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Multi-dimensional (ω, c)-almost periodic type functions and applications

Multi-dimensional (ω, c)-almost periodic type functions and applications Nonauton. Dyn. Syst. 2021; 8:136–151 Research Article Open Access M. Kostić* Multi-dimensional (ω, c)-almost periodic type functions and applications https://doi.org/10.1515/msds-2020-0130 Received February 1, 2021; accepted April 19, 2021 Abstract: In this paper, we analyze various classes of multi-dimensional (ω, c)-almost periodic type func- tions with values in complex Banach spaces. The main structural properties and characterizations for the introduced classes of functions are presented. We provide certain applications of our abstract theoretical results to the abstract Volterra integro-dierential equations, as well. Keywords: Multi-dimensional (ω, c)-almost periodic functions, multi-dimensional (ω , c ) -periodic func- j j j2N tions, multi-dimensional c-almost periodic functions, (I , a, ω, c)-almost periodic type functions, abstract Volterra integro-dierential equations MSC: 42A75, 43A60, 47D99 1 Introduction and preliminaries The notion of an almost periodic function was introduced by H. Bohr [7] around 1924-1926 and later general- ized by many other mathematicians (for more details about the subject, we refer the reader to the research monographs [6], [9]-[11], [17]-[19], [21]-[23] and [25]). Let I be either R or [0,∞), and let f : I ! X be a given continuous function, where X is a complex Banach space equipped with the normk·k. For any ε > 0, a num- ber τ > 0 is called a ε-period for f (·) if and only ifkf (t + τ)− f (t)k ≤ ε, t 2 I. The set consisting of all ε-periods for f (·) is denoted by ϑ(f , ε). The function f (·) is said to be almost periodic if and only if for each ε > 0 the set ϑ(f , ε) is relatively dense in [0,∞), i.e., there exists l > 0 such that any subinterval of [0,∞) of length l meets ϑ(f , ε). The notion of a periodic function has recently been reconsidered by E. Alvarez, A. Gómez and M. Pinto [2] in the following way: A continuous function f : I ! X is said to be (ω, c)-periodic (ω > 0, c 2 C \f0g) if and only if f (x + ω) = cf (x) for all x 2 I. We know that a continuous function f : I ! X is (ω, c)-periodic if −·/ω −·/ω and only if the function g(·)  c f (·) is periodic and g(x + ω) = g(x) for all x 2 I; here, c denotes the principal branch of the exponential function (see also the research articles [3]-[4] by E. Alvarez, S. Castillo, M. Pinto for more details about the subject). In our joint paper [16] with M. T. Khalladi, A. Rahmani, M. Pinto and D. Velinov, we have recently extended the notion of (ω, c)-periodicity by examining various classes of (ω, c)-almost periodic type functions. On the other hand, in our recent paper [15], we have recently introduced the class of c-almost periodic functions depending on one real variable. The multi-dimensional c-almost periodic type functions have re- cently been investigated in [16]. The main aim of this paper is to continue the above-mentioned research studies by introducing and analyzing various notions of (ω, c)-periodicity and (ω, c)-almost periodicity for vector-valued functions depending on several real variables; we provide certain applications to the abstract partial dierential equations, as well. For the sake of simplicity and better exposition, we will consider the *Corresponding Author: M. Kostić: Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovića 6, 21125 Novi Sad, Serbia, E-mail: marco.s@verat.net Open Access. © 2021 M. Kostić, published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. Multi-dimensional (ω, c)-almost periodic type functions.. . Ë 137 corresponding Stepanov classes of multi-dimensional (ω, c)-almost periodic type functions and the corre- sponding (Stepanov) classes of multi-dimensional semi-c-periodic functions somewhere else. The organization and main ideas of the paper can be briey described as follows. In Subsection 1.1, we recall the basic denitions and results about c-almost periodic functions in R . The main aim of Section 2 is to introduce and analyze the classes of multi-dimensional (ω, c)-periodic functions and multi-dimensional (ω , c ) -periodic functions. The main structural results obtained in this section are Proposition 2.4- j j j2N Proposition 2.5 and Proposition 2.8-Proposition 2.11. The corresponding classes of asymptotically (ω, c)- almost periodic type functions are introduced in Denition 2.15; Subsection 2.1 investigates (ω , c ; r , I ) - j j j j2N j n almost periodic type functions. In Denition 2.16, we introduce the notion of (ω , c ; r , I ) -almost peri- j j j j2N j n odicity, (ω , c ; r , I ) -uniform recurrence and (ω , c ) -almost automorphy. The main structrural pro- j j j j2N j j j2N j n n lations of function spaces introduced in Denition 2.16 are stated in Proposition 2.19; we also discuss the convolution invariance of function spaces introduced here before we move ourselves to the third section of ′ ′ paper, which is reserved for the study of (I , a, ω, c)-uniform recurrence of type 1 (2) and the (I , a, ω, c)- almost periodicity of type 1 and type 2. Here we continue our investigation from [16, Section 3] and prove two negative results, Theorem 3.3 and Theorem 3.5, saying that the introduction of Denition 3.1 is basically an unsatisfactory way to extend the notion of (ω, c)-almost periodicity. In the nal section of paper, we provide certain application to the abstract Volterra integro-dierential equations in Banach spaces. The author would like to thank Prof. A. Chávez, M. T. Khalladi, M. Pinto, A. Rahmani and D. Velinov for many valuable comments and suggestions. Numerous research articles concerning multi-dimensional almost periodic type functions are written in my collaboration with these mathematicians. Notation and terminology. We assume henceforth that (X,k·k) and (Y ,k·k ) are complex Banach spaces, n 2 N, B is a certain collection of subsets of X satisfying that for each x 2 X there exists B 2 B such that x 2 B. We will use the principal branch of the exponential function to take the powers of complex numbers. n n If t 2 R and ϵ > 0, then we set B(t , ϵ) := ft 2 R : jt − t j ≤ ϵg, wherej ·j denotes the Euclidean norm 0 0 0 n n in R . Set N := f1,· · ·, ng and I := ft 2 I : jtj ≤ Mg (I  R ; M > 0). 1.1 c-Almost periodic functions In [19], we have recently introduced the following notion: ′ n ′ Denition 1.1. Suppose that; ≠ I  I  R , F : I × X ! Y is a continuous function and I + I  I. Then we say that: (i) F(·;·) is Bohr (B, I , c)-almost periodic if and only if for every B 2 B and ϵ > 0 there exists l > 0 such that ′ ′ for each t 2 I there exists τ 2 B(t , l)\ I such that 0 0 F(t + τ; x) − cF(t; x) ≤ ϵ, t 2 I, x 2 B. ′ ′ (ii) F(·;·) is (B, I , c)-uniformly recurrent if and only if for every B 2 B there exists a sequence (τ ) in I such that lim jτ j = +∞ and k!+∞ k lim sup F(t + τ ; x) − cF(t; x) = 0. k!+∞ t2I;x2B ′ ′ If X 2 B, then it is also said that F(·;·) is Bohr (I , c)-almost periodic ((I , c)-uniformly recurrent). The most important case is that in which I = I, when we also say that the function F(·;·) is Bohr (B, c)- almost periodic [(B, c)-uniformly recurrent]; if X 2 B, then it is also said that F(·;·) is Bohr c-almost peri- ′ ′ odic (c-uniformly recurrent). The classes of Bohr (B, I )-almost periodic functions, (B, I )-uniformly recur- ′ ′ rent functions, Bohr I -almost periodic functions and I -uniformly recurrent functions, obtained by plugging c = 1, are thoroughly investigated in [8]. If c = −1, then we also say that F(·;·) is (B, I )-almost anti-periodic, ′ ′ ′ (B, I )-uniformly anti-recurrent, I -almost anti-periodic or I -uniformly anti-recurrent. We know the following: 138 Ë M. Kostić ′ n ′ ′ ′ (i) Suppose that ; ≠ I  I  R , I + I = I and F : I × X ! Y is Bohr (B, I , c)-almost periodic ((B, I , c)- uniformly recurrent). If F(·;·) ≠ 0, thenjcj = 1. ′ n ′ ′ (ii) Suppose that; ≠ I  I  R and I + I = I. If the function F : I ! R is (B, I , c)-uniformly recurrent and F ≠ 0, then c = ±1. Furthermore, if F(t) ≥ 0 for all t 2 I, then c = 1. ′ n ′ ′ (iii) Suppose that l 2 N, ; ≠ I  I  R , I + I  I and F : I × X ! Y is Bohr (B, I , c)-almost periodic ′ ′ ′ ′ l ((B, I , c)-uniformly recurrent). Then lI  I, I + lI  I and F(·;·) is Bohr (B, lI , c )-almost periodic ′ l ((B, lI , c )-uniformly recurrent). (iv) Suppose that p 2 Z \f0g, q 2 N, (p, q) = 1, jcj = 1 and arg(c) = πp/q. (1.1) ′ n ′ ′ Suppose, further, that ; ≠ I  I  R , I + I  I and F : I × X ! Y is Bohr (B, I , c)-almost periodic ((B, I , c)-uniformly recurrent). Then the following holds: ′ ′ (a) If p is even, then F(·;·) is Bohr (B, qI )-almost periodic ((B, qI )-uniformly recurrent). ′ ′ (b) If p is odd, then F(·;·) is Bohr (B, qI )-almost anti-periodic ((B, qI )-uniformly anti-recurrent). ′ n ′ ′ ′ (v) Let jcj = 1 and arg(c)/π 2̸ Q. If ; ≠ I  I  R , I + I  I, lI = I for all l 2 N and F : I × X ! Y ′ ′ is a bounded, Bohr (B, I , c)-almost periodic ((B, I , c)-uniformly recurrent) function, then the function ′ ′ ′ ′ ′ F(·;·) is Bohr (B, I , c )-almost periodic ((B, I , c )-uniformly recurrent) for all c 2 S . n ′ n If F : R ! Y is an almost periodic function (X = f0g, B = fXg, I = I = R ), then the mean value M(F) := lim F(t) dt T!+∞ (2T ) s+K n n exists and it does not depend on s 2 R , where K := ft = (t , t ,· · ·, t ) 2 R : jt j ≤ T for 1 ≤ i ≤ ng. The 1 2 n T i Bohr-Fourier coecient F 2 X is dened by −ihλ,·i n F := M e F(·) , λ 2 R . We know that the Bohr spectrum of F(·), dened by σ(F) := λ 2 R : F ≠ 0 , n n is at most a countable set. By AP(R : X) and AP (R : X) we denote respectively the Banach space consisting of all almost periodic functions F : R ! X and its subspace consisting of all almost periodic functions F : R ! X such that σ(F)  Λ. We also need the following denition from [8]: Denition 1.2. Suppose that D  I  R and the set D is unbounded. By C (I × X : Y ) we denote 0,D,B the vector space consisting of all continuous functions Q : I × X ! Y such that, for every B 2 B, we have lim Q(t; x) = 0, uniformly for x 2 B. If X = f0g, then we abbreviate C (I×X : Y ) to C (I : Y ). 0,D,B 0,D,B t2D,jtj!+∞ 2 (ω, c)-Periodic functions and (ω , c ) -periodic functions j j j2N Let us recall that a continuous function F : I ! X is said to be Bloch (p, k)-periodic, or Bloch periodic with n n period p and Bloch wave vector or Floquet exponent k, where p 2 R and k 2 R , if and only if F(t + p) = ihk,pi e F(t), t 2 I (we assume here that p + I  I). In [8, Example 2.15(viii)], we have observed that the Bloch ′ −ihk,·i ′ (p, k)-periodicity of function F(·) implies the Bohr (B, I )-almost periodicity of function e F(·) with I being the intersection of I and the one-dimensional submanifold generated by the vector p as well as that the orthogonality of vectors k and p implies that the function F(·) is Bohr (B, I )-almost periodic. For more details about the Bloch periodic functions, we refer the reader to the research articles [13] by M. Hasler and [14] by M. Hasler, G. M. N’Guérékata. Multi-dimensional (ω, c)-almost periodic type functions.. . Ë 139 Following the recent research analyses of E. Alvarez, A. Gómez, M. Pinto [2] and E. Alvarez, S. Castillo, M. Pinto [3]-[4], we generalize the notion of Bloch (p, k)-periodicity in the following way: Denition 2.1. Let ω 2 R \f0g, c 2 C \f0g and ω + I  I. A continuous function F : I ! X is said to be (ω, c)-periodic if and only if F(t + ω) = cF(t), t 2 I. ihk,pi If F : I ! X is a Bloch (p, k)-periodic function, then F(·) is (p, c)-periodic with c = e ; conversely, if jcj = 1 and F : I ! X is (ω, c)-periodic, then we can always nd a point k 2 R such that the function F(·) is Bloch (p, k)-periodic. In the case that jcj ≠ 1, we have the following: if F : I ! X is (ω, c)-periodic, then F(t + mω) = c F(t), t 2 I, m 2 N, so that the existence of a point t 2 I such that F(t ) ≠ 0 implies 0 0 lim jjF(t + mω)jj = +∞, provided thatjcj > 1, and lim jjF(t + mω)jj = 0, provided thatjcj < 1. m!∞ m!∞ 0 0 If c = 1, resp. c = −1, then we also say that the function F(·) is ω-periodic, resp. ω-anti-periodic. It is k k clear that, if F(·) is (ω, c)-periodic, k 2 N and c = 1, resp. c = −1, then F(·) is (kω)-periodic, resp. (kω)- anti-periodic. In [14, Denition 2.1], the authors have assumed that any Bloch (p, k)-periodic is bounded a priori, which is a slightly redundant condition as the following example shows: Example 2.2. There exists a continuous, unbounded function F : R ! R which satises F(t+(1, 1,···, 1)) = n 2 F(t) for all t 2 R . We can simply construct such a function, with n = 2, as follows. Let F : f(t , t ) 2 R : 0 1 2 0 ≤ t + t ≤ 2g be any continuous function satisfying that: 1 2 F t , t = F t + 1, t + 1 , provided t , t 2 R and t + t = 0, (2.1) 0 1 2 1 2 1 2 1 2 p p the set (4k 2,−4k 2) : k 2 N is unbounded, and (2.2) p p F (4k + 2) 2,−(4k + 2) 2 = 1, k 2 N. (2.3) Due to condition (2.1), we can extend the function F (·) to a continuous function F : R ! R which satises F(t + 1, t + 1) = F(t , t ) for all t , t 2 R. Clearly, this function is unbounded due to condition (2.2). 1 2 1 2 1 2 The following denition is also meaningful: Denition 2.3. Let ω 2 R\f0g, c 2 C\f0g and ω e + I  I (1 ≤ j ≤ n). A continuous function F : I ! X is j j j j said to be (ω , c ) -periodic if and only if F(t + ω e ) = c F(t), t 2 I, j 2 N . j j j2N j j j It is clear that, if F : I ! X is (ω , c ) -periodic, then F(t + mω e ) = c F(t), t 2 I, m 2 N, j 2 N , so j j j2N j j n j that the existence of a point t 2 I such that F(t ) ≠ 0 implies lim jjF(t + mω e )jj = +∞, provided that m!∞ 0 0 0 j j jc j > 1, and lim jjF(t + mω e )jj = 0, provided thatjc j < 1, for some j 2 N . m!∞ n j 0 j j j If c = 1 for all j 2 N , resp. c = −1 for all j 2 N , then we also say that the function F(·) is (ω ) - n n j j j j2N periodic, resp. (ω ) -anti-periodic. It is clear that, if F(·) is (ω , c ) –periodic, k 2 N and c = 1 for all j j2N j j j2N n n j j 2 N , resp. c = −1 for all j 2 N , then F(·) is (kω ) -periodic, resp. (kω ) -anti-periodic. n n j j2N j j2N j n n The classes of (ω, c)-periodic functions and (ω , c ) -periodic functions are closed under the operation j j j2N of the pointwise convergence of functions, as easily approved. In the scalar-valued case, the following holds: If the function F : I ! C \ f0g is (ω, c)-periodic, resp. (ω , c ) -periodic, then the function (1/F)(·) is j j j2N (ω, 1/c)-periodic, resp. (ω , 1/c ) -periodic. It is also clear that we have the following: j j j2N Proposition 2.4. (i) Let ω, a 2 R \ f0g, c 2 C \ f0g, α 2 C, ω + I  I and a + I  I. If the function ˇ ˇ F : I ! X is (ω, c)-periodic, then −ω − I  −I and the function F : −I ! X, dened by F(x) := F(−x), x 2 I, is (−ω, c)-periodic. Moreover,kF(·)k is (ω,jcj)-periodic, the function F : I ! X dened by F (t) := F(t +a), a a t 2 I is (ω, c)-periodic and the function αF(·) is (ω, c)-periodic. (ii) Let ω 2 R\f0g, c 2 C\f0g, α 2 C, ω e +I  I (1 ≤ j ≤ n) and a +I  I. If a continuous function F : I ! X j j j j is (ω , c ) -periodic, then−ω e −I  −I (1 ≤ j ≤ n) and the function F : −I ! X is (−ω , c ) -periodic. j j j2N j j j j j2N n n 140 Ë M. Kostić Moreover,kF(·)k is (ω ,jc j) -periodic, the function F : I ! X dened above is (ω , c ) -periodic and j j j2N j j j2N n n the function αF(·) is (ω , c ) -periodic. j j j2N Proposition 2.5. Let ω 2 R\f0g, c 2 C\f0g and ω e + I  I (1 ≤ j ≤ n). If a continuous function F : I ! X j j j j is (ω , c ) -periodic, then ω + I  I, where ω := ω e , and the function F(·) is (ω, c)-periodic with j j j2N j j n j=1 c =: c . j=1 The converse statement is not true in general case n > 1, as the following simple counterexample shows: Example 2.6. Consider the function F : R ! R from Example 2.2. Then there do not exist numbers ω , ω 2 1 2 R \ f0g and numbers c , c 2 C \ f0g such that the function F(·) is (ω , c ) -periodic. If we assume the 1 2 j j j2N contrary, then we would have F(t + ω , 0) = c F(t , 0) for all t 2 R. If jc j ≤ 1, then the contradiction is 1 1 1 1 1 1 p p obvious since (2.2) implies the unboundedness of function F(·, 0), because F(8k, 0) = F(4k 2,−4k 2) for all k 2 N. Ifjc j > 1, then the contradiction is obvious due to condition (2.3), which implies that the function F(·, 0) cannot tends to plus innity as the time variable tends to plus innity (see also [15, Remark 2.4]). Sometimes the converse in Proposition 2.5 is possible to be made: 2 ′ 2 Example 2.7. (cf. also [19, Example 2.8]) Suppose that I := f(x, y) 2 R : x + y ≥ 0g and I := f(x, y) 2 R : −x−y ′ −1 x + y = 1g. Set F(x, y) := 2 , (x, y) 2 I. Then, for every (a, b) 2 I , we have F((x, y) + (a, b)) = 2 F(x, y), −1 ′ (x, y) 2 I, so that F(·,·) is (ω, 2 )-periodic provided that ω 2 I . Similarly, we have that F(·;·) is (ω , c ) - j j j2N −ω −ω 1 2 periodic for every ω > 0 and ω > 0, with c = 2 and c = 2 . 1 2 1 2 Concerning the boundedness of (ω , c ) -periodic functions, we will state only one result: j j j2N Proposition 2.8. Suppose that ω 2 R \f0g, c 2 C \f0g, M > 0, ω e + I  I (1 ≤ j ≤ n), the set I is closed, j j j j the function F : I ! X is (ω , c ) -periodic,jc j ≤ 1 for all j 2 N and, for every t = (t , t ,· · ·, t ) 2 I, there n n j j j2N j 1 2 exist a point η = (η , η ,· · ·, η ) 2 I and integers k 2 N (1 ≤ j ≤ n) such that t = k ω + η (1 ≤ j ≤ n). Then 1 2 M j j j j j the function F(·) is bounded. Proof. Let a point t = (t , t ,···, t ) 2 I be xed, and let η 2 I and integers k 2 N (1 ≤ j ≤ n) satisfy the above 1 2 j P Q n n j requirements. Then we have t = η + k ω e so that F(t) = c F(η). Since I is closed, I is compact j j j M j=1 j=1 j and there exist a nite constant M > 0 such thatkF(x)k ≤ M for all x 2 I . ThenkF(t)k ≤ M sincejc j ≤ 1 1 1 M 1 j for all j 2 N . In the following extension of [2, Proposition 2], we prole the class of (ω, c)-periodic functions in the following way: Proposition 2.9. Let ω = (ω , ω ,· · ·, ω ) 2 R \ f0g, ω + I  I, c 2 C \ f0g and S := fi 2 N : ω ≠ 0g. n n 1 2 i jSj Denote by A the collection of all tuples a = (a , a ,· · ·, a ) 2 R such that a = 1. Then a continuous 1 2 jSj i i2S function F : I ! X is (ω, c)-periodic if and only if, for every (some) a 2 A, the function G : I ! X, dened by a t i i i2S ω G t , t ,· · ·, t := c F t , t ,· · ·, t , t = t , t ,· · ·, t 2 I, (2.4) a n n n 1 2 1 2 1 2 is (ω, 1)-periodic. Proof. Let a point t = (t , t ,· · ·, t ) 2 I be xed. Then it is clear that G (t + ω) = G (t) if and only if 1 2 n a a P P a (t +ω ) a t i i i  i i − − i2S i2S ω ω i i c F t + ω , t + ω ,· · ·, t + ω = c F t , t ,· · ·, t n n n 1 1 2 2 1 2 if and only if F(t + ω) = cF(t). We illustrate Proposition 2.9 with the following example: Multi-dimensional (ω, c)-almost periodic type functions. .. Ë 141 n n Example 2.10. (see also [14, pp. 22-23]) Suppose that ω = (ω , ω ,· · ·, ω ) 2 R \f0g, k 2 R \f0g, a 2 A, 1 2 −2 (b ) is any sequence of complex numbers such thatjb j = O(l ), hk, ωi = 2π/3 and l l a t i i ilht,ki n i2S ω F t , t ,· · ·, t = c b e , t = t , t ,· · ·, t 2 R . 1 2 n 1 2 n l21+3N Then F(·) is (3ω, c)-almost periodic. Similarly, we can prove the following: Proposition 2.11. Let ω 2 R \ f0g, c 2 C \ f0g, ω e + I  I (1 ≤ j ≤ n) and the function F : I ! X is j j j j continuous. For each j 2 N , we dene the function G : I ! X by G t , t ,· · ·, t := c F t , t ,· · ·, t , t = t , t ,· · ·, t 2 I. (2.5) j 1 2 n 1 2 n 1 2 n Then F(·) is (ω , c ) -periodic if and only if, for every t = (t , t ,· · ·, t ) 2 I and j 2 N , we have n n j j j2N 1 2 G t , t ,· · ·, t + ω ,· · ·, t = G t , t ,· · ·, t ,· · ·, t . j 1 2 j j n j 1 2 j n Therefore, we have the following: Example 2.12. Let c 2 C \f0g for all j 2 N . Then the function j n Y j 2π F(t ,· · ·, t ) := c sin t , t = (t ,· · ·, t ) 2 R 1 n j 1 n j=1 is (2π, c ) -periodic. j j2N If ω 2 R \ f0g, c 2 C \ f0g for i = 1, 2, ω + I  I, the function G : I ! C is (ω, c )-periodic and the i 1 function H : I ! X is (ω, c )-periodic, then the function F(·) := G(·)H(·) is (ω, c c )-periodic. For the class 2 1 2 of (ω , c ) -periodic functions, we can clarify the following result (cf. also [16, Proposition 2.2]): j j j2N Proposition 2.13. Let ω 2 R \ f0g, c 2 C \ f0g and ω e + I  I (1 ≤ j ≤ n, 1 ≤ i ≤ 2). Suppose that j j,i j j the function G : I ! C is (ω , c ) -periodic and the function H : I ! X is (ω , c ) -periodic. Set j j,1 j2N j j,2 j2N n n c := c c , 1 ≤ j ≤ n. Then the function F(·) := G(·)H(·) is (ω , c ) -periodic. j j,1 j,2 j j j2N Concerning the convolution invariance of spaces introduced in Denition 2.1 and Denition 2.3, we will state and prove the following result: n jSj Proposition 2.14. Suppose that ω 2 R \f0g, c 2 C\f0g, S = fi 2 N : ω ≠ 0g, a = (a , a ,···, a ) 2 R n i 1 2 jSj and a = 1, resp. ω 2 R \f0g and c 2 C \f0g (1 ≤ j ≤ n). Suppose, further, that F : R ! X is (ω, c)- i j j i2S periodic and the function G (·), dened through (2.4) is bounded, resp. F : R ! X is (ω , c ) -periodic and a j j j2N − a t /ω i i i i2S for each j 2 N the function G (·), dened through (2.5) is bounded. If the function c h(t ,· · ·, t ) n j 1 n 1 n −t /ω 1 n j j belongs to the space L (R ), resp. for each j 2 N the function c h(t ,···, t ) belongs to the space L (R ), n 1 n then the function (h F)(t) := h(y)F(t − y) dy, t 2 R is (ω, c)-periodic, resp. (ω , c ) -periodic. j j j2N Proof. We will consider only (ω, c)-periodicity. By Proposition 2.9, it suces to show that P P a (t +ω ) a t i i i i i − − i2S ω i2S ω i i c (h F)(t + ω) = c (h F)(t) (2.6) * * 142 Ë M. Kostić for every xed point t = (t ,· · ·, t ) 2 R . Note rst that the value (h F)(t) is well dened, since we have n * − a t /ω 1 n i i i i2S assumed that the function c h(t ,···, t ) belongs to the space L (R ), as well as that the function G (·), dened through (2.4), is bounded and − a t /ω i i i i2S c (h F)(t) " # " # P P − a y /ω − a (t −y )/ω i i i i i i i i2S i2S = c h(y ,· · ·, y ) · c F(t − y) dy. (2.7) 1 n Keeping in mind (2.7) and the dominated convergence theorem, we get that the function (h F)(·) is continu- ous. Similarly, by plugging t + ω in place of t in (2.7), we get that (2.6) holds because a = 1. i2S Concerning asymptotically (ω, c)-periodic type functions, we will use the following denition, only: n n Denition 2.15. Suppose that D  I  R , the set D is unbounded, ω 2 R \ f0g, c 2 C \ f0g, ω + I  I, ω 2 R \f0g, c 2 C \f0g, ω e + I  I (1 ≤ j ≤ n, 1 ≤ i ≤ 2) and F : I ! X. Then we say that the function F(·) j j j j is D-asymptotically (ω, c)-periodic, resp. D-asymptotically (ω , c ) -periodic, if and only if there exists a j j j2N (ω, c)-periodic, resp. (ω , c ) -periodic, function F : I ! X and a function Q 2 C (I : X) such that j j j2N 0 0,D,B F(t) = F (t) + Q(t), t 2 I. 2.1 (ω , c ; r , I ) -Almost periodic type functions j j j j2N Following our idea from [16, Denition 2.1], we can introduce and analyze several various generalizations of the class of multi-dimensional (ω , c ) -periodic functions with the help of Proposition 2.11. For example, j j j2N suppose that ω 2 R\f0g, c 2 C\f0g and ω e +I  I (1 ≤ j ≤ n); if a function F : I ! X is (ω , c ) -periodic, j j j j j j j2N then for each j 2 N and for every k 2 N we have F(t + kω e ) = c F(t), t 2 I, j 2 N and G (t + kω e ) = G (t), n n j j j j j j t 2 I, j 2 N . Set W := fj 2 N : ω > 0g and W := fj 2 N : ω < 0g, as well as I := fx ≥ 0 : t + xe 2 Ig n + n − n j j j,t j if j 2 W , resp. I := fx ≥ 0 : t − xe 2 Ig if j 2 W (t 2 I), and G (x) := G (t + xe ), x 2 I if j 2 W , + − + j,t j j,t j j j,t resp. G (x) := G (t − xe ), x 2 I if j 2 W (t 2 I). Then we can generalize the class of (ω , c ) -periodic j,t j j j,t j j j2N functions as follows: Denition 2.16. Suppose that ω 2 R\f0g, c 2 C\f0g, ω e +I  I (1 ≤ j ≤ n) and F : I ! X is a continuous j j j j ′ ′ function. Let r 2 C \ f0g for 1 ≤ j ≤ n, and let ; ≠ I  I  R, I + I  I for 1 ≤ j ≤ n, t 2 I. Set j j,t j,t j,t j,t j,t ′ ′ I := fI : t 2 Ig. Then we say that the function F(·) is: j j,t (i) (ω , c ; r , I ) -almost periodic if and only if, for every j 2 N and t 2 I, the function G (·) dened j j j j2N j,t j n above is (I , r )-almost periodic; j,t (ii) (ω , c ; r , I ) -uniformly recurrent if and only if, for every j 2 N and t 2 I, the function G (·) is j j j j2N j,t j n (I , r )-uniformly recurrent. j,t Suppose that I = R . Then we say that the function F(·) is: (iii) (ω , c ) -almost automorphic if and only if, for every j 2 N , for every t 2 R and for every real j j j2N sequence (b ), there exist a subsequence (a ) of (b ) and a function F : R ! X such that k k k j,t * * lim G t + x + a e = F (x) and lim F x − a = G t + xe , (2.8) j k j j,t j,t k j j k!+∞ k!+∞ pointwise for x 2 R; if, moreover, the convergence in (2.8) is uniform in the variable x on compact subsets of R, then we say that the function F(·) is compactly (ω , c ) -almost automorphic. j j j2N Remark 2.17. (i) It is clear that I = R is equivalent to saying that I + ηe  I for all η 2 R \f0g and j 2 N . (ii) It is clear that (i) implies (ii) and that the almost periodicity of the function G (x) := G (t + xe ), x 2 R j,t j j for all j 2 N and t 2 I implies (iii), which is equivalent to saying that the function G (·) dened above j,t is almost automorphic for all j 2 N and t 2 I. n Multi-dimensional (ω, c)-almost periodic type functions. .. Ë 143 n ′ Now we will provide an illustrative example in which we have I = R , ω = c = 1 and I = I = [0,∞) for j j j,t j,t all j 2 N and t 2 R : Example 2.18. (i) Suppose that r = 1 for all j 2 N . Then the function F t ,· · ·, t := sin t + sin 2t , t = (t ,· · ·, t ) 2 R n n 1 j j 1 j=1 is (ω , c ; r , I ) -almost periodic but not (ω , c ) -periodic. j j j j2N j j j2N j n n (ii) ([12], [15], [19]) Suppose that r = −1 for all j 2 N . Then the function j n " # n ∞ Y X 2 j n F t ,· · ·, t := sin t · sin , t = (t ,· · ·, t ) 2 R 1 n j 1 n n 3 j=1 n=1 ′ ′ is (ω , c ; r , I ) -uniformly recurrent but not (ω , c ; r , I ) -almost periodic. j j j j j2N j j j j j2N n n (iii) Suppose that r = 1 for all j 2 N . Then the function F t ,· · ·, t := sin p  , t = (t ,· · ·, t ) 2 R 1 n 1 n 2 + sin t + sin 2t j j j=1 is (ω , c ) -almost automorphic but not (ω , c ; r , I ) -almost periodic. j j j2N j j j j2N n j n The function spaces introduced in Denition 2.16 are translation invariant and closed under the pointwise multiplications with complex scalars. Furthermore, if the function F(·) is (ω , c ; r , I ) -almost periodic, j j j j2N j n then it can be easily proved that the functionkF(·)k is (ω ,jc j;jr j, I ) -almost periodic. Suppose now that j j j j2N j n the function F : I ! C \f0g is (ω , c ; r , I ) -almost periodic, jF(t)j ≥ m > 0 for all t 2 I, and jc j = 1 for j j j j2N j j n all j 2 N . Then the function (1/F)(·) is (ω , 1/c ; r , I ) -almost periodic, which can be simply proved as j j j j2N j n follows (cf. also [16, Proposition 2.5]). Let j 2 N , t 2 I and ϵ > 0 be xed; without loss of generality, we may t /ω ′ j j assume that j 2 W . Let τ 2 I andjG (x + τ) − r G (x)j < ϵ, x ≥ 0. After multiplication with c , we get j,t j j,t j,t x+τ c F t , t ,· · ·, t + x,· · ·, t − r F t , t ,· · ·, t + (x + τ),· · ·, t ≤ ϵ, x ≥ 0. (2.9) 1 2 n 1 2 n j j j Hence, for every x ≥ 0, we have: t +x+τ t +x j j ω ω j j c c j −1 j − r F t , t ,· · ·, t + x + τ,· · ·, t F t , t ,· · ·, t + x,· · ·, t 1 2 j n 1 2 j n j −1 = c F t , t ,· · ·, t + x,· · ·, t − r F t , t ,· · ·, t + x + τ,· · ·, t n n 1 2 j j 1 2 j −2 −1 · ≤ m jr j ϵ, jF(t , t ,· · ·, t + x + τ,· · ·, t )j ·jF(t , t ,· · ·, t + x,· · ·, t )j 1 2 j n 1 2 j n where we have employed (2.9) in the last estimate. This simply implies the required. By a careful examination of the notion introduced in Denition 2.16 and the paragraph preceding it, we may deduce that the (ω , c ; r , I ) -almost periodicity of the function F : I ! X implies the j j j j2N j n (−ω , c ; r , I ) -almost periodicity of the function F(·). We leave all details concerning the proof of this j j j j2N j n fact to the interested readers. Using the statements (i)-(v) from Subsection 1.1 and corresponding denitions, we may deduce the fol- lowing proposition: Proposition 2.19. (i) Suppose that, for every j 2 N and t 2 I, we have I + I = I and the function j,t j,t j,t F : I ! X is (ω , c ; r , I ) -uniformly recurrent. Then, for every j 2 N , we have r = ±1; if, additionally, j j j j2N j j n F(t) ≥ 0 for all t 2 I, then, for every j 2 N , we have r = 1. j 144 Ë M. Kostić ′ ′ (iii) Suppose that l 2 N, and F : I ! X is (ω , c ; r , I ) -almost periodic ((ω , c ; r , I ) -uniformly recur- j j j j2N j j j j2N j n j n ′ ′ l ′ rent). Then, for every j 2 N and t 2 I, we have lI  I , I + lI  I and F(·) is (ω , c ; r , lI ) - j,t j,t j,t j j j2N j,t j,t j j n l ′ ′ ′ almost periodic ((ω , c ; r , lI ) -uniformly recurrent), where lI := flI : t 2 Ig for all j 2 N . j j j2N n j j n j j,t ′ ′ (iv) Suppose that (1.1) holds and F : I ! X is (ω , c ; r , I ) -almost periodic ((ω , c ; r , I ) -uniformly j j j j2N j j j j2N j n j n recurrent). Then the following holds: (a) If p is even, then F(·) is (ω , c ; 1, I ) -almost periodic j j j2N j n ′ ′ ′ ((ω , c ; 1, qI ) -uniformly recurrent), where qI := fqI : t 2 Ig for all j 2 N . j j j2N n j n j j,t (b) If p is odd, then F(·) is (ω , c ;−1, I ) -almost periodic j j j2N j n ((ω , c ;−1, qI ) -uniformly recurrent). j j j2N j n ′ ′ (v) Let jcj = 1 and arg(c)/π 2̸ Q. If, for every j 2 N and t 2 I, lI = I for all l 2 N and F : I ! X is a j,t j,t ′ ′ bounded, (ω , c ; r , I ) -almost periodic ((ω , c ; r , I ) -uniformly recurrent) function, then the func- j j j j2N j j j j2N j n j n ′ ′ ′ ′ ′ ′ tion F(·) is (ω , c ; r , I ) -almost periodic ((ω , c ; r , I ) -uniformly recurrent) for all (r ,· · ·, r ) 2 j j j2N j j j2N n j j n j j n 1 (S ) . Concerning the convolution invariance of spaces introduced in Denition 2.16, the following important fact should be said: we have introduced the notion of (I , r )-almost periodicity, for example, by requiring that, j,t for every j 2 N and t 2 I, the function G (·) is (I , r )-almost periodic. Unfortunately, sometimes we need n j,t j j,t to assume that, for every j 2 N , the function G (·) is (I , r )-almost periodic uniformly in the variable t 2 I, n j,t j j,t in a certain sense. For simplicity, let us assume that I = R , which immediately implies that, for every j 2 N and t 2 R , we have I = [0,∞). Assume, further, that for each j 2 N there exists a set A  [0,∞) such j,t n j ′ n that A = I for every t 2 R , as well as that for each ϵ > 0 there exists l > 0 such that for each x 2 A we j 0 j j,t have the existence of a number x 2 B(x , l)\ A such that 0 j G (x + τ) − r G (x) ≤ ϵ, x ≥ 0, t 2 R , j,t j j,t i.e., t +x+τ t +x j j − − ω ω j j n c F t + (x + τ)e − r c F t + xe ≤ ϵ, x ≥ 0, t 2 R . (2.10) j j j j j 1 n n If h 2 L (R ) and F(·) is a bounded, continuous function, then the function (h F)(·) is well dened on R , bounded and continuous. If we assume, in addition to the all above, that jc j = 1 for all j 2 N , then the estimate (2.10) will be invariant under the action of convolution h ·, since t +x+τ t +x j j − − ω ω j j c (h * F) t + (x + τ)e − r c (h * F) t + xe j j j j j t +x+τ t +x j j − − ω ω j j ≤ jh(y)j c F t + (x + τ)e − y − r c F t + xe − y dy j j j j j t −y +x+τ t −y +x j j j j − − ω ω j j = jh(y)j c F t + (x + τ)e − y − r c F t + xe − y dy j j j j j ≤ ϵ, x ≥ 0, t 2 R , where we have employed (2.10) in the last estimate. We close this section with the observation that a similar result can be established for the (ω , c ; r , I ) -uniform recurrence and the (ω , c ) -almost automor- j j j j2N j j j2N j n n phy. Multi-dimensional (ω, c)-almost periodic type functions. .. Ë 145 3 Further generalizations of (ω, c)-periodicity and (ω , c ) -periodicity j j j2N ′ n ′ n Unless stated otherwise, in this section we will assume that ; ≠ I  I  R , I + I  I, ω 2 R \ f0g and jSj c 2 C \f0g. Dene S := fi 2 N : ω ≠ 0g and A to be the collection of all tuples a = (a , a ,· · ·, a ) 2 R i 1 2 jSj such that a = 1. Let a 2 A. i2S Following our analyses from [16, Section 3], we introduce the next notion: Denition 3.1. We say that a continuous function F : I ! X is: ′ ′ (i) (I , a, ω, c)-uniformly recurrent of type 1, resp. (I , a, ω, c)-uniformly recurrent of type 2, if and only if there exists a sequence (α = (α ,· · ·, α )) in I such that lim jα j = +∞ and k k,1 k,n k!+∞ k P a α i k,i i2S ω lim sup F t + α − c F(t) = 0, (3.1) k!+∞ t2I resp. a α i k,i i2S ω lim sup c F t + α − F(t) = 0; (3.2) k!+∞ t2I ′ ′ (ii) (I , a, ω, c)-almost periodic of type 1, resp. (I , a, ω, c)-almost periodic of type 2, if and only if for each ′ ′ ϵ > 0 and t 2 I there exist a nite number l > 0 and a point τ 2 B(t , l)\ I such that 0 0 a τ i i i2S ω sup F t + τ − c F(t) < ϵ, t2I resp. a τ i i i2S ω sup c F t + τ − F(t) < ϵ. (3.3) t2I ′ ′ If jcj = 1, then the concept (I , a, ω, c)-uniform recurrence of type 1 and the concept (I , a, ω, c)-uniform P a α i k,i i2S recurrence of type 2 coincide, as easily approved by multiplying (3.1) with c ; this also holds for ′ ′ the concepts (I , a, ω, c)-almost periodicity of type 1 and (I , a, ω, c)-almost periodicity of type 2, but then a t i i i2S ω we can say a little bit more. Speaking-matter-of-factly, we can multiply the both sides of (3.3) with c in order to see that F(·) is (I , a, ω, c)-almost periodic of type 2 (1) if and only if the function G (·), dened ′ ′ n through (2.4), is I -almost periodic; in the usually considered case I = I = R , this is equivalent to saying that the function F(·) is almost periodic. The function spaces introduced in Denition 2.16 are translation invariant and closed under the pointwise multiplications with complex scalars; if I = R and F(·) is a bounded, continuous function which belongs to 1 n any of the above introduced function spaces, then for each h 2 L (R ) the function (h * F)(·) is also bounded and belongs to the same space. Furthermore, if the function F(·) is (I , a, ω, c)-uniformly recurrent of type 1, ′ ′ ′ resp. (I , a, ω, c)-uniformly recurrent of type 2 [(I , a, ω, c)-almost periodic of type 1, resp. (I , a, ω, c)-almost ′ ′ periodic of type 2], then F(·) is (I , a, ω,jcj)-uniformly recurrent of type 1, resp. (I , a, ω,jcj)-uniformly recur- ′ ′ rent of type 2 [(I , a, ω,jcj)-almost periodic of type 1, resp. (I , a, ω,jcj)-almost periodic of type 2]. Concern- ing the invariance of function spaces under the operation of uniform convergence, we will only state that the ′ n ′ assumptions a ω > 0 for all j 2 S, jcj ≤ 1, I  [0,∞) and the sequence (F ) of (I , a, ω, c)-uniformly recur- j j k rent functions of type 1 [(I , a, ω, c)-almost periodic functions of type 1] uniformly converges to a function ′ ′ F : I ! X imply that the function F(·) is likewise (I , a, ω, c)-uniformly recurrent of type 1 [(I , a, ω, c)-almost periodic of type 1]. For the sequel, we need the following extension of [16, Lemma 3.4]: 146 Ë M. Kostić ′ n ′ Lemma 3.2. Suppose that ; ≠ I  I  R , I = −I, I + I = I and the function F : I ! X is continuous. ′ ′ Then F(·) is (I , a, ω, c)-uniformly recurrent of type 1 [(I , a, ω, c)-almost periodic of type 1] if and only if F(·) is ′ ′ (I , a, ω, 1/c)-uniformly recurrent of type 2 [(I , a, ω, 1/c)-almost periodic of type 2]. Proof. We will present the main details of the proof provided that F(·) is (I , a, ω, c)-uniformly recurrent of type 1. Then there exists a sequence (α = (α ,···, α )) in I such that lim jα j = +∞ and (3.1) holds. k k,1 k,n k!+∞ k Since we have assumed I = −I and I + I = I, the proof simply follows from the next computation (k 2 N): P a α i k,i i2S ω ˇ ˇ sup (1/c) F t + α − F(t) t2I P a α i k,i i2S ω = sup (1/c) F −t − α − F(−t) t2I P a α i k,i i2S ω = sup c F t − F(t + α ) t2−(I+I ) P a α i k,i i2S ω = sup c F t − F(t + α ) t2−I P a α i k,i i2S ω = sup c F t − F(t + α ) . t2I Concerning [16, Theorem 3.2(i)] and its proof, we will rst state and prove the following result: ′ n ′ ′ n Theorem 3.3. Suppose that; ≠ I  I  R , I is unbounded, F : I ! X is continuous, I + I = I, ω 2 R \f0g, jcj > 1, S = N and any component of a tuple a 2 A is positive. Suppose further that, for every t 2 I and j 2 N , n n we have ω t ≥ 0. Then the following assertions are equivalent: j j (i) The function F(·) is (I , a, ω, c)-uniformly recurrent of type 1. (ii) The function F(·) is (I , a, ω, c)-uniformly recurrent of type 2. (iii) There exists a sequence (α = (α ,···, α )) in I such that lim jα j = +∞ and the function G : I ! k k,1 k,n k!+∞ k X, dened through (2.4), satises G (t + α ) = G (t) for all t 2 I and k 2 N. a a (iv) There exists a sequence (α = (α ,· · ·, α )) in I such that lim jα j = +∞ and k k,1 k,n k!+∞ k a α i k,i i2S ω F t + α = c F(t), t 2 I, k 2 N. (3.4) (v) There exists a point ω 2 I \f0g such that a ω i i i2S ω F t + ω = c F(t), t 2 I. (3.5) Proof. If F(·) is (I , a, ω, c)-uniformly recurrent of type 1, then our assumptions a > 0 and α /ω > 0 (k 2 N, j k,j i P a α P a α n j k,j n j k,j − − j=1 ω j=1 ω j j j 2 N ) imply thatjc j ≤ 1, so that (3.1) implies (3.2) after multiplication with c ; hence, (i) ′ ′ implies (ii). Suppose now that F(·) is (I , a, ω, c)-uniformly recurrent of type 2 and the sequence (α ) in I satises (3.2). Let k 2 N be xed. Then (3.2) implies the existence of a nite real number M ≥ 1 such that − a t /ω j j j j=1 sup G t + α − G (t) ≤ Mjcj . a a t2I Since we have assumed that a > 0 and t /ω > 0 for all j 2 N , the above estimate yields j j j −1 − minfa :j2N g maxfω :j2N g jt j+···+jt j j n j n 1 n G t + α − G (t) ≤ Mjcj , a a for all t 2 I, which implies that lim kG (t + α ) − G (t)k = 0. On the other hand, (3.2) implies a a jtj!∞ k lim G t + α + α − G t + α = 0. a m a m m!+∞ Multi-dimensional (ω, c)-almost periodic type functions. .. Ë 147 Therefore, the function t 7! G (t + α ) − G (t), t 2 I is I -uniformly recurrent and tends to zero asjtj ! +∞. a a Since we have assumed that I + I = I, we may apply [19, Corollary 2.11] in order to see that G (t + α ) = G (t) a a for all t 2 I, which implies (iii). The implications (iii)) (iv), (iv)) (i) and (iv)) (v) are trivial. To complete the proof, it suces to show that (v) implies (iv). This follows by plugging α := kω for all k 2 N since (3.5) implies inductively a kω i i i2S ω F t + kω = c F(t), t 2 I, k 2 N. ′ ′ Remark 3.4. (i) Since I is unbounded, it is clear that the (I , a, ω, c)-almost periodicity of type 1 implies the ′ ′ (I , a, ω, c)-uniform recurrence of type 1 for F(·) as well as that the (I , a, ω, c)-almost periodicity of type 1 ′ ′ implies the (I , a, ω, c)-almost periodicity of type 2 for F(·), which further implies the (I , a, ω, c)-uniform recurrence of type 2 for F(·). (ii) Let (α ) be a sequence from (iv). Then it is clear that (iv) implies that for each number k 2 N the function ′ ′ F(·) is (I , a, ω, c)-almost periodic of type 1, where I := fmα : m 2 Ng. Keeping Theorem 3.3 and this k k observation in mind, we have extended so the rst part of [16, Theorem 3.2(i)], where we have assumed that I = [0,∞). Concerning the statement of [16, Theorem 3.2(i)] with the interval I = R, we would like to note that it can be ′ n ′ extended to the higher dimensions as follows. Suppose that I = I [ I , where; ≠ I  I  R , I + I = I 0 1 0 0 0 0 0 and the function F : I ! X is (I , a, ω, c)-uniformly recurrent of type 2, wherejcj > 1, S = N , any component of a tuple a 2 A is positive and, for every t 2 I and j 2 N , we have ω t ≥ 0. Then the restriction of function 0 j j F(·) to the interval I is (I , a, ω, c)-uniformly recurrent of type 2, as well, so that we can apply Theorem 3.3 in order to conclude that (3.4) holds for every t 2 I and k 2 N. To show the validity of this condition for all t 2 I and k 2 N, we may assume additionally that: (a) For every t 2 I , there exists m 2 N such that, for every m 2 N with m ≥ m , we have t + α 2 I . 1 0 0 0 Applying (3.4) twice, with t + α and t the rst time, and with t + α + α and t + α the second time, we easily m m k k get that (3.4) holds for every t 2 I. Therefore, we have proved the following: ′ n ′ ′ Theorem 3.5. Suppose that ; ≠ I  I  R , I is unbounded, I + I = I , I = I [ I , condition (a) holds 0 0 0 0 1 0 0 0 and F : I ! X is continuous. Suppose that ω 2 R \f0g, jcj > 1, S = N and any component of a tuple a 2 A is positive. Suppose further that, for every t 2 I and j 2 N , we have ω t ≥ 0. Then the following assertions are 0 j j equivalent: (i) The function F(·) is (I , a, ω, c)-uniformly recurrent of type 1. (ii) The function F(·) is (I , a, ω, c)-uniformly recurrent of type 2. (iii) There exists a sequence (α = (α ,···, α )) in I such that lim jα j = +∞ and the function G : I ! k k,1 k,n k!+∞ k X, dened through (2.4), satises G (t + α ) = G (t) for all t 2 I and k 2 N. a a (iv) There exists a sequence (α = (α ,· · ·, α )) in I such that lim jα j = +∞ and (3.4) holds. k k,1 k,n k!+∞ k (v) There exists a point ω 2 I \f0g such that (3.5) holds. Suppose now that jcj < 1, S := N , any component of a tuple a 2 A is positive and, for every t 2 I and j 2 N , we have ω t ≥ 0. Applying Lemma 3.2, we can simply extend the statement of [16, Theorem 3.2(ii)] to j j the higher dimensions, provided that condition (a) holds with I = −I . Details can be left to the interested 1 0 readers. ′ n In the case that a ω > 0 for all j 2 S = N , jcj < 1, I = I = [0,∞) , then it can be simply proved (cf. [16, j j Proposition 3.6, Corollary 3.8]) that the function F : I ! X is (I , a, ω, c)-almost periodic of type 1 if and only if there exists a nite constant M ≥ 1 such that a t /ω i i i i2S kF(t)k ≤ Mjcj , t 2 I; the statement of [16, Proposition 3.11] can be also extended to the higher dimensions provided that the func- tion F(·) is bounded, a ω > 0 for all j 2 S = N andjcj < 1. Without any essential changes of the proof of [16, j j Proposition 3.12], we may deduce the following: 148 Ë M. Kostić ′ n Proposition 3.6. Suppose that a ω > 0 for all j 2 S = N ,jcj < 1 and I = I = [0,∞) . Then a continuous func- j j ′ −a t /ω i i i i2S tion F : I ! X is (I , a, ω, c)-almost periodic of type 2 if and only if the function t 7! G(t)  c F(t), t 2 I is bounded, continuous and satises that for each ϵ > 0, t 2 I and N > 0 there exist a nite number l > 0 and a point τ 2 B(t , l)\ I such that G(t + τ) − G(t) ≤ ϵ, t 2 I . In connection with Proposition 3.6, we want to note that the notion of a complex-valued Levitan N-almost periodic function was introduced by B. M. Levitan in 1937 (see [21]-[22] and references cited therein) and later studied on topological groups by B. Ya. Levin [20] in 1949. Let us recall that a continuous function f : [0,∞) ! X is said to be Levitan N-almost periodic if and only if for each ϵ > 0 and N > 0 the set of all positive reals τ > 0 such thatkf (t + τ)− f (t)k ≤ ϵ, t 2 [0, N ] is relatively dense in [0,∞). The study of vector-valued Levitan N-almost periodic functions on topological (semi-)groups and multi-dimensional vector-valued Levitan N- almost periodic functions will be carried out somewhere else. In our previous research studies of the multi-dimensional almost periodicity, we have also analyzed the invariance of almost periodicity under the actions of the nite convolution products and the innite convo- lution products. In the one-dimensional case, this theme is crucially important for giving the most intriguing applications in the qualitative analysis of almost periodic type solutions for various classes of the abstract Volterra integro-dierential equations. In the multi-dimensional case, the results obtained so far are not so easily applicable and, because of that, we will skip all related details with regards to this question. 4 Applications In this section, we will present several illustrative examples and applications of our results to the abstract Volterra integro-dierential equations in Banach spaces. We feel it is our duty to say that the the points [1,3,4,5] have been also considered in [8] for multi-dimensional almost periodic functions. 1. We start with the observation that all established results on the convolution invariance of introduced function spaces can be applied to the Gaussian semigroup jyj −(n/2) − n 4t (G(t)F)(x) := 4πt F(x − y)e dy, t > 0, f 2 Y , x 2 R ; see [5, Example 3.7.6] for more details. Suppose, for example, that I = R ,jcj = 1 and F(·) is a bounded, (ω, c)- periodic function. Then, due to Proposition 2.14, we have that for each t > 0 the functionR 3 x 7! u(x, t ) 0 0 (G(t )F)(x) 2 C is likewise bounded and (ω, c)-periodic. A similar result can be given for the Poisson semi- group; see [5, Example 3.7.9] and [8] for more details. Concerning the strongly continuous semigroups, we would like to note that our recent consideration from [19, Example 1.1] can be used to justify the intoduction of analyzed function spaces, as well. 2. In [19, Example 1.2], we have recently observed an interesting feature of the famous d’Alembert formula, which has been used by S. Zaidman [25, Example 5] for almost periodic functions of one real variable, a long time ago (see also [14, Example 2.3]). Suppose that a > 0. Then it is well known that the regular solution of the wave equation u = a u in domain f(x, t) : x 2 R, t > 0g, equipped with the initial conditions tt xx 2 1 u(x, 0) = f (x) 2 C (R) and u (x, 0) = g(x) 2 C (R), is given by the d’Alembert formula x+at 1 1 u(x, t) = f (x − at) + f (x + at) + g(s) ds, x 2 R, t > 0. 2 2a x−at [1] In the above-mentioned example, if the function x 7! (f (x), g (x)), x 2 R is c-almost periodic, where [1] g (·)  g(s) ds; then the conclusion is: the solution u(x, t) can be extended to the whole real line in the time variable and this solution is c-almost periodic in (x, t) 2 R . Multi-dimensional (ω, c)-almost periodic type functions. .. Ë 149 2.1. We assume here that there exist numbers ω 2 R \ f0g and c 2 C \ f0g such that the function x 7! [1] (f (x), g (x)), x 2 R is (ω, c)-periodic. Then it is clear that the solution u(x, t) can be extended to the whole real line in the time variable and now we will prove that u x + ω, t = cu(x, t), x, t 2 R, i.e., the function u(·;·) is ((ω, 0), c)-periodic. But, the last equality simply follows from the next calculation: h i u x + ω, t = f x − at + ω + f x + at + ω h i [1] [1] + g x + at + ω − g x − at + ω 2a h i = cf x − at + cf x + at h i [1] [1] + g x + at − g x − at = cu(x, t), x, t 2 R. 2a k−1 2.2. We assume here that there exist numbers ω 2 R \f0g, k 2 N and c 2 C \f0g such that c = 1 and [1] the function x 7! (f (x), g (x)), x 2 R is (ω, c)-periodic. Set 1 + k k − 1 ω := ω and ω := ω. 1 2 2 2a k k Then (ω , ω ) ≠ (0, 0), ω − aω = ω, ω + aω = kω, c = c, f (x + ω) = cf (x) = c f (x) = f (x + kω), 1 2 1 2 1 2 [1] [1] k [1] [1] g (x + ω) = cg (x) = c g (x) = g (x + kω) for all x 2 R, and we can simply show as above that u x + ω , t + ω = cu(x, t), x, t 2 R, 1 2 i.e., the function u(·;·) is ((ω , ω ), c)-periodic. 1 2 [1] 2.3. Let the assumptions of the previous point hold. Assume, further, that the function x 7! (f (x), g (x)), [1] 2 2 x 2 R satises lim f (x) = lim g (x) = 0. Set B := f(x, t) 2 R : x = ±atg. If D is any subset of R x!±∞ 0 x!±∞ satisfying that lim dist((x, t); B) = +∞, j(x,t)j!+∞,(x,t)2D then the solution given by the d’Alembert formula, with the functions f (·) and g(·) replaced therein with the functions (f + f )(·) and (g + g )(·), is D-asymptotically ((ω , ω ), c)-periodic. 0 0 1 2 n n 3. Let ω 2 R \f0g andjcj = 1. Equipped with the sup-norm, the space B (R : X) consisting of all X- ω,c valued, bounded and (ω, c)-periodic functions becomes a Banach space. In a series of our previous research studies, we have analyzed the following Hammerstein integral equation of convolution type on R : y(t) = g(t) + k(t − s)G(s, y(s)) ds, t 2 R . (4.1) n 1 n n Suppose now that g : R ! X is bounded and (ω, c)-periodic, k 2 L (R ), G : R × X ! X is continuous and satises that for each bounded subset of X we have that the set fG(t, x) : t 2 R , x 2 Bg is bounded as well as that G(t + ω, x) = cG(t, x) for all t 2 R and x 2 X. If there exists a nite real constant L ≥ 1 such that G(t, x) − G t, y ≤ L x − y , t 2 R ; x, y 2 X and L jk(y)j dy < 1, then we can apply the Banach contraction principle and Proposition 2.14 in order to see that there exists a unique solution of the integral equation (4.1) which belongs to the space B (R : X). ω,c 4. Of concern is the system of abstract partial dierential equations u (s, t) = Au(s, t) + f (s, t), u (s, t) = Bu(s, t) + f (s, t); u(0, 0) = x, s ≥ 0, t ≥ 0. (4.2) s 1 t 2 Motivated by the recent results of S. M. A. Alsulami given in [1, Section 2.1], we have recently considered, in [8], the case in which A and B are two complex matrices of format n × n, AB = BA, and A, resp. B, generate an exponentially decaying, strongly continuous semigroup (T (s)) , resp. (T (t)) . The follow- 1 s≥0 2 t≥0 ing assumptions have been made there: the functions f (s, t) and f (s, t) are continuously dierentiable, 1 2 the compatibility condition (f ) − Af = (f ) − Bf holds (s, t ≥ 0), D := f(s, t) 2 [0,∞) : c s ≤ t ≤ 2 2 1 t 1 1 c s for some positive real numbers c and c g, and 2 1 2 150 Ë M. Kostić (i) There is a nite real constant M > 0 such thatjf (v, 0)j +jf (0, ω)j ≤ M, for v, ω ≥ 0; 1 2 2 n (ii) The mappings g : R ! C are continuous, bounded (i = 1, 2) and satisfy that, for every ϵ > 0, there exists l > 0 such that any subinterval I of R of length l > 0 contains a number τ 2 I such that, for every s, t ≥ 0, we havejg (s + τ, t) − g (s, t)j ≤ ϵ andjg (s, t + τ) − g (s, t)j ≤ ϵ; 1 1 2 2 2 n 2 n (iii) The function q : [0,∞) ! C is bounded, q 2 C ([0,∞) : C ) and f (s, t) = g (s, t) + q (s, t) for i i 0,D i i i (s, t) 2 [0,∞) and i = 1, 2. The conclusion is: there is a unique classical solution u(s, t) of (4.2) (cf. also [1, Denition 2.13]), and more- 2 2 n over, there exist a continuous function u (s, t) on [0,∞) and a function u 2 C ([0,∞) : C ) such that ap 0 0,D u(s, t) = u (s, t) + u (s, t) for all (s, t) 2 [0,∞) , as well as for every ϵ > 0, there exists l > 0 such that ap any subinterval I of [0,∞) of length l > 0 contains a number τ 2 I such that, for every s, t ≥ 0, we have ju (s + τ, t) − u (s, t)j ≤ ϵ andju (s, t + τ) − u (s, t)j ≤ ϵ. ap ap ap ap If we replace condition (ii) with condition: 2 n (ii)’ The mappings g : R ! C are continuous, bounded (i = 1, 2) and satisfy that there exist positive real numbers ω > 0 and ω > 0 as well as complex numbers c and c such thatjc j = jc j = 1 and, for every 1 2 1 2 1 2 s, t 2 R, we have g (s + ω , t) = c g (s, t) and g (s, t + ω ) = c g (s, t), 1 1 1 1 2 2 2 2 and accept all remaining assumptions, then we similarly may deduce that there exist a continuous function 2 2 n u (s, t) on [0,∞) and a function u 2 C ([0,∞) : C ) such that u(s, t) = u (s, t) + u (s, t) for all (s, t) 2 h 0 0,D h 0 [0,∞) , as well as that, for every s, t ≥ 0, we have u (s + ω , t) = c u (s, t) and u (s, t + ω ) = c u (s, t). h 1 1 h h 2 2 h 5. Finally, it is worth recalling that the existence and uniqueness of almost periodic solutions for a class of boundary value problems for hyperbolic equations have been investigated by B. I. Ptashnic and P. I. Shta- balyuk in [24]. In the region D = (0, T )× R (T > 0, p 2 N), these authors have analyzed the almost periodic type solutions of the following initial value problem: 2n X X ∂ u(t, x) Lu  a = 0, (4.3) α p 2n−2s ∂t ∂x · · · ∂x s=0 jαj=2s j−1 j−1 ∂ u ∂ u = φ (x), = φ (x) (1 ≤ j ≤ n). (4.4) j j+n j−1 j−1 ∂t ∂t t=0 t=T p p Suppose that any of the functions φ (x),· · ·, φ (x) is almost periodic in R and M = fμ : k 2 Z g is the 1 2n p union of all Bohr-Fourier spectrum of functions φ (x),···, φ (x). Under certain assumptions, the solutions 1 2n u(t, x) of problem (4.3)-(4.4) have been found in the form ihμ ,xi u(t, x) = u (t)e , μ 2 M , (4.5) k k k2Z where the functions u (t) have the form [24, (8), p. 670] (cf. [24, Theorem 1, Theorem 2] for more details con- cerning the existence and uniqueness of solutions to (4.3)-(4.4)). p n Suppose now that ω 2 R \f0g and C 2 R. We want to observe here that the assumption φ 2 AP (R : C) j Λ for all j 2 N , where 2n Λ := (x ,· · ·, x ) 2 R ; x ω + · · · + x ω = C , 1 p 1 1 p p iC implies that the solution u(t, x) of problem (4.3)-(4.4) is (ω, e )-periodic in the space variable x. This follows from the computation (t 2 (0, T ), x 2 R ): X X ihμ ,x+ωi ihμ ,xi ihμ ,ωi k k k u(t, x + ω) = u (t)e = u (t)e e k k p p k2Z k2Z iC ihμ ,xi = e u (t)e = cu(t, x). k2Z Acknowledgement: The author is partially supported by grant 451-03-68/2020/14/200156 of Ministry of Science and Technological Development, Republic of Serbia. Multi-dimensional (ω, c)-almost periodic type functions.. . Ë 151 Conict of interest: The author states that there is no conict of interest. Data Availability Statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study. References [1] S. M. A. Alsulami, On Evolution Equations In Banach Spaces And Commuting Semigroups. PhD. Thesis, Ohio University, [2] E. Alvarez, A. Gómez, M. Pinto, (ω, c)-Periodic functions and mild solution to abstract fractional integro-dierential equa- tions, Electron. J. Qual. Theory Dier. Equ. 16 (2018), 1–8. [3] E. Alvarez, S. Castillo, M. Pinto, (ω, c)-Pseudo periodic functioins, rst order Cauchy problem and Lasota-Wazewska model with ergodic and unbounded oscillating production of red cells, Bound. Value Probl. 106 (2019), 1–20. [4] E. Alvarez, S. Castillo, M. Pinto, (ω, c)-Asymptotically periodic functions, rst-order Cauchy problem, and Lasota-Wazewska model with unbounded oscillating production of red cells, Math. Methods Appl. Sci. 43 (2020), 305–319. [5] W. Arendt, C. J. K. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Birkhäuser/Springer Basel AG, Basel, 2001. [6] A. S. Besicovitch, Almost Periodic Functions, Dover Publ, New York, 1954. [7] H. Bohr, Zur theorie der fastperiodischen Funktionen I; II; III, Acta Math. 45 (1924), 29–127; H6 (1925), 101–214; HT (1926), 237–281. [8] A. Chávez, K. Khalil, M. Kostić, M. Pinto, (R, B)-Multi-almost periodic type functions and applications, preprint. arXiv:2012.00543. [9] T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, Springer-Verlag, New York, [10] A. M. Fink, Almost Periodic Dierential Equations, Springer-Verlag, Berlin, 1974. [11] G. M. N’Guérékata, Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer Acad. Publ, Dordrecht, [12] A. Haraux, P. Souplet, An example of uniformly recurrent function which is not almost periodic, J. Fourier Anal. Appl. 10 (2004), 217–220. [13] M. F. Hasler, Bloch-periodic generalized functions, Novi Sad J. Math. 46 (2016), 135–143. [14] M. F. Hasler, G. M. N’Guérékata, Bloch-periodic functions and some applications, Nonlinear Studies 21 (2014), 21–30. [15] M. T. Khalladi, M. Kostić, A. Rahmani, M. Pinto, D.Velinov, c-Almost periodic type functions and applications, Nonauton. Dyn. Syst. 7 (2020), 176–193. [16] M. T. Khalladi, M. Kostić, A. Rahmani, M. Pinto, D.Velinov, (ω, c)-Almost periodic type functions and applications, Filomat, submitted. hal-02549066. [17] M. Kostić, Almost Periodic and Almost Automorphic Type Solutions to Integro-Dierential Equations, W. de Gruyter, Berlin, [18] M. Kostić, Selected Topics in Almost Periodicity, Book manuscript, 2021. [19] M. Kostić, Multi-dimensional c-almost periodic type functions and applications, preprint. arXiv:2012.15735. [20] B. Ya. Levin, On the almost periodic functions of Levitan, Ukrainian Math. J. 1 (1949), 49–100. [21] M. Levitan, Almost Periodic Functions, G.I.T.T.L., Moscow, 1953 (in Russian). [22] B. M. Levitan, V. V. Zhikov, Almost Periodic Functions and Dierential Equations, Cambridge Univ. Press, London, 1982. [23] A. A. Pankov, Bounded and Almost Periodic Solutions of Nonlinear Operator Dierential Equations, Kluwer Acad. Publ., Dor- drecht, 1990. [24] B. I. Ptashnic, P. I. Shtabalyuk, A boundary value problem for hyperbolic equations in a class of functions that are almost periodic with respect to space variables, Dier. Uravn. 22 (1986), 669–678 (in Russian). [25] S. Zaidman, Almost-Periodic Functions in Abstract Spaces, Pitman Research Notes in Math, Vol. 126, Pitman, Boston, 1985. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Nonautonomous and Stochastic Dynamical Systems de Gruyter

Multi-dimensional (ω, c)-almost periodic type functions and applications

Kostić, M.
Nonautonomous and Stochastic Dynamical Systems , Volume 8 (1): 16 – Jan 1, 2021
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Abstract

Nonauton. Dyn. Syst. 2021; 8:136–151 Research Article Open Access M. Kostić* Multi-dimensional (ω, c)-almost periodic type functions and applications https://doi.org/10.1515/msds-2020-0130 Received February 1, 2021; accepted April 19, 2021 Abstract: In this paper, we analyze various classes of multi-dimensional (ω, c)-almost periodic type func- tions with values in complex Banach spaces. The main structural properties and characterizations for the introduced classes of functions are presented. We provide certain applications of our abstract theoretical results to the abstract Volterra integro-dierential equations, as well. Keywords: Multi-dimensional (ω, c)-almost periodic functions, multi-dimensional (ω , c ) -periodic func- j j j2N tions, multi-dimensional c-almost periodic functions, (I , a, ω, c)-almost periodic type functions, abstract Volterra integro-dierential equations MSC: 42A75, 43A60, 47D99 1 Introduction and preliminaries The notion of an almost periodic function was introduced by H. Bohr [7] around 1924-1926 and later general- ized by many other mathematicians (for more details about the subject, we refer the reader to the research monographs [6], [9]-[11], [17]-[19], [21]-[23] and [25]). Let I be either R or [0,∞), and let f : I ! X be a given continuous function, where X is a complex Banach space equipped with the normk·k. For any ε > 0, a num- ber τ > 0 is called a ε-period for f (·) if and only ifkf (t + τ)− f (t)k ≤ ε, t 2 I. The set consisting of all ε-periods for f (·) is denoted by ϑ(f , ε). The function f (·) is said to be almost periodic if and only if for each ε > 0 the set ϑ(f , ε) is relatively dense in [0,∞), i.e., there exists l > 0 such that any subinterval of [0,∞) of length l meets ϑ(f , ε). The notion of a periodic function has recently been reconsidered by E. Alvarez, A. Gómez and M. Pinto [2] in the following way: A continuous function f : I ! X is said to be (ω, c)-periodic (ω > 0, c 2 C \f0g) if and only if f (x + ω) = cf (x) for all x 2 I. We know that a continuous function f : I ! X is (ω, c)-periodic if −·/ω −·/ω and only if the function g(·)  c f (·) is periodic and g(x + ω) = g(x) for all x 2 I; here, c denotes the principal branch of the exponential function (see also the research articles [3]-[4] by E. Alvarez, S. Castillo, M. Pinto for more details about the subject). In our joint paper [16] with M. T. Khalladi, A. Rahmani, M. Pinto and D. Velinov, we have recently extended the notion of (ω, c)-periodicity by examining various classes of (ω, c)-almost periodic type functions. On the other hand, in our recent paper [15], we have recently introduced the class of c-almost periodic functions depending on one real variable. The multi-dimensional c-almost periodic type functions have re- cently been investigated in [16]. The main aim of this paper is to continue the above-mentioned research studies by introducing and analyzing various notions of (ω, c)-periodicity and (ω, c)-almost periodicity for vector-valued functions depending on several real variables; we provide certain applications to the abstract partial dierential equations, as well. For the sake of simplicity and better exposition, we will consider the *Corresponding Author: M. Kostić: Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovića 6, 21125 Novi Sad, Serbia, E-mail: marco.s@verat.net Open Access. © 2021 M. Kostić, published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. Multi-dimensional (ω, c)-almost periodic type functions.. . Ë 137 corresponding Stepanov classes of multi-dimensional (ω, c)-almost periodic type functions and the corre- sponding (Stepanov) classes of multi-dimensional semi-c-periodic functions somewhere else. The organization and main ideas of the paper can be briey described as follows. In Subsection 1.1, we recall the basic denitions and results about c-almost periodic functions in R . The main aim of Section 2 is to introduce and analyze the classes of multi-dimensional (ω, c)-periodic functions and multi-dimensional (ω , c ) -periodic functions. The main structural results obtained in this section are Proposition 2.4- j j j2N Proposition 2.5 and Proposition 2.8-Proposition 2.11. The corresponding classes of asymptotically (ω, c)- almost periodic type functions are introduced in Denition 2.15; Subsection 2.1 investigates (ω , c ; r , I ) - j j j j2N j n almost periodic type functions. In Denition 2.16, we introduce the notion of (ω , c ; r , I ) -almost peri- j j j j2N j n odicity, (ω , c ; r , I ) -uniform recurrence and (ω , c ) -almost automorphy. The main structrural pro- j j j j2N j j j2N j n n lations of function spaces introduced in Denition 2.16 are stated in Proposition 2.19; we also discuss the convolution invariance of function spaces introduced here before we move ourselves to the third section of ′ ′ paper, which is reserved for the study of (I , a, ω, c)-uniform recurrence of type 1 (2) and the (I , a, ω, c)- almost periodicity of type 1 and type 2. Here we continue our investigation from [16, Section 3] and prove two negative results, Theorem 3.3 and Theorem 3.5, saying that the introduction of Denition 3.1 is basically an unsatisfactory way to extend the notion of (ω, c)-almost periodicity. In the nal section of paper, we provide certain application to the abstract Volterra integro-dierential equations in Banach spaces. The author would like to thank Prof. A. Chávez, M. T. Khalladi, M. Pinto, A. Rahmani and D. Velinov for many valuable comments and suggestions. Numerous research articles concerning multi-dimensional almost periodic type functions are written in my collaboration with these mathematicians. Notation and terminology. We assume henceforth that (X,k·k) and (Y ,k·k ) are complex Banach spaces, n 2 N, B is a certain collection of subsets of X satisfying that for each x 2 X there exists B 2 B such that x 2 B. We will use the principal branch of the exponential function to take the powers of complex numbers. n n If t 2 R and ϵ > 0, then we set B(t , ϵ) := ft 2 R : jt − t j ≤ ϵg, wherej ·j denotes the Euclidean norm 0 0 0 n n in R . Set N := f1,· · ·, ng and I := ft 2 I : jtj ≤ Mg (I  R ; M > 0). 1.1 c-Almost periodic functions In [19], we have recently introduced the following notion: ′ n ′ Denition 1.1. Suppose that; ≠ I  I  R , F : I × X ! Y is a continuous function and I + I  I. Then we say that: (i) F(·;·) is Bohr (B, I , c)-almost periodic if and only if for every B 2 B and ϵ > 0 there exists l > 0 such that ′ ′ for each t 2 I there exists τ 2 B(t , l)\ I such that 0 0 F(t + τ; x) − cF(t; x) ≤ ϵ, t 2 I, x 2 B. ′ ′ (ii) F(·;·) is (B, I , c)-uniformly recurrent if and only if for every B 2 B there exists a sequence (τ ) in I such that lim jτ j = +∞ and k!+∞ k lim sup F(t + τ ; x) − cF(t; x) = 0. k!+∞ t2I;x2B ′ ′ If X 2 B, then it is also said that F(·;·) is Bohr (I , c)-almost periodic ((I , c)-uniformly recurrent). The most important case is that in which I = I, when we also say that the function F(·;·) is Bohr (B, c)- almost periodic [(B, c)-uniformly recurrent]; if X 2 B, then it is also said that F(·;·) is Bohr c-almost peri- ′ ′ odic (c-uniformly recurrent). The classes of Bohr (B, I )-almost periodic functions, (B, I )-uniformly recur- ′ ′ rent functions, Bohr I -almost periodic functions and I -uniformly recurrent functions, obtained by plugging c = 1, are thoroughly investigated in [8]. If c = −1, then we also say that F(·;·) is (B, I )-almost anti-periodic, ′ ′ ′ (B, I )-uniformly anti-recurrent, I -almost anti-periodic or I -uniformly anti-recurrent. We know the following: 138 Ë M. Kostić ′ n ′ ′ ′ (i) Suppose that ; ≠ I  I  R , I + I = I and F : I × X ! Y is Bohr (B, I , c)-almost periodic ((B, I , c)- uniformly recurrent). If F(·;·) ≠ 0, thenjcj = 1. ′ n ′ ′ (ii) Suppose that; ≠ I  I  R and I + I = I. If the function F : I ! R is (B, I , c)-uniformly recurrent and F ≠ 0, then c = ±1. Furthermore, if F(t) ≥ 0 for all t 2 I, then c = 1. ′ n ′ ′ (iii) Suppose that l 2 N, ; ≠ I  I  R , I + I  I and F : I × X ! Y is Bohr (B, I , c)-almost periodic ′ ′ ′ ′ l ((B, I , c)-uniformly recurrent). Then lI  I, I + lI  I and F(·;·) is Bohr (B, lI , c )-almost periodic ′ l ((B, lI , c )-uniformly recurrent). (iv) Suppose that p 2 Z \f0g, q 2 N, (p, q) = 1, jcj = 1 and arg(c) = πp/q. (1.1) ′ n ′ ′ Suppose, further, that ; ≠ I  I  R , I + I  I and F : I × X ! Y is Bohr (B, I , c)-almost periodic ((B, I , c)-uniformly recurrent). Then the following holds: ′ ′ (a) If p is even, then F(·;·) is Bohr (B, qI )-almost periodic ((B, qI )-uniformly recurrent). ′ ′ (b) If p is odd, then F(·;·) is Bohr (B, qI )-almost anti-periodic ((B, qI )-uniformly anti-recurrent). ′ n ′ ′ ′ (v) Let jcj = 1 and arg(c)/π 2̸ Q. If ; ≠ I  I  R , I + I  I, lI = I for all l 2 N and F : I × X ! Y ′ ′ is a bounded, Bohr (B, I , c)-almost periodic ((B, I , c)-uniformly recurrent) function, then the function ′ ′ ′ ′ ′ F(·;·) is Bohr (B, I , c )-almost periodic ((B, I , c )-uniformly recurrent) for all c 2 S . n ′ n If F : R ! Y is an almost periodic function (X = f0g, B = fXg, I = I = R ), then the mean value M(F) := lim F(t) dt T!+∞ (2T ) s+K n n exists and it does not depend on s 2 R , where K := ft = (t , t ,· · ·, t ) 2 R : jt j ≤ T for 1 ≤ i ≤ ng. The 1 2 n T i Bohr-Fourier coecient F 2 X is dened by −ihλ,·i n F := M e F(·) , λ 2 R . We know that the Bohr spectrum of F(·), dened by σ(F) := λ 2 R : F ≠ 0 , n n is at most a countable set. By AP(R : X) and AP (R : X) we denote respectively the Banach space consisting of all almost periodic functions F : R ! X and its subspace consisting of all almost periodic functions F : R ! X such that σ(F)  Λ. We also need the following denition from [8]: Denition 1.2. Suppose that D  I  R and the set D is unbounded. By C (I × X : Y ) we denote 0,D,B the vector space consisting of all continuous functions Q : I × X ! Y such that, for every B 2 B, we have lim Q(t; x) = 0, uniformly for x 2 B. If X = f0g, then we abbreviate C (I×X : Y ) to C (I : Y ). 0,D,B 0,D,B t2D,jtj!+∞ 2 (ω, c)-Periodic functions and (ω , c ) -periodic functions j j j2N Let us recall that a continuous function F : I ! X is said to be Bloch (p, k)-periodic, or Bloch periodic with n n period p and Bloch wave vector or Floquet exponent k, where p 2 R and k 2 R , if and only if F(t + p) = ihk,pi e F(t), t 2 I (we assume here that p + I  I). In [8, Example 2.15(viii)], we have observed that the Bloch ′ −ihk,·i ′ (p, k)-periodicity of function F(·) implies the Bohr (B, I )-almost periodicity of function e F(·) with I being the intersection of I and the one-dimensional submanifold generated by the vector p as well as that the orthogonality of vectors k and p implies that the function F(·) is Bohr (B, I )-almost periodic. For more details about the Bloch periodic functions, we refer the reader to the research articles [13] by M. Hasler and [14] by M. Hasler, G. M. N’Guérékata. Multi-dimensional (ω, c)-almost periodic type functions.. . Ë 139 Following the recent research analyses of E. Alvarez, A. Gómez, M. Pinto [2] and E. Alvarez, S. Castillo, M. Pinto [3]-[4], we generalize the notion of Bloch (p, k)-periodicity in the following way: Denition 2.1. Let ω 2 R \f0g, c 2 C \f0g and ω + I  I. A continuous function F : I ! X is said to be (ω, c)-periodic if and only if F(t + ω) = cF(t), t 2 I. ihk,pi If F : I ! X is a Bloch (p, k)-periodic function, then F(·) is (p, c)-periodic with c = e ; conversely, if jcj = 1 and F : I ! X is (ω, c)-periodic, then we can always nd a point k 2 R such that the function F(·) is Bloch (p, k)-periodic. In the case that jcj ≠ 1, we have the following: if F : I ! X is (ω, c)-periodic, then F(t + mω) = c F(t), t 2 I, m 2 N, so that the existence of a point t 2 I such that F(t ) ≠ 0 implies 0 0 lim jjF(t + mω)jj = +∞, provided thatjcj > 1, and lim jjF(t + mω)jj = 0, provided thatjcj < 1. m!∞ m!∞ 0 0 If c = 1, resp. c = −1, then we also say that the function F(·) is ω-periodic, resp. ω-anti-periodic. It is k k clear that, if F(·) is (ω, c)-periodic, k 2 N and c = 1, resp. c = −1, then F(·) is (kω)-periodic, resp. (kω)- anti-periodic. In [14, Denition 2.1], the authors have assumed that any Bloch (p, k)-periodic is bounded a priori, which is a slightly redundant condition as the following example shows: Example 2.2. There exists a continuous, unbounded function F : R ! R which satises F(t+(1, 1,···, 1)) = n 2 F(t) for all t 2 R . We can simply construct such a function, with n = 2, as follows. Let F : f(t , t ) 2 R : 0 1 2 0 ≤ t + t ≤ 2g be any continuous function satisfying that: 1 2 F t , t = F t + 1, t + 1 , provided t , t 2 R and t + t = 0, (2.1) 0 1 2 1 2 1 2 1 2 p p the set (4k 2,−4k 2) : k 2 N is unbounded, and (2.2) p p F (4k + 2) 2,−(4k + 2) 2 = 1, k 2 N. (2.3) Due to condition (2.1), we can extend the function F (·) to a continuous function F : R ! R which satises F(t + 1, t + 1) = F(t , t ) for all t , t 2 R. Clearly, this function is unbounded due to condition (2.2). 1 2 1 2 1 2 The following denition is also meaningful: Denition 2.3. Let ω 2 R\f0g, c 2 C\f0g and ω e + I  I (1 ≤ j ≤ n). A continuous function F : I ! X is j j j j said to be (ω , c ) -periodic if and only if F(t + ω e ) = c F(t), t 2 I, j 2 N . j j j2N j j j It is clear that, if F : I ! X is (ω , c ) -periodic, then F(t + mω e ) = c F(t), t 2 I, m 2 N, j 2 N , so j j j2N j j n j that the existence of a point t 2 I such that F(t ) ≠ 0 implies lim jjF(t + mω e )jj = +∞, provided that m!∞ 0 0 0 j j jc j > 1, and lim jjF(t + mω e )jj = 0, provided thatjc j < 1, for some j 2 N . m!∞ n j 0 j j j If c = 1 for all j 2 N , resp. c = −1 for all j 2 N , then we also say that the function F(·) is (ω ) - n n j j j j2N periodic, resp. (ω ) -anti-periodic. It is clear that, if F(·) is (ω , c ) –periodic, k 2 N and c = 1 for all j j2N j j j2N n n j j 2 N , resp. c = −1 for all j 2 N , then F(·) is (kω ) -periodic, resp. (kω ) -anti-periodic. n n j j2N j j2N j n n The classes of (ω, c)-periodic functions and (ω , c ) -periodic functions are closed under the operation j j j2N of the pointwise convergence of functions, as easily approved. In the scalar-valued case, the following holds: If the function F : I ! C \ f0g is (ω, c)-periodic, resp. (ω , c ) -periodic, then the function (1/F)(·) is j j j2N (ω, 1/c)-periodic, resp. (ω , 1/c ) -periodic. It is also clear that we have the following: j j j2N Proposition 2.4. (i) Let ω, a 2 R \ f0g, c 2 C \ f0g, α 2 C, ω + I  I and a + I  I. If the function ˇ ˇ F : I ! X is (ω, c)-periodic, then −ω − I  −I and the function F : −I ! X, dened by F(x) := F(−x), x 2 I, is (−ω, c)-periodic. Moreover,kF(·)k is (ω,jcj)-periodic, the function F : I ! X dened by F (t) := F(t +a), a a t 2 I is (ω, c)-periodic and the function αF(·) is (ω, c)-periodic. (ii) Let ω 2 R\f0g, c 2 C\f0g, α 2 C, ω e +I  I (1 ≤ j ≤ n) and a +I  I. If a continuous function F : I ! X j j j j is (ω , c ) -periodic, then−ω e −I  −I (1 ≤ j ≤ n) and the function F : −I ! X is (−ω , c ) -periodic. j j j2N j j j j j2N n n 140 Ë M. Kostić Moreover,kF(·)k is (ω ,jc j) -periodic, the function F : I ! X dened above is (ω , c ) -periodic and j j j2N j j j2N n n the function αF(·) is (ω , c ) -periodic. j j j2N Proposition 2.5. Let ω 2 R\f0g, c 2 C\f0g and ω e + I  I (1 ≤ j ≤ n). If a continuous function F : I ! X j j j j is (ω , c ) -periodic, then ω + I  I, where ω := ω e , and the function F(·) is (ω, c)-periodic with j j j2N j j n j=1 c =: c . j=1 The converse statement is not true in general case n > 1, as the following simple counterexample shows: Example 2.6. Consider the function F : R ! R from Example 2.2. Then there do not exist numbers ω , ω 2 1 2 R \ f0g and numbers c , c 2 C \ f0g such that the function F(·) is (ω , c ) -periodic. If we assume the 1 2 j j j2N contrary, then we would have F(t + ω , 0) = c F(t , 0) for all t 2 R. If jc j ≤ 1, then the contradiction is 1 1 1 1 1 1 p p obvious since (2.2) implies the unboundedness of function F(·, 0), because F(8k, 0) = F(4k 2,−4k 2) for all k 2 N. Ifjc j > 1, then the contradiction is obvious due to condition (2.3), which implies that the function F(·, 0) cannot tends to plus innity as the time variable tends to plus innity (see also [15, Remark 2.4]). Sometimes the converse in Proposition 2.5 is possible to be made: 2 ′ 2 Example 2.7. (cf. also [19, Example 2.8]) Suppose that I := f(x, y) 2 R : x + y ≥ 0g and I := f(x, y) 2 R : −x−y ′ −1 x + y = 1g. Set F(x, y) := 2 , (x, y) 2 I. Then, for every (a, b) 2 I , we have F((x, y) + (a, b)) = 2 F(x, y), −1 ′ (x, y) 2 I, so that F(·,·) is (ω, 2 )-periodic provided that ω 2 I . Similarly, we have that F(·;·) is (ω , c ) - j j j2N −ω −ω 1 2 periodic for every ω > 0 and ω > 0, with c = 2 and c = 2 . 1 2 1 2 Concerning the boundedness of (ω , c ) -periodic functions, we will state only one result: j j j2N Proposition 2.8. Suppose that ω 2 R \f0g, c 2 C \f0g, M > 0, ω e + I  I (1 ≤ j ≤ n), the set I is closed, j j j j the function F : I ! X is (ω , c ) -periodic,jc j ≤ 1 for all j 2 N and, for every t = (t , t ,· · ·, t ) 2 I, there n n j j j2N j 1 2 exist a point η = (η , η ,· · ·, η ) 2 I and integers k 2 N (1 ≤ j ≤ n) such that t = k ω + η (1 ≤ j ≤ n). Then 1 2 M j j j j j the function F(·) is bounded. Proof. Let a point t = (t , t ,···, t ) 2 I be xed, and let η 2 I and integers k 2 N (1 ≤ j ≤ n) satisfy the above 1 2 j P Q n n j requirements. Then we have t = η + k ω e so that F(t) = c F(η). Since I is closed, I is compact j j j M j=1 j=1 j and there exist a nite constant M > 0 such thatkF(x)k ≤ M for all x 2 I . ThenkF(t)k ≤ M sincejc j ≤ 1 1 1 M 1 j for all j 2 N . In the following extension of [2, Proposition 2], we prole the class of (ω, c)-periodic functions in the following way: Proposition 2.9. Let ω = (ω , ω ,· · ·, ω ) 2 R \ f0g, ω + I  I, c 2 C \ f0g and S := fi 2 N : ω ≠ 0g. n n 1 2 i jSj Denote by A the collection of all tuples a = (a , a ,· · ·, a ) 2 R such that a = 1. Then a continuous 1 2 jSj i i2S function F : I ! X is (ω, c)-periodic if and only if, for every (some) a 2 A, the function G : I ! X, dened by a t i i i2S ω G t , t ,· · ·, t := c F t , t ,· · ·, t , t = t , t ,· · ·, t 2 I, (2.4) a n n n 1 2 1 2 1 2 is (ω, 1)-periodic. Proof. Let a point t = (t , t ,· · ·, t ) 2 I be xed. Then it is clear that G (t + ω) = G (t) if and only if 1 2 n a a P P a (t +ω ) a t i i i  i i − − i2S i2S ω ω i i c F t + ω , t + ω ,· · ·, t + ω = c F t , t ,· · ·, t n n n 1 1 2 2 1 2 if and only if F(t + ω) = cF(t). We illustrate Proposition 2.9 with the following example: Multi-dimensional (ω, c)-almost periodic type functions. .. Ë 141 n n Example 2.10. (see also [14, pp. 22-23]) Suppose that ω = (ω , ω ,· · ·, ω ) 2 R \f0g, k 2 R \f0g, a 2 A, 1 2 −2 (b ) is any sequence of complex numbers such thatjb j = O(l ), hk, ωi = 2π/3 and l l a t i i ilht,ki n i2S ω F t , t ,· · ·, t = c b e , t = t , t ,· · ·, t 2 R . 1 2 n 1 2 n l21+3N Then F(·) is (3ω, c)-almost periodic. Similarly, we can prove the following: Proposition 2.11. Let ω 2 R \ f0g, c 2 C \ f0g, ω e + I  I (1 ≤ j ≤ n) and the function F : I ! X is j j j j continuous. For each j 2 N , we dene the function G : I ! X by G t , t ,· · ·, t := c F t , t ,· · ·, t , t = t , t ,· · ·, t 2 I. (2.5) j 1 2 n 1 2 n 1 2 n Then F(·) is (ω , c ) -periodic if and only if, for every t = (t , t ,· · ·, t ) 2 I and j 2 N , we have n n j j j2N 1 2 G t , t ,· · ·, t + ω ,· · ·, t = G t , t ,· · ·, t ,· · ·, t . j 1 2 j j n j 1 2 j n Therefore, we have the following: Example 2.12. Let c 2 C \f0g for all j 2 N . Then the function j n Y j 2π F(t ,· · ·, t ) := c sin t , t = (t ,· · ·, t ) 2 R 1 n j 1 n j=1 is (2π, c ) -periodic. j j2N If ω 2 R \ f0g, c 2 C \ f0g for i = 1, 2, ω + I  I, the function G : I ! C is (ω, c )-periodic and the i 1 function H : I ! X is (ω, c )-periodic, then the function F(·) := G(·)H(·) is (ω, c c )-periodic. For the class 2 1 2 of (ω , c ) -periodic functions, we can clarify the following result (cf. also [16, Proposition 2.2]): j j j2N Proposition 2.13. Let ω 2 R \ f0g, c 2 C \ f0g and ω e + I  I (1 ≤ j ≤ n, 1 ≤ i ≤ 2). Suppose that j j,i j j the function G : I ! C is (ω , c ) -periodic and the function H : I ! X is (ω , c ) -periodic. Set j j,1 j2N j j,2 j2N n n c := c c , 1 ≤ j ≤ n. Then the function F(·) := G(·)H(·) is (ω , c ) -periodic. j j,1 j,2 j j j2N Concerning the convolution invariance of spaces introduced in Denition 2.1 and Denition 2.3, we will state and prove the following result: n jSj Proposition 2.14. Suppose that ω 2 R \f0g, c 2 C\f0g, S = fi 2 N : ω ≠ 0g, a = (a , a ,···, a ) 2 R n i 1 2 jSj and a = 1, resp. ω 2 R \f0g and c 2 C \f0g (1 ≤ j ≤ n). Suppose, further, that F : R ! X is (ω, c)- i j j i2S periodic and the function G (·), dened through (2.4) is bounded, resp. F : R ! X is (ω , c ) -periodic and a j j j2N − a t /ω i i i i2S for each j 2 N the function G (·), dened through (2.5) is bounded. If the function c h(t ,· · ·, t ) n j 1 n 1 n −t /ω 1 n j j belongs to the space L (R ), resp. for each j 2 N the function c h(t ,···, t ) belongs to the space L (R ), n 1 n then the function (h F)(t) := h(y)F(t − y) dy, t 2 R is (ω, c)-periodic, resp. (ω , c ) -periodic. j j j2N Proof. We will consider only (ω, c)-periodicity. By Proposition 2.9, it suces to show that P P a (t +ω ) a t i i i i i − − i2S ω i2S ω i i c (h F)(t + ω) = c (h F)(t) (2.6) * * 142 Ë M. Kostić for every xed point t = (t ,· · ·, t ) 2 R . Note rst that the value (h F)(t) is well dened, since we have n * − a t /ω 1 n i i i i2S assumed that the function c h(t ,···, t ) belongs to the space L (R ), as well as that the function G (·), dened through (2.4), is bounded and − a t /ω i i i i2S c (h F)(t) " # " # P P − a y /ω − a (t −y )/ω i i i i i i i i2S i2S = c h(y ,· · ·, y ) · c F(t − y) dy. (2.7) 1 n Keeping in mind (2.7) and the dominated convergence theorem, we get that the function (h F)(·) is continu- ous. Similarly, by plugging t + ω in place of t in (2.7), we get that (2.6) holds because a = 1. i2S Concerning asymptotically (ω, c)-periodic type functions, we will use the following denition, only: n n Denition 2.15. Suppose that D  I  R , the set D is unbounded, ω 2 R \ f0g, c 2 C \ f0g, ω + I  I, ω 2 R \f0g, c 2 C \f0g, ω e + I  I (1 ≤ j ≤ n, 1 ≤ i ≤ 2) and F : I ! X. Then we say that the function F(·) j j j j is D-asymptotically (ω, c)-periodic, resp. D-asymptotically (ω , c ) -periodic, if and only if there exists a j j j2N (ω, c)-periodic, resp. (ω , c ) -periodic, function F : I ! X and a function Q 2 C (I : X) such that j j j2N 0 0,D,B F(t) = F (t) + Q(t), t 2 I. 2.1 (ω , c ; r , I ) -Almost periodic type functions j j j j2N Following our idea from [16, Denition 2.1], we can introduce and analyze several various generalizations of the class of multi-dimensional (ω , c ) -periodic functions with the help of Proposition 2.11. For example, j j j2N suppose that ω 2 R\f0g, c 2 C\f0g and ω e +I  I (1 ≤ j ≤ n); if a function F : I ! X is (ω , c ) -periodic, j j j j j j j2N then for each j 2 N and for every k 2 N we have F(t + kω e ) = c F(t), t 2 I, j 2 N and G (t + kω e ) = G (t), n n j j j j j j t 2 I, j 2 N . Set W := fj 2 N : ω > 0g and W := fj 2 N : ω < 0g, as well as I := fx ≥ 0 : t + xe 2 Ig n + n − n j j j,t j if j 2 W , resp. I := fx ≥ 0 : t − xe 2 Ig if j 2 W (t 2 I), and G (x) := G (t + xe ), x 2 I if j 2 W , + − + j,t j j,t j j j,t resp. G (x) := G (t − xe ), x 2 I if j 2 W (t 2 I). Then we can generalize the class of (ω , c ) -periodic j,t j j j,t j j j2N functions as follows: Denition 2.16. Suppose that ω 2 R\f0g, c 2 C\f0g, ω e +I  I (1 ≤ j ≤ n) and F : I ! X is a continuous j j j j ′ ′ function. Let r 2 C \ f0g for 1 ≤ j ≤ n, and let ; ≠ I  I  R, I + I  I for 1 ≤ j ≤ n, t 2 I. Set j j,t j,t j,t j,t j,t ′ ′ I := fI : t 2 Ig. Then we say that the function F(·) is: j j,t (i) (ω , c ; r , I ) -almost periodic if and only if, for every j 2 N and t 2 I, the function G (·) dened j j j j2N j,t j n above is (I , r )-almost periodic; j,t (ii) (ω , c ; r , I ) -uniformly recurrent if and only if, for every j 2 N and t 2 I, the function G (·) is j j j j2N j,t j n (I , r )-uniformly recurrent. j,t Suppose that I = R . Then we say that the function F(·) is: (iii) (ω , c ) -almost automorphic if and only if, for every j 2 N , for every t 2 R and for every real j j j2N sequence (b ), there exist a subsequence (a ) of (b ) and a function F : R ! X such that k k k j,t * * lim G t + x + a e = F (x) and lim F x − a = G t + xe , (2.8) j k j j,t j,t k j j k!+∞ k!+∞ pointwise for x 2 R; if, moreover, the convergence in (2.8) is uniform in the variable x on compact subsets of R, then we say that the function F(·) is compactly (ω , c ) -almost automorphic. j j j2N Remark 2.17. (i) It is clear that I = R is equivalent to saying that I + ηe  I for all η 2 R \f0g and j 2 N . (ii) It is clear that (i) implies (ii) and that the almost periodicity of the function G (x) := G (t + xe ), x 2 R j,t j j for all j 2 N and t 2 I implies (iii), which is equivalent to saying that the function G (·) dened above j,t is almost automorphic for all j 2 N and t 2 I. n Multi-dimensional (ω, c)-almost periodic type functions. .. Ë 143 n ′ Now we will provide an illustrative example in which we have I = R , ω = c = 1 and I = I = [0,∞) for j j j,t j,t all j 2 N and t 2 R : Example 2.18. (i) Suppose that r = 1 for all j 2 N . Then the function F t ,· · ·, t := sin t + sin 2t , t = (t ,· · ·, t ) 2 R n n 1 j j 1 j=1 is (ω , c ; r , I ) -almost periodic but not (ω , c ) -periodic. j j j j2N j j j2N j n n (ii) ([12], [15], [19]) Suppose that r = −1 for all j 2 N . Then the function j n " # n ∞ Y X 2 j n F t ,· · ·, t := sin t · sin , t = (t ,· · ·, t ) 2 R 1 n j 1 n n 3 j=1 n=1 ′ ′ is (ω , c ; r , I ) -uniformly recurrent but not (ω , c ; r , I ) -almost periodic. j j j j j2N j j j j j2N n n (iii) Suppose that r = 1 for all j 2 N . Then the function F t ,· · ·, t := sin p  , t = (t ,· · ·, t ) 2 R 1 n 1 n 2 + sin t + sin 2t j j j=1 is (ω , c ) -almost automorphic but not (ω , c ; r , I ) -almost periodic. j j j2N j j j j2N n j n The function spaces introduced in Denition 2.16 are translation invariant and closed under the pointwise multiplications with complex scalars. Furthermore, if the function F(·) is (ω , c ; r , I ) -almost periodic, j j j j2N j n then it can be easily proved that the functionkF(·)k is (ω ,jc j;jr j, I ) -almost periodic. Suppose now that j j j j2N j n the function F : I ! C \f0g is (ω , c ; r , I ) -almost periodic, jF(t)j ≥ m > 0 for all t 2 I, and jc j = 1 for j j j j2N j j n all j 2 N . Then the function (1/F)(·) is (ω , 1/c ; r , I ) -almost periodic, which can be simply proved as j j j j2N j n follows (cf. also [16, Proposition 2.5]). Let j 2 N , t 2 I and ϵ > 0 be xed; without loss of generality, we may t /ω ′ j j assume that j 2 W . Let τ 2 I andjG (x + τ) − r G (x)j < ϵ, x ≥ 0. After multiplication with c , we get j,t j j,t j,t x+τ c F t , t ,· · ·, t + x,· · ·, t − r F t , t ,· · ·, t + (x + τ),· · ·, t ≤ ϵ, x ≥ 0. (2.9) 1 2 n 1 2 n j j j Hence, for every x ≥ 0, we have: t +x+τ t +x j j ω ω j j c c j −1 j − r F t , t ,· · ·, t + x + τ,· · ·, t F t , t ,· · ·, t + x,· · ·, t 1 2 j n 1 2 j n j −1 = c F t , t ,· · ·, t + x,· · ·, t − r F t , t ,· · ·, t + x + τ,· · ·, t n n 1 2 j j 1 2 j −2 −1 · ≤ m jr j ϵ, jF(t , t ,· · ·, t + x + τ,· · ·, t )j ·jF(t , t ,· · ·, t + x,· · ·, t )j 1 2 j n 1 2 j n where we have employed (2.9) in the last estimate. This simply implies the required. By a careful examination of the notion introduced in Denition 2.16 and the paragraph preceding it, we may deduce that the (ω , c ; r , I ) -almost periodicity of the function F : I ! X implies the j j j j2N j n (−ω , c ; r , I ) -almost periodicity of the function F(·). We leave all details concerning the proof of this j j j j2N j n fact to the interested readers. Using the statements (i)-(v) from Subsection 1.1 and corresponding denitions, we may deduce the fol- lowing proposition: Proposition 2.19. (i) Suppose that, for every j 2 N and t 2 I, we have I + I = I and the function j,t j,t j,t F : I ! X is (ω , c ; r , I ) -uniformly recurrent. Then, for every j 2 N , we have r = ±1; if, additionally, j j j j2N j j n F(t) ≥ 0 for all t 2 I, then, for every j 2 N , we have r = 1. j 144 Ë M. Kostić ′ ′ (iii) Suppose that l 2 N, and F : I ! X is (ω , c ; r , I ) -almost periodic ((ω , c ; r , I ) -uniformly recur- j j j j2N j j j j2N j n j n ′ ′ l ′ rent). Then, for every j 2 N and t 2 I, we have lI  I , I + lI  I and F(·) is (ω , c ; r , lI ) - j,t j,t j,t j j j2N j,t j,t j j n l ′ ′ ′ almost periodic ((ω , c ; r , lI ) -uniformly recurrent), where lI := flI : t 2 Ig for all j 2 N . j j j2N n j j n j j,t ′ ′ (iv) Suppose that (1.1) holds and F : I ! X is (ω , c ; r , I ) -almost periodic ((ω , c ; r , I ) -uniformly j j j j2N j j j j2N j n j n recurrent). Then the following holds: (a) If p is even, then F(·) is (ω , c ; 1, I ) -almost periodic j j j2N j n ′ ′ ′ ((ω , c ; 1, qI ) -uniformly recurrent), where qI := fqI : t 2 Ig for all j 2 N . j j j2N n j n j j,t (b) If p is odd, then F(·) is (ω , c ;−1, I ) -almost periodic j j j2N j n ((ω , c ;−1, qI ) -uniformly recurrent). j j j2N j n ′ ′ (v) Let jcj = 1 and arg(c)/π 2̸ Q. If, for every j 2 N and t 2 I, lI = I for all l 2 N and F : I ! X is a j,t j,t ′ ′ bounded, (ω , c ; r , I ) -almost periodic ((ω , c ; r , I ) -uniformly recurrent) function, then the func- j j j j2N j j j j2N j n j n ′ ′ ′ ′ ′ ′ tion F(·) is (ω , c ; r , I ) -almost periodic ((ω , c ; r , I ) -uniformly recurrent) for all (r ,· · ·, r ) 2 j j j2N j j j2N n j j n j j n 1 (S ) . Concerning the convolution invariance of spaces introduced in Denition 2.16, the following important fact should be said: we have introduced the notion of (I , r )-almost periodicity, for example, by requiring that, j,t for every j 2 N and t 2 I, the function G (·) is (I , r )-almost periodic. Unfortunately, sometimes we need n j,t j j,t to assume that, for every j 2 N , the function G (·) is (I , r )-almost periodic uniformly in the variable t 2 I, n j,t j j,t in a certain sense. For simplicity, let us assume that I = R , which immediately implies that, for every j 2 N and t 2 R , we have I = [0,∞). Assume, further, that for each j 2 N there exists a set A  [0,∞) such j,t n j ′ n that A = I for every t 2 R , as well as that for each ϵ > 0 there exists l > 0 such that for each x 2 A we j 0 j j,t have the existence of a number x 2 B(x , l)\ A such that 0 j G (x + τ) − r G (x) ≤ ϵ, x ≥ 0, t 2 R , j,t j j,t i.e., t +x+τ t +x j j − − ω ω j j n c F t + (x + τ)e − r c F t + xe ≤ ϵ, x ≥ 0, t 2 R . (2.10) j j j j j 1 n n If h 2 L (R ) and F(·) is a bounded, continuous function, then the function (h F)(·) is well dened on R , bounded and continuous. If we assume, in addition to the all above, that jc j = 1 for all j 2 N , then the estimate (2.10) will be invariant under the action of convolution h ·, since t +x+τ t +x j j − − ω ω j j c (h * F) t + (x + τ)e − r c (h * F) t + xe j j j j j t +x+τ t +x j j − − ω ω j j ≤ jh(y)j c F t + (x + τ)e − y − r c F t + xe − y dy j j j j j t −y +x+τ t −y +x j j j j − − ω ω j j = jh(y)j c F t + (x + τ)e − y − r c F t + xe − y dy j j j j j ≤ ϵ, x ≥ 0, t 2 R , where we have employed (2.10) in the last estimate. We close this section with the observation that a similar result can be established for the (ω , c ; r , I ) -uniform recurrence and the (ω , c ) -almost automor- j j j j2N j j j2N j n n phy. Multi-dimensional (ω, c)-almost periodic type functions. .. Ë 145 3 Further generalizations of (ω, c)-periodicity and (ω , c ) -periodicity j j j2N ′ n ′ n Unless stated otherwise, in this section we will assume that ; ≠ I  I  R , I + I  I, ω 2 R \ f0g and jSj c 2 C \f0g. Dene S := fi 2 N : ω ≠ 0g and A to be the collection of all tuples a = (a , a ,· · ·, a ) 2 R i 1 2 jSj such that a = 1. Let a 2 A. i2S Following our analyses from [16, Section 3], we introduce the next notion: Denition 3.1. We say that a continuous function F : I ! X is: ′ ′ (i) (I , a, ω, c)-uniformly recurrent of type 1, resp. (I , a, ω, c)-uniformly recurrent of type 2, if and only if there exists a sequence (α = (α ,· · ·, α )) in I such that lim jα j = +∞ and k k,1 k,n k!+∞ k P a α i k,i i2S ω lim sup F t + α − c F(t) = 0, (3.1) k!+∞ t2I resp. a α i k,i i2S ω lim sup c F t + α − F(t) = 0; (3.2) k!+∞ t2I ′ ′ (ii) (I , a, ω, c)-almost periodic of type 1, resp. (I , a, ω, c)-almost periodic of type 2, if and only if for each ′ ′ ϵ > 0 and t 2 I there exist a nite number l > 0 and a point τ 2 B(t , l)\ I such that 0 0 a τ i i i2S ω sup F t + τ − c F(t) < ϵ, t2I resp. a τ i i i2S ω sup c F t + τ − F(t) < ϵ. (3.3) t2I ′ ′ If jcj = 1, then the concept (I , a, ω, c)-uniform recurrence of type 1 and the concept (I , a, ω, c)-uniform P a α i k,i i2S recurrence of type 2 coincide, as easily approved by multiplying (3.1) with c ; this also holds for ′ ′ the concepts (I , a, ω, c)-almost periodicity of type 1 and (I , a, ω, c)-almost periodicity of type 2, but then a t i i i2S ω we can say a little bit more. Speaking-matter-of-factly, we can multiply the both sides of (3.3) with c in order to see that F(·) is (I , a, ω, c)-almost periodic of type 2 (1) if and only if the function G (·), dened ′ ′ n through (2.4), is I -almost periodic; in the usually considered case I = I = R , this is equivalent to saying that the function F(·) is almost periodic. The function spaces introduced in Denition 2.16 are translation invariant and closed under the pointwise multiplications with complex scalars; if I = R and F(·) is a bounded, continuous function which belongs to 1 n any of the above introduced function spaces, then for each h 2 L (R ) the function (h * F)(·) is also bounded and belongs to the same space. Furthermore, if the function F(·) is (I , a, ω, c)-uniformly recurrent of type 1, ′ ′ ′ resp. (I , a, ω, c)-uniformly recurrent of type 2 [(I , a, ω, c)-almost periodic of type 1, resp. (I , a, ω, c)-almost ′ ′ periodic of type 2], then F(·) is (I , a, ω,jcj)-uniformly recurrent of type 1, resp. (I , a, ω,jcj)-uniformly recur- ′ ′ rent of type 2 [(I , a, ω,jcj)-almost periodic of type 1, resp. (I , a, ω,jcj)-almost periodic of type 2]. Concern- ing the invariance of function spaces under the operation of uniform convergence, we will only state that the ′ n ′ assumptions a ω > 0 for all j 2 S, jcj ≤ 1, I  [0,∞) and the sequence (F ) of (I , a, ω, c)-uniformly recur- j j k rent functions of type 1 [(I , a, ω, c)-almost periodic functions of type 1] uniformly converges to a function ′ ′ F : I ! X imply that the function F(·) is likewise (I , a, ω, c)-uniformly recurrent of type 1 [(I , a, ω, c)-almost periodic of type 1]. For the sequel, we need the following extension of [16, Lemma 3.4]: 146 Ë M. Kostić ′ n ′ Lemma 3.2. Suppose that ; ≠ I  I  R , I = −I, I + I = I and the function F : I ! X is continuous. ′ ′ Then F(·) is (I , a, ω, c)-uniformly recurrent of type 1 [(I , a, ω, c)-almost periodic of type 1] if and only if F(·) is ′ ′ (I , a, ω, 1/c)-uniformly recurrent of type 2 [(I , a, ω, 1/c)-almost periodic of type 2]. Proof. We will present the main details of the proof provided that F(·) is (I , a, ω, c)-uniformly recurrent of type 1. Then there exists a sequence (α = (α ,···, α )) in I such that lim jα j = +∞ and (3.1) holds. k k,1 k,n k!+∞ k Since we have assumed I = −I and I + I = I, the proof simply follows from the next computation (k 2 N): P a α i k,i i2S ω ˇ ˇ sup (1/c) F t + α − F(t) t2I P a α i k,i i2S ω = sup (1/c) F −t − α − F(−t) t2I P a α i k,i i2S ω = sup c F t − F(t + α ) t2−(I+I ) P a α i k,i i2S ω = sup c F t − F(t + α ) t2−I P a α i k,i i2S ω = sup c F t − F(t + α ) . t2I Concerning [16, Theorem 3.2(i)] and its proof, we will rst state and prove the following result: ′ n ′ ′ n Theorem 3.3. Suppose that; ≠ I  I  R , I is unbounded, F : I ! X is continuous, I + I = I, ω 2 R \f0g, jcj > 1, S = N and any component of a tuple a 2 A is positive. Suppose further that, for every t 2 I and j 2 N , n n we have ω t ≥ 0. Then the following assertions are equivalent: j j (i) The function F(·) is (I , a, ω, c)-uniformly recurrent of type 1. (ii) The function F(·) is (I , a, ω, c)-uniformly recurrent of type 2. (iii) There exists a sequence (α = (α ,···, α )) in I such that lim jα j = +∞ and the function G : I ! k k,1 k,n k!+∞ k X, dened through (2.4), satises G (t + α ) = G (t) for all t 2 I and k 2 N. a a (iv) There exists a sequence (α = (α ,· · ·, α )) in I such that lim jα j = +∞ and k k,1 k,n k!+∞ k a α i k,i i2S ω F t + α = c F(t), t 2 I, k 2 N. (3.4) (v) There exists a point ω 2 I \f0g such that a ω i i i2S ω F t + ω = c F(t), t 2 I. (3.5) Proof. If F(·) is (I , a, ω, c)-uniformly recurrent of type 1, then our assumptions a > 0 and α /ω > 0 (k 2 N, j k,j i P a α P a α n j k,j n j k,j − − j=1 ω j=1 ω j j j 2 N ) imply thatjc j ≤ 1, so that (3.1) implies (3.2) after multiplication with c ; hence, (i) ′ ′ implies (ii). Suppose now that F(·) is (I , a, ω, c)-uniformly recurrent of type 2 and the sequence (α ) in I satises (3.2). Let k 2 N be xed. Then (3.2) implies the existence of a nite real number M ≥ 1 such that − a t /ω j j j j=1 sup G t + α − G (t) ≤ Mjcj . a a t2I Since we have assumed that a > 0 and t /ω > 0 for all j 2 N , the above estimate yields j j j −1 − minfa :j2N g maxfω :j2N g jt j+···+jt j j n j n 1 n G t + α − G (t) ≤ Mjcj , a a for all t 2 I, which implies that lim kG (t + α ) − G (t)k = 0. On the other hand, (3.2) implies a a jtj!∞ k lim G t + α + α − G t + α = 0. a m a m m!+∞ Multi-dimensional (ω, c)-almost periodic type functions. .. Ë 147 Therefore, the function t 7! G (t + α ) − G (t), t 2 I is I -uniformly recurrent and tends to zero asjtj ! +∞. a a Since we have assumed that I + I = I, we may apply [19, Corollary 2.11] in order to see that G (t + α ) = G (t) a a for all t 2 I, which implies (iii). The implications (iii)) (iv), (iv)) (i) and (iv)) (v) are trivial. To complete the proof, it suces to show that (v) implies (iv). This follows by plugging α := kω for all k 2 N since (3.5) implies inductively a kω i i i2S ω F t + kω = c F(t), t 2 I, k 2 N. ′ ′ Remark 3.4. (i) Since I is unbounded, it is clear that the (I , a, ω, c)-almost periodicity of type 1 implies the ′ ′ (I , a, ω, c)-uniform recurrence of type 1 for F(·) as well as that the (I , a, ω, c)-almost periodicity of type 1 ′ ′ implies the (I , a, ω, c)-almost periodicity of type 2 for F(·), which further implies the (I , a, ω, c)-uniform recurrence of type 2 for F(·). (ii) Let (α ) be a sequence from (iv). Then it is clear that (iv) implies that for each number k 2 N the function ′ ′ F(·) is (I , a, ω, c)-almost periodic of type 1, where I := fmα : m 2 Ng. Keeping Theorem 3.3 and this k k observation in mind, we have extended so the rst part of [16, Theorem 3.2(i)], where we have assumed that I = [0,∞). Concerning the statement of [16, Theorem 3.2(i)] with the interval I = R, we would like to note that it can be ′ n ′ extended to the higher dimensions as follows. Suppose that I = I [ I , where; ≠ I  I  R , I + I = I 0 1 0 0 0 0 0 and the function F : I ! X is (I , a, ω, c)-uniformly recurrent of type 2, wherejcj > 1, S = N , any component of a tuple a 2 A is positive and, for every t 2 I and j 2 N , we have ω t ≥ 0. Then the restriction of function 0 j j F(·) to the interval I is (I , a, ω, c)-uniformly recurrent of type 2, as well, so that we can apply Theorem 3.3 in order to conclude that (3.4) holds for every t 2 I and k 2 N. To show the validity of this condition for all t 2 I and k 2 N, we may assume additionally that: (a) For every t 2 I , there exists m 2 N such that, for every m 2 N with m ≥ m , we have t + α 2 I . 1 0 0 0 Applying (3.4) twice, with t + α and t the rst time, and with t + α + α and t + α the second time, we easily m m k k get that (3.4) holds for every t 2 I. Therefore, we have proved the following: ′ n ′ ′ Theorem 3.5. Suppose that ; ≠ I  I  R , I is unbounded, I + I = I , I = I [ I , condition (a) holds 0 0 0 0 1 0 0 0 and F : I ! X is continuous. Suppose that ω 2 R \f0g, jcj > 1, S = N and any component of a tuple a 2 A is positive. Suppose further that, for every t 2 I and j 2 N , we have ω t ≥ 0. Then the following assertions are 0 j j equivalent: (i) The function F(·) is (I , a, ω, c)-uniformly recurrent of type 1. (ii) The function F(·) is (I , a, ω, c)-uniformly recurrent of type 2. (iii) There exists a sequence (α = (α ,···, α )) in I such that lim jα j = +∞ and the function G : I ! k k,1 k,n k!+∞ k X, dened through (2.4), satises G (t + α ) = G (t) for all t 2 I and k 2 N. a a (iv) There exists a sequence (α = (α ,· · ·, α )) in I such that lim jα j = +∞ and (3.4) holds. k k,1 k,n k!+∞ k (v) There exists a point ω 2 I \f0g such that (3.5) holds. Suppose now that jcj < 1, S := N , any component of a tuple a 2 A is positive and, for every t 2 I and j 2 N , we have ω t ≥ 0. Applying Lemma 3.2, we can simply extend the statement of [16, Theorem 3.2(ii)] to j j the higher dimensions, provided that condition (a) holds with I = −I . Details can be left to the interested 1 0 readers. ′ n In the case that a ω > 0 for all j 2 S = N , jcj < 1, I = I = [0,∞) , then it can be simply proved (cf. [16, j j Proposition 3.6, Corollary 3.8]) that the function F : I ! X is (I , a, ω, c)-almost periodic of type 1 if and only if there exists a nite constant M ≥ 1 such that a t /ω i i i i2S kF(t)k ≤ Mjcj , t 2 I; the statement of [16, Proposition 3.11] can be also extended to the higher dimensions provided that the func- tion F(·) is bounded, a ω > 0 for all j 2 S = N andjcj < 1. Without any essential changes of the proof of [16, j j Proposition 3.12], we may deduce the following: 148 Ë M. Kostić ′ n Proposition 3.6. Suppose that a ω > 0 for all j 2 S = N ,jcj < 1 and I = I = [0,∞) . Then a continuous func- j j ′ −a t /ω i i i i2S tion F : I ! X is (I , a, ω, c)-almost periodic of type 2 if and only if the function t 7! G(t)  c F(t), t 2 I is bounded, continuous and satises that for each ϵ > 0, t 2 I and N > 0 there exist a nite number l > 0 and a point τ 2 B(t , l)\ I such that G(t + τ) − G(t) ≤ ϵ, t 2 I . In connection with Proposition 3.6, we want to note that the notion of a complex-valued Levitan N-almost periodic function was introduced by B. M. Levitan in 1937 (see [21]-[22] and references cited therein) and later studied on topological groups by B. Ya. Levin [20] in 1949. Let us recall that a continuous function f : [0,∞) ! X is said to be Levitan N-almost periodic if and only if for each ϵ > 0 and N > 0 the set of all positive reals τ > 0 such thatkf (t + τ)− f (t)k ≤ ϵ, t 2 [0, N ] is relatively dense in [0,∞). The study of vector-valued Levitan N-almost periodic functions on topological (semi-)groups and multi-dimensional vector-valued Levitan N- almost periodic functions will be carried out somewhere else. In our previous research studies of the multi-dimensional almost periodicity, we have also analyzed the invariance of almost periodicity under the actions of the nite convolution products and the innite convo- lution products. In the one-dimensional case, this theme is crucially important for giving the most intriguing applications in the qualitative analysis of almost periodic type solutions for various classes of the abstract Volterra integro-dierential equations. In the multi-dimensional case, the results obtained so far are not so easily applicable and, because of that, we will skip all related details with regards to this question. 4 Applications In this section, we will present several illustrative examples and applications of our results to the abstract Volterra integro-dierential equations in Banach spaces. We feel it is our duty to say that the the points [1,3,4,5] have been also considered in [8] for multi-dimensional almost periodic functions. 1. We start with the observation that all established results on the convolution invariance of introduced function spaces can be applied to the Gaussian semigroup jyj −(n/2) − n 4t (G(t)F)(x) := 4πt F(x − y)e dy, t > 0, f 2 Y , x 2 R ; see [5, Example 3.7.6] for more details. Suppose, for example, that I = R ,jcj = 1 and F(·) is a bounded, (ω, c)- periodic function. Then, due to Proposition 2.14, we have that for each t > 0 the functionR 3 x 7! u(x, t ) 0 0 (G(t )F)(x) 2 C is likewise bounded and (ω, c)-periodic. A similar result can be given for the Poisson semi- group; see [5, Example 3.7.9] and [8] for more details. Concerning the strongly continuous semigroups, we would like to note that our recent consideration from [19, Example 1.1] can be used to justify the intoduction of analyzed function spaces, as well. 2. In [19, Example 1.2], we have recently observed an interesting feature of the famous d’Alembert formula, which has been used by S. Zaidman [25, Example 5] for almost periodic functions of one real variable, a long time ago (see also [14, Example 2.3]). Suppose that a > 0. Then it is well known that the regular solution of the wave equation u = a u in domain f(x, t) : x 2 R, t > 0g, equipped with the initial conditions tt xx 2 1 u(x, 0) = f (x) 2 C (R) and u (x, 0) = g(x) 2 C (R), is given by the d’Alembert formula x+at 1 1 u(x, t) = f (x − at) + f (x + at) + g(s) ds, x 2 R, t > 0. 2 2a x−at [1] In the above-mentioned example, if the function x 7! (f (x), g (x)), x 2 R is c-almost periodic, where [1] g (·)  g(s) ds; then the conclusion is: the solution u(x, t) can be extended to the whole real line in the time variable and this solution is c-almost periodic in (x, t) 2 R . Multi-dimensional (ω, c)-almost periodic type functions. .. Ë 149 2.1. We assume here that there exist numbers ω 2 R \ f0g and c 2 C \ f0g such that the function x 7! [1] (f (x), g (x)), x 2 R is (ω, c)-periodic. Then it is clear that the solution u(x, t) can be extended to the whole real line in the time variable and now we will prove that u x + ω, t = cu(x, t), x, t 2 R, i.e., the function u(·;·) is ((ω, 0), c)-periodic. But, the last equality simply follows from the next calculation: h i u x + ω, t = f x − at + ω + f x + at + ω h i [1] [1] + g x + at + ω − g x − at + ω 2a h i = cf x − at + cf x + at h i [1] [1] + g x + at − g x − at = cu(x, t), x, t 2 R. 2a k−1 2.2. We assume here that there exist numbers ω 2 R \f0g, k 2 N and c 2 C \f0g such that c = 1 and [1] the function x 7! (f (x), g (x)), x 2 R is (ω, c)-periodic. Set 1 + k k − 1 ω := ω and ω := ω. 1 2 2 2a k k Then (ω , ω ) ≠ (0, 0), ω − aω = ω, ω + aω = kω, c = c, f (x + ω) = cf (x) = c f (x) = f (x + kω), 1 2 1 2 1 2 [1] [1] k [1] [1] g (x + ω) = cg (x) = c g (x) = g (x + kω) for all x 2 R, and we can simply show as above that u x + ω , t + ω = cu(x, t), x, t 2 R, 1 2 i.e., the function u(·;·) is ((ω , ω ), c)-periodic. 1 2 [1] 2.3. Let the assumptions of the previous point hold. Assume, further, that the function x 7! (f (x), g (x)), [1] 2 2 x 2 R satises lim f (x) = lim g (x) = 0. Set B := f(x, t) 2 R : x = ±atg. If D is any subset of R x!±∞ 0 x!±∞ satisfying that lim dist((x, t); B) = +∞, j(x,t)j!+∞,(x,t)2D then the solution given by the d’Alembert formula, with the functions f (·) and g(·) replaced therein with the functions (f + f )(·) and (g + g )(·), is D-asymptotically ((ω , ω ), c)-periodic. 0 0 1 2 n n 3. Let ω 2 R \f0g andjcj = 1. Equipped with the sup-norm, the space B (R : X) consisting of all X- ω,c valued, bounded and (ω, c)-periodic functions becomes a Banach space. In a series of our previous research studies, we have analyzed the following Hammerstein integral equation of convolution type on R : y(t) = g(t) + k(t − s)G(s, y(s)) ds, t 2 R . (4.1) n 1 n n Suppose now that g : R ! X is bounded and (ω, c)-periodic, k 2 L (R ), G : R × X ! X is continuous and satises that for each bounded subset of X we have that the set fG(t, x) : t 2 R , x 2 Bg is bounded as well as that G(t + ω, x) = cG(t, x) for all t 2 R and x 2 X. If there exists a nite real constant L ≥ 1 such that G(t, x) − G t, y ≤ L x − y , t 2 R ; x, y 2 X and L jk(y)j dy < 1, then we can apply the Banach contraction principle and Proposition 2.14 in order to see that there exists a unique solution of the integral equation (4.1) which belongs to the space B (R : X). ω,c 4. Of concern is the system of abstract partial dierential equations u (s, t) = Au(s, t) + f (s, t), u (s, t) = Bu(s, t) + f (s, t); u(0, 0) = x, s ≥ 0, t ≥ 0. (4.2) s 1 t 2 Motivated by the recent results of S. M. A. Alsulami given in [1, Section 2.1], we have recently considered, in [8], the case in which A and B are two complex matrices of format n × n, AB = BA, and A, resp. B, generate an exponentially decaying, strongly continuous semigroup (T (s)) , resp. (T (t)) . The follow- 1 s≥0 2 t≥0 ing assumptions have been made there: the functions f (s, t) and f (s, t) are continuously dierentiable, 1 2 the compatibility condition (f ) − Af = (f ) − Bf holds (s, t ≥ 0), D := f(s, t) 2 [0,∞) : c s ≤ t ≤ 2 2 1 t 1 1 c s for some positive real numbers c and c g, and 2 1 2 150 Ë M. Kostić (i) There is a nite real constant M > 0 such thatjf (v, 0)j +jf (0, ω)j ≤ M, for v, ω ≥ 0; 1 2 2 n (ii) The mappings g : R ! C are continuous, bounded (i = 1, 2) and satisfy that, for every ϵ > 0, there exists l > 0 such that any subinterval I of R of length l > 0 contains a number τ 2 I such that, for every s, t ≥ 0, we havejg (s + τ, t) − g (s, t)j ≤ ϵ andjg (s, t + τ) − g (s, t)j ≤ ϵ; 1 1 2 2 2 n 2 n (iii) The function q : [0,∞) ! C is bounded, q 2 C ([0,∞) : C ) and f (s, t) = g (s, t) + q (s, t) for i i 0,D i i i (s, t) 2 [0,∞) and i = 1, 2. The conclusion is: there is a unique classical solution u(s, t) of (4.2) (cf. also [1, Denition 2.13]), and more- 2 2 n over, there exist a continuous function u (s, t) on [0,∞) and a function u 2 C ([0,∞) : C ) such that ap 0 0,D u(s, t) = u (s, t) + u (s, t) for all (s, t) 2 [0,∞) , as well as for every ϵ > 0, there exists l > 0 such that ap any subinterval I of [0,∞) of length l > 0 contains a number τ 2 I such that, for every s, t ≥ 0, we have ju (s + τ, t) − u (s, t)j ≤ ϵ andju (s, t + τ) − u (s, t)j ≤ ϵ. ap ap ap ap If we replace condition (ii) with condition: 2 n (ii)’ The mappings g : R ! C are continuous, bounded (i = 1, 2) and satisfy that there exist positive real numbers ω > 0 and ω > 0 as well as complex numbers c and c such thatjc j = jc j = 1 and, for every 1 2 1 2 1 2 s, t 2 R, we have g (s + ω , t) = c g (s, t) and g (s, t + ω ) = c g (s, t), 1 1 1 1 2 2 2 2 and accept all remaining assumptions, then we similarly may deduce that there exist a continuous function 2 2 n u (s, t) on [0,∞) and a function u 2 C ([0,∞) : C ) such that u(s, t) = u (s, t) + u (s, t) for all (s, t) 2 h 0 0,D h 0 [0,∞) , as well as that, for every s, t ≥ 0, we have u (s + ω , t) = c u (s, t) and u (s, t + ω ) = c u (s, t). h 1 1 h h 2 2 h 5. Finally, it is worth recalling that the existence and uniqueness of almost periodic solutions for a class of boundary value problems for hyperbolic equations have been investigated by B. I. Ptashnic and P. I. Shta- balyuk in [24]. In the region D = (0, T )× R (T > 0, p 2 N), these authors have analyzed the almost periodic type solutions of the following initial value problem: 2n X X ∂ u(t, x) Lu  a = 0, (4.3) α p 2n−2s ∂t ∂x · · · ∂x s=0 jαj=2s j−1 j−1 ∂ u ∂ u = φ (x), = φ (x) (1 ≤ j ≤ n). (4.4) j j+n j−1 j−1 ∂t ∂t t=0 t=T p p Suppose that any of the functions φ (x),· · ·, φ (x) is almost periodic in R and M = fμ : k 2 Z g is the 1 2n p union of all Bohr-Fourier spectrum of functions φ (x),···, φ (x). Under certain assumptions, the solutions 1 2n u(t, x) of problem (4.3)-(4.4) have been found in the form ihμ ,xi u(t, x) = u (t)e , μ 2 M , (4.5) k k k2Z where the functions u (t) have the form [24, (8), p. 670] (cf. [24, Theorem 1, Theorem 2] for more details con- cerning the existence and uniqueness of solutions to (4.3)-(4.4)). p n Suppose now that ω 2 R \f0g and C 2 R. We want to observe here that the assumption φ 2 AP (R : C) j Λ for all j 2 N , where 2n Λ := (x ,· · ·, x ) 2 R ; x ω + · · · + x ω = C , 1 p 1 1 p p iC implies that the solution u(t, x) of problem (4.3)-(4.4) is (ω, e )-periodic in the space variable x. This follows from the computation (t 2 (0, T ), x 2 R ): X X ihμ ,x+ωi ihμ ,xi ihμ ,ωi k k k u(t, x + ω) = u (t)e = u (t)e e k k p p k2Z k2Z iC ihμ ,xi = e u (t)e = cu(t, x). k2Z Acknowledgement: The author is partially supported by grant 451-03-68/2020/14/200156 of Ministry of Science and Technological Development, Republic of Serbia. Multi-dimensional (ω, c)-almost periodic type functions.. . Ë 151 Conict of interest: The author states that there is no conict of interest. Data Availability Statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study. References [1] S. M. A. Alsulami, On Evolution Equations In Banach Spaces And Commuting Semigroups. PhD. Thesis, Ohio University, [2] E. Alvarez, A. Gómez, M. Pinto, (ω, c)-Periodic functions and mild solution to abstract fractional integro-dierential equa- tions, Electron. J. Qual. Theory Dier. Equ. 16 (2018), 1–8. [3] E. Alvarez, S. Castillo, M. Pinto, (ω, c)-Pseudo periodic functioins, rst order Cauchy problem and Lasota-Wazewska model with ergodic and unbounded oscillating production of red cells, Bound. Value Probl. 106 (2019), 1–20. [4] E. Alvarez, S. Castillo, M. Pinto, (ω, c)-Asymptotically periodic functions, rst-order Cauchy problem, and Lasota-Wazewska model with unbounded oscillating production of red cells, Math. Methods Appl. Sci. 43 (2020), 305–319. [5] W. Arendt, C. J. K. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Birkhäuser/Springer Basel AG, Basel, 2001. [6] A. S. Besicovitch, Almost Periodic Functions, Dover Publ, New York, 1954. [7] H. 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Journal

Nonautonomous and Stochastic Dynamical Systems – de Gruyter

Published: Jan 1, 2021

Keywords: Multi-dimensional ( ω,c )-almost periodic functions; multi-dimensional ( ωj,cj ) j ∈ℕ n -periodic functions; multi-dimensional c -almost periodic functions; ( I ′ , a , ω , c )-almost periodic type functions; abstract Volterra integro-differential equations; 42A75; 43A60; 47D99

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