In this paper, we analyze multi-dimensional (R X , B)-almost periodic type functions and multi-dimensional Bohr B-almost periodic type functions. The main structural characterizations and composition principles for the introduced classes of almost periodic functions are established. Several applications of our abstract theoretical results to the abstract Volterra integrodifferential equations in Banach spaces are provided, as well.
Examples and applications to the abstractVolterra integro-differential equations 3.1. Application to nonautonomous retarded functional evolution equations 4. Appendix 4.1. n-Parameter strongly continuous semigroups 4.2. Multivariate trigonometric polynomials and approximations of periodic functions of several real variables References 2010 Mathematics Subject Classification. 42A75, 43A60, 47D99. Key words and phrases. (R, B)-Multi-almost periodic type functions, (R X , B)-multi-almost periodic type functions, Bohr B-almost periodic type functions, composition principles, abstract Volterra integro-differential equations. Marko Kostić is partially supported by grant 451-03-68/2020/14/200156 of Ministry of Science and Technological Development, Republic of Serbia. Manuel Pinto is partially supported by Fondecyt 1170466.The Euler Gamma function is denoted by Γ(•). If t 0 ∈ R n and ǫ > 0, then we set B(t 0 , ǫ)Now we are ready to briefly explain the organization and main ideas of this paper. In Subsection 1.1, we recall the basic facts and definitions about vectorvalued almost periodic functions of several real variables; in Subsection 1.2, we recall some applications of vector-valued almost periodic functions of several real variables made so far. Definition 2.1 and Definition 2.2 introduce the notion of (R, B)-multi-almost periodicity and the notion of (R X , B)-multi-almost periodicity for a continuous function F : I × X → Y, respectively. The convolution invariance of space consisting of all (R X , B)-multi-almost periodic functions is stated in Proposition 2.5, while the supremum formula for the class of (R, B)-multi-almost periodic functions is stated in Proposition 2.6.The notion of Bohr B-almost periodicity and the notion of B-uniform recurrence for a continuous function F : I × X → Y are introduced in Definition 2.9, provided that the region I satisfies the semigroup property I + I ⊆ I. Numerous illustrative examples of Bohr B-almost periodic functions and B-uniformly recurrent functions are presented in Example 2.12 and Example 2.13. In Definition 2.14, we introduce the notion of Bohr (B, I ′ )-almost periodicity and (B, I ′ )-uniform recurrence, provided that ∅ = I ′ ⊆ I ⊆ R n , F : I × X → Y is a continuous function and I + I ′ ⊆ I. After that, we provide several examples of Bohr (B, I ′ )-almost periodic functions and (B, I ′ )-uniformly recurrent functions in Example 2.15. The relative compactness of range F (I × B) for a Bohr B-almost periodic function F : I × X → Y,where B ∈ B, is analyzed in Proposition 2.16. The Bochner criterion for Bohr B-almost periodic functions is sta...
