤qμ({xn}), for any bounded sequence {xn},xn∈En. 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 ∥(λIn-Bn)-1∥≤C,λ∈Ψ,n∈ℕ.Proof.
Any pointλ∈Ψ belongs to Pσ(Bn)∪ρ(Bn). This means that one gets for a sequence ∥xn∥=1,xn∈En, two cases: (λIn-Bn)xn=0 or ∥(λIn-Bn)xn∥≥γλ,n∥xn∥ with some γλ,n>0,λ∈Ψ. We are going to show that in reality we have ∥(λIn-Bn)xn∥≥γΨ∥xn∥,λ∈Ψ.
Assume in contradiction that there are sequences{λn},λn∈Ψ,{xn},∥xn∥=1, such that(3.19)(λnIn-Bn)xn→P0asn∈N.
Then, λn→λ0∈Ψ,n∈ℕ′⊆ℕ. One has for r̃=inf{|ξ|:ξ∈Ψ}(3.20)μ({xn})≤|λn|r̃μ({xn})≤μ({Bnxn})r̃≤qr̃μ({xn}),
which means because of q/r̃<1, that μ({xn})=0, that is, {xn} is 𝒫-compact. Now xn→𝒫x0,n∈ℕ′′⊆ℕ′ and Bnxn→𝒫Bx0,λnxn→𝒫λ0x0,n∈ℕ′′, that is, λ0x0=Bx0 with ∥x0∥=1, which contradicts our assumption σ(B)∩Ψ=∅. The proposition is proved.Proposition 3.22.
LetBn→𝒫𝒫B and μ({Bnxn})≤qμ({xn}) for any bounded sequence {xn},xn∈En. Assume that any λ0∈σ(B),|λ0|>q, is an isolated eigenvalue with the finite dimensional projector Q(λ0). Then, there are sequence {λn},λn∈σ(Bn), and sequence of projectors Qn(λ0)∈B(En) such that λn→λ0 and Qn(λ0)→𝒫𝒫Q(λ0) converge compactly.Proof.
Note first that forΓr={λ:|λ-λ0|=r}⊂C∖{λ:|λ|≤q}, where r can be taken small enough, we have by Proposition 3.21(3.21)(λIn-Bn)-1→PP(λI-B)-1asλ∈Γr,n∈N.
Therefore, Qn(λ0)→𝒫𝒫Q(λ0). To show compact convergence of these projectors one can note that
(3.22)μ({(λ0In-Bn)xn})≥|λ0|μ({xn})-μ({Bnxn})≥|λ0|μ({xn})-qμ({xn})≥γμ({xn}),
where γ=|λ0|-q>0. This means that
(3.23)μ({(λ0In-Bn)kxn})≥γkμ({xn})foranyk∈N.
By functional calculus
(3.24)(λ0In-Bn)kQn(λ0)xn=12πi∫Γr(λ0-λ)k(λIn-Bn)-1xndλ.
From this representation using (3.23), one has
(3.25)γkμ({Qn(λ0)xn})≤‖(λ0In-Bn)kQn(λ0)xn‖≤C2πrk‖xn‖.
It is clear that from r/γ<1 it follows that (r/γ)k→0 as k→∞. This means that Qn(λ0)→𝒫𝒫Q(λ0) compactly. The proposition is proved.Theorem 3.23.
Let the conditions(A) and (B1) be satisfied, and let the analytic C0-semigroup etA,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({Bnxn})≤qμ({xn}), for any bounded sequence {xn},xn∈En, with q<1, where Bn=e1An. Then, conclusions of Theorem 3.16, that is, (3.14), are hold and Pn(σn+)→𝒫𝒫P(σ+) converge compactly.Proof.
Because of the spectral mapping theorem, the spectrum of operatorB=e1A which is located outside of unit disc 𝕋 consists of finite number of points of the set Pσ(e1A)={ζ:ζ=eλ,λ∈Pσ(A)∩{ξ:Reξ≥0}}. Moreover, since q<1, for any Ψ which contains 𝕋 and Ψ⊂ρ(B),B=e1A, one has ∥(λIn-Bn)-1∥≤constant as λ∈Ψ by Proposition 3.22. Now from Theorem 3.12 it follows that Bn have discrete dichotomy. By Theorem 2.5, the operator 𝒟n is invertible, and by Proposition 2.4 and by Theorem 2.3 we get that the semigroups etAn,t∈ℝ+, have an exponential dichotomy uniformly in index n∈ℕ and (2.7), that is, (3.14) holds. By Theorem 3.9  Qn(λ0)→𝒫𝒫Q(λ0) compactly, which implies that conditions of Theorem 3.16 are satisfied. The theorem is proved.


### 3.4. Discretisation in Time Variable andΔcc≠∅

Consider now the discretization of the problem (3.9) in time by the following scheme:(3.26)Vn(t+τn)-Vn(t)τn=Aun*,nVn(t+τn)+Fun*,n(Vn(t)),t=kτn,
with initial data Vn(0)=vn0. The solution of such problem is given by formula(3.27)Vn(t+τn)=(In-τnAun*,n)-1Vn(t)+τn(In-τnAun*,n)-1Fun*,n(Vn(t))=(In-τnAun*,n)-kVn(0)+τn∑j=0k(In-τnAun*,n)-(k-j+1)Fun*,n(Vn(jτn)),t=kτn,
where Vn(0)=vn0.The problem (3.12) also can be discretized following (3.26) approach, so we have(3.28)Vn(t+τn)-Vn(t)τn=Aun*,nVn(t+τn)+Fun*,n(Vn(t)),t=kτn,(3.29)(In-Pn)Vn(0)=(In-Pn)vn0,PnVn(T)=PnvnT.
The solution of problem (3.28) could be obtained by using the formulas(3.30)(In-Pn)Vn(t+τn)=(In-τnAun*,n)-1(In-Pn)Vn(t)+τn(In-τnAun*,n)-1(In-Pn)Fun*,n(Vn(jτn)),(In-τnAun*,n)PnVn(t+τn)=PnVn(t)+τnPnFun*,n(Vn(kτn)),t=kτn.
So one has a representation of solution of problem (3.28) as(3.31)Vn(t;vn0,vnT)=(In-Pn)Vn(t)+PnVn(t)=(In-τnAun*,n)-k+1(In-Pn)vn0+τn∑j=0k-1(In-τnAun*,n)-(k-j)(In-Pn)Fun*,n(Vn(jτn))+(In-τnAun*,n)K-kPnvnT-τn∑j=kK-1(In-τnAun*,n)K-1-jPnFun*,n(Vn(jτn)),t=kτn.
From (3.31), it is clear that corresponding estimates on powers of operators (In-τnAun*,n)-k+1(In-Pn),(In-τnAun*,n)K-kPn play the main role in approximation of solutions of (1.4) in the vicinity of u*.Theorem 3.24.
Let operatorsAn,A be generators of analytic C0-semigroups and let condition (B1) be satisfied. Assume also that the analytic C0-semigroup etAu*,t∈ℝ+, is hyperbolic and for ρ>0 such that {λ:-ρ≤Reλ≤ρ}⊂ρ(A), operators λIn-Aun*,n are Fredholm operators of ind 0, and operators λIn-Aun*,n,λI-Au* are regularly consistent for any Reλ≥-ρ. Then, Pn(σ+)→𝒫𝒫P(σ+) compactly and
(3.32)‖(In-τnAun*,n)-knPn‖En≤M2e-γt,t≥0,‖(In-τnAun*,n)kn(In-Pn)‖En≤M2eγt,t≤0,
for some γ>0.Proof.
From Theorem3.16 it follows that the analytic C0-semigroups etAun*,n have dichotomy uniformly in n∈ℕ. Now one can see as in [20] that ∥(In-τnAun*,n)-kn-etAun*,n∥≤Mτneωt/t, where t=knτn=1. Using perturbation dichotomy theorem from [1, p. 254], one gets that (3.32) holds. The Theorem is proved.Theorem 3.25.
LetΔcc(An,A)≠∅ and resolvents of An,A be compact operators. Assume also that the analytic C0-semigroup etAu*,t∈ℝ+, is hyperbolic and condition (B1) is satisfied. Then,
(3.33)‖(In-τnAun*,n)-knPn‖En≤M2r[t],t=knτn≥0,‖(In-τnAun*,n)kn(In-Pn)‖En≤M2r-[t],-t=-knτn≤0,
where r<1.Proof.
Compact convergence of resolvents(λIn-An)-1→𝒫𝒫(λI-A)-1 implies [18] that
(3.34)(λIn-Aun*,n)-1→PP(λI-Au*)-1compactly.
We set Bn=(In-τnAun*,n)-kn,τnkn=1, and B=e1Au*. Then, Bn→𝒫𝒫B, since the operators Aun*,n,Au* are consistent. Note that by condition (B1) one has [21, 22] ∥τnknAun*,n(In-τnAun*,n)-kn∥B(En)≤constant, which implies Bn=Aun*,n-1τnknAun*,n(In-τnAun*,n)-kn→𝒫𝒫B compactly, since Aun*,n-1→𝒫𝒫Au*-1 compactly. Now by applying Proposition 3.15 one gets discrete dichotomy for Bn 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({Bnxn})≤qμ({xn}), for any bounded sequence {xn},xn∈En, with q<1 and Bn=eAun*,n. Assume also that the analytic C0-semigroup etAu*,t∈ℝ+, is hyperbolic.Then,
(3.35)‖(In-τnAun*,n)-knPn‖En≤M2r[t],t=knτn≥0,‖(In-τnAun*,n)kn(In-Pn)‖En≤M2r-[t],-t=-knτn≤0,
where r<1.Proof.
By Theorem3.23 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.


## 3.1. General Approximation Scheme

LetEn and E be Banach spaces and {pn} a sequence of linear bounded operators pn:E→En,pn∈B(E,En),n∈ℕ={1,2,…}, with the property:(3.1)‖pnx‖En⟶‖x‖Easn→∞foranyx∈E.Definition 3.1.
The sequence of elements{xn},xn∈En,n∈ℕ, is said to be 𝒫-convergent to x∈E if and only if ∥xn-pnx∥En→0 as n→∞, and we write this xn→𝒫x.Definition 3.2.
The sequence of bounded linear operatorsBn∈B(En), 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 {xn},xn∈En,n∈ℕ, such that xn→𝒫x one has Bnxn→𝒫Bx. We write then Bn→𝒫𝒫B.In the case of unbounded operators as it occurs for general infinitesimal generators of PDE’s, the notion ofcompatibility turns out to be useful.Definition 3.3.
The sequence of closed linear operators{An},An∈𝒞(En),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 {xn},xn∈D(An)⊆En,n∈ℕ, such that xn→𝒫x and Anxn→𝒫Ax. We write that (An,A) are compatible.For analyticC0-semigroups, the following ABC Theorem holds.Theorem 3.4 (see [16]).
Let operatorsA and An generate analytic C0-semigroups. The following conditions (A) and (B1) are equivalent to condition (C1).(A)
Compatibility. There existsλ∈ρ(A)∩∩nρ(An) such that the resolvents converge (λI-An)-1→𝒫𝒫(λI-A)-1.(B1)
Stability. There are some constantsM1≥1 and ω1∈ℝ such that(3.2)‖(λIn-An)-1‖≤M1|λ-ω1|,Reλ>ω1,n∈N.(C1)
Convergence. For any finiteμ>0 and some 0<θ<π/2, one has(3.3)max⁡η∈Σ(θ,μ)‖eηAnun0-pneηAu0‖⟶0asn⟶∞wheneverun0→Pu0.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 stabilityΔs=Δs({An}),An∈𝒞(Bn), is defined as the set of all λ∈C such that λ∈ρ(An) for almost all n and such that the sequence {∥(λIn-An)-1∥}n∈ℕ is bounded. The region of convergence Δc=Δc({An}),An∈𝒞(En), is defined as the set of all λ∈C such that λ∈Δs({An}) and such that the sequence of operators {(λIn-An)-1}n∈ℕ is 𝒫𝒫-convergent to some operator S(λ)∈B(E).Definition 3.6.
A sequence of operators{Bn},Bn∈B(En),n∈ℕ, is said to be stably convergent to an operator B∈B(E) if and only if Bn→𝒫𝒫B and ∥Bn-1∥B(En)=O(1), n→∞. We will write this as: Bn→𝒫𝒫B stably.Definition 3.7.
A sequence of operators{Bn}, Bn∈B(En), is called regularly convergent to the operator B∈B(E) if and only if Bn→𝒫𝒫B and the following implication holds
(3.4)‖xn‖En=O(1),{Bnxn}isP-compact⟹{xn}isP-compact.
We write this as: Bn→𝒫𝒫B regularly.Theorem 3.8 (see [15]).
LetCn,Qn∈B(En), C,Q∈B(E) and ℛ(Q)=E. Assume also that Cn→𝒫𝒫C compactly and Qn→𝒫𝒫Q stably. Then, Qn+Cn→𝒫𝒫Q+C converge regularly.Theorem 3.9 (see [15]).
ForQn∈B(En) and Q∈B(E), the following conditions are equivalent:(i)
Qn→𝒫𝒫Q regularly, Qn are Fredholm operators of index 0 and 𝒩(Q)={0};(ii)
Qn→𝒫𝒫Q stably and ℛ(Q)=E;(iii)
Qn→𝒫𝒫Q stably and regularly;(iv)
if one of conditions (i)–(iii) holds, then there existQn-1∈B(En), Q-1∈B(E), and Qn-1→𝒫𝒫Q-1 regularly and stably.Theorem 3.10.
Let operatorsλIn-Bn∈B(En) be Fredholm operators with ind(λIn-Bn)=0 for any λ∈𝕋,n∈ℕ. Assume also that B∈B(E) has the property 𝕋∩σ(B)=∅ and λIn-Bn→𝒫𝒫λI-B regularly for any λ∈𝕋. Then λIn-Bn→𝒫𝒫λI-B stably for any λ∈𝕋 and supλ∈𝕋∥(λIn-Bn)-1∥<∞.Proof.
Assume that there are some sequences{λn},λn∈𝕋, and {xn},xn∈En, such that ∥xn∥=1 and (λnIn-Bn)xn→𝒫0 as n→∞. Since 𝕋 is compact, one can find ℕ′⊂ℕ such that λn→λ0∈𝕋 as n∈ℕ′. In the meantime, λ0In-Bn→𝒫𝒫λ0I-B regularly for such λ0∈𝕋, and (λ0In-Bn)xn=(λ0In-λnIn)xn+(λnIn-Bn)xn→𝒫0 for n∈ℕ′. Therefore, there is ℕ′′⊂ℕ′ such that xn→𝒫x0≠0 as n∈ℕ′′. But in such case (λ0In-Bn)xn→𝒫𝒫(λ0I-B)x0=0 as n∈ℕ′′, which contradicts our assumption 𝕋∩σ(B)=∅.Definition 3.11.
The operatorBn∈B(En) has a uniform exponential discrete dichotomy with data (M,r,Pn) if Pn∈B(En) is a projector in En and M,r are constants with 0≤r<1 such that the following properties hold(i)
BnkPn=PnBnk and ∥Pn∥≤constant for all k,n∈ℕ;(ii)
∥Bnk(In-Pn)∥≤Mrk for all k,n∈ℕ;(iii)
B̂n:=Bn|ℛ(Pn):ℛ(Pn)↦ℛ(Pn) is a homeomorphism that satisfies
(3.5)‖Bn̂-kPn‖≤Mrk,k,n∈N.Theorem 3.12.
The following conditions are equivalent:(i)
λIn-Bn→𝒫𝒫λI-B stably and λ∈ρ(B) for any λ∈𝕋;(ii)
operator𝒟=I-ℬ is invertible and 𝒟n→𝒫𝒫𝒟 stably, where (ℬu)(k)=Bu(k-1),k∈ℕ;(iii)
Bn→𝒫𝒫B and λI-B be invertible for any λ∈𝕋 and the operators Bn have an exponential discrete dichotomy with data (M,r,Pn) uniformly in n∈ℕ.Proof.
The equivalence (i)⇔(iii) follows from Theorem 2.7. Indeed, by formula (2.19), one gets from (i) that Pn→𝒫𝒫P and ∥Pn∥≤constant. By Theorem 3.10, one has supλ∈𝕋∥(λIn-Bn)-1∥<∞, and by Theorem 2.7 (ii) we have (iii). Conversely, from condition (iii), it follows by Theorem 2.7 (i) that λIn-Bn→𝒫𝒫λI-B stably for all λ∈𝕋.
To prove (ii)⇒(i), we note that (ii) means that for any u(·)∈lp(ℤ;E) one has Σk=-∞∞∥pnu(k)-Bnpnu(k-1)-pnu(k)+pnBu(k-1)∥Enp→0 as n→∞, that is, Bn→𝒫𝒫B. Assume now that In-Bn is not uniformly invertible in l∞(ℤ;En), that is, for some sequence ∥xn∥=1, one has (λ0In-Bn)xn→𝒫0 as n→∞ for some λ0=1∈𝕋. This means that for stationary sequence un(k)=xn,k∈ℤ,n∈𝕋, one has (𝒟nun)(k)=un(k)-Bnun(k-1)=xn-Bnxn→𝒫0 for any k∈ℕ and n→∞. But, 𝒟n→𝒫𝒫𝒟 stably, that is, ∥𝒟nun∥l∞(ℤ;En)≥γ∥un∥l∞(ℤ;En) which contradicts (λ0In-Bn)xn→𝒫0 as n→∞. Now we show that ℛ(λ0In-Bn)=En. For any yn∈En,∥yn∥=1, consider vn(k)=yn,k∈ℤ,n∈ℕ. The solution of 𝒟nun=vn is a sequence un(k), which is stationary too, that is, (λ0In-Bn)xn=yn, where xn=un(k),k∈ℤ,n∈ℕ. To show (i)⇒(ii), we note that ∥𝒟n∥B(lp(ℤ;En))≤constant,n∈ℕ. Now for any u(·)∈lp(ℤ;E) and for any ϵ>0, one can find K∈ℕ such that (Σk=K∞+Σk=-K-∞)∥u(k)∥p≤ϵ. In the meantime, Σk=-KK∥pnu(k)-Bnpnu(k-1)-pnu(k)+pnBu(k-1)∥Enp→0 as n→∞, since Bn→𝒫𝒫B. So we have 𝒟n→𝒫𝒫𝒟. The convergence 𝒟n-1→𝒫𝒫𝒟-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{Bn},Bn:En→En,n∈ℕ, converges compactly to an operator B:E→E if Bn→𝒫𝒫B and the following compactness condition holds
(3.6)‖xn‖En=O(1)⟹{Bnxn}isP-compact.Definition 3.14.
The region of compact convergence of resolvents,Δcc=Δcc(An,A), where An∈𝒞(En) and A∈𝒞(E) is defined as the set of all λ∈Δc∩ρ(A) such that (λIn-An)-1→𝒫𝒫(λI-A)-1 compactly.Proposition 3.15.
Assume that operatorsB,Bn are compact, 𝕋⊂ρ(B) and Δcc(Bn,B)≠∅. Then, λIn-Bn→𝒫𝒫λI-B stably for any λ∈𝕋 and supλ∈𝕋∥(λIn-Bn)-1∥<∞.Proof.
The proof follows from Theorems3.8, 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 pnα=(-An)-αpn(-A)α∈B(Eα,Enα) which satisfy property (3.1), but for the spaces Eα,Enα. The operators An 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 xn→𝒫αx if and only if ∥xn-pnαx∥Enα→0 as n→∞. One can see that ∥xn-pnαx∥Enα=∥(-An)αxn-pn(-A)αx∥ and ∥pnαx∥Enα=∥pn(-A)αx∥En→∥(-A)αx∥E=∥x∥Eα for any x∈D((-A)α) and n→∞.
Consider in Banach spacesEnα the family of parabolic problems(3.7)un′(t)=Anun(t)+fn(un(t)),t≥0,un(0)=un0∈Enα,
where un0→𝒫αu0, operators (An,A) are compatible, fn(·):Enα→En are globally bounded and globally Lipschitz continuous both uniformly in n∈ℕ, and continuously Fréchet differentiable.
Under the above assumptions, the mild solutionun(·) of (3.7) is defined for all t≥0 (see [1, 17]), and we define it by un(·)=Tn(·)un0:ℝ+→En. The nonlinear semigroup Tn(·) satisfies the variation of constants formula(3.8)Tn(t)un0=etAnun0+∫0te(t-s)Anfn(Tn(s)un0)ds,t≥0.
Recall that a hyperbolic equilibrium pointu* 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 Δcc≠∅, the operators An-1fn(·)→𝒫𝒫A-1f(·) compactly. From [15], it follows that equations un=-An-1fn(un) have solutions {un*},Anun*+fn(un*)=0, such that un*→𝒫u*.
Now, consider the problems (3.7) near hyperbolic equilibrium points un*. In this case one gets(3.9)vn′(t)=Aun*,nvn(t)+Fun*,n(vn(t)),vn(0)=vn0,t≥0,
where Aun*,n=An+fn′(un*),Fun*,n(vn(t))=fn(vn(t)+un*)-fn(un*)-fn′(un*)vn(t). From now on, we consider hyperbolic point u* and hyperbolic points un*→𝒫αu*.
We decomposeEnα using the projection(3.10)Pn(σn+)∶=Pn(σn+,Aun*,n)∶=12πi∫∂U(σn+)(ζIn-Aun*,n)-1dζ
defined by the set σn+, which is enclosed in a contour consisting of a part of iℝ and the contour from condition (B1) for operators Aun*,n.
There are some positiveM2,γ>0, because of analyticity of C0-semigroup etAun*,n and condition Δcc≠∅, which is applied to operator Aun*,n, such that [7, 18](3.11)‖etAun*,nzn‖Enα≤M2e-γt‖zn‖Enα,t≥0,‖etAun*,nvn‖Enα≤M2eγt‖vn‖Enα,t≤0,
for all vn∈Pn(σn+)Enα and zn∈(In-Pn(σn+))Enα. One has to note that the neighborhood of any part of iℝ does not intersect σ(Aun*,n). Moreover, the condition Δcc≠∅ implies that Pn(σn+)→𝒫𝒫P(σ+) compactly, and therefore, as was shown in [14], dimPn(σn+)=dimP(σ+) for n≥n0. One can consider for the T=T(vn0)=sup{t≥0:vn(t,vn0)∈𝒰Enα(0;ρ)} with vn0∈𝒰Enα(0;ρ) the problem
(3.12)vn′(t)=Aun*,nvn(t)+Fun*,n(vn(t)),0≤t≤T,(In-Pn(σn+))vn(0)=(In-Pn(σn+))vn0,Pn(σn+)vn(T)=Pn(σn+)vnT.
The mild solution of problem (3.12) is given by the formula (for 0≤t≤T)
(3.13)vn(t)=e(t-T)Aun*,nPn(σn+)vnT+etAun*,n(In-Pn(σn+))vn0+∫0te(t-s)Aun*,n(In-Pn(σn+))Fun*,n(vn(s))ds+∫tTe(t-s)Aun*,nPn(σn+)Fun*,n(vn(s))ds.
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 operatorsAn,A be generators of analytic C0-semigroups and let condition (B1) be satisfied. Assume also that the semigroup etAu* is hyperbolic, σ(Au*)∩{λ:Reλ≥0}=Pσ(Au*),dimP(σ+)<∞, and for ρ>0 such that {λ:-ρ≤Reλ≤ρ}⊂ρ(Au*), operators λIn-Aun*,n are Fredholm operators of ind 0, and operators λIn-Aun*,n,λI-Au* are regularly consistent for any Reλ≥-ρ. Then, Pn(σn+)→𝒫𝒫P(σ+) compactly and
(3.14)‖etAun*,n(In-Pn(σn+))‖En≤M2e-γt,t≥0,‖etAun*,nPn(σn+)‖En≤M2eγt,t≤0,
where γ>0.Proof.
The condition(B1) implies that (λIn-Aun*,n)-1→𝒫𝒫(λI-Au*)-1 as -ρ≤Reλ≤ρ for |λ| big enough. For the other -ρ≤Reλ≤ρ, the convergence (λIn-Aun*,n)-1→𝒫𝒫(λI-Au*)-1 follows from analogy of Theorem 3.9 for closed operators. Now the compact convergence Pn(σn+)→𝒫𝒫P(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, Theorem3.16 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 operatorsAn,A be generators of analytic C0-semigroups and let the condition (B1) be satisfied. Assume also that the C0-semigroup etAu* is hyperbolic, σ(Au*)∩{λ:Reλ≥0}=Pσ(Au*),dimP(σ+)<∞. Assume also that Δcc(An,A)≠∅ and resolvents of An,A are compact operators. Then, (3.14) holds.Proof.
We setBn=e1Aun*,n and B=e1Au* and apply Proposition 3.15. It is known [3] that Δcc≠∅ is equivalent to compact convergence Bn→𝒫𝒫B. Then, the condition (i) of Theorem 3.12 is satisfied and one gets discrete dichotomy for Bn. From the other side, since we have exactly the form Bn=e1Aun*,n, this means by Theorem 2.7 and Proposition 2.4 that operators ℒn=-d/dt+Aun*,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

LetΛ⊆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 pointof B if λI-B is a Fredholm operator of index zero and Q(λ) is of finite rank. Denote (3.15)W(λ,δ;Bn)=⋃|λn-λ|<δλn∈σ(Bn),W(λn,Bn).
It is clear that 𝒲(λ,δ;Bn)=Qn(λ)En, where(3.16)Qn(λ)=12πi∫|ζ-λ|=δ(ζIn-Bn)-1dζ.Definition 3.19.
The functionμ(·) is said to be the measure of noncompactness if for any bounded sequence {xn},xn∈En, one has
(3.17)μ({xn})=inf⁡{ϵ>0:∀N′⊆N,∃N′′⊆N′,x′∈Esuchthat‖xn-pnx′‖≤ϵ,n∈N′′}.Definition 3.20.
We say that the operatorsBn∈B(En) are jointly condensing with constant q>0 with respect to measure μ(·) if for any bounded sequence {xn},xn∈En, one has
(3.18)μ({Bnxn})≤qμ({xn}).It is known [19, page 82], that outside a closed disc of radius q centered at zero each operator Bn has only isolated points of spectrum, each of which can only be an eigenvalue of finite multiplicity.Proposition 3.21.
LetBn→𝒫𝒫B for Bn∈B(En),B∈B(E) and μ({Bnxn})≤qμ({xn}), for any bounded sequence {xn},xn∈En. 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 ∥(λIn-Bn)-1∥≤C,λ∈Ψ,n∈ℕ.Proof.
Any pointλ∈Ψ belongs to Pσ(Bn)∪ρ(Bn). This means that one gets for a sequence ∥xn∥=1,xn∈En, two cases: (λIn-Bn)xn=0 or ∥(λIn-Bn)xn∥≥γλ,n∥xn∥ with some γλ,n>0,λ∈Ψ. We are going to show that in reality we have ∥(λIn-Bn)xn∥≥γΨ∥xn∥,λ∈Ψ.
Assume in contradiction that there are sequences{λn},λn∈Ψ,{xn},∥xn∥=1, such that(3.19)(λnIn-Bn)xn→P0asn∈N.
Then, λn→λ0∈Ψ,n∈ℕ′⊆ℕ. One has for r̃=inf{|ξ|:ξ∈Ψ}(3.20)μ({xn})≤|λn|r̃μ({xn})≤μ({Bnxn})r̃≤qr̃μ({xn}),
which means because of q/r̃<1, that μ({xn})=0, that is, {xn} is 𝒫-compact. Now xn→𝒫x0,n∈ℕ′′⊆ℕ′ and Bnxn→𝒫Bx0,λnxn→𝒫λ0x0,n∈ℕ′′, that is, λ0x0=Bx0 with ∥x0∥=1, which contradicts our assumption σ(B)∩Ψ=∅. The proposition is proved.Proposition 3.22.
LetBn→𝒫𝒫B and μ({Bnxn})≤qμ({xn}) for any bounded sequence {xn},xn∈En. Assume that any λ0∈σ(B),|λ0|>q, is an isolated eigenvalue with the finite dimensional projector Q(λ0). Then, there are sequence {λn},λn∈σ(Bn), and sequence of projectors Qn(λ0)∈B(En) such that λn→λ0 and Qn(λ0)→𝒫𝒫Q(λ0) converge compactly.Proof.
Note first that forΓr={λ:|λ-λ0|=r}⊂C∖{λ:|λ|≤q}, where r can be taken small enough, we have by Proposition 3.21(3.21)(λIn-Bn)-1→PP(λI-B)-1asλ∈Γr,n∈N.
Therefore, Qn(λ0)→𝒫𝒫Q(λ0). To show compact convergence of these projectors one can note that
(3.22)μ({(λ0In-Bn)xn})≥|λ0|μ({xn})-μ({Bnxn})≥|λ0|μ({xn})-qμ({xn})≥γμ({xn}),
where γ=|λ0|-q>0. This means that
(3.23)μ({(λ0In-Bn)kxn})≥γkμ({xn})foranyk∈N.
By functional calculus
(3.24)(λ0In-Bn)kQn(λ0)xn=12πi∫Γr(λ0-λ)k(λIn-Bn)-1xndλ.
From this representation using (3.23), one has
(3.25)γkμ({Qn(λ0)xn})≤‖(λ0In-Bn)kQn(λ0)xn‖≤C2πrk‖xn‖.
It is clear that from r/γ<1 it follows that (r/γ)k→0 as k→∞. This means that Qn(λ0)→𝒫𝒫Q(λ0) compactly. The proposition is proved.Theorem 3.23.
Let the conditions(A) and (B1) be satisfied, and let the analytic C0-semigroup etA,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({Bnxn})≤qμ({xn}), for any bounded sequence {xn},xn∈En, with q<1, where Bn=e1An. Then, conclusions of Theorem 3.16, that is, (3.14), are hold and Pn(σn+)→𝒫𝒫P(σ+) converge compactly.Proof.
Because of the spectral mapping theorem, the spectrum of operatorB=e1A which is located outside of unit disc 𝕋 consists of finite number of points of the set Pσ(e1A)={ζ:ζ=eλ,λ∈Pσ(A)∩{ξ:Reξ≥0}}. Moreover, since q<1, for any Ψ which contains 𝕋 and Ψ⊂ρ(B),B=e1A, one has ∥(λIn-Bn)-1∥≤constant as λ∈Ψ by Proposition 3.22. Now from Theorem 3.12 it follows that Bn have discrete dichotomy. By Theorem 2.5, the operator 𝒟n is invertible, and by Proposition 2.4 and by Theorem 2.3 we get that the semigroups etAn,t∈ℝ+, have an exponential dichotomy uniformly in index n∈ℕ and (2.7), that is, (3.14) holds. By Theorem 3.9  Qn(λ0)→𝒫𝒫Q(λ0) compactly, which implies that conditions of Theorem 3.16 are satisfied. The theorem is proved.


## 3.4. Discretisation in Time Variable andΔcc≠∅

Consider now the discretization of the problem (3.9) in time by the following scheme:(3.26)Vn(t+τn)-Vn(t)τn=Aun*,nVn(t+τn)+Fun*,n(Vn(t)),t=kτn,
with initial data Vn(0)=vn0. The solution of such problem is given by formula(3.27)Vn(t+τn)=(In-τnAun*,n)-1Vn(t)+τn(In-τnAun*,n)-1Fun*,n(Vn(t))=(In-τnAun*,n)-kVn(0)+τn∑j=0k(In-τnAun*,n)-(k-j+1)Fun*,n(Vn(jτn)),t=kτn,
where Vn(0)=vn0.The problem (3.12) also can be discretized following (3.26) approach, so we have(3.28)Vn(t+τn)-Vn(t)τn=Aun*,nVn(t+τn)+Fun*,n(Vn(t)),t=kτn,(3.29)(In-Pn)Vn(0)=(In-Pn)vn0,PnVn(T)=PnvnT.
The solution of problem (3.28) could be obtained by using the formulas(3.30)(In-Pn)Vn(t+τn)=(In-τnAun*,n)-1(In-Pn)Vn(t)+τn(In-τnAun*,n)-1(In-Pn)Fun*,n(Vn(jτn)),(In-τnAun*,n)PnVn(t+τn)=PnVn(t)+τnPnFun*,n(Vn(kτn)),t=kτn.
So one has a representation of solution of problem (3.28) as(3.31)Vn(t;vn0,vnT)=(In-Pn)Vn(t)+PnVn(t)=(In-τnAun*,n)-k+1(In-Pn)vn0+τn∑j=0k-1(In-τnAun*,n)-(k-j)(In-Pn)Fun*,n(Vn(jτn))+(In-τnAun*,n)K-kPnvnT-τn∑j=kK-1(In-τnAun*,n)K-1-jPnFun*,n(Vn(jτn)),t=kτn.
From (3.31), it is clear that corresponding estimates on powers of operators (In-τnAun*,n)-k+1(In-Pn),(In-τnAun*,n)K-kPn play the main role in approximation of solutions of (1.4) in the vicinity of u*.Theorem 3.24.
Let operatorsAn,A be generators of analytic C0-semigroups and let condition (B1) be satisfied. Assume also that the analytic C0-semigroup etAu*,t∈ℝ+, is hyperbolic and for ρ>0 such that {λ:-ρ≤Reλ≤ρ}⊂ρ(A), operators λIn-Aun*,n are Fredholm operators of ind 0, and operators λIn-Aun*,n,λI-Au* are regularly consistent for any Reλ≥-ρ. Then, Pn(σ+)→𝒫𝒫P(σ+) compactly and
(3.32)‖(In-τnAun*,n)-knPn‖En≤M2e-γt,t≥0,‖(In-τnAun*,n)kn(In-Pn)‖En≤M2eγt,t≤0,
for some γ>0.Proof.
From Theorem3.16 it follows that the analytic C0-semigroups etAun*,n have dichotomy uniformly in n∈ℕ. Now one can see as in [20] that ∥(In-τnAun*,n)-kn-etAun*,n∥≤Mτneωt/t, where t=knτn=1. Using perturbation dichotomy theorem from [1, p. 254], one gets that (3.32) holds. The Theorem is proved.Theorem 3.25.
LetΔcc(An,A)≠∅ and resolvents of An,A be compact operators. Assume also that the analytic C0-semigroup etAu*,t∈ℝ+, is hyperbolic and condition (B1) is satisfied. Then,
(3.33)‖(In-τnAun*,n)-knPn‖En≤M2r[t],t=knτn≥0,‖(In-τnAun*,n)kn(In-Pn)‖En≤M2r-[t],-t=-knτn≤0,
where r<1.Proof.
Compact convergence of resolvents(λIn-An)-1→𝒫𝒫(λI-A)-1 implies [18] that
(3.34)(λIn-Aun*,n)-1→PP(λI-Au*)-1compactly.
We set Bn=(In-τnAun*,n)-kn,τnkn=1, and B=e1Au*. Then, Bn→𝒫𝒫B, since the operators Aun*,n,Au* are consistent. Note that by condition (B1) one has [21, 22] ∥τnknAun*,n(In-τnAun*,n)-kn∥B(En)≤constant, which implies Bn=Aun*,n-1τnknAun*,n(In-τnAun*,n)-kn→𝒫𝒫B compactly, since Aun*,n-1→𝒫𝒫Au*-1 compactly. Now by applying Proposition 3.15 one gets discrete dichotomy for Bn 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({Bnxn})≤qμ({xn}), for any bounded sequence {xn},xn∈En, with q<1 and Bn=eAun*,n. Assume also that the analytic C0-semigroup etAu*,t∈ℝ+, is hyperbolic.Then,
(3.35)‖(In-τnAun*,n)-knPn‖En≤M2r[t],t=knτn≥0,‖(In-τnAun*,n)kn(In-Pn)‖En≤M2r-[t],-t=-knτn≤0,
where r<1.Proof.
By Theorem3.23 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 conditionμ(Bnxn)≤qμ(xn) with q<1 in Theorems 3.23 and 3.26 can be checked for instance in case of compact convergence of operators An-1fn′(un*)→𝒫𝒫A-1f′(u*). We present here an example where the analogy of such condition is naturally satisfied.Example 4.1.
Consider inL2(ℝ) an operator
(4.1)(Av)(x)=v′′(x)+av′(x)+bv(x),x∈(-∞,∞).
Since we took E=L2(R), we can take (pnv)(x)=(1/h)∫-h/2h/2v(x+y)dy, and the main condition ∥pnv∥Lh2(Z)→∥v∥L2(R) is satisfied [15].
As in Section 5.4 of [1], one can see that σess(-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,(4.2)Anvn(x)=vn(x+h)-2vn(x)+vn(x-h)h2+bvn(x),
that is, ωess(An)≤ω1<0 uniformly in h>0. Moreover, it is easy to see that
(4.3)‖etAn‖≤Meω2t,t≥0,withω2<0,
that is,
(4.4)μ({etAnxn})≤γμ({xn})withγ<1forsomet=t0>0.
Now to get to the range of Theorem 3.23, let us denote B=et0A,Bn=et0An. For analytic C0-semigroups, the spectrums of operators A and B are strictly related, which is also concerned to point spectrum Pσ(B)=et0Pσ(A). This means that operators An have for almost all n the spectrums σ(An)∩{λ:Reλ>0} which approximate the spectrum σ(A)∩{λ:Reλ>0}.
Let us now consider inL2(ℝ) the case of perturbed operator with smooth function b(x)(4.5)Ãv(x)=v′′(x)+b(x)v(x)
and its approximation, say like
(4.6)(Ãnvn)(x)=vn(x+h)-2vn(x)+vn(x-h)h2+b(x)vn(x),
with condition b(x)→b as x→±∞ for simplicity. The operator ((Ã-A)v)(x)=(b(x)-b)v(x) is an additive perturbation. We assume that C0-semigroup etÃ,t∈ℝ+, is hyperbolic. The perturbation Ã-A is a relatively compact perturbation like in [23]. The same must happen to An because of Ãn=An+(Ãn-An) and
(4.7)et0Ãn=et0An+∫0t0Anαe(t0-s)AnAn-α(Ãn-An)esÃnds.
The crucial point is that such perturbation gives us from (4.4) the estimate
(4.8)μ({etÃnxn})≤γμ({xn})withγ<1forsomet=t0>0,
since the integral part in (4.7) could be estimated by any small ϵ>0 as μ(∫0t0-ϵ+∫t0-ϵt0)≤cϵ1-α.
Then any point of spectrum ofÃ which is located to the right of b belongs to Pσ(Ã) and it is of finite dimensional generalized eigenspace. The same is true for Ãn with Bn=etÃn, since we have (4.8). Using property (4.8), we get from Theorems 3.8 and 3.9 the regular consistence of operators λIn-Ãn,λI-Ã for any λ∈iℝ and any Reλ>b.
If as beforeP is a dichotomy projector, then one has dimP<∞ and PÃ=ÃP. We can also state that Pn→P compactly by Proposition 3.22. This means that say from dichotomy of Ã we get dichotomy for Ãn uniformly in n by Theorem 3.16.
The similar situation for concrete differential operator was considered in [24].



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*Source: 582740-2012-02-14.xml*