Almost Periodic Functions Defined on Rn With

Punjab University Journal of Mathematics (ISSN 1016-2526) Vol. 39 (2007) pp. 19-27

Almost Periodic Functions Defined on Rn With Values in Fuzzy Setting Muhammad Amer Latif P.O. Box. 220871, Riyadh 11311, Saudi Arabia. E-mail: m amer [email protected]. Muhammad Iqball Bhatti Department of Mathematics University of Engineering and Technology Lahore - Pakistan E-mail: [email protected] Abstract. In this paper we develop the theory of almost periodic functions defined on Rn with values in fuzzy setting. AMS (MOS) Subject Classification: Primary 05C38, 15A15; Secondary 05A15, 15A18. Keywords and Phrases: Almost Periodic Functions, Fuzzy subset, Fuzzy-numbertype spaces, Relatively dense set

1. Introduction The theory of almost periodic functions was developed in its main features by Bohr [5] in three rather long papers in the Acta Mathematica (Volumes 45, 46 and 47) under the common title “Zur theorie der Fast Periodische Funktionen” in 1923; the first of these deals with the almost periodic functions of a real variable, while the third takes up the case of a complex variable. Afterwards theory was continuously getting established by several mathematicians like Besicovitch [3], Bochner [4], Amerio and Prouse [1], Levitan [8], Levitan and Zhikov [9], Corduneaue [6], Fink [7] and Zaidman [11] etc. In 1933, Bochner defined and studied the almost periodic functions with values in Banach spaces. He showed that these functions include certain earlier generalizations of the notion of almost periodic functions. The theory of almost periodic functions was further developed by replacing Banach spaces by complete Hausdroff locally convex spaces and Fréchet spaces by N’Guérékata [10]. The theory of almost periodicity as known in Banach spaces, is studied in fuzzy setting that is based on the work of Bede and Gal [2]. The theory of almost periodic functions defined on Rn with values in Banach spaces is given in monograph of Zaidman [11]. However the theory of almost periodic functions defined on R n with values in fuzzy-number-type spaces was not yet developed. It is the main goal of this present paper to develop this theory in section 3. To this end we first recall the following: 19

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Muhammad Amer Latif and Muhammad Iqball Bhatti

2. Preliminaries Definition 1. Let us denote by RF the class of fuzzy subsets of real axis R ( i.e. u : R −→ [0, 1]), satisfying the following properties: (i) ∀u ∈ RF , u is normal i.e.with u(x) = 1. (ii) ∀u ∈ RF , u is convex fuzzy set i.e. u(tx + (1 − t)y) ≥ min {u(x), u(y)} , ∀t ∈ [0, 1] . (iii) ∀ u ∈ RF , u is upper semi-continuous on R. (iv) {x ∈ R : u(x) > 0} is compact. The set RF is called the space of fuzzy real numbers. Remark 2. It is clear that R ⊂ RF , because any real number x0 ∈ R, can be described as the fuzzy number whose value is 1 for x = x0 and zero otherwise. We will collect some other definitions and notations needed in the sequel. For 0 < r ≤ 1 and u ∈ RF , we define r [u] = {x ∈ R : u(x) ≥ r} 0 [u] = {x ∈ R : u(x) > 0} r Now it is well known that for each r ∈ [0, 1] , [u] , is bounded closed interval. For u, v ∈ RF and λ ∈ R, we have the sum u ⊕ v and the product λ ¯ u are defined r r r r r r r by [u ⊕ v] = [u] + [v] , [λ ¯ u] = λ [u] , ∀r ∈ [0, 1], where [u] + [v] means the r usual addition of two intervals as subsets of R and λ [u] means the usual product between a scalar and a subset of R. Now we define D : RF × RF −→¡ R ∪ ©¯ {0} by ¯ ¯ ¯ª¢ r¯ ¯ r D(u, v) = sup max ¯ur− − v− , u+ + ur+ ¯ ¤ r∈[0,1] £ r r¤ £ r r , v+ then (D, RF ) is a metric space and it where [u] = ur− , ur+ , [v] = v− possesses the following properties: (i) D(u ⊕ w, v ⊕ w) = D(u, v), ∀u, v, w ∈ RF . (ii) D(λ¯ u, λ ¯ v) = λD(u, v), ∀u, v ∈ RF , ∀λ ∈ R. (iii) D(u ⊕ v, w ⊕ e) ≤ D(u, w) + D(v, e), ∀u, v, w, e ∈ RF and (RF , D) is a complete metric space. Also we have the following theorem. Theorem 3. (i) If we denote e 0 = X{0} then e 0 ∈ RF is neutral element with respect to e e ⊕, i.e. u ⊕ 0 = 0 ⊕ u, for all u ∈ RF . (ii) With respect to e 0 none of u ∈ RF \R has opposite in RF with respect to ⊕. (iii) For any a, b ∈ R with a, b ≥ 0 or a, b ≤ 0, any u ∈ RF , we have (a+b)¯u = a ¯ u ⊕ b¯ u.∀a, b ∈ R the above property does not hold. (iv) For any λ ∈ Rand any u, v ∈ RF , we have λ ¯ (u ⊕ v) = λ ¯ u ⊕ λ ¯ v. (v) For any λ, µ ∈ R and any u ∈ R, we have λ ¯ (λ ¯ v) = (λ.µ) ¯ v. (vi) If we denotekukF = D(u, e 0), ∀u ∈ RF then k.kF has the properties of a usual norm on RF , i.e. kukF = 0 if and only if u = e 0, kλ ¯ ukF = |λ| . kukF and ku ⊕ vkF ≤ kukF + kvkF , |kukF + kvkF | ≤ D(u, v). Remark 4. The propositions (ii) and (iii) in theorem show us that (RF , ⊕, ¯) is not a linear space over R and consequently (RF , k.kF ) cannot be a normed space. However, the properties of D and those in theorem (iv)-(vi), have as an effect that most of the metric properties of a functions defined on R with values in a Banach space, can be extended to functions f : R −→ RF , called fuzzy functions.

Almost Periodic Functions Defined on Rn ...

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We now recall the following definitions and theorems Definition 5. A function f : R −→ RF is said to be continuous at x0 ∈ R if for every ε > 0 we can find δ > 0 such that D(f (x), f (x0 )) < ε, whenever |x − x0 | < δ. f is said to be continuous on R if it is continuous at every x ∈ R. Definition 6. Let f : R −→ RF be continuous on R. We say that f is B-almost periodic if ∀ε > 0, ∃ l > 0 such that any interval [a, a + l] of length l contains at least one point τ with D(f (t + τ ), f (t)) < ε, ∀t ∈ R. Definition 7. We say that f is normal if for any sequence Fn : R −→ RF of the form Fn (x) = f (x + hn ), n ∈ N, where (hn )n is a sequence of real numbers, one can extract a subsequence of (Fn )n , converging uniformly on R i.e. for every sequence (hn )n of real numbers there exists a subsequence (hnk )nk , and F : R −→ RF which may depend on (hn )n , such that D(Fnk (x), F (x)) −→ 0 as k −→ ∞ , uniformly with respect to x . Theorem 8. If f : R −→ RF is B-almost periodic then f is bounded i.e.∃, M > 0 with D(f (x), f (y)) < M, ∀x, y ∈ R. Theorem 9. If f : R −→ RF is B-almost-periodic then f is uniformly continuous on R. Theorem 10. If fn : R −→ RF , n ∈ N are B-almost periodic and fn −→ f as n −→ ∞ uniformly on R, then f is B-almost periodic. Theorem 11. If f : R −→ RF is B-almost periodic, then the set of values of f is relatively compact in the complete metric space (RF , D). Theorem 12. If f : R −→ RF is B-almost periodic, then λ ¯ f, λ ∈ R, Fh (x) = f (x + h), and G(x) = kf (x)kF , x ∈ R are B-almost periodic functions. Theorem 13. The sum ⊕ of two B-almost periodic functions is B-almost periodic. Remark 14. Let us denote AP (RF ) = {f : R −→ RF : f is B-almost periodic} , and for f ∈ AP (RF ), let us definekf k = sup {kf (x)kF : x ∈ R}. By theorem 8 we get kf k < +∞. Also by theorems 3, 12 and 13 AP (RF , ⊕, ¯), is not a linear space, and consequently AP (RF , k.kF ) is not a normed space. However, endowed with the metric D∗ : AP (RF ) × AP (RF ) −→ R+ ∪ {0} defined by D∗ (f, g) = supD(f (x), g(x)), f, g ∈ AP (RF ) x∈R

becomes a complete metric space. Indeed if we denote Cb (RF ) = {f : R −→ RF : f is continuous and bounded on R} , then because (RF , D) is a complete metric space, it follows that (Cb ( RF ), D∗ ) is a complete metric space. Then theorems 8 and 11 show that AP (RF ) is a closed subset of Cb (RF ), i.e. (AP ( RF ), D∗ ) is a complete metric space. For all of the above C.f [5]. 3. Almost Periodic Functions Defined On Rn With Values In Fuzzy-Number-Type Space Now we recall the following facts about the Euclidean n-dimensional space R n

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Let Rn the usual Euclidean n-dimensional space. The elements x of Rn are the n-tuples x = (x1 , x2 , ..., xn ) and a norm of x ∈ Rn is given by kxk = (x1 + x2 + 1 ... + xn ) 2 . A closed ball B(x0 ; r) in Rn with center x and radius r > 0 is defined by the set B(x0 ; r) = {x ∈ Rn : kx − x0 k ≤ r} . A set P is said to be relatively dense in Rn if there exists a number r > 0 such that P ∩ B(x0 ; r) 6= ∅, for all x ∈ Rn . We also have the following two important theorems for our further discussion. Theorem 15. A subset P of Rn is relatively dense in Rn if and only if, for some r > 0, we have the relation Rn = ∪ B(p; r). p∈P

Theorem 16. A subset P of R is relatively dense if and only if there exists a compact set K in such that K + P = Rn (vector sum of K and P ). Now we define almost periodic functions defined on Rn and taking values in RF but before that we define continuity and uniform continuity of functions defined on R n and taking values in RF then we define the almost periodicity of functions defined on Rn with values in the fuzzy-number-type spaces. n

Definition 17. A function f : Rn −→ RF is said to be continuous at x0 ∈ Rn if for every ε > 0 we can find δ > 0 such that D(f (x), f (x0 )) < ε, whenever kx − x0 k < δ. f is said to be continuous on Rn if it is continuous at every x ∈ Rn . Definition 18. A function f : Rn −→ RF is said to be uniformly continuous on Rn if such that D(f (x1 ), f (x2 )) < ε, whenever kx1 − x2 k < δ, ∀x1 , x2 ∈ Rn . Definition 19. A continuous function f : Rn −→ RF , is said to be B-almost periodic, if for every ε > 0, we can find a relatively dense set which we denote by T (f ; ε) in Rn such that D(f (t+τ ), f (t)) < ε, ∀t ∈ Rn , τ ∈ T (f ; ε). Hence to any ε > 0, we may associate a number, r = r(ε) > 0, in such manner that in any closed ball B(x; r) there exists at least one element of the set T (f ; ε). Theorem 20. If f : Rn −→ RF is B-almost periodic function then given any any ε > 0, there are two positive numbers r1 = r1 (ε) and r2 = r2 (ε) such that any ball B(x; r1 ) in Rn contains a ball of radius r2 which is contained in T (f ; ε). Proof. Consider the set T (f ; 2ε ) which is relatively dense in Rn ( f : Rn −→ RF is assumed to be almost periodic) and the associated number R = R( 2ε ) such that B(a; R) ∩ T (f ; 2ε ) 6= ∅, ∀a ∈ Rn . Using now the uniform continuity of f over Rn we find a number δ1 = δ1 ( 2ε ) such that if y ∈ Rn and kyk ≤ δ1 it follows that D(f (x + y), f (x)) < 2ε , ∀ x ∈ Rn we say that r1 = R + 2δ and r1 = δ1 form required numbers. In fact, given x ∈ Rn , take z ∈ Rn such that kzk = r1 . Then ∃ y ∈ B(x0 + z; R) ∩ T (f ; 2ε ). Hence ky − x0 k ≤ R + δ1 < r1 so that y ∈ B(x0 ; r1 ). Furthermore ∀y °∈ Rn , kyk ≤ δ1 , ° 0 0 ° ° °x + y − x0 ° ≤ R +δ1 +δ1 = R+2δ1 = r1 . Hence x + y ∈ B(x0 ; r1 ). Therefore 0

the whole ball ∈ B(x ; δ1 ) is contained in ∈ B(x0 ; r1 ). Finally any vector in this 0 ball belong toT (f ; ε), this is because x + y with kyk ≤ δ1 is such a vector, we have ∀ y ∈ Rn 0 D(f (x + y + x ), f (x)) 0 = D(f (x + y) ⊕ f (x + y + x ), f (x) ⊕ f (x + y)) 0 = D(f (x + y) ⊕ f (x + y + x ), f (x + y) ⊕ f (x))


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0

≤ D(f (x + y), f (x + y + x )) + D(f (x + y), f (x)) ≤ 2ε + 2ε . 0 Here we have used the fact that x ∈ T (f ; 2ε ), kyk ≤ δ1 and uniform continuity of f over Rn is proved. The following result shows the boundedness of B-almost periodic. ¤ Theorem 21. If a continuous function f : Rn −→ RF , is B-almost periodic, then f is bounded i.e. ∃M > 0 with D(f (x), f (y)) < M, ∀x, y ∈ R. Proof. Because D(f (x), f (y)) ≤ D(f (x), e 0) + D(e 0, f (y)) = kf (x)kF + kf (y)kF it is sufficient to prove that M1 with kf (x)kF ≤ M1 . Take any ε > 0 and associative relatively set T (f ; ε). Therefore, for some r = r(ε) > 0, Rn = ∪ B(τ ; r) and τ ∈T (f ;ε) ° ° 0 0 °0 ° consequently ∀t ∈ Rn , ∃ τ ∈ T (f ; ε) such that °t − y ° ≤ r. Then, if t is t − y, we 0

have ° t 0=°t − y where τ ∈ T (f ; ε). Therefore ° ° °f (t )° F 0 = D(f (t ), e 0) = D(f (t) ⊕ f (t + τ ), f (t) ⊕ e 0) ≤ D(f (t), f (t + τ )) + D(f (t) ⊕ e 0) < ε + kf (t)kF < ε + sup (kf (t)kF ) t∈B(0;r)

° 0 ° ° ° For instance, if we take ε = 1, °f (t )° < ε + F

an upper bound for f over Rn .

sup (kf (t)kF ) which gives us

ktk≤r(1)

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Next theorem shows that the range of B-almost periodic functions is relatively compact. Theorem 22. If f : Rn −→ RF is B-almost periodic, then the set of values of f is relatively compact in the complete metric space (RF , D) . Proof. In complete metric spaces, the relatively compact sets coincides with totally bounded sets, it is sufficient to show that the values of the functions can be embedded in a finite number of spheres of radius 2ε. Take any ε > 0 and the and the number r = r(ε) > 0 . The range of f when t runs over the compact ball B(0; r) = {t ∈ Rn : ktk ≤ r} is compact in RF . Therefore, there are ν points ν

f (t1 ), f (t2 ), ..., f (tν ) where kti k ≤ r, 1 ≤ i ≤ ν and f (t) ∈ ∪ B(f (ti ); ε), ∀t, ktk ≤ r. 0

0

i=1

Take now any arbitrary t ∈ Rn , it can be written as t = t + τ where ktk ≤ r and 0 τ ∈ T (f ; ε), hence, there is an i ∈ {1, 2, 3, ..., ν} such that D(f (t ), f (ti )) ≤ ε. It follows that 0 D(f (t ), f (ti )) = D(f (t + τ ), f (ti )) = D(f (t + τ ) ⊕ f (ti ), f (ti ) ⊕ f (t)) ≤ D(f (t + τ ), f (ti )) + D(f (ti ) ⊕ f (t)) ≤ ε + ε = 2ε. ν

Thus Rf ⊂ ∪ B(f (ti ); 2ε).

¤

i=1

Remark 23. Let f : Rn −→ RF is B-almost periodic, and let us consider the sequence (f (tn ))n of values. Denote A = {f (tn ) : n ∈ N} ,and take the closure A ⊂

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f (Rn ) ⊂ RF . It follows that A is compact, so is A sequentially compact too, which by A ⊂ A implies that the sequence has a convergent subsequence in R F. Theorem 24. If f : Rn −→ RF is B-almost periodic, then it is uniformly continuous over Rn . 0

Proof. Let ε > 0 be given, we can find r = r( 3ε ) > 0 such that for any t ∈ Rn can be 0 represented as t = t + τ where τ ∈ T (f ; 3ε ). Next we note the uniform continuity of the function f ¡ on the closed ball B(0; 2r) = {t ∈ Rn : ktk ≤ 2r}. Therefore, ¢ ε there is a δ > 0, δ < r( 3 ) such that, if kt1 k ≤ 2r, kt2 k ≤ 2r and kt1 − t2 k < δ, 0 0 then have D(f (t1 ), f (t2 )) ≤ 3ε . Take now any pair t1 , t2 ∈ Rn , such that ° ° 0 we 0 0 ° ° °t1 − t2 ° < δ. We can write, for some τ ∈ T (f ; 3ε ), the decomposition t1 = t1 + τ , ° ° 0 0 ° °0 where kt1 k ≤ r. Then define t2 as t2 − τ . It follows that kt1 − t2 k = °t1 − t2 ° < δ. Also kt2 k = kt1 − t2 k + kt1 k < δ + r ≤ 2r. From the above we derive that D(f (t1 ), f (t2 )) ≤ 3ε , and accordingly, as τ ∈ T (f ; 3ε ), we find that 0 0 D(f (t2 ), f (t1 ) = D(f (t2 + τ ), f (t1 + τ )) = D(f (t1 ) ⊕ f (t2 + τ ), f (t1 + τ ) ⊕ f (t1 )) = D(f (t1 ) ⊕ f (t1 + τ ), f (t1 ) ⊕ f (t2 + τ )) ≤ D(f (t1 ), f (t1 + τ )) + D(f (t1 ), f (t2 + τ )) = D(f (t1 ), f (t1 + τ )) + D(f (t1 ) ⊕ f (t2 + τ ), f (t1 ) ⊕ f (t2 + τ )) ≤ D(f (t1 ), f (t1 + τ )) + D(f (t1 ), f (t1 + τ )) + D(f (t1 + τ ), f (t2 + τ )) ¤ = 3ε + 3ε + 3ε = ε. The next result shows that the set AP (RF ) is closed with respect to uniform convergence. Theorem 25. If fk : Rn −→ RF , n ∈ N are B-almost periodic and f k −→ ∞ uniformly on Rn , then is B-almost periodic.

k

−→ f as

Proof. Let ε > 0. Since fk (t) −→ f (t) uniformly over Rn as k −→ ∞, so we can find a natural number k0 such that∀k ≥ k0, we have D(fk (t), f (t)) < 3ε . Since f n k : R −→ RF is almost periodic for k = 1, 2, 3, ... so for already chosen ε > 0, we can find a relatively dense set T (fk ; 3ε ) such that D(fk (t + τ ), f k (t)) < 3ε , ∀τ ∈ T (fk ; 3ε ), t ∈ Rn , k = 1, 2, 3, .... Now D(fk (t + τ ), f k (t)) = D(f (t + τ ) ⊕ fk (t + τ ), fk (t + τ ) ⊕ f (t)) ≤ D(f (t + τ ), fk (t + τ )) + D(fk (t + τ ), f (t)) = D(f (t + τ ), fk (t + τ )) + D(fk (t + τ ) ⊕ fk (t), fk (t) ⊕ f (t)) ≤ D(f (t + τ ), fk (t + τ )) + D(fk (t + τ ), fk (t)) + D(fk (t), f (t)) = 3ε + 3ε + 3ε = ε, ∀τ ∈ T (fk ; 3ε ), t ∈ Rn , k = 1, 2, 3, .... which implies that τ ∈ T (f ; ε), and hence T (f ; ε) is a relatively dense set in R n so f is proved to be almost periodic function. ¤ We now give the following simple theorem. Theorem 26. If f : Rn −→ RF is B-almost periodic, then λ ¯ f, λ ∈ R, Fh (x) = f (x+h), and G(x) = kf (x)kF , x ∈ Rn are B-almost periodic functions. Proof. Because D(λ ¯ f (t + τ ), λ ¯ f (t)) = |λ| D(f (t + τ ), f (t)), for all, it follows that λf is almost periodic whenever f is B-almost periodic. And since


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|kf (t + τ )kF − kf (t)kF | ≤ D(f (t + τ ), f (t)), then it is immediate that G(x) = kf (x)kF , x ∈ Rn , is B-almost periodic. Now at the last let h be fixed and for every ε > 0, let r = r(ε) > 0 be attached to f in the definition of B-almost periodic. By D(f (t + τ ), f (t)) < ε, ∀τ ∈ T (f ; ε) and ∀t ∈ Rn , we get ( by taking t = u + h ), D(f (u + h + τ ), f (u + h)) < ε, ∀τ ∈ T (f ; ε) and ∀t ∈ Rn . D(Fh (u + τ ), F (u)) < ε, ∀τ ∈ T (f ; ε) and ∀t ∈ Rn . This implies that T (f ; ε) ⊂ T (Fh ; ε), therefore F is B-almost periodic. ¤ We now define normal functions and prove some results. Definition 27. We say that f is normal if for any sequence Fk : Rn −→ RF of the form Fk (x) = f (x + hk ), k ∈ N, where (hk )k is a sequence of real numbers, one can extract a subsequence of (Fk )k , converging uniformly on Rn i.e. for every sequence (hk )k of real numbers there exists a subsequence (hkl )k , and F : Rn −→ RF which l may depend on (hk )k , such that D(Fkl (x), F (x)) −→ 0 as l −→ ∞, uniformly with respect to x. We now apply this to prove the following theorem. Theorem 28. The sum of two B-almost periodic functions are B-almost periodic. Proof. Let f and g be two B-almost periodic functions, and let (hk )k an arbitrary sequence in Rn . From the sequence (fhk )k of translates, we can choose a uniformly convergent subsequence on Rn say (flk )k . From the sequence (gkk )k ,we choose a subsequence uniformly convergent on Rn , say (glk )k . Then the sequence (flk + gkk )k , which is a subsequence of the sequence (fhk + ghk )k , is uniformly convergent on Rn . ¤ To prove the equivalence between the normal functions and B-almost periodic functions we need the following lemma. Lemma 29. Let f : Rn −→ RF be a B-almost periodic function and a sequence (xk )k ⊂ Rn be given then to any ε > 0 we may associate a subsequence (xki )ki such that the inequality sup D(f (x + xkp ), f (x + xkq )) < ε, ∀p, q ∈ N, is satisfied. x∈Rn

Proof. We know that any vector xk ∈ Rn , can be written as xk = yk +zk , where zk ∈ T (f ; 2² ) and kyk k ≤ r( 2ε ) = r > 0 ( r is independent of k ). Let y be the limit point of the sequence (yk )k then kyk ≤ r. Since f : Rn −→ RF be a B-almost periodic function so it is uniformly continuous over Rn so we can find δ > 0 such that kx1 − x2 k < δ =⇒ kf (x1 ) − f (x2 )k < 2ε . Then in the ball {x ∈ Rn : kxk ≤ δ} we find an infinite sequence of yk , s which we denote by (xki )ki . Take now two vectors xkp and xkq then xkp = ykp +zkp , xkq = ykq + zkq . We deduce that sup D(f (x + xkp ), f (x + xkq )) = x∈Rn

sup D(f (x + ykp + zkp ), f (x + ykq + zkq ))

x∈Rn

= sup D(f (x + ykp + zkp − ykq − zkq ), f (x)) x∈Rn

= sup D(f (x + ykp + zkp − ykq − zkq ) ⊕ f (x + ykp − ykq ), f (x) ⊕ f (x + ykp − ykq )) x∈Rn

≤ sup D(f (x + ykp + zkp − ykq − zkq ), f (x + ykp − ykq )) + D(f (x), f (x + ykp − ykq )) x∈Rn

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≤ 2ε + 2ε = ε. The last inequality is a consequence ° ° of °fact that° ° ° ykp − ykq ∈ T (f ; 2ε ) and °ykp − ykq ° ≤ °ykp − y ° + °y − ykq ° ≤ δ + δ = 2δ.

¤

Theorem 30. Any B-almost periodic f : Rn −→ RF , is normal.

Proof. Let (xk )k ⊂ Rn be a given sequence. Then by Lemma 29 we can find a subsequence (xki ,1 )i such that sup D(f (x + xkp ,1 ), f (x + xkq ,1 )) < 1, ∀p, q ∈ N. x∈Rn

D(f (x + xkp ,1 ), f (x + xkq ,1 )) < 1, ∀p, q ∈ N, ∀x ∈ Rn . Next we choose a subsequence (xki ,2 )i ⊂ (xki ,1 )i with the property D(f (x + xkp ,2 ), f (x + xkq ,2 )) < 21 , ∀p, q ∈ N, ∀x ∈ Rn . We can choose a further subsequence (xki ,3 )i ⊂ (xki ,2 )i with the property. D(f (x + xkp ,3 ), f (x + xkq ,3 )) < 31 , ∀p, q ∈ N, ∀x ∈ Rn . And so on. Consider now a diagonal sequence, (f (x + xki ,i )i of translated functions. Now if p, q ∈ N, with p ≤ q, then we have sup D(f (x + xkp ,p ), f (x + xkq ,q )) < 13 , ∀p, q ∈ N. This proves that the sequence x∈Rn

(f (x + xki ,i ))i is uniformly convergent over Rn this also proves the normality of f. ¤ The converse of this theorem also holds true as proved in the following theorem. Theorem 31. Any normal function f : Rn −→ RF is B-almost periodic. Proof. On contrary suppose that f is not B-almost periodic function then there exists an ε > 0 such that the set T (f ; ε) is not relatively dense in Rn . This implies that for all r(ε) = r > 0 there is a ball B(a; r) which contains no element of the set T (f ; ε). Consider now an arbitrary element x1 ∈ Rn and take r1 > kx1 k, hence there exists a ball B(x2 ; r2 ) which is disjoint with T (f ; ε). Since x2 − x1 ∈ B(x2 ; r2 ) =⇒ x2 − x1 ∈ / T (f ; ε). Next take r1 > kx1 k + kx2 k and find a ball B(x3 ; r3 ) which is disjoint of T (f ; ε). Now both the vectors x2 − x1 and x2 − x3 belong to B(x3 ; r3 ) but x2 − x1 ∈ / T (f ; ε) and x2 − x3 ∈ / T (f ; ε). Continuing this procedure, we can find an infinite sequence (xk )k ⊂ Rn such that ∀k, l ∈ N, k 6= l =⇒ xk − xl ∈ / T (f ; ε). It follows that, by replacing x by x − xk D(f (x + xk ), f (x + xl )) = D(f (x + xk − xl ), f (x)) > ε, ∀x ∈ Rn . This shows that the sequence, (f (x + xk ))k , x ∈ Rn contains no subsequence which converges uniformly over Rn . A contradiction to the fact that f is normal function. So our assumption that f is not B-almost periodic function is wrong. Therefore f is proved to be almost periodic. ¤ Remark 32. Let us denote AP (RF ) = {f : Rn −→ RF : f is B-almost periodic} , and for f ∈ AP (RF ), let us define kf k = sup {kf (x)kF : x ∈ Rn }. By theorem 20 we get kf k < +∞. AP (RF ), is a complete metric space with respect to the metric D∗ : AP (RF ) × AP (RF ) −→ R+ ∪ {0} defined by D∗ (f, g) = sup D(f (x), g(x)), f, g ∈ AP (RF ) x∈Rn

Let us denote Cb (RF ) = {f : Rn −→ RF : f is continuous and bounded on Rn } Then because


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(RF , D) is a complete metric space, it follows that (Cb ( RF ), D∗ ) is a complete metric space. Then theorems 20 and 23 show that AP (RF ) is a closed subset of Cb ( RF ), i.e. (AP ( RF ), D∗ ) is a complete metric space. References 1. L. Amerio and G. Prouse, Almost periodic and functional equations, Van Nostrand, New York, 1971. 2. B. Bede and S. G. Gal, Almost periodic fuzzy-number-valued functions, Fuzzy Sets and System 47(3) (2004), 385–403. 3. A. S. Besicovitch, Almost periodic functions, Dover Publications Inc., 1954. 4. S. Bochner, A new approach to almost periodicity, Proc. Nat. Acad. Sci. 48 (1962), 2039–2043. 5. H. Bohr, Almost periodic functions, Chelsea Publishing company, New York, 1947. 6. C. Corduneanu, Almost periodic functions, Inter-Science Publishers, New York, London, Toronto, 1989. 7. A. M. Fink, Almost periodic differential equations, Springer-Verlag, 1974. 8. B. M. Levitan, Almost periodic functions, Goz. Izd., Moscow, 1953. 9. B. M. Levitan and V. V. Zhikov, Almost periodic functions and differential equations, Cambridge Univ. Press, Cambridge, England, 1982. 10. Gaston M. N’Gu´ er´ ekata, Topics in almost automorphy, Springer and Verllage, New York, 2005. 11. S. Zaidman, Almost periodic functions with values in abstract spaces, Pitman Advanced Publishing Program, Boston, London, Melbourne, 1985.