# Almost Periodic Functions and Their Applications: A Survey of Results and Perspectives

Authors: Wei-Shih Du; Marko Kostić; Manuel Pinto
Journal: Journal of Mathematics
 (2021)

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## Abstract

The main aim of this survey article is to present several known results about vector-valued almost periodic functions and their applications. We separately consider almost periodic functions depending on one real variable and almost periodic functions depending on two or more real variables. We address several open problems and possibilities for further investigations of almost periodic functions, quoting more than two hundred references about the subject under our consideration.


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## Body

## 1. Introduction

The class of almost periodic functions was introduced by the Danish mathematician H. Bohr [1] (1925), the younger brother of the Nobel Prize-winning physicist N. Bohr, and later generalized by many others. Let I=ℝ or I=0,∞, let X be a complex Banach space, and let f:I⟶X be continuous. Given ε>0, we call τ>0 an ε-period for f⋅ if and only if(1)ft+τ−ft≤ε,t∈I.Byϑf,ε, we denote the set of all ϵ-periods for f⋅. We say that f⋅ is almost periodic if and only if for each ϵ>0, the set ϑf,ε is relatively dense in 0,∞, which means that there exists l>0 such that any subinterval of 0,∞ of length l meets ϑf,ε. There are many research monographs concerning the theory of almost periodic functions and their applications; at the very beginning, we would like to cite the important research monograph [2] by Levitan, only.The class of almost automorphic functions was introduced by the American mathematician Bochner [3]. A continuous function f:ℝ⟶X is said to be almost automorphic if and only if for every real sequence bn, there exist a subsequence an of bn and a map g:ℝ⟶X such that(2)limn⟶∞ft+an=gt,limn⟶∞gt−an=ft,pointwise for t∈ℝ. Any almost periodic function is almost automorphic, but the converse statement is not true in general (see the research monograph [4] by N’Guerekata for more details). The theories of almost periodic functions and almost automorphic functions are still very active fields of investigations of numerous authors, full of open problems, conjectures, hypotheses, and possibilities for further expansions.As mentioned in the abstract, this survey article aims to present several known results about vector-valued almost periodic functions and their applications (there is no need to say that it would be very difficult to summarize so many important research results obtained in the theory of almost periodic functions within only one research report, and because of that, we feel it is our duty to say that this survey article does not intend to be exhaustively complete). We divide our further exposition into two individual sections; inSection 1, we analyze the almost periodic functions of one real variable and their applications, while in Section 2, we analyze the almost periodic functions of several real variables and their applications. The material is basically taken from the introductory part, notes, and appendices to the second and the third sections of the forthcoming research monograph [5].


## 2. Almost Periodic Functions of One Real Variable and Their Applications

From the application point of view, almost periodic functions of one real variable are much important than the almost periodic functions of two or more real variables. There is enormous literature devoted to the study of almost periodicity in the time variable and the almost automorphy in the time variable of solutions for various kinds of the abstract differential equations of the first order. The notion of an almost periodic strongly continuous semigroup was introduced by Bart and Goldberg in [6], but some particular results concerning the almost periodicity of individual orbits of strongly continuous semigroups were already given by Foias and Zaidman [7], Zhikov [8, 9], and Perov and Hai [10]; also, see the survey article [11] by Phóng as well as the reference list of [12] and the articles [13, 14] obtained in a collaboration of Phóng and Lyubich. The notion of an asymptotically almost periodic strongly continuous semigroup was introduced by Ruess and Summers [15] in 1986 (see also [16–18]), while the notion of an (asymptotically) Stepanov almost periodic strongly continuous semigroup was introduced by Henríquez [19] in 1990. Concerning the study of the existence and uniqueness of almost periodic solutions of nondegenerate semilinear Cauchy problems, it seems that the fractional powers of operators have been employed for the first time by Bahaj and Sidki in [20]. For the periodic solutions of abstract first-order differential equations, we refer the reader to the research monographs [21] by Burton, [22] by Liu, Guerekata, and Minh, and [23] by Yoshizawa.The notion of almost periodic cosine operator functions was introduced by Cioranescu [24] and after that received considerable attention of many authors. The existence and uniqueness of almost periodic-type solutions of the (abstract) second-order differential equations have been investigated in many research articles by now, using the theory of cosine operator functions or other methods (see, e.g., [25–33]). For example, Diagana, Hassan, and Messaoudi recently analyzed, in [34], the existence of asymptotically almost periodic mild solutions of the abstract Volterra integrodifferential equation(3)u″t+A2ut−∫−∞tgt−sA2usds=ft,ut,t≥0,accompanied with the initial conditions u−t=u0t for t≥0 and u′0=u1. The main strategy used is a transformation of such a system into a first-order linear evolution equation whose solutions are governed by exponentially decaying strongly continuous semigroups; an interesting application was made in the study of Kirchhoff plate equation with infinite memory. Regarding the abstract second-order differential equations in Hilbert spaces, it should also be noted that the existence and uniqueness of periodic solutions for the following equations,(4)utt+A+γIut=Ft,ut,t≥0,γ∈ℝ,utt+A2ut=Ft,ut,u′t,t≥0,uttt+2αutt+Aut=gt+Ft,uta,t≥0,were analyzed by Strashkraby, Vejvoda (1973), Lovicar (1977), and Masudy (1966), respectively (A is a positive self-adjoint operator in a Hilbert space H). For more details about the existence and uniqueness of almost periodic-type solutions of the abstract first-order Cauchy problems and the abstract second-order Cauchy problems, we refer the reader to the reference lists in [5, 12]. We recall the following problem proposed in [12].Problem: let a closed multivalued linear operatorA be the integral generator of a bounded C-cosine function Ctt≥0. Suppose that x∈X satisfies that the mapping t⟼Ctx, t≥0, is asymptotically Stepanov almost periodic. Is it true that the mapping t⟼CtCx, t≥0, is almost periodic?Chronologically, the study of almost periodic solutions of the abstract Volterra integrodifferential equations was initiated by Prüss in [35], Section 11.4, where the author analyzed the almost periodic solutions, Stepanov almost periodic solutions, and asymptotically almost periodic solutions of the following abstract integrodifferential equation:(5)u′t=∫0∞A0su′t−sds+∫0∞dA1sut−s+ft,t∈ℝ.Here,A0∈L10,∞:LY,X, t⟼A1t∈LY,X, t≥0, is locally of bounded variation, and X and Y are Banach spaces such that Y is densely and continuously embedded into X. Almost immediately after that, Vu [36] investigated the almost periodicity of the abstract Cauchy problem(6)u′t=Aut+∫0∞dBuτut−τ+ft,t∈ℝ,where A is a closed linear operator acting on a Banach space X, Btt≥0 is a family of closed linear operators on X, and f:ℝ⟶X is continuous.It is very difficult and unpleasant to say precisely who was the first to study the almost periodic solutions of the abstract fractional differential equations. Recently, Mu, Zhoa, and Peng [37] investigated the periodic solutions and S-asymptotically periodic solutions to fractional evolution equation Dt,+γut=−Aut+gt,t∈ℝ, and its semilinear analogue Dt,+γut=−Aut+gt,ut,t∈ℝ, where Dt,+γ denotes the Weyl–Liouville fractional derivative of order γ∈0,1, A is the infinitesimal generator of an exponentially decaying strongly continuous semigroup of operators, and g:ℝ×X⟶X satisfies certain assumptions (also, see the article [38] by Agarwal, Andrade, and Cuevas as well as the recent articles [39] by Bedi, Kumar, Abdeljawad, and Khan and [40] by Brindle and Guérékata, where the authors analyzed S-asymptotically ω-periodic mild solutions for fractional differential equations with Hilfer derivatives and Riemann–Liouville derivatives). Later, Kostić extended the results of Mu, Zhoa, and Peng to the abstract fractional differential inclusion Dt,+γut∈−Aut+gt,t∈ℝ, and its semilinear analogue(7)Dt,+γut∈−Aut+gt,ut,t∈ℝ,where A is a closed multivalued linear operator satisfying condition (P); here, we follow the terminology employed in [41], where we have obeyed the multivalued approach to the abstract degenerate Volterra integrodifferential equations. (P) There exist finite constants c,M>0 and β∈0,1 such that Ψ≔Ψc≔λ∈ℂ:ℜλ≥−cℑλ+1⊆ρA and Rλ:A≤M1+λ−β,λ∈Ψ. The obtained results enable one to consider the almost periodic-type solutions of the following fractional Poisson heat equations,(8)∂∂tmxvt,x=Δ−bvt,x+ft,mxvt,x,t∈ℝ,x∈Ω,vt,x=0,t,x∈0,∞×∂Ω,Dtγmxvt,x=Δvt,x+bvt,x,t≥0,x∈Ω,vt,x=0,t,x∈0,∞×∂Ω,mxv0,x=u0x,x∈Ω,and the following fractional semilinear equation with higher-order differential operators in the Hölder space X=CαΩ¯:(9)Dtγut,x=−∑β≤2maβt,xDβut,x−σut,x+ft,ut,x,t≥0,x∈Ω,u0,x=u0x,x∈Ω.See [12] for more details. Let us also recall that Ponce [42] investigated the bounded mild solutions of the following nondegenerate fractional integrodifferential equation: (10)Dt,+γut=Aut+∫−∞tat−sAusds+ft,ut,t∈ℝ,where A is a closed linear operator, a∈L10,∞ is a scalar-valued kernel, and f⋅,⋅ satisfies some Lipschitz-type conditions. In particular, almost periodic solutions of (10) have been analyzed. Abbas, Kavitha, and Murugesu recently analyzed Stepanov-like (weighted) pseudo-almost automorphic solutions to the following fractional-order abstract integrodifferential equation:(11)Dtαut=Aut+Dtα−1ft,ut,Kut,t∈ℝ,where(12)Kut=∫−∞tkt−shs,usds,t∈ℝ,1<α<2, A is a sectorial operator with the domain and range in X, of negative sectorial type ω<0, the function kt is exponentially decaying, and the functions f:ℝ×X×X⟶X and h:ℝ×X⟶X are Stepanov-like weighted pseudo-almost automorphic in time for each fixed element of X×X and X, respectively, satisfying some extra conditions [43]. For more details about almost periodic-type solutions of the abstract fractional differential equations, see the reference list of [12] and the articles [44–48].As mentioned from the above, many results concerning the existence and uniqueness of almost periodic-type solutions and almost automorphic-type solutions to the abstract (semilinear) fractional nondegenerate differential equations have been given recently by numerous authors. In almost all these results (in the linear setting, the quite exceptional are some examples and results presented by Zaidman ([49], Examples 4, 5, 7, and 8; pp. 32–34), which have been employed by many authors so far, for various purposes), the basic key is to investigate the invariance of certain kinds of generalized almost periodicity and generalized almost automorphicity under the actions of the infinite convolution products(13)t⟼∫−∞tRt−sfsds,t∈ℝ,and(14)t⟼∫0∞Rt−sfsds,t≥0.Here, it is commonly assumed thatRtt≥0⊆LX,Y is a nondegenerate strongly continuous operator family between the Banach spaces X and Y which exponentially or, at least, polynomially decays as t⟶+∞. In [12], we have investigated the case in which Rtt>0⊆LX,Y is a degenerate strongly continuous operator family which decays exponentially or polynomially as t⟶+∞, but we have allowed Rtt>0 to have a removable singularity at zero, by that we basically mean that there exists a number ζ∈0,1 such that the operator family tζRtt≥0 is well defined and strongly continuous at the point t=0. The integral generator of Rtt≥0 is not single-valued any longer, and this is the main reason why we have employed the multivalued linear approach to the abstract degenerate integrodifferential equations in [12]. The well-posedness of the abstract degenerate Cauchy problem,(15)But=ft+∫0tat−sAusds,t∈0,τ,where 0<τ≤∞, t⟼ft, t∈0,τ is a continuous mapping, a∈Lloc10,τ, and A,B are closed linear operators, has been thoroughly analyzed in the monograph [41].We will say just a few words about periodic solutions of the abstract degenerate Volterra integrodifferential equations. In [50], Barbu and Favini analyzed the 1-periodic solutions of the abstract degenerate differential equation d/dtBut=Aut, t≥0, accompanied with the initial condition Bu0=Bu1, by using Grisvard’s sum of operators method and some results from the investigation of Prüss [51] in the nondegenerate case. The authors reduced the above problem to v′t∈Avt, t≥0, v0=v1, where the multivalued linear operator A is given by A=AB−1. The main problem is whether the inclusion 1∈ρA holds or not; recall that Prüss [51] proved that 1∈ρA if and only if 2πiℤ⊆ρA and sup2πin−A−1:n∈ℤ<∞, provided that A generates a nondegenerate strongly continuous semigroup. Applications are given to the Poisson heat equation in H−1Ω and L2Ω, as well as to some systems of ordinary differential equations. On the contrary, Lizama and Ponce [52] analyzed the existence of 2π-periodic solutions to the following abstract inhomogeneous linear equation:(16)ddtBut=Aut+∫−∞tat−sAusds+ft,t≥0,subjected with the initial condition Bu0=Bu2π. The authors also considered the maximal regularity of (16) in periodic Besov, Triebel–Lizorkin, and Lebesgue vector-valued function spaces.Concerning the classical theory of partial differential equations with integer-order derivatives, we would like to recommend for the reader the references and works quoted in the introductory part of the fourth section of the monograph [53] by Ptashnic, where the following have been especially emphasized:(1)
Theω-periodic solutions in time for the linear wave equation and the following weakly nonlinear wave equation uttt,x−uxxt,x=εft,x,u,ut,ux,ε,t≥0,0≤x≤π, accompanied with the boundary conditions ut,0=ut,π=0, were analyzed by Vejvoda [54] in 1964 (ε>0 is a sufficiently small real parameter). If ω∈2πℚ and ω>0, then the existence of ω-periodic solutions for both classes of wave equations was proved; on the contrary, if ω∉2πℚ and ω>0, then the situation is much more complicated, and the author proved the existence of ω-periodic solutions for a corresponding linear wave equation, only, provided that ω=2πα and there exist positive real numbers c>0 and γ>0 such that α−m/k>c/kγ.Only one year later, in 1965, Gavlova investigated the existence and uniqueness of periodic solutions for the following weakly nonlinear telegraph equation:utt−uxx+2aut+2bux+cu=ht,x+εft,u,ut,ux,ε, accompanied with the boundary conditions ut,0=ut,π=0, where a,b,c∈ℝ are certain constants and ε>0 is a sufficiently small real parameter.(2)
In 1972, Azis and Gorak investigated the existence and uniqueness of periodic solutions in the time variable and space variable for the following quasi-linear hyperbolic second-order equationuxy+ax,yux+bx,yuy+cx,yu=fx,y,u,ux,uy; in 1971, Krylovoi and Vejvoda investigated the existence and uniqueness of ω-periodic solutions in the time variable for the following equation: utt+uxxxx=gt,x+εft,x,u,ux,uxx,ut,ε, accompanied with the boundary conditions ut,0=ut,2π=uxxt,0=uxxt,π=0.Six years later, in 1977, Kopachkovoi and Vejvoda analyzed the existence and uniqueness ofω-periodic solutions in the time variable for the following nonlinear equation: utt+uxxxx−εuxx∫0πu2x,ξdξ=gt,x+ε2Fut,x, which appears in the study of beam vibrations with the effect of elongation. Also, see the important research monograph [55] by Vejvoda (with Herrmann and Lovicar as contributors).Furthermore, the Bohr almost periodic solutions to boundary value problems for systems of partial differential equations that arise in solving certain problems for inhomogeneous media have been investigated in the research articles [56] by Berselli and Bisconti, [57] by Berselli and Romito, and [58] by Vetchanin and Mikishanina. Concerning the existence and uniqueness of Bohr almost periodic solutions of the Navier–Stokes-type equations, the reader may consult the reference list of [5].The study of differential equations with discontinuous arguments was initiated by Myshkis [59] in 1977. The analysis of asymptotically antiperiodic solutions for nonlinear differential first-order equations with piecewise constant argument carried out by Dimbour and Valmorin [60] has recently been reconsidered and extended for asymptotically Bloch periodic solutions for nonlinear fractional differential inclusions with a piecewise constant argument by Kostić and Velinov in [61]. We have analyzed the following fractional differential Cauchy inclusion with a piecewise constant argument:(17)Dtγut∈Aut+A0ut+gt,ut,t>0;u0=u0,where A is a multivalued linear operator satisfying certain assumptions, A0∈LX, g:0,∞×X⟶X is a given function, and Dtγut denotes the Caputo fractional derivative of order γ, taken in a weak sense. It is also worth noting that Chávez, Castillo, and Pinto [62] analyzed the existence of a unique almost automorphic solution for the following differential equation with a piecewise constant argument:(18)y′t=Atyt+Btyt+ft,yt,yt,t∈ℝ,where At and Bt are almost automorphic p×p complex matrices and f:ℝ×ℂp×ℂp⟶ℂp is an almost automorphic function satisfying a condition of Lipschitz type. The study carried out in [62] leans heavily on the use of results on discontinuous almost automorphic functions, exponential dichotomies, and the Banach fixed-point theorem. The almost periodic solutions of (18) were considered for the first time by Yuan and Hong in [63]; for more details about differential equations with a piecewise constant argument (DEPCA), the reader may consult articles [64] by Cooke and Wiener, [65] by Shah and Wiener, and [66] by Wiener, as well as articles [67–73], the list of publication of Pinto (https://www.zbmath.org/?q=ai(percent/sign)3Apinto.manuel), and the list of references cited therein.There is a vast amount of articles in the existing literature which consider almost automorphic-type solutions for various classes of integrodifferential equations. Let us only mention our analysis (the joint work of the second-named author with Prof. Guérékata [74]) of the following abstract multiterm fractional differential inclusion:(19)Dtαnut+∑i=1n−1AiDtαiut∈ADtαut+ft,t≥0,uk0=uk,k=0,…,αn−1,where n∈ℕ\1, A1,…,An−1 are bounded linear operators on a Banach space X, A is a closed multivalued linear operator on X, 0≤α1<⋯<αn, 0≤α<αn, f⋅ is an X-valued function, and Dtα denotes the Caputo fractional derivative of order α. Many excellent examples have been presented in monograph [75] by Diagana; also, see the following monographs:(1)
[76] by Amerio and Prouse for almost periodic solutions of functional equations(2)
[77] by Argabright and de Lamadrid for almost periodic measures(3)
[78, 79] by Baake and Grimm for applications of almost periodic functions in crystallography(4)
[80] by Bezandry and Diagana for almost periodic solutions of stochastic differential equations(5)
[81] by Böttcher, Karlovich, and Spitkovsky for factorization of almost periodic matrix functions (cf. also article [82] by Böttcher for the issues regarding the corona theorem for almost periodic functions of several real variables and articles [83] by Boggiatto, Ferández, and Galbis and [84] by Kim for issues concerning Gabor systems and almost periodic functions)(6)
[85] by Chang, Guerekata, and Ponce for Bloch p,k-periodic functions, antiperiodic functions, and their applications(7)
[86] by Cheban for asymptotically almost periodic solutions of linear and nonlinear equations(8)
[87] by Emel’yanov for weakly almost periodic C0-semigroups(9)
[88] by Hino, Naito, Minh, and Shin and [89] by Guérékata for spectral analysis of almost periodic functions and Massera-type theorems [90](10)
[91] by Hsu for weakly almost periodic functions(11)
[92] by Stamov for almost periodic solutions of impulsive differential equations (see also research monographs [93] by Bainov and Simeonov, [94] by Perestyuk, Plotnikov, Somoilenko, and Skripnik, [95] by Stamova and Stamov, and [96] by Song, Gno, and Shi for more details on the subject)Concerning the existence and uniqueness of almost periodic-type solutions of inhomogeneous evolution equations of first order, the notions of hyperbolic evolution systems and Green’s functions are incredible important; for more details on the subject, we refer the reader to Acquistapace [97], Acquistapace and Terreni [98], Chang and Chen [99], Diagana [75], Khalil [100], Schnaubelt [101], Zhikov [102, 103], and the list of references in [12]. The almost periodic- and almost automorphic-type solutions of the abstract Cauchy problems,(20)u′t=Atut+ft,t∈ℝ,u′t=Atut+ft,t>0;u0=x,and their semilinear analogues have been investigated in a great number of research papers. Without going into full details, we will only refer the readers to research monographs [75] by Diagana and [12] by Kostić, articles [104] by Baroun, Maniar, and Schnaubelt and [105] by Baroun, Ezzinbi, Khalil, and Maniar, and the list of references therein. Concerning the applications of evolution systems in the theory of the second-order nonautonomous differential equations, mention should be made of paper [106] by Zakora.The almost periodic and almost automorphic functions on time scales and their applications to the abstract Volterra integrodifferential equations have recently been considered by numerous mathematicians (for time-scale calculus, we warmly recommend monograph [107] by Bochner and Peterson). It would be really troublesome to quote here all relevant references concerning the almost periodic traveling wave solutions and the almost automorphic traveling wave solutions for various classes of nonlinear partial differential equations. For more details about the above problematic, we refer the reader to the references cited in [5].The definitions and basic properties ofω,c-periodic and ω,c-pseudo-periodic functions were introduced and analyzed by Alvarez, Gómez, and Pinto in [108, 109], motivated by some known results regarding the qualitative properties of the solution to Mathieu’s linear differential equation y″t+a−2qcos2tyt=0, arising in modeling of railroad rails and seasonally forced population dynamics (ω>0 and c∈ℂ\0). The linear delayed equations can have ω,c-periodic solutions as well (see, e.g., [109], Example 2.5). The notions of antiperiodicity and Bloch periodicity are special cases of the notion of an ω,c-periodicity, which has also been analyzed in [110].The authors of [109] analyzed the existence and uniqueness of mild ω,c-periodic solutions to abstract semilinear integrodifferential equation (10). Furthermore, Alvarez, Castillo, and Pinto analyzed in [108] the existence and uniqueness of mild ω,c-pseudo-periodic solutions to the abstract semilinear differential equation of the first order:(21)u′t=Aut+ft,ut,t∈ℝ,where A generates a strongly continuous semigroup. The authors proved the existence of positive ω,c-pseudo-periodic solutions to the Lasota–Wazewska equation with ω,c-pseudo-periodic coefficients:(22)y′t=−δyt+hte−atyt−τ,t≥0.This equation describes the survival of red blood cells in blood of an animal (see, e.g., Wazewska-Czyzewska and Lasota [111]). Concerning the applications to time-varying impulsive differential equations, mention should be made of article [112] by Wang, Ren, and Zhou; also, cf. article [113] by Mophou, Guérékata, and Milce and article [114] by Li, Wang, and Fečkan. For further information about (weighted) pseudo-almost periodic solutions and (weighted) pseudo-almost automorphic solutions of various types of abstract Volterra integrodifferential equations, we refer the reader to [115–122] and [123–130].Before we explain the main results and applications of multidimensional-type functions, we will single out a few important topics for our readers.Almost periodic functions of complex variables: the theory of almost periodic functions of one complex variable, initiated already by Bohr in the third part of [1], is still very popular and attracts the attention of many mathematicians (see, e.g., [131–134]). Suppose that −∞≤α<β≤+∞ and the function f:Ω≡z∈ℂ:α<ℜz<β⟶X is analytic. Then, we say that f⋅ is almost periodic if and only if for any ε>0 and every reduced strip z∈ℂ:α′<ℜz<β′, where α<α′<β′<β, there exists a number l>0 such that each subinterval of length l of ℝ contains a number τ satisfying the inequality(23)fz+iτ−fz≤ε,forα′<ℜz<β′.In particular, this definition implies that, for any fixedσ∈α,β, the function fσt≔fσ+it, t∈ℝ, is almost periodic. Moreover, the definition implies that the almost periodicity should be uniform on various straight lines, with the meaning being clear. The Fourier series of these functions can be obtained from a certain exponential series with complex coefficients; the associated series is called the Dirichlet series of f⋅. As for the functions of one real variable, Bohr’s notion of almost periodicity of f⋅ in a vertical strip Ω is equivalent to the relative compactness of the set of its vertical translates, f⋅+ih:h∈ℝ, with the topology of the uniform convergence on reduced strips. Mean motions and zeros of generalized almost periodic analytic functions have been analyzed by Borchsenius and Jessen in [135], where the reader can find several important applications to the Riemann zeta function (also, see [136] and the references therein for further information about applications of results from the theory of almost periodic analytic functions to the Riemann zeta function). For more details about subharmonic almost periodic functions and holomorphic almost periodic functions, we refer the reader to [131, 137–140] and references cited therein.Cn-almost periodic functions: the notion of Cn-almost periodicity was introduced by Adamczak [141] in 1997 and later received great attention of many other authors. In this article, we will only say a few words about generalized Cn-almost periodic functions and possibilities for further expansions. Several different classes of Stepanov-like Cn-pseudo-almost automorphic functions have been analyzed by Diagana, Nelson, and N’Guérékata in [142]. For example, let 1≤p<∞, let n∈ℕ, and let f∈LlocpI:X. Then, we say that (see [5] for the notion)(i)
the functionf⋅ is Stepanov-p-Cn-almost periodic, f∈Cn−APSpI:X for short, if and only if for each k=0,1,…,n, we have that fk∈APSpI:X.(ii)
the functionf∈Llocp0,∞:X is asymptotically Stepanov-p-Cn-almost periodic, f∈Cn−AAPSp0,∞:X for short, if and only if for each k=0,1,…,n, we have that fk∈AAPSp0,∞:X. The following definitions have been analyzed in [12].(iii)
the functionf⋅ is equi-Weyl-p-Cn-almost periodic, f∈e−Cn−WappI:X for short, if and only if for each k=0,1,…,n, we have that fk∈e−WappI:X.(iv)
the functionf⋅ is Weyl-p-Cn-almost periodic, f∈Cn−WappI:X for short, if and only if for each k=0,1,…,n, we have that fk∈WappI:X.(v)
the functionf⋅ is Besicovitch-Doss-p-Cn-almost periodic, f∈Cn−BpI:X for short, if and only if for each k=0,1,…,n, we have that fk∈BpI:X.Using the same idea, we can introduce and analyze a great number ofCn-almost automorphic function spaces [12]. For example, the function(24)ft=∑n=1∞sinntn4,t∈ℝ,is C2-almost periodic but not C3-almost automorphic. Furthermore, for any real-valued function g∈C3−AAℝ:ℂ satisfying inft∈ℝg″′t>0, we have that the function(25)ft=∑n=1∞gntn4,t∈ℝ,belongs to the space C2-AAS1ℝ:ℂ\C3-AAS1ℝ:ℂ; see, e.g., [142], Example 2.23. It is clear that we can slightly generalize the notion of all the aforementioned function spaces by using the definitions and results from the theory of Lpx-spaces.Before proceeding further, we also want to mention research articles [2, 124, 143–147] by the second-named author as well as to recall the following question proposed in [12]: is it true that the classes of Besicovitch-p-almost periodic functions and Besicovitch-Doss-p-almost periodic functions coincide in vector-valued case (1≤p<∞)?Nemytskii operators between Stepanov almost periodic function spaces: letp and q be two real numbers belonging to the interval 1,∞, and let T>0. It is said that f:0,T×X⟶Y is a Carathéodory function if and only if the following holds:(i)
The mappingt⟼ft,x, t∈0,T, is measurable for any fixed element x∈X(ii)
For a.e.t∈0,T, the function ft,⋅ is continuous from X and YNow, consider the Nemytskii operatorNf:Lp0,T:X⟶Lq0,T:Y by(26)Nfωt≔ft,ωt,t∈0,T,ω∈Lp0,T:X.The well-known result of Lucchetti and Patrone ([148], Theorem 3.1) states that the Nemytskii operator is well defined between these spaces if and only if there exist a>0 and b∈Lp0,T such that, for all x∈X and a.e. t∈0,T, we have(27)ft,x≤axp/q+bt.In this case, the Nemytskii operator is continuous.Concerning the Nemytskii operator between the spaces of almost periodic functionsAPℝ:X and APℝ:Y, it should be noted that we have the equivalence of the following statements (see, e.g. Blot, Cieutat, Guérékata, and Pennequin [149]):(i)
The Nemytskii operatorNf:APℝ:X⟶APℝ:Y is continuous.(ii)
For each compact setK⊆X and for each ε>0, the set(28)τ∈ℝ:supt∈ℝsupx∈K‖ft+τ−ft,x‖≤ε,is relatively dense in ℝ.(iii)
For allx∈X, f⋅,x∈APℝ:Y, and for each compact set K⊆X and for each ε>0, there exists δ>0 such that, for each x1,x2∈K and for each t∈ℝ, we have the implication: x1−x2≤δ⟹ft,x1−ft,x2≤ε.A similar statement holds for the continuity of the Nemytskii operator between the spaces of almost automorphic functionsAAℝ:X and AAℝ:Y; see, e.g., the recent paper ([150], Theorem 2.3) by Cieutat. Several necessary and sufficient conditions clarifying the continuity of Nemytskii operators between almost periodic and almost automorphic spaces in the sense of Stepanov approach can be found in [150], Section 4.Geometric properties of generalized almost periodic function spaces of Orlicz type: in his fundamental paper [151], Hillmann investigated the Besicovitch–Orlicz spaces of almost periodic functions. After that, numerous mathematicians working in the field of almost periodic functions have investigated the geometric properties of generalized almost periodic function spaces of Orlicz type. Here, we will describe the results of Morsli and Smaali established in [152] and the results of Bedouhene, Djabri, and Boulahia established in [153], only; for more details on the subject, we refer the reader to the list of references quoted in these papers and [5].Assume that the functionφ:ℝ×0,∞⟶0,∞ satisfies the following conditions:(i)
For everyt∈ℝ, we have φt,0=0(ii)
For everyt∈ℝ, the mapping u⟼φt,u, u≥0, is convex(iii)
φt+1,u=φt,u for all t∈ℝ and u≥0(iv)
For everyu>0, we have inft∈ℝφt,u=ϕu>0Iff:ℝ⟶0,+∞ is a measurable function, then it is well known that the function(29)f⟼ρφf≔limsupt⟶+∞12t∫−ttφtftdt,f∈Mℝ,is convex and pseudo-modular.In [152], the authors defined the Besicovitch–Musielak–Orlicz space associated to φ⋅,⋅ by(30)Bφℝ≔f∈Mℝ:limα⟶0+ρφαf=0.We have(31)Bφℝ=f∈Mℝ:∃α>0,ρφαf<∞.The spaceBφℝ is equipped with the pseudo-norm fφ≔k>0:ρφf/k≤1.The authors introduced two different types of Besicovitch–Musielak–Orlicz spaces of almost periodic functions,B˜a.p.φℝ and Ba.p.φℝ, as follows: A function f:ℝ⟶ℂ is said to belong to the space Ba.p.φℝ, resp. B˜a.p.φℝ, if and only if there exists a sequence fn of trigonometric polynomials such that, for every k>0, resp. there exists k>0 such that limn⟶+∞ρφkfn−f=0. Then, we clearly have Ba.p.φℝ⊆B˜a.p.φℝ⊆Bφℝ.Ifφt,x=x, then by Ba.p.1ℝ, B˜a.p.1ℝ, and B1ℝ, we denote the respective spaces.Let us recall that a functionφ:ℝ×0,∞⟶0,∞ is strictly convex if and only if φt,λu+1−λv<λφt,u+1−λφt,v for a.e. t∈ℝ and for all λ∈0,1, 0≤u<v<∞. On the contrary, a normed linear space E,⋅ is said to be strictly convex if and only if(32)x+y2<1,provided thatx=y=1andx≠y.It is said that the functionφ⋅,⋅ satisfies the Δ2-condition if and only if there exist a number k>1 and a measurable nonnegative function h⋅ such that ρφh<∞ and φt,2u≤kφt,u for almost all t∈ℝ and all u≥ht.Letf∈Ba.p.φℝ. Then, due to [152], Proposition 1, we have φ⋅,f⋅∈Ba.p.1ℝ so that the limit limT⟶+∞1/2T∫−TTφt,ftdt always exists and is finite. The main result of paper is [152], Theorem 1, which states that the space B˜a.p.φℝ is strictly convex if and only if φ⋅,⋅ is strictly convex and satisfies the Δ2-condition.Ergodicity in Stepanov–Orlicz spaces was investigated in [153]. Let us recall that a convex function ϕ:ℝ⟶0,∞ is said to be an Orlicz function if and only if it is nondecreasing, even, and continuous on ℝ and satisfies ϕ0=0, ϕu>0 for u>0, and limu⟶+∞ϕu=+∞. In the newly arisen situation, we say that the function ϕ⋅ satisfies the Δ2-condition if and only if there exist real numbers k>1 and u0>0 such that ϕ2u≤kφu for u≥u0. For any Orlicz function ϕ:ℝ⟶0,∞, it can be simply proved that f∈PAP0ℝ:X if and only if ϕf∈PAP0ℝ:X. Here, PAP0ℝ:X stands for the space consisting of all pseudo-ergodic components, i.e., the bounded continuous functions Φ:ℝ⟶X, such that(33)liml⟶∞12l∫−llΦsds=0.For any vector-valued measurable functionf:ℝ⟶X, we define the positive function(34)ρSϕf≔supx∈ℝ∫xx+1ϕfsds.The Stepanov–Orlicz function space generated byϕ is defined by(35)BSϕℝ,X≔f∈Mℝ:X;∃α>0ρSϕαf<∞.We know that the vector spaceBSϕℝ,X equipped with the Luxemburg norm(36)fSϕ≔infk>0:supx∈ℝ∫xx+1ϕfs/kds≤1,is a Banach space. It is also worth noting that the Morse–Transue space type(37)BS˜ϕℝ,X≔f∈Mℝ,X;∃α>0ρSϕαf<∞,equipped with the Luxemburg norm, is a closed subspace of BSϕℝ,X, which is commonly called the Besicovitch–Orlicz class. We know that BSϕℝ,X=BS˜ϕℝ,X if and only if ϕ⋅ satisfies the Δ2-condition.Furthermore, for anyp∈C+ℝ, we define the Musielak–Orlicz modular-type space(38)BSp⋅ℝ,X≔f∈Mℝ:X;∃α>0supx∈ℝ∫xx+1fs/kpsds≤1.For any functionf∈BSp⋅ℝ,X, the notion of BSp⋅ℝ,X-ergodicity in the norm sense and the notion of BSp⋅ℝ,X-ergodicity in the modular sense are introduced in [153], Definition 3.1, and [153], Definition 3.2, respectively. Due to [153], Proposition 3.4, these concepts are equivalent.Letϕ:ℝ⟶0,∞ be an Orlicz function. In [153], Definition 3.6, the authors introduced the notions of norm ergodicity in Stepanov–Orlicz sense, modular ergodicity in Stepanov–Orlicz sense, and strongly modular ergodicity in Stepanov–Orlicz sense for a given function f∈BSϕℝ,X. After that, the authors further explored these notions in [153], Theorems 3.8, 3.10, and 3.11, and provided several illustrative examples in [153], Section 4.Density theorems for almost periodic functions in Hilbert spaces: in this section, we will inscribe a few relevant results obtained by Haraux and Komornik in [154]; these results have been obtained in their investigation of the oscillatory properties of the wave equation. Denote XT the vector space of all square-integrable functions with zero mean by XT:(39)XT≔f∈Lloc2ℝ:ℂ;ft+T≡ft,∫0Tftdt=0,where T>0. If the set A=T1,…,TN is a given set of positive real numbers, we define X≔XT1+⋯+XTN.IfV is a certain collection of complex-valued functions and I is an interval in ℝ, then we set VI≔fI:f∈V. In [154], Theorem 1, the authors proved that there exists a positive real number TA such that, for any interval I⊆ℝ, we have(40)XIis dense inL2Iif and only ifI<TA,where I denotes the length of interval I; furthermore, the orthogonal complement of XI in L2I is finite-dimensional if I=TA and infinite-dimensional if I>TA. Suppose that I=TA and the orthogonal complement of XI in L2I is p-dimensional for some integer p∈ℕ. If Pp−1 denotes the vector space consisting of all complex polynomials of degree ≤p−1 (also including the zero polynomial), then in [154], Theorem 3(a), it is stated that YI is dense in L2I, where Y≔Pp−1+X; furthermore, YI=L2I if and only if p=1, which is equivalent to saying that Pi/Pj∈ℚ for 1≤i≤j≤N. Due to [154], Theorem 3(b), there exists a real-valued function h∈L2I such that the functions h,h′,⋅⋅⋅,hp−1 span XI; furthermore, if we extend the function h⋅ by zero to the whole real line and denote the obtained function by H⋅, then we know that the function H⋅ is a nonzero finite linear combination of Dirac measures.Almost periodicity in chaos: in this section, we will only draw the attention of the readers to the results presented in the tenth section of the recent research monograph [155] by Akhmet. In [155], Section 10, the author investigated the dynamical properties of the following system:(41)y′=Ay+Gt,y+Hxt,t∈ℝ,where G:ℝ×ℝn⟶ℝn is continuous in both variables and almost periodic in variable t uniformly for y∈ℝn, the function H:ℝm⟶ℝn is continuous, and all eigenvalues of the constant n×n real matrix A have negative real parts. Roughly speaking, if the perturbation part Hxt is chaotic in a certain sense, then system (41) has the interesting feature of chaos with infinitely many almost periodic motions. The obtained results are well illustrated with several numerical tests involving the coupled Duffing oscillators, for which it is well known that they play an important role in modeling of the enhanced signal propagation. The most important notion used in [155], Section 10, is the notion of the Li–Yorke chaotic set with infinitely many almost periodic motions, which is introduced in [155], Definition 10.1, for the equicontinuous families of uniformly bounded functions x:ℝ⟶Λ, where Λ is a nonempty compact subset of ℝm. We would like to note here that this notion can be introduced in the infinite-dimensional setting, even for other types of chaos such as distributional chaos or mean Li–Yorke chaos [156].Almost periodicity in mathematical biology: there exist numerous research articles concerning almost periodic- and almost automorphic-type solutions for various classes of ordinary and partial differential equations appearing in mathematical biology (see, e.g., the recent article [157] by Abbas, Dama, Pinto, and Sepulveda, monograph [5], and the references quoted therein). In this section, we will present the main details of the investigation [158] carried out by Ding, Liang, and Xiao and the investigation [159] carried out by Zhang, Yang, and Wang. The nonlinear functional differential equation(42)x′t=−axt+p1+xnt−τ,n>0,was proposed by Mackey and Glass [160] for modeling of hematopoiesis describing the process of production of all types of blood cells generated by a remarkable self-regulated system that is responsive to the demands put upon it. The authors of [158] studied the following modification of (42):(43)x′t=−atxt+ptxlt−τt1+xlt−τt,n>0,where a,p,τ:ℝ⟶0,∞ are almost periodic functions, 0<m≤1, and l>0. The authors of [158] employed a fixed-point theorem in cones to achieve their aims. The authors of [159] considered the existence and global exponential convergence of positive almost periodic solutions for the generalized model of hematopoiesis, described by the following nonlinear functional differential equation:(44)x′t=−atxt+∑i=1mbit1+xnt−τit,n>0,where a,bi,τi:ℝ⟶0,∞ are continuous functions for i=1,2,…,m; clearly, this equation is a generalization of (42). This model has been proposed by Gyori and Ladas to describe the dynamics of hematopoiesis, i.e., blood cell production. In any reasonable biological interpretation of model (44), only positive functions x⋅ can be accepted as solutions. The main results of [159] are Theorems 3.1 and 3.2, in which the authors assumed that a,bi,τi:ℝ⟶0,∞ are almost periodic functions for i=1,2,…,m. Set(45)a−=inft∈ℝat,a+=supt∈ℝat,bi−=inft∈ℝbit>0,bi+=supt∈ℝbit,r=max1≤q≤nsupt∈ℝτit>0,M1≔∑i=1mbi+a−,M2≔∑i=1mbi−a+1+M1n,and suppose that n∑i=1mbi+<a−.Then, there exists a unique positive almost periodic solution of (44) in the closed set B∗=f∈APℝ:ℝ;M2≤f∞≤M1. If we denotex∗⋅ this solution by x∗⋅, then any solution xt;t0,φ of equation (44) equipped with the initial condition(46)xt0=φ,φ∈C+,φ0>0,converges exponentially to x∗t as t⟶+∞; see [159] for the notion and more details.Interpolation by periodic and almost periodic functions: the problems of interpolation by periodic and almost periodic functions were intensively studied by a group of Polish mathematicians during the 1960s. Probably, the first fundamental result in this direction was obtained in 1961 by Mycielski [161], who proved that there exists a sequence tn of positive real numbers such that, for every sequence εn in 0,1, there exists a continuous periodic function f:ℝ⟶ℂ such that ftn=εn for all n∈ℕ, answering a question proposed earlier by Marczewski and Ryll-Nardzewski. Two years later, this result was extended by Lipiński in [162], who proved that there exists a sequence tn of positive real numbers such that, for every bounded real function g⋅ defined on the set tn:n∈ℕ, there exists a continuous periodic function f:ℝ⟶ℂ such that ftn=gtn for all n∈ℕ. The essential thing in the aforementioned results is a rapid increase of the sequence tn as n⟶+∞: in [161], we concretely have that tn=3+αn, where α>0. Let us note that Ryll-Nardzewski showed that, for every sequence εn in 0,1, there exists a continuous periodic function f:ℝ⟶ℂ such that f3n=εn for all n∈ℕ as well as that there does not exist a sequence tn of positive real numbers with tn=O2n, n∈ℕ, satisfying the above property. Interpolation by almost periodic functions was investigated for the first time by Hartman [163] in 1961 and later reconsidered in a series of his joint research papers with Ryll-Nardzewski [164–166] during the period 1964–1967. In [164], the authors analyzed the following properties for the subset Λ of the real line ℝ (and the abelian topological groups):I: Λ satisfies property I if and only if any bounded, uniformly continuous function g:Λ⟶ℂ can be extended to an almost periodic function f:ℝ⟶ℂI0: Λ satisfies property I0 if and only if any bounded function g:Λ⟶ℂ can be extended to an almost periodic function f:ℝ⟶ℂThe authors first proved that there are no sequenceεn in 0,1 and an almost periodic function f:ℝ⟶ℂ such that fnα=εn for all n∈ℕ, provided that α>0 is not an integer; this essentially follows from the equality(47)limN⟶∞1N∑n=1Nfnα=limT⟶∞12T∫−TTftdt,which is valid for these values of number α>0. The main results concerning properties I and I0 and extensions to uniformly continuous almost periodic functions were proved in [164], Theorems 1 and 2, while the third main result of this paper, [164], Theorem 3, analyzes a similar problem for extensions to Stepanov almost periodic functions. In [167], Strzelecki proved that any sequence tn of positive real numbers such that tn+1/tn>1+δ, n∈ℕ, where δ>0 is a fixed real number, has property I0; later, this result was extended in [165], Theorem 5. Interpolation by Levitan almost periodic functions was considered by Hartman in [168].In the list of [5], we have also quoted some references concerning subjects such as the Bohr compactifications, almost periodic functions on C∗-algebras, semiholomorphic almost periodic functions, and certain interplays between the almost periodicity and the representation theory.


## 3. Almost Periodic Functions of Several Real Variables and Their Applications

The notion of almost periodicity can be simply generalized to the case in whichI=ℝn. Suppose that F:ℝn⟶X is a continuous function. Then, we say that F⋅ is almost periodic if and only if for each ε>0, there exists l>0 such that, for each t0∈ℝn, there exists τ∈Bt0,l such that(48)Ft+τ−Ft≤ε,t∈ℝn.This is equivalent to saying that, for any sequencebn in ℝn, there exists a subsequence an of bn such that F⋅+an converges in Cbℝn:X. Any trigonometric polynomial in ℝn is almost periodic, and it is also well known that F⋅ is almost periodic if and only if there exists a sequence of trigonometric polynomials in ℝn which converges uniformly to F⋅; let us recall that a trigonometric polynomial in ℝn is any linear combination of functions such as t⟼eiλ,t, t∈ℝn, where λ∈ℝn and ⋅,⋅ denotes the inner product in ℝn. Any almost periodic function F:ℝn⟶X is almost periodic with respect to each of the variables, but the converse statement is not true since the function t1,t2⟼cost1t2, t1,t2∈ℝ, is almost periodic with respect to both variables t1 and t2 but not almost periodic with respect to t1,t2. Furthermore, for any almost periodic function F⋅, we have that, for each ε>0, there exists l>0 such that, for each t0∈t,t,…,t:t∈ℝ, there exists τ∈Bt0,l∩t,t,…,t:t∈ℝ such that (48) holds. Any almost periodic function F⋅ is bounded, the mean value(49)MF≔limT⟶+∞12Tn∫s+KTFtdt,exists, and it does not depend on s∈ℝn; here, KT≔t=t1,t2,…,tn∈ℝn:ti≤T,for1≤i≤n. The Bohr–Fourier coefficient Fλ∈X is defined by(50)Fλ≔Me−iλ,⋅F⋅,λ∈ℝn,where ⋅,⋅ denotes the usual inner product in ℝn. The Bohr spectrum of F⋅, defined by σF≔λ∈ℝn:Fλ≠0, is at most a countable set.The almost periodic functions of two real variables are also investigated by Besicovitch in the classic [169]. Here, we would like to note that the results established in [169] can be straightforwardly generalized to the almost periodic functions of several real variables. For example, if ti is a fixed variable from the set t1,…,tn, then the function ti⟼Ft1,…,ti,…,tn, ti∈ℝ, is almost periodic for every fixed real number t1,…,ti−1,ti+1,…,tn so that the mean value(51)MtiFt1,…,tn≔limTi⟶+∞12Ti∫−TiTiFt1,…,ti,…,tndti,exists. Considering MtiFt1,…,tn as a function of the variables t1,…,ti−1,ti+1,…,tn, it can be easily shown that it is almost periodic in ℝn−1. Therefore, we can calculate the repeated mean value(52)Mtj∘MtiFt1,…,tn≔limTj⟶+∞12Tj∫−TjTjMtiFt1,…,tndtj,for any fixed real numbers from the set t1,…,tn\ti,tj. If we fix these numbers in advance, we can apply ([169], Corollary, p. 63) to the almost periodic function(53)Fijti,tj≔Ft1,…,ti,…,tj,…,tn,ti,tj∈ℝ2,in order to see that(54)Mtj∘MtiFt1,…,tn≡Mti∘MtjFt1,…,tn.Inductively, we easily get that, for every finite tuple of different variablesti1,…,til, where 1≤i1<i2<⋯<il≤n, and for every permutation σ:i1,…,il⟶i1,…,il, we have(55)Mti1∘⋯∘MtilFt1,…,tn=Mtσi1∘⋯∘MtσilFt1,…,tn.ByAPℝn:X and APΛℝn:X, we denote, respectively, the Banach space consisting of all almost periodic functions F:ℝn⟶X, equipped with the sup-norm, and its subspace consisting of all almost periodic functions F:ℝn⟶X such that σF⊆Λ. As is well known, for every almost periodic function F∈APΛℝn:X, we can always find a sequence Pk of trigonometric polynomials in ℝn which uniformly converges to F⋅ on ℝn and satisfies that σPk⊆Λ for all k∈ℕ; see, e.g., [170], Chapter 1, Section 2.3. The Wiener algebra APWℝn:X is defined as the set of all functions F:ℝn⟶X such that its Fourier series converges absolutely; APWΛℝn:X≡APWℝn:X∩APΛℝn:X. It is well known that the Wiener algebra is a Banach algebra with respect to the Wiener norm F≔∑λ∈ℝnFλ, F∈APWℝn:X, as well as that APWℝn:X is dense in APℝn:X.The theory of almost periodic functions of several real variables has not attracted so much attention compared with the theory of almost periodic functions of one real variable by now. In the following, we will remind the readers of several important investigations of multidimensional almost periodic functions carried out so far:1.
Problems of Nehari type and contractive extension problems for matrix-valued (Wiener) almost periodic functions of several real variables have been considered by Rodman, Spitkovsky, and Woerdeman in [171], where the authors proved a generalization of the famous Sarason’s theorem. In their analysis, the notion of a half-space in ℝn plays an important role: a nonempty subset S⊆ℝn is said to be a half-space if and only if the following four conditions hold:(i)
ℝn=S∪−S(ii)
0=S∩−S(iii)
S+S⊆S(iv)
α⋅S⊆S for α≥0For any half-spaceS, we can always find a linear bijective mapping D:ℝn⟶ℝn such that S=DEn, where En is a very special half-space defined on [172], p. 3190. In [172], Theorem 1.3, Rodman and Spitkovsky proved that if S is a half-space and Λ⊆S, 0∈Λ, and Λ+Λ⊆Λ, then APΛℝn:ℂ and APWΛℝn:ℂ are Hermitian rings. See also [173].(2)
Let us recall that a subsetΛ of ℝn is called discrete if and only if any point λ∈Λ is isolated in Λ. By VΛ, we denote the vector space of all finite complex-valued trigonometric polynomials ∑λ∈Λcλe−πiλ⋅ whose frequencies λ belong to Λ. The space of mean-periodic functions with the spectrum Λ, denoted by CΛ, is defined as the closure of the space VΛ in the Fréchet space Cℝn. Clearly, APΛℝn:ℂ is contained in CΛ, but the converse statement is not true, in general. The problem of describing the structure of closed discrete sets Λ for which the equality APΛℝn:ℂ=CΛ holds was proposed by Kahane in 1957. For more details about this interesting problem, we refer the reader to the survey article [174] by Meyer; for more details about mean-periodic functions, see also the lectures by Kahane [175].(3)
In 1971, Basit [176] observed that there exists a complex-valued almost periodic function f:ℝ2⟶ℂ such that the function F:ℝ2⟶ℂ, defined by Fx,y≔∫0xft,ydt, x,y∈ℝ2, is bounded but not almost periodic. This result was recently reconsidered by Alsulami in [177], Theorem 2.2, who proved that, for a complex-valued almost periodic function f:ℝ2⟶ℂ, the boundedness of the function F⋅ in the whole plane implies its almost periodicity, provided that there exists a complex-valued almost periodic function g:ℝ2⟶ℂ such that fxx,y=gyx,y is a continuous function in the whole plane. This result was proved with the help of an old result of Loomis which states that, for a bounded complex-valued function f:ℝn⟶ℂ, the almost periodicity of all its partial derivatives of the first order implies the almost periodicity of f⋅ itself. Let us observe that the aforementioned result of Alsulami can be straightforwardly extended, with the same proof, to the almost periodic functions f:ℝn⟶ℂ; in actual fact, if the function f:ℝn⟶ℂ is almost periodic, the function Fx1,x2,…,xn≔∫0x1ft,x2,…,xndt, x1,x2,…,xn∈ℝn is bounded, and there exist almost periodic functions Gi:ℝn⟶ℂ such that Fxix1,x2,…,xn=Gix1x1,x2,…,xn is a continuous function on ℝn, for 2≤i≤n, then the function F:ℝn⟶ℂ is almost periodic.(4)
In [178–183], Khasanov investigated the approximations of uniformly almost periodic functions of two variables by partial sums of Fourier sums and Marcinkiewicz sums in the uniform metric, provided certain conditions.(5)
In [184, 185], Latif and Bhatti investigated several important questions concerning almost periodic functions defined on ℝn with values in locally convex spaces and fuzzy-number-type spaces (almost periodic functions defined on ℝn with values in p-Fréchet spaces, where 0<p<1, were investigated in [186] by N’Guérékata, Latif, and Bhatti).Concerning applications made so far, we recall the following:(1
)The problem of the existence of almost periodic solutions for the system of linear partial differential equations∑j=1nLijuj=fi,1≤i≤n, on ℝm, where Lij is an arbitrary linear partial differential operator on ℝm, was analyzed by Sell [187, 188]. He extended the results obtained by Sibuya, where the author analyzed the almost periodic solutions of Poisson’s equation.(2)
The almost periodic solutions of the (semilinear) systems of ordinary differential equations were analyzed by Fink in [189], Chapter 8, with the help of fixed-point theorems. Furthermore, Liu Bao-Ping and Pao investigated the almost periodic plane wave solutions of certain classes of coupled nonlinear reaction-diffusion equations [190]; in their approach, a solution ut,x of such a system, where t∈ℝ and x∈ℝn, is almost periodic in ℝn+1 and satisfies that ut,x is almost periodic in the time variable t∈ℝ and periodic in each spatial variable (see [190], Theorem 2).(3)
In his doctoral dissertation [191], Alsulami considered the question whether the boundedness of solutions of the following system of partial first-order differential equations(56)uss,t=Aus,t+f1s,t,uts,t=Bus,t+f2s,t,s,t∈ℝ2,implies the almost periodicity of solutions to (56). He analyzed this question in the finite-dimensional setting and the infinite-dimensional setting, using two different techniques; in both cases, A and B are bounded linear operators acting on the pivot space X.(4)
In [192–194], Spradlin provided several interesting results and applications regarding almost periodic functions of several real variables. The existence of positive homoclinic-type solutions of the equation(57)−Δu+u=Htfu,whereH⋅ is almost periodic and the first integral of f⋅ satisfies certain superquadraticity and critical growth conditions, has been analyzed in [194], Theorem 1.2. The equations of type(58)−ε2Δu+Htu=fu,arise in the study of the nonlinear Schrödinger equationsε>0. A qualitative analysis of solutions of (58) has been carried out in [193], provided the almost periodicity of function H⋅ and several other nontrivial assumptions.(5)
The existence and uniqueness of almost periodic solutions for a class of boundary value problems for hyperbolic equations were investigated by Ptashnic and Shtabalyuk in [195] (also, cf. the sixth chapter in monograph [53] by Ptashnic). In the region Dp=0,T×ℝp (T>0, p∈ℕ), they have analyzed the well-posedness of the following initial value problem:(59)Lu≡∑s=0n∑α=2saα∂2nut,x∂t2n−2s∂x1α1…∂xpαp=0,(60)∂j−1u∂tj−1|t=0=φjx,∂j−1u∂tj−1|t=T=φj+nx,1≤j≤n.The basic assumption employed in [195] is that equation (59) is Petrovsky-hyperbolic, i.e., for each μ=μ1,μ2,…,μp∈ℝp, all λ-zeroes of the equation(61)∑s=0n∑α=2saαλ2n−2sμ1α1μ2α2…μpαp=0,are real. The basic function space used is the Banach spaceCBqDp¯ consisting of all q-times continuously differentiable functions ut,x in Dp¯ that are Bohr almost periodic in variables x1,x2,…,xp, uniformly in t∈0,T, equipped with the norm(62)uCBqDp¯≔sup0≤s≤qsupt,x∈Dp¯∂sut,x∂ts0∂x1s1…∂xpsp,and byCBqℝp, the authors designated the subspace of CBqDp¯ consisting of those functions which do not depend on variable t. The existence and uniqueness of solutions of initial value problems (59) and (60) have been investigated in the space CB2nDp¯, under the assumption that φjx∈CBrℝp and r∈ℕ is sufficiently large. If Mp=μk:k∈ℤp is the union of spectrum of all functions φ1x,…,φ2nx, the solutions ut,x of problems (59) and (60) have been found in the form(63)ut,x=∑k∈ℤpukteiμk,x,μk∈Mp,where the functionsukt satisfy certain conditions and have the form given in equation ([195], (8), p. 670). The uniqueness of solutions of problems (59) and (60) has been considered in [195], Theorem 1, while the existence of solutions of problems (59) and (60) has been considered in [195], Theorem 2.(6)
The class of vector-valued remotely almost periodic functions defined onℝn was introduced by Yang and Zhang in [196]. In the same paper, the authors provided several applications in the study of the existence and uniqueness of remotely almost periodic solutions for parabolic boundary value problems. A function F:ℝn⟶X is said to be remotely almost periodic if and only if for each ε>0, the set of all vectors τ∈ℝn for which(64)limsupt⟶+∞Ft+τ−Ft<ε,is relatively dense in ℝn (the vector τ is called a remotely ϵ-translation vector of F⋅); furthermore, if ∅≠Ω⊆ℝm, then a continuous function F:ℝn×Ω⟶X is said to be remotely almost periodic in t∈ℝn and uniform on compact subsets of Ω if and only if F⋅,y is remotely almost periodic for each y∈Ω and is uniformly continuous on ℝn×K for any compact subset K⊆Ω. The following statements hold in the scalar-valued case (see, e.g., [196], Propositions 2.1–2.3):(i)
IfF⋅, resp. F⋅;⋅, is remotely almost periodic and the function ∂F/∂ti⋅, resp. ∂F/∂ti⋅;⋅, is uniformly continuous on ℝn, then the function ∂F/∂ti, resp. ∂F/∂ti⋅;⋅, is remotely almost periodic, as well 1≤i≤n.(ii)
If the functionsF1⋅,…,Fk⋅ are remotely almost periodic k∈ℕ, then for each ε>0, the set of their common ϵ-translation vectors is relatively dense in ℝn.(iii)
If the functionsH1⋅,…,Hk⋅ are remotely almost periodic k∈ℕ and H1t,…,Hkt∈Ω for all t∈ℝ, then for every remotely almost periodic function F:ℝ×Ω⟶ℂ, we have that the function(65)t⟼FH1t,…,Hkt,t,t∈ℝ,is remotely almost periodic.In [196], Propositions 2.4–2.6, the authors examined the existence and uniqueness of remotely almost periodic solutions of multidimensional heat equations, while the main results of the third section of this paper are concerned with the existence and uniqueness of remotely almost periodic-type solutions of certain types of parabolic boundary value problems.The boundedness and almost periodicity in time for certain classes of evolution variational inequalities, positive boundary value problems for symmetric hyperbolic systems, and nonlinear Schrödinger equations have been investigated in the third and fourth section of the important research monograph [170] by Pankov (for almost periodic properties of Schrödinger equations and Schrödinger-type operators, see the reference list of [5]). Spatially, Besicovitch almost periodic solutions for certain classes of nonlinear second-order elliptic equations, first-order hyperbolic systems, single higher-order hyperbolic equations, and nonlinear Schrödinger equations have been investigated in the fifth section of this monograph. For more details about the applications of Stepanov multidimensional almost periodic functions and Weyl multidimensional almost periodic functions, as well as to some interplays between the multidimensional almost periodic functions, calculus of variations, and homogenization theory, we refer the reader to notes and appendices to the third section of [5].It is worth mentioning that Spradlin constructed, in [192], an almost periodic infinitely differentiable function G:ℝn⟶ℝ with no local minimum (it can be simply shown that this situation cannot occur in the one-dimensional case because any almost periodic function g:ℝ⟶ℝ must have infinitely many local minima); this important peculiarity of almost periodic functions of several real variables was perceived twenty five years ago. The construction of an almost periodic function G:ℝn⟶ℝ with no local minimum, established in [192], is very complicated, and the proof of the main result of this paper ([192], Theorem 1.0) contains almost eight pages including some preliminaries. It can be easily proved, by observing that the function Gx,y is strictly positive, that the function x,y⟼Hx,y≡∫0xGt,ydt is bounded and not almost periodic in the plane. As already mentioned, the existence of a complex-valued almost periodic function Hx,y with these properties was clarified by Basit (1971) with very obscure evidence, not including the smoothness of Gx,y or its nonnegativity.At the end of paper [192], Spradlin proposed the following questions:(1)
The almost periodic functionF:ℝ2⟶ℝ constructed in the proof of [192], Theorem 1.0, has an absolute maximum at the point 0,0. Does there exist an almost periodic function F:ℝn⟶ℝ with no local minimum or maximum?(2)
Does there exist a real analytic almost periodic functionF:ℝn⟶ℝ with no local minimum or maximum?(3)
Is it true that a continuously differentiable almost periodic functionF:ℝn⟶ℝ has a critical point?(4)
Does there exist a quasi-periodic functionF:ℝn⟶ℝ with no local minimum (local minimum or maximum)?To the best of authors’ knowledge, all these questions are still open. Concerning open problems, we also want to remind our readers of article [197] by Basit.Now, we would like to say something more about the following intriguing topics.Multivariate trigonometric polynomials and approximations of periodic functions of several real variables: without any doubt, trigonometric polynomials of several real variables, sometimes also called multivariate trigonometric polynomials, present the best-explored class of almost periodic functions of several real variables. Multivariate trigonometric polynomials have an invaluable importance in the theory of approximations of periodic functions of several real variables, especially in the two-dimensional case. For the basic source of information about this subject, the reader may consult research monographs [198] by Dumitrescu, [199] by Dung, Temlyakov, and Ullrich, and [200, 201] by Temlyakov, research article [202] by Temlyakov, and the list of references quoted in [5].In this part, we will briefly explain the main results and ideas of papers [203] by Babayev, [204] by Pfister and Bresler, and [205] by Kämmerer, Potts, and Volkmer. If f:ℝ⟶ℝ belongs to the space C2π of all real continuous functions of period 2π, then it is well known that the Vallee Poussin singular integral Vk⋅, defined by(66)Vkx≔12π2k!!2k−1!!∫−ππftcos2kt−x2dt,x∈ℝk∈ℕ,has the property that limk⟶+∞Vkx=fx, uniformly for x∈ℝ. This result of Vallee Poussin improves the classical Weierstrass second theorem on the density of trigonometric polynomials in the spaces of continuous functions. Two-dimensional Vallee Poussin singular integral Vk,m⋅, defined for each x∈ℝ by k,m∈ℕ,(67)Vk,mx,y≔12π22k!!2k−1!!2m!!2m−1!!∫−ππft,τcos2kt−x2cos2kτ−y2dτ,has been introduced in [203], Definition 2. In the same paper, the author showed that limk⟶+∞limm⟶+∞Vk,mx,y=fx,y, uniformly for x,y∈ℝ2, as well as that Vk,mx,y is a trigonometric polynomial in variables x and y, for all k,m∈ℕ (see [203], Theorem 2). For proving the last fact, the author used a lemma clarifying that the product of two trigonometric polynomials of two variables is also the trigonometric polynomial of two variables whose order equals the sum of order of cofactors as well as that any even trigonometric polynomial Tx,y, i.e., a trigonometric polynomial Tx,y which satisfies that T−x,−y=Tx,y, T−x,y=Tx,y, and Tx,−y=Tx;y identically for x,y∈ℝ2, may be represented in the form(68)Tx,y=A+∑k=1m∑l=1naklcoskxcosly+bklcoskx+cklcosly,x,y∈ℝ2,which does not contain the sines of multiple arcs (see [203], Lemmas 3 and 4). We would like to note that the obtained results continue to hold in the vector-valued case.In [204], Pfister and Bresler investigated bounding multivariate trigonometric polynomials and gave some applications to the problems of filter bank design. Denote;(69)Tln≔spaneik,λ:λ∈0,2πn,k∈ℤn,k≔sup1≤i≤nki≤l,l∈ℕ,ΘN≔2πkN:k=0,1,…,N−1,N∈ℕ.For anyN∈ℕ and for any real-valued trigonometric polynomial(70)Pλ≔∑k1=−ll∑k2=−ll…∑kn=−llck1k2…kneik,λ∈Tln,i.e., the multivariate trigonometric polynomial P⋅ for which ck1,k2,…,kn=c−k1,−k2,…,−kn∗ (k≤l;star denotes complex conjugation), we define(71)P∞≔maxλ∈0,2πnPλandPNn,∞≔maxλ∈ΘNnPλ.Then, two well-known results of the approximation theory state that(72)P∞≤P2l+1n,∞1+4π−1+2π−1ln2l+1n,and in the one-dimensional case,(73)P∞≤PN,∞1−2l/N.In [204], Theorem 1, the authors showed that the assumptions N≥2l+1 and α=2l/N yield the existence of a positive real constant CN,ln∈0,1−α−n/2 such that(74)P∞≤CN,lnPNn,∞,P∈Tln,and CN,lnPNn,∞−P∞=Oln/N, P∈Tln. In order to achieve their aims, the authors used the de la Vallee Poussin kernels and the tensor products of one-dimensional Dirichlet kernels.In [205], the authors investigated certain algorithms for the approximation of multivariate periodic functions by trigonometric polynomials, which are based on the use of a single one-dimensional fast Fourier transform and the so-called method of sampling of multivariate functions on rank-1 lattices. In their analysis, the authors used periodic Sobolev spaces of generalized mixed smoothness and presented some advantages of their method compared to the method based on the trigonometric interpolations on generalized sparse grids. Some numerical results and tests are presented up to dimension n=10, as well.Almost periodic pseudo-differential operators and Gevrey classes: almost periodic pseudo-differential operators have been analyzed by numerous mathematicians including Coburn, Moyer, and Singer [206], Dedik [207], Iannacci, Bersani, Dell’Acqua, and Santucci [208], Pankov [209], Shubin [210–213], and Wahlberg [214]. In this part, we will present the main ideas and results of research study [215] by Oliaro, Rodino, and Wahlberg, only. It is well known that Shubin proved that almost periodic pseudo-differential operators act continuously on the space of smooth almost periodic functions as well as that the operator norm on L2 equals that on the Hilbert space B2ℝn of Besicovitch almost periodic functions whose Fourier coefficients are square summable. It is also well known that Shubin introduced, for every exponent p∈1,∞ and for every real number t∈ℝ, the space Wtpℝn of almost periodic functions and proved the continuity of any almost periodic pseudo-differential operator A:Wt2ℝn⟶Wt−m2ℝn, with arbitrary t∈ℝ, provided that the symbol of A belongs to the class Sρ,δm (0≤δ<ρ≤1). In the papers of Shubin, some regularity results for formally hypoelliptic almost periodic pseudo-differential operators have been examined on the space W−∞2ℝn≔∪t∈ℝWt2ℝn.In [215], the authors sought for ultradistributional analogues of the aforementioned results, working with almost periodic functions that are Gevrey regular of order s≥1 (the difference between the real analytic case s=1 and the pure ultradistributional case s>1 should be emphasized here). If ∅≠Ω⊆ℝn, then the space of all Gevrey functions of order s≥1, denoted by GsΩ, is defined as a collection of all infinitely differentiable functions F:ℝn⟶ℂ such that, for each compact set K⊆ℝn, there exists a finite real constant CK>0 such that DαFt≤CK1+αα!s for all t∈K and α∈ℕ0n. It is natural to ask whether an almost periodic function F:ℝn⟶ℂ which belongs to the space GsΩ obeys the property of the existence of a global real constant C>0 such that DαFt≤C1+αα!s for all t∈ℝn and α∈ℕ0n. An instructive counterexample in the one-dimensional setting, with s>1, is given in [215], Example 2.1, showing that this is not true in general: set gsx≔exp−x1/1−s, x>0, gsx≔0, x≤0, ψsx≔gsxgs1−x, x∈ℝ, ψs,nx≔ψsnx, x∈ℝ, and φs,nx≔∑k∈ℤψsx−2n2k+1,x∈ℝ (n∈ℕ). It has been shown that the function(75)Fsx≔∑n=1∞n−1/4φs,nx,x∈ℝ,is well defined, as well as that the above series is uniformly convergent in the variable x∈ℝ, so that the function Fs⋅ is actually semiperiodic since the function φs,n⋅ is of period 2n+1 (n∈ℕ). We also have that Fs∈Gsℝ as well as that Fs∉Gapsℝ; see the notion explained in the following. Albeit not explicitly constructed in [215], it is our strong belief that this example can be transferred to the multidimensional setting without any serious difficulties, as well (more to the point, the case s=1 has not been considered in [215], Example 2.1, and deserves further analyses).After providing this counterexample, the authors paid special attention to the analysis of almost periodic functionsF:ℝn⟶ℂ belonging to the space Gsℝn and obeying the property that there exists a real constant C>0 such that DαFt≤C1+αα!s for all t∈ℝn and α∈ℕ0n. The union of these functions, denoted by Gapsℝn, is equipped with the usual inductive limit topology as a union of Banach spaces. Then, the authors introduced the corresponding classes of symbols and pseudo-differential operators and continued their nontrivial analysis; see [215] for more details.The theory of almost periodic-type functions is far from being completed, and finally, we would like to mention some topics that are not very well explored in the existing literature by now:(1)
Almost anything has been said about the almost periodic properties and the almost automorphic properties of various types of fractional integrals and fractional derivatives of vector-valued periodic functions (see Area, Losada, and Nieto [216] and Jonnalagadda [217]).(2)
The notion ofc-periodicity and the notion of c-almost periodicity require several further investigations within the theory of vector-valued generalized functions [218].(3)
Applications of the multidimensional almost periodic-type functions in the classical theory of partial differential equations and applications of the multidimensional almost periodic-type functions to the boundary value problems are still not examined to a satisfactory extent.(4)
The Stepanov, Weyl, and Besicovitch classes of multidimensional almost automorphic functions have not been analyzed before. See also the recent research studies [219, 220, 221].(5)
The results about the invariance of certain kinds of generalized almost periodicity and generalized almost automorphicity under the actions of infinite convolution products (11) and (12) can be simply transferred to the multidimensional setting. It is not clear how we can apply these results in mathematical physics and applied science.


## 4. Conclusions

In this survey article, we have collected several known results about vector-valued almost periodic functions, separately considering the almost periodic functions of one real variable and the almost periodic functions of several real variables. We have tried to present the most representative applications of almost periodic functions to the abstract Volterra integrodifferential equations in Banach spaces as well as to remind our readers of some landmarks, pioneering investigations of almost periodic functions. We have proposed some open problems and perspectives for further investigations of almost periodicity.



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