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Advances in Numerical Analysis
VolumeΒ 2011Β (2011), Article IDΒ 582740, 22 pages
Research Article

The Exponential Dichotomy under Discretization on General Approximation Scheme

1Department de Mathemàtica Aplicada, University de València, 46100 Burjassot, Valencia, Spain
2Scientific Research Computer Center, Moscow State University, Vorobjevy Gory, Moscow 119899, Russia

Received 31 August 2011; Accepted 15 November 2011

Academic Editor: William J.Β Layton

Copyright Β© 2011 Javier Pastor and Sergey Piskarev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper is devoted to the numerical analysis of abstract parabolic problem π‘’ξ…ž(𝑑)=𝐴𝑒(𝑑); 𝑒(0)=𝑒0, 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 𝐡(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 π’ž(𝐸). For π΅βˆˆπ’ž(𝐸), let 𝜎(𝐡) be its spectrum and 𝜌(𝐡), its resolvent set. In the following let 𝐴∢𝐷(𝐴)βŠ†πΈβ†’πΈ be a closed linear operator, such thatβ€–β€–(πœ†πΌβˆ’π΄)βˆ’1‖‖𝐡(𝐸)≀𝑀||πœ†||1+foranyReπœ†β‰₯0.(1.1) Under condition (1.1), the spectrum of 𝐴 is on the left: sup{Reπœ†βˆΆπœ†βˆˆπœŽ(𝐴)}<0, so the fractional power operators (βˆ’π΄)𝛼,π›Όβˆˆβ„+, (see [1, 2]) associated to 𝐴 and 𝐸𝛼, can be constructed and the corresponding fractional power spaces too, that is, πΈπ›ΌβˆΆ=𝐷((βˆ’π΄)𝛼) endowed with the graph norm β€–π‘₯‖𝐸𝛼=β€–(βˆ’π΄)𝛼π‘₯‖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 space 𝐸𝛼𝑒′(𝑒𝑑)=𝐴𝑒(𝑑)+𝑓(𝑒(𝑑)),𝑑β‰₯0,(0)=𝑒0βˆˆπΈπ›Ό,(1.2) where 𝑓(β‹…)βˆΆπΈπ›ΌβŠ†πΈβ†’πΈ,0≀𝛼<1, is assumed to be continuous, bounded, and continuously FrΓ©chet differentiable function. More precisely, we assume that the following condition holds.(F1) For any πœ–>0, there is 𝛿>0 such that ‖𝑓′(𝑀)βˆ’π‘“β€²(𝑧)‖𝐡(𝐸𝛼,𝐸)β‰€πœ– as β€–π‘€βˆ’π‘§β€–πΈπ›Όβ‰€π›Ώ for all 𝑀,π‘§βˆˆπ’°πΈπ›Ό(π‘’βˆ—;𝜌), where π‘’βˆ— is a hyperbolic equilibrium point of (1.2).

By means of the change of variables 𝑣(β‹…)=𝑒(β‹…)βˆ’π‘’βˆ— in the problem (1.2), where π‘’βˆ— is the hyperbolic equilibrium, we obtain the problemπ‘£ξ…žξ€·(𝑑)=𝐴+π‘“ξ…žξ€·π‘’βˆ—π‘£ξ€·π‘£ξ€Έξ€Έ(𝑑)+𝑓(𝑑)+π‘’βˆ—ξ€Έξ€·π‘’βˆ’π‘“βˆ—ξ€Έβˆ’π‘“ξ…žξ€·π‘’βˆ—ξ€Έπ‘£(𝑑),𝑣(0)=𝑒0βˆ’π‘’βˆ—=𝑣0.(1.3) Such problem can be written in the formπ‘£ξ…ž(𝑑)=π΄π‘’βˆ—π‘£(𝑑)+πΉπ‘’βˆ—(𝑣(𝑑)),𝑣(0)=𝑣0,𝑑β‰₯0,(1.4) where π΄π‘’βˆ—=𝐴+π‘“ξ…ž(π‘’βˆ—), πΉπ‘’βˆ—(𝑣(𝑑))=𝑓(𝑣(𝑑)+π‘’βˆ—)βˆ’π‘“(π‘’βˆ—)βˆ’π‘“ξ…ž(π‘’βˆ—)𝑣(𝑑). We note that from condition (F1) it follows that the function πΉπ‘’βˆ—(𝑣(𝑑))=𝑓(𝑣(𝑑)+π‘’βˆ—)βˆ’π‘“(π‘’βˆ—)βˆ’π‘“ξ…ž(π‘’βˆ—)𝑣(𝑑) for small ‖𝑣0‖𝐸𝛼 is of order π‘œ(‖𝑣(𝑑)‖𝐸𝛼). Since 𝑓′(π‘’βˆ—)∈𝐡(𝐸𝛼,𝐸),0≀𝛼<1, the operator π΄π‘’βˆ—=𝐴+𝑓′(π‘’βˆ—) is the generator of an analytic 𝐢0-semigroup [3]. It can happen that the spectrum of operator π΄π‘’βˆ— can be split into two parts 𝜎+ and πœŽβˆ’.

We assume that the part 𝜎+ of the spectrum of operator 𝐴+𝑓′(π‘’βˆ—), 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 π΄π‘’βˆ— has the dichotomy property were studied say in [4–6]. In case of hyperbolic equilibrium point π‘’βˆ—, there is no spectrum of π΄π‘’βˆ— on 𝑖ℝ. Let π‘ˆ(𝜎+)βŠ‚{πœ†βˆˆπ‚βˆΆReπœ†>0} be an open connected neighborhood of 𝜎+ which has a closed rectifiable curve πœ•π‘ˆ(𝜎+) as a boundary. We decompose 𝐸𝛼 using the Riesz projectionπ‘ƒξ€·πœŽ+ξ€Έξ€·πœŽβˆΆ=𝑃+,π΄π‘’βˆ—ξ€Έ1∢=ξ€œ2πœ‹π‘–πœ•π‘ˆ(𝜎+)ξ€·πœπΌβˆ’π΄π‘’βˆ—ξ€Έβˆ’1π‘‘πœ(1.5) defined by 𝜎+. Due to this definition and analyticity of the 𝐢0-semigroup π‘’π‘‘π΄π‘’βˆ—,π‘‘βˆˆβ„+, we have positive constants 𝑀1,𝛽>0, such thatβ€–β€–π‘’π‘‘π΄π‘’βˆ—π‘§β€–β€–πΈπ›Όβ‰€π‘€1π‘’βˆ’π›½π‘‘β€–π‘§β€–πΈπ›Όβ€–β€–π‘’,𝑑β‰₯0,π‘‘π΄π‘’βˆ—π‘£β€–β€–πΈπ›Όβ‰€π‘€1𝑒𝛽𝑑‖𝑣‖𝐸𝛼,𝑑≀0,(1.6) for all π‘£βˆˆπ‘ƒ(𝜎+)𝐸𝛼 and π‘§βˆˆ(πΌβˆ’π‘ƒ(𝜎+))𝐸𝛼. Since πΉπ‘’βˆ—(𝑣(𝑑))=π‘œ(‖𝑣(𝑑)‖𝐸𝛼) for small 𝑣(β‹…), 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 π‘’βˆ—.

If 𝑣0 is close to 0, that is, say 𝑣0βˆˆπ’°πΈπ›Ό(0;𝜌) with small 𝜌>0, then the mild solution 𝑣(𝑑;𝑣0) of (1.4) can stay in the ball 𝒰𝐸𝛼(0;𝜌), for some time. We denote now the maximal time of staying 𝑣(𝑑;𝑣0) in 𝒰𝐸𝛼(0;𝜌) by 𝑇=𝑇(𝑣0)=sup{𝑑β‰₯0βˆΆβ€–π‘£(𝑑;𝑣0)β€–πΈπ›Όβ‰€πœŒor𝑣(𝑑;𝑣0)βˆˆπ’°πΈπ›Ό(0;𝜌)}. Now coming back to solution of (1.4) for any two 𝑣0,π‘£π‘‡βˆˆπ’°πΈπ›Ό(0;𝜌) we consider the boundary value problem𝑣′(𝑑)=π΄π‘’βˆ—π‘£(𝑑)+πΉπ‘’βˆ—(ξ€·ξ€·πœŽπ‘£(𝑑)),0≀𝑑≀𝑇,πΌβˆ’π‘ƒ+π‘£ξ€·ξ€·πœŽξ€Έξ€Έ(0)=πΌβˆ’π‘ƒ+𝑣0ξ€·πœŽ,𝑃+ξ€Έπ‘£ξ€·πœŽ(𝑇)=𝑃+𝑣𝑇.(1.7) A mild solution of problem (1.7) as was shown in [7] satisfies the integral equation𝑣(𝑑)=𝑒(π‘‘βˆ’π‘‡)π΄π‘’βˆ—π‘ƒξ€·πœŽ+𝑣𝑇+π‘’π‘‘π΄π‘’βˆ—ξ€·ξ€·πœŽπΌβˆ’π‘ƒ+𝑣0+ξ€œπ‘‘0𝑒(π‘‘βˆ’π‘ )π΄π‘’βˆ—ξ€·ξ€·πœŽπΌβˆ’π‘ƒ+πΉξ€Έξ€Έπ‘’βˆ—(+ξ€œπ‘£(𝑠))𝑑𝑠𝑇𝑑𝑒(π‘‘βˆ’π‘ )π΄π‘’βˆ—π‘ƒξ€·πœŽ+ξ€ΈπΉπ‘’βˆ—(𝑣(𝑠))𝑑𝑠,0≀𝑑≀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 𝑒(β‹…).

2. Preliminaries

Let 𝕋(π‘Ÿ)={πœ†βˆΆπœ†βˆˆπ‚,|πœ†|=𝐫},𝕋=𝕋(1).

Definition 2.1. A 𝐢0-semigroup 𝑒𝑑𝐴,𝑑β‰₯0, defined on a Banach space 𝐸 is called hyperbolic if 𝜎(𝑒𝑑𝐴)βˆ©π•‹=βˆ… for all 𝑑>0. The generator 𝐴 is called hyperbolic if 𝜎(𝐴)βˆ©π‘–β„=βˆ….

Let us denote by Ξ₯(ℝ;𝐸) any of the spaces 𝐿𝑝(ℝ;𝐸),1≀𝑝<∞,𝐢0(ℝ;𝐸) or Stepanov’s space 𝑆𝑝(ℝ;𝐸),1≀𝑝<∞. We consider in the Banach space Ξ₯(ℝ;𝐸) (we call this space as Palmer’s space, see [8], where Fredholm property was mentioned for the first time) the linear differential operator𝑑ℒ=βˆ’π‘‘π‘‘+𝐴∢𝐷(β„’)βŠ†Ξ₯(ℝ;𝐸)⟢Ξ₯(ℝ;𝐸),(2.1) where 𝐴 generates 𝐢0-semigroup, domain of β„’ is assumed to consist of the functions 𝑒(β‹…)∈Ξ₯(ℝ;𝐸) such that for some function 𝑔(β‹…)∈Ξ₯(ℝ;𝐸) one has 𝑒(𝑑)=𝑒(π‘‘βˆ’π‘ )π΄βˆ«π‘’(𝑠)βˆ’π‘‘π‘ π‘’(π‘‘βˆ’πœ‚)𝐴𝑔(πœ‚)π‘‘πœ‚,𝑠≀𝑑,π‘‘βˆˆβ„, and ℒ𝑒(β‹…)=𝑔(β‹…). Let us note [9] that the operator β„’ is the generator of 𝐢0-semigroup 𝑒𝑑ℒ on a Banach space Ξ₯(ℝ;𝐸), which is defined for all 𝑣(β‹…)∈Ξ₯(ℝ;𝐸) by formula𝑒𝑑ℒ𝑣(𝑠)=𝑒𝑑𝐴𝑣(π‘ βˆ’π‘‘)foranyπ‘ βˆˆβ„,𝑑β‰₯0.(2.2)

Definition 2.2. A 𝐢0-semigroup 𝑒𝑑𝐴,𝑑β‰₯0, has an exponential dichotomy on ℝ with exponential dichotomy data (𝑀β‰₯1,𝛽>0) if there exists projector π‘ƒβˆΆπΈβ†’πΈ such that(i)𝑒𝑑𝐴𝑃=𝑃𝑒𝑑𝐴 for all 𝑑β‰₯0,(ii)the restriction 𝑒𝑑𝐴|β„›(𝑃),𝑑β‰₯0, is invertible on 𝑃(𝐸) andβ€–β€–π‘’βˆ’π‘‘π΄β€–β€–π‘ƒπ‘₯β‰€π‘€π‘’βˆ’π›½π‘‘β€–β€–π‘’β€–π‘ƒπ‘₯β€–,𝑑β‰₯0,π‘₯∈𝐸,𝑑𝐴(β€–β€–πΌβˆ’π‘ƒ)π‘₯β‰€π‘€π‘’βˆ’π›½π‘‘β€–(πΌβˆ’π‘ƒ)π‘₯β€–,𝑑β‰₯0,π‘₯∈𝐸.(2.3)

Theorem 2.3 (see [10]). The operator β„’ in the Banach space Ξ₯(ℝ;𝐸) is invertible if and only if the condition πœŽξ€·π‘’1π΄ξ€Έβˆ©π•‹=βˆ…(2.4) holds. If condition (2.4) is satisfied, then ξ€·β„’βˆ’1π‘“ξ€Έξ€œ(𝑑)=βˆžβˆ’βˆžπΊ(π‘‘βˆ’π‘ )𝑓(𝑠)𝑑𝑠,π‘‘βˆˆβ„,𝑓(β‹…)∈Ξ₯(ℝ;𝐸),(2.5) where the Green’s function 𝐺(πœ‚)=βˆ’π‘’πœ‚π΄π‘ƒβˆ’π‘’,πœ‚β‰₯0,πœ‚π΄π‘ƒ+𝑀,πœ‚<0,(2.6)‖𝐺(πœ‚)‖≀+π‘’βˆ’π›Ύ+πœ‚π‘€,πœ‚β‰₯0,βˆ’π‘’π›Ύβˆ’πœ‚,πœ‚<0,(2.7) where 𝑀+ξ‚΅1=2𝑀(β„’)1+ξ‚Ά2(β„’)2,π‘€βˆ’ξ‚΅1=2𝑀(β„’)1βˆ’ξ‚Ά2(β„’)2,𝛾+ξ‚΅1=ln1+2ξ‚Ά(β„’),π›Ύβˆ’ξ‚΅1=βˆ’ln1βˆ’2ξ‚Ά,(β„’)(2.8) and Γ¦ (β„’)=1+𝐢(Ξ₯)(𝑀+𝑀2β€–β„’βˆ’1β€–).

We note that the constant 𝐢(Ξ₯) is defined as 𝐢(Ξ₯)=1 if Ξ₯(ℝ;𝐸)=𝐿∞(ℝ,𝐸) or Ξ₯(ℝ;𝐸)=𝐢0(ℝ,𝐸) and 𝐢(Ξ₯)=21βˆ’1/𝑝 if Ξ₯(ℝ;𝐸)=𝐿𝑝(ℝ,𝐸) or Ξ₯(ℝ;𝐸)=𝑆𝑝(ℝ,𝐸),π‘βˆˆ[1,∞).

Denote by Ξ₯(β„€;𝐸) the Banach space of 𝐸-valued sequences with corresponding discrete norm which is consistent with the norm of Ξ₯(ℝ;𝐸). For any 𝑒(β‹…)∈Ξ₯(β„€;𝐸), that is, {𝑒(π‘˜)}π‘˜βˆˆβ„€, and 𝐡=𝑒1𝐴∈𝐡(𝐸), we define an operator β„¬βˆΆπ·(ℬ)βŠ†π‘™π‘(β„€;𝐸)→𝑙𝑝(β„€;𝐸) by formula(ℬ𝑒)(π‘˜)=𝐡𝑒(π‘˜βˆ’1),π‘˜βˆˆβ„€,𝑒(β‹…)βˆˆπ‘™π‘(β„€;𝐸).(2.9) Now we define an operator π’Ÿ=πΌβˆ’β„¬βˆΆπ·(π’Ÿ)=𝐷(ℬ)βŠ†π‘™p(β„€;𝐸)→𝑙𝑝(β„€;𝐸) as(π’Ÿπ‘’)(π‘˜)=𝑒(π‘˜)βˆ’π΅π‘’(π‘˜βˆ’1),𝑒(β‹…)∈𝐷(ℬ),π‘˜βˆˆβ„€.(2.10)

Proposition 2.4 (see [10]). Let an operator β„’=βˆ’π‘‘/𝑑𝑑+𝐴∢𝐷(β„’)βŠ†Ξ₯(ℝ;𝐸)β†’Ξ₯(ℝ;𝐸) be invertible. Then, an operator π’ŸβˆΆπ·(π’Ÿ)βŠ†Ξ₯(β„€;𝐸)β†’Ξ₯(β„€;𝐸) is invertible too and β€–β€–π’Ÿβˆ’1‖‖≀1+𝐢(Ξ₯)𝑀+𝑀2β€–β€–β„’βˆ’1β€–β€–ξ€Έ.(2.11) Conversely, if π’Ÿ is invertible, then β„’βˆΆπ·(β„’)βŠ†Ξ₯(ℝ;𝐸)β†’Ξ₯(ℝ;𝐸) is invertible and β€–β€–β„’βˆ’1‖‖≀𝐢(Ξ₯)𝑀+𝑀2β€–β€–π’Ÿβˆ’1β€–β€–ξ€Έ.(2.12)

Theorem 2.5 (see [11]). The difference operator (π’Ÿπ‘’)(π‘˜)=u(π‘˜)βˆ’π΅π‘’(π‘˜βˆ’1),π‘˜βˆˆβ„•,𝑒(β‹…)βˆˆπ‘™π‘(ℝ;𝐸),1β‰€π‘β‰€βˆž,(2.13) is invertible if and only if 𝜎(𝐡)βˆ©π•‹=βˆ….(2.14) If condition (2.14) is satisfied, then the inverse operator has the form ξ€·π’Ÿβˆ’1𝑣(π‘˜)=Ξ£π‘šβˆˆβ„€Ξ“(π‘˜βˆ’π‘š)𝑣(π‘š),(2.15) where 𝑣(β‹…)βˆˆπ‘™π‘(β„€;𝐸),1β‰€π‘β‰€βˆž, and the function Ξ“(β‹…)βˆΆβ„€β†’π΅(𝐸) is defined by 𝐡Γ(π‘˜)=π‘˜(πΌβˆ’π‘ƒ),π‘˜β‰₯0,βˆ’π΅0βˆ’π‘˜π‘ƒ,π‘˜β‰€βˆ’1,(2.16) where 𝐡0 is the restriction of 𝐡 on β„›(𝑃).

Definition 2.6. The operator 𝐡∈𝐡(𝐸) has an exponential discrete dichotomy with data (𝑀,π‘Ÿ,𝑃) if π‘ƒβˆˆπ΅(𝐸) is a projector in 𝐸 and 𝑀,π‘Ÿ are constants with 0β‰€π‘Ÿ<1 such that the following properties hold(i)π΅π‘˜π‘ƒ=π‘ƒπ΅π‘˜ for all π‘˜βˆˆβ„•,(ii)β€–π΅π‘˜(πΌβˆ’π‘ƒ)β€–β‰€π‘€π‘Ÿπ‘˜ for all π‘˜βˆˆβ„•,(iii)𝐡∢=𝐡|β„›(𝑃)βˆΆβ„›(𝑃)↦ℛ(𝑃) is a homeomorphism that satisfies β€–ξπ΅βˆ’π‘˜π‘ƒβ€–β‰€π‘€π‘Ÿπ‘˜,π‘˜βˆˆβ„•.

The following result relates the constants in an exponential discrete dichotomy to a bound on the resolvent (πœ†πΌβˆ’π΅)βˆ’1 for πœ†βˆˆπ•‹.

Theorem 2.7. For 𝐡∈𝐡(𝐸), the following conditions are equivalent:(i)πœ†βˆˆπœŒ(𝐡) for all πœ†βˆˆπ•‹ and β€–(πœ†πΌβˆ’π΅)βˆ’1‖≀𝛽<βˆžβˆ€πœ†βˆˆπ•‹;(ii)𝐡 has an exponential dichotomy with data (𝑀,π‘Ÿ,𝑃). More precisely, one shows that (i) implies (ii) with 𝑀=2𝛽2/(π›½βˆ’1) and π‘Ÿ=1βˆ’1/2𝛽. Conversely, (ii) implies (i) with 𝛽=𝑀((1+π‘Ÿ)/(1βˆ’π‘Ÿ)).

Proof. First assume (i) and without loss of generality let 𝛽>1. For π‘§βˆˆπ‚,π³β‰ πŸŽ, we have ||||π‘§π‘§βˆ’||||=|||||𝑧|1βˆ’|𝑧|.(2.17) Hence, if |1βˆ’|𝑧||𝛽<1, the classical perturbation estimate shows that π‘§βˆˆπœŒ(𝐡) and β€–β€–(π‘§πΌβˆ’π΅)βˆ’1‖‖≀𝛽||||.1βˆ’π›½1βˆ’|𝑧|(2.18) We define 𝑃 as the Riesz projector defined by the formula 1πΌβˆ’π‘ƒ=ξ€œ2πœ‹π‘–|𝑧|=1(π‘§πΌβˆ’π΅)βˆ’1𝑑𝑧.(2.19) Since 𝐡 commutes with the resolvent, condition (i) of Definition 2.6 holds. Further, using (2.18) and Cauchy’s theorem, we can shift the contour 1πΌβˆ’π‘ƒ=ξ€œ2πœ‹π‘–|𝑧|=π‘Ÿ(π‘§πΌβˆ’π΅)βˆ’1||||<1𝑑𝑧,if1βˆ’π‘Ÿπ›½.(2.20) Now we claim π΅π‘˜(1πΌβˆ’π‘ƒ)=ξ€œ2πœ‹π‘–|𝑧|=π‘Ÿπ‘§π‘˜(π‘§πΌβˆ’π΅)βˆ’1||||<1𝑑𝑧if1βˆ’π‘Ÿπ›½.(2.21) For π‘˜=0 this follows from (2.20). If (2.21) holds for some π‘˜, then we obtain π΅π‘˜+11𝑃=ξ€œ2πœ‹π‘–|𝑧|=π‘Ÿ(π΅βˆ’π‘§πΌ+𝑧𝐼)π‘§π‘˜(π‘§πΌβˆ’π΅)βˆ’1=1π‘‘π‘§ξ€œ2πœ‹π‘–|𝑧|=π‘Ÿπ‘§π‘˜+1(π‘§πΌβˆ’π΅)βˆ’11π‘‘π‘§βˆ’ξ€œ2πœ‹π‘–|𝑧|=π‘Ÿπ‘§π‘˜π‘‘π‘§,(2.22) and thus the assertion holds for π‘˜+1. Equations (2.21) and (2.18) immediately lead to the first dichotomy estimate for 1βˆ’1/𝛽<π‘Ÿβ‰€1β€–β€–π΅π‘˜π‘ƒβ€–β€–β‰€12πœ‹2πœ‹π‘Ÿπ‘Ÿπ‘˜π›½=1βˆ’π›½(1βˆ’π‘Ÿ)π›½π‘Ÿπ‘˜+11βˆ’π›½(1βˆ’π‘Ÿ)forπ‘˜β‰₯0.(2.23) For the second dichotomy estimate, we use the resolvent equation (π‘§πΌβˆ’π΅)βˆ’1=1z𝐼+π‘§βˆ’1𝐡(π‘§πΌβˆ’π΅)βˆ’1.(2.24) For |1βˆ’π‘Ÿ|<1/𝛽, (2.20) and (2.24) lead to 1πΌβˆ’π‘ƒ=ξ€œ2πœ‹π‘–|𝑧|=1ξ‚€1π‘§πΌβˆ’(π‘§πΌβˆ’π΅)βˆ’11𝑑𝑧=βˆ’ξ€œ2πœ‹π‘–|𝑧|=π‘Ÿ1𝑧𝐡(π‘§πΌβˆ’π΅)βˆ’1𝑑𝑧.(2.25) This shows that the following equality holds for π‘˜=1πΌβˆ’π‘ƒ=βˆ’π΅π‘˜1ξ€œ2πœ‹π‘–|𝑧|=π‘Ÿπ‘§βˆ’π‘˜(π‘§πΌβˆ’π΅)βˆ’1𝑑𝑧,forπ‘˜β‰₯1.(2.26) If (2.26) is known for some π‘˜, then use (2.24) and find πΌβˆ’π‘ƒ=βˆ’π΅π‘˜1ξ€œ2πœ‹π‘–|𝑧|=π‘Ÿξ€·π‘§βˆ’(π‘˜+1)+π‘§βˆ’(π‘˜+1)𝐡(π‘§πΌβˆ’π΅)βˆ’1𝑑𝑧=βˆ’π΅π‘˜+11ξ€œ2πœ‹π‘–|𝑧|=π‘Ÿπ‘§βˆ’(π‘˜+1)(π‘§πΌβˆ’π΅)βˆ’1𝑑𝑧.(2.27) We apply (2.26) to π‘₯∈𝐸, using that πΌβˆ’π‘ƒ commutes with 𝐡 as well as the estimate (2.18) β€–β€–β€–β€–1(πΌβˆ’π‘ƒ)π‘₯β€–=ξ€œ2πœ‹π‘–|𝑧|=π‘Ÿπ‘§βˆ’π‘˜(π‘§πΌβˆ’π΅)βˆ’1π‘‘π‘§π΅π‘˜β€–β€–β€–β‰€π‘Ÿ(πΌβˆ’π‘ƒ)π‘₯βˆ’π‘˜π›½2πœ‹2πœ‹π‘Ÿβ€–β€–π΅1βˆ’π›½(π‘Ÿβˆ’1)π‘˜β€–β€–.(πΌβˆ’π‘ƒ)π‘₯(2.28) Summarizing, we have shown for all π‘˜β‰₯1,1β‰€π‘Ÿ<1+1/𝛽,π‘’βˆˆπΈβ€–β€–β‰€(πΌβˆ’π‘ƒ)π‘’π›½π‘Ÿβˆ’π‘˜+1‖‖𝐡1βˆ’π›½(π‘Ÿβˆ’1)π‘˜β€–β€–.(πΌβˆ’π‘ƒ)𝑒(2.29) For π‘˜=1, this estimate shows that 𝐡=𝐡|𝒩(𝑃)βˆΆπ’©(𝑃)↦𝒩(𝑃) is one-to-one with a bound for the inverse. To show that 𝐡 is onto, we take π‘“βˆˆπ’©(𝑃) and set 1𝑣=βˆ’ξ€œ2πœ‹π‘–|𝑧|=1π‘§βˆ’1(π‘§πΌβˆ’π΅)βˆ’1𝑓𝑑𝑧.(2.30) From this equation, we have (πΌβˆ’π‘ƒ)𝑣=0, and using (2.24), we find 1𝐡𝑣=ξ€œ2πœ‹π‘–|𝑧|=1ξ€·π‘§βˆ’1πΌβˆ’(π‘§πΌβˆ’π΅)βˆ’1𝑓𝑑𝑧=(πΌβˆ’π‘ƒ)𝑓=𝑓.(2.31) Therefore, 𝐡 is a linear homeomorphism on 𝒩(𝑃) satisfying β€–β€–ξπ΅βˆ’π‘˜β€–β€–β‰€(πΌβˆ’π‘ƒ)π‘’π›½π‘Ÿβˆ’π‘˜+1β€–β€–11βˆ’π›½(π‘Ÿβˆ’1)(πΌβˆ’π‘ƒ)𝑒for1β‰€π‘Ÿ<1+𝛽.(2.32) This proves exponential dichotomy.
To arrive at the specific constants, we choose π‘Ÿ=1βˆ’1/2𝛽 in (2.23) and obtain the bound (2π›½βˆ’1)π‘Ÿπ‘˜. In order to have the same rate in the opposite direction, we apply (2.32), with π‘Ÿ=(1βˆ’1/2𝛽)βˆ’1<1+1/𝛽. In (2.32) we then find the upper bound π‘€π‘Ÿβˆ’π‘˜ with the constant 𝑀=2𝛽2/(π›½βˆ’1). Since 𝑀>2π›½βˆ’1, our assertion follows.
Now we assume exponential dichotomy and prove condition (i). For |πœ†|=1 the equation (πœ†πΌβˆ’π΅)𝑒=𝑓 is equivalent to the system𝐡(πœ†πΌβˆ’π΅π‘ƒ)𝑃𝑒=𝑃𝑓,πœ†πΌβˆ’(πΌβˆ’π‘ƒ)𝑒=(πΌβˆ’π‘ƒ)𝑓,(2.33) which we can rewrite as ξ€·πΌβˆ’πœ†βˆ’1𝐡𝑃𝑃𝑒=πœ†βˆ’1𝐡𝑃𝑓,πΌβˆ’πœ†βˆ’1𝐡(πΌβˆ’π‘ƒ)𝑒=βˆ’βˆ’1(πΌβˆ’π‘ƒ)𝑓.(2.34) Both equations have a unique solution given by a geometric series 𝑃𝑒=βˆžξ“π‘˜=0πœ†βˆ’(π‘˜+1)π΅π‘˜π‘ƒπ‘“,(πΌβˆ’π‘ƒ)𝑒=βˆ’βˆžξ“π‘˜=0ξπ΅βˆ’(π‘˜+1)πœ†π‘˜(πΌβˆ’π‘ƒ)𝑓.(2.35) The exponential dichotomy then implies the estimates 𝑀‖𝑃𝑒‖≀‖(π‘Ÿ1βˆ’π‘Ÿβ€–π‘“β€–,πΌβˆ’π‘ƒ)𝑒‖≀𝑀1βˆ’π‘Ÿβ€–π‘“β€–.(2.36) By the triangle inequality, we obtain condition (i) with 𝛽=𝑀((1+π‘Ÿ)/(1βˆ’π‘Ÿ)).

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 𝐸𝑛 and 𝐸 be Banach spaces and {𝑝𝑛} a sequence of linear bounded operators π‘π‘›βˆΆπΈβ†’πΈπ‘›,π‘π‘›βˆˆπ΅(𝐸,𝐸𝑛),π‘›βˆˆβ„•={1,2,…}, with the property:‖‖𝑝𝑛π‘₯β€–β€–πΈπ‘›βŸΆβ€–π‘₯‖𝐸asπ‘›β†’βˆžforanyπ‘₯∈𝐸.(3.1)

Definition 3.1. The sequence of elements {π‘₯𝑛},π‘₯π‘›βˆˆπΈπ‘›,π‘›βˆˆβ„•, is said to be 𝒫-convergent to π‘₯∈𝐸 if and only if β€–π‘₯π‘›βˆ’π‘π‘›π‘₯‖𝐸𝑛→0 as π‘›β†’βˆž, and we write this π‘₯π‘›π’«βˆ’βˆ’β†’π‘₯.

Definition 3.2. The sequence of bounded linear operators π΅π‘›βˆˆπ΅(𝐸𝑛), π‘›βˆˆβ„•, is said to be 𝒫𝒫-convergent to the bounded linear operator 𝐡∈𝐡(𝐸) if for every π‘₯∈𝐸 and for every sequence {π‘₯𝑛},π‘₯π‘›βˆˆπΈπ‘›,π‘›βˆˆβ„•, such that π‘₯π‘›π’«βˆ’βˆ’β†’π‘₯ one has 𝐡𝑛π‘₯π‘›π’«βˆ’βˆ’β†’π΅π‘₯. We write then π΅π‘›π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π΅.

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 {𝐴𝑛},π΄π‘›βˆˆπ’ž(𝐸𝑛),π‘›βˆˆβ„•, is called compatible with a closed linear operator π΄βˆˆπ’ž(𝐸) if and only if for each π‘₯∈𝐷(𝐴) there is a sequence {π‘₯𝑛},π‘₯π‘›βˆˆπ·(𝐴𝑛)βŠ†πΈπ‘›,π‘›βˆˆβ„•, such that π‘₯π‘›π’«βˆ’βˆ’β†’π‘₯ and 𝐴𝑛π‘₯π‘›π’«βˆ’βˆ’β†’π΄π‘₯. We write that (𝐴𝑛,𝐴) are compatible.

For analytic 𝐢0-semigroups, the following ABC Theorem holds.

Theorem 3.4 (see [16]). Let operators 𝐴 and 𝐴𝑛 generate analytic 𝐢0-semigroups. The following conditions (𝐴) and (𝐡1) are equivalent to condition (𝐢1).(𝐴) Compatibility. There exists πœ†βˆˆπœŒ(𝐴)βˆ©βˆ©π‘›πœŒ(𝐴𝑛) such that the resolvents converge (πœ†πΌβˆ’π΄π‘›)βˆ’1π’«π’«βˆ’βˆ’βˆ’βˆ’β†’(πœ†πΌβˆ’π΄)βˆ’1.(𝐡1)Stability. There are some constants 𝑀1β‰₯1 and πœ”1βˆˆβ„ such thatβ€–β€–ξ€·πœ†πΌπ‘›βˆ’π΄π‘›ξ€Έβˆ’1‖‖≀𝑀1||πœ†βˆ’πœ”1||,Reπœ†>πœ”1,π‘›βˆˆβ„•.(3.2)(𝐢1) Convergence. For any finite πœ‡>0 and some 0<πœƒ<πœ‹/2, one hasmaxπœ‚βˆˆΞ£(πœƒ,πœ‡)β€–β€–π‘’πœ‚π΄π‘›π‘’0π‘›βˆ’π‘π‘›π‘’πœ‚π΄π‘’0β€–β€–βŸΆ0asπ‘›βŸΆβˆžwhenever𝑒0π‘›π’«βˆ’βˆ’β†’π‘’0.(3.3)Here one used the sector of angle 2πœƒ and radius 𝜌 given by Ξ£(πœƒ,πœ‡)={π‘§βˆˆΞ£(πœƒ)∢|𝑧|β‰€πœ‡}, and Ξ£(πœƒ)={π‘§βˆˆβ„‚βˆΆ|arg𝑧|β‰€πœƒ}.

It is natural to assume in semidiscretization that conditions like (A) and (𝐡1) are satisfied.

Definition 3.5. The region of stability Δ𝑠=Δ𝑠({𝐴𝑛}),π΄π‘›βˆˆπ’ž(𝐡𝑛), is defined as the set of all πœ†βˆˆπ‚ such that πœ†βˆˆπœŒ(𝐴𝑛) for almost all 𝑛 and such that the sequence {β€–(πœ†πΌπ‘›βˆ’π΄π‘›)βˆ’1β€–}π‘›βˆˆβ„• is bounded. The region of convergence Δ𝑐=Δ𝑐({𝐴𝑛}),π΄π‘›βˆˆπ’ž(𝐸𝑛), is defined as the set of all πœ†βˆˆπ‚ such that πœ†βˆˆΞ”π‘ ({𝐴𝑛}) and such that the sequence of operators {(πœ†πΌπ‘›βˆ’π΄π‘›)βˆ’1}π‘›βˆˆβ„• is 𝒫𝒫-convergent to some operator 𝑆(πœ†)∈𝐡(𝐸).

Definition 3.6. A sequence of operators {𝐡𝑛},π΅π‘›βˆˆπ΅(𝐸𝑛),π‘›βˆˆβ„•, is said to be stably convergent to an operator 𝐡∈𝐡(𝐸) if and only if π΅π‘›π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π΅ and β€–π΅π‘›βˆ’1‖𝐡(𝐸𝑛)=𝑂(1), π‘›β†’βˆž. We will write this as: π΅π‘›π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π΅ stably.

Definition 3.7. A sequence of operators {𝐡𝑛}, π΅π‘›βˆˆπ΅(𝐸𝑛), is called regularly convergent to the operator 𝐡∈𝐡(𝐸) if and only if π΅π‘›π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π΅ and the following implication holds β€–β€–π‘₯𝑛‖‖𝐸𝑛𝐡=𝑂(1),𝑛π‘₯𝑛π‘₯is𝒫-compactβŸΉπ‘›ξ€Ύis𝒫-compact.(3.4) We write this as: π΅π‘›π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π΅ regularly.

Theorem 3.8 (see [15]). Let 𝐢𝑛,π‘„π‘›βˆˆπ΅(𝐸𝑛), 𝐢,π‘„βˆˆπ΅(𝐸) and β„›(𝑄)=𝐸. Assume also that πΆπ‘›π’«π’«βˆ’βˆ’βˆ’βˆ’β†’πΆ compactly and π‘„π‘›π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π‘„ stably. Then, 𝑄𝑛+πΆπ‘›π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π‘„+𝐢 converge regularly.

Theorem 3.9 (see [15]). For π‘„π‘›βˆˆπ΅(𝐸𝑛) and π‘„βˆˆπ΅(𝐸), the following conditions are equivalent:(i)π‘„π‘›π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π‘„ regularly, 𝑄𝑛 are Fredholm operators of index 0 and 𝒩(𝑄)={0};(ii)π‘„π‘›π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π‘„ stably and β„›(𝑄)=𝐸;(iii)π‘„π‘›π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π‘„ stably and regularly; (iv)if one of conditions (i)–(iii) holds, then there exist π‘„π‘›βˆ’1∈𝐡(𝐸𝑛), π‘„βˆ’1∈𝐡(𝐸), and π‘„π‘›βˆ’1π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π‘„βˆ’1 regularly and stably.

Theorem 3.10. Let operators πœ†πΌπ‘›βˆ’π΅π‘›βˆˆπ΅(𝐸𝑛) be Fredholm operators with 𝑖𝑛𝑑(πœ†πΌπ‘›βˆ’π΅π‘›)=0 for any πœ†βˆˆπ•‹,π‘›βˆˆβ„•. Assume also that 𝐡∈𝐡(𝐸) has the property π•‹βˆ©πœŽ(𝐡)=βˆ… and πœ†πΌπ‘›βˆ’π΅π‘›π’«π’«βˆ’βˆ’βˆ’βˆ’β†’πœ†πΌβˆ’π΅ regularly for any πœ†βˆˆπ•‹. Then πœ†πΌπ‘›βˆ’π΅π‘›π’«π’«βˆ’βˆ’βˆ’βˆ’β†’πœ†πΌβˆ’π΅ stably for any πœ†βˆˆπ•‹ and supπœ†βˆˆπ•‹β€–(πœ†πΌπ‘›βˆ’π΅π‘›)βˆ’1β€–<∞.

Proof. Assume that there are some sequences {πœ†π‘›},πœ†π‘›βˆˆπ•‹, and {π‘₯𝑛},π‘₯π‘›βˆˆπΈπ‘›, such that β€–π‘₯𝑛‖=1 and (πœ†π‘›πΌπ‘›βˆ’π΅π‘›)π‘₯π‘›π’«βˆ’βˆ’β†’0 as π‘›β†’βˆž. Since 𝕋 is compact, one can find β„•β€²βŠ‚β„• such that πœ†π‘›β†’πœ†0βˆˆπ•‹ as π‘›βˆˆβ„•ξ…ž. In the meantime, πœ†0πΌπ‘›βˆ’π΅π‘›π’«π’«βˆ’βˆ’βˆ’βˆ’β†’πœ†0πΌβˆ’π΅ regularly for such πœ†0βˆˆπ•‹, and (πœ†0πΌπ‘›βˆ’π΅π‘›)π‘₯𝑛=(πœ†0πΌπ‘›βˆ’πœ†π‘›πΌπ‘›)π‘₯𝑛+(πœ†π‘›πΌπ‘›βˆ’π΅π‘›)π‘₯π‘›π’«βˆ’βˆ’β†’0 for π‘›βˆˆβ„•ξ…ž. Therefore, there is β„•ξ…žξ…žβŠ‚β„•ξ…ž such that π‘₯π‘›π’«βˆ’βˆ’β†’π‘₯0β‰ 0 as π‘›βˆˆβ„•ξ…žξ…ž. But in such case (πœ†0πΌπ‘›βˆ’π΅π‘›)π‘₯π‘›π’«π’«βˆ’βˆ’βˆ’βˆ’β†’(πœ†0πΌβˆ’π΅)π‘₯0=0 as π‘›βˆˆβ„•ξ…žξ…ž, which contradicts our assumption π•‹βˆ©πœŽ(𝐡)=βˆ….

Definition 3.11. The operator π΅π‘›βˆˆπ΅(𝐸𝑛) has a uniform exponential discrete dichotomy with data (𝑀,π‘Ÿ,𝑃𝑛) if π‘ƒπ‘›βˆˆπ΅(𝐸𝑛) is a projector in 𝐸𝑛 and 𝑀,π‘Ÿ are constants with 0β‰€π‘Ÿ<1 such that the following properties hold(i)π΅π‘˜π‘›π‘ƒπ‘›=π‘ƒπ‘›π΅π‘˜π‘› and β€–π‘ƒπ‘›β€–β‰€π‘π‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘ for all π‘˜,π‘›βˆˆβ„•;(ii)β€–π΅π‘˜π‘›(πΌπ‘›βˆ’π‘ƒπ‘›)β€–β‰€π‘€π‘Ÿπ‘˜ for all π‘˜,π‘›βˆˆβ„•;(iii)ξπ΅π‘›βˆΆ=𝐡𝑛|β„›(𝑃𝑛)βˆΆβ„›(𝑃𝑛)↦ℛ(𝑃𝑛) is a homeomorphism that satisfies β€–β€–ξ‚Šπ΅π‘›βˆ’π‘˜π‘ƒπ‘›β€–β€–β‰€π‘€π‘Ÿπ‘˜,π‘˜,π‘›βˆˆβ„•.(3.5)

Theorem 3.12. The following conditions are equivalent:(i)πœ†πΌπ‘›βˆ’π΅π‘›π’«π’«βˆ’βˆ’βˆ’βˆ’β†’πœ†πΌβˆ’π΅ stably and πœ†βˆˆπœŒ(𝐡) for any πœ†βˆˆπ•‹;(ii)operator π’Ÿ=πΌβˆ’β„¬ is invertible and π’Ÿπ‘›π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π’Ÿ stably, where (ℬ𝑒)(π‘˜)=𝐡𝑒(π‘˜βˆ’1),π‘˜βˆˆβ„•;(iii)π΅π‘›π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π΅ and πœ†πΌβˆ’π΅ be invertible for any πœ†βˆˆπ•‹ and the operators 𝐡𝑛 have an exponential discrete dichotomy with data (𝑀,π‘Ÿ,𝑃𝑛) uniformly in π‘›βˆˆβ„•.

Proof. The equivalence (i)⇔(iii) follows from Theorem 2.7. Indeed, by formula (2.19), one gets from (i) that π‘ƒπ‘›π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π‘ƒ and β€–π‘ƒπ‘›β€–β‰€π‘π‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘. By Theorem 3.10, one has supπœ†βˆˆπ•‹β€–(πœ†πΌπ‘›βˆ’π΅π‘›)βˆ’1β€–<∞, and by Theorem 2.7 (ii) we have (iii). Conversely, from condition (iii), it follows by Theorem 2.7 (i) that πœ†πΌπ‘›βˆ’π΅π‘›π’«π’«βˆ’βˆ’βˆ’βˆ’β†’πœ†πΌβˆ’π΅ stably for all πœ†βˆˆπ•‹.
To prove (ii)β‡’(i), we note that (ii) means that for any 𝑒(β‹…)βˆˆπ‘™π‘(β„€;𝐸) one has Ξ£βˆžπ‘˜=βˆ’βˆžβ€–π‘π‘›π‘’(π‘˜)βˆ’π΅π‘›π‘π‘›π‘’(π‘˜βˆ’1)βˆ’π‘π‘›π‘’(π‘˜)+𝑝𝑛𝐡𝑒(π‘˜βˆ’1)‖𝑝𝐸𝑛→0 as π‘›β†’βˆž, that is, π΅π‘›π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π΅. Assume now that πΌπ‘›βˆ’π΅π‘› is not uniformly invertible in π‘™βˆž(β„€;𝐸𝑛), that is, for some sequence β€–π‘₯𝑛‖=1, one has (πœ†0πΌπ‘›βˆ’π΅π‘›)π‘₯π‘›π’«βˆ’βˆ’β†’0 as π‘›β†’βˆž for some πœ†0=1βˆˆπ•‹. This means that for stationary sequence 𝑒𝑛(π‘˜)=π‘₯𝑛,π‘˜βˆˆβ„€,π‘›βˆˆπ•‹, one has (π’Ÿπ‘›π‘’π‘›)(π‘˜)=𝑒𝑛(π‘˜)βˆ’π΅π‘›π‘’π‘›(π‘˜βˆ’1)=π‘₯π‘›βˆ’π΅π‘›π‘₯π‘›π’«βˆ’βˆ’β†’0 for any π‘˜βˆˆβ„• and π‘›β†’βˆž. But, π’Ÿπ‘›π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π’Ÿ stably, that is, β€–π’Ÿπ‘›π‘’π‘›β€–π‘™βˆž(β„€;𝐸𝑛)β‰₯π›Ύβ€–π‘’π‘›β€–π‘™βˆž(β„€;𝐸𝑛) which contradicts (πœ†0πΌπ‘›βˆ’π΅π‘›)π‘₯π‘›π’«βˆ’βˆ’β†’0 as π‘›β†’βˆž. Now we show that β„›(πœ†0πΌπ‘›βˆ’π΅π‘›)=𝐸𝑛. For any π‘¦π‘›βˆˆπΈπ‘›,‖𝑦𝑛‖=1, consider 𝑣𝑛(π‘˜)=𝑦𝑛,π‘˜βˆˆβ„€,π‘›βˆˆβ„•. The solution of π’Ÿπ‘›π‘’π‘›=𝑣𝑛 is a sequence 𝑒𝑛(π‘˜), which is stationary too, that is, (πœ†0πΌπ‘›βˆ’π΅π‘›)π‘₯𝑛=𝑦𝑛, where π‘₯𝑛=𝑒𝑛(π‘˜),π‘˜βˆˆβ„€,π‘›βˆˆβ„•. To show (i)β‡’(ii), we note that β€–π’Ÿπ‘›β€–π΅(𝑙𝑝(β„€;𝐸𝑛))β‰€π‘π‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘,π‘›βˆˆβ„•. Now for any 𝑒(β‹…)βˆˆπ‘™π‘(β„€;𝐸) and for any πœ–>0, one can find πΎβˆˆβ„• such that (Ξ£βˆžπ‘˜=𝐾+Ξ£βˆ’βˆžπ‘˜=βˆ’πΎ)‖𝑒(π‘˜)β€–π‘β‰€πœ–. In the meantime, Ξ£πΎπ‘˜=βˆ’K‖𝑝𝑛𝑒(π‘˜)βˆ’π΅π‘›π‘π‘›π‘’(π‘˜βˆ’1)βˆ’π‘π‘›π‘’(π‘˜)+𝑝𝑛𝐡𝑒(π‘˜βˆ’1)‖𝑝𝐸𝑛→0 as π‘›β†’βˆž, since π΅π‘›π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π΅. So we have π’Ÿπ‘›π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π’Ÿ. The convergence π’Ÿπ‘›βˆ’1π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π’Ÿβˆ’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 β€œpreserve” the property of compactness.

Definition 3.13. A sequence of operators {𝐡𝑛},π΅π‘›βˆΆπΈπ‘›β†’πΈπ‘›,π‘›βˆˆβ„•, converges compactly to an operator π΅βˆΆπΈβ†’πΈ if π΅π‘›π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π΅ and the following compactness condition holds β€–β€–π‘₯𝑛‖‖𝐸𝑛𝐡=𝑂(1)βŸΉπ‘›π‘₯𝑛is𝒫-compact.(3.6)

Definition 3.14. The region of compact convergence of resolvents, Δ𝑐𝑐=Δ𝑐𝑐(𝐴𝑛,𝐴), where π΄π‘›βˆˆπ’ž(𝐸𝑛) and π΄βˆˆπ’ž(𝐸) is defined as the set of all πœ†βˆˆΞ”π‘βˆ©πœŒ(𝐴) such that (πœ†πΌπ‘›βˆ’π΄π‘›)βˆ’1π’«π’«βˆ’βˆ’βˆ’βˆ’β†’(πœ†πΌβˆ’π΄)βˆ’1 compactly.

Proposition 3.15. Assume that operators 𝐡,𝐡n are compact, π•‹βŠ‚πœŒ(𝐡) and Δ𝑐𝑐(𝐡𝑛,𝐡)β‰ βˆ…. Then, πœ†πΌπ‘›βˆ’π΅π‘›π’«π’«βˆ’βˆ’βˆ’βˆ’β†’πœ†πΌβˆ’π΅ stably for any πœ†βˆˆπ•‹ and supπœ†βˆˆπ•‹β€–(πœ†πΌπ‘›βˆ’π΅π‘›)βˆ’1β€–<∞.

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 𝑝𝛼𝑛=(βˆ’π΄π‘›)βˆ’π›Όπ‘π‘›(βˆ’π΄)π›Όβˆˆπ΅(𝐸𝛼,𝐸𝛼𝑛) which satisfy property (3.1), but for the spaces 𝐸𝛼,𝐸𝛼𝑛. The operators 𝐴𝑛 and 𝐴 are supposed to be related by condition (1.1), and conditions (𝐴) and (𝐡1) of Theorem 3.4 are assumed to hold. So we say that π‘₯π‘›π’«π›Όβˆ’βˆ’βˆ’β†’π‘₯ if and only if β€–π‘₯π‘›βˆ’π‘π›Όπ‘›π‘₯‖𝐸𝛼𝑛→0 as π‘›β†’βˆž. One can see that β€–π‘₯π‘›βˆ’π‘π›Όπ‘›π‘₯‖𝐸𝛼𝑛=β€–(βˆ’π΄π‘›)𝛼π‘₯π‘›βˆ’π‘π‘›(βˆ’π΄)𝛼π‘₯β€– and ‖𝑝𝛼𝑛π‘₯‖𝐸𝛼𝑛=‖𝑝𝑛(βˆ’π΄)𝛼π‘₯‖𝐸𝑛→‖(βˆ’π΄)𝛼π‘₯‖𝐸=β€–π‘₯‖𝐸𝛼 for any π‘₯∈𝐷((βˆ’π΄)𝛼) and π‘›β†’βˆž.
Consider in Banach spaces 𝐸𝛼𝑛 the family of parabolic problems𝑒′𝑛(𝑑)=𝐴𝑛𝑒𝑛(𝑑)+𝑓𝑛𝑒𝑛(𝑒𝑑),𝑑β‰₯0,𝑛(0)=𝑒0π‘›βˆˆπΈπ›Όπ‘›,(3.7) where 𝑒0π‘›π’«π›Όβˆ’βˆ’βˆ’β†’π‘’0, operators (𝐴𝑛,𝐴) are compatible, 𝑓𝑛(β‹…)βˆΆπΈπ›Όπ‘›β†’πΈπ‘› are globally bounded and globally Lipschitz continuous both uniformly in π‘›βˆˆβ„•, and continuously FrΓ©chet differentiable.
Under the above assumptions, the mild solution 𝑒𝑛(β‹…) of (3.7) is defined for all 𝑑β‰₯0 (see [1, 17]), and we define it by 𝑒𝑛(β‹…)=𝑇𝑛(β‹…)𝑒0π‘›βˆΆβ„+→𝐸𝑛. The nonlinear semigroup 𝑇𝑛(β‹…) satisfies the variation of constants formula𝑇𝑛(𝑑)𝑒0𝑛=𝑒𝑑𝐴𝑛𝑒0𝑛+ξ€œπ‘‘0𝑒(π‘‘βˆ’π‘ )𝐴𝑛𝑓𝑛𝑇𝑛(𝑠)𝑒0𝑛𝑑𝑠,𝑑β‰₯0.(3.8)
Recall that a hyperbolic equilibrium point π‘’βˆ— is a solution of equation 𝐴𝑒+𝑓(𝑒)=0, or equivalently π‘’βˆ—=βˆ’π΄βˆ’1𝑓(π‘’βˆ—). Since the operator 𝐴 has a compact resolvent, the operator π΄βˆ’1𝑓(β‹…) is compact. In case of Ξ”π‘π‘β‰ βˆ…, the operators π΄π‘›βˆ’1𝑓𝑛(β‹…)π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π΄βˆ’1𝑓(β‹…) compactly. From [15], it follows that equations 𝑒𝑛=βˆ’π΄π‘›βˆ’1𝑓𝑛(𝑒𝑛) have solutions {π‘’βˆ—π‘›},π΄π‘›π‘’βˆ—π‘›+𝑓𝑛(π‘’βˆ—π‘›)=0, such that π‘’βˆ—π‘›π’«βˆ’βˆ’β†’π‘’βˆ—.
Now, consider the problems (3.7) near hyperbolic equilibrium points π‘’βˆ—π‘›. In this case one getsπ‘£ξ…žπ‘›(𝑑)=π΄π‘’βˆ—π‘›,𝑛𝑣𝑛(𝑑)+πΉπ‘’βˆ—π‘›,𝑛𝑣𝑛(𝑑),𝑣𝑛(0)=𝑣0𝑛,𝑑β‰₯0,(3.9) where π΄π‘’βˆ—π‘›,𝑛=𝐴𝑛+π‘“ξ…žπ‘›(π‘’βˆ—π‘›),πΉπ‘’βˆ—π‘›,𝑛(𝑣𝑛(𝑑))=𝑓𝑛(𝑣𝑛(𝑑)+π‘’βˆ—π‘›)βˆ’π‘“π‘›(π‘’βˆ—π‘›)βˆ’π‘“ξ…žπ‘›(π‘’βˆ—π‘›)𝑣𝑛(𝑑). From now on, we consider hyperbolic point π‘’βˆ— and hyperbolic points π‘’βˆ—π‘›π’«π›Όβˆ’βˆ’βˆ’β†’π‘’βˆ—.
We decompose 𝐸𝛼𝑛 using the projectionπ‘ƒπ‘›ξ€·πœŽ+π‘›ξ€ΈβˆΆ=π‘ƒπ‘›ξ€·πœŽ+𝑛,π΄π‘’βˆ—π‘›,𝑛1∢=ξ€œ2πœ‹π‘–πœ•π‘ˆ(𝜎+𝑛)ξ€·πœπΌπ‘›βˆ’π΄π‘’βˆ—π‘›,π‘›ξ€Έβˆ’1π‘‘πœ(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 (𝐡1) for operators π΄π‘’βˆ—π‘›,𝑛.
There are some positive 𝑀2,𝛾>0, because of analyticity of 𝐢0-semigroup π‘’π‘‘π΄π‘’βˆ—π‘›,𝑛 and condition Ξ”π‘π‘β‰ βˆ…, which is applied to operator π΄π‘’βˆ—π‘›,𝑛, such that [7, 18]β€–β€–π‘’π‘‘π΄π‘’βˆ—π‘›,𝑛𝑧𝑛‖‖𝐸𝛼𝑛≀𝑀2π‘’βˆ’π›Ύπ‘‘β€–β€–π‘§π‘›β€–β€–πΈπ›Όπ‘›β€–β€–π‘’,𝑑β‰₯0,π‘‘π΄π‘’βˆ—π‘›,𝑛𝑣𝑛‖‖𝐸𝛼n≀𝑀2𝑒𝛾𝑑‖‖𝑣𝑛‖‖𝐸𝛼𝑛,𝑑≀0,(3.11) for all π‘£π‘›βˆˆπ‘ƒπ‘›(𝜎+𝑛)𝐸𝛼𝑛 and π‘§π‘›βˆˆ(πΌπ‘›βˆ’π‘ƒπ‘›(𝜎+𝑛))𝐸𝛼𝑛. One has to note that the neighborhood of any part of 𝑖ℝ does not intersect 𝜎(π΄π‘’βˆ—π‘›,𝑛). Moreover, the condition Ξ”π‘π‘β‰ βˆ… implies that 𝑃𝑛(𝜎+𝑛)π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π‘ƒ(𝜎+) compactly, and therefore, as was shown in [14], dim𝑃𝑛(𝜎+𝑛)=dim𝑃(𝜎+) for 𝑛β‰₯𝑛0. One can consider for the 𝑇=𝑇(𝑣0𝑛)=sup{𝑑β‰₯0βˆΆπ‘£π‘›(𝑑,𝑣0𝑛)βˆˆπ’°πΈπ›Όπ‘›(0;𝜌)} with 𝑣0π‘›βˆˆπ’°πΈπ›Όπ‘›(0;𝜌) the problem 𝑣′𝑛(𝑑)=π΄π‘’βˆ—π‘›,𝑛𝑣𝑛(𝑑)+πΉπ‘’βˆ—π‘›,𝑛𝑣𝑛(𝐼𝑑),0≀𝑑≀𝑇,π‘›βˆ’π‘ƒπ‘›ξ€·πœŽ+𝑛𝑣𝑛𝐼(0)=π‘›βˆ’π‘ƒπ‘›ξ€·πœŽ+𝑛𝑣0𝑛,π‘ƒπ‘›ξ€·πœŽ+𝑛𝑣𝑛(𝑇)=π‘ƒπ‘›ξ€·πœŽ+𝑛𝑣𝑇𝑛.(3.12) The mild solution of problem (3.12) is given by the formula (for 0≀𝑑≀𝑇) 𝑣𝑛(𝑑)=𝑒(π‘‘βˆ’π‘‡)π΄π‘’βˆ—π‘›,π‘›π‘ƒπ‘›ξ€·πœŽ+𝑛𝑣𝑇𝑛+π‘’π‘‘π΄π‘’βˆ—π‘›,π‘›ξ€·πΌπ‘›βˆ’π‘ƒπ‘›ξ€·πœŽ+𝑛𝑣0𝑛+ξ€œπ‘‘0𝑒(π‘‘βˆ’π‘ )π΄π‘’βˆ—π‘›,π‘›ξ€·πΌπ‘›βˆ’π‘ƒπ‘›ξ€·πœŽ+π‘›πΉξ€Έξ€Έπ‘’βˆ—π‘›,π‘›ξ€·π‘£π‘›ξ€Έξ€œ(𝑠)𝑑𝑠+𝑇𝑑𝑒(π‘‘βˆ’π‘ )π΄π‘’βˆ—π‘›,π‘›π‘ƒπ‘›ξ€·πœŽ+π‘›ξ€ΈπΉπ‘’βˆ—π‘›,𝑛𝑣𝑛(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 𝐴𝑛,𝐴 be generators of analytic 𝐢0-semigroups and let condition (𝐡1) be satisfied. Assume also that the semigroup π‘’π‘‘π΄π‘’βˆ— is hyperbolic, 𝜎(π΄π‘’βˆ—)∩{πœ†βˆΆReπœ†β‰₯0}=π‘ƒπœŽ(π΄π‘’βˆ—),dim𝑃(𝜎+)<∞, and for 𝜌>0 such that {πœ†βˆΆβˆ’πœŒβ‰€Reπœ†β‰€πœŒ}βŠ‚πœŒ(π΄π‘’βˆ—), operators πœ†πΌπ‘›βˆ’π΄π‘’βˆ—π‘›,𝑛 are Fredholm operators of ind 0, and operators πœ†πΌπ‘›βˆ’π΄π‘’βˆ—π‘›,𝑛,πœ†πΌβˆ’π΄π‘’βˆ— are regularly consistent for any Reπœ†β‰₯βˆ’πœŒ. Then, 𝑃𝑛(𝜎+𝑛)π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π‘ƒ(𝜎+) compactly and β€–β€–π‘’π‘‘π΄π‘’βˆ—π‘›,π‘›ξ€·πΌπ‘›βˆ’π‘ƒπ‘›ξ€·πœŽ+𝑛‖‖𝐸𝑛≀𝑀2π‘’βˆ’π›Ύπ‘‘β€–β€–π‘’,𝑑β‰₯0,π‘‘π΄π‘’βˆ—π‘›,π‘›π‘ƒπ‘›ξ€·πœŽ+𝑛‖‖𝐸𝑛≀𝑀2𝑒𝛾𝑑,𝑑≀0,(3.14) where 𝛾>0.

Proof. The condition (𝐡1) implies that (πœ†πΌπ‘›βˆ’π΄π‘’βˆ—π‘›,𝑛)βˆ’1π’«π’«βˆ’βˆ’βˆ’βˆ’β†’(πœ†πΌβˆ’π΄π‘’βˆ—)βˆ’1 as βˆ’πœŒβ‰€Reπœ†β‰€πœŒ for |πœ†| big enough. For the other βˆ’πœŒβ‰€Reπœ†β‰€πœŒ, the convergence (πœ†πΌπ‘›βˆ’π΄π‘’βˆ—π‘›,𝑛)βˆ’1π’«π’«βˆ’βˆ’βˆ’βˆ’β†’(πœ†πΌβˆ’π΄π‘’βˆ—)βˆ’1 follows from analogy of Theorem 3.9 for closed operators. Now the compact convergence 𝑃𝑛(𝜎+𝑛)π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π‘ƒ(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 𝐢0-semigroup with condition 𝜎(𝐴)∩{πœ†βˆΆReπœ†β‰₯0}=π‘ƒπœŽ(𝐴),dim𝑃(𝜎+)<∞ and corresponding conditions on approximation of operators. The structure of operator like 𝐴+π‘“ξ…ž(π‘’βˆ—) is not necessary.

Theorem 3.18. Let the operators 𝐴𝑛,𝐴 be generators of analytic 𝐢0-semigroups and let the condition (𝐡1) be satisfied. Assume also that the 𝐢0-semigroup π‘’π‘‘π΄π‘’βˆ— is hyperbolic, 𝜎(π΄π‘’βˆ—)∩{πœ†βˆΆReπœ†β‰₯0}=π‘ƒπœŽ(π΄π‘’βˆ—),dim𝑃(𝜎+)<∞. Assume also that Δ𝑐𝑐(𝐴𝑛,𝐴)β‰ βˆ… and resolvents of 𝐴𝑛,𝐴 are compact operators. Then, (3.14) holds.

Proof. We set 𝐡𝑛=𝑒1π΄π‘’βˆ—π‘›,𝑛 and 𝐡=𝑒1π΄π‘’βˆ— and apply Proposition 3.15. It is known [3] that Ξ”π‘π‘β‰ βˆ… is equivalent to compact convergence π΅π‘›π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π΅. Then, the condition (i) of Theorem 3.12 is satisfied and one gets discrete dichotomy for 𝐡𝑛. From the other side, since we have exactly the form 𝐡𝑛=𝑒1π΄π‘’βˆ—π‘›,𝑛, this means by Theorem 2.7 and Proposition 2.4 that operators ℒ𝑛=βˆ’π‘‘/𝑑𝑑+π΄π‘’βˆ—π‘›,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 Ξ›βŠ†π‚ be some open connected set, and let 𝐡∈𝐡(𝐸). For an isolated point πœ†βˆˆπœŽ(𝐡), the corresponding maximal invariant space (or generalized eigenspace) will be denoted by 𝒲(πœ†;𝐡)=𝑄(πœ†)𝐸, where βˆ«π‘„(πœ†)=(1/2πœ‹π‘–)|πœβˆ’πœ†|=𝛿(πœπΌβˆ’π΅)βˆ’1π‘‘πœ and 𝛿 is small enough so that there are no points of 𝜎(𝐡) in the disc {𝜁∢|πœβˆ’πœ†|≀𝛿} different from πœ†. The isolated point πœ†βˆˆπœŽ(𝐡) is a Riesz point of 𝐡 if πœ†πΌβˆ’π΅ is a Fredholm operator of index zero and 𝑄(πœ†) is of finite rank. Denote π’²ξ€·πœ†,𝛿;𝐡𝑛=ξšπœ†π‘›ξ€·π΅π‘›ξ€Έ,π‘›βˆˆπœŽ|πœ†βˆ’πœ†|<π›Ώπ’²ξ€·πœ†π‘›,𝐡𝑛.(3.15) It is clear that 𝒲(πœ†,𝛿;𝐡𝑛)=𝑄𝑛(πœ†)𝐸𝑛, where𝑄𝑛(1πœ†)=ξ€œ2πœ‹π‘–|πœβˆ’πœ†|=π›Ώξ€·πœπΌπ‘›βˆ’π΅π‘›ξ€Έβˆ’1π‘‘πœ.(3.16)

Definition 3.19. The function πœ‡(β‹…) is said to be the measure of noncompactness if for any bounded sequence {π‘₯𝑛},π‘₯π‘›βˆˆπΈπ‘›, one has πœ‡π‘₯𝑛=infπœ–>0βˆΆβˆ€β„•ξ…žβŠ†β„•,βˆƒβ„•ξ…žξ…žβŠ†β„•ξ…ž,π‘₯ξ…žβ€–β€–π‘₯∈𝐸suchthatπ‘›βˆ’π‘π‘›π‘₯β€²β€–β€–β‰€πœ–,π‘›βˆˆβ„•ξ…žξ…žξ‚‡.(3.17)

Definition 3.20. We say that the operators π΅π‘›βˆˆπ΅(𝐸𝑛) are jointly condensing with constant π‘ž>0 with respect to measure πœ‡(β‹…) if for any bounded sequence {π‘₯𝑛},π‘₯π‘›βˆˆπΈπ‘›, one has πœ‡π΅ξ€·ξ€½π‘›π‘₯𝑛π‘₯ξ€Ύξ€Έβ‰€π‘žπœ‡ξ€·ξ€½π‘›ξ€Ύξ€Έ.(3.18)

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

Proposition 3.21. Let π΅π‘›π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π΅ for π΅π‘›βˆˆπ΅(𝐸𝑛),𝐡∈𝐡(𝐸) and πœ‡({𝐡𝑛π‘₯𝑛})β‰€π‘žπœ‡({π‘₯𝑛}), for any bounded sequence {π‘₯𝑛},π‘₯π‘›βˆˆπΈπ‘›. Assume that 𝜎(𝐡)∩Ψ=βˆ…, where bounded closed set Ξ¨βŠ‚π‚β§΅{πœ†βˆΆ|πœ†|β‰€π‘ž} and 𝜎(𝐡)⧡{πœ†βˆΆ|πœ†|≀πͺ} consists only of discrete spectrum. Then, there is a constant 𝐢>0 such that β€–(πœ†πΌπ‘›βˆ’π΅π‘›)βˆ’1‖≀𝐢,πœ†βˆˆΞ¨,π‘›βˆˆβ„•.

Proof. Any point πœ†βˆˆΞ¨ belongs to π‘ƒπœŽ(𝐡𝑛)βˆͺ𝜌(𝐡𝑛). This means that one gets for a sequence β€–π‘₯𝑛‖=1,π‘₯π‘›βˆˆπΈπ‘›, two cases: (πœ†πΌπ‘›βˆ’π΅π‘›)π‘₯𝑛=0 or β€–(πœ†πΌπ‘›βˆ’π΅π‘›)π‘₯𝑛‖β‰₯π›Ύπœ†,𝑛‖π‘₯𝑛‖ with some π›Ύπœ†,𝑛>0,πœ†βˆˆΞ¨. We are going to show that in reality we have β€–(πœ†πΌπ‘›βˆ’π΅π‘›)π‘₯𝑛‖β‰₯𝛾Ψ‖π‘₯𝑛‖,πœ†βˆˆΞ¨.
Assume in contradiction that there are sequences {πœ†π‘›},πœ†π‘›βˆˆΞ¨,{π‘₯𝑛},β€–π‘₯𝑛‖=1, such thatξ€·πœ†π‘›πΌπ‘›βˆ’π΅π‘›ξ€Έπ‘₯π‘›π’«βˆ’βˆ’β†’0asπ‘›βˆˆβ„•.(3.19) Then, πœ†π‘›β†’πœ†0∈Ψ,π‘›βˆˆβ„•ξ…žβŠ†β„•. One has for Μƒπ‘Ÿ=inf{|πœ‰|βˆΆπœ‰βˆˆΞ¨}πœ‡π‘₯𝑛≀||πœ†ξ€Ύξ€Έπ‘›||πœ‡π‘₯Μƒπ‘Ÿξ€·ξ€½π‘›β‰€πœ‡π΅ξ€Ύξ€Έξ€·ξ€½π‘›π‘₯π‘›ξ€Ύξ€Έβ‰€π‘žΜƒπ‘Ÿπœ‡π‘₯Μƒπ‘Ÿξ€·ξ€½π‘›,ξ€Ύξ€Έ(3.20) which means because of π‘ž/Μƒπ‘Ÿ<1, that πœ‡({π‘₯𝑛})=0, that is, {π‘₯𝑛} is 𝒫-compact. Now π‘₯π‘›π’«βˆ’βˆ’β†’π‘₯0,π‘›βˆˆβ„•ξ…žξ…žβŠ†β„•ξ…ž and 𝐡𝑛π‘₯π‘›π’«βˆ’βˆ’β†’π΅π‘₯0,πœ†π‘›π‘₯π‘›π’«βˆ’βˆ’β†’πœ†0π‘₯0,π‘›βˆˆβ„•ξ…žξ…ž, that is, πœ†0π‘₯0=𝐡π‘₯0 with β€–π‘₯0β€–=1, which contradicts our assumption 𝜎(𝐡)∩Ψ=βˆ…. The proposition is proved.

Proposition 3.22. Let π΅π‘›π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π΅ and πœ‡({𝐡𝑛π‘₯𝑛})β‰€π‘žπœ‡({π‘₯𝑛}) for any bounded sequence {π‘₯𝑛},π‘₯π‘›βˆˆπΈπ‘›. Assume that any πœ†0∈𝜎(𝐡),|πœ†0|>π‘ž, is an isolated eigenvalue with the finite dimensional projector 𝑄(πœ†0). Then, there are sequence {πœ†π‘›},πœ†π‘›βˆˆπœŽ(𝐡𝑛), and sequence of projectors 𝑄𝑛(πœ†0)∈𝐡(𝐸𝑛) such that πœ†π‘›β†’πœ†0 and 𝑄𝑛(πœ†0)π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π‘„(πœ†0) converge compactly.

Proof. Note first that for Ξ“π‘Ÿ={πœ†βˆΆ|πœ†βˆ’πœ†0|=π‘Ÿ}βŠ‚π‚β§΅{πœ†βˆΆ|πœ†|≀πͺ}, where π‘Ÿ can be taken small enough, we have by Proposition 3.21ξ€·πœ†πΌπ‘›βˆ’π΅π‘›ξ€Έβˆ’1π’«π’«βˆ’βˆ’βˆ’βˆ’β†’(πœ†πΌβˆ’π΅)βˆ’1asπœ†βˆˆΞ“π‘Ÿ,π‘›βˆˆβ„•.(3.21) Therefore, 𝑄𝑛(πœ†0)π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π‘„(πœ†0). To show compact convergence of these projectors one can note that πœ‡πœ†ξ€·ξ€½ξ€·0πΌπ‘›βˆ’π΅π‘›ξ€Έπ‘₯𝑛β‰₯||πœ†ξ€Ύξ€Έ0||πœ‡π‘₯ξ€·ξ€½π‘›π΅ξ€Ύξ€Έβˆ’πœ‡ξ€·ξ€½π‘›π‘₯𝑛β‰₯||πœ†ξ€Ύξ€Έ0||πœ‡π‘₯𝑛π‘₯ξ€Ύξ€Έβˆ’π‘žπœ‡ξ€·ξ€½π‘›π‘₯ξ€Ύξ€Έβ‰₯π›Ύπœ‡ξ€·ξ€½π‘›ξ€Ύξ€Έ,(3.22) where 𝛾=|πœ†0|βˆ’π‘ž>0. This means that πœ‡ξ€·πœ†ξ‚€ξ‚†0πΌπ‘›βˆ’π΅π‘›ξ€Έπ‘˜π‘₯𝑛β‰₯π›Ύπ‘˜πœ‡π‘₯𝑛foranyπ‘˜βˆˆβ„•.(3.23) By functional calculus ξ€·πœ†0πΌπ‘›βˆ’π΅π‘›ξ€Έπ‘˜π‘„π‘›ξ€·πœ†0ξ€Έπ‘₯𝑛=1ξ€œ2πœ‹π‘–Ξ“π‘Ÿξ€·πœ†0ξ€Έβˆ’πœ†π‘˜ξ€·πœ†πΌπ‘›βˆ’π΅π‘›ξ€Έβˆ’1π‘₯π‘›π‘‘πœ†.(3.24) From this representation using (3.23), one has π›Ύπ‘˜πœ‡π‘„ξ€·ξ€½π‘›ξ€·πœ†0ξ€Έπ‘₯π‘›β‰€β€–β€–ξ€·πœ†ξ€Ύξ€Έ0πΌπ‘›βˆ’π΅π‘›ξ€Έπ‘˜π‘„π‘›ξ€·πœ†0ξ€Έπ‘₯π‘›β€–β€–β‰€πΆπ‘Ÿ2πœ‹π‘˜β€–β€–π‘₯𝑛‖‖.(3.25) It is clear that from π‘Ÿ/𝛾<1 it follows that (π‘Ÿ/𝛾)π‘˜β†’0 as π‘˜β†’βˆž. This means that 𝑄𝑛(πœ†0)π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π‘„(πœ†0) compactly. The proposition is proved.

Theorem 3.23. Let the conditions (𝐴) and (𝐡1) be satisfied, and let the analytic 𝐢0-semigroup 𝑒𝑑𝐴,π‘‘βˆˆβ„+, be hyperbolic such that the set 𝜎(𝐴)∩{πœ†βˆΆReπœ†β‰₯0} consists of a finite number of points π‘ƒπœŽ(𝐴) and dim𝑃(𝜎+)<∞. Assume also that πœ‡({𝐡𝑛π‘₯𝑛})β‰€π‘žπœ‡({π‘₯𝑛}), for any bounded sequence {π‘₯𝑛},π‘₯π‘›βˆˆπΈπ‘›, with π‘ž<1, where 𝐡𝑛=𝑒1𝐴𝑛. Then, conclusions of Theorem 3.16, that is, (3.14), are hold and 𝑃𝑛(πœŽπ‘›+)π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π‘ƒ(𝜎+) converge compactly.

Proof. Because of the spectral mapping theorem, the spectrum of operator 𝐡=𝑒1𝐴 which is located outside of unit disc 𝕋 consists of finite number of points of the set π‘ƒπœŽ(𝑒1𝐴)={𝜁∢𝜁=π‘’πœ†,πœ†βˆˆπ‘ƒπœŽ(𝐴)∩{πœ‰βˆΆReπœ‰β‰₯0}}. Moreover, since π‘ž<1, for any Ξ¨ which contains 𝕋 and Ξ¨βŠ‚πœŒ(𝐡),𝐡=𝑒1𝐴, one has β€–(πœ†πΌπ‘›βˆ’π΅π‘›)βˆ’1β€–β‰€π‘π‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘ as πœ†βˆˆΞ¨ by Proposition 3.22. Now from Theorem 3.12 it follows that 𝐡𝑛 have discrete dichotomy. By Theorem 2.5, the operator π’Ÿπ‘› is invertible, and by Proposition 2.4 and by Theorem 2.3 we get that the semigroups 𝑒𝑑𝐴𝑛,π‘‘βˆˆβ„+, have an exponential dichotomy uniformly in index π‘›βˆˆβ„• and (2.7), that is, (3.14) holds. By Theorem 3.9  𝑄𝑛(πœ†0)π’«π’«βˆ’βˆ’βˆ’βˆ’β†’π‘„(πœ†0) compactly, which implies that conditions of Theorem 3.16 are satisfied. The theorem is proved.

3.4. Discretisation in Time Variable and Ξ”π‘π‘β‰ βˆ…

Consider now the discretization of the problem (3.9) in time by the following scheme:𝑉𝑛𝑑+πœπ‘›ξ€Έβˆ’π‘‰π‘›(𝑑)πœπ‘›=π΄π‘’βˆ—π‘›,𝑛𝑉𝑛𝑑+πœπ‘›ξ€Έ+πΉπ‘’βˆ—π‘›,𝑛𝑉𝑛(𝑑),𝑑=π‘˜πœπ‘›,(3.26) with initial data 𝑉𝑛(0)=𝑣0𝑛. The solution of such problem is given by formula𝑉𝑛𝑑+πœπ‘›ξ€Έ=ξ€·πΌπ‘›βˆ’πœπ‘›π΄π‘’βˆ—π‘›,π‘›ξ€Έβˆ’1𝑉𝑛(𝑑)+πœπ‘›ξ€·πΌπ‘›βˆ’πœπ‘›π΄π‘’βˆ—π‘›,π‘›ξ€Έβˆ’1πΉπ‘’βˆ—π‘›,𝑛𝑉𝑛=𝐼(𝑑)π‘›βˆ’πœπ‘›π΄π‘’βˆ—π‘›,π‘›ξ€Έβˆ’π‘˜π‘‰π‘›(0)+πœπ‘›π‘˜ξ“π‘—=0ξ€·πΌπ‘›βˆ’πœπ‘›π΄π‘’βˆ—π‘›,π‘›ξ€Έβˆ’(π‘˜βˆ’π‘—+1)πΉπ‘’βˆ—π‘›,π‘›ξ€·π‘‰π‘›ξ€·π‘—πœπ‘›ξ€Έξ€Έ,𝑑=π‘˜πœπ‘›,(3.27) where 𝑉𝑛(0)=𝑣0𝑛.

The problem (3.12) also can be discretized following (3.26) approach, so we have𝑉𝑛𝑑+πœπ‘›ξ€Έβˆ’π‘‰π‘›(𝑑)πœπ‘›=π΄π‘’βˆ—π‘›,𝑛𝑉𝑛𝑑+πœπ‘›ξ€Έ+πΉπ‘’βˆ—π‘›,𝑛𝑉𝑛(𝑑),𝑑=π‘˜πœπ‘›,𝐼(3.28)π‘›βˆ’π‘ƒπ‘›ξ€Έπ‘‰π‘›ξ€·πΌ(0)=π‘›βˆ’π‘ƒπ‘›ξ€Έπ‘£0𝑛,𝑃𝑛𝑉𝑛(𝑇)=𝑃𝑛𝑣𝑇𝑛.(3.29) The solution of problem (3.28) could be obtained by using the formulasξ€·πΌπ‘›βˆ’π‘ƒπ‘›ξ€Έπ‘‰π‘›ξ€·π‘‘+πœπ‘›ξ€Έ=ξ€·πΌπ‘›βˆ’πœπ‘›π΄π‘’βˆ—π‘›,π‘›ξ€Έβˆ’1ξ€·πΌπ‘›βˆ’π‘ƒπ‘›ξ€Έπ‘‰π‘›(𝑑)+πœπ‘›ξ€·πΌπ‘›βˆ’πœπ‘›π΄π‘’βˆ—π‘›,π‘›ξ€Έβˆ’1ξ€·πΌπ‘›βˆ’π‘ƒπ‘›ξ€ΈπΉπ‘’βˆ—π‘›,π‘›ξ€·π‘‰π‘›ξ€·π‘—πœπ‘›,ξ€·πΌξ€Έξ€Έπ‘›βˆ’πœπ‘›π΄π‘’βˆ—π‘›,𝑛𝑃𝑛𝑉𝑛𝑑+πœπ‘›ξ€Έ=𝑃𝑛𝑉𝑛(𝑑)+πœπ‘›π‘ƒπ‘›πΉπ‘’βˆ—π‘›,π‘›ξ€·π‘‰π‘›ξ€·π‘˜πœπ‘›ξ€Έξ€Έ,𝑑=π‘˜πœπ‘›.(3.30) So one has a representation of solution of problem (3.28) as𝑉𝑛𝑑;𝑣0𝑛,𝑣𝑇𝑛=ξ€·πΌπ‘›βˆ’π‘ƒπ‘›ξ€Έπ‘‰π‘›(𝑑)+𝑃𝑛𝑉𝑛=𝐼(𝑑)π‘›βˆ’πœπ‘›π΄π‘’βˆ—π‘›,π‘›ξ€Έβˆ’π‘˜+1ξ€·πΌπ‘›βˆ’π‘ƒπ‘›ξ€Έπ‘£0𝑛+πœπ‘›π‘˜βˆ’1𝑗=0ξ€·πΌπ‘›βˆ’πœπ‘›π΄π‘’βˆ—π‘›,π‘›ξ€Έβˆ’(π‘˜βˆ’π‘—)ξ€·πΌπ‘›βˆ’π‘ƒπ‘›ξ€ΈπΉπ‘’βˆ—π‘›,π‘›ξ€·π‘‰π‘›ξ€·π‘—πœπ‘›+ξ€·πΌξ€Έξ€Έπ‘›βˆ’πœπ‘›π΄π‘’βˆ—π‘›,π‘›ξ€ΈπΎβˆ’π‘˜π‘ƒπ‘›π‘£π‘‡π‘›βˆ’πœπ‘›πΎβˆ’1𝑗=π‘˜ξ€·πΌπ‘›βˆ’πœπ‘›Aπ‘’βˆ—π‘›,π‘›ξ€ΈπΎβˆ’1βˆ’π‘—π‘ƒπ‘›πΉπ‘’βˆ—π‘›,π‘›ξ€·π‘‰π‘›ξ€·π‘—πœπ‘›ξ€Έξ€Έ,𝑑=π‘˜πœπ‘›.(3.31) From (3.31), it is clear that corresponding estimates on powers of operators (πΌπ‘›βˆ’πœπ‘›π΄π‘’βˆ—π‘›,𝑛)βˆ’π‘˜+1(πΌπ‘›βˆ’π‘ƒπ‘›),(πΌπ‘›βˆ’πœπ‘›π΄π‘’βˆ—π‘›,𝑛)πΎβˆ’π‘˜π‘ƒπ‘› play the main role in approximation of solutions of (1.4) in the vicinity of π‘’βˆ—.

Theorem 3.24. Let operators 𝐴𝑛,𝐴 be generators of analytic 𝐢0-semigroups and let condition (𝐡1) be satisfied. Assume also that the analytic 𝐢0-semigroup π‘’π‘‘π΄π‘’βˆ—,π‘‘βˆˆβ„+, is hyperbolic and for 𝜌>0 such that {πœ†βˆΆβˆ’πœŒβ‰€Reπœ†β‰€πœŒ}βŠ‚πœŒ(𝐴), operators πœ†πΌπ‘›βˆ’π΄π‘’βˆ—π‘›,𝑛 are Fredholm operators of ind 0, and operators πœ†πΌπ‘›βˆ’π΄