${\mathcal{R}}$-bounded in ${\mathcal{L}}(X)$; see e.g. the proof of [38,
Lemma 3.2]. Finally, one can easily check that the condition $(ii)$ of [38,
Theorem 3.4] implies the assumptions of Corollary 7.1(2) for $\alpha=2$ (with
$P=I$).

Analogously, one can check that Theorem 7.1(a) extends [38, Theorems 4.4 and
4.7]; see also references therein, as well corresponding results in [48, 23,
24, 54, 40].

To illustrate Theorem 6.12, we remark on the integro-differential equations
with infinite delays, which have been studied in [47]. Namely, in the problem
studied in [47] (i.e., (AEE) with $P=0$ and $B=I$) the infinite delay operator
is given by

$c\ast u:=d(\cdot)A\ast u,$

where $d\in L^{1}_{loc}(\mathbb{R}_{+})\subset L^{1}_{loc}(\mathbb{R})$ is
such that the Fourier transform of $d$ exists at points $k\in\mathbb{Z}$ and
the sequence $({\mathcal{F}}d(k))_{k\in\mathbb{Z}}$ satisfies some further
assumptions corresponding to a $\gamma$-regularity ($\gamma=1,2)$; see [47,
Theorems 2.12 and 3.9]). It is readily seen that, these assumptions imply
$\hat{c}\in\mathfrak{M}^{\gamma}_{\mathcal{R}}(\mathbb{Z};{\mathcal{L}}(Z,X))$
with $Z:=D_{A}$ ($\gamma=1,2$). Therefore, Theorem 6.12 might also be seen as
an extension of those results.

###  7.2. Differential equations

We conclude with some remarks on (AEE) with $c\equiv 0$, that is,

(28) $\partial{\mathcal{P}}\partial u+{\mathcal{B}}\partial
u+{\mathcal{A}}u=f\qquad(\textrm{in }{\mathcal{D}}^{\prime}(X)).$

The particular form of (28) with ${\mathcal{P}}=0$ and ${\mathcal{B}}=I$ was,
in a sense, a prototype for further periodic extensions done in the
literature; see Arendt and Bu [7, Theorem 2.3 and Corollary 2.4] for the
corresponding result on the $L^{p}$-well-posedness and its $p$-independence.
For the studies of the other forms of (28) in connection to their well-
posedness in the context of the classical Lebesgue-Bochner, Besov, and
Triebel-Lizorkin spaces (i.e., corresponding to $\Phi=L^{p}$) see for instance
[7, Section 2], [8, 55, 19], as well as the references provided for the
integro-differential equations.

Theorem 6.9 (for $c\equiv 0$), in particular, gives an extension of those
results and also provides periodic variant of extrapolation results known in
the euclidean setting; see, e.g. [60, 12, 27, 29, 37, 51] and references
therein. We leave the formulation of this result to the interested reader.
Finally, in articles mentioned in this section the reader can find several
concrete models of equations arising in the applied mathematics. Such
equations with periodic boundary conditions have been studied therein in the
context of the well-posedness with respect to the classical Lebesgue-Bochner
$L^{p}$, Besov $B^{s,q}_{p}$, and Triebel-Lizorkin $F^{s,q}_{p}$ spaces. The
same equations could serve as an illustration of our abstract results,
Theorems 6.7, 6.9, and 6.12.

Declaration of interest: none.

Data availability: no data was used for the research described in the article.

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