††2010 Mathematics Subject Classification. 42A75, 43A60, 47D99.
Key words and phrases. $({\mathrm{R}},{\mathcal{B}})$-Multidimensional-almost
automorphic functions, asymptotically $({\mathrm{R}},{\mathcal{B}})$-multi-
almost automorphic functions, abstract Volterra integrodifferential 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.

#  $({\mathrm{R}},{\mathcal{B}})$-Multidimensional-almost automorphic
functions with applications to integral and partial differential equations

A. Chávez  Departamento de Matemáticas, Facultad de Ciencias Físicas Y
Matemáticas, Universidad Nacional de Trujillo, Trujillo, Perú
<EMAIL_ADDRESS>,  K. Khalil  Faculty of Sciences Semlalia, Cadi Ayyad
University, B.P. 2390, 40000 Marrakesh, Morocco<EMAIL_ADDRESS>,
M. Kostić  Faculty of Technical Sciences, University of Novi Sad, Trg D.
Obradovića 6, 21125 Novi Sad, Serbia<EMAIL_ADDRESS>and  M. Pinto
Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile,
Santiago de Chile, Chile<EMAIL_ADDRESS>
###### Abstract.

In this paper, we analyze the classes of $({\mathrm{R}},{\mathcal{B}})$-multi-
almost automorphic functions and asymptotically
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic functions. We provide
plenty valuable applications to the abstract Volterra integrodifferential
equations in Banach spaces and partial differential equations.

##  1\. Introduction and preliminaries

S. Bochner introduced the concept of almost automorphy in the literature ([6,
7]) as a generalization of the almost periodicity. He discovered the class of
almost automorphic functions when he was studying problems related to
differential geometry ([8]), and also he used the almost automorphy of
functions to prove some results about almost periodic functions. Afterward,
Veech [42, 43] and many other researchers deeply investigated this concept. In
connection with the study of the asymptotic behavior of differential
equations, the concept of almost automorphy has been intensively appeared. In
fact, it is well known that some almost periodic systems do not carry
necessarily almost periodic dynamics ([28, 35, 39]), whereas, these systems
may have bounded oscillating solutions, these oscillations belong to a boarder
class than the class of almost periodic functions, namely almost automorphic
functions.

It is well known that, (usually mild) solutions to nonautonomous evolution
differential equations satisfy a concrete integral equation in which the
integral kernels are expressed using two parameter evolution families
$\\{U(t,s)\\}_{t\geq s\geq 0}$, see [36]. Thanks to the bi-almost automorhy of
the evolution operator $\\{U(t,s)\\}_{t\geq s\geq 0}$ we may prove the
existence of almost automorphic solutions, see [9, 10, 11, 17, 46] for more
details.

For $X$ a Banach space, roughly speaking, a continuous function
$f:\mathbb{R}\times\mathbb{R}\to X$ is bi-almost automorphic if it behaves
like an almost automorphic function, but in which the sequences
$\\{s_{n}\\}\in\mathbb{R}^{2}$ (translations) are considered in the set
$\mathrm{R}=\\{(w,w)\,:\,w\in\mathbb{R}\\}$ and not in the full domain
$\mathbb{R}^{2}$.

The notion of a (positively) bi-almost automorphic function was introduced by
T. J. Xiao et al. in [46, (2009)]. Three years later, Z. Chen and W. Lin
employed this notion in their investigation of nonautonomous stochastic
evolution equations [17], see also [9]-[10] and [19, Appendix A.3], where the
authors analyzed the notion of bi-almost automorphic sequences. Besides the
above-mentioned papers, we would like to mention a recent research study [11]
by A. Chávez, M. Pinto and U. Zavaleta, where the authors systematically
analyzed the notion of bi-almost automorphy in the study of abstract nonlinear
integral equations that are simultaneously of advanced and delayed type, as
well as the research studies [14] by Y.-K. Chang, S. Zheng, [27] by Z. Hu, Z.
Jin, [44] by Z. Xia and [45] by Z. Xia, D. Wang.

Now we would like to mention some works in which bi-almost automorphic
functions had appeared

  * •

In [46], the authors introduced the notion of a continuous bi-almost
automorphic function and obtained conditions for studying pseudo almost
automorphic mild solutions of the following equations in $\mathbb{R}$ :

$\displaystyle x^{\prime}(t)$ $\displaystyle=$ $\displaystyle
A(t)x(t)+f(t,x(t))$ $\displaystyle x^{\prime}(t)$ $\displaystyle=$
$\displaystyle A(t)x(t)+f(t,x(t-h))$ $\displaystyle x^{\prime}(t)$
$\displaystyle=$ $\displaystyle A(t)x(t)+f(t,x\left(\alpha(t,x(t))\right)\,.$

  * •

In [17], the authors introduced the notion of square-mean bi-almost
automorphic functions for a stochastic processes and analyze the existence of
square-mean almost automorphic solutions of the following non-autonomous
linear stochastic evolution equation :

$\displaystyle dx(t)=A(t)x(t)dt+f(t)dt+\gamma(t)dW(t),$

for $f,\gamma$ stochastic process and $W$ a two-sided standard one-dimensional
Brownian motion.

  * •

The notion of a discrete bi-almost automorphic function can be found in [9,
10], where the authors have been used it in the study of the non-autonomous
difference equation that appears in the study of the following hybrid system
of a differential equation in finite dimensional Banach spaces:

$\displaystyle x^{\prime}(t)=A(t)x(t)+B(t)x([t])+f(t,x(t),x([t])).$

  * •

We also mention that, in [11] the authors have used the notion of bi-almost
automorphy and the one of $\lambda$-boundedness in order to study the
following nonlinear abstract integral equations of advanced and delayed type:

$\displaystyle y(t)$ $\displaystyle=$ $\displaystyle
f(t,y(t),y(a_{0}(t)))+\int_{-\infty}^{t}C_{1}(t,s,y(s),y(a_{1}(s)))ds$
$\displaystyle+\int_{t}^{+\infty}C_{2}(t,s,y(s),y(a_{2}(s)))ds\;.$

Observing the previous works (and references cited therein), we emphasize that
the notion of bi-almost automorphic function is crucial in the study of the
almost automorphic dynamics in differential, integrodifferential and
difference equations.

One other important notion that brings light to our present work is that of
$\mathbb{Z}$-almost automorphic functions (see [9, 10]). This notion, roughly
speaking, says that a function (not necessarily continuous) $f:\mathbb{R}\to
X$ (defined on $\mathbb{R}$) is $\mathbb{Z}$-almost automorphic if it behaves
like an almost automorphic function but only with integer sequences (a subset
of $\mathbb{R}$) translations in the variable $t\in\mathbb{R}$, but not with
sequences of the full domain $\mathbb{R}$ (see Example 2.6 below).

The main objectives in this work are twofold: the first one is to develop the
basic theory of $({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic
functions, and the second one is provide concrete applications; in fact we
apply the results to multi-dimensional integral equations of Volterra type,
integrodifferential equations and partial differential equations such as the
classical heat equation and Poisson equation. In the last application (of
Poisson equation) we introduce a new generalization of Sibuya’s result for
almost periodic functions [40] using the uniform continuity of the solution,
our result is new even for the classical almost automorphic functions.

The organization and main ideas of this paper can be briefly described as
follows. The main purpose of paper is to investigate
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic functions and
asymptotically $({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic
functions. We will not consider here the notion of a positively
$({\mathrm{R}},{\mathcal{B}})$-multi-almost periodicity and their
generalizations ([46]).

We assume heneceforth that $(X,\|\cdot\|)$, $(Y,\|\cdot\|_{Y})$ and
$(Z,\|\cdot\|_{Z})$ are complex Banach space. By $L(X,Y)$ we denote the Banach
algebra of all bounded linear operators from $X$ into $Y$ with $L(X,X)$ being
denoted $L(X)$. If $A:D(A)\subseteq X\mapsto X$ is a closed linear operator,
then its nullspace (or kernel) and range will be denoted respectively by
$N(A)$ and $R(A)$. By $[D(A)]$ we denote the Banach space
$(D(A),\|\cdot\|_{[D(A)]}),$ where $\|\cdot\|_{[D(A)]}$ is the graph norm
defined by $\|x\|_{[D(A)]}:=\|x\|+\|Ax\|$ for all $x\in D(A).$

For given real numbers $s\in{\mathbb{R}}$ and $\theta\in(0,\pi]$, we define
$\lceil s\rceil:=\inf\\{l\in{\mathbb{Z}}:s\leq l\\}$ and
$\Sigma_{\theta}:=\\{z\in{\mathbb{C}}\setminus\\{0\\}:|\arg(z)|<\theta\\}.$
The Euler Gamma function is denoted by $\Gamma(\cdot)$. We also set
$g_{\zeta}(t):=t^{\zeta-1}/\Gamma(\zeta),$ $\zeta>0.$ The convolution operator
$\ast$ is defined by $f\ast g(t):=\int_{0}^{t}f(t-s)g(s)\,ds.$

The symbol $C(I:X),$ where $I={\mathbb{R}}$ or $I=[0,\infty),$ stands for the
space of all $X$-valued continuous functions on the interval $I$. By
$C_{b}(I:X)$ (respectively, $BUC(I:X)$) we denote the subspaces of $C(I:X)$
consisting of all bounded (respectively, all bounded uniformly continuous
functions). Both $C_{b}(I:X)$ and $BUC(I:X)$ are Banach spaces with the sup-
norm. This also holds for the space $C_{0}(I:X)$ consisting of all continuous
functions $f:I\rightarrow X$ such that $\lim_{|t|\rightarrow+\infty}f(t)=0.$

Suppose that $f:{\mathbb{R}}\rightarrow X$ is continuous. Then it is said that
$f(\cdot)$ is almost automorphic if and only if for every real sequence
$(b_{n})$ there exist a subsequence $(a_{n})$ of $(b_{n})$ and a map
$g:{\mathbb{R}}\rightarrow X$ such that

(1.1) $\displaystyle\lim_{n\rightarrow\infty}f\bigl{(}t+a_{n}\bigr{)}=g(t)\
\mbox{ and }\ \lim_{n\rightarrow\infty}g\bigl{(}t-a_{n}\bigr{)}=f(t),$

pointwise for $t\in{\mathbb{R}}.$ If this is the case, then $f\in
C_{b}({\mathbb{R}}:X)$ and the limit function $g(\cdot)$ is bounded on
${\mathbb{R}}$ but not necessarily continuous on ${\mathbb{R}}.$ Furthermore,
if the convergence of limits appearing in (1.1) is uniform on compact subsets
of ${\mathbb{R}},$ then it is said that $f(\cdot)$ is compactly almost
automorphic. Recall that an almost automorphic function $f(\cdot)$ is
compactly almost automorphic if and only if it is uniformly continuous [21,
Lemma 3.7]. Concerning two-parameter almost automorphic functions, we want
only to recall that the authors of [19] and [29] have used the following
notion: A jointly continuous function $F:{\mathbb{R}}\times X\rightarrow Y$ is
said to be almost automorphic if and only if for every sequence of real
numbers $(s_{n}^{\prime})$ there exists a subsequence $(s_{n})$ such that

$G(t;x):=\lim_{n\rightarrow\infty}F\bigl{(}t+s_{n};x\bigr{)}$

is well defined for each $t\in{\mathbb{R}}$ and $x\in X,$ and

$\lim_{n\rightarrow\infty}G\bigl{(}t-s_{n};x\bigr{)}=F(t;x)$

for each $t\in{\mathbb{R}}$ and $x\in X.$ This definition can be modified in
the following sense which will be important in our consideration of case [L2]
below (a general notion with a non-empty collection of subsets ${\mathcal{B}}$
of certain subsets of $X$ can be also introduced): A jointly continuous
function $F:{\mathbb{R}}\times X\rightarrow Y$ is said to be almost
automorphic, uniformly on bounded subsets of $X$ if and only if for every
bounded subset $B$ of $X$ and for every sequence of real numbers
$(s_{n}^{\prime}),$ there exists a subsequence $(s_{n})$ such that

$G(t;x):=\lim_{n\rightarrow\infty}F\bigl{(}t+s_{n};x\bigr{)}$

is well defined for each $t\in{\mathbb{R}}$ and $x\in X,$ uniformly for $x\in
B$ and

$\lim_{n\rightarrow\infty}G\bigl{(}t-s_{n};x\bigr{)}=F(t;x)$

for each $t\in{\mathbb{R}}$ and $x\in X,$ uniformly for $x\in B.$

If the above limits converge uniformly on compact intervals of $\mathbb{R}$,
then we say that $F$ is compactly almost automorphic uniformly on bounded sets
of $X$.

##  2\. $({\mathrm{R}},{\mathcal{B}})$-Multi-almost automorphic type functions

Throughout this paper, we assume that $n\in{\mathbb{N}},$ ${\mathcal{B}}$ is a
non-empty collection of subsets of $X$ and ${\mathrm{R}}$ is a non-empty
collection of sequences in ${\mathbb{R}}^{n};$ usually, ${\mathcal{B}}$
denotes the collection of all bounded subsets of $X$ or all compact subsets of
$X.$ Henceforth we will always assume that there exist $x\in X$ and
$B\in{\mathcal{B}}$ such that $x\in B.$

In this section, we investigate $({\mathrm{R}},{\mathcal{B}})$-multi-almost
automorphic functions and asymptotically $({\mathrm{R}},{\mathcal{B}})$-multi-
almost automorphic functions.

######  Definition 2.1.

Suppose that $F:{\mathbb{R}}^{n}\times X\rightarrow Y$ is a continuous
function. Then we say that the function $F(\cdot;\cdot)$ is
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic if and only if for
every $B\in{\mathcal{B}}$ and for every sequence $({\bf
b}_{k}=(b_{k}^{1},b_{k}^{2},\cdot\cdot\cdot,b_{k}^{n}))\in{\mathrm{R}}$ there
exist a subsequence $({\bf
b}_{k_{l}}=(b_{k_{l}}^{1},b_{k_{l}}^{2},\cdot\cdot\cdot,b_{k_{l}}^{n}))$ of
$({\bf b}_{k})$ and a function $F^{\ast}:{\mathbb{R}}^{n}\times X\rightarrow
Y$ such that

(2.1) $\displaystyle\lim_{l\rightarrow+\infty}F\bigl{(}{\bf
t}+(b_{k_{l}}^{1},\cdot\cdot\cdot,b_{k_{l}}^{n});x\bigr{)}=F^{\ast}({\bf
t};x)$

and

(2.2) $\displaystyle\lim_{l\rightarrow+\infty}F^{\ast}\bigl{(}{\bf
t}-(b_{k_{l}}^{1},\cdot\cdot\cdot,b_{k_{l}}^{n});x\bigr{)}=F({\bf t};x),$

pointwise for all $x\in B$ and ${\bf t}\in{\mathbb{R}}^{n}.$ If the above
limits converge uniformly on compact subsets of ${\mathbb{R}}^{n}$, then we
say that $F(\cdot;\cdot)$ is compactly $({\mathrm{R}},{\mathcal{B}})$-multi-
almost automorphic. By
$AA_{({\mathrm{R}},{\mathcal{B}})}({\mathbb{R}}^{n}\times X:Y)$ and
$AA_{({\mathrm{R}},{\mathcal{B}},{\bf c})}({\mathbb{R}}^{n}\times X:Y)$ we
denote the spaces consisting of all $({\mathrm{R}},{\mathcal{B}})$-multi-
almost automorphic functions and compactly
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic functions,
respectively.

Notice that, in particular, if the function $F({\bf t};x)=F({\bf t})$ (i.e.,
$F:{\mathbb{R}}^{n}\rightarrow Y$), then we say that $F$ is
${\mathrm{R}}$-multi-almost automorphic and that space is denoted by
$AA_{{\mathrm{R}}}({\mathbb{R}}^{n}:Y)$.

######  Remark 2.2.

The following special cases are very important:

  * L1.

Let ${\mathrm{R}}:=\\{b:{\mathbb{N}}\rightarrow{\mathbb{R}}^{n}\,;\,\mbox{ for
all }j\in{\mathbb{N}}\mbox{ we have
}b_{j}\in\\{(a,a,a,\cdot\cdot\cdot,a)\in{\mathbb{R}}^{n}:a\in{\mathbb{R}}\\}\\}.$
In the case where $n=2$ and ${\mathcal{B}}$ is the collection of all bounded
subsets of $X,$ we say that the function $F(\cdot;\cdot)$ is bi-almost
automorphic. The concept of bi-almost automorphy was introduced in literature
in [11, Definition 2.7] to study the asymptotic behavior of nonautonomous
evolution equations in Banach spaces, see also [5, 9] and references therein.
The research study [9] by A. Chávez, S. Castillo and M. Pinto where the
authors have used the notion of bi-almost automorphy of the Green functions in
their investigation of almost automorphic solutions of abstract differential
equations with piecewise constant arguments. Here, the pivot space is denoted
in general by $X$ which equals to the pivot product space in [9]. Furthermore,
in [5], the authors proved the existence and uniqueness of $\mu$-pseudo almost
automorphic solutions to a class of nonautonomous evolution equations with
inhomogeneous boundary conditions, using the notion of bi-almost automorphic
Green functions. In addition, the authors established sufficient weak
conditions on the initial data of the equation insuring the bi-almost
automorphy of the associated Green function, see [5]. In general case with the
use of this collection of sequences in ${\mathbb{R}}^{n}$, we have that the
function $F(\cdot;\cdot)$ is $({\mathrm{R}},{\mathcal{B}})$-multi-almost
automorphic if and only if for every $B\in{\mathcal{B}}$ and for every real
sequence $(b_{k})$ there exist a subsequence $(a_{k})$ of $(b_{k})$ and a
function $F^{\ast}:{\mathbb{R}}^{n}\times X\rightarrow Y$ such that

$\displaystyle\lim_{k\rightarrow+\infty}\Bigl{\|}F\bigl{(}{\bf
t}+(a_{k},\cdot\cdot\cdot,a_{k});x\bigr{)}-F^{\ast}({\bf t};x)\Bigr{\|}_{Y}=0$

and

$\displaystyle\lim_{k\rightarrow+\infty}\Bigl{\|}F^{\ast}\bigl{(}{\bf
t}-(a_{k},\cdot\cdot\cdot,a_{k});x\bigr{)}-F({\bf t};x)\Bigr{\|}_{Y}=0,$

pointwise for all $x\in B$ and ${\bf t}\in{\mathbb{R}}^{n}.$

  * L2.

${\mathrm{R}}=\\{b:{\mathbb{N}}\rightarrow{\mathbb{R}}^{n}\,;\,\mbox{ for all
}j\in{\mathbb{N}}\mbox{ we have
}b_{j}\in\\{(a,0,0,\cdot\cdot\cdot,0)\in{\mathbb{R}}^{n}:a\in{\mathbb{R}}\\}\\}.$
Without going into full details, we want only to consider here the case in
which $X\in{\mathcal{B}}$ (the choice in which ${\mathcal{B}}$ is a collection
of all bounded or compact subsets of $X$ is a bit complicated but the obtained
conclusions are similar; the difficulty actually lies in the fact that a
bounded (compact) set in the space ${\mathbb{R}}^{n-1}\times X$ is not
necessarily a direct product of a bounded (compact) set in
${\mathbb{R}}^{n-1}$ and a bounded (compact) set in $X$); then it can be
simply approved that the function $F(\cdot;\cdot)$ is (compactly)
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic if and only if the
function ${\mathbb{F}}:{\mathbb{R}}\times{\mathcal{X}}\rightarrow Y,$ given by
${\mathcal{F}}(t;{\mathcal{x}}):=F((t,{\bf t^{\prime}});x),$
$t\in{\mathbb{R}},$ ${\mathcal{x}}=({\bf
t^{\prime}};x)\in{\mathcal{X}}={\mathbb{R}}^{n-1}\times X.$ This implies that
the notion introduced in Definition 2.12 cannot be viewed as some special case
of the notion of almost automorphic function from ${\mathbb{R}}\times X$ into
$Y.$

  * L3.

${\mathrm{R}}$ is a collection of all sequences $b(\cdot)$ in
${\mathbb{R}}^{n}.$ This is the limit case in our analysis because, in this
case, any $({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic function is
automatically $({\mathrm{R}}_{1},{\mathcal{B}})$-multi-almost automorphic for
any other collection ${\mathcal{R}}_{1}$ of sequences $b(\cdot)$ in
${\mathbb{R}}^{n}.$

In what follows we provide some examples illustrating the concept of
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic functions.

######  Example 2.3.

Let the scalar function $\varphi:\mathbb{R}\longrightarrow\mathbb{R}$ be
almost automorphic and let $(T(t))_{t\geq 0}\subset L(X)$ be uniformly bounded
(i.e. $\sup_{t\in\mathbb{R}}\|T(t)\|_{L(X)}\leq M$) family of strongly
continuous operators on $X$. Define a new function $G:\mathbb{R}^{2}\times
X\to X$ by

$G(t,s;x)=e^{\int_{s}^{t}\varphi(\tau)d\tau}T(t-s)x,\quad(t,s)\in\mathbb{R}^{2},x\in
X.$

The function $G$ yields a bi-almost automorphic function in the sense of
Remark 2.2. For a detailed proof, we refer to [17, Example 7.1] (see also [46,
Example 4.1]) where, in this case, $(T(t))_{t\geq 0}$ presents an
exponentially stable strongly continuous semigroup of bounded linear operators
([34]). More generally, we have

  1. (1)

For $j\in\\{1,2,\cdots,n-1\\}$ let $\varphi_{j}:\mathbb{R}\to\mathbb{R}$ be a
family of almost automorphic functions and $T_{j}$ a family of semigroups that
decays exponentially to $0$ as $t\to+\infty$. Then, $F:\mathbb{R}^{n}\times
X\to X$ defined as follows

$F(t_{1},t_{2},\cdots,t_{n};x):=\sum_{j=1}^{n-1}T_{j}(t_{j+1}-t_{j})e^{\int_{t_{j}}^{t_{j+1}}\varphi_{j}(\xi)d\xi}x\,,$

is $({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic, in which
$\mathrm{R}=\\{b:{\mathbb{N}}\rightarrow{\mathbb{R}}^{n}\,;\,\mbox{ for all
}j\in{\mathbb{N}}\mbox{ we have
}b_{j}\in\\{(a,a,a,\cdot\cdot\cdot,a)\in{\mathbb{R}}^{n}:a\in{\mathbb{R}}\\}\\},$
and ${\mathcal{B}}$ denotes the collection of all bounded subsets of $X.$

  2. (2)

For $j\in\\{1,2,\cdots,n\\}$ let $\varphi_{j}:\mathbb{R}\to\mathbb{R}$ be a
family of almost automorphic functions and $T_{j}$ a family of semigroups that
decays exponentially to $0$ as $t\to+\infty$. Thus, $F:\mathbb{R}^{2n}\times
X\to X$ defined as follows

$F(t_{1},t_{2},\cdots,t_{n};x):=\sum_{j=1}^{n}T_{j}(t_{2j}-t_{2j-1})e^{\int_{t_{2j-2}}^{t_{2j}}f_{j}(\xi)d\xi}x\,,$

is $({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic, in which
$\mathrm{R}=\\{b:{\mathbb{N}}\rightarrow{\mathbb{R}}^{n}\,;\,\mbox{ for all
}j\in{\mathbb{N}}\mbox{ we have
}b_{j}\in\\{(a_{1},a_{1},a_{2},a_{2},\cdot\cdot\cdot,a_{n},a_{n})\in{\mathbb{R}}^{2n}:a_{i}\in{\mathbb{R}}\\}\\},$
and ${\mathcal{B}}$ denotes the collection of all bounded subsets of $X.$

######  Example 2.4.

The proof of this example shows haw to go in the proof of the next example.
For $j=1,2,\cdots,n$, let $f_{j}$ be a family of almost automorphic function
from $\mathbb{R}$ to $\mathbb{R}$. The function
$F:\mathbb{R}^{2n}\to\mathbb{R}$ defined by

$(s_{1},s_{2},\cdots,s_{n},t_{1},t_{2},\cdots,t_{n})\to
F(s_{1},s_{2},\cdots,s_{n},t_{1},t_{2},\cdots,t_{n})=\prod_{j=1}^{n}\int_{s_{j}}^{t_{j}}f_{j}(\xi)d\xi\,,$

is $\mathrm{R}$-multi-almost automorphic function, where
$\mathrm{R}=\\{b:{\mathbb{N}}\rightarrow{\mathbb{R}}^{n}\,;\,\mbox{ for all
}j\in{\mathbb{N}}\mbox{ we have
}b_{j}\in\\{(a_{1},a_{2},\cdots,a_{n},a_{1},a_{2},\cdots,a_{n})\in{\mathbb{R}}^{2n}:a_{i}\in{\mathbb{R}}\\}\\}$.

######  Example 2.5.

This example generalizes the previous one to multidimensional almost
automorphic functions. Let $\mathrm{R}\subset\mathbb{R}^{n}$ be the set of
sequences such that all its subsequences are also in $\mathrm{R}$ and
$\mathrm{R}^{\prime}\subset\mathbb{R}^{m}$ be the set of sequences such that
all its subsequences are also in $\mathrm{R}^{\prime}$. For $i=1,2,\cdots,p$
let $f_{i}:\mathbb{R}^{n}\to\mathbb{R}$ be $\mathrm{R}$-almost automorphic and
for $j=1,2,\cdots,q$ let $g_{j}:\mathbb{R}^{m}\to\mathbb{R}$ be
$\mathrm{R}^{\prime}$-almost automorphic. Define the new functions
$F:\mathbb{R}^{n}\to\mathbb{R}^{q}$ by $F({\bf t})=\sum_{i=1}^{p}f_{i}({\bf
t})e_{i}$ and $G:\mathbb{R}^{m}\to\mathbb{R}^{q}$ by $G({\bf
s})=\sum_{j=1}^{q}g_{j}({\bf s})e_{j}$, where $\\{e_{i},\,i=1,\cdots,p\\}$ is
the canonical basis of $\mathbb{R}^{p}$ and $\\{e_{j},\,j=1,\cdots,q\\}$ is
the canonical basis of $\mathbb{R}^{q}$. Now define the function $F\bigotimes
G:\mathbb{R}^{n}\times\mathbb{R}^{m}\to M_{p\times q}(\mathbb{R})$ by

(2.3) $F\bigotimes G\,({\bf t},{\bf s})=\begin{pmatrix}f_{1}({\bf
t})g_{1}({\bf s})&f_{1}({\bf t})g_{2}({\bf s})&\cdots&f_{1}({\bf t})g_{q}({\bf
s})\\\ f_{2}({\bf t})g_{1}({\bf s})&f_{2}({\bf t})g_{2}({\bf
s})&\cdots&f_{2}({\bf t})g_{q}({\bf s})\\\ \vdots&\vdots&\ddots&\vdots\\\
f_{p}({\bf t})g_{1}({\bf s})&f_{p}({\bf t})g_{2}({\bf s})&\cdots&f_{p}({\bf
t})g_{q}({\bf s})\end{pmatrix}$

It is not difficult to see that $F\bigotimes G$ is
$\mathrm{R}\times\mathrm{R}^{\prime}$-almost automorphic, where
$\mathrm{R}\times\mathrm{R}^{\prime}=\\{({\bf b},{\bf b^{\prime}})\,:\,{\bf
b}\in\mathrm{R}\,,\,\,{\bf b^{\prime}}\in\mathrm{R}^{\prime}\\}$.

Observe that, for $p=q$ the function
$F\widehat{\bigotimes}G:\mathbb{R}^{n}\times\mathbb{R}^{m}\times\mathbb{R}^{p}\to\mathbb{R}^{p}$
defined by

$F\widehat{\bigotimes}G\,({\bf t},{\bf s},x)=F\bigotimes G\,({\bf t},{\bf
s})\cdot x\,\,$

is $(\mathrm{R}\times\mathrm{R}^{\prime},\mathcal{B})$-almost automorphic with
$\mathcal{B}$ being the collection of bounded subsets of $\mathbb{R}^{p}$.

######  Example 2.6.

In order to have more examples, we need to modify the continuity of $F$ in
Definition 2.1 accordingly. This example comes from the theory of differential
equations with piecewise constant argument [9, 10]. In the mentioned papers,
we found the following definition [10, Definition 2.3] :

######  Definition 2.7.

Let $f:\mathbb{R}\to\mathbb{C}^{p}$ be a bounded and piecewise continuous
function. $f$ is said to be $\mathbb{Z}$-almost automorphic if for any
sequence of integer numbers $\\{s_{n}\\}$, there exist a subsequence
$\\{s^{{}^{\prime}}_{n}\\}\subseteq\\{s_{n}\\}$ and a function
$\tilde{f}:\mathbb{R}\to\mathbb{C}^{p}$ such that the following pointwise
limits hold:

$\lim_{n\to\infty}f(t+s_{n}^{{}^{\prime}})=\tilde{f}(t)\,,$
$\lim_{n\to\infty}f(t+s_{n}^{{}^{\prime}})=\tilde{f}(t)\,.$

Let $A(t)$ be an almost automorphic matrix, and $\phi(t)$ a fundamental matrix
solution of the system $x^{\prime}(t)=A(t)x(t)$, then the Green matrix

$\Phi(t,s)=\phi(t)\phi^{-1}(s)\,,$

is Bi-almost automorphic [10, Lemma 3.2]. Moreover the functions

$\Phi(t,[t]),\,\,\,\,\int_{[t]}^{t}\Phi(t,\eta)B(\eta)d\eta\,,$

are $\mathbb{Z}$-almost automorphic, for $B$ an almost automorphc function,
see [10, Lemma 3.3]. Also, if $g\in
AA(\mathbb{R}\times\mathbb{C}^{p}\times\mathbb{C}^{p};\mathbb{C}^{p})$ and is
uniformly continuous on compact subsets of
$\mathbb{C}^{p}\times\mathbb{C}^{p}$ and $\psi\in
AA(\mathbb{R};\mathbb{C}^{p})$, then the composition map
$g(t,\psi(t),\psi([t]))$ is $\mathbb{Z}$-almost automorphic. Whit this
preliminary exposition we construct the piecewise continuous function:
$F:\mathbb{R}^{3}\times\mathbb{C}^{p}\to\mathbb{C}^{p}$, such that

$\displaystyle F(t,s,u,\zeta)$ $\displaystyle=$
$\displaystyle||\zeta||_{\mathbb{C}^{p}}\left(\int_{[t]}^{t}\Phi(t,\eta)B(\eta)d\eta+g(s,\psi(s),\psi([s]))\right.$
$\displaystyle\left.+\sum_{j=1}^{p}\sin(\dfrac{1}{2+\cos(u)+\cos(\sqrt{2}u)})e_{j}\right),$

where $\\{e_{j},\,j=1,\cdots,p\\}$ is the canonical basis of $\mathbb{R}^{p}$.
Then, $F$ is $({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic, with
$\mathrm{R}=\mathbb{Z}\times\mathbb{Z}\times\mathbb{R}$ and $\mathcal{B}$ is
the collection of bounded subsets of $\mathbb{C}^{p}$ .

######  Remark 2.8.

The first systematic study of almost automorphic functions on topological
groups was conducted by W. A. Veech in [42]-[43] (see also the papers [38] by
A. Reich and [41] by R. Terras). Following P. Milnes [32], who considered only
the scalar-valued case, we say that a continuous function $f:G\rightarrow Y,$
where $G$ is a semitopological group, is almost automorphic if and only if for
any sequence $(n_{i}^{\prime})$ in $G$ there exists a subsequence $(n_{i})$ of
$(n_{i}^{\prime})$ such that the joint limit
$\lim_{i,j}f(n_{i}n_{j}^{-1}t)=f(t)$ exists for all $t\in G.$ It is clear that
${\mathbb{R}}^{n}\times X$ is a semitopological group as well as that the
notion introduced in Definition 2.1 can be extended in this rather general
framework. Observing that the element $x$ in the equations (2.6)-(2.7) is
fixed, we have that any almost automorphic function $F:{\mathbb{R}}^{n}\times
X\rightarrow Y$ in the sense of the above definition is
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic, where ${\mathrm{R}}$
is a collection of all sequences $b(\cdot)$ in ${\mathbb{R}}^{n}$ and
${\mathcal{B}}$ is the collection of all subsets of $X$. For more details
about almost periodic functions on topological groups, see the research
monograph [31] by B. M. Levitan.

######  Remark 2.9.

From Remark 2.8, we may observe that for a given continuous function $f:G\to
Y$ to be almost automorphic, it is important to start with any sequence in the
full domain $G$; in this sense the functions presented in Examples 2.3 and 2.5
are not almost automorphic in the classical sense, but they are
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic for concrete not empty
sets $\mathrm{R}$. This clarifies the differences between this new class of
functions and the classical ones known in the literature ([47]).

###  2.1. Further properties of $({\mathrm{R}},{\mathcal{B}})$-multi-almost
automorphic functions

In order to give more interesting results about
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic functions, we
introduce the following concept.

######  Definition 2.10.

Let $F:{\mathbb{R}}^{n}\times X\rightarrow Y$. Then, the function
$F(\cdot;\cdot)$ is called $({\mathrm{R}},{\mathcal{B}})$-uniformly continuous
if and only if for every $B\in{\mathcal{B}}$ and for every two sequences
$({\bf a}_{k}=(a_{k}^{1},a_{k}^{2},\cdots,a_{k}^{n})),({\bf
b}_{k}=(b_{k}^{1},b_{k}^{2},\cdot\cdot\cdot,b_{k}^{n}))\in{\mathrm{R}}$ such
that $|{\bf a}_{k}-{\bf b}_{k}|\rightarrow 0$ we have

(2.4) $\displaystyle\sup_{x\in B}\|F\bigl{(}{\bf
a}_{k};x\bigr{)}-F\bigl{(}{\bf b}_{k};x\bigr{)}\|_{Y}\rightarrow 0,$

as $k\rightarrow+\infty$.

If the function $F$ is uniformly continuous (in the classical sense) in ${\bf
t}$ uniformly on bounded subsets of $X$. Then, in particular, $F$ is
$({\mathrm{R}},{\mathcal{B}})$-uniformly continuous with ${\mathrm{R}}$ is the
collection of all sequences of $\mathbb{R}^{n}$ and ${\mathcal{B}}$ is the
collection of all bounded subsets of $X$. As in the usual considerations of
almost automorphic functions, we have:

######  Proposition 2.11.

Suppose that $F:{\mathbb{R}}^{n}\times X\rightarrow Y$ is a continuous
function. Then the function $F(\cdot;\cdot)$ is
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic if and only if for
every $B\in{\mathcal{B}}$ and for every sequence $({\bf
b}_{k}=(b_{k}^{1},b_{k}^{2},\cdot\cdot\cdot,b_{k}^{n}))\in{\mathrm{R}}$ there
exist a subsequence $({\bf
b}_{k_{l}}=(b_{k_{l}}^{1},b_{k_{l}}^{2},\cdot\cdot\cdot,b_{k_{l}}^{n}))$ of
$({\bf b}_{k})$ and a function $F^{\ast}:{\mathbb{R}}^{n}\times X\rightarrow
Y$ such that

(2.5)
$\displaystyle\lim_{l\rightarrow+\infty}\lim_{l\rightarrow+\infty}F\Bigl{(}{\bf
t}-\bigl{(}b_{k_{l}}^{1},\cdot\cdot\cdot,b_{k_{l}}^{n}\bigr{)}+\bigl{(}b_{k_{m}}^{1},\cdot\cdot\cdot,b_{k_{m}}^{n}\bigr{)};x\Bigr{)}=F({\bf
t};x),$

pointwise for all $x\in B$ and ${\bf t}\in{\mathbb{R}}^{n}.$

It is clear that we have the following simple results:

######  Proposition 2.12.

Suppose that $F:{\mathbb{R}}^{n}\times X\rightarrow Y$ is a continuous
function. If ${\mathcal{B}}^{\prime}$ is a certain collection of subsets of
$X$ which contains ${\mathcal{B}},$ ${\mathrm{R}}^{\prime}$ is a certain
collection of sequences in ${\mathbb{R}}^{n}$ which contains ${\mathrm{R}}$
and $F(\cdot;\cdot)$ is (compactly)
$({\mathrm{R}}^{\prime},{\mathcal{B}}^{\prime})$-multi-almost automorphic,
then $F(\cdot;\cdot)$ is (compactly) $({\mathrm{R}},{\mathcal{B}})$-multi-
almost automorphic.

######  Proposition 2.13.

Suppose that $F:{\mathbb{R}}^{n}\times X\rightarrow Y$ is
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic and $\phi:Y\rightarrow
Z$ is continuous. Then $\phi\circ F:{\mathbb{R}}^{n}\times X\rightarrow Z$ is
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic.

In the next result we give a very useful characterization of compactly
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic functions. A similar
result was already established in [21, Lemma 3.7] for the particular case of
$(\mathbb{R},\\{0_{X}\\})$-one-dimensional almost automorphic functions.

######  Theorem 2.14.

Let $F:{\mathbb{R}}^{n}\times X\rightarrow Y$ be continuous. Then, $F$ is
compactly $({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic if and only
$F$ is $({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic and
$({\mathrm{R}},{\mathcal{B}})$-uniformly continuous.

###### Proof.

Let $({\mathrm{R}},{\mathcal{B}})$ be fixed. Assume that $F$ is
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic and
$({\mathrm{R}},{\mathcal{B}})$-uniformly continuous. Then for every sequence
$({\bf b}_{k}=(b_{k}^{1},b_{k}^{2},\cdot\cdot\cdot,b_{k}^{n}))\in{\mathrm{R}}$
there exist a subsequence $({\bf
b}_{k_{l}}=(b_{k_{l}}^{1},b_{k_{l}}^{2},\cdot\cdot\cdot,b_{k_{l}}^{n}))$ of
$({\bf b}_{k})$ and a function $F^{\ast}:{\mathbb{R}}^{n}\times X\rightarrow
Y$ such that for each ${\bf t}\in{\mathbb{R}}^{n}$, we have

(2.6) $\displaystyle\lim_{l\rightarrow+\infty}F\bigl{(}{\bf t}+{\bf
b}_{k_{l}};x\bigr{)}=F^{\ast}({\bf t};x)$

and

(2.7) $\displaystyle\lim_{l\rightarrow+\infty}F^{\ast}\bigl{(}{\bf t}-{\bf
b}_{k_{l}};x\bigr{)}=F({\bf t};x),$

uniformly for all $x\in B$. Let $x\in B$ and define the sequence

$F_{l}(\cdot;x)=F(\cdot+{\bf b}_{k_{l}};x),\quad l\in\mathbb{N}.$

Then, from the $({\mathrm{R}},{\mathcal{B}})$-uniform continuity it follows
that the family $(F_{l}(\cdot;x))_{l}$ is equicontinuous. Hence, the
convergence (2.6) implies that $F^{\ast}({\cdot};x)$ is continuous. So, the
convergence (2.6) holds uniformly in compact subsets of $\mathbb{R}^{n}$.
Indeed, let $\varepsilon>0$ and let $I_{c}\subset\mathbb{R}^{n}$ be any
compact subset. That is, there exist $\delta>0$ and $\\{{\bf
t}_{1},\cdots,{\bf t}_{m}\\}\subset I_{c}$ such that
$I_{c}\subset\cup_{i=1}^{m}B({\bf t}_{i},\delta)$. Thus, for all ${\bf t}\in
I_{c}$, there exists a certain $i_{0}\in{1,\cdots,m}$ satisfying $|{\bf
t}-{\bf t}_{i_{0}}|<\delta$. Then, for all $l$, we get

$\displaystyle\|F_{l}({\bf t};x)-F^{\ast}({\bf t};x)\|$
$\displaystyle\leq\|F_{l}({\bf t};x)-F_{l}({\bf t}_{i_{0}};x)\|+\|F_{l}({\bf
t}_{i_{0}};x)-F^{\ast}({\bf t}_{i_{0}};x)\|$ $\displaystyle+\|F^{\ast}({\bf
t}_{i_{0}};x)-F^{\ast}({\bf t};x)\|.$

The equicontinuity of $(F_{l}(\cdot;x))_{l}$ along with the continuity of
$F^{\ast}(\cdot;x)$ yield for $l$ large enough that

$\sup_{x\in B}\sup_{{\bf t}\in I_{c}}\|F_{l}({\bf t};x)-F^{\ast}({\bf
t};x)\|\rightarrow 0\quad\text{ as }l\rightarrow\infty.$

Similarly, we can prove that the limit in (2.7) holds uniformly in compact
subsets of $\mathbb{R}^{n}$ using the $({\mathrm{R}},{\mathcal{B}})$-uniform
continuity of $F^{\ast}(\cdot;x)$ which is a consequence of (2.6). This proves
the sufficiency.

Reciprocally, assume that $F$ is compactly
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic. Then, it is
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic. Therefore, it
suffices to prove that $F$ is $({\mathrm{R}},{\mathcal{B}})$-uniformly
continuous. Let $({\bf a}_{k}=(a_{k}^{1},a_{k}^{2},\cdots,a_{k}^{n})),({\bf
b}_{k}=(b_{k}^{1},b_{k}^{2},\cdot\cdot\cdot,b_{k}^{n}))\in{\mathrm{R}}$ be any
two sequences such that $|{\bf a}_{k}-{\bf b}_{k}|\rightarrow 0$. Then, we
will prove that

$\displaystyle\sup_{x\in B}\|F\bigl{(}{\bf a}_{k};x\bigr{)}-F\bigl{(}{\bf
b}_{k};x\bigr{)}\|_{Y}\rightarrow 0,$

as $k\rightarrow\infty$. Take any subsequences $({\bf a}_{k_{l}})_{l}$ and
$({\bf b}_{k_{l}})_{l}$ of $({\bf a}_{k})_{k}$ and $({\bf b}_{k})_{k}$
respectively. By the compact $({\mathrm{R}},{\mathcal{B}})$-multi-almost
automorphy, there exist another two subsequences (still denoted by) $({\bf
a}_{k_{l}})_{l}$ and $({\bf b}_{k_{l}})_{l}$ and a continuous function
$F^{\ast}$ such that (2.6) holds uniformly in compact subsets of
$\mathbb{R}^{n}$. Define ${\bf c}_{k_{l}}:={\bf a}_{k_{l}}-{\bf b}_{k_{l}},$
$l\in\mathbb{N}$ (by assumption $|{\bf c}_{k_{l}}|\rightarrow 0$ as
$l\rightarrow\infty$) and let $I_{c}\subset\mathbb{R}^{n}$ be compact such
that ${\bf c}_{k_{l}}\in I_{c}$ for all $l\in\mathbb{N}$. Then, we have

$\displaystyle\|F\bigl{(}{\bf a}_{k_{l}};x\bigr{)}-F\bigl{(}{\bf
b}_{k_{l}};x\bigr{)}\|_{Y}\leq$ $\displaystyle\|F\bigl{(}{\bf c}_{k_{l}}+{\bf
b}_{k_{l}};x\bigr{)}-F^{\ast}\bigl{(}{\bf
c}_{k_{l}};x\bigr{)}\|_{Y}+\|F^{\ast}\bigl{(}0;x\bigr{)}-F^{\ast}\bigl{(}{\bf
c}_{k_{l}};x\bigr{)}\|_{Y}$ $\displaystyle+\|F\bigl{(}{\bf
b}_{k_{l}};x\bigr{)}-F^{\ast}\bigl{(}0;x\bigr{)}\|_{Y}$
$\displaystyle\leq\sup_{{\bf t}\in I_{c}}\|F\bigl{(}{\bf t}+{\bf
b}_{k_{l}};x\bigr{)}-F^{\ast}\bigl{(}{\bf
t};x\bigr{)}\|_{Y}+\|F^{\ast}\bigl{(}0;x\bigr{)}-F^{\ast}\bigl{(}{\bf
c}_{k_{l}};x\bigr{)}\|_{Y}$ $\displaystyle+\|F\bigl{(}{\bf
b}_{k_{l}};x\bigr{)}-F^{\ast}\bigl{(}0;x\bigr{)}\|_{Y}.$

So, using the compact almost automorphy of $F$ and the continuity of
$F^{\ast}\bigl{(}\cdot;x\bigr{)}$, we obtain that

$\sup_{x\in B}\|F\bigl{(}{\bf a}_{k_{l}};x\bigr{)}-F\bigl{(}{\bf
b}_{k_{l}};x\bigr{)}\|_{Y}\rightarrow 0\quad\text{ as }l\rightarrow\infty.$

Since a subsequence of any subsequence of $\left(\sup_{x\in B}\|F\bigl{(}{\bf
a}_{k};x\bigr{)}-F\bigl{(}{\bf b}_{k};x\bigr{)}\|_{Y}\right)_{k}$ converges.
Then (the whole sequence),

$\sup_{x\in B}\|F\bigl{(}{\bf a}_{k};x\bigr{)}-F\bigl{(}{\bf
b}_{k};x\bigr{)}\|_{Y}\rightarrow 0\quad\text{ as }k\rightarrow\infty.$

This proves the statement. ∎

In [29, Lemma 3.9.9], we have clarified the supremum formula for almost
automorphic functions. This formula can be extended in our framework as
follows:

######  Proposition 2.15.

(The supremum formula) Let $F:{\mathbb{R}}^{n}\times X\rightarrow Y$ be
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic. Suppose that there
exist $a\geq 0$ and a sequence $b(\cdot)$ in ${\mathrm{R}}$ whose any
subsequence is unbounded. Then we have

(2.8) $\displaystyle\sup_{{\bf t}\in{\mathbb{R}}^{n},x\in X}\bigl{\|}F({\bf
t};x)\Bigr{\|}_{Y}=\sup_{{\bf t}\in{\mathbb{R}}^{n},|t|\geq a,x\in
X}\bigl{\|}F({\bf t};x)\Bigr{\|}_{Y}.$

###### Proof.

We will include all relevent details of the proof for the sake of
completeness. Let $\epsilon>0,$ $a\geq 0$ and $x\in X$ be given. Then (2.8)
will hold if we prove that

(2.9) $\displaystyle\bigl{\|}F({\bf t};x)\Bigr{\|}_{Y}\leq\epsilon+\sup_{{\bf
t}\in{\mathbb{R}}^{n},|t|\geq a}\bigl{\|}F({\bf t};x)\Bigr{\|}_{Y}.$

By assumption, there exists $B\in{\mathcal{B}}$ with $x\in B.$ Let $b(\cdot)$
be a sequence in ${\mathrm{R}}$ whose any subsequence is unbounded. Then we
have (2.5), and consequently, there exist two integers $l_{0}\in{\mathbb{N}}$
and $m_{0}\in{\mathbb{N}}$ such that

$\bigl{\|}F({\bf t};x)\Bigr{\|}_{Y}\leq\epsilon+\Bigl{\|}F\Bigl{(}{\bf
t}-\bigl{(}b_{k_{l}}^{1},\cdot\cdot\cdot,b_{k_{l}}^{n}\bigr{)}+\bigl{(}b_{k_{m}}^{1},\cdot\cdot\cdot,b_{k_{m}}^{n}\bigr{)};x\Bigr{)}\Bigr{\|}_{Y},\quad
l\geq l_{0},\ m\geq m_{0}.$

In particular,

$\bigl{\|}F({\bf t};x)\Bigr{\|}_{Y}\leq\epsilon+\Bigl{\|}F\Bigl{(}{\bf
t}-\bigl{(}b_{k_{l_{0}}}^{1},\cdot\cdot\cdot,b_{k_{l_{0}}}^{n}\bigr{)}+\bigl{(}b_{k_{m}}^{1},\cdot\cdot\cdot,b_{k_{m}}^{n}\bigr{)};x\Bigr{)}\Bigr{\|}_{Y},\quad
m\geq m_{0}.$

Since the sequence $((b_{k_{m}}^{1},\cdot\cdot\cdot,b_{k_{m}}^{n})_{m\geq
m_{0}}$ is unbounded, (2.9) follows immediately. ∎

Now we will prove also the following result.

######  Proposition 2.16.

Suppose that for each integer $j\in{\mathbb{N}}$ the function
$F_{j}(\cdot;\cdot)$ is (compactly) $({\mathrm{R}},{\mathcal{B}})$-multi-
almost automorphic. If the sequence $(F_{j}(\cdot;\cdot))$ converges uniformly
to a function $F(\cdot;\cdot)$, then the function $F(\cdot;\cdot)$ is
(compactly) $({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic.

###### Proof.

The proof is very similar to the proof of [25, Theorem 2.1.10] but we will
provide all relevant details. Let the set $B\in{\mathcal{B}}$ and the sequence
$({\bf b}_{k}=(b_{k}^{1},b_{k}^{2},\cdot\cdot\cdot,b_{k}^{n}))\in{\mathrm{R}}$
be given. Using the diagonal procedure, we get the existence of a subsequence
$({\bf
b}_{k_{l}}=(b_{k_{l}}^{1},b_{k_{l}}^{2},\cdot\cdot\cdot,b_{k_{l}}^{n}))$ of
$({\bf b}_{k})$ such that for each integer $j\in{\mathbb{N}}$ there exists a
function $F^{\ast}_{j}:{\mathbb{R}}^{n}\times X\rightarrow Y$ such that

(2.10) $\displaystyle\lim_{l\rightarrow+\infty}\Bigl{\|}F_{j}\bigl{(}{\bf
t}+(b_{k_{l}}^{1},\cdot\cdot\cdot,b_{k_{l}}^{n});x\bigr{)}-F^{\ast}_{j}({\bf
t};x)\Bigr{\|}_{Y}=0$

and

(2.11)
$\displaystyle\lim_{l\rightarrow+\infty}\Bigl{\|}F^{\ast}_{j}\bigl{(}{\bf
t}-(b_{k_{l}}^{1},\cdot\cdot\cdot,b_{k_{l}}^{n});x\bigr{)}-F_{j}({\bf
t};x)\Bigr{\|}_{Y}=0,$

pointwise for all $x\in B$ and ${\bf t}\in{\mathbb{R}}^{n}.$ Moreover, the
convergence of the above limits is uniform on compact subsets of
${\mathbb{R}}^{n}$ if for each integer $j\in{\mathbb{N}}$ the function
$F_{j}(\cdot;\cdot)$ is compactly $({\mathrm{R}},{\mathcal{B}})$-multi-almost
automorphic. Fix now an element $x\in B,$ a number ${\bf
t}\in{\mathbb{R}}^{n}$ and positive real number $\epsilon>0.$ Since

$\displaystyle\Bigl{\|}F_{i}^{\ast}({\bf t};x)$ $\displaystyle-
F_{j}^{\ast}({\bf t};x)\Bigr{\|}_{Y}\leq\Bigl{\|}F_{i}^{\ast}({\bf
t};x)-F_{i}({\bf
t}+(b_{k_{l}}^{1},\cdot\cdot\cdot,b_{k_{l}}^{n});x)\Bigr{\|}_{Y}$
$\displaystyle+\Bigl{\|}F_{i}({\bf
t}+(b_{k_{l}}^{1},\cdot\cdot\cdot,b_{k_{l}}^{n});x)-F_{j}({\bf
t}+(b_{k_{l}}^{1},\cdot\cdot\cdot,b_{k_{l}}^{n});x)\Bigr{\|}_{Y}$
$\displaystyle+\Bigl{\|}F_{j}({\bf
t}+(b_{k_{l}}^{1},\cdot\cdot\cdot,b_{k_{l}}^{n});x)-F_{j}^{\ast}({\bf
t};x)\Bigr{\|}_{Y},$

and (2.10)-(2.11) hold, we can find a number $l_{0}\in{\mathbb{N}}$ such that
for all integers $l\geq l_{0}$ we have:

$\displaystyle\Bigl{\|}F_{i}^{\ast}({\bf t};x)-F_{i}({\bf
t}+(b_{k_{l}}^{1},\cdot\cdot\cdot,b_{k_{l}}^{n});x)\Bigr{\|}_{Y}+\Bigl{\|}F_{j}({\bf
t}+(b_{k_{l}}^{1},\cdot\cdot\cdot,b_{k_{l}}^{n});x)-F_{j}^{\ast}({\bf
t};x)\Bigr{\|}_{Y}<2\epsilon/3.$

Since the sequence $(F_{j}(\cdot;\cdot))$ converges uniformly to a function
$F(\cdot;\cdot),$ there exists $N(\epsilon)\in{\mathbb{N}}$ such that for all
integers $i,\ j\in{\mathbb{N}}$ with $\min(i,j)\geq N(\epsilon)$ we have

$\Bigl{\|}F_{i}({\bf
t}+(b_{k_{l}}^{1},\cdot\cdot\cdot,b_{k_{l}}^{n});x)-F_{j}({\bf
t}+(b_{k_{l}}^{1},\cdot\cdot\cdot,b_{k_{l}}^{n});x)\Bigr{\|}_{Y}<\epsilon/3.$

This implies that $(F_{j}^{\ast}({\bf t};x))$ is a Cauchy sequence in $Y$ and
therefore convergent to an element $F^{\ast}({\bf t};x),$ say (the above
arguments also shows that $\lim_{j\rightarrow+\infty}F_{j}^{\ast}({\bf
t};x)=F^{\ast}({\bf t};x)$ uniformly on compact subsets of ${\mathbb{R}}^{n},$
for fixed element $x$). It remains to be proved that (2.6)-(2.7) hold true.
Toward this end, observe that for each $j\in{\mathbb{N}}$ we have:

$\displaystyle\Bigl{\|}F\bigl{(}{\bf t}$
$\displaystyle+(b_{k_{l}}^{1},\cdot\cdot\cdot,b_{k_{l}}^{n});x\bigr{)}-F^{\ast}({\bf
t};x)\Bigr{\|}_{Y}$ $\displaystyle\leq\Bigl{\|}F\bigl{(}{\bf
t}+(b_{k_{l}}^{1},\cdot\cdot\cdot,b_{k_{l}}^{n});x\bigr{)}-F_{j}\bigl{(}{\bf
t}+(b_{k_{l}}^{1},\cdot\cdot\cdot,b_{k_{l}}^{n});x\bigr{)}\Bigr{\|}_{Y}$
$\displaystyle+\Bigl{\|}F_{j}\bigl{(}{\bf
t}+(b_{k_{l}}^{1},\cdot\cdot\cdot,b_{k_{l}}^{n});x\bigr{)}-F^{\ast}_{j}({\bf
t};x)\Bigr{\|}_{Y}+\Bigl{\|}F^{\ast}_{j}({\bf t};x)-F^{\ast}({\bf
t};x)\Bigr{\|}_{Y}.$

It can be simply shown that there exists a number
$j_{0}(\epsilon)\in{\mathbb{N}}$ such that for all integers $j\geq j_{0}$ we
have that the first and the third addend in the above estimate is less or
greater than $\epsilon/3.$ For the second addend, take any integer
$l\in{\mathbb{N}}$ such that

$\Bigl{\|}F_{j}\bigl{(}{\bf
t}+(b_{k_{l}}^{1},\cdot\cdot\cdot,b_{k_{l}}^{n});x\bigr{)}-F^{\ast}_{j}({\bf
t};x)\Bigr{\|}_{Y}<\epsilon/3.$

This completes the proof in a routine manner. ∎

For the sequel, we need the following general definition:

######  Definition 2.17.

Suppose that ${\mathbb{D}}\subseteq\Omega\subseteq{\mathbb{R}}^{n}$ and the
set ${\mathbb{D}}$ is unbounded. By
$C_{0,{\mathbb{D}},{\mathcal{B}}}(\Omega\times X:Y)$ we denote the vector
space consisting of all continuous functions $Q:\Omega\times X\rightarrow Y$
such that, for every $B\in{\mathcal{B}},$ we have
$\lim_{t\in{\mathbb{D}},|t|\rightarrow+\infty}Q({\bf t};x)=0,$ uniformly for
$x\in B.$

######  Remark 2.18.

In [30, Definition 16-(i)], we have used the following notion: Let
$I=[0,\infty)$ or $I={\mathbb{R}}.$ By $C_{0,{\mathcal{B}}}(I\times X:Y)$ we
denote the space of all continuous functions $Q:I\times X\rightarrow Y$ such
that $\lim_{|t|\rightarrow\infty}Q(t,x)=0$ uniformly for $x$ in any subset of
$B\in{\mathcal{B}}.$ The notion introduced in Definition 2.17 properly
generalizes this notion (see also case [L2.] considered above).

Now we are ready to introduce the following notion:

######  Definition 2.19.

Suppose that the set ${\mathbb{D}}\subseteq{\mathbb{R}}^{n}$ is unbounded,
$i=1,2$ and $F:{\mathbb{R}}^{n}\times X\rightarrow Y$ is a continuous
function. Then we say that $F(\cdot;\cdot)$ is ${\mathbb{D}}$-asymptotically
(compactly) $({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic if and
only if there exist a function $G(\cdot;\cdot)$ which is (compactly)
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic and a function $Q\in
C_{0,{\mathbb{D}},{\mathcal{B}}}({\mathbb{R}}^{n}\times X:Y)$ such that
$F({\bf t};x)=G({\bf t};x)+Q({\bf t};x)$ for all ${\bf t}\in{\mathbb{R}}^{n}$
and $x\in X.$

It is said that $F(\cdot;\cdot)$ is asymptotically (compactly)
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic if and only if
$F(\cdot;\cdot)$ is ${\mathbb{R}}^{n}$-asymptotically (compactly)
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic.

The proof of the following proposition can be given as for the usually
considered almost automorphic functions ([25]):

######  Proposition 2.20.

  * (i)

Suppose that $c\in{\mathbb{C}}$ and $F(\cdot;\cdot)$ is (compactly)
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic. Then
$cF(\cdot;\cdot)$ is (compactly) $({\mathrm{R}},{\mathcal{B}})$-multi-almost
automorphic. The same holds for ${\mathbb{D}}$-asymptotically (compactly)
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic functions.

  * (ii)

    * (a)

Suppose that $\tau\in{\mathbb{R}}^{n}$ and $F(\cdot;\cdot)$ is (compactly)
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic. Then
$F(\cdot+\tau;\cdot)$ is (compactly) $({\mathrm{R}},{\mathcal{B}})$-multi-
almost automorphic. Furthermore, if $F(\cdot;\cdot)$ is
${\mathbb{D}}$-asymptotically (compactly)
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic, then
$F(\cdot+\tau;\cdot)$ is $({\mathbb{D}}-\tau)$-asymptotically (compactly)
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic.

    * (b)

Suppose that $x_{0}\in X$ and $F(\cdot;\cdot)$ is (compactly)
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic. Then
$F(\cdot;\cdot+x_{0})$ is (compactly)
$({\mathrm{R}},{\mathcal{B}}_{x_{0}})$-multi-almost automorphic. Furthermore,
if $F(\cdot;\cdot)$ is ${\mathbb{D}}$-asymptotically (compactly)
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic, then
$F(\cdot;\cdot+x_{0})$ is ${\mathbb{D}}$-asymptotically (compactly)
$({\mathrm{R}},{\mathcal{B}}_{x_{0}})$-multi-almost automorphic.

    * (c)

Suppose that $\tau\in{\mathbb{R}}^{n},$ $x_{0}\in X$ and $F(\cdot;\cdot)$ is
(compactly) $({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic. Then
$F(\cdot+\tau;\cdot+x_{0})$ is (compactly)
$({\mathrm{R}},{\mathcal{B}}_{x_{0}})$-multi-almost automorphic, where
${\mathcal{B}}_{x_{0}}\equiv\\{-x_{0}+B:B\in{\mathcal{B}}\\}.$ Furthermore, if
$F(\cdot;\cdot)$ is ${\mathbb{D}}$-asymptotically (compactly)
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic, then
$F(\cdot+\tau;\cdot+x_{0})$ is $({\mathbb{D}}-\tau)$-asymptotically
(compactly) $({\mathrm{R}},{\mathcal{B}}_{x_{0}})$-multi-almost automorphic.

  * (iii)

    * (a)

Suppose that $c\in{\mathbb{C}}\setminus\\{0\\}$ and $F(\cdot;\cdot)$ is
(compactly) $({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic. Then
$F(c\cdot;\cdot)$ is (compactly) $({\mathrm{R}}_{c},{\mathcal{B}})$-multi-
almost automorphic, where ${\mathrm{R}}_{c}\equiv\\{c^{-1}{\bf b}:{\bf
b}\in{\mathrm{R}}\\}.$ Furthermore, if $F(\cdot;\cdot)$ is
${\mathbb{D}}$-asymptotically (compactly)
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic, then
$F(c\cdot;\cdot)$ is ${\mathbb{D}}/c$-asymptotically (compactly)
$({\mathrm{R}}_{c},{\mathcal{B}})$-multi-almost automorphic.

    * (b)

Suppose that $c_{2}\in{\mathbb{C}}\setminus\\{0\\},$ and $F(\cdot;\cdot)$ is
(compactly) $({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic. Then
$F(\cdot;c_{2}\cdot)$ is (compactly)
$({\mathrm{R}},{\mathcal{B}}_{c_{2}})$-multi-almost automorphic, where
${\mathcal{B}}_{c_{2}}\equiv\\{c_{2}^{-1}B:B\in{\mathcal{B}}\\}.$ Furthermore,
if $F(\cdot;\cdot)$ is ${\mathbb{D}}$-asymptotically (compactly)
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic, then
$F(\cdot;c_{2}\cdot)$ is ${\mathbb{D}}$-asymptotically (compactly)
$({\mathrm{R}},{\mathcal{B}}_{c_{2}})$-multi-almost automorphic.

    * (c)

Suppose that $c_{1}\in{\mathbb{C}}\setminus\\{0\\},$
$c_{2}\in{\mathbb{C}}\setminus\\{0\\},$ and $F(\cdot;\cdot)$ is (compactly)
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic. Then
$F(c_{1}\cdot;c_{2}\cdot)$ is (compactly)
$({\mathrm{R}}_{c_{1}},{\mathcal{B}}_{c_{2}})$-multi-almost automorphic.
Furthermore, if $F(\cdot;\cdot)$ is ${\mathbb{D}}$-asymptotically (compactly)
$({\mathrm{R}}_{c_{1}},{\mathcal{B}})$-multi-almost automorphic, then
$F(c_{1}\cdot;c_{2}\cdot)$ is ${\mathbb{D}}/c_{1}$-asymptotically (compactly)
$({\mathrm{R}}_{c_{1}},{\mathcal{B}}_{c_{2}})$-multi-almost automorphic.

  * (iv)

Suppose that $\alpha,\ \beta\in{\mathbb{C}}$ and, for every sequence which
belongs to ${\mathrm{R}},$ we have that any its subsequence belongs to
${\mathrm{R}}.$ If $F(\cdot;\cdot)$ and $G(\cdot;\cdot)$ are (compactly)
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic, then $\alpha
F(\cdot;\cdot)+\beta G(\cdot;\cdot)$ is also (compactly)
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic. The same holds for
${\mathbb{D}}$-asymptotically (compactly)
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic functions.

  * (v)

If $X\in{\mathcal{B}}$ and $F(\cdot;\cdot)$ is asymptotically
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic, then $F(\cdot;\cdot)$
is bounded in case _[L3]_.

Using Proposition 2.20(iv) and the supremum formula clarified in Proposition
2.15 (see also the estimate (2.9)), we can simply deduce that the
decomposition in Definition 2.19 is unique:

######  Proposition 2.21.

Suppose that there exist a function $G_{i}(\cdot;\cdot)$ which is
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic and a function
$Q_{i}\in C_{0,{\mathbb{R}}^{n},{\mathcal{B}}}({\mathbb{R}}^{n}\times X:Y)$
such that $F({\bf t};x)=G_{i}({\bf t};x)+Q_{i}({\bf t};x)$ for all ${\bf
t}\in{\mathbb{R}}^{n}$ and $x\in X$ ($i=1,2$). Then we have $G_{1}\equiv
G_{2}$ and $Q_{1}\equiv Q_{2},$ provided that the collection ${\mathrm{R}}$
satisfies the following two conditions:

  * D1.

There exists a sequence in ${\mathrm{R}}$ whose any subsequence is unbounded.

  * D2.

For every sequence which belongs to ${\mathrm{R}},$ we have that any its
subsequence belongs to ${\mathrm{R}}.$

Furthermore, arguing as in the proof of [19, Lemma 4.28], we may deduce the
following:

######  Lemma 2.22.

Suppose that there exist an $({\mathrm{R}},{\mathcal{B}})$-multi-almost
automorphic function $G(\cdot;\cdot)$ and a function $Q\in
C_{0,{\mathbb{R}}^{n},{\mathcal{B}}}({\mathbb{R}}^{n}\times X:Y)$ such that
$F({\bf t};x)=G({\bf t};x)+Q({\bf t};x)$ for all ${\bf t}\in{\mathbb{R}}^{n}$
and $x\in X.$ Then we have

$\overline{\bigl{\\{}G({\bf t};x):{\bf t}\in{\mathbb{R}}^{n},\ x\in
X\bigr{\\}}}\subseteq\overline{\bigl{\\{}F({\bf t};x):{\bf
t}\in{\mathbb{R}}^{n},\ x\in X\bigr{\\}}},$

provided that condition _[D1.]_ holds.

###  2.2. Integrals and derivatives of $({\mathrm{R}},{\mathcal{B}})$-multi-
almost automorphic functions

Concerning the integral invariance of space consisting of all
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic functions, we would
like to state the following result:

######  Proposition 2.23.

Suppose that $h\in L^{1}({\mathbb{R}}^{n})$ as well as the function
$F(\cdot;\cdot)$ is bounded and $({\mathrm{R}},{\mathcal{B}})$-multi-almost
automorphic. Then the function

$(h\ast F)({\bf t};x):=\int_{{\mathbb{R}}^{n}}h(\sigma)F({\bf
t}-\sigma;x)\,d\sigma,\quad{\bf t}\in{\mathbb{R}}^{n},\ x\in X$

is bounded and $({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic.

###### Proof.

Since $h\in L^{1}({\mathbb{R}}^{n})$ and the function $F(\cdot;\cdot)$ is
bounded $({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic, it is clear
that the function $(h\ast F)(\cdot;\cdot)$ is well defined and bounded as well
as that the limit function $F^{\ast}(\cdot;\cdot)$ from Definition 2.1 is
bounded and measurable. The continuity of function $(h\ast F)(\cdot;\cdot)$
follows from the dominated convergence theorem. The remainder of proof can be
deduced as in the proof of [19, Theorem 4.5, pp. 113–115]; let us only note
that the limit function for $(h\ast F)(\cdot;\cdot)$ will be $(h\ast
F^{\ast})(\cdot;\cdot)$. ∎

######  Proposition 2.24.

Suppose that conditions _[D1.]-[D2.]_ hold and for each integer
$j\in{\mathbb{N}}$ the function $F_{j}(\cdot;\cdot)$ is asymptotically
(compactly) $({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic. If the
sequence $(F_{j}(\cdot;\cdot))$ converges uniformly to a function
$F(\cdot;\cdot)$, then the function $F(\cdot;\cdot)$ is asymptotically
(compactly) $({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic.

###### Proof.

Due to Proposition 2.21, we know that there exist a uniquely determined
function $G(\cdot;\cdot)$ which is $({\mathrm{R}},{\mathcal{B}})$-multi-almost
automorphic and a uniquely determined function $Q\in
C_{0,{\mathbb{R}}^{n},{\mathcal{B}}}({\mathbb{R}}^{n}\times X:Y)$ such that
$F({\bf t};x)=G({\bf t};x)+Q({\bf t};x)$ for all ${\bf t}\in{\mathbb{R}}^{n}$
and $x\in X$. Furthermore, we have

$F_{j}({\bf t};x)-F_{m}({\bf t};x)=\bigl{[}G_{j}({\bf t};x)-G_{m}({\bf
t};x)\bigr{]}+\bigl{[}Q_{j}({\bf t};x)-Q_{m}({\bf t};x)\bigr{]},$

for all ${\bf t}\in{\mathbb{R}}^{n},$ $x\in X$ and $j,\ m\in{\mathbb{N}}.$ Due
to Proposition 2.20(iv), we have that the function
$F_{j}(\cdot;\cdot)-F_{m}(\cdot;\cdot)$ is asymptotically
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic as well as that the
function $G_{j}(\cdot;\cdot)-G_{m}(\cdot;\cdot)$ is
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic ($j,\
m\in{\mathbb{N}}$). Keeping in mind this fact, Lemma 2.22 and the
argumentation used in the proof of [19, Theorem 4.29], we get that

$\displaystyle 3\sup_{{\bf t}\in{\mathbb{R}}^{n},x\in X}$
$\displaystyle\Bigl{\|}F_{j}({\bf t};x)-F_{m}({\bf t};x)\Bigr{\|}_{Y}$
$\displaystyle\geq\sup_{{\bf t}\in{\mathbb{R}}^{n},x\in X}\Bigl{\|}G_{j}({\bf
t};x)-G_{m}({\bf t};x)\Bigr{\|}_{Y}+\sup_{{\bf t}\in{\mathbb{R}}^{n},x\in
X}\Bigl{\|}Q_{j}({\bf t};x)-Q_{m}({\bf t};x)\Bigr{\|}_{Y},$

for any $j,\ m\in{\mathbb{N}}.$ This implies that the sequences
$(G_{j}(\cdot;\cdot))$ and $(Q_{j}(\cdot;\cdot))$ converge uniformly to the
functions $G(\cdot;\cdot)$ and $Q(\cdot;\cdot),$ respectively. Due to
Proposition 2.16, we get that the function $G(\cdot;\cdot)$ is
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic. The final conclusion
follows from the obvious equality $F=G+Q$ and the fact that
$C_{0,{\mathbb{R}}^{n},{\mathcal{B}}}({\mathbb{R}}^{n}\times X:Y)$ is a Banach
space. ∎

Concerning the partial derivatives of (asymptotically)
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic functions, we would
like to state the following result (by $(e_{1},e_{2},\cdot\cdot\cdot,e_{n})$
we denote the standard basis of ${\mathbb{R}}^{n}$):

######  Proposition 2.25.

  * (i)

Suppose that the function $F(\cdot;\cdot)$ is (compactly)
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic, _[D2.]_ holds, the
partial derivative

$\frac{\partial F(\cdot;\cdot)}{\partial t_{i}}:=\lim_{h\rightarrow
0}\frac{F(\cdot+he_{i};\cdot)-F(\cdot;\cdot)}{h},\quad{\bf
t}\in{\mathbb{R}}^{n},\ x\in X$

exists and it is uniformly continuous on ${\mathcal{B}},$ i.e.,

$\displaystyle(\forall B\in{\mathcal{B}})$ $\displaystyle\
(\forall\epsilon>0)\ (\exists\delta>0)\ (\forall{\bf t^{\prime}},\ {\bf
t^{\prime\prime}}\in{\mathbb{R}}^{n})\ (\forall x\in B)$
$\displaystyle\Biggl{(}\bigl{|}{\bf t^{\prime}}-{\bf
t^{\prime\prime}}\bigr{|}<\delta\Rightarrow\Bigl{\|}\frac{\partial F({\bf
t^{\prime}};x)}{\partial t_{i}}-\frac{\partial F({\bf
t^{\prime\prime}};x)}{\partial t_{i}}\Bigr{\|}<\epsilon\Biggr{)}.$

Then the function $\frac{\partial F(\cdot;\cdot)}{\partial t_{i}}$ is
(compactly) $({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic.

  * (ii)

Suppose that the function $F(\cdot;\cdot)$ is asymptotically (compactly)
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic, _[D1.]-[D2.]_ hold,
the partial derivative $\frac{\partial F({\bf t};x)}{\partial t_{i}}$ exists
for all ${\bf t}\in{\mathbb{R}}^{n}$, $x\in X$ and it is uniformly continuous
on ${\mathcal{B}}.$ Then the function $\frac{\partial F(\cdot;\cdot)}{\partial
t_{i}}$ is asymptotically (compactly) $({\mathrm{R}},{\mathcal{B}})$-multi-
almost automorphic.

###### Proof.

We will prove only (i) because (ii) follows similarly, by appealing to
Proposition 2.24 instead of Proposition 2.16. The proof immediately follows
from the fact that the sequence $(F_{j}(\cdot;\cdot)\equiv
j[F(\cdot+j^{-1}e_{i};\cdot)-F(\cdot;\cdot)])$ of (compactly)
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic functions converges
uniformly to the function $\frac{\partial F(\cdot;\cdot)}{\partial t_{i}}$ as
$j\rightarrow+\infty.$ This can be shown as in one-dimesional case, by
observing that

$F_{j}(\cdot;\cdot)-\frac{\partial F(\cdot;\cdot)}{\partial
t_{i}}=j\int^{1/j}_{0}\Biggl{[}\frac{\partial F(\cdot+se_{i};\cdot)}{\partial
t_{i}}-\frac{\partial F(\cdot;\cdot)}{\partial t_{i}}\Biggr{]}\,ds.$

∎

###  2.3. Composition theorems for $({\mathrm{R}},{\mathcal{B}})$-multi-almost
automorphic type functions

Suppose that $F:{\mathbb{R}}^{n}\times X\rightarrow Y$ and
$G:{\mathbb{R}}^{m}\times Y\rightarrow Z$ are given functions, where
$m\in{\mathbb{N}}.$ The main aim of this subsection is to analyze the
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic properties of the
following multi-dimensional Nemytskii operator
$H:{\mathbb{R}}^{n}\times{\mathbb{R}}^{m}\times X\rightarrow Z$, given by

$H({\bf t};{\bf s};x):=G\bigl{(}{\bf s};F({\bf t};x)\bigr{)},\quad{\bf
t}\in{\mathbb{R}}^{n},\ {\bf s}\in{\mathbb{R}}^{m},\ x\in X.$

If $m=n,$ then we will also analyze the $({\mathrm{R}},{\mathcal{B}})$-multi-
almost automorphic properties of the following multi-dimensional Nemytskii
operator $W:{\mathbb{R}}^{n}\times X\rightarrow Z,$ given by

$W({\bf t};x):=G\bigl{(}{\bf t};F({\bf t};x)\bigr{)},\quad{\bf
t}\in{\mathbb{R}}^{n},\ x\in X.$

We will first state the following generalization of [19, Theorem 4.16]; the
proof is similar to the proof of the above-mentioned theorem but we will
present it for the sake of completeness:

######  Theorem 2.26.

Suppose that $F:{\mathbb{R}}^{n}\times X\rightarrow Y$ is
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic and
$G:{\mathbb{R}}^{n}\times X\rightarrow Y$ is
$({\mathrm{R}}^{\prime},{\mathcal{B}}^{\prime})$-multi-almost automorphic,
where ${\mathrm{R}}^{\prime}$ is a collection of all sequences
$b:{\mathbb{N}}\rightarrow{\mathbb{R}}^{n}$ from ${\mathrm{R}}$ and all their
subsequences, as well as

(2.12) $\displaystyle{\mathcal{B}}^{\prime}:=\Biggl{\\{}\bigcup_{{\bf
t}\in{\mathbb{R}}^{n}}F(t;B):B\in{\mathcal{B}}\Biggr{\\}}.$

If there exists a finite constant $L>0$ such that

(2.13) $\displaystyle\bigl{\|}G({\bf t};x)-G({\bf t};y)\bigr{\|}_{Z}\leq
L\|x-y\|_{Y},\quad{\bf t}\in{\mathbb{R}}^{n},\ x,\ y\in Y,$

then the function $W(\cdot;\cdot)$ is $({\mathrm{R}},{\mathcal{B}})$-multi-
almost automorphic.

###### Proof.

Let the set $B\in{\mathcal{B}}$ and the sequence $({\bf
b}_{k}=(b_{k}^{1},b_{k}^{2},\cdot\cdot\cdot,b_{k}^{n}))\in{\mathrm{R}}$ be
given. By definition, there exist a subsequence $({\bf
b}_{k_{l}}=(b_{k_{l}}^{1},b_{k_{l}}^{2},\cdot\cdot\cdot,b_{k_{l}}^{n}))$ of
$({\bf b}_{k})$ and a function $F^{\ast}:{\mathbb{R}}^{n}\times X\rightarrow
Y$ such that (2.6)-(2.7) hold true. Set $B^{\prime}:=\bigcup_{{\bf
t}\in{\mathbb{R}}^{n}}F(t;B)$ and $b^{\prime}:=({\bf b}_{k_{l}}).$ Then there
exist a subsequence $({\bf
b}_{k_{l_{m}}}=(b_{k_{l_{m}}}^{1},b_{k_{l_{m}}}^{2},\cdot\cdot\cdot,b_{k_{l_{m}}}^{n}))$
of $({\bf b}_{k_{l}})$ and a function $G^{\ast}:{\mathbb{R}}^{n}\times
X\rightarrow Y$ such that

(2.14) $\displaystyle\lim_{m\rightarrow+\infty}\Bigl{\|}G\bigl{(}{\bf
t}+(b_{k_{l_{m}}}^{1},\cdot\cdot\cdot,b_{k_{l_{m}}}^{n});y\bigr{)}-G^{\ast}({\bf
t};y)\Bigr{\|}_{Z}=0$

and

(2.15) $\displaystyle\lim_{m\rightarrow+\infty}\Bigl{\|}G^{\ast}\bigl{(}{\bf
t}-(b_{k_{l_{m}}}^{1},\cdot\cdot\cdot,b_{k_{l_{m}}}^{n});y\bigr{)}-G({\bf
t};y)\Bigr{\|}_{Z}=0,$

pointwise for all $y\in B^{\prime}$ and ${\bf t}\in{\mathbb{R}}^{n}.$ It
suffices to show that

(2.16) $\displaystyle\lim_{m\rightarrow+\infty}\Bigl{\|}G\bigl{(}{\bf
t}+(b_{k_{l_{m}}}^{1},\cdot\cdot\cdot,b_{k_{l_{m}}}^{n});F\bigl{(}{\bf
t}+(b_{k_{l_{m}}}^{1},\cdot\cdot\cdot,b_{k_{l_{m}}}^{n});x\bigr{)}\bigr{)}-G^{\ast}({\bf
t};F^{\ast}({\bf t};x))\Bigr{\|}_{Z}=0$

and

(2.17) $\displaystyle\lim_{m\rightarrow+\infty}\Bigl{\|}G^{\ast}\bigl{(}{\bf
t}-(b_{k_{l_{m}}}^{1},\cdot\cdot\cdot,b_{k_{l_{m}}}^{n});F^{\ast}\bigl{(}{\bf
t}-(b_{k_{l_{m}}}^{1},\cdot\cdot\cdot,b_{k_{l_{m}}}^{n});x\bigr{)}\bigr{)}-G({\bf
t};F({\bf t};x))\Bigr{\|}_{Z}=0.$

We will prove only (2.16) since the proof of (2.17) is quite analogous. For
simplicity, denote
${\bf\tau_{m}}:=(b_{k_{l_{m}}}^{1},\cdot\cdot\cdot,b_{k_{l_{m}}}^{n})$ for all
$m\in{\mathbb{N}}$. We have (${\bf t}\in{\mathbb{R}}^{n},$ $x\in B,$
$m\in{\mathbb{N}}$):

$\displaystyle\Bigl{\|}G\bigl{(}{\bf t}+{\bf\tau_{m}};F\bigl{(}{\bf
t}+{\bf\tau_{m}};x\bigr{)}\bigr{)}-G^{\ast}({\bf t};F^{\ast}({\bf
t};x))\Bigr{\|}_{Z}$ $\displaystyle\leq\Bigl{\|}G\bigl{(}{\bf
t}+{\bf\tau_{m}};F\bigl{(}{\bf
t}+{\bf\tau_{m}};x\bigr{)}\bigr{)}-G^{\ast}({\bf
t}+{\bf\tau_{m}};F^{\ast}({\bf t};x))\Bigr{\|}_{Z}$
$\displaystyle+\Bigl{\|}G^{\ast}({\bf t}+{\bf\tau_{m}};F^{\ast}({\bf
t};x))-G^{\ast}({\bf t};F^{\ast}({\bf t};x))\Bigr{\|}_{Z}$ $\displaystyle\leq
L\Bigl{\|}F\bigl{(}{\bf t}+{\bf\tau_{m}};x\bigr{)}-F^{\ast}({\bf
t};x)\Bigr{\|}_{Y}+\Bigl{\|}G^{\ast}({\bf t}+{\bf\tau_{m}};F^{\ast}({\bf
t};x))-G^{\ast}({\bf t};F^{\ast}({\bf t};x))\Bigr{\|}_{Z}.$

Since $x\in B$ and $F^{\ast}({\bf t};x)\in B^{\prime}$ for all ${\bf
t}\in{\mathbb{R}}^{n}$, (2.16) follows by applying (2.6) and (2.16), which
completes the proof of the theorem. ∎

######  Remark 2.27.

Suppose that $F:{\mathbb{R}}^{n}\times X\rightarrow Y$ is compactly
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic,
$G:{\mathbb{R}}^{n}\times X\rightarrow Y$ is compactly
$({\mathrm{R}}^{\prime},{\mathcal{B}}^{\prime})$-multi-almost automorphic and
additionally satisfies that the limit equations (2.6)-(2.7) for
$G(\cdot;\cdot)$ and $G^{\ast}(\cdot;\cdot)$ hold uniformly on $K\times
B^{\prime},$ where $K$ is an arbitrary compact subset of ${\mathbb{R}}^{n}$
and $B^{\prime}$ is an arbitrary set in ${\mathcal{B}}^{\prime}.$ By the above
proof, the function $W(\cdot;\cdot)$ is then compactly
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic.

A slight modification of the proof of Theorem 2.30 (cf. also the proof of [19,
Theorem 4.17]) shows that the following result holds true:

######  Theorem 2.28.

Suppose that $F:{\mathbb{R}}^{n}\times X\rightarrow Y$ is
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic and
$G:{\mathbb{R}}^{n}\times X\rightarrow Y$ is
$({\mathrm{R}}^{\prime},{\mathcal{B}}^{\prime})$-multi-almost automorphic,
where ${\mathrm{R}}^{\prime}$ is a collection of all sequences
$b:{\mathbb{N}}\rightarrow{\mathbb{R}}^{n}$ from ${\mathrm{R}}$ and all their
subsequences, as well as ${\mathcal{B}}^{\prime}$ be given by (2.12). Set

${\mathcal{B}}^{{}^{\prime}*}:=\Biggl{\\{}\bigcup_{{\bf
t}\in{\mathbb{R}}^{n}}F^{*}(t;B):B\in{\mathcal{B}}\Biggr{\\}}.$

If

$\displaystyle(\forall B\in{\mathcal{B}})\ (\forall\epsilon>0)\
(\exists\delta>0)$ $\displaystyle\Bigl{(}x,\
y\in{\mathcal{B}}^{\prime}\cup{\mathcal{B}}^{{}^{\prime}*}\mbox{ and }\
\bigl{\|}x-y\bigr{\|}_{Y}<\delta\Rightarrow\bigl{\|}G({\bf t};x)-G({\bf
t};y)\bigr{\|}_{Z}<\epsilon,\ {\bf t}\in{\mathbb{R}}^{n}\Bigr{)},$

then the function $W(\cdot;\cdot)$ is $({\mathrm{R}},{\mathcal{B}})$-multi-
almost automorphic.

Now we proceed with the analysis of composition theorems for asymptotically
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic functions. Our first
result corresponds to Theorem 2.30 and [19, Theorem 4.34]:

######  Theorem 2.29.

Suppose that $F_{0}:{\mathbb{R}}^{n}\times X\rightarrow Y$ is
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic, $Q_{0}\in
C_{0,{\mathbb{R}}^{n},{\mathcal{B}}}({\mathbb{R}}^{n}\times X:Y)$ and $F({\bf
t};x)=F_{0}({\bf t};x)+Q_{0}({\bf t};x)$ for all ${\bf t}\in{\mathbb{R}}^{n}$
and $x\in X.$ Suppose further that $G_{1}:{\mathbb{R}}^{n}\times X\rightarrow
Y$ is $({\mathrm{R}}^{\prime},{\mathcal{B}}^{\prime})$-multi-almost
automorphic, where ${\mathrm{R}}^{\prime}$ is a collection of all sequences
$b:{\mathbb{N}}\rightarrow{\mathbb{R}}^{n}$ from ${\mathrm{R}}$ and all their
subsequences as well as ${\mathcal{B}}^{\prime}$ is defined by (2.12) with the
function $F(\cdot;\cdot)$ replaced therein by the function
$F_{0}(\cdot;\cdot),$ $Q_{1}\in
C_{0,{\mathbb{R}}^{n},{\mathcal{B}}_{1}}({\mathbb{R}}^{n}\times Y:Z),$ where

(2.18) $\displaystyle{\mathcal{B}}_{1}:=\Biggl{\\{}\bigcup_{{\bf
t}\in{\mathbb{R}}^{n}}F(t;B):B\in{\mathcal{B}}\Biggr{\\}},$

and $G({\bf t};x)=G_{1}({\bf t};x)+Q_{1}({\bf t};x)$ for all ${\bf
t}\in{\mathbb{R}}^{n}$ and $x\in X.$ If there exists a finite constant $L>0$
such that the estimate (2.13) holds with the function $G(\cdot;\cdot)$
replaced therein by the function $G_{1}(\cdot;\cdot),$ then the function
$W(\cdot;\cdot)$ is asymptotically $({\mathrm{R}},{\mathcal{B}})$-multi-almost
automorphic.

###### Proof.

Using the above assumptions and Theorem 2.30, we have that the function $({\bf
t};x)\mapsto G_{1}({\bf t};F_{0}({\bf t};x)),$ ${\bf t}\in{\mathbb{R}}^{n},$
$x\in X$ is $({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic.
Furthermore, we have the following decomposition

$\displaystyle W({\bf t};x)=G_{1}({\bf t};F_{0}({\bf t};x))+\Bigl{[}G_{1}({\bf
t};F({\bf t};x))-G_{1}({\bf t};F_{0}({\bf t};x))\Bigr{]}+Q_{1}({\bf t};F({\bf
t};x)),$

for any ${\bf t}\in{\mathbb{R}}^{n}$ and $x\in X.$ Since

$\Bigl{\|}G_{1}({\bf t};F({\bf t};x))-G_{1}({\bf t};F_{0}({\bf
t};x))\Bigr{\|}_{Z}\leq L\bigl{\|}Q_{0}({\bf t};x)\bigr{\|}_{Y},\quad{\bf
t}\in{\mathbb{R}}^{n},\ x\in X,$

we have that the function $({\bf t};x)\mapsto G_{1}({\bf t};F({\bf
t};x))-G_{1}({\bf t};F_{0}({\bf t};x)),$ ${\bf t}\in{\mathbb{R}}^{n},$ $x\in
X$ belongs to the space
$C_{0,{\mathbb{R}}^{n},{\mathcal{B}}}({\mathbb{R}}^{n}\times X:Z).$ The same
holds for the function $({\bf t};x)\mapsto Q_{1}({\bf t};F({\bf t};x)),$ ${\bf
t}\in{\mathbb{R}}^{n},$ $x\in X$ because of our choice of the collection
${\mathcal{B}}_{1}$ in (2.18). The proof of the theorem is thereby complete. ∎

Similarly we can prove the following result which corresponds to Theorem 2.28
and [19, Theorem 4.35]:

######  Theorem 2.30.

Suppose that $F_{0}:{\mathbb{R}}^{n}\times X\rightarrow Y$ is
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic, $Q_{0}\in
C_{0,{\mathbb{R}}^{n},{\mathcal{B}}}({\mathbb{R}}^{n}\times X:Y)$ and $F({\bf
t};x)=F_{0}({\bf t};x)+Q_{0}({\bf t};x)$ for all ${\bf t}\in{\mathbb{R}}^{n}$
and $x\in X.$ Suppose further that $G_{1}:{\mathbb{R}}^{n}\times X\rightarrow
Y$ is $({\mathrm{R}}^{\prime},{\mathcal{B}}^{\prime})$-multi-almost
automorphic, where ${\mathrm{R}}^{\prime}$ is a collection of all sequences
$b:{\mathbb{N}}\rightarrow{\mathbb{R}}^{n}$ from ${\mathrm{R}}$ and all their
subsequences as well as ${\mathcal{B}}^{\prime}$ is defined by (2.12) with the
function $F(\cdot;\cdot)$ replaced therein by the function
$F_{0}(\cdot;\cdot),$ $Q_{1}\in
C_{0,{\mathbb{R}}^{n},{\mathcal{B}}_{1}}({\mathbb{R}}^{n}\times Y:Z),$ where
${\mathcal{B}}_{1}$ is given through (2.18), and $G({\bf t};x)=G_{1}({\bf
t};x)+Q_{1}({\bf t};x)$ for all ${\bf t}\in{\mathbb{R}}^{n}$ and $x\in X.$ Set

${\mathcal{B}}_{2}:=\Biggl{\\{}\bigcup_{{\bf
t}\in{\mathbb{R}}^{n}}F_{0}(t;B):B\in{\mathcal{B}}\Biggr{\\}}\cup\Biggl{\\{}\bigcup_{{\bf
t}\in{\mathbb{R}}^{n}}F_{0}^{\ast}(t;B):B\in{\mathcal{B}}\Biggr{\\}}.$

If

$\displaystyle(\forall B\in{\mathcal{B}})\ (\forall\epsilon>0)\
(\exists\delta>0)$ $\displaystyle\Bigl{(}x,\
y\in{\mathcal{B}}_{1}\cup{\mathcal{B}}_{2}\mbox{ and }\
\bigl{\|}x-y\bigr{\|}_{Y}<\delta\Rightarrow\bigl{\|}G_{1}({\bf
t};x)-G_{1}({\bf t};y)\bigr{\|}_{Z}<\epsilon,\ {\bf
t}\in{\mathbb{R}}^{n}\Bigr{)},$

then the function $W(\cdot;\cdot)$ is asymptotically
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic.

Composition theorems for (asymptotically) compactly
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic functions will be
considered somewhere else (see e.g., [19, Lemma 4.36, Lemma 4.37, Lemma 4.38]
and [2] for the one-dimensional case).

###  2.4. Convolution of $({\mathrm{R}},\mathcal{B})$-multi-almost automorphic
functions

For ${\bf t}=(t_{1},t_{2},\cdots,t_{n})$, let us introduce the notation
$\mathcal{I}_{{\bf
t}}=(-\infty,t_{1}]\times(-\infty,t_{2}]\times\cdots\times(-\infty,t_{n}]$ and
also impose the following condition:

  * (E1)

$\sup_{{\bf t}}\displaystyle\int_{\mathcal{I}_{\bf t}}K({\bf
t-\eta})d\eta\,<\,+\infty\,.$

######  Theorem 2.31.

Let $f:\mathbb{R}^{n}\longrightarrow X$ be a ${\mathrm{R}}$-multi-almost
automorphic function and $K$ is a kernel that satisfies the condition (E1).
Then

$F({\bf t})=\int_{\mathcal{I}_{\bf t}}K({\bf t}-\eta)f(\eta)d\eta$

yields a ${\mathrm{R}}$-multi-almost automorphic function.

###### Proof.

First of all observe that, using condition (E1) and the Lebesgue’s dominated
convergence Theorem we have that $F$ is a continuous function of ${\bf t}$. On
the other hand, since $f$ is ${\mathrm{R}}$-multi-AA, given a sequence
$b_{n}\in\mathrm{R}$, there exist a subsequence $c_{n}$ of $b_{n}$ and a
function $\tilde{f}$ such that the following pointwise limits holds:

$\lim_{n\to\infty}f({\bf t}+c_{n})=\tilde{f}({\bf t}),$

and

$\lim_{n\to\infty}\tilde{f}({\bf t}-c_{n})=f({\bf t}).$

Now, let us take the (well-defined) function:

$F^{*}({\bf t}):=\int_{\mathcal{I}_{\bf t}}K({\bf
t}-\eta)\tilde{f}(\eta)d\eta\,.$

Then, we have

$\displaystyle\|F({\bf t}+c_{n})-F^{*}(t)\|$ $\displaystyle=$
$\displaystyle\Big{|}\int_{\mathcal{I}_{{\bf t}+c_{n}}}K({\bf
t}+c_{n}-\eta)f(\eta)d\eta-\int_{\mathcal{I}_{\bf t}}K({\bf
t}-\eta)\tilde{f}(\eta)d\eta\Big{|}$ $\displaystyle\leq$
$\displaystyle\int_{\mathcal{I}_{\bf t}}|K({\bf
t}-\eta)|\|f(\eta+c_{n})+\tilde{f}(\eta)\|d\eta\,.$

From which, using condition (E1) and Lebesgue’s dominated convergence Theorem,
we have

$\lim_{n\to\infty}F({\bf t}+c_{n})=F^{*}(\bf t);$

similarly, we obtain

$\lim_{n\to\infty}F^{*}({\bf t}-c_{n})=F(\bf t).$

∎

Let $\mathbb{D}$ be an unbounded subset of $\mathbb{R}^{n}$ and $K$ a kernel
function defined on $\mathbb{R}^{n}$. For the invariance of the space of
$\mathrm{R}$-multi-asymptotically almost automorphic functions under some
particular integral operators, we need the following additional conditions:

  * (E2)

$\displaystyle\lim_{|{\bf t}|\to+\infty,{\bf
t}\in\mathbb{D}}\int_{\mathcal{I}_{\bf t}\cap\mathbb{D}^{c}}K({\bf
t-\eta})d\eta\,=0\,.$

  * (E3)

for every $r>0$ (big enough), it is satisfied that

$\displaystyle\lim_{|{\bf t}|\to+\infty,{\bf
t}\in\mathbb{D}}\int_{\mathcal{I}_{\bf t}\cap\mathbb{D}\cap B(0,r)}K({\bf
t-\eta})d\eta\,=0\,.$

######  Theorem 2.32.

Assume that $(E1),\,(E2)$ and $(E3)$ be fulfilled. Let $\mathbb{D}_{{\bf
t}}=\mathcal{I}_{{\bf t}}\cap\mathbb{D}$ be such that $int(\mathbb{D}_{{\bf
t_{0}}})\not=\emptyset$ for some ${\bf t_{0}}\in\mathbb{R}^{n}$. Then, the
integral

$\Gamma f({\bf t})=\int\limits_{\mathbb{D}_{{\bf t}}}K({\bf
t}-\eta)f(\eta)d\eta\,,$

defines an operator $\Gamma$ from the space of $\mathbb{D}$\- asymptotically
${\mathrm{R}}$-multi-almost automorphic functions to itself.

###### Proof.

Let $f=f_{a}+f_{0}$, be a $\mathbb{D}$-asymptotically ${\mathrm{R}}$-multi-
almost automorphic function, where $f_{a}$ is ${\mathrm{R}}$-multi-almost
automorphic and $f_{0}$ is in $C_{0,\mathbb{D}}(\mathbb{R}^{n};Y)$, then

$\displaystyle\int\limits_{\mathbb{D}_{{\bf t}}}K({\bf t}-\eta)f(\eta)d\eta$
$\displaystyle=$ $\displaystyle\int\limits_{\mathbb{D}_{{\bf t}}}K({\bf
t}-\eta)\left(f_{a}(\eta)+f_{0}(\eta)\right)d\eta$ $\displaystyle=$
$\displaystyle\int\limits_{\mathcal{I}_{\bf t}}K({\bf
t}-\eta)f_{a}(\eta)d(\eta)-\int\limits_{\mathcal{I}_{\bf
t}\cap\mathbb{D}^{c}}K({\bf t}-\eta)f_{a}(\eta)d(\eta)+$ $\displaystyle+$
$\displaystyle\int\limits_{\mathbb{D}_{\bf t}}K({\bf t}-\eta)f_{0}(\eta)d\eta$
$\displaystyle=$ $\displaystyle\Gamma_{1}f_{a}({\bf t})+\Gamma_{2}f({\bf
t})\,,$

where

$\Gamma_{1}f_{a}({\bf t}):=\int\limits_{\mathcal{I}_{{\bf t}}}K({\bf
t}-\eta)f_{a}(\eta)d(\eta)\,,$

and

$\Gamma_{2}f({\bf t}):=\int\limits_{\mathbb{D}_{\bf t}}K({\bf
t}-\eta)f_{0}(\eta)d\eta-\int\limits_{\mathcal{I}_{{\bf
t}}\cap\mathbb{D}^{c}}K({\bf t}-\eta)f_{a}(\eta)d(\eta)\,.$

From Theorem 2.31 we have that $\Gamma_{1}f_{a}$ is $\mathrm{R}$-multi-almost
automorphic. Now let us prove the following decay at infinity across
$\mathbb{D}$:

$\lim_{{\bf t}\in\mathbb{D}:|{\bf t}|\to+\infty}\Gamma_{2}f({\bf t})=0\,.$

First note that since $||f_{a}||_{\infty}<\infty$ and condition (E2) hods,
then:

$\lim_{{\bf t}\in\mathbb{D}:|{\bf t}|\to+\infty}\int\limits_{\mathcal{I}_{\bf
t}\cap\mathbb{D}^{c}}K({\bf t}-\eta)f_{a}(\eta)d(\eta)=0\;,$

therefore, the second integral in the representation of $\Gamma_{2}f$ vanishes
at infinity, it rest to work with the first integral. Let $\epsilon>0$, since
$f_{0}\in C_{0,\mathbb{D}}(\Omega;Y)$ there exists $r>0$ (big enough) such
that for all ${\bf t}\in\mathbb{D}$ with $|{\bf t}|>r$ we have $|f_{0}({\bf
t})|<\epsilon$, moreover because of condition (E3), for this $r$ and for
$|{\bf t}|$ big enough we have:

$\int\limits_{B(0,r)\cap\mathbb{D}_{{\bf t}}}K({\bf t}-\eta)d\eta<\epsilon\,;$

therefore, since

$\int\limits_{\mathbb{D}_{{\bf t}}}K({\bf
t}-\eta)f_{0}(\eta)d\eta=\int\limits_{B(0,r)\cap\mathbb{D}_{{\bf t}}}K({\bf
t}-\eta)f_{0}(\eta)d\eta+\int\limits_{B(0,r)^{c}\cap\mathbb{D}_{{\bf
t}}}K({\bf t}-\eta)f_{0}(\eta)d\eta\,,$

we conclude

$\Big{|}\int\limits_{\mathbb{D}_{{\bf t}}}K({\bf
t}-\eta)f_{0}(\eta)d\eta\Big{|}<\left(||f_{0}||_{\infty}+\int\limits_{B(0,r)^{c}\cap\mathbb{D}_{{\bf
t}}}|K({\bf t}-\eta)|d\eta\right)\epsilon\,.$

∎

######  Example 2.33.

In $\mathbb{R}^{2}$, let us consider the set $\mathbb{D}$ formed by the union
of lines passing throughout a fixed point $p\in\mathbb{R}^{2}$, then we have

$\int_{\mathbb{D}_{\bf t}}K({\bf t}-\eta)f(\eta)d\eta=0\,,$

for every ${\bf t}\in\mathbb{R}^{2}$. More generally, if $\mathbb{D}$ consists
of sets contained in euclidean spaces of dimension less than $n$, and after
the canonical embedding of it into $\mathbb{R}^{n}$, we have that

$\int_{\mathbb{D}_{\bf t}}K({\bf t}-\eta)f(\eta)d\eta=0\,,$

for every ${\bf t}\in\mathbb{R}^{n}$. This clarifies the necessity of the
condition $int(\mathbb{D}_{\bf t_{0}})\not=\emptyset$ for some ${\bf
t_{0}}\in\mathbb{R}^{n}$ (in previous Theorem), in order to have no nontrivial
functions in the integral representation of $\Gamma f$.

######  Example 2.34.

In the special case in which
$\mathbb{D}=[\alpha_{1},+\infty)\times[\alpha_{2},+\infty)\times\cdots\times[\alpha_{n},+\infty)$
we have $\mathbb{D}_{{\bf
t}}=[\alpha_{1},t_{1}]\times[\alpha_{2},t_{2}]\times\cdots\times[\alpha_{n},t_{n}]:=[{\bf\alpha},{\bf
t}]$ and under the hypothesis (E1), (E2) and (E3) and assuming the notation

$\int_{{\bf\alpha}}^{{\bf
t}}:=\int_{\alpha_{1}}^{t_{1}}\cdots\int_{\alpha_{n}}^{t_{n}}\,,$

we have that the integral function

$\Gamma f({\bf t})=\int_{{\bf\alpha}}^{{\bf t}}K({\bf t}-\eta)f(\eta)d\eta\,,$

is $\mathbb{D}$\- asymptotically ${\mathrm{R}}$ \- multi-almost automorphic
function, provided that $f$ is.

######  Example 2.35.

Let $\alpha,\beta$ be positive real numbers and consider the kernel function
$K_{e}(x-s,y-t)=\exp(-\alpha(x-s))\exp(-\beta(y-t))$ and ${\bf t}=(x,y)$.

Nevertheless if $\mathbb{D}$ is the first quadrant
$[0,+\infty)\times[0,+\infty)$ and $f$ an asymptotically $\mathbb{D}$-multi-
almost automorphic function, then

$F({\bf t})=\iint\limits_{\mathbb{D}_{{\bf t}}}K_{e}(x-s,y-t)f(s,t)dsdt\,,$

is not $\mathbb{D}$\- asymptotically ${\mathrm{R}}$ \- multi-almost
automorphic. In fact, let us consider the asymptotically $\mathbb{D}$-multi-
almost automorphic function $f({\bf t})=1+e^{-(\alpha s+\beta t)}$ where ${\bf
t}=(s,t)$; then

$\displaystyle F({\bf t})$ $\displaystyle=$
$\displaystyle\iint\limits_{\mathbb{D}_{{\bf t}}}K_{e}(x-s,y-t)(1+e^{-(\alpha
s+\beta t)})dsdt$ $\displaystyle=$
$\displaystyle\iint\limits_{\mathbb{D}_{{\bf
t}}}K_{e}(x-s,y-t)dsdt+\iint\limits_{\mathbb{D}_{{\bf
t}}}K_{e}(x-s,y-t)e^{-(\alpha s+\beta t)}dsdt$ $\displaystyle=$
$\displaystyle\iint\limits_{\mathcal{I}_{{\bf
t}}}K_{e}(x-s,y-t)dsdt-\iint\limits_{\mathcal{I}_{{\bf
t}}\cap\mathbb{D}^{c}}K_{e}(x-s,y-t)dsdt+$ $\displaystyle+$
$\displaystyle\iint\limits_{\mathbb{D}_{{\bf t}}}K_{e}(x-s,y-t)e^{-(\alpha
s+\beta t)}dsdt$ $\displaystyle=$ $\displaystyle F_{1}({\bf t})+F_{2}({\bf
t}).$

where,

$F_{1}({\bf t}):=\iint\limits_{\mathcal{I}_{{\bf t}}}K_{e}(x-s,y-t)dsdt\,,$

and

$F_{2}({\bf t}):=\iint\limits_{\mathbb{D}_{{\bf t}}}K_{e}(x-s,y-t)e^{-(\alpha
s+\beta t)}dsdt-\iint\limits_{\mathcal{I}_{{\bf
t}}\cap\mathbb{D}^{c}}K_{e}(x-s,y-t)dsdt\,.$

We see that $F_{1}$ is $\mathrm{R}$-multi-almost automorphic (note that
$K_{e}$ satisfy condition (E1)). On the other hand $F_{2}$ is not
$\mathrm{R}$-multi-almost automorphic and for $(x_{0},y)\in\mathbb{D}$ with
fixed $x_{0}\in(0,+\infty)$, we have

$\lim_{|(x_{0},y)|\to+\infty}F_{2}(x_{0},y)\not=0\,.$

Now let us prove a convolution type result for compact $\mathrm{R}$-multi-
almost automorphic function. Let us first introduce the following definition
of $\mathrm{R}$-multi-bi almost autmorphy:

######  Definition 2.36.

A jointly continuous function $G:\mathbb{R}^{n}\times\mathbb{R}^{n}\times X\to
Y$ is $(\mathrm{R},\mathcal{B})$-multi-bi-almost automorphic if for any
$B\in\mathcal{B}$ and any sequence $\\{s_{n}\\}\subset\mathrm{R}$, there exist
a subsequence $\\{s^{\prime}_{n}\\}\subseteq\\{s_{n}\\}$ and a function
$G^{*}:\mathbb{R}^{n}\times\mathbb{R}^{n}\times X\to Y$ such that

(2.19) $\lim_{n\to+\infty}G({\bf t}+s_{n},{\bf s}+s_{n},x)=G^{*}({\bf t},{\bf
s},x)\,$

and

(2.20) $\lim_{n\to+\infty}G^{*}({\bf t}-s_{n},{\bf s}-s_{n},x)=G({\bf t},{\bf
s},x)\,,$

pointwise for $({\bf t},{\bf s})\in\mathbb{R}^{n}\times\mathbb{R}^{n}$ and any
$x\in B$. If the limits in (2.19) and (2.20) are uniform on compact sets of
$\mathbb{R}^{n}\times\mathbb{R}^{n}$, then we say that $G$ is compactly
$(\mathrm{R},\mathcal{B})$-multi-bi-almost automorphic.

######  Remark 2.37.

Let us note that, this generalization of the notion of Bi-almost
automorphicity is different from the one given by the notion of
$\mathrm{R}$-multi almost automorphicity, in which

${\mathrm{R}}=\\{b:{\mathbb{N}}\rightarrow{\mathbb{R}}^{n}\,;\,\mbox{ for all
}j\in{\mathbb{N}}\mbox{ we have
}b_{j}\in\\{(a,a,a,\cdot\cdot\cdot,a)\in{\mathbb{R}}^{n}:a\in{\mathbb{R}}\\}\\}.$

Next, we introduce some abstract results about the existence of compactly
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic (resp. asymptotically
$({\mathrm{R}},{\mathcal{B}})$-multi-almost automorphic) solutions.

######  Theorem 2.38.

Let $Z$ be a Banach space and $G:\mathbb{R}^{l}\times\mathbb{R}^{l}\times X\to
Z$ be compact $(\mathrm{R},\mathcal{B})$-multi-bi-almost automorphic such that
there exists $\lambda:\mathbb{R}^{l}\longrightarrow[0,+\infty)$ satisfying

(2.21) $\|G({\bf t},{\bf s};x)-G({\bf t},{\bf s};y)\|_{Z}\leq\lambda({\bf
t}-{\bf s})\|x-y\|\quad\text{ for all }x,y\in X,$

with $\sup_{{\bf t}\in\mathbb{R}^{l}}\int_{\mathbb{R}^{l}}\lambda({\bf t}-{\bf
s})d{\bf s}<\infty$. Then, the operator $\Gamma$ defined by

$\Gamma u({\bf t})=\int_{\mathbb{R}^{l}}G({\bf t},{\bf s};u({\bf s}))d{\bf
s}\,,$

leaves invariant the space of $\mathrm{R}$-compact multi almost automorphic
functions.

###### Proof.

Let $u$ be compact $(\mathrm{R},\mathcal{B})$-multi-almost automorphic. Since
the function $G$ is compact $\mathrm{R}$-multi-bi-almost automorphic, we can
ensure that for given any sequence $\\{\bf s^{\prime}_{n}\\}\subset\mathrm{R}$
there exist a subsequence $\\{\bf s_{n}\\}\subseteq\\{\bf s^{\prime}_{n}\\}$
and a continuous function $G^{\ast}:\mathbb{R}^{l}\times\mathbb{R}^{l}\times
X\to Z$ such that on compact subsets of $\mathbb{R}^{l}\times\mathbb{R}^{l}$
the following uniform limits hold

$\lim\limits_{n\to+\infty}G({\bf t}+{\bf s_{n}},{\bf s}+{\bf
s_{n}};x)=:G^{\ast}({\bf t},{\bf
s};x),\quad\lim\limits_{n\to+\infty}G^{\ast}({\bf t}-{\bf s_{n}},{\bf
s-s_{n}};x)=G({\bf t},{\bf s};x),$

uniformly for $x\in B$. Moreover, there exists a function
$\tilde{u}:\mathbb{R}^{l}\longrightarrow X$ such that on compact subsets of
$\mathbb{R}^{l}$ the following uniform limits also hold

$\lim\limits_{n\to+\infty}u({\bf t}+{\bf s_{n}})=:\tilde{u}({\bf
t}),\quad\lim\limits_{n\to+\infty}\tilde{u}({\bf t}-{\bf s_{n}})=u({\bf t}).$

Let $v({\bf t}):=\Gamma u({\bf t})$ and $\tilde{v}({\bf
t}):=\displaystyle\int_{\mathbb{R}^{l}}G^{\ast}({\bf t},{\bf s};\tilde{u}({\bf
s}))d{\bf s}$. Then, it is clear by the Lebesgue’s dominate convergence
theorem that

$\displaystyle\|v({\bf t}+{\bf s_{n}})-\tilde{v}({\bf t})\|=$
$\displaystyle\|\int_{\mathbb{R}^{l}}G({\bf t}+{\bf s_{n}},{\bf s}+{\bf
s_{n}};u({\bf s}+{\bf s_{n}}))d{\bf s}-\int_{\mathbb{R}^{l}}G^{\ast}({\bf
t},{\bf s};\tilde{u}({\bf s}))d{\bf s}\|$
$\displaystyle\leq\|\int_{\mathbb{R}^{l}}G({\bf t}+{\bf s_{n}},{\bf s}+{\bf
s_{n}};u({\bf s}+{\bf s_{n}}))d{\bf s}-\int_{\mathbb{R}^{l}}G({\bf t}+{\bf
s_{n}},{\bf s}+{\bf s_{n}};\tilde{u}({\bf s}))d{\bf s}\|$
$\displaystyle+\|\int_{\mathbb{R}^{l}}G({\bf t}+{\bf s_{n}},{\bf s}+{\bf
s_{n}};\tilde{u}({\bf s}))d{\bf s}-\int_{\mathbb{R}^{l}}G^{\ast}({\bf t},{\bf
s};\tilde{u}({\bf s}))d{\bf s}\|$
$\displaystyle\leq\int_{\mathbb{R}^{l}}\lambda({\bf t}-{\bf s})\|u({\bf
s}+{\bf s_{n}}))-\tilde{u}({\bf s}))\|d{\bf s}$
$\displaystyle+\int_{\mathbb{R}^{l}}\sup_{x\in B}\|G({\bf t}+{\bf s_{n}},{\bf
s}+{\bf s_{n}};x)-G^{\ast}({\bf t},{\bf s};x)\|d{\bf s}.$

What we have proved is that $v=\Gamma u$ is an
$(\mathrm{R},\mathcal{B})$-multi almost automorphic function. Henceforth, in
view of Theorem (2.38), we prove that $v$ is
$(\mathrm{R},\mathcal{B})$-uniformly continuous. Let any two sequences $({\bf
a}_{k}=(a_{k}^{1},a_{k}^{2},\cdots,a_{k}^{n})),({\bf
b}_{k}=(b_{k}^{1},b_{k}^{2},\cdot\cdot\cdot,b_{k}^{n}))\in{\mathrm{R}}$ such
that $|{\bf a}_{k}-{\bf b}_{k}|\rightarrow 0$. Then, we have

(2.22) $\displaystyle\sup_{x\in B}\|G\bigl{(}{\bf a}_{k},{\bf
a}_{k};x\bigr{)}-G\bigl{(}{\bf b}_{k},{\bf b}_{k};x\bigr{)}\|_{Y}\rightarrow
0\quad\text{ as }k\rightarrow+\infty.$

It is clear that (2.22) implies that for all ${\bf t},{\bf
s}\in\mathbb{R}^{l}$, we have

(2.23) $\displaystyle\sup_{x\in B}\|G\bigl{(}{\bf t}+{\bf a}_{k},{\bf s}+{\bf
a}_{k};x\bigr{)}-G\bigl{(}{\bf t}+{\bf b}_{k},{\bf s}+{\bf
b}_{k};x\bigr{)}\|_{Y}\rightarrow 0\quad\text{ as }k\rightarrow+\infty.$

Furthermore, we have

(2.24) $\displaystyle\|u\bigl{(}{\bf a}_{k}\bigr{)}-u\bigl{(}{\bf
b}_{k}\bigr{)}\|_{Y}\rightarrow 0,$

as $k\rightarrow+\infty$. Then,

$\displaystyle\|v\bigl{(}{\bf a}_{k}\bigr{)}-v\bigl{(}{\bf b}_{k}\bigr{)}\|=$
$\displaystyle\|\int_{\mathbb{R}^{l}}G({\bf a}_{k},s;u({\bf s}))d{\bf
s}-\int_{\mathbb{R}^{l}}G({\bf b}_{k},{\bf s};u({\bf s}))d{\bf s}\|$
$\displaystyle\leq\|\int_{\mathbb{R}^{l}}G({\bf a}_{k},{\bf s}+{\bf
a}_{k};u({\bf s}+{\bf a}_{k}))d{\bf s}-\int_{\mathbb{R}^{l}}G({\bf b}_{k},{\bf
s}+{\bf b}_{k};u({\bf s}+{\bf b}_{k}))d{\bf s}\|$
$\displaystyle\leq\|\int_{\mathbb{R}^{l}}G({\bf a}_{k},{\bf s}+{\bf
a}_{k};u({\bf s}+{\bf a}_{k}))d{\bf s}-\int_{\mathbb{R}^{l}}G({\bf a}_{k},{\bf
s}+{\bf a}_{k};u({\bf s}+{\bf b}_{k}))d{\bf s}\|$
$\displaystyle+\|\int_{\mathbb{R}^{l}}G({\bf a}_{k},{\bf s}+{\bf a}_{k};u({\bf
s}+{\bf b}_{k}))d{\bf s}-\int_{\mathbb{R}^{l}}G({\bf b}_{k},{\bf s}+{\bf
b}_{k};u({\bf s}+{\bf b}_{k}))d{\bf s}\|$
$\displaystyle\leq\int_{\mathbb{R}^{l}}\lambda({\bf s})\|u({\bf s}+{\bf
a}_{k})-u({\bf s}+{\bf b}_{k})\|d{\bf s}$
$\displaystyle+\int_{\mathbb{R}^{l}}\sup_{x\in B}\|G({\bf a}_{k},{\bf s}+{\bf
a}_{k};x)-G({\bf b}_{k},{\bf s}+{\bf b}_{k};x)\|d{\bf s}.$

∎

From the proof of Theorem 2.38, one may deduce that the following result holds
immediately.

######  Corollary 2.39.

Let $Z$ be a Banach space and $G:\mathbb{R}^{l}\times\mathbb{R}^{l}\times X\to
Z$ be compact $(\mathrm{R},\mathcal{B})$-multi-bi-almost automorphic such that
there exists $\lambda:\mathbb{R}^{l}\longrightarrow[0,+\infty)$ satisfying

(2.25) $\|G({\bf t},{\bf s};x)-G({\bf t},{\bf s};y)\|_{Z}\leq\lambda({\bf
t}-{\bf s})\|x-y\|\quad\text{ for all }x,y\in X,$

with $\sup_{{\bf t}\in\mathbb{R}^{l}}\int_{\mathbb{R}^{l}}\lambda({\bf t}-{\bf
s})d{\bf s}<\infty$. Then, the operator $\Gamma$ defined by

$\Gamma u({\bf t})=\int_{\mathbb{R}^{l}}G({\bf t},{\bf s};u({\bf s}))d{\bf
s}\,,$

leaves invariant the space of $\mathrm{R}$-compact multi almost automorphic
functions.

######  Theorem 2.40.

Let $Z$ be a Banach space and $G:\mathbb{R}^{l}\times\mathbb{R}^{l}\times X\to
Z$ be compact $(\mathrm{R},\mathcal{B})$-multi-bi-almost automorphic such that
there exists $\lambda:\mathbb{R}^{l}\longrightarrow[0,+\infty)$ satisfying

(2.26) $\|G({\bf t},{\bf s};x)-G({\bf t},{\bf s};y)\|_{Z}\leq\lambda({\bf
t}-{\bf s})\|x-y\|\quad\text{ for all }x,y\in X,$

with $\sup_{{\bf t}\in\mathbb{R}^{l}}\int_{\mathbb{R}^{l}}\lambda({\bf t}-{\bf
s})d{\bf s}<\infty$. Assume that $\sup_{{\bf
t}\in\mathbb{R}^{l}}\int_{\mathbb{R}^{l}}\lambda({\bf t}-{\bf s})d{\bf s}<1$.
Then, $\Gamma$ has a unique compact $(\mathrm{R},\mathcal{B})$-multi-almost
automorphic solution defined by

$u({\bf t})=\int_{\mathbb{R}^{l}}G({\bf t},{\bf s};u({\bf s}))d{\bf
s},\quad{\bf t}\in\mathbb{R}^{l}.$

###### Proof.

Let $u,v$ be compact $(\mathrm{R},\mathcal{B})$-multi-almost automorphic
functions. Then,

$\displaystyle\|\Gamma u({\bf t})-\Gamma v({\bf t})\|_{Y}$ $\displaystyle\leq$
$\displaystyle\int_{\mathbb{R}^{l}}\|G({\bf t},{\bf s};u({\bf s}))-G({\bf
t},{\bf s};v({\bf s}))\|_{Z}\ d{\bf s}$ $\displaystyle\leq$
$\displaystyle\int_{\mathbb{R}^{l}}\lambda({\bf t}-{\bf s})\|u({\bf s})-v({\bf
s})\|_{Y}\ d{\bf s}$ $\displaystyle\leq$
$\displaystyle\int_{\mathbb{R}^{l}}\lambda({\bf t}-{\bf s})\ d{\bf
s}\|u-v\|_{\infty},\quad{\bf t}\in\mathbb{R}^{l}.$

Hence,

$\|\Gamma u-\Gamma v\|_{\infty}\leq\sup_{{\bf
t}\in\mathbb{R}^{l}}\int_{\mathbb{R}^{l}}\lambda({\bf t}-{\bf s})\ d{\bf
s}\|u-v\|_{\infty}.$

So, the result follows in view of the Banach contraction principle. ∎

Similarly to Theorem 2.40, we prove the existence and uniqueness results of
$\mathbb{D}$-asymptotically ${\mathrm{R}}$-multi-almost automorphic solutions
to equation. Moreover, we have the following existence result.

  * (E4)

Let $F:\mathbb{R}^{n}\times X\longrightarrow Y$ and $L_{F}\geq 0$ such that

$\|F({\bf t};x)-F({\bf t};y)\|_{Y}\leq L_{F}\|x-y\|,\quad{\bf
t}\in\mathbb{R}^{l},\,x,y\in X.$

######  Proposition 2.41.

Let $F,\ G:\mathbb{R}^{n}\times X\longrightarrow Y$ be
$\mathbb{D}$-asymptotically $({\mathrm{R}},\mathcal{B})$-multi-almost
automorphic such that the conditions $(E1)$-$(E4)$ (for $F$ and $G$ with
$L_{G}$-Lipschitz constant) are satisfied, and let $\mathbb{D}_{{\bf
t}}:=\mathcal{I}_{{\bf t}}\cap\mathbb{D}$ with $int(\mathbb{D}_{{\bf
t_{0}}})\not=\emptyset$ for some ${\bf t_{0}}\in\mathbb{R}^{n}$. Then, the
integral operator

$\Gamma u({\bf t})=G({\bf t};u({\bf t}))+\int\limits_{\mathbb{D}_{{\bf
t}}}K({\bf t}-\eta)F(\eta,u(\eta))d\eta\,,$

maps the space of $\mathbb{D}$-asymptotically ${\mathrm{R}}$-multi-almost
automorphic functions into itself.

###### Proof.

Let $u$ be $\mathbb{D}$-asymptotically ${\mathrm{R}}$-multi-almost
automorphic. Then, Theorem 2.30 yields that $\eta\longmapsto F(\eta,u(\eta)),\
G(\eta,u(\eta))$ are $\mathbb{D}$-asymptotically ${\mathrm{R}}$-multi-almost
automorphic functions. Hence, from Theorem 2.32, we deduce that ${\bf
t}\longmapsto\Gamma u({\bf t})$ as sum of two $\mathbb{D}$-asymptotically
${\mathrm{R}}$-multi-almost automorphic functions. ∎

######  Theorem 2.42.

Let $F,\ G:\mathbb{R}^{n}\times X\longrightarrow Y$ be
$\mathbb{D}$-asymptotically $({\mathrm{R}},\mathcal{B})$-multi-almost
automorphic such that the conditions $(E1)$-$(E4)$ are all satisfied, and let
$\mathbb{D}_{{\bf t}}:=\mathcal{I}_{{\bf t}}\cap\mathbb{D}$ with
$int(\mathbb{D}_{{\bf t_{0}}})\not=\emptyset$ for some ${\bf
t_{0}}\in\mathbb{R}^{n}$. Assume that $L_{G}+L_{F}\sup_{{\bf
t}\in\mathbb{R}^{l}}\int\limits_{\mathbb{D}_{{\bf t}}}K({\bf t}-\eta)<1$.
Then, the integral operator

$\Gamma u({\bf t})=G({\bf t};u({\bf t}))+\int\limits_{\mathbb{D}_{{\bf
t}}}K({\bf t}-\eta)F(\eta,u(\eta))d\eta\,,$

has a unique $\mathbb{D}$-asymptotically ${\mathrm{R}}$-multi-almost
automorphic solution i.e., $\Gamma u=u$.

###### Proof.

Let $u,v$ be $\mathbb{D}$-asymptotically ${\mathrm{R}}$-multi-almost
automorphic. Then,

$\displaystyle\|\Gamma u({\bf t})-\Gamma v({\bf t})\|_{Y}$ $\displaystyle\leq$
$\displaystyle\|G({\bf t};u({\bf t}))-G({\bf t};v({\bf
t}))\|+\int\limits_{\mathbb{D}_{{\bf t}}}K({\bf
t}-\eta)\|F(\eta,u(\eta))-F(\eta,v(\eta))\|d\eta$ $\displaystyle\leq$
$\displaystyle L_{G}\|u({\bf t})-v({\bf
t})\|+L_{F}\int\limits_{\mathbb{D}_{{\bf t}}}K({\bf
t}-\eta)\|u({\eta})-v({\eta})\|_{Y}\ d\eta$ $\displaystyle\leq$
$\displaystyle\left(L_{G}+L_{F}\sup_{{\bf
t}\in\mathbb{R}^{l}}\int\limits_{\mathbb{D}_{{\bf t}}}K({\bf t}-\eta)\
d\eta\right)\|u-v\|_{\infty},\quad{\bf t}\in\mathbb{R}^{l}.$

Hence,

$\|\Gamma u-\Gamma v\|_{\infty}\leq\left(L_{G}+L_{F}\sup_{{\bf
t}\in\mathbb{R}^{l}}\int\limits_{\mathbb{D}_{{\bf t}}}K({\bf t}-\eta)\
d\eta\right)\|u-v\|_{\infty}.$

So, the result follows in view of the Banach contraction principle. ∎

##  3\. Applications to the abstract multidimensional Volterra integral
equations and partial differential equations

In this section, we present different applications to a wide class of
nonlinear multi-dimensional Volterra integral equations, linear and nonlinear
partial differential equations, and integrodifferential equations.

###  3.1. Almost automorphic solutions to nonlinear multidimensional Volterra
integral equations with infinite delay

Let ${\bf t}=(x,y)$, we aim to study the following integral equation of
Volterra type with infinite delay:

(3.1) $f({\bf t})=g({\bf t})+\int_{-\infty}^{x}\int_{-\infty}^{y}K({\bf
t}-\eta)h(\eta,f(\eta))d\eta$

where $g:\mathbb{R}^{2}\to Y$ is compact $\mathrm{R}$-multi-almost automorphic
and $h:\mathbb{R}^{2}\times X\to Y$ is compactly
$(\mathrm{R},\mathcal{B})$-multi almost automorphic, and $\mathcal{B}$
consists of bounded subsets of $X$. Assume that $h$ is $L_{h}$-Lipschitzian in
the second variable. Define $G({\bf t},\eta;x):=K({\bf
t}-\eta)\chi_{\mathcal{I}_{{\bf t}}}(\eta)h(\eta,x)$. By construction, the map
$G$ is compactly $(\mathrm{R},\mathcal{B})$-multi-bi-almost automorphic and it
satisfies (2.25) with $\lambda(\cdot)=L_{h}\ K(\cdot)\chi_{\mathcal{I}_{{\bf
t}}}(\cdot)$ which is assumed to be integrable (if the kernel $K$ is
integrable) such that $L_{h}\sup_{{\bf
t}}\int_{-\infty}^{x}\int_{-\infty}^{y}K({\bf t}-\eta)d\eta<1$. Therefore,
since $g$ is compactly $\mathrm{R}$-multi-almost automorphic, it follows in
view of Theorem 2.40, that the integral operator in the equation (3.1) has a
unique compact $\mathrm{R}$-multi-almost automorphic solution.

######  Example 3.1.

Consider the following application to the two-dimensional nonlinear Volterra
integral equation of the second kind with infinite delay, see [3, 4] for some
examples in the absence of delay (i.e., $x,y\geq 0$), namely

(3.2)
$u(x,y)=g(x,y)+\int_{-\infty}^{x}\int_{-\infty}^{y}K(x,y,s,t,u(s,t))dsdt,\quad(x,y)\in\mathbb{R}^{2},$

where $u$ is the (unknown) state function called the solution of two-
dimensional integral Volterra equation, the kernel $K$ is a known nonlinear
function in $u$ and $g$ is also a known function. This kind of integral
equations have significant applications in various domains of applied science
and engineering, see the references mentioned in the papers [3, 4] about the
modeling overview of this kind of integral equations. Moreover, this kind of
multi-dimensional integral operators appears a solution of nonlinear elliptic
differential equations, see [18, Chapter 10]. Let us consider

$\displaystyle K(x,y,s,t,u(s,t))$ $\displaystyle:=$ $\displaystyle\gamma
e^{-|x-s|-|y-t|}\cos(s)\sin(t)\ln(1+\mid u(s,t)\mid)$ $\displaystyle=$
$\displaystyle k(s-x,t-y)h((s,t),u(s,t))\ln(1+\mid u(s,t)\mid),$

$k(\eta_{1},\eta_{2})=e^{-|\eta_{1}|-|\eta_{2}|}$ is the kernel and
$h(\eta_{1},\eta_{2},u(\eta_{1},\eta_{2}))=\gamma\cos(\eta_{1})\sin(\eta_{2})\ln(1+\mid
u(\eta_{1},\eta_{2})\mid)$ which is $\gamma$-Lipschitzian in the second
variable and $\gamma$ is chosen sufficiently small. Moreover, we take

$g(x,y)=\left[\sin(x)+\sin(\pi x)\right]\left[\cos(x)+\cos(\pi
x)\right]+\dfrac{1}{\sqrt{1+x^{2}+y^{2}}}.$

Hence, under the above considerations, equation (3.2) has the same form as
(3.1). Further on, all the assumptions and hypotheses of Section 3.1 are
satisfied. Consequently, equation (3.2) has a unique compact almost
automorphic solution.

###  3.2. Asymptotically almost automorphic solutions to abstract Volterra
integral equations

Let ${\bf t}=(x,y)$, $\mathbb{D}=[0,+\infty)\times[0,+\infty)$ and
$\mathbb{D}_{{\bf t}}=[0,x]\times[0,y]$. Here we prove the existence and
uniqueness of $\mathbb{D}$-asymptotically ${\mathrm{R}}$-multi-almost
automorphic solutions to the following abstract integral equation of Volterra
type, namely:

(3.3) $f({\bf t})=g({\bf t};f({\bf t}))+\int_{0}^{x}\int_{0}^{y}K({\bf
t}-\eta)h(\eta,f(\eta))d\eta,$

where $g:\mathbb{R}^{2}\times X\to Y$ and $h:\mathbb{R}^{2}\times X\to Y$ are
two $\mathbb{D}$-asymptotically $({\mathrm{R}},\mathcal{B})$-multi-almost
automorphic functions with respect to ${\bf t}$ and respectively $L_{g}$,
$L_{h}$ Lipschitzian with respect to $x$, whereas $\mathcal{B}$ consists of
the collection of bounded subsets of $X$. For the kernel $K$ it is assumed
that it satisfies conditions $(E1)$, $(E2)$ and $(E3)$. Hence, under the
assumption that $L_{g}+L_{h}\sup_{{\bf
t}\in\mathbb{R}^{l}}\int_{0}^{x}\int_{0}^{y}K({\bf t}-\eta)d\eta<1$ it yields
from Theorem 2.42 that the integral equation (3.3) has a unique
$\mathbb{D}$-asymptotically ${\mathrm{R}}$-multi-almost automorphic solution.

######  Example 3.2.

Consider the following nonlinear wave partial differential equation:

(3.4) $\dfrac{\partial^{2}u}{\partial t^{2}}=\dfrac{\partial^{2}u}{\partial
x^{2}}-\delta\cos(t)e^{-x}\sin(u),\,t,x\geq 0,$

with the initial conditions given on the $x=\pm t$ i.e., $u(x,x)=e^{-x}$ and
$u(t,-t)=-e^{-t}$. We argue similarly as in the paper [33] and we use the
transformation around the lines $y=x+t$ and $s=x-t$. Define the new state
function by

$v(y,s)=\dfrac{1}{4}u\left(\dfrac{1}{2}(Y+S),\dfrac{1}{2}(Y-S)\right).$

Then, a further calculation yields that equation (3.4) is equivalent to the
following integral two-dimensional Volterra nonlinear operators of the form

$\displaystyle v(y,s)$ $\displaystyle=$ $\displaystyle
g(y,s,u(y,s))-\delta\int_{0}^{y}\int_{0}^{s}\sin(v(Y,S))\cos\left(\dfrac{1}{2}(Y-S)\right)e^{-\dfrac{1}{2}(Y+S)}dYdS,$

where

$g(y,s,v(y,s))=e^{-\frac{y}{2}}-e^{-\frac{s}{2}}.$

Then the kernel $K(Y,S)=-e^{-\dfrac{1}{2}(Y+S)}$ satisfies the assumptions
$(E1)$, $(E2)$ and $(E3)$. The function $g$ is $\mathbb{D}$-asymptotically-
multi-almost automorphic and
$h(y,s,v(y,s))=\delta\cos(\dfrac{1}{2}(Y-S))\sin(v(Y,S))$, which is
$\delta$-Lipschitzian with respect to $v$ and $\mathbb{D}$-asymptotically-
multi-almost automorphic with respect to $(Y,S)$. Thus, by taking $\delta$
small enough, we obtain that the equations (3.4) has a unique
$\mathbb{D}$-asymptotically-multi-almost automorphic solution.

###  3.3. Poisson equation: $\mathrm{R}$-Compact multi-almost automorphic
solutions.

Let us denote by $\Delta$ the Laplace operator and $\mathrm{R}$ be the set of
sequences such that all its subsequences are also in $\mathrm{R}$. In this
subsection we shall prove the following main Theorem which extend (and
improve) Sibuya’s result for multi-almost periodic functions ([40]).

######  Theorem 3.3.

Let $f$ be an $\mathrm{R}$-multi-almost automorphic function and let $u\in
C_{b}(\mathbb{R}^{n})$. Assume that $u$ solves the (Poisson) equation i.e.,

(3.5) $\Delta u=f$

in the distributional sense. Then, $u$ is $\mathrm{R}$-compact almost
automorphic.

First, let us define what we mean by solution in the distributional sense. Let
$C^{\infty}_{c}(\mathbb{R}^{n})$ denote the space of test functions in
$\mathbb{R}^{n}$ i.e., functions of class $C^{\infty}$ with compact support in
$\mathbb{R}^{n}$.

######  Definition 3.4.

A function $u:\mathbb{R}^{n}\to\mathbb{R}$ is a solution of equation (3.5) in
the sense of distributional sense, if it satisfies

(3.6)
$\int_{\mathbb{R}^{n}}u(x)\Delta\phi(x)dx=\int_{\mathbb{R}^{n}}f(x)\phi(x)dx\,\,,\,\,\,\forall\phi\in
C^{\infty}_{c}(\mathbb{R}^{n})\,.$

For the proof, we follows Sibuya’s paper [40].

###### Proof.

(Proof of Theorem (3.3)) Let $\\{\zeta_{n}\\}$ be a sequence in $\mathrm{R}$.
Since $f$ is $\mathrm{R}$-multi almost automorphic, there exist a subsequence
$\\{\omega_{n}\\}\subset\\{\zeta_{n}\\}$ and a measurable bounded function
$g:\mathbb{R}^{n}\to\mathbb{R}$ such that the following limits hold:

(3.7) $\lim_{n\to+\infty}f(x+\omega_{n})=g(x)\quad\text{ and
}\lim_{n\to+\infty}g(x-\omega_{n})=f(x)\,$

pointwise in $x\in\mathbb{R}^{n}$. Moreover, since $f$ is bounded, it follows
in view of the formula (3.6), the sequence
$(u_{n}(\cdot):=u(\cdot+\omega_{n}))_{n}$ converges pointwise in
$x\in\mathbb{R}^{n}$. Furthermore, we point out that the uniform continuity of
$u$ yields (see [18] on $\mathbb{R}^{n}$ that the family $(u_{n}(\cdot))_{n}$
is equicontinuous. Therefore, we obtain the following uniform limit on compact
subsets of $\mathbb{R}^{n}$, i.e.,

(3.8) $\lim_{n\to+\infty}u(x+\omega_{n}):=v(x).$

Note that the function $v$ is also uniformly continuous and bounded. Now, we
prove that

(3.9) $\lim_{n\to+\infty}v(x-\omega_{n})=u(x),$

holds uniformly on compact subsets of $\mathbb{R}^{n}$. Notice first that $v$
is solution of the equation

$\Delta v=g\,,$

in the sense of distributions. Arguing as above and since $v$ is uniformly
continuous, there exist a subsequence $\\{\xi_{n}\\}\subset\\{\omega_{n}\\}$
and a bounded uniformly continuous functions $w$ such that the following limit
holds uniformly on compact subsets of $\mathbb{R}^{n}$

(3.10) $\lim_{n\to+\infty}v(x-\omega_{n})=w(x).$

Moreover $w$ satisfies the equation

$\Delta w=f,$

in the sense of distributions. Therefore, the function $u-w$ is a bounded and
harmonic. Hence, there exists a constant $C$ such that

$u=w+C.$

In order to prove that $C=0$, it suffices to see that

$\inf u\leq\inf v\leq\inf w$

and

$\sup u\geq\sup v\geq\sup w\,.$

This is consequence of (3.8), (3.9) and (3.10) respectively, which proves the
result.

∎

###  3.4. heat equation: : $\mathrm{R}$-almost automorphic solutions.

In this subsection we study the initial value problem for the homogeneous heat
equation with nonlocal diffusion

(3.11) $\displaystyle u_{t}-\Delta u$ $\displaystyle=$ $\displaystyle
0\,\,\,\,{\rm in}\,\,\,\,\mathbb{R}^{n}\times(0,+\infty),$ $\displaystyle
u(x,0^{+})$ $\displaystyle=$ $\displaystyle g(x)\,\,\,\,{\rm
in}\,\,\,\,\mathbb{R}^{n}\times\\{0\\},$

######  Proposition 3.5.

Let the initial profile $g$ of equation (3.11) be a $\mathrm{R}$-multi almost
automorphic function. Then its solution $u$ is $\mathrm{R}$-multi-almost
automorphic in space variable.

###### Proof.

From [23], we have that the representation formula of the solution for initial
value problem (3.11) is given by

(3.12) $u(x,t)=\int_{\mathbb{R}^{n}}\Phi(x-y,t)g(y)dy\,,$

where $\Phi(x,t)$ is the fundamental solution which satisfies

(3.13) $\int_{\mathbb{R}^{n}}\Phi(\xi,t)d\xi=1\,.$

The rest of the proof is similar to that of Proposition 2.23. ∎

Note that, if $f$ is $\mathrm{R}$-multi-almost periodic (compactly
${\mathrm{R}}$-multi-almost automorphic function), then its solution is also
$\mathrm{R}$-multi-almost periodic (compactly ${\mathrm{R}}$-multi-almost
automorphic function respectively).

###  3.5. Heat conduction in materials with memory: $\mathrm{R}$-multi-almost
automorphic solutions.

In this subsection we suppose that $\mathrm{R}\subset\mathbb{R}$ is a subset
such that conditions [D1] and [D2] holds, also we denote by
$\mathrm{R}AAA(\mathbb{R}^{+};X)$ and by $\mathrm{R}AAA(\mathbb{R}^{+}\times
X;X)$ the Banach space of asymptotically $\mathrm{R}$-almost automorphic
functions and asymptotically $(\mathrm{R},\mathcal{B})$-almost automorphic
functions respectively (see Theorem 2.24).

The following integrodifferential equation (with nonlocal initial data) models
the heat conduction in materials with memory

(3.14) $\displaystyle u^{\prime}(t)$ $\displaystyle=$ $\displaystyle
Au(t)+\int_{0}^{t}B(t-s)u(s)ds+f(t,u(t)),\ t\geq 0\;,$ (3.15) $\displaystyle
u(0)$ $\displaystyle=$ $\displaystyle u_{0}+g(u)\,;$

where $u_{0}\in X$, $A$ and $\\{B(t)\\}_{t\geq 0}$ are linear, closed and
densely defined operators on the Banach space $X$.

Results about existence and uniqueness of the asymptotically almost
automorphic solution to the integrodifferential equation (3.14)-(3.15) have
been studied in [11, 20]. This subsection is an adaptation of the results on
asymptotically almost automorphic solutions to integrodifferential equations
in [11] to the case of asymptotically $\mathrm{R}$-almost automorphic ones,
and we give details for the reader convenience. Let us start with the
following definition

######  Definition 3.6.

A family {$R(t)\\}_{t\geq 0}$ of continuous linear operators on $X$ is called
a resolvent operator for the equation (3.14) if, and only if:

  1. (R1)

$R(0)=I$, is the identity operator on $X$.

  2. (R2)

for all $x\in X$, the operator $t\rightarrow R(t)x$ is a continuous function
on $[0,+\infty)$.

  3. (R3)

For all $t\geq 0$ the operator $R(t)$ is continuous on $Y$, and for all $y\in
Y$, the application $t\rightarrow R(t)y$ belongs to $C([0,+\infty);Y)\cap
C^{1}([0,+\infty);X)$ and satisfies

$\frac{d}{dt}R(t)y=AR(t)y+\int_{0}^{t}B(t-s)R(s)yds=R(t)Ay+\int_{0}^{t}R(t-s)B(s)yds,\
t\geq 0,$

where $Y=D(A)=B(t)$ for all $t\geq 0$ and is equipped with the graph norm.
Details on resolvent operators and same conditions on their existence can be
found in [26, 37].

We assume that the resolvent operator $\\{R(t)\\}_{t\geq 0}$ of equation
(3.14) exists; under this situation, we can define a mild solution of
equations (3.14)-(3.15) as follows:

######  Definition 3.7.

A function $u\in C(\mathbb{R}^{+};X)$ is a mild solution of the nonlocal
integrodifferential equation (3.14)-(3.15) if

$u(t)=R(t)\left(u_{0}+g(u)\right)+\int_{0}^{t}R(t-s)f(s,u(s))ds,\ t\geq 0\,.$

######  Definition 3.8.

(Uniform exponential stability). The resolvent operator $\\{R(t)\\}_{t\geq 0}$
has uniform exponential stability, if there exist positive constants $M\geq
1,\,\delta>0$ such that for all $t\geq 0$ we have $\|R(t)\|\leq Me^{-\delta
t}$.

We say that the resolvent operator $\\{R(t)\\}_{t\geq 0}$ has property (R), if
it has uniform exponential stability.

The proof of the following two local theorems are adaptations of the ones
contained in [11].

######  Theorem 3.9.

Let $f\in\mathrm{R}AAA(\mathbb{R}^{+}\times X;X)$,
$g:\mathrm{R}AAA(\mathbb{R}^{+};X)\to X$, the resolvent operator has property
(R), $\rho>0$, and the set
$\Delta_{0}=\\{y\in\mathrm{R}AAA(\mathbb{R}^{+},X):||y-y_{0}||_{\infty}\leq\rho\\},$
where

$y_{0}(t)=R(t)(u_{0}+g(0))+\int_{0}^{t}R(t-s)f(s,0)ds.$

Suppose also that $||y_{0}||_{\infty}\leq\rho$, there exist positive constants
$L_{f},L_{g}$ such that:

$||f(t,x(t))-f(t,y(t))||\leq L_{f}||x(t)-y(t)||,\ x,y\in\Delta_{0},t\geq 0,$

and $g$ is $L_{g}$-Lipschitz in $\Delta_{0}$. If the constants
$L_{g},L_{f},\rho,M,\delta$ satisfy the inequality:

(3.16) $\delta L_{g}+L_{f}<\dfrac{\rho\delta}{M(\rho+||y_{0}||_{\infty})}\,;$

then, equation (3.14)-(3.15) has a unique mild solution in $\Delta_{0}$.

Note that, under hypothesis of theorem 3.9, we have the well defined operator
$\Gamma:\mathrm{R}AAA(\mathbb{R}^{+};X)\to\mathrm{R}AAA(\mathbb{R}^{+};X)$,
where

(3.17) $\Gamma
u(t)=R(t)\left(u_{0}+g(u)\right)+\int_{0}^{t}R(t-s)f(s,u(s))ds.$

######  Theorem 3.10.

Let $f\in\mathrm{R}AAA(\mathbb{R}^{+}\times X;X)$,
$g:\mathrm{R}AAA(\mathbb{R}^{+};X)\to X$, the resolvent operator satisfies
property (R), $\rho>0$ and the set
$\Delta_{0}=\\{y\in\mathrm{R}AAA(\mathbb{R}^{+};X):||y-y_{0}||_{\infty}\leq\rho\\},$
where

$y_{0}(t)=R(t)(u_{0}+g(0))+\int_{0}^{t}R(t-s)f(s,0)ds\,.$

Suppose, further that there exist positive constants $L_{f},L_{g}$ such that:

$\|f(t,x(t))-f(t,y(t))\|\leq L_{f}\|x(t)-y(t)\|,\ x,y\in\Delta_{0},\ t\geq 0,$

and $g$ is $L_{g}$-Lipschitz in $\Delta_{0}$. If $\Gamma$ is the operator
defined in (3.17) and the inequality

$0<\theta=\left(1-ML_{g}-\dfrac{M}{\delta}L_{f}\right)^{-1}||\Gamma
y_{0}-y_{0}||_{\infty}\leq\rho\,$

holds. Then equation (3.14)-(3.15) has a unique mild solution
$y\in\Delta_{0}$.

After introducing this two local abstract theorems for the integrodifferential
equation (3.14)-(3.15), let us apply the results to the heat conduction in
materials with memory.

Let $\Omega\subset\mathbb{R}^{3}$ be any open bounded set with smooth boundary
$\partial\Omega$. The heat conduction in materials with memory is described
with the following partial integrodifferential equation:

$\displaystyle\frac{\partial^{2}\theta}{\partial^{2}t}(x,t)+\beta(0)\frac{\partial\theta}{\partial
t}(x,t)$ $\displaystyle=$
$\displaystyle\alpha(0)\Delta\theta(x,t)-\int_{-\infty}^{t}\beta^{\prime}(t-s)\frac{\partial\theta}{\partial
t}(x,s)ds$ $\displaystyle+$
$\displaystyle\int_{-\infty}^{t}\alpha^{\prime}(t-s)\Delta\theta(x,s)ds+a(t)b(\theta(t)),$

where $\alpha,\beta\in C^{2}([0,+\infty[;\mathbb{R})$, are the thermal
relaxation function of the heat flux and the energy relaxation function
respectively, with $\alpha(0),\beta(0)$ positive and $\Delta$ is the Laplace
operator in $\Omega$. For isotropic materials, in which the temperature
$\theta(x,t)$ is know for all $t\leq 0$ and does not depend on the position
$x\in\Omega$, equation (3.5) can be rewritten as follows:

$\displaystyle\theta^{{}^{\prime\prime}}(t)+\beta(0)\theta^{{}^{\prime}}(t)$
$\displaystyle=$
$\displaystyle\alpha(0)\Delta\theta(t)-\int_{0}^{t}\beta^{\prime}(t-s)\theta^{{}^{\prime}}(s)ds+$
$\displaystyle+$
$\displaystyle\int_{0}^{t}\alpha^{\prime}(t-s)\Delta\theta(s)ds+a(t)b(\theta(t)).$

Introducing the function $\eta(t)=\theta^{\prime}(t)$, the previous equation
is written as the system:

$\displaystyle\theta^{\prime}(t)$ $\displaystyle=$ $\displaystyle\eta(t).$
$\displaystyle\eta^{\prime}(t)$ $\displaystyle=$
$\displaystyle-\beta(0)\eta(t)+\alpha(0)\Delta\theta(t)-\int_{0}^{t}\beta^{\prime}(t-s)\eta(s)ds+\int_{0}^{t}\alpha^{\prime}(t-s)\Delta\theta(s)ds+a(t)b(\theta(t))\,,$

that is :

$\displaystyle\begin{bmatrix}{\theta(t)}\\\
{\eta(t)}\end{bmatrix}^{{}^{\prime}}=\begin{bmatrix}{0}&{I}\\\
{\alpha(0)\Delta}&{-\beta(0)I}\end{bmatrix}\begin{bmatrix}{\theta(t)}\\\
{\eta(t)}\end{bmatrix}+\int_{0}^{t}\begin{bmatrix}{0}&{0}\\\
{\alpha^{\prime}(t-s)\Delta}&{-\beta^{\prime}(t-s)I}\end{bmatrix}\begin{bmatrix}{\theta(s)}\\\
{\eta(s)}\end{bmatrix}ds+\begin{bmatrix}{0}\\\
{a(t)b(\theta(t))}\end{bmatrix}.$

Now, consider the ambient space $X=H_{0}^{1}(\Omega)\times L^{2}(\Omega)$ and
identify the operators:

$\displaystyle A=\begin{bmatrix}{0}&{I}\\\
{\alpha(0)\Delta}&{-\beta(0)I}\end{bmatrix},\,B(t-s)=\begin{bmatrix}{0}&{0}\\\
{\alpha^{\prime}(t-s)\Delta}&{-\beta^{\prime}(t-s)I}\end{bmatrix}.$

Furthermore for $\begin{bmatrix}{\theta}\\\ {\eta}\end{bmatrix}$ $\in X$
identify:

$\displaystyle u=\begin{bmatrix}{\theta}\\\
{\eta}\end{bmatrix},\,\,f(t,u)=\begin{bmatrix}{0}\\\
{a(t)b(\theta)}\end{bmatrix}\,.$

Then, for each initial condition $u(0)=u_{0}=\begin{bmatrix}{\theta_{0}}\\\
{\eta_{0}}\end{bmatrix}\in X$ we have the following integrodifferential
equation with nonlocal initial condition:

(3.20) $\displaystyle u^{\prime}(t)$ $\displaystyle=$ $\displaystyle
Au(t)+\int_{0}^{t}B(t-s)u(s)ds+f(t,u(t)),\ t\geq 0,$ (3.21) $\displaystyle
u(0)$ $\displaystyle=$ $\displaystyle u_{0}+h(u),$

with $D(A)=(H^{2}(\Omega)\cap H_{0}^{1}(\Omega))\times H_{0}^{1}(\Omega)$.

It follows from [16] (see also [26]), that $A$ generates a semigroup
$\\{T(t)\\}_{t\geq 0}$ with $||T(t)||\leq Me^{-\gamma t}$ for all $t\geq 0$
and $M,\gamma$ positive constants . Let $B(t)=F(t)A$, where:

$\displaystyle F(t)=\begin{bmatrix}{0}&{0}\\\
{-\beta^{\prime}(t)I+\beta(0)\frac{\alpha^{\prime}(t)}{\alpha(0)}I}&{\frac{\alpha^{\prime}(t)}{\alpha(0)}I}\end{bmatrix}.$

It is well known that under the following conditions:

$R_{1})$. $\alpha^{\prime}(t)e^{\gamma
t},\alpha^{{}^{\prime\prime}}(t)e^{\gamma t},\beta^{\prime}(t)e^{\gamma
t},\beta^{{}^{\prime\prime}}(t)e^{\gamma t}$ are bounded and uniformly
continuous functions.

$R_{2})$. Let $p,q>1$ such that $1/p+1/q=1$ and for all $t\geq 0$,

$\max\\{||F_{21}(t)||,||F_{22}(t)||\\}\leq\frac{\gamma e^{-\gamma t}}{pM},\ \
\max\\{||F_{21}^{\prime}(t)||,||F_{22}^{\prime}(t)||\\}\leq\frac{\gamma^{2}e^{-\gamma
t}}{(pM)^{2}}\,.$

R.C. Grimmer has shown that the equation (3.20) has a resolvent operator
$\\{R(t)\\}_{t\geq 0}$ which satisfies condition (R) ; that is, $\forall t\geq
0:\ ||R(t)||\leq Me^{-\frac{\gamma t}{q}}$ (see [26] for original references).

Now, let us consider the following conditions

  1. (C1)

$y_{0}(t)=R(t)(u_{0}+h(0))+\int_{0}^{t}R(t-s)f(s,0)ds.$

  2. (C2)

$h(u)=\begin{bmatrix}{h_{1}(\xi)}\\\ {h_{2}(\eta)}\end{bmatrix}$, where
$h_{1}:C(\mathbb{R}^{+};H_{0}^{1}(\Omega))\to H_{0}^{1}(\Omega)$ and
$h_{2}:C(\mathbb{R}^{+};L^{2}(\Omega))\to L^{2}(\Omega)$ are Lipschitz with
the same Lipschitz’s constant $\dfrac{\rho}{pM(\rho+||y_{0}||_{\infty})}$.

  3. (C3)

$b:H_{0}^{1}(\Omega)\to L^{2}(\Omega)$ is also Lipschitz, with Lipschitz’s
constant

$\dfrac{\gamma\rho}{qM||a||(\rho+||y_{0}||_{\infty})}.$

Finally, we have the following theorem as an application to the heat
conduction in materials with memory

######  Theorem 3.11.

Let $a\in\mathrm{R}AAA(\mathbb{R}^{+};\mathbb{R})$ and consider a real number
$\rho>0$ such that:

(3.22) $\rho\geq
M\left(||u_{0}||+||h(0)||+\dfrac{q}{\gamma}||a||\;||b(0)||\right).$

Suppose that the conditions $(C1)-(C3)$ holds. Then, equation (3.20)-(3.21)
has a unique mild solution $y\in\mathrm{R}AAA(\mathbb{R}^{+};X)$ such that
$||y-y_{0}||_{\infty}\leq\rho$.

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