JFSA Journal of Function Spaces and Applications 1758-4965 0972-6802 Hindawi Publishing Corporation 965746 10.1155/2013/965746 965746 Research Article Asymptotic Almost Periodic Functions with Range in a Topological Vector Space Khan Liaqat Ali Alsulami Saud M. Narayan Sivaram K. Department of Mathematics King Abdulaziz University P.O. Box 80203 Jeddah 21589 Saudi Arabia kau.edu.sa 2013 30 10 2013 2013 30 05 2013 13 09 2013 2013 Copyright © 2013 Liaqat Ali Khan and Saud M. Alsulami. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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  published an important article devoted to extension of the theory of almost periodic functions on the real line with values in a Banach space E. His results were further developed by several mathematicians, see, for example .

The concept of asymptotic almost periodicity was first considered by Fréchet [8, 9] in 1941 for functions f:+E with the range E restricted to a finite dimensional space. The semigroup case of + turns out to be significantly different from the group case of . If E is a Banach space or a locally convex case and +=[0,) replaced by Ja=[a,), with a, this notion has been extensively studied in recent years (see ). In this paper, we generalize the concept of asymptotic almost periodicity to the case of E, 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, E denotes a nontrivial Hausdorff topological vector space (in short, a TVS) with a base 𝒲 of closed balanced shrinkable neighborhoods of 0. (A neighborhood W of 0 in E is called shrinkable  if rW¯intW for 0r<1.) By [15, Theorem 4 and 5], every Hausdorff TVS has a base of shrinkable neighborhoods of 0, and also the Minkowski functional ρW of any such neighborhood W is continuous and satisfies (1)W¯={xE:ρW(x)1},intW={xE:ρW(x)<1}. We mention that, for any neighborhood W of 0 in E, ρW need not be absolutely homogeneous or subadditive; however, the following useful properties hold [15, 16].

ρW is positively homogeneous; it is absolutely homogeneous if W is balanced.

If V is a balanced neighborhood of 0 in E with V+VW, then (2)ρW(x+y)ρV(x)+ρV(y)x,yE.

A complete metrizable TVS is called an F-space.

Notations. Let X be a completely regular Hausdorff space, and let C(X,E) be the set of all continuous functions E-valued functions on X. Furthermore, let (3)Cb(X,E)={fC(X,E):f(X)is  bounded  in  E},Cpc(X,E)={fC(X,E):f(X)is  precompact  in  E},C0(X,E)={fC(X,E):fvanishes  at  infinityon  X}. Clearly, Cpc(X,E)Cb(X,E)C(X,E), C0(X,E)Cb(X,E), and all these sets are vector spaces over 𝕂 with the pointwise operations of addition and scalar multiplication. The uniform topology u on Cb(X,E) is defined as the linear topology which has a base of neighborhoods of 0 consisting of all sets of the form (4)Nb(X,W)={fCb(X,E):f(X)W}, where W𝒲. The compact-open topology k on C(X,E) is defined as the linear topology which has a base of neighborhoods of 0 consisting of all sets of the form (5)N(K,W)={fC(X,E):f(K)W}, where KX is compact and W𝒲. Clearly, ku on Cb(X,E). (For details, see .)

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

Theorem 1.

Let X be a locally compact space. A subset 𝒜 of C0(X,E) is precompact if and only if the following conditions hold:

𝒜 is equicontinuous:

𝒜(x)={f(x):f𝒜} is precompact in E for each xX;

𝒜 vanishes at infinity on X; that is, given W𝒲, there exists a compact set KX such that    f(y)W for all f𝒜 and yXK;

Before stating the next result, we introduce the following notation: for any 𝒜Cpc(X,E) and W𝒲, let (6)Gx(𝒜,W)={yX:f(y)-f(x)Wf𝒜},xX.

Theorem 2.

Let X be a k-space. Then, for any 𝒜Cpc(X,E), the following are equivalent:

𝒜 is precompact;

consider the following:

(i) 𝒜 is equicontinuous;

(ii) 𝒜(x) is precompact in E for each xX;

(iii) given W𝒲, there exists a compact set KX such that {Gx(𝒜,W):xK} covers X;

(i) 𝒜(X)={f(x):xX,f𝒜} is precompact in E;

(ii) given W𝒲, there exists a finite set FX such that {Gx(𝒜,W):xF} covers X;

(i) 𝒜(x) is precompact in E for each xX;

(ii) given W𝒲, there exists a finite set FX such that {Gx(𝒜,W):xF} covers X.

Definition 3.

A subset P of is called relatively dense in if there exists a number >0 such that every interval of length in contains at least one point of P.

Definition 4.

A function f:E is called almost periodic if it is continuous and, for each W𝒲, there exists a number =(W,f)>0 such that each interval of length in contains a point τ=τ(W,f) such that (7)f(t+τ)-f(t)Wt. A number τ for which (7) holds is called a (W,f)-translation number of f. The above property says that, for each W𝒲, the function f has a set of (W,f)-translation numbers P=P(W,f) which is relatively dense in . The set of all almost periodic functions f:E is denoted by AP(,E). For any f:E and a fixed ω, the ω-translate of f is defined as the function fω:E defined by (8)fω(t)=f(t+ω),t. We will denote H(f)={fω:ω}, the set of all translates of f.

Theorem 5 (see [<xref ref-type="bibr" rid="B12">4</xref>, <xref ref-type="bibr" rid="B18">7</xref>]).

Let E be a TVS. Let fAP(,E). Then,

f has totally bounded range f(); hence f is bounded;

f is uniformly continuous on .

Remark 6.

Clearly, by the above theorem, AP(,E)Cb(,E). We shall see below that AP(,E) is a vector space.

Theorem 7 (see [<xref ref-type="bibr" rid="B12">4</xref>, <xref ref-type="bibr" rid="B18">7</xref>]).

Let E be a TVS. If {fn} is a sequence in AP(,E) such that fnuf, then fAP(,E).

Theorem 8 (Bochner’s criterion [<xref ref-type="bibr" rid="B12">4</xref>, <xref ref-type="bibr" rid="B18">7</xref>]).

Let E be a TVS and f:E a continuous function.

If the set of translates H(f)={fω:ω} is u-sequentially compact in Cb(,E), then f is almost periodic.

Conversely, if f is almost periodic and, in addition, E is an F-space, then the set of translates H(f)={fω:ω} is u-compact in Cb(,E).

Thus, for E being an F-space, a continuous function f:E is almost periodic if and only if the set H(f)={fω:ω} is u-compact in Cb(,E).

Theorem 9 (see [<xref ref-type="bibr" rid="B12">4</xref>, <xref ref-type="bibr" rid="B18">7</xref>]).

Let E be a TVS. If f:E is almost periodic, then the functions (i) λf(λ𝕂), (ii) f¯(t)f(-t), and (iii) fω(t)=f(t+ω)(ω) are also almost periodic.

Theorem 10 (see[<xref ref-type="bibr" rid="B12">4</xref>, <xref ref-type="bibr" rid="B18">7</xref>]).

Let E be an F-space.

If f,gAP(,E), then f+gAP(,E); hence AP(,E) is a vector space.

(AP(,E),u) is complete.

3. Main Results

For a fixed a, let Ja=[0,)={t:ta}. A subset P of Ja is said to be relatively dense in Ja if there exists >0 such that, for each tJa, the closed interval [t,t+] contains at least one member of P.

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

Definition 11.

Consider a fixed a. A continuous function f:JaE will be called asymptotically almost periodic if, given any W𝒲, there exists r=r(W,f)a and a relatively dense set P=P(W,f) in Jr such that, for each τP and every tJr with t+τr, (9)f(t+τ)-f(t)W.

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

Definition 12.

Consider a fixed a. A subset 𝒜 of C(Ja,E) is called

equialmost periodic if, given W𝒲, there exists a relatively dense set P=P(W,𝒜) in such that, for each τP and every t, (10)f(t+τ)-f(t)Wf𝒜;

equiasymptotically almost periodic if, given W𝒲, there exists r=r(W,𝒜)a and a relatively dense set P=P(W,𝒜) in Jr such that, for each τP and every tJr with t+τr, (11)f(t+τ)-f(t)Wf𝒜.

Lemma 13.

(a) For any a, let T=Ja, and let 𝒜 be a precompact subset of (C(T,E),k). If 𝒜 is also equi-asymptotically almost periodic, then 𝒜 is uniformly equicontinuous on T and 𝒜(T) is precompact in E.

(b) Let T=, and let 𝒜 be a precompact subset of (C(T,E),k). If 𝒜 is equi-almost periodic, then 𝒜 is uniformly equicontinuous on T and 𝒜(T) is precompact in E.

Proof.

(a) Suppose T=Ja with 𝒜 being a precompact subset of (C(T,E),k) and also equi-asymptotically almost periodic.

First, 𝒜 is uniformly equicontinuous on Ja as follows. Let W𝒲. Choose balanced V𝒲 such that V+V+VW. There exist ra, >0, and a relatively dense set P in Jr such that, for each τP and every tJr with t+τr, (12)f(t+τ)-f(t)Vf𝒜, while [t,t+]P for any tJr. Let d=max{r,}, and choose τk[kd,(k+1)d]P,k=1,2, By Theorem 1, 𝒜 is equicontinuous on Ja, and hence 𝒜 is uniformly equi-continuous on the closed interval [a,5d]. Then there exists δ(0,d/2) such that (13)f(t1)-f(t2)Vwheneverf𝒜,t1,t2[a,5d]  with  |t1-t2|<δ. Now, let t1,t2>4d with |t1-t2|<δ. Choose k such that t1,t2[kd,(k+2)d] and put s1=t1-τk-2,  s2=t2-τk-2. Then it is easy to see that s1,s2[d,4d] and |s1-s2|<δ, and so by (12) and (13), (14)f(t1)-f(t2)=(f(s1+τk-2)-f(s1))+(f(s1)-f(s2))+(f(s2+τk-2)-f(s2))V+V+VW for any f𝒜. This shows that 𝒜 is uniformly equicontinuous on Ja.

Next, 𝒜(Ja) is precompact in E as follows. Let W,V,r,,P, and d be as above. By the equicontinuity of 𝒜, we can choose a finite (open) cover {Gi:1in} of [a,3d] and tiGi,i=1,,n, such that (15)f(t)-f(ti)Vfor  tGi,i{1,,n}  and  all  f𝒜. If t>3d, choose k such that t[kd,(k+1)d]. Putting s=t-tk-2, we have s[d,3d]. Then sGi for some i{1,,n}, and therefore, for any f𝒜, by (12) and (15), (16)f(t)-f(ti)=(f(s+τk-2)-f(s))+(f(s)-f(ti))V+VW. That is, 𝒜(Ja)i=1n(𝒜(ti)+W). By Theorem 1, for each i{1,,n}, 𝒜(ti) is precompact in E. Thus 𝒜(Ja) is precompact in E.

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

Theorem 14.

Let E be a TVS, and let T=Ja, where a. Then the following are equivalent for a subset 𝒜 of C(T,E).

Consider the following:

𝒜 is precompact in (C(T,E),k);

𝒜 is equi-asymptotically almost periodic.

The set H+(𝒜)={fω:f𝒜,ω+} of translates is a precompact subset of (Cb(T,E),u).

Proof.

( 1 ) ( 2 ) First, we consider the case T=Ja and assume that 𝒜 is a k-precompact and equi-asymptotically almost periodic subset of C(Ja,E). By Lemma 13, 𝒜(Ja) is precompact in E. Since H+(𝒜) is contained in (Cpc(Ja,E),u), H+(𝒜)(t) is precompact in E for each tJa. To show that H+(𝒜) is precompact in (Cpc(Ja,E),u), we need to verify that the finite covering condition 3(ii) of Arzelà-Ascoli Theorem 2 holds for H+(𝒜) in this setting. [Let W𝒲; choose a balanced V𝒲 such that V+V+VW. As in Lemma 13, choose ra,  l>0, and a relatively dense set P in Jr such that, for each τP and every tJr with t+τr, (17)f(t+τ)-f(t)Vf𝒜, while [t,t+l]P for any tJr. Furthermore, we put d=max{r,l}, set τ0=0, and fix τk[kd,(k+1)d]P,  k=1,2,. By Lemma 13, 𝒜 is uniformly equicontinuous on Ja; H+(𝒜) is also clearly uniformly equicontinuous. In particular, we obtain a finite cover {Si:1in} of [d,3d] by (relatively open) subsets of Jr and siSi,i=1,,n, such that, for every f𝒜 and all ω+,  (18)fω(s)-fω(si)Vwhenever  sSi,  i{1,,n}. Put Gi=k=0(Si+τk), i=1,,n. Now, taking tGi for i{1,,n}, choose sSi and k{0} so that t=s+τk. Using (17) and (18), we obtain (19)fω(t)-fω(si)=(f(s+ω+τk)-f(s+ω))+(fω(s)-fω(si))V+VW for every f𝒜 and all ω+.

Next, for any 𝒜Cpc(Ja,E) and W𝒲, recall the notation (6): (20)Gs(𝒜,W)={tJa:f(t)-f(s)W  f𝒜},sJa. It is easy to verify that Jd=[d,)i=1nGi, and so {Gsi(H+(𝒜),W):i=1,,n} covers Jd. By the equicontinuity of H+(𝒜), it is possible to trivially cover [a,d] by finitely many sets of this same prescribed form, we see that 3(ii) of Arzelà-Ascoli Theorem 2 is satisfied. Thus H+(𝒜) is precompact in (Cb(Ja,E),u).

( 2 ) ( 1 ) Assume that (6) holds.

(1)(i) This is clearly satisfied.

(1)(ii) In view of the Arzelà-Ascoli Theorem 1, H+(𝒜) is a precompact subset of (Cb(Ja,E),u). Furthermore, the Arzelà-Ascoli Theorem 2 can be used in showing that H+(𝒜) is equi-asymptotically almost periodic as follows. Fixing W𝒲, we use (3)(ii) of Arzelà-Ascoli Theorem 2 to obtain a finite cover {Gi:1in} of Jb, where b=max{a,1} and GiJb, i=1,,n, and tiGi, i=1,,n, such that, for every f𝒜 and all ω+,  (21)fω(t)-fω(ti)Wwhenever  tGi,i{1,,n}. Putting r==max{t1,,tn}, we note that ra and >0. Now, let P=(i=1n(Gi-ti))Jr. Then P is relatively dense in Jr. (For any tJr,  t+b, we have t+Gi for some i{1,,n}. Then t(t+)-tit+.) Now, given tJr and τP, choose i{1,,n} and sGi such that τ=s-ti. Since t-ti0, we then have by (21) (22)f(t+τ)-f(t)=f((t-ti)+s)-f((t-ti)+ti)=ft-ti(s)-ft-ti(ti)W for all f𝒜. This proves that 𝒜 is equi-asymptotically almost periodic.

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

Theorem 15.

Let E be a TVS, and let T=. Then the following are equivalent for a subset 𝒜 of C(T,E).

(i) 𝒜 is precompact in (C(T,E),k);

(ii) 𝒜 is equi-almost periodic.

The set H(𝒜)={fω:f𝒜,ω} of translates is a precompact subset of (Cb(T,E),u).

Proof.

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

( 1 ) ( 2 ) Suppose 𝒜 is a k-precompact and equi-almost periodic subset of C(,E). For this part, we can easily adapt the arguments of Theorem 14 to show that H(𝒜) is a precompact subset of (Cpc(,E),u).

( 2 ) ( 1 ) In view of the Arzelà-Ascoli Theorem 1, 𝒜 is a precompact subset of (Cpc(,E),u). Further, 𝒜 is equi-almost periodic. In fact, fix W𝒲. Since H(𝒜) is a precompact, we use (3)(ii) of Arzelà-Ascoli Theorem 2 to obtain a finite cover {Gi:1in} of and tiGi,i=1,,n, such that, for every f𝒜 and all ω,(23)fω(t)-fω(ti)Wwhenever  tGi,i{1,,n}. In this case, take >2max{|t1|,,|tn|}, and put P=i=1n(Gi-ti). Then P is relatively dense in . In fact, for any t, choose i{1,,n} such that t+/2Gi. Then (24)(t+2)-ti[t,t+]P. Also, given t and any τP, we can adapt the arguments of Theorem 14 till proving (22) and obtain (25)f(t+τ)-f(t)Wf𝒜. Consequently, 𝒜 is equi-almost periodic.

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

Theorem 16.

Let E be a TVS, and let a. For any fC(Ja,E), f is asymptotically almost periodic if and only if the set H+(f)={fω:ω0} is a precompact subset of (Cb(Ja,E),u).

We next consider an alternate view of asymptotically almost periodic functions. Fixing a, if g:E is a continuous almost periodic function and fC0(Ja,E), then the function f=gJa+h is clearly asymptotically almost periodic on the interval Ja. On the other hand, it is also known that every asymptotically almost periodic function in C(Ja,E) can be so represented in case E is finite dimensional. We shall use Theorem 14 to study it in a more general situation. Recall that a TVS    E is said to be quasicomplete if every bounded Cauchy net in E converges. Clearly completeness implies quasicompleteness.

Theorem 17.

Assume that E is a quasi-complete TVS, and fix a. Then fC(Ja,E) is asymptotically almost periodic if and only if there is a unique almost periodic function gC(,E) and a unique function hC0(Ja,E) such that (26)f=gJa+h.

Proof.

Since E is a quasi-complete TVS, it follows that both (Cpc(Ja,E),u) and (Cpc(,E),u) 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 fC(Ja,E). Then, for each pair (n,W)×𝒲, there exist r(n,W)max{a,n} and a relatively dense set P(n,W) in Jr(n,W) such that, for any τP(n,W) and every tJr(n,W), (27)f(t+τ)-f(t)1nW. Let us equip D=×𝒲 with the usual product order; that is, given (m,V),(n,W)D,(m,V)(n,W) if and only if mn and WV (or equivalently ρVρW). Also, for each α=(n,W)D, we choose ταP(n,W). By Theorem 14, since {τα}αD is a net in +, there is a subnet {τα(λ)}λD of {τα}α for which the net of translates {fτα(λ)} converges uniformly on Ja to some φ(Cpc(Ja,E),u).

We claim that φ has an almost periodic extension gC(,E). For a fixed bJa=(-,a), there exists λbD so that  b+τα(λ)a whenever λD and λλb; put Db={λD:λλb}. For each λDb, let fτα(λ)*:JbX be an extension of fτα(λ) defined by (28)fτα(λ)*(s)=f(s+τα(λ)),sJb. We can easily see that {fτα(λ)*:λDb} is a bounded net in (Cpc(Ja,E),u). Now, let ε>0 and W𝒲 be given. Choose V𝒲 with V+V+VW and choose n with nmax{a-b,3/ε}, let τP(n,V), and take λ0Db for which the following conditions are satisfied.

If λ,μDb and λ,μλ0, then (29)fτα(λ)(t)-fτα(μ)(t)ε3VtJa,or  ρV[fτα(λ)(t)-fτα(μ)(t)]ε3tJa.

In case λDb and λλ0, b+τα(λ)r(n,V). For any sJb, if λ,μλ0 (as in case (i)), we then have by (2), (28), and (29) that (30)ρW[fτα(λ)*(s)-fτα(λ)*(s)]ρV(f(s+τα(λ))-f(s+τα(λ)+τ))+ρV(fτα(λ)(s+τ)-fτα(μ)(s+τ))+ρV(f(s+τα(μ)+τ)-f(s+τα(μ)))ε3+ε3+ε3=ε.

So {fτα(λ)}λDb is a Cauchy net in (Cpc(Ja,E),u). Since it is also bounded and (Cpc(Ja,E),u) is quasi-complete, it converges uniformly on Jb to a function φb(Cpc(Ja,E),u). Clearly, we have that φbJa=φ. If c with cb and if λc is any element in D for which c+τα(λ)a for all λD such that λλ0, then the corresponding net {fτα(λ)}λλc of extensions from Ja to Jc will converge in Cpc(Jc,E) to a function φc. Clearly φc=φb on Jb. Define a function g:E by (31)g(t)={φ(t),  tJaφt(t),  tJa. Then g is well defined and continuous on and gJa=φ.

We claim that g is also almost periodic. Let ε>0 and W𝒲. Choose a balanced U𝒲 with U+U+UW. Choose n such that n3/ε and the set P=P(n,W){τ:-τP(n,W)} is relatively dense in . Next, given t and τP, first choose bJa with bmin{t,t+τ}, and then take λD such that

b+τα(λ)r(n,W)+|τ|,

φb(s)-f(s+τα(λ))(ε/3)U for every sJb.

Then, by (2), (31), and (b), (32)ρW[g(t+τ)-g(t)]ρU(φb(t+τ)-f(t+τ+τα(λ)))+ρU(f(τ+τα(λ)+τ)-f(τ+τα(λ)))+ρU(f(τ+τα(λ))-φb(t))ε3+ε3+ε3=ε, and so g is almost periodic.

It remains to show that h=f-gJa vanishes at infinity on Ja. Fix W𝒲. Choose a balanced V𝒲 with V+VW, and choose sufficiently large n such that there exists λD such that α(λ)(n,V) and (33)ftα(λ)(t)-φ(t)VtJa; we may assume that α(λ)=(m,V), where m. Thus, if tJr(m,V), then (34)h(t)=f(t)-φ(t)=(f(t)-f(t+τα(λ)))+(f(t+τα(λ))-φ(t))V+VW.

Finally, we show that the functions g and h in the representation (26) are unique. First observe that an almost periodic function θC(,E) must be identically zero on if θJaC0(Ja,E). Therefore, given almost periodic functions g and φ in C(,E), we only need to verify that g+φ is also almost periodic. For this, choosing any net {ωα}α in , we apply Theorem 15 (twice) to obtain a subnet {ωα(λ)}λ such that the corresponding nets of translates {gωα(λ)} both converge in (Cpc(,E),u). Another application of Theorem 15 now gives us that g+φ is almost periodic.

Scope of Applications. (1) The importance of such a work has been highlighted in (; , p. 19-20). If, for a given Banach space E, a linear operator A:𝒟(A)EE is the infinitesimal generator of a C0-semigroup (T(t))t>0 of bounded linear operators from E to E, then, for any x0𝒟(A), the unique strong solution of the abstract Cauchy problem (CP)dx(t)dt=A[x(t)],t>0,x(0)=x0E, associated with A, is given by the motion x(·)=T(·)x0 of (T(t))t>0 through x0. See also .

In the qualitative study of the solution, one of the problems is to determine its asymptotic behaviour as t. In this regard, a useful concept is the so-called positive ω-limit set (35)ω+(x0)={yE:0tn  such  that  T(tn)x0.y} of all possible limit points. The basic result is that: if the orbit γ+(x0)={T(t)x0:t0} of the motion is relatively compact, then ω+(x0) is nonempty, compact, connected, and invariant. A qualitative much stronger mode of asymptotic behaviour results if not only γ+(x0) is relatively compact, but also the set H+(T(·)x0)={Tω(·)x0:ω>0}, Tω(t)x0=T(t+ω)x0, of all translates of the motion is a relatively compact subset of the space (Cb(+,E),·). This observation raises the following problem.

(*) Characterize those fCb(+,E) for which H+(f) is relatively compact in (Cb(+,E),·).

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) p-Fréchet space E, 0<p<1, (CP) has a unique solution x(t)=T(t)(x), with (36)T(t)(x)=limn(1-tnA)-1(x), with the limit being taken in the p-norm of E.

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.

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