Multi-dimensional ρ-almost periodic type functions and applications

Asymptotically ρ-almost periodic type functions in general metric

“…It is worth noting that we do not assume the continuity of a function F (•; •) here. The notion of D-quasi-asymptotical Bohr (B, I , ρ)-almost periodicity and the notion of D-quasi-asymptotical (B, I , ρ)-uniform recurrence, introduced and analyzed in [16], are obtained by plugging that P τ,M = L ∞ (I τ,M : Y ) for all τ ∈ I and M > 0; similarly, the notion of D-asymptotical Bohr (B, I , ρ)-almost periodicity of type 1 and the notion of D-asymptotical (B, I , ρ)-uniform recurrence of type 1, introduced and analyzed in [6], are obtained in the same way; see also [ Remark 4.1. The notion introduced in the former part of this paper can be understood in a more general setting.…”

“…1. It is clear that Proposition 2.2 can be applied in the analysis of inhomogeneous heat equation in R n , and to the Poisson semigroup in R n ; let us only note here that the use of multivalued binary relations is inevitable here and that a fairly complete analysis of this problematic cannot be obtained by assuming that ρ is a function, only ( [6,16]).…”

“…For example, suppose that (F k (•; •) : I × X → Y ) is a sequence of functions and there exists a function F : I × X → Y such that lim k→+∞ F k (t; x) = F (t; x), uniformly on I × B for each set B of collection B. Concerning the binary relation ρ on Y , we assume that D(ρ) is closed, ρ is single-valued on R(F ) and continuous on D(ρ) in the usual sense ( [6]); we assume the same conditions for the sequence (ρ j ) j∈Nn of binary relations on Y, if considered. Then we have the following:…”

“…In M. Fečkan et al [6], we have initiated the study of multi-dimensional ρ-almost periodic type functions. In our previous research studies, we have also analyzed the Stepanov and Weyl classes of multi-dimensional ρ-almost periodic type functions ( [16]).…”

“…For more details about binary relations, we refer the reader to the classical university textbooks; see also [6]. We will always assume henceforth that (X, • ) and (Y, • Y ) are complex Banach spaces, n ∈ N, ∅ = Λ ⊆ R n , B is a non-empty collection of non-empty subsets of X satisfying that for each x ∈ X there exists B ∈ B such that x ∈ B.…”