Multi-dimensional Almost Automorphic Type Functions and Applications

Abstract

In this paper, we introduce and analyze several new classes of multi-dimensional almost automorphic functions which generalize the classical one of Bochner. We develop the basic theory for the introduced classes, investigating the themes like composition principles, convolution invariance and the invariance under the actions of convolution products. We present several illustrative examples and applications to the abstract Volterra integro-differential equations and partial differential equations, providing also a mini appendix about almost automorphic functions on semi-topological groups.

Appendix: Almost Automorphic Functions on Semi-topological Groups

The first systematic study of almost automorphic functions on topological groups was conducted by Veech (1965, 1967) (see also the papers by Reich (1970) and Terras (1972)). Following Milnes (1977), who considered only the scalar-valued case, we say that a continuous function $f : G\rightarrow Y,$ where G is a (semi-)topological group, is almost automorphic if and only if for any sequence $(n_{i}')$ in G there exists a subsequence $(n_{i})$ of $(n_{i}')$ 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 semi-topological group as well as that the notion of $({\mathrm R}_{\mathrm X},{\mathcal B})$-multi-almost automorphy can be extended in this rather general framework. For more details about almost periodic functions on topological groups, see the research monograph by Levitan (1953) and the reference list given in the forthcoming monograph (Kostić 2021).

In this section, we will briefly explain the main ideas and results about almost automorphic functions on semi-topological groups established by Milnes (1977); we will also remind the readers of some known results about almost automorphic functions on topological groups obtained by other authors (there is a vast literature about topological groups and their generalizations; we will only refer the reader to the recent book (Morris 2019) edited by S. A. Morris and references cited therein).

Let G be a topological space which is also a multiplicative group. Then we say that G is a semi-topological space if and only if the mappings $s\mapsto st$ and $s\mapsto ts$ from G into G are continuous for all $t\in G$; furthermore, G is called a topological group if, in addition to the above, we have that the mapping $(s,t) \mapsto st^{-1}$ from $G\times G$ into G is continuous. By ${{\mathcal {J}}}$ we denote the topology on G and by $C_{b}(G:Y)$ we denote the space of all bounded continuous functions $f : G \rightarrow Y$ equipped with the sup-norm $\Vert \cdot \Vert _{\infty }.$ We say that:

• (i) a subset D of a semi-topological group G is left relatively dense if and only if there exists a finite set of elements $\{s_{i} : 1\le i\le N\}$ in G such that $G\subseteq \bigcup _{i=1}^{N}(s_{i}D);$

• (ii) a topological group G is totally bounded if and only if for every non-empty neighbourhood V in G we have the existence of a finite set of elements $\{s_{i} : 1\le i\le N\}$ in G such that $G\subseteq \bigcup _{i=1}^{N}(s_{i}V).$

For any $s\in G$, the left (right) translate $f_{s}$ ($f^{s}$) of f is defined through $f_{s}(\cdot ):=f(s\cdot )$ ($f_{s}(\cdot ):=f(\cdot s)$). A subspace C of $C_{b}(G:Y)$ is called translation invariant if and only if $f_{s}$ and $f^{s}$ belong to C for every $f\in C.$ If $f : G \rightarrow Y$ and $g : G \rightarrow Y$ are given functions and $(\alpha _{i})_{i\in I}$, resp. $(n_{i})_{i\in {{\mathbb {N}}}},$ is a net in G, resp. a sequence in G, then we write $T_{\alpha }f=g$ if and only if the net of left translations $f_{\alpha _{i}},$ resp. $f_{n_{i}}$, converges pointwise on G. The right uniformly continuous subspace $RUC_{b}( G : Y)$ of $C_{b}(G:Y)$ is defined as the set of all functions $f\in C_{b}(G:Y)$ such that $\Vert f^{\alpha _{i}}-f^{s}\Vert$ tends to zero whenever $(\alpha _{i})_{i\in I}$ is a net in G converging to $s\in G;$ the left continuous subspace $LUC_{b}( G : Y)$ of $C_{b}(G:Y)$ is defined similarly.

Definition 4.1

Let G be a semi-topological group. Then we say that a continuous function $f: G \rightarrow Y$ is left almost automorphic if and only if every net $\alpha ' \subseteq G$ has a subnet $\alpha \subseteq G$ such that $T_{\alpha }f=g$ and $T_{\alpha ^{-1}}g=f,$ where $\alpha ^{-1}= (\alpha _{i}^{-1});$ the notion of right almost automorphy is introduced similarly, with the analogous conditions involving right translates. By LAA(G : Y) and RAA(G : Y) we denote the family of all left almost automorphic functions on G and the right almost automorphic functions on G, respectively.

A function $f\in C_{b}(G : Y)$ is called almost periodic if and only if the set of all left translations $\{f_{s} : s\in G\}$ is relatively compact in $C_{b}(G : Y).$ Any almost periodic function $f\in C_{b}(G : Y)$ is left almost automorphic and satisfies that the convergence in $T_{\alpha }f=g$ is uniform on G,  along with the convergence in $T_{\alpha ^{-1}}g=f.$ We know that LAA(G : Y) and RAA(G : Y) are translation invariant spaces as well as that the limit $T_{\alpha }f=g$ need not be continuous on G.

Suppose, for the time being, that $Y={{\mathbb {C}}}.$ Then we know that, if G is a Hausdorff topological group that is complete in a left invariant metric or locally compact and $f\in C_{b}(G: {\mathbb C}),$ then we can always find a net $(\alpha _{i})_{i\in I}$ such that $T_{\alpha }f=g$ is discontinuous on G if and only if $f\notin RUC_{b}( G : {{\mathbb {C}}}).$ In what follows, it will be said that the Bohr topology ${\mathrm B}$ on a semi-topological group G is that topology which has the property that a subbase of ${\mathrm B}$-neighbourhoods of a point $s\in G$ forms the sets $\{t\in G : | f(t)-f(s)|<\epsilon \},$ where $f : G \rightarrow {{\mathbb {C}}}$ is almost periodic and $\epsilon >0;$ a function $f : G \rightarrow {{\mathbb {C}}}$ is said to be Bohr continuous if and only if the function $f(\cdot )$ is continuous for the Bohr topology. Due to Milnes (1977, Theorem 8), a necessary and sufficient condition for a topological group G to be totally bounded is that every continuous complex-valued function on G is Bohr continuous.

For the scalar-valued functions, Milnes (1977, Theorem 13) states that for any continuous function $f : G \rightarrow {{\mathbb {C}}}$, where G is a semi-topological group, the following conditions are mutually equivalent:

• 1. (2.) $f(\cdot )$ is left (right) almost automorphic.

• 3. $f(\cdot )$ is Bohr continuous.

• 4. For every $\epsilon >0$ and for every finite set $N\subseteq G$, there exists a left relatively dense subset $D\subseteq G \ni D^{-1}D \subseteq \{s\in G : \sup _{r,t\in N}|f(rst)-f(rt)|<\epsilon \}.$

• 5. (6.) For every $\epsilon >0$ and $t\in G$, there exists a left relatively dense subset $D\subseteq G \ni D^{-1}D \subseteq \{s\in G : \sup _{r,t\in N}|f(ts)-f(t)|<\epsilon \}$ ($D\subseteq G \ni D^{-1}D \subseteq \{s\in G : \sup _{r,t\in N}|f(st)-f(t)|<\epsilon \}$).

• 7. For every net $\alpha \subseteq G$, there exists a subnet $\alpha \subseteq G$ such that the joint limit $\lim _{i,j}f(s\alpha _{i}\alpha _{j}^{-1}t)=f(st)$ for all $s,\ t\in G.$

• 8. (9.) For every net $\alpha \subseteq G$, there exists a subnet $\alpha \subseteq G$ such that the joint limit $\lim _{i,j}f(\alpha _{i}\alpha _{j}^{-1}t)=f(t)$ for all $t\in G$ ($\lim _{i,j}f(t\alpha _{i}\alpha _{j}^{-1})=f(t)$ for all $t\in G$).

• 10. For every sequence $\mathbf{n}'\subseteq G$, there exists a subnet $\mathbf{n}\subseteq G$ such that the joint limit $\lim _{i,j}f(sn_{i}n_{j}^{-1}t)=f(st)$ for all $s,\ t\in G$

• 11. (12.) For every sequence $\mathbf{n}'\subseteq G$, there exists a subnet $\mathbf{n}\subseteq G$ such that the joint limit $\lim _{i,j}f(n_{i}n_{j}^{-1}t)=f(t)$ for all $t\in G$ ($\lim _{i,j}f(tn_{i}n_{j}^{-1})=f(t)$ for all $t\in G$).

Although it would be very unpleasant to clarify the validity or non-validity of above conditions for the vector-valued functions $f : G \rightarrow Y,$ especially for those Banach spaces Y which are not separable (see e.g., the proof of (Milnes 1977, Theorem 10)), we would like to note that some equivalence relations clarified above hold for the vector-valued functions $f : G \rightarrow Y$ on topological groups G. For example, B. Basit has proved, in Basit (1974, Theorem 1.2), that a bounded continuous function $f : G \rightarrow Y$ is almost automorphic if and only if $f(\cdot )$ is Levitan almost periodic (see (Basit 1974, Definition 1.1)), which immediately implies the equivalence of [1. (2.)] and [8. (9.)] in this framework. Keeping this in mind, it seems reasonable to further explore the following notion (more details will appear somewhere else; see also the research study of Weyl multi-dimensional almost automorphic functions carried out in Kostić (2021), where this approach has been essentially followed):

Definition 4.2

Suppose that $F : {{\mathbb {R}}}^{n} \times X \rightarrow Y$ is a continuous function as well as that for each $B\in {{\mathcal {B}}}$ and $(\mathbf{b}_{k}=(b_{k}^{1},b_{k}^{2},\ldots ,b_{k}^{n})) \in {\mathrm R}$ we have $W_{B,(\mathbf{b}_{k})} : B\rightarrow P(P({{\mathbb {R}}}^{n}))$ and ${\mathrm P}_{B,(\mathbf{b}_{k})}\in P(P({{\mathbb {R}}}^{n} \times B)).$ Then we say that $F(\cdot ;\cdot )$ is:

• (i) jointly $({\mathrm R},{{\mathcal {B}}})$-multi-almost automorphic if and only if for every $B\in {{\mathcal {B}}}$ and for every sequence $(\mathbf{b}_{k}=(b_{k}^{1},b_{k}^{2},\ldots ,b_{k}^{n})) \in {\mathrm R}$ there exists a subsequence $(\mathbf{b}_{k_{l}}=(b_{k_{l}}^{1},b_{k_{l}}^{2},\ldots , b_{k_{l}}^{n}))$ of $(\mathbf{b}_{k})$ such that

  $$\begin{aligned} \lim _{(l,m)\rightarrow +\infty } F\Bigl (\mathbf{t} -\bigl (b_{k_{l}}^{1},\ldots , b_{k_{l}}^{n}\bigr )+\bigl (b_{k_{m}}^{1},\ldots , b_{k_{m}}^{n}\bigr );x\Bigr )=F(\mathbf{t};x) , \end{aligned}$$

  (4.1)

  pointwisely for all $x\in B$ and $\mathbf{t}\in {\mathbb R}^{n};$

• (ii) jointly $({\mathrm R},{{\mathcal {B}}},W_{{{\mathcal {B}}},{\mathrm R}})$-multi-almost automorphic if and only if for every $B\in {{\mathcal {B}}}$ and for every sequence $(\mathbf{b}_{k}=(b_{k}^{1},b_{k}^{2},\ldots ,b_{k}^{n})) \in {\mathrm R}$ there exists a subsequence $(\mathbf{b}_{k_{l}}=(b_{k_{l}}^{1},b_{k_{l}}^{2},\ldots , b_{k_{l}}^{n}))$ of $(\mathbf{b}_{k})$ such that (4.1) holds pointwisely for all $x\in B$ and $\mathbf{t}\in {{\mathbb {R}}}^{n}$ as well as that for each $x\in B$ the convergence in (4.1) is uniform in $\mathbf{t}$ for any set of the collection $W_{B,(\mathbf{b}_{k})}(x);$

• (iii) jointly $({\mathrm R},{{\mathcal {B}}},{\mathrm P}_{{{\mathcal {B}}},{\mathrm R}})$-multi-almost automorphic if and only if for every $B\in {{\mathcal {B}}}$ and for every sequence $(\mathbf{b}_{k}=(b_{k}^{1},b_{k}^{2},\ldots ,b_{k}^{n})) \in {\mathrm R}$ there exists a subsequence $(\mathbf{b}_{k_{l}}=(b_{k_{l}}^{1},b_{k_{l}}^{2},\ldots , b_{k_{l}}^{n}))$ of $(\mathbf{b}_{k})$ such that (4.1) holds pointwisely for all $x\in B$ and $\mathbf{t}\in {{\mathbb {R}}}^{n}$ as well as that the convergence in (4.1) is uniform in $(\mathbf{t};x)$ for any set of the collection ${\mathrm P}_{B,(\mathbf{b}_{k})}.$

Arguing as above, it can be simply verified that any $({\mathrm R},{{\mathcal {B}}})$-multi-almost periodic function $F : {\mathbb R}^{n} \times X \rightarrow Y$ is jointly $({\mathrm R},{\mathcal B},{\mathrm P}_{{{\mathcal {B}}},{\mathrm R}})$-multi-almost automorphic with ${\mathrm P}_{{{\mathcal {B}}},{\mathrm R}}=\{ {{\mathbb {R}}}^{n} \times B\}.$ We also have the following:

Proposition 4.3

Suppose that $F : {{\mathbb {R}}}^{n} \rightarrow Y$ is a c-uniformly recurrent function (see Definition 1.2-(ii)), where the sequence $(\mathbf{\tau }_{k})$ satisfies $\lim _{k\rightarrow +\infty } |\mathbf{\tau }_{k}|=+\infty$ and (1.3). Let ${\mathrm R}$ denote the collection consisting of the sequence $({\varvec{\tau }}_{k})$ and all its subsequences. Then the function $F(\cdot )$ is jointly $({\mathrm R},{\mathrm P}_{{\mathrm R}})$-multi-almost automorphic with ${\mathrm P}_{{\mathrm R}}$ being the singleton $\{ {\mathbb R}^{n}\}.$

Proof

Let $({\varvec{\tau }}_{k}')$ be any subsequence of $({\varvec{\tau }}_{k})$. Then we have (1.3) with the sequence $(\mathbf{\tau }_{k})$ replaced with the sequence $({\varvec{\tau }}_{k}')$ therein; therefore, we also have

$$\begin{aligned} \lim _{k\rightarrow +\infty }\sup _{\mathbf{t}\in {{\mathbb {R}}}^{n}} \bigl \Vert F(\mathbf{t}-{\varvec{\tau }}_{k}';x)-c^{-1}F(\mathbf{t};x)\bigr \Vert _{Y} =0. \end{aligned}$$

(4.2)

The final conclusion simply follows from the above estimates, the corresponding definition of joint $({\mathrm R},{\mathrm P}_{{\mathrm R}})$-multi-almost automorphy and the decomposition:

$$\begin{aligned}&\sup _{\mathbf{t}\in {{\mathbb {R}}}^{n}} \bigl \Vert F\bigl (\mathbf{t} -\mathbf{\tau }_{l}'+{\varvec{\tau }}_{m}';x\bigr )-F(\mathbf{t};x)\bigr \Vert _{Y} \\ {}&\le \sup _{\mathbf{t}\in {{\mathbb {R}}}^{n}} \bigl \Vert F\bigl (\mathbf{t} -{\varvec{\tau }}_{l}'+{\varvec{\tau }}_{m}';x\bigr )-cF\bigl (\mathbf{t}-{\varvec{\tau }}_{l}';x\bigr )\bigr \Vert _{Y}+\sup _{\mathbf{t}\in {{\mathbb {R}}}^{n}}\bigl \Vert cF\bigl (\mathbf{t} -{\varvec{\tau }}_{l}';x\bigr )-F(\mathbf{t};x)\bigr \Vert _{Y} \\ {}&=\sup _{\mathbf{t}\in {{\mathbb {R}}}^{n}} \bigl \Vert F\bigl (\mathbf{t} -{\varvec{\tau }}_{l}'+{\varvec{\tau }}_{m}';x\bigr )-cF\bigl (\mathbf{t}-{\varvec{\tau }}_{l}';x\bigr )\bigr \Vert _{Y}+|c|\sup _{\mathbf{t}\in {{\mathbb {R}}}^{n}}\bigl \Vert F\bigl (\mathbf{t} -{\varvec{\tau }}_{l}';x\bigr )-c^{-1}F(\mathbf{t};x)\bigr \Vert _{Y}. \end{aligned}$$

$\square$

Furthermore, we can similarly introduce and analyze the notions of joint $({\mathrm R}_{X},{{\mathcal {B}}})$-multi-almost automorphy, joint $({\mathrm R}_{X},{{\mathcal {B}}},W_{{{\mathcal {B}}},{\mathrm R}_{X}})$-multi-almost automorphy and joint $({\mathrm R}_{X},{{\mathcal {B}}},{\mathrm P}_{{{\mathcal {B}}},{\mathrm R}_{X}})$-multi-almost automorphy (see Definition 4.2).

The results on approximations of almost automorphic functions, proved by Veech (1965, 1967) on topological groups, continue to hold on semi-topological groups without any essential changes. For example, by Milnes (1977, Theorem 18), we know that a continuous function $f : G \rightarrow Y$ is almost automorphic if and only if there exists a uniformly bounded sequence $(f_{k})$ of almost periodic functions $f_{k} : G \rightarrow {{\mathbb {C}}}$ ($k\in {{\mathbb {N}}}$) such that, for every $s\in G$ and $\epsilon >0,$ we have the existence of a Bohr neighbourhood V of s and an integer $k_{0} \in {{\mathbb {N}}}$ such that, for very integer $k\ge k_{0},$ we have $|f_{k}(t)-f(t)|<\epsilon$ for all $t\in V.$ See also Veech (1965, Subsection 6.2) for some elementary facts regarding analytic almost automorphic functions defined on the additive group of integers ${{\mathbb {Z}}}.$

The complete characterization of those semi-topological groups for which the equality $AP(G : {{\mathbb {C}}})=AA(G: {{\mathbb {C}}})$ holds is given in Milnes (1977, Theorem 23). In Milnes (1977, Theorem 25), P. Milnes has shown that, if G is arbitrary semi-topological group and $f : G \rightarrow Y$ is almost automorphic, then $f(\cdot )$ is almost periodic if and only if $T_{\alpha }f \in AA(G: {{\mathbb {C}}})$ whenever it exists, extending thus a result of W. A. Veech known on topological groups before that. It is also worth noting that Terras (1972) has constructed an almost automorphic function $f : {{\mathbb {Z}}}\rightarrow {{\mathbb {R}}}$ for which the limit $\lim _{N\rightarrow +\infty }\frac{1}{2N+1}\sum _{i=-N}^{N}f(i)$ does not exist. It is well known that this example can be transferred to the continuous setting as well as that there exists an almost automorphic function $f : {{\mathbb {R}}}\rightarrow {{\mathbb {R}}}$ such that the limit

$$\begin{aligned} {{\mathcal {M}}}(f):=\lim _{t\rightarrow +\infty }\frac{1}{2t}\int ^{t}_{-t}f(s)\, ds \end{aligned}$$

does not exist.

Concerning the notion of almost automorphy and the notion of almost periodicity for functions defined on (semi-)topological groups, it should be noted that some definitions for introducing these notions do not require a priori the continuity or measurability of function $f : G \rightarrow Y$ under consideration; see the research articles by Davies (1967) and Veech (1969) for some results obtained in this direction.

Concerning differences of almost periodic and almost automorphic functions defined on topological groups, with values in general locally convex spaces, we refer the reader to the research articles by Basit and Emam (1983) and Dimitrova and Dimitrov (2003). Mention should be made of paper by Péraire (1993), as well.

We close the paper with the observation that the Stepanov multi-dimensional almost automorphic type functions and their applications have recently been considered in Kostić et al. (2021).