Eurasian mathematical journal

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ISSN (Print): 2077-9879 ISSN (Online): 2617-2658

Eurasian

Mathematical Journal

2020, Volume 11, Number 4

Founded in 2010 by

the L.N. Gumilyov Eurasian National University in cooperation with

the M.V. Lomonosov Moscow State University

the Peoples’ Friendship University of Russia (RUDN University) the University of Padua

Starting with 2018 co-funded

by the L.N. Gumilyov Eurasian National University and

the Peoples’ Friendship University of Russia (RUDN University)

Supported by the ISAAC

(International Society for Analysis, its Applications and Computation) and

by the Kazakhstan Mathematical Society

Published by

the L.N. Gumilyov Eurasian National University

Nur-Sultan, Kazakhstan

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Editorial Board

Editors–in–Chief

V.I. Burenkov, M. Otelbaev, V.A. Sadovnichy

Vice–Editors–in–Chief

K.N. Ospanov, T.V. Tararykova

Editors

Sh.A. Alimov (Uzbekistan), H. Begehr (Germany), T. Bekjan (China), O.V. Besov (Russia), N.K. Bliev (Kazakhstan), N.A. Bokayev (Kazakhstan), A.A. Borubaev (Kyrgyzstan), G. Bourdaud (France), A. Caetano (Portugal), M. Carro (Spain), A.D.R. Choudary (Pakistan), V.N. Chubarikov (Russia), A.S. Dzumadildaev (Kazakhstan), V.M. Filippov (Russia), H. Ghazaryan (Armenia), M.L. Goldman (Russia), V. Goldshtein (Israel), V. Guliyev (Azerbaijan), D.D. Haroske (Germany), A. Hasanoglu (Turkey), M. Huxley (Great Britain), P. Jain (India), T.Sh. Kalmenov (Kazakhstan), B.E. Kangyzhin (Kazakhstan), K.K. Kenzhibaev (Kazakhstan), S.N. Kharin (Kazakhstan), E. Kissin (Great Britain), V. Kokilashvili (Georgia), V.I. Korzyuk (Belarus), A. Kufner (Czech Republic), L.K. Kussainova (Kazakhstan), P.D. Lamberti (Italy), M. Lanza de Cristoforis (Italy), F. Lan- zara (Italy), V.G. Maz’ya (Sweden), K.T. Mynbayev (Kazakhstan), E.D. Nursultanov (Kazakhstan), R. Oinarov (Kazakhstan), I.N. Parasidis (Greece), J. Peˇcari´c (Croatia), S.A. Plaksa (Ukraine), L.- E. Persson (Sweden), E.L. Presman (Russia), M.A. Ragusa (Italy), M.D. Ramazanov (Russia), M. Reissig (Germany), M. Ruzhansky (Great Britain), M.A. Sadybekov (Kazakhstan), S. Sagitov (Sweden), T.O. Shaposhnikova (Sweden), A.A. Shkalikov (Russia), V.A. Skvortsov (Poland), G. Sin- namon (Canada), E.S. Smailov (Kazakhstan), V.D. Stepanov (Russia), Ya.T. Sultanaev (Russia), D. Suragan (Kazakhstan), I.A. Taimanov (Russia), J.A. Tussupov (Kazakhstan), U.U. Umirbaev (Kazakhstan), Z.D. Usmanov (Tajikistan), N. Vasilevski (Mexico), Dachun Yang (China), B.T. Zhu- magulov (Kazakhstan)

Managing Editor

A.M. Temirkhanova

c

The L.N. Gumilyov Eurasian National University

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Aims and Scope

The Eurasian Mathematical Journal (EMJ) publishes carefully selected original research papers in all areas of mathematics written by mathematicians, principally from Europe and Asia. However papers by mathematicians from other continents are also welcome.

From time to time the EMJ publishes survey papers.

The EMJ publishes 4 issues in a year.

The language of the paper must be English only.

The contents of the EMJ are indexed in Scopus, Web of Science (ESCI), Mathematical Reviews, MathSciNet, Zentralblatt Math (ZMATH), Referativnyi Zhurnal – Matematika, Math-Net.Ru.

The EMJ is included in the list of journals recommended by the Committee for Control of Education and Science (Ministry of Education and Science of the Republic of Kazakhstan) and in the list of journals recommended by the Higher Attestation Commission (Ministry of Education and Science of the Russian Federation).

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Authors may nominate a member of the Editorial Board whom they consider appropriate for the article. However, assignment to that particular editor is not guaranteed.

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Manuscripts are accepted for review on the understanding that the same work has not been already published (except in the form of an abstract), that it is not under consideration for publication elsewhere, and that it has been approved by all authors.

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For information on Ethics in publishing and Ethical guidelines for journal publication see http://www.elsevier.com/publishingethics and http://www.elsevier.com/journal-authors/ethics.

Submission of an article to the EMJ implies that the work described has not been published previously (except in the form of an abstract or as part of a published lecture or academic thesis or as an electronic preprint, see http://www.elsevier.com/postingpolicy), that it is not under consideration for publication elsewhere, that its publication is approved by all authors and tacitly or explicitly by the responsible authorities where the work was carried out, and that, if accepted, it will not be published elsewhere in the same form, in English or in any other language, including electronically without the written consent of the copyright-holder. In particular, translations into English of papers already published in another language are not accepted.

No other forms of scientific misconduct are allowed, such as plagiarism, falsification, fraudulent data, incorrect interpretation of other works, incorrect citations, etc. The EMJ follows the Code of Conduct of the Committee on Publication Ethics (COPE), and follows the COPE Flowcharts for Resolving Cases of Suspected Misconduct (http://publicationethics.org/files/u2/NewCode.pdf).

To verify originality, your article may be checked by the originality detection service CrossCheck http://www.elsevier.com/editors/plagdetect.

The authors are obliged to participate in peer review process and be ready to provide corrections, clarifications, retractions and apologies when needed. All authors of a paper should have significantly contributed to the research.

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The Editorial Board of the EMJ will monitor and safeguard publishing ethics.

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The procedure of reviewing a manuscript, established by the Editorial Board of the Eurasian Mathematical Journal

1. Reviewing procedure

1.1. All research papers received by the Eurasian Mathematical Journal (EMJ) are subject to mandatory reviewing.

1.2. The Managing Editor of the journal determines whether a paper fits to the scope of the EMJ and satisfies the rules of writing papers for the EMJ, and directs it for a preliminary review to one of the Editors-in-chief who checks the scientific content of the manuscript and assigns a specialist for reviewing the manuscript.

1.3. Reviewers of manuscripts are selected from highly qualified scientists and specialists of the L.N. Gumilyov Eurasian National University (doctors of sciences, professors), other universities of the Republic of Kazakhstan and foreign countries. An author of a paper cannot be its reviewer.

1.4. Duration of reviewing in each case is determined by the Managing Editor aiming at creating conditions for the most rapid publication of the paper.

1.5. Reviewing is confidential. Information about a reviewer is anonymous to the authors and is available only for the Editorial Board and the Control Committee in the Field of Education and Science of the Ministry of Education and Science of the Republic of Kazakhstan (CCFES). The author has the right to read the text of the review.

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1.13. Originals reviews are stored in the Editorial Office for three years from the date of publication and are provided on request of the CCFES.

1.14. No fee for reviewing papers will be charged.

2. Requirements for the content of a review

2.1. In the title of a review there should be indicated the author(s) and the title of a paper.

2.2. A review should include a qualified analysis of the material of a paper, objective assessment and reasoned recommendations.

2.3. A review should cover the following topics:

- compliance of the paper with the scope of the EMJ;

- compliance of the title of the paper to its content;

- compliance of the paper to the rules of writing papers for the EMJ (abstract, key words and phrases, bibliography etc.);

- a general description and assessment of the content of the paper (subject, focus, actuality of the topic, importance and actuality of the obtained results, possible applications);

- content of the paper (the originality of the material, survey of previously published studies on the topic of the paper, erroneous statements (if any), controversial issues (if any), and so on);

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- exposition of the paper (clarity, conciseness, completeness of proofs, completeness of biblio- graphic references, typographical quality of the text);

- possibility of reducing the volume of the paper, without harming the content and understanding of the presented scientific results;

- description of positive aspects of the paper, as well as of drawbacks, recommendations for corrections and complements to the text.

2.4. The final part of the review should contain an overall opinion of a reviewer on the paper and a clear recommendation on whether the paper can be published in the Eurasian Mathematical Journal, should be sent back to the author for revision or cannot be published.

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The web-page of the EMJ is www.emj.enu.kz. One can enter the web-page by typing Eurasian Mathematical Journal in any search engine (Google, Yandex, etc.). The archive of the web-page contains all papers published in the EMJ (free access).

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Tel.: +7-7172-709500 extension 33312 13 Kazhymukan St

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EURASIAN MATHEMATICAL JOURNAL ISSN 2077-9879

Volume 11, Number 4 (2020), 08 – 24

ALMOST PERIODIC AT INFINITY FUNCTIONS

FROM HOMOGENEOUS SPACES AS SOLUTIONS TO DIFFERENTIAL EQUATIONS WITH UNBOUNDED OPERATOR COEFFICIENTS

A.G. Baskakov, V.E. Strukov, I.I. Strukova

Communicated by A.A. Shkalikov

Key words: almost periodic at infinity function, vanishing at infinity function, homogeneous space, Banach module, differential equations, asymptotically finite-dimensional operator semigroup.

AMS Mathematics Subject Classification: 26A99, 34A99, 46B25.

Abstract. By using the subspace of functions from homogeneous spaces with integrals decreasing at infinity we define new classes of functions almost periodic at infinity. We obtain spectral criteria for a function to be almost periodic at infinity and asymptotically almost periodic (with respect to the chosen subspace). These results are used for deriving criteria for almost periodicity at infinity of bounded solutions to differential equations with unbounded operator coefficients. In addition, for the new class of asymptotically finite-dimensional operator semigroups we prove the almost periodicity at infinity of their orbits.

DOI: https://doi.org/10.32523/2077-9879-2020-11-4-08-24

1 Introduction

Many authors studied the problem of almost periodicity of bounded functions on the real axis R = (−∞,+∞) and semi-axis R+ = [0,∞) presenting solutions to differential and functional equations (see [1–10]). One of the first articles in this direction was written by Bochner and von Neumann [17].

They established the almost periodicity of bounded solutions to difference equations. In [24] Sobolev obtained a criterion for almost periodicity of solutions to parabolic equations with a selfadjoint operator on a Hilbert space.

In this article, a family is introduced of subspaces of vanishing at infinity (in certain senses) functions belonging to homogeneous spaces. We give four equivalent definitions (see Theorem 5.1) of functions almost periodic at infinity with respect to a chosen subspace of functions vanishing at infinity. These functions are used in studying bounded solutions to differential equations with unbounded operator coefficients.

While studying differential equations we use the method developed in [4, 5]. It is based on showing that the set of nonbelonging (see Definition 19) of a bounded solution to the differential equation

˙

x(t)−Ax(t) =ψ(t), t∈R, (1.1)

is included in σ(A)∩(iR), with σ(A)standing for the spectrum of an operatorA, whereA :D(A)⊂ X → X is a generator of a strongly continuous operator semigroup, X is a complex Banach space andψ :R→X is a continuous almost periodic function. This method has become the most common for studying the almost periodicity of bounded solutions to differential equations (see also [1, 3, 22]).

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Almost periodic at infinity solutions to differential equations 9 As in [5, 6], the derivation of the criterion of almost periodicity at infinity of bounded solutions to (1.1) with the function ψ almost periodic at infinity (with respect to a subspace) is based on the description of the Beurling spectrum of a bounded solution to this equation (Theorem 4.1) and a criterion for almost periodicity of a vector (Theorem 4.1). The main result of this article appears in Theorems 6.3 and 6.4: we obtain a sufficient condition for almost periodicity at infinity of a mild solution to (1.1) with the function ψ almost periodic at infinity with respect to some subspace of functions vanishing at infinity.

Section 7 introduces the new class of asymptotically finite-dimensional operator semigroups. We establish a theorem on the almost periodicity at infinity of the orbits of the operator semigroup under consideration.

2 Homogeneous function spaces

Let X be a complex Banach space, End X be a Banach algebra of bounded linear operators on X. Let J be either R⁺ = [0,∞) or R = (−∞,∞). By L¹_loc(J, X) we denote the space of Bochner measurable locally integrable on J (classes of) functions with values in X. By S^p(J, X), p ∈[1,∞), we denote the Stepanov space of functions [21]x∈L¹_loc(J, X) with the norm

kxk_S^p = sup

s∈J





1

Z

0

kx(s+t)k^p_Xdt





1/p

, p∈[1,∞).

The Stepanov function spaces play an important role when studying differential equations in Banach spaces (see [9]). By C_b(J, X) we denote the space of bounded continuous functions with the norm kxk∞= sup

t∈J

kx(t)k_X, x∈C_b(J, X).

In the case X = C we omit ’X’ in designations of the spaces mentioned above. For example, C_b(R) = C_b(R,C).

Definition 1. A Banach space F(R, X) of functions defined on R, with values in X is called homogeneous, if the following conditions are satisfied:

(a) the space F(R, X) is injectively (that means injectivity of the inclusion operator) and continuously embedded in S¹(R, X);

(b) the group of shift operatorsS(t) for t∈R, where

(S(t)x)(s) = x(s+t), s, t∈R, x∈F(R, X), (2.1) is defined and bounded on F(R, X);

(c) for any f ∈L¹(R)and x∈F(R, X) their convolution defined by (f∗x)(t) =

Z

R

f(τ)x(t−τ)dτ = Z

R

f(τ)(S(−τ)x)(t)dτ, t∈R, (2.2) belongs to F(R, X) and satisfies the inequality kf∗xk ≤Ckfkkxk for some C ≥1 independent of x and f (usually, C= 1);

(d) the inclusionϕx∈F(R, X)holds for any x∈F(R, X)and any infinitely differentiable function ϕ ∈ C_b(R) which has a compact support suppϕ; in addition, the inequality kϕxk ≤ kϕkkxk holds and the mappingt 7→ϕS(t)x:R→F(R, X) is continuous.

By F0(R, X) we denote the least closed subspace of F(R, X) containing all functions ϕx, x∈ F(R, X), whereϕ∈Cb(R, X) is infinitely differentiable and suppϕis a compact set.

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10 A.G. Baskakov, V.E. Strukov, I.I. Strukova

Definition 2. A Banach space F(R+, X) of functions from S¹(R+, X) is called homogeneous, if there exists a homogeneous space F(R, X) associated with the space F(R+, X) such that for any function x∈F(R+, X) there is an extension y∈F(R, X) with the following properties:

1) y(t) = x(t)for all t∈R+;

2) kyk ≤Ckxk, where C >0 is independent of x and y;

3) y∈F0(R⁻, X);

4) S(t)x∈F(R+, X) for all t∈R+ and x∈F(R+, X);

5) for any other extensionz ∈F(R, X)possessing properties 1)–4) the conditiony−z ∈F0(R, X) holds.

Further in this article we denote a homogeneous space by F(J, X), while in case X =C we use the notationF(J)(see [26]).

Definition 3. A closed subspace of F(J, X) defined as {x ∈ F(J, X) : the function t 7→ S(t)x : J→F(J, X)is continuous} is denoted by Fc(J, X).

For example, (L^p(J, X))_c={x∈L^p(J, X) :function t7→S(t)x:J→L^p(J, X)is continuous}.

Example 1. The following Banach spaces of functions defined on J and with values in a Banach space X are homogeneous. All of them are linear subspaces ofL¹_loc(J, X).

1. The spaces L^p = L^p(J, X), p ∈ [1,∞), of Lebesgue measurable and integrable with power p ∈ [1,∞) (classes of) functions with the norm kxk_p = R

J

kx(t)k^p_Xdt

!1/p

, p ∈ [1,∞). It should be noted that(L^p(J, X))_c =L^p(J, X), (L^p(J, X))₀ =L^p(J, X).

2. The space L^∞=L^∞(J, X)of essentially bounded (classes of) functions with the normkxk∞ = supvrai

t∈J

kx(t)k_X.Note that (L^∞(J, X))_c =C_b,u(J, X).

3. The Stepanov spaces S^p =S^p(J, X), p∈[1,∞).

4. The Wiener amalgam spaces (L^p, l^q) = (L^p(J, X), l^q(J, X)), p∈ [1,∞), q ∈[1,∞], (see [9]) of functions in L¹_loc(J, X) with the norm kxk_p,q =

P

k∈Z

R1

0 kx(s+k)k^pdsq/p1/q

< ∞, p, q ∈ [1,∞). These spaces are homogeneous with respect to the following equivalent norm:

sup

t∈[0,1]

X

k∈Z

Z 1 0

kx(s+t+k)k^pds

^q/p!1/q

<∞, p, q ∈[1,∞).

5. The space C_b = C_b(J, X) of bounded continuous functions. Note that C_b(J, X) is a closed subspace of L^∞(J, X) and (Cb(J, X))c=Cb,u(J, X), (Cb(J, X))0 =C0(J, X).

6. The subspace C_b,u = C_b,u(J, X) of C_b(J, X) of uniformly continuous functions belonging to C_b. It should be noted that (C_b,u(J, X))_c = C_b,u(J, X), (C_b,u(J, X))₀ =C₀(J, X).

7. The subspace C₀ =C₀(J, X) of C_b,u(J, X)of continuous vanishing at infinity functions. These functions satisfy the condition lim

|t|→∞kx(t)k= 0, t∈J.

8. The subspace Csl,∞ = Csl,∞(J, X) of C_b,u(J, X) of continuous slowly varying at infinity functions. These functions satisfy the condition lim

|t|→∞kx(t+τ)−x(t)k= 0, t, τ ∈J (see [15, 27]).

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Almost periodic at infinity solutions to differential equations 11 9. The subspace Cω,∞ = Cω,∞(J, X) of C_b,u(J, X) of continuous ω-periodic at infinity functions, ω ∈ R+. These functions satisfy the condition

|t|→∞lim kx(t+ω)−x(t)k= 0, t∈J (see [15, 27]).

10. The subspace AP∞ =AP∞(J, X) of C_b,u(J, X) of continuous almost periodic at infinity functions (see [9, 10]).

11. The subspaces C^k = C^k(J, X), k ∈ N, of k times continuously differentiable functions with bounded k-th derivative and the norm kxk_(k) =kxk∞+kx^(k)k∞.

12. The H¨older spaces C^k,α=C^k,α(J, X), k ∈N∪ {0}, α∈(0,1], C^k,α =

x∈C^k :kx^(k)k_C^0,α = sup

t6=s∈J

|x^(k)(t)−x^(k)(s)|

|t−s|^α <∞

,

kxk_C^k,α =kxk_C^k +kx^(k)k_C^0,α.

13. The subspace V = V(J, X) of functions from L^∞(J, X) with bounded variation kxk_V = sup

t∈J

V_t^t+1(x) + sup

t∈J

kxk_X which is used as a norm.

Definition 1 implies that each of the homogeneous spaces F(R, X) mentioned above is endowed with the structure of a BanachL¹(R)-module by using convolution (2.2), whereS is a group of shifts defined by (2.1). Thus, there is an opportunity to use classic notions and results of the theory of Banach L¹(R)-modules given below. In particular, in this terminology the spaces F_c(R, X) coincide with the spaces of S-continuous vectors (see Definition 16).

3 Almost periodic at infinity functions

LetF(J, X)be a homogeneous space of functions that satisfies Definition 1 forJ=Rand Definition 2 for J = R⁺. In the Banach space F(J, X) we consider the (semi-) group S : J → EndF(J, X) of shift operators (S(t)x)(τ) = x(t+τ) for t, τ ∈J.

Definition 4. A function x from F(J, X) is called integrally vanishing at infinity, if the following condition holds:

α→∞lim 1 αsup

t∈J α

Z

0

kx(t+s)kds= 0.

The set of functions from F(J, X) integrally vanishing at infinity is denoted by F0,int =F0,int(J, X).

Definition 4 directly implies that F0,int(J, X) is a closed subspace ofF(J, X).

Definition 5. By Fmid = Fmid(J, X) we denote a closed subspace of F(J, X) with the following properties:

1) S(t)x∈Fmid for every t ∈Jand x∈Fmid; 2) F0 ⊂Fmid ⊂F0,int;

3) eλx∈Fmid for every λ∈R, whereeλ(t) =e^iλt for t∈R.

The spaces of this kind are called the subspaces of F(J, X) of functions vanishing at infinity.

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In particular, such subspaces include the set of closed subspaces F0,p for p∈[1,∞)of a homogeneous space F(J, X) defined by

F0,p =F0,p(J, X) = {x∈F(J, X) : lim

t→∞

1 αsup

t∈J α

Z

0

kx(s+t)k^pds= 0}.

Note thatF0,1(J, X) = F0,int(J, X).

Definition 6. A functionx∈Fc(J, X)is calledslowly varying at infinity with respect to the subspace Fmid(J, X) if and only if the condition (S(α)x−x)∈Fmid(J, X) is fulfilled for everyα ∈J.

Note that in articles [9, 10, 11, 12] slowly varying at infinity with respect to the subspaceC₀(R, X) functions are defined and their properties are studied. Slowly varying and periodic at infinity (with respect to F0(R, X)) functions in F(R, X) were studied in [26]). Particularly, in [11] it was established that the solutions of the heat equation are slowly varying at infinity (with respect to time and the subspace C₀(R, X)).

By (Fmid)sl,∞ we denote the set of functions in F(J, X) slowly varying at infinity with respect to the subspace Fmid(J, X).

Definition 6 directly implies that the space (Fmid)sl,∞ ⊂F(J, X) of functions slowly varying at infinity with respect to the subspace Fmid(J, X) is a closed subspace ofF(J, X).

Here are some examples of functions from (C₀)sl,∞(J) = (C₀)sl,∞(J,C)slowly varying at infinity:

1) x₁(t) = sin ln(1 +|t|), t∈J;

2) x2(t) =c+x0(t) fort ∈J,where c∈Cand x0 ∈C0(J);

3) x₃(t) = arctg t, t∈J;

4) any continuously differentiable function x ∈ C_b(J) with the property

|t|→∞lim x(t) = 0.˙

Let us state four definitions of functions fromF(J, X)almost periodic at infinity with respect to the subspaceFmid(J, X)of vanishing at infinity functions fromF(J, X). After that we are going to prove them to be equivalent.

To start with, let us introduce a definition of continuous almost periodic at infinity function (see [10, 16]), which is based on the notion of ε-periodicity at infinity.

Definition 7. Assume that ε >0.A number ω > 0 is called an ε-period at infinity of x∈Cb(J, X) if there exists a number a(ε)>0such that sup

|t|≥a(ε)

kx(t+ω)−x(t)k< ε.

The set of ε-periods at infinity of x∈C_b(J, X)is denoted by Ω∞(ε, x).

Definition 8. A subset Ω of J is called relatively dense in J whenever there exists l > 0 with [t, t+l]∩Ω6=∅ for every t∈J.

Definition 9. A function x∈C_b(J, X)is calledalmost periodic at infinity whenever for every ε >0 the set Ω∞(ε, x)of its ε-periods at infinity is relatively dense in J.

Definitions 7 and 9 imply that every function x ∈ C_b(J, X) almost periodic in Bohr’s sense (see [7, 21]) is almost periodic at infinity.

The set of classical almost periodic functions is denoted by AP(J, X). The set of continuous almost periodic at infinity functions is denoted by AP∞(J, X). These functions were considered in (see [9, 10, 16]).

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Almost periodic at infinity solutions to differential equations 13 Definition 10. Assume thatε >0.A numberω >0 is called anε-period at infinity of x∈F(J, X) with respect to the subspace Fmid(J, X)whenever there exists a function x₀ ∈Fmid(J, X)such that kS(ω)x−x−x₀k< ε.

The set ofε-periods at infinity ofx∈F(J, X)with respect to the subspaceFmid(J, X)is denoted byΩ∞(x;Fmid;ε).

Definition 11. A function x from Fc(J, X) is called almost periodic at infinity with respect to the subspace Fmid(J, X)whenever for any ε >0 the set Ω∞(x;Fmid;ε)of its ε-periods at infinity (with respect toFmid(J, X)) is relatively dense in J.

Definitions 10 and 11 imply that every function x ∈ Cb(J, X) almost periodic in Bohr’s sense (see [7, 21]) is almost periodic at infinity with respect to every subspace (C_b)_mid of vanishing at infinity functions from C_b(J, X).

The set of functions fromF(J, X)almost periodic at infinity with respect toFmid(J, X)is denoted by AP∞F(J, X;Fmid). Definition 6 directly implies that Ω∞(x;Fmid;ε) = J for all x∈ (Fmid)sl,∞

and ε >0,hence, x∈AP∞F(J, X;Fmid). Consequently, (Fmid)sl,∞ ⊂AP∞F(J, X;Fmid).

Definition 12. A set M ⊂ F(J, X) is called precompact at infinity with respect to the subspace Fmid(J, X) whenever for any ε > 0 there exist finitely many functions b₁, . . . , b_N ∈ M called an ε-net at infinity such that for every x∈ M there exist a function b_k, k ∈ {1, . . . , N}, and a function α_ε ∈Fmid(J, X) such that kx−b_k−α_εk< ε.

Definition 13. A function x ∈ Fc(J, X) is called almost periodic at infinity with respect to the subspace Fmid(J, X) whenever the set of shifts S(t)x of x for t ∈ J is precompact at infinity with respect toFmid(J, X).

For F(J, X) = C_b,u(J, X) Definition 13 corresponds to Bochner’s criterion (see [21]) of almost periodicity.

Observe that the functions of the form x(t) =

N

X

k=1

x_k(t)e^iλ^k^t, x₁, . . . , x_N ∈(Fmid)sl,∞, λ₁, . . . , λ_N ∈R,

called generalized trigonometric polynomials, are almost periodic at infinity with respect to Fmid(J, X) in the sense of Definition 13.

Definition 14. A function x ∈ Fc(J, X) is called almost periodic at infinity with respect to the subspace Fmid(J, X) whenever given ε > 0 we can indicate a finite collection λ₁, . . . , λ_N of real numbers and functions x₁, . . . , x_N from(Fmid)_sl,∞ such thatsup

t∈J

kx(t)− P^N

k=1

x_k(t)e^iλ^k^tk< ε.

Definition 15. A function x∈Fc(J, X) is calledasymptotically almost periodic with respect to the subspace Fmid(J, X)whenever x=x₁+x₀ with x₁ ∈AP(J, X), x₀ ∈Fmid(J, X).

The set AAPF(J, X;Fmid) of asymptotically almost periodic functions with respect to Fmid(J, X) is a closed subspace of F(J, X).Moreover,

AAPF(J, X;Fmid) = AP(J, X)⊕Fmid(J, X).

Thus, the strict inclusions

AP(J, X)⊂AAPF(J, X;Fmid)⊂AP_∞F(J, X;Fmid) hold.

It should be noted that for F =C_b(R⁺, X)the space AAPF(R⁺, X;C₀) was considered in [22].

It was denoted byAAP(R+, X)there.

The fourth definition of a function almost periodic at infinity with respect toFmid(J, X)is given after introducing some definitions and results of the spectral theory of Banach L¹(R)-modules.

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4 Almost periodic vectors from Banach L

¹

( R )-modules

LetX be a complex Banach space and EndX be a Banach algebra of linear operators on X. By L¹(R) we denote the algebra of complex Lebegue’s measurable (classes of) functions summable on R with the convolution as multiplication:

(f∗g)(t) = Z

R

f(t−s)g(s)ds, t∈R, f, g∈L¹(R).

We endow X with the structure of a nondegenerate Banach L¹(R)-module (see [5, 7, 8, 13, 20]) associated with some bounded isometric representation T : R → End X. This means that the following properties hold.

Assumption 1. A Banach L¹(R)-module X satisfies the conditions below:

1) if f x = 0 for every function f ∈ L¹(R) then the vector x ∈ X is null (a nondegeneracy property of X);

2) for every f ∈L¹(R) and x∈X the following equations hold true:

T(t)(f x) = (T(t)f)x=f(T(t)x) for t∈R,

which means that the module structure on X is associated with the representation T :R→EndX. If T :R→EndX is a strongly continuous isometric representation then the formula

T(f)x=f x= Z

R

f(t)T(−t)xdt, f ∈L¹(R), x∈X,

endows X with the structure of a Banach L¹(R)-module satisfying the conditions of Assumption 1.

This structure is associated with the representationT.

Remark 1. With every nondegenerate Banach L¹(R)-module X we associate a unique representation T : R → End X (see [13]). In order to emphasise that sometimes we use the notation (X, T).

Definition 16. A vector from a BanachL¹(R)-module(X, T)is called continuous (with respect to the representation T) or T-continuous if a functionϕx :R→X defined by ϕx(t) = T(t)x for t∈R is continuous at t= 0 (hence, it is continuous onR).

A set of all continuous vectors from a Banach L¹(R)-module X denoted by Xc or (X, T)_c is a closed submodule in X, i.e., Xc is a closed linear subspace in X invariant under shift operators T(f), T(t)for f ∈L¹(R) and t∈R.

Every homogeneous space F(R, X) is endowed with the structure of a Banach L¹(R)-module using the convolution (2.2), where S : R → End F(R, X) is the group of shifts defined by (2.1).

However, formula (2.2) does not define the structure of L¹(R)-module for F(R+, X). Nevertheless, the quotient spaceF(R+, X)/Fmid(R+, X)is endowed with the structure of a BanachL¹(R)-module.

In what follows we use some definitions and results of the spectral theory of Banach modules (see [2, 4, 5, 6, 7, 8, 14]) which stem from Domar [18] and Reiter [23].

Given a function f fromL¹(R), the Fourier transformfb:R→Cis defined as f(λ) =b

Z

R

f(t)e^−iλtdt, λ∈R.

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Almost periodic at infinity solutions to differential equations 15 Definition 17. By the Beurling spectrum of a vector x ∈X we call the set of numbers Λ(x) inR defined by

Λ(x) = {λ₀ ∈R:f x6= 0 for every functionf ∈L¹(R)withfb(λ₀)6= 0}.

The definition implies that Λ(x) = R\{µ₀ ∈ R : there exists a function f ∈ L¹(R) such that fb(µ₀)6= 0andf x= 0}.

The Beurling spectrum of vectors in a Banach L¹(R)-module X has the properties listed below.

Lemma 4.1. For every x∈X and f ∈L¹(R) the following properties hold true:

1) the set Λ(x) is closed and Λ(x) =∅ if and only if x= 0 ; 2) Λ(f x)⊆(suppfb)S

Λ(x);

3) f x = 0 when (suppf)b T

Λ(x) = ∅ and f x = x if the set Λ(x) is compact and fb= 1 in its neighbourhood;

4) the set Λ(x) is a singleton ((Λ(x) ={λ₀})) if and only if x6= 0 and T(t)x=e^iλ⁰^tx for t∈R. Remark 2. As we indicated above, a homogeneous space F(R, X) is a Banach L¹(R)-module. If a function x ∈F(R, X) has the property Λ(x) ={λ₀} then it can be represented as x(t) = x₀e^iλ⁰^t for t∈R, where x₀ ∈X.

Further in this paper we use the following concept of an almost periodic vector in a Banach space X (see [5, 6, 7, 8]) with a strongly continuous isometric representation T :R→EndX.

Definition 18. A vector x0 ∈ X is called almost periodic (with respect to a representation T) if one of the following conditions is met:

1) for every ε >0the set Ω(x₀, ε) ={ω ∈R:kT(ω)x₀−x₀k< ε} of ε-periods of the vectorx₀ is relatively dense in R;

2) the orbit {T(t)x₀, t∈R} of x₀ is precompact in X;

3) t 7→ ϕ(t) = T(t)x₀, t ∈ R, is a continuous almost periodic function, i.e., ϕ ∈ AP(R, X) (see [21]);

4) for any ε > 0 there are eigenvalues λ₁, . . . , λ_N and associated eigenvectors x₁, . . . , x_N of the representation T, i.e., T(t)x_k =e^iλ^k^tx_k for t∈R and k = 1, . . . , N such that kx₀−

N

P

k=1

x_kk< ε.

The set AP(X) = AP(X, T) of almost periodic vectors (with respect to the representation T) is a closed submodule of X. Observe that AP(C_b,u(R, X), S) = AP(R, X) and AP(X)⊂Xc.

If X0 is a closed submodule of X invariant under operators T(t),t∈R, then the quotient space X/X0 is also a Banach L¹(R)-module whose structure for all f ∈ L¹(R) and equivalence classes ex = x +X0, x ∈ X, is defined as fxe = f x+X0 = f x.f This structure is associated with the representation

Te:R→EndX/X0, Te(t)xe=^T(t)x=T(t)x+X0, x∈X.

Definition 19. For any vector x ∈ X the Beurling spectrum Λ(ex) of the equivalence class xe = x+X0 is called a set of nonbelonging of x in the submodule X0 and denoted by Λ(x,X0). the set Λ(x, AP(X)) is calleda set of nonalmost periodicity of x.

The following statement established in [5] is important for further.

Theorem 4.1. If the set Λ(x) for x∈X lacks limit points on R then x∈AP(X).

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Theorem 4.2. Assume that the set Λ(x, AP(X)) for x ∈ X is no-more-than countable. In this case x is almost periodic whenever every limit point ofΛ(x, AP(X)) is ergodic for x. In particular, if the set Λ(x, AP(X)) lacks limit points then x∈AP(X).

Theorem 4.2 was established in [4]. It can also be found in [2, 21].

Let us recall that the notationFmid =Fmid(J, X)stands for the subspace ofF(J, X)of functions vanishing at infinity, where F0 ⊂ Fmid ⊂F0,int. The definitions of the subspaces Fmid(R, X) and (Fmid)sl,∞ imply that they are closed subspaces of F(J, X) invariant under shift operators. Thus, they are closed submodules of F(R, X).

By X(J), J ∈ {R+,R}, we denote the quotient space F(J, X)/Fmid(J, X), which is a Banach space under the norm kexk = inf

y∈x+Fmid

kyk, where ex = x+Fmid(J, X) is the equivalence class containing the function x∈F(J, X). The Banach space X(J) is a Banach algebra with multiplication rule defined by exey=fxy, ex,ey∈X(J).

On X(R) we define the strongly continuous group of isometries Se : R → EndX(R) by the formula

S(t)e ex=S(t)x, x] ∈F(R, X), t∈R.

The quotient spaceX(R)is endowed with the structure of a BanachL¹(R)-module using the formula fex=f]∗x, f ∈L¹(R), x∈F(R, X).

Remark 3. Assume that J=R+.Each function x∈F(R+, X)can be extended to the function yon R so that y satisfied all 5 conditions of Definition 2. Note that the equivalence class ex∈X(R) does not depend on the certain extension and, consequently, the Banach space X(R+) is isometrically embedded in X(R) as a closed submodule. In this case the group Se is correctly defined on X(R+) and the space X(R+) is endowed with the structure of a Banach L¹(R)-module as well.

5 Criteria of almost periodicity at infinity

Now let us give the fourth definition of function fromF(J, X)almost periodic at infinity with respect to the subspace Fmid(J, X).

Definition 20. A function x from Fc(J, X) is called almost periodic at infinity with respect to the subspace Fmid(J, X) of functions from F(J, X) vanishing at infinity whenever the equivalence class xe = x+Fmid(J, X) is an almost periodic vector from X(J) with respect to the isometric representation Se:R→EndX(J).

The functions in C_b,u(J, X) almost periodic at infinity (with respect to the subspace C₀(J, X)) appeared for the first time in [9, 10]. In those papers the definition similar to Definition 20 was used. The main results of those papers deal with the asymptotic behaviour of bounded operator semigroups.

On the Banach spaceAP(X) =AP(X, T)of almost periodic vectors there exists a unique linear operator J ∈End AP(X) with the properties:

1) kJ k= 1;

2) J(T(t)x) =Jxfor t ∈Jand x∈X.

For every vector x from AP(X) let us consider a function xb_B : R→AP(X), bx_B(λ) =J(T_λx), where T_λ(t) = T(t)e^−iλt for t ∈ R. This function is called the Bohr transform of the vector x. Its support suppxbB is at most countable set, i.e., suppxbB ={λ1, λ2, . . .}, and

T(t)xk =e^iλ^k^txk, t ∈R, k ≥1,

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Almost periodic at infinity solutions to differential equations 17 wherex_k, k ≥1, are eigenvectors of the representation T (they are also eigenvectors of the generator iA of the operator group T, i.e., iAx_k =iλ_kx_k, k ≥ 1). Moreover, Λ(x_k) ={λ_k} for k ≥1. The set Λ_B(x) ={λ₁, λ₂, . . .} is called the Bohr spectrum of the vector x∈AP(X).

The series

x∼X

k≥1

x_k (5.1)

is called the Fourier series of the vector x ∈ AP(X). Note that if the series converges then x= P

k≥1

x_k.

Lemma 5.1. Given a functionf from L¹(R) and an almost periodic vector x∈AP(X, T) with the Fourier series of form (5.1), the vector f xis almost periodic with the Fourier series of the form

f x∼X

k≥1

fb(λ_k)x_k.

The following lemma uses the bounded approximative identity (b.a.i.) (f_n), n ≥ 1, in L¹(R) constructed below (see [5, 6, 7, 8]).

Let us consider a functionfb₀ in the spaceLb¹(R)of Fourier transforms of functions in L¹(R)(with pointwise multiplication) with compact support suppfb₀ on the interval [−1,1] such that fb₀(0) = 1.

Then the sequence (f_n), n≥1, of functions of the form

f_n(t) =nf₀(nt), t∈R, (5.2)

is a bounded approximation of the identity in L¹(R). Let us note that kf_nk=kf₀k for n≥1.

Lemma 5.1 implies the following

Lemma 5.2. Let x be an almost periodic vector from AP(X, T) with the Bohr spectrum ΛB(x) = {λ₁, λ₂, ...}, where (λ_n), n≥1, satisfies the condition lim

n→∞|λ_n|=∞. In this case

n→∞lim kx−f_nxk= lim

n→∞kx− X

|λk|<n

fb₀(λ_k n )x_kk,

where x_k, k ≥1, are taken from expansion into its Fourier series (5.1).

Theorem 5.1. All definitions of almost periodic at infinity functions from F(J, X) with respect to the subspace Fmid(J, X) (Definitions 11, 13, 14, 20) are equivalent.

Proof. We assume J = R (the result for J = R+ follows from Remark 3). Let us consider the quotient space X(R) = F(R, X)/Fmid(R, X) and the group of isometries T =Se:R→EndX(R) defined above. For this representation Definition 14 corresponds to property 4) of Definition 18.

Since all properties of Definition 18 are equivalent, it suffices to show that the first three properties are equivalent to Definitions 11,13 and 20 respectively.

Givenx∈F(R, X),take the equivalence classexinX(R),constructed from the functionx.Then for each ε >0the setΩ∞(x;Fmid;ε)∪(−Ω∞(x;Fmid;ε)) coincides with the set Ω(x, ε)e of ε-periods of ex. Hence, the corresponding definitions are equivalent.

The equivalence of Definition 13 and property 2) of Definition 18 follows directly from the definition of the quotient module X(R).

In order to verify the equivalence of the approximative Definition 14 and property 4) of Defini- tion 18, it suffices to establish that the Beurling spectrum Λ(ey) of the equivalence class ye∈ X(R)

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with ye=y+Fmid is a singleton (Λ(ey) ={λ₀}) if and only if we can represent y ∈F(R, X) in the formy(t) =y₀(t)e^iλ⁰^t for t∈R, where y₀ ∈(Fmid)_sl,∞.

If Λ(y) =e {λ₀}, then S(t)e ey = e^iλ⁰^tey for every t ∈ R (see property 4) of Lemma 4.1). Hence, Λ(ye₀) = {0}, where y₀(s) = y(s)e^−iλ⁰^s for s ∈ R. Therefore, S(t)e ye₀ = ye₀ for every t ∈ R. Thus, S(t)y₀−y₀ ∈Fmid(R, X) for t∈R, i.e., y₀ ∈(Fmid)_sl,∞.

Conversely, if y(t) =y₀(t)e^iλ⁰^t for t ∈ R, where y₀ ∈ Fmid(R, X), then S(t)e ye= e^iλ⁰^tey for t ∈ R and so property 4) of Lemma 4.1 implies thatΛ(y) =e {λ₀} (see Remark 2).

Observe that the closedness of the space AP∞F(J, X;Fmid) follows, for instance, from the approximative Definition 14.

Note the following inclusions:

Fmid(J, X)⊂(Fmid)sl,∞ ⊂AP∞F(J, X;Fmid), AP(J, X)⊂AP_∞F(J, X;Fmid).

In the results mentioned below by X we denote the quotient space F(J, X)/Fmid(J, X) and by Y — the quotient spaceF(J, X)/AP∞F(J, X;Fmid).

Theorem 5.2. If the set Λ_∞(x,Fmid) lacks limit points then x∈AP(J, X;Fmid).

Proof. As we mentioned before, the quotient space X is a Banach L¹(R)-module whose structure is endowed by the representation Se : R → EndX mentioned above. Therefore, the set Λ(ex) = Λ∞(x) lacks limit points in R and, consequently, Theorem 4.1 implies that the equivalence class ex=x+Fmid(J, X)is an almost periodic vector inAP(X), which means thatx∈AP∞F(J, X;Fmid) (see Definition 20).

Definition 21. The setΛ(ex), whereex=x+Fmid(J, X)withx∈F(J, X)is calleda spectrum of the function x at infinity with respect to the subspace Fmid =Fmid(J, X) and denoted by Λ_∞(x;Fmid).

The set Λ(x, AP∞F(J, X;Fmid)) = Λ(ex) for xe = x+AP∞F(J, X;Fmid) ∈ Y is called a set of nonalmost periodicity of x (see Definition 19).

Theorem 5.3. Assume that the set Λ(x, AP∞F(J, X;Fmid)) for x ∈ Fc(J, X) is no more than countable. In this casexis almost periodic at infinity with respect to the subspaceFmid =Fmid(J, X) whenever every limit point of Λ(x, AP∞F(J, X;Fmid)) is ergodic for the equivalence class xe=x+ AP∞F(J, X;Fmid)) from the factor-module Y. In particular, if the set Λ(x, AP∞F(J, X;Fmid)) lacks limit points then x∈AP∞F(J, X;Fmid).

The proof of Theorem 5.3 follows directly from Theorem 4.2 by using the fact that the quotient space Y is endowed by the structure of a Banach L¹(R)-module.

These results are of great importance in the derivation of spectral criteria for almost periodicity at infinity of solutions to differential equations.

We also derived the following spectral criterion of asymptotic almost periodicity with respect to the subspace Fmid(J, X).

Theorem 5.4. A function x ∈ AP∞F(J, X;Fmid) is asymptotically almost periodic with respect to the subspace Fmid(J, X) (i.e., x ∈ AAPF(J, X;Fmid)) whenever every point of the spectrum Λ∞(x;Fmid) of the function x at infinity with respect to the subspace Fmid(J, X) is ergodic for x.

Proof. The necessity of the statement of the theorem follows from the fact that every point of the set Λ∞(x)is egrodic forx₁ ∈AP(J, X;Fmid)andx₀ ∈Fmid(J, X),wherex=x₁+x₀ (see Definition 15).

The sufficiency follows from Definition 20 and Theorem 4.2 by using Lemma 2 from [10].

An analogous result for uniformly continuous functions was obtained in [10].

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Almost periodic at infinity solutions to differential equations 19

6 Almost periodic at infinity solutions to differential equations

In this section, as above, the notation Fmid = Fmid(J, X) stands for the subspace of F(J, X) of functions vanishing at infinity. Observe the inclusions F0(J, X) ⊂ Fmid(J, X) ⊂ F0,int(J, X) and nonseparability of the Banach space (Fmid)_sl,∞ of functions slowly varying at infinity with respect toFmid(J, X) of functions vanishing at infinity.

Take the linear operator A : D(A) ⊂ X → X with domain D(A) which is a generator of a strongly continuous operator semigroupU :R⁺ →End X.

Consider the differential equation

˙

x(t)−Ax(t) = ψ(t), t∈J, ψ ∈F(J, X). (6.1) Refer as a classical solution to differential equation (6.1) to a differentiable function x : J →X such that x(t)∈D(A)for every t ∈Jsatisfying (6.1) for all t∈J.

Let us state two definitions of a mild solution to (6.1) with ψ ∈F(J, X).

Definition 22. A continuous function x : J → X is called a mild solution to the differential equation (6.1) whenever a functionz :J→X of the formz(t) =

t

R

0

x(s)ds fort ∈J has the following properties:

1) z(t)∈D(A)for every t ∈J; 2) x(t)−x(0) =Az(t) +

t

R

0

ψ(s)ds fort ∈J.

Definition 23. A function x:J→X is called a mild solution to (6.1) whenever x(t) =U(t−s)x(s) +

t

Z

s

U(t−τ)ψ(τ)dτ (6.2)

for all s, t ∈Jsuch that s≤t.

For continuous functions the equivalence of these definitions of a mild solution to (6.1) is proved in [2, Chapter 3].

In the Banach space F =F(R, X) consider the linear operator L = d

dt −A:D(L)⊂F(R, X)→F(R, X).

Definition 24. A functionx∈F(R, X) lies in thedomain D(L)of the operatorL whenever there exists a function ψ ∈F(R, X) such that equation (6.2) holds for alls≤t in R.

For x ∈ D(L) put Lx = ψ provided that x and ψ satisfy (6.2). For F(R, X) = C_b,u(R, X) this definition of the operator L appeared for the first time in [4, 5] (see also [21, 9, 14]).

By S(f) : F(R, X) → F(R, X) we denote the operator S(f)x = f ∗x of convolution of x ∈ F(R, X) with f ∈L¹(R). The following commutation theorem was established in [14].

Theorem 6.1. For every f from L¹(R) and every xfrom D(L) the function S(f)x belongs to D(L) and satisfies the condition LS(f)x=S(f)Lx.

Remark 4. Given ϕ ∈ AP∞F(R+, X;Fmid) every its extension ϕ₁ on R satisfying the conditions 1)-5) of Definition 2 with the property lim

t→−∞ϕ₁(t) = 0 lies in the space AP∞F(R, X;Fmid). It follows from the inclusions F0(J, X)⊂Fmid(J, X), ±Ω_∞(ϕ, ε)⊂Ω∞(ϕ1, ε), where ε >0.

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The following result is of great importance for the derivation of a spectral criterion for almost periodicity at infinity of mild solutions to differential equation (6.1).

Theorem 6.2. For every mild solutionx∈F(J, X) to differential equation (6.1) withψ ∈F(J, X) we have

Λ∞(x,Fmid)⊂(i⁻¹σ(A)∩R)∪Λ∞(ψ,Fmid). (6.3) Proof. To start with assume that J =R. Choose a number λ₀ so that iλ₀ lies outside ∆ = (σ(A)∩ (iR))∪Λ∞(ψ,Fmid). Consider f in L¹(R) such that fb(λ₀) 6= 0 and suppfb∩∆ = ∅. Theorem 6.1 implies that f∗x lies in the domainD(L)of the operator L =−_dt^d +A and

L(f ∗x) =f ∗ Lx=f∗ψ. (6.4)

Inclusion (6.3) implies that the resolvent of A is defined in some neighbourhood iV₀ ⊂iR of iλ₀, furthermore, V₀ ∩Λ_∞(ψ,Fmid) = ∅. Then from the definition of Λ_∞(ψ,Fmid) it follows that the functionψ₀ =f∗ψbelongs to the subspaceFmid. Choose a numberδ₀ >0so that[λ₀−δ₀;λ₀+δ₀]⊂ V₀.

Consider an infinitely differentiable function cϕ₀ : R → C such that cϕ₀(λ₀) 6= 0 and suppcϕ₀ ⊂ [λ₀−δ₀;λ₀+δ₀]. It is the Fourier transform of some functionϕ₀ ∈L¹(R), while

Fb(λ) =

ϕb₀R(iλ, A), λ∈[λ₀−δ₀;λ₀+δ₀], 0, λ /∈[λ₀−δ₀;λ₀+δ₀], is the Fourier transform of a summable functionF :R→EndX.

Equation (6.4) implies that

F ∗ L(y∗x) = ϕ₀∗x=F ∗ϕ∗ψ ∈Fmid(R, X).

Since cϕ₀(λ₀)6= 0 it follows that λ₀ lies outside the set Λ∞(x,Fmid). Thus, inclusion (6.3) is proved for functions defined on R.

Now we assume that J = R+. Consider a continuously differentiable function ϕ : R → R with the properties suppϕ⊂[1,∞)and ϕ≡1on[2,∞). By ϕx we denote the function equal to zero on (−∞,0]and the product of ϕ and x onR+. Then ϕx ∈ D(L) and L(ϕx) = ˙ϕx+ϕψ, where ϕψ is regarded as the identical zero on R⁻. Sinceϕx˙ ∈Fmid(R, X) and ϕ= 0 on(−∞,0], we have

Λ∞(ψ,Fmid(R⁺, X)) = Λ∞(ϕψ,Fmid(R, X)),

Λ∞(x,Fmid(R⁺, X)) = Λ∞(ϕx,Fmid(R, X)),

where Fmid(R, X) is the subspace of functions in F(R, X), which are extensions of functions in Fmid(R+, X) to(−∞,0]with the property lim

t→−∞y₁(t) = 0 for every extension y₁ of y.

Theorem 6.3. Assume that a function ψ in (6.1) belongs to AP∞F(J, X;Fmid) and the set ((i⁻¹σ(A))∩R)∪Λ∞(ψ,Fmid)

lacks limit points on R. Then each bounded mild solution x to (6.1) is almost periodic at infinity with respect to Fmid(J, X) (x∈AP∞F(J, X;Fmid)).

The statement of Theorem 6.3 immediately follows from Theorem 6.2 and the spectral criterion of almost periodicity at infinity (Theorem 5.2).

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Almost periodic at infinity solutions to differential equations 21 Theorem 6.4. Assume that a function ψ in (6.1) belongs to Fmid = Fmid(J, X) and the set σ(A) ∩ (iR) lacks limit points on iR. Then each bounded mild solution x to (6.1) belongs to AP∞F(J, X;Fmid)and for each ε >0we can find numbers iλ₁, . . . , iλ_m ∈σ(A)∩(iR)and functions x₁, . . . , x_m in (Fmid)sl,∞ such that

sup

t∈J

kx(t)−

m

X

k=1

xk(t)e^iλ^k^tk< ε,

furthermore, x(t) =

m

P

k=1

x_k(t)e^iλ^k^t, t∈R, whenever σ(A)∩(iR) ={iλ₁, . . . , iλ_m} is a finite set.

Proof. Since x belongs toFmid(J, X)we have Λ∞(ψ,Fmid) =∅. Therefore, Λ∞(x,Fmid)⊂(i⁻¹σ(A))∩R.

Suppose thatΛ∞(x,Fmid) = {λ1, λ2, . . .}is a countable set. Then lim

n→∞|λn|=∞. Regarding the quotient space X =F(J, X)/Fmid(J, X) as a Banach L¹(R)-module, the definition of Λ∞(x,Fmid) directly implies thatΛ(x,e S) =e {λ₁, λ₂, . . .}. Theorem 5.2 yields that the equivalence class ex∈X is an almost periodic vector in the L¹(R)-module (X,S).e

Lemma 5.2 implies that the bounded approximation of the identity(f_n), n≥1,specified in (5.2) satisfies the condition

n→∞lim kex− X

|λ_k|<n

fb₀(λ_k

n )ex_kk= 0. (6.5)

SinceΛ(xe_k) = {λ_k},it follows (see the proof of Theorem 5.1) that we can represent each function x_k as x_k(t) = x⁰_k(t)e^iλ^k^t for t ∈ J, where x⁰_k ∈ (Fmid)_sl,∞ and |λ_k| < n. Therefore, (6.5) implies the existence of some sequence(y_n) inFmid such that

n→∞lim sup

t∈J

kx(t)−

n

X

k=1

fb₀(λ_k

n )x⁰_k(t)e^iλ^k^t−y_n(t)k= 0.

It remains to note that by property 3) of the subspace Fmid (see Definition 5) we can represent each functiony_n, n≥1,in the formy_n(t) =x⁰_n,k(t)e^iλ^k^tfort ∈J,where|λ_k|< nandx⁰_n,k ∈Fmid. Corollary 6.1. Consider a finite-dimensional Banach space X, an operator A in End X and a function ψ ∈ Fmid(J, X). Then every bounded solution x : J → X to differential equation (6.1) belongs to AP∞F(J, X;Fmid) and can be represented in the form

x(t) =

m

X

k=1

x_k(t)e^iλ^k^t, x_k ∈(Fmid)sl,∞, t∈J,

provided that σ(A)∩(iR) = {iλ₁, . . . , iλ_m}.

Thus, our results are of some interest even for systems of ordinary differential equations.

7 Asymptotically finite-dimensional operator semigroups

Let us consider a bounded strongly continuous operator semigroupT :R+ →End X and a subspace of vanishing at infinity functions Fmid =Fmid(R⁺, X) (see Definition 5.)

By X₀ =X₀(T;Fmid) we denote a closed subspace of X defined by

X₀(T;Fmid) ={x₀ ∈X :a functiont7→T(t)x:R+ →X belongs toFmid(R+, X)}.

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Definition 25. The semigroup T is called asymptotically finite-dimensional with respect to the subspace Fmid(R+, X), if the quotient spaceX/X₀(T;Fmid)is finite-dimensional.

Asymptotically finite-dimensional semigroups were defined and studied in [19, 25].

Theorem 7.1. Let T : R⁺ → End X be an asymptotic finite-dimensional semigroup of operators with respect to the subspace Fmid(R+, X) of vanishing at infinity functions. Then each function t7→T(t)x:R+→X, x∈X, can be represented as

T(t)x=

m

X

k=1

x_k(t)e^iλ^k^t, t∈R+,

where x_k ∈(Fmid)sl,∞, k = 1, . . . , m, and {iλ₁, . . . , iλ_m} ⊂σ(A)∩(iR).

Proof. The subspace X0 =X0(T;Fmid) is closed and invariant underT(t) for t ≥0. Therefore, the quotient group of operators

Te:R⁺ →EndX, Te(t)ex=T^(t)x, ex=x+X0,

is well defined, where X = X/X₀. The quotient space X is finite-dimensional, and therefore the operators Te(t) for t≥0 are invertible. The boundedness of the semigroup T implies the uniform boundedness of the semigroup of inverse operators Te(t) =Te(t)⁻¹ for t <0. Its generator Aebelongs to EndX, while its spectrum σ(A) =e {iλ₁, . . . , iλ_m} ⊂(iR)∩σ(A) is a finite set of cardinality m not exceeding dimX.

Since Te is a bounded operator group acting in the finite-dimensional space X, for any vector x∈X and the equivalence class ex∈X containing xthere exist vectors x₁, . . . , x_n∈X such that

Te(t)ex=

m

X

k=1

ex_ke^iλ^k^t, t≥0,

Te(t)exk=e^iλ^k^txek, t≥0, 1≤k ≤m.

Consequently, the inclusion T(t)x_k −e^iλ^k^tx_k ∈ X₀ holds for any t ≥ 0. Hence, the function s7→T(t+s)x_k−e^iλ^k^(t+s)x_k :R+ →X belongs toFmid(R+, X) for all t ≥0.

Put x_k(s) = T(s)x_k for s≥ 0 and k = 1, . . . , m. with x₀(s) = T(s)x for s ≥0 we infer that the function x₀(s) =x(s)−

m

P

k=1

x_k(s)e^iλ^k^s for s≥0 belongs toFmid(R+, X).

Acknowledgments

The research was supported by the Russian Foundation for Basic Research (project 19-01-00732 A).