#### Abstract

The notion of asymptotic almost periodicity was *…*first introduced by Fréchet in 1941 in the case of *…*finite dimensional range spaces. Later, its extension to the case of Banach range spaces and locally convex range spaces has been considered by several authors. In this paper, we have generalized the concept of asymptotic almost periodicity to the case where the range space is a general topological vector space, not necessarily locally convex. Our results thus widen the scope of applications of asymptotic almost periodicity.

#### 1. Introduction

The theory of almost periodic functions was mainly created and published during 1924–1926 by the Danish mathematician Harold Bohr. In 1933, Bochner [1] published an important article devoted to extension of the theory of almost periodic functions on the real line with values in a Banach space . His results were further developed by several mathematicians, see, for example [2–7].

The concept of asymptotic almost periodicity was first considered by Fréchet [8, 9] in 1941 for functions with the range restricted to a finite dimensional space. The semigroup case of turns out to be significantly different from the group case of . If is a Banach space or a locally convex case and replaced by , with , this notion has been extensively studied in recent years (see [10–14]). In this paper, we generalize the concept of asymptotic almost periodicity to the case of , a general topological vector space.

#### 2. Preliminaries

In this section, we give prerequisites on topological vector spaces and almost periodic functions for our main results of Section 3.

Throughout this paper, denotes a nontrivial Hausdorff topological vector space (in short, a TVS) with a base of closed balanced shrinkable neighborhoods of . (A neighborhood of in is called *shrinkable* [15] if for .) By [15, Theorem 4 and 5], every Hausdorff TVS has a base of shrinkable neighborhoods of , and also the Minkowski functional of any such neighborhood is continuous and satisfies
We mention that, for any neighborhood of in , need not be absolutely homogeneous or subadditive; however, the following useful properties hold [15, 16].(a) is positively homogeneous; it is absolutely homogeneous if is balanced.(b)If is a balanced neighborhood of in with , then

A complete metrizable TVS is called an *-space*.

*Notations.* Let be a completely regular Hausdorff space, and let be the set of all continuous functions -valued functions on . Furthermore, let
Clearly, , , and all these sets are vector spaces over with the pointwise operations of addition and scalar multiplication. The *uniform topology * on is defined as the linear topology which has a base of neighborhoods of consisting of all sets of the form
where . The *compact-open topology * on is defined as the linear topology which has a base of neighborhoods of consisting of all sets of the form
where is compact and . Clearly, on . (For details, see [16].)

We state the following two general versions of the Arzelà-Ascoli theorem [16, 17] for reference purpose.

Theorem 1. *Let be a locally compact space. A subset of is precompact if and only if the following conditions hold:*(i)* is equicontinuous:*(ii)* is precompact in for each ;*(iii)* vanishes at infinity on ; that is, given , there exists a compact set such that for all and ;*

Before stating the next result, we introduce the following notation: for any and , let

Theorem 2. *Let be a -space. Then, for any , the following are equivalent:*(1) * is precompact;*(2)* consider the following:* *(i) is equicontinuous;* *(ii) is precompact in for each ;* *(iii) given , there exists a compact set such that covers ;*(3)* (i) is precompact in ;* *(ii) given , there exists a finite set such that covers ;*(4)* (i) is precompact in for each ;* *(ii) given , there exists a finite set such that covers . *

*Definition 3. *A subset of is called *relatively dense* in if there exists a number such that every interval of length in contains at least one point of .

*Definition 4. *A function is called *almost periodic* if it is continuous and, for each , there exists a number such that each interval of length in contains a point such that
A number for which (7) holds is called a *-translation number *of . The above property says that, for each , the function has a set of -translation numbers which is relatively dense in . The set of all almost periodic functions is denoted by . For any and a fixed , the -*translate* of is defined as the function defined by
We will denote , the set of all translates of .

Theorem 5 (see [4, 7]). *Let be a TVS. Let . Then,*(a)* has totally bounded range ; hence is bounded; *(b)* is uniformly continuous on .*

*Remark 6. *Clearly, by the above theorem, . We shall see below that * is a vector space. *

Theorem 7 (see [4, 7]). *Let be a TVS. If is a sequence in such that , then . *

Theorem 8 (Bochner’s criterion [4, 7]). * Let be a TVS and a continuous function. *(a)*If the set of translates is -sequentially compact in , then is almost periodic. *(b)*Conversely, if is almost periodic and, in addition, is an -space, then the set of translates is -compact in . **Thus, for being an -space, a continuous function is almost periodic if and only if the set is -compact in . *

Theorem 9 (see [4, 7]). * Let be a TVS. If is almost periodic, then the functions (i) , (ii) , and (iii) are also almost periodic.*

Theorem 10 (see[4, 7]). *Let be an -space. *(a)*If , then ; hence is a vector space.*(b)* is complete. *

#### 3. Main Results

For a fixed , let . A subset of is said to be *relatively dense* in if there exists such that, for each , the closed interval contains at least one member of .

We define the notion of asymptotic almost periodicity in the case where the range is in a general topological vector space as follows.

*Definition 11. *Consider a fixed . A continuous function will be called *asymptotically almost periodic *if, given any , there exists and a relatively dense set in such that, for each and every with ,

In this section, we obtain extension of some results of [13] to our general setting.

*Definition 12. *Consider a fixed . A subset of is called(i)*equialmost periodic* if, given , there exists a relatively dense set in such that, for each and every ,
(ii)*equiasymptotically almost periodic* if, given , there exists and a relatively dense set in such that, for each and every with ,

Lemma 13. *(a) For any , let , and let be a precompact subset of . If is also equi-asymptotically almost periodic, then is uniformly equicontinuous on and is precompact in .**(b) Let , and let be a precompact subset of . If is equi-almost periodic, then is uniformly equicontinuous on and is precompact in .*

*Proof. *(a) Suppose with being a precompact subset of and also equi-asymptotically almost periodic.

First, is uniformly equicontinuous on as follows. Let . Choose balanced such that . There exist , , and a relatively dense set in such that, for each and every with ,
while for any . Let , and choose By Theorem 1, is equicontinuous on , and hence is uniformly equi-continuous on the closed interval . Then there exists such that
Now, let with . Choose such that and put , . Then it is easy to see that and , and so by (12) and (13),
for any . This shows that is uniformly equicontinuous on .

Next, is precompact in as follows. Let , and be as above. By the equicontinuity of , we can choose a finite (open) cover of and , such that
If , choose such that . Putting , we have . Then for some , and therefore, for any , by (12) and (15),
That is, . By Theorem 1, for each , is precompact in . Thus is precompact in .

(b) Suppose , and let be a precompact subset of and also equi-almost periodic. Then, with minor changes in the above proof, it follows that is uniformly equicontinuous on and is precompact in .

Theorem 14. *Let be a TVS, and let , where . Then the following are equivalent for a subset of .*(1)*Consider the following:(i) is precompact in ;(ii) is equi-asymptotically almost periodic.*(2)

*The set of translates is a precompact subset of .*

*Proof. * First, we consider the case and assume that is a -precompact and equi-asymptotically almost periodic subset of . By Lemma 13, is precompact in . Since is contained in , is precompact in for each . To show that is precompact in , we need to verify that the finite covering condition 3(ii) of Arzelà-Ascoli Theorem 2 holds for in this setting. [Let ; choose a balanced such that . As in Lemma 13, choose , and a relatively dense set in such that, for each and every with ,
while for any . Furthermore, we put , set , and fix . By Lemma 13, is uniformly equicontinuous on ; is also clearly uniformly equicontinuous. In particular, we obtain a finite cover of by (relatively open) subsets of and , such that, for every and all
Put , . Now, taking for , choose and so that . Using (17) and (18), we obtain
for every and all .

Next, for any and , recall the notation (6):
It is easy to verify that , and so covers . By the equicontinuity of , it is possible to trivially cover by finitely many sets of this same prescribed form, we see that 3(ii) of Arzelà-Ascoli Theorem 2 is satisfied. Thus is precompact in .

Assume that (6) holds.

(1)(i) This is clearly satisfied.

(1)(ii) In view of the Arzelà-Ascoli Theorem 1, is a precompact subset of . Furthermore, the Arzelà-Ascoli Theorem 2 can be used in showing that is equi-asymptotically almost periodic as follows. Fixing , we use (ii) of Arzelà-Ascoli Theorem 2 to obtain a finite cover of , where and , , and , , such that, for every and all
Putting , we note that and . Now, let . Then is relatively dense in . (For any , we have for some . Then .) Now, given and , choose and such that . Since , we then have by (21)
for all . This proves that is equi-asymptotically almost periodic.

We next obtain an analogue of the above result for (instead of ) and equi-almost periodic (instead of equi-asymptotically almost periodic) as follows.

Theorem 15. *Let be a TVS, and let . Then the following are equivalent for a subset of .*(1)*(i) is precompact in ;* *(ii) is equi-almost periodic.*(2)*The set of translates is a precompact subset of .*

*Proof. *This follows from Theorem 14 with minor changes, outlined as follows.

Suppose is a -precompact and equi-almost periodic subset of . For this part, we can easily adapt the arguments of Theorem 14 to show that is a precompact subset of .

In view of the Arzelà-Ascoli Theorem 1, is a precompact subset of . Further, is equi-almost periodic. In fact, fix . Since is a precompact, we use (3)(ii) of Arzelà-Ascoli Theorem 2 to obtain a finite cover of and , such that, for every and all
In this case, take , and put . Then is relatively dense in . In fact, for any , choose such that . Then
Also, given and any , we can adapt the arguments of Theorem 14 till proving (22) and obtain
Consequently, is equi-almost periodic.

From Theorem 14, we can deduce the following extension of Fréchet’s theorem.

Theorem 16. *Let be a TVS, and let . For any , is asymptotically almost periodic if and only if the set is a precompact subset of .*

We next consider an alternate view of asymptotically almost periodic functions. Fixing , if is a continuous almost periodic function and , then the function is clearly asymptotically almost periodic on the interval . On the other hand, it is also known that every asymptotically almost periodic function in can be so represented in case is finite dimensional. We shall use Theorem 14 to study it in a more general situation. Recall that a TVS is said to be *quasicomplete* if every bounded Cauchy net in converges. Clearly completeness implies quasicompleteness.

Theorem 17. *Assume that is a quasi-complete TVS, and fix . Then is asymptotically almost periodic if and only if there is a unique almost periodic function and a unique function such that
*

*Proof. *Since is a quasi-complete TVS, it follows that both and are quasi-complete [13, 16, 18]. Therefore, precompactness can be considered equivalent to relative compactness in either of these two spaces.

Consider an arbitrary asymptotically almost periodic function . Then, for each pair , there exist and a relatively dense set in such that, for any and every ,
Let us equip with the usual product order; that is, given if and only if and (or equivalently ). Also, for each , we choose . By Theorem 14, since is a net in , there is a subnet of for which the net of translates converges uniformly on to some .

We claim that has an almost periodic extension . For a fixed , there exists so that whenever and ; put . For each , let be an extension of defined by
We can easily see that is a bounded net in . Now, let and be given. Choose with and choose with , let , and take for which the following conditions are satisfied.(i)If and , then
(ii)In case and , . For any , if (as in case (i)), we then have by (2), (28), and (29) that
So is a Cauchy net in . Since it is also bounded and is quasi-complete, it converges uniformly on to a function . Clearly, we have that . If with and if is any element in for which for all such that , then the corresponding net of extensions from to will converge in to a function . Clearly on . Define a function by
Then is well defined and continuous on and .

We claim that is also almost periodic. Let and . Choose a balanced with . Choose such that and the set is relatively dense in . Next, given and , first choose with , and then take such that(a),(b) for every .

Then, by (2), (31), and (b),
and so is almost periodic.

It remains to show that vanishes at infinity on . Fix . Choose a balanced with , and choose sufficiently large such that there exists such that and
we may assume that , where . Thus, if , then

Finally, we show that the functions and in the representation (26) are unique. First observe that an almost periodic function must be identically zero on if . Therefore, given almost periodic functions and in , we only need to verify that is also almost periodic. For this, choosing any net in , we apply Theorem 15 (twice) to obtain a subnet such that the corresponding nets of translates both converge in . Another application of Theorem 15 now gives us that is almost periodic.

*Scope of Applications*. (1) The importance of such a work has been highlighted in ([12]; [13], p. 19-20). If, for a given Banach space , a linear operator is the infinitesimal generator of a -semigroup of bounded linear operators from to , then, for any , the unique strong solution of the abstract Cauchy problem
associated with , is given by the motion of through . See also [10].

In the qualitative study of the solution, one of the problems is to determine its asymptotic behaviour as . In this regard, a useful concept is the so-called positive -limit set of all possible limit points. The basic result is that: if the orbit of the motion is relatively compact, then is nonempty, compact, connected, and invariant. A qualitative much stronger mode of asymptotic behaviour results if not only is relatively compact, but also the set , , of all translates of the motion is a relatively compact subset of the space . This observation raises the following problem. Characterize those for which is relatively compact in .

Clearly, our results contain a complete solution of problem in the general setting.

(2) In [3, 19], it has been obtained that in the (nonlocally convex) -Fréchet space , , has a unique solution , with with the limit being taken in the -norm of .

Our results thus widen the scope of applications of asymptotic almost periodicity to the nonlocally convex setting.

#### Acknowledgments

This research was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Project no. 97/130/1432. The authors, therefore, acknowledge with thanks DSR technical and financial support. The authors are also grateful to the referee for several helpful comments.