On the statistical type convergence and fundamentality in metric spaces

Transcription

1 Caspian Journal of Applied Mathematics, Ecology and Economics V. 2, No, 204, July ISSN On the statistical type convergence and fundamentality in metric spaces B.T. Bilalov, T.Y. Nazarova Abstract. The concept of F-fundamentality, generated by some filter F is introduced in metric spaces. Its equivalence to the concept of F-convergence is proved in metric spaces. This convergence generalizes many kinds of convergence, including the well-known statistical convergence. Key Words and Phrases: F-convergence, F- fundamentality, statistical convergence 2000 Mathematics Subject Classifications: 40A05, 54A20, 54E35.. Introduction The idea of statistical convergence (stat-convergence was first proposed by A.Zigmund [] in his famous monograph where he talked about almost convergence. The first definition of it was given by H. Fast [2] and H. Steinhaus [3]. Later, this concept has been generalized in many directions. It is impossible to list all the related papers. More details on this matter and its applications can be found in [4-5, 24]. It should be noted that the methods of non-convergent sequences have long been known and they include e.g. Cesaro method, Abel method, etc. These methods are used in different areas of mathematics. For the applicability of these methods it is very important that the considered space has a linear structure. Therefore, the study of statistical convergence in metric spaces is of special scientific interest. Different aspects of this problem have been studied in [6, 7]. Statistical convergence is currently actively used in many areas of mathematics such as summation theory [7, 8, 9], number theory [, 3], trigonometric series [], probability theory [8], measure theory [2], optimization [20], approximation theory [2, 22], fuzzy theory [26], etc. It should be noted that the concept of statistical fundamentality (stat-fundamentality was first introduced by J.A. Fridy [4] who proved its equivalence to stat-convergence with respect to numerical sequences. This issue was raised in [0] concerning uniform space (X;U. It was proved that if the sequence {x n } n N X is stat-convergent, then it is stat-fundamental. The problem of the validity of converse statement was also raised in [0]. Corresponding author c 203 CJAMEE All rights reserved.

2 Stat-convergence was generalized by many mathematicians (see [0, 5, 23, 24, 25]. Theconcepts of I-convergence and I -convergence were introduced in [23]. Thesekinds of convergences generalizes many previously known convergences, including the well-known stat-convergence. In present paper we introduce the concepts of F-convergence and F- fundamentality, generated by some filter F 2 N. Their equivalence is proved. F- convergence generalizes many kinds of convergence, related to such concepts as statistical density, logarithmic density, uniform density, etc. More details on these concepts can be found in [23] Needful information We will use the standard notation. N will be a set of all positive integers; R will be a set of real numbers; χ M ( is the characteristic function of M; (X;ρ is a metric space. O ε (a is an open ball centered at a with radius ε, i.e. O ε (a {x X : ρ(x;a < ε}. 2 M will be a set of all subsets M; M will stand for the closure of M; A = carda is the number of elements of A. M C = N\M. will be a quantifier which means and. Let us recall the definition of asymptotic (statistical density of A N. Assume δ n (A = n χ A (k, and let δ (A = liminf δ n(a, δ (A = limsupδ n (A. δ (A and δ (A are called lower and upper asymptotic density of the set A, respectively. If δ (A = δ (A = δ(a, then δ(a is called asymptotic (or statistical density of A. It should be noted that the statistical convergence is determined by means of this concept, namely, the consequence {x n } n N X is called statistically convergent to x, if δ(a ε = 0, for ε > 0, where A ε {n N : ρ(x n ;x ε}. Let us also recall the definitions of the ideal and the filter. A family of sets I 2 N is called an ideal if: α I; β A;B I A B I ; γ (A I B A B I. A family F 2 N is called a filter on X, if : i / F ; ii from A;B F A B F; iii from A F (A B B F. Filter, satisfying the condition iv if A A 2... A n F, n N {n m } m N N; n < n 2 <... : m= ((n m,n m+ ] A m F is called a monotone closed filter. Filter F satisfying the following condition is called a right filter: v F C F, for any finite subset F N. An ideal I is called non-trivial if I I X. I 2 N is a non-trivial ideal if and only if F = F (I = {X\A : A I} is a filter on X. A non-trivial ideal I 2 N is called admissible if and only if I {{x} : x X}. k=

3 86 B.T. Bilalov, T.Y. Nazarova In the sequel, we assume that (X;ρ is a metric space with metric ρ, and I 2 N is some non-trivial ideal. Definition [23]. The sequence {x n } n N X is called I-convergent to x X (I- lim x n = x, if A ε I, ε > 0, where A ε = {n N : ρ(x n ;x ε}. Let I d {A N : d(a = 0}. I d is an ideal on N. I d -convergence means the statistical convergence. It should be noted that if I is an admissible ideal, then the usual convergence in X implies I-convergence in X. Definition 2. The sequence {x n } n N X is called I -convergent to x X, if M F (I (i.e. N\M I, M = {m < m 2 <... < m k <...} : lim ρ(x m k ;x = 0. k In the following discussion we will need the following interesting results of [23]. Theorem [23]. Let I be an admissible ideal. If I - lim x n = x I- lim x n = x. The converse iot always true, it depends on the structure of space (X;ρ, namely, we have Theorem 2 [23]. Let (X;ρ be a metric space. (i If X hao accumulation point, then I-convergence and I -convergence coincide for each admissible ideal I 2 N ; (ii If X has an accumulation point ξ, then there exists an admissible ideal I 2 N and a sequence {y n } n N X: I- lim y n = ξ, but I - lim y n doeot exist. 3. Main results Let (X;ρ be some complete metric space and F 2 N be some filter. Accept the following Definition 3. Let F 2 N be some filter. The sequence {x n } n N X is called F-convergent to x X (F- lim x n = x, if A ε F, ε > 0, where A ε {n N : x n O ε (x}. Let us introduce the concept of F-fundamentality. Definition 4. The sequence {x n } n N X is called F-fundamental, if ε > 0, n ε N : nε F, where nε {n N : x n O ε (x nε }. Assume that F- lim x n = x. Let ε > 0 be an arbitrary number. Consequently, A ε/2 F. From the condition i in the definition of filter above it follows that A ε/2. Take n ε A ε/2 : ρ(x nε ;x < ε 2. From the relation ρ(x n ;x nε ρ(x n ;x+ ρ(x;x nε < ε 2 +ρ(x n;x, it directly follows that { n N : ρ(x n ;x < ε } {n N : ρ(x n ;x nε < ε}. 2 Hence nε F, i.e. the sequence {x n } n N is F-fundamental in X. Now, vice versa, let {x n } n N X be F-fundamental. From F- fundamentality it follows that n j N : K j F, where K j {n N : ρ ( x n ;x nj 2 j }, j =,2.

4 By the definition of filter we obtain K K 2 F. Put M O (x n O 2 (x n2. It is obvious that x n M, n (K K 2 K (. Thus, n 3 N : K 3 F, where K 3 { n : ρ(x n ;x n3 2 2}. Let K (2 = K ( K 3. It is clear that K (2 F. Now let M 2 M O 2 2 (x n3. Denote by d ρ (M the diameter of the set M, i.e. d ρ (M = sup ρ(x;y. x,y M Continuing in the same way, we obtain the nested sequence of closed sets {M n } n N X: M M 2...; whose diameters tend to zero: i.e. d ρ (M n 2 n+ 0, n. Moreover, K (n F, where K (n {n N : x n M n }. Take x n M n, n N. We have ρ( x n ; x n+p d ρ (M n 0, n, p N. Hence, the sequence { x n } n N is fundamental in X. Let x lim n = x. It is absolutely clear that x n M n, i.e. n M n ion-empty. From d ρ (M n 0, n, it directly follows that n M n {x}, i.e. n M n consists of one element. Let us show that F- lim x n = x. Take ε > 0. Take n ε N : d ρ (M nε < ε. Let y M nε be an arbitrary element. So ρ(y,x d ρ (M nε < ε. Consequently, M nε O ε (x. We have K (nε F, where K (nε = {n N : x n M nε }. So, K (nε {n N : x n O ε (x}, it is clear that {n N : x n O ε (x} F Flim x n = x. Thus, we have proved the following theorem. Theorem 3. Let (X;ρ be complete metric space and F 2 N be some filter. The sequence {x n } n N X is F-convergent in X if and only if it is F-fundamental in X. It is easy to see that F- lim x n is unique if it exists. In fact, let F- lim x n has two values y y 2. Take ε ( 0, 2 ρ(y ;y 2. Let A k {n N : ρ(x n ;y k < ε}, k =,2. It is clear that A k F, k =,2 A A 2 F. As A A 2 = / F, the obtained contradiction proves that y = y 2. Let us consider the sequence { K (n, constructed in the proof of Theorem. We }n N have K ( K (2... K (n F, n N. Then from the condition iv in the definition of filter we have {n m : n < n 2 <...} : m=( (nm,n m+ ] K (m F. 87 Assume N 0 {k N : k (n m,n m+ ] K C (m, m N } [,n ], where M C N\M. Define y k = { x, k N0 ; x k, otherwise, where F- lim x n = x. Take ε > 0. If k N 0, then ρ(y k ;x = ρ(x;x < ε. If k / N 0, then m : n m < k n m+ k / K c (m k K (m x k M m (M M 2...

5 88 B.T. Bilalov, T.Y. Nazarova is a sequence from Theorem ρ(x k ;x d ρ (M m < ε for sufficiently great values of m (as x M m, m N. Hence, we have lim y k = x. Let us show that K k {k N : x k = y k } F. In fact, it is clear that m=( (nm,n m+ ] K (m K, holds. Then from the condition iii in the definition of filter we get K F. Thus, if F- lim x n = x, then K F: lim y n = x and x n = y n, n K. Conversely, let lim y n = x and K {n : x n = y n } F. Take ε > 0. Then n ε N : ρ(y n (;x < ε, n n ε. We have {n N : n n ε } K {n N : ρ(x n ;x < ε}. It is clear that {n N : n n ε } K F. Then from the condition iii in the definition of filter it follows {n N : ρ(x n ;x < ε} F. Thus, the following theorem is true. Theorem 4. Let (X;ρ be a metric space and F 2 N be some filter. Then: if F is a monotone close and F- lim x n = x, then {y n } n N X : lim y n = x {n N : x n = y n } F; 2if F is aright filter and lim y n = x ({n N : x n = y n } F, then F- lim x n = x. The Theorems ;2 imply the following Corollary. Let (X;ρ be a complete metric space, F 2 N be some monotone close and right filter. Then the following statements are equivalent to each other: α F- lim x n = x; β {x n } n N is F-fundamental; γ lim y n = x ({n N : x n = y n } F. The Theorem 2 immediately implies the following Corollary 2. Let (X;ρ be a metric space and F 2 N be a right filter. If Flim x n = x, then {n k : n < n 2 <...} F: lim x n k = x. k 4. Filters I. An ordinary convergence. F {M N : M C N\M is a finite set}. F-convergence, generated by this filter, coincides with the ordinary convergence. II. Statistical convergence. Assume F δ {M N : δ(m = }. F δ is a filter. It is easy to see that F δ is a right filter. Let us show that F δ is a monotone close filter. Let K K 2... (δ(k n =, n N. It is clear that δ(kn c = 0, n N. Therefore {n k } k N N; n < n 2 <... : n I n K c m < m, n n m. Let N 0 = Ñ0 I n, where Ñ0 {k N : n m < k n m+ (k Km c }. It is obvious that δ(n 0 = δ (Ñ0. Take n N. Then m N : n m < n n m+. Without loss of generality, we may suppose that n > n. Let us show that ( I n Ñ0 (I n Km c. (

6 89 ( Let k I n Ñ0 m 0 m : n m0 < k n m0 + ( k Km c 0 k K c m. So, the inclusion ( is true. Consequently I n Ñ0 n n I n Km c < m. (2 From (2 it directly follows that δ (Ñ0 = 0. As a result, δ(n 0 = 0 δ(n c 0 = N c 0 F δ. In the sequel, it should be pointed out that N c 0 {k N : n m < k n m+ (k K m }. Thus, F δ is a monotone close filter. Fulfilment of condition v by F δ is obvious. Then, the statement of Corollary is true with respect to F δ -convergence. So,we get the validity of Statement. Filter F δ, generated by statistical density, is a monotone close and right filter. III. Logarithmic convergence. Let M N. Assume l n (M = k= χ M (k, k where = n k= k. If lim l n(m = l(m, then l(m is called a logarithmic density of the set M. Let F l {M N : l(m = }. The following lemma is true. Lemma. If l(m k =, k =,2 l(m M 2 =. Proof. We have Consequently M M 2 = (M M 2 \[(M 2 \M (M \M 2 ]. M M 2 I n = [(M M 2 I n ]\[((M 2 \M (M \M 2 I n ]. (3 From ((M 2 \M I n (M c I n, we get k= k χ M 2 \M (k k= k χ M c (k. (4 It is absolutely clear that, if l(m =, then l(m c = 0. Then from (4 we obtain l(m 2 \M = 0. Similarly, we have l(m \M 2 = 0. So ((M 2 \M (M \M 2 I n = ((M 2 \M I n ((M \M 2 I n. It is clear that l((m 2 \M (M \M 2 = 0. (5

7 90 B.T. Bilalov, T.Y. Nazarova It is easy to see that l(m M 2 =. From (3 we have k= k χ M M 2 (k = k= k χ M M 2 (k k= k χ (M 2 \M (M \M 2 (k. Taking into account (5 we get l(m M 2 =. Lemma is proved. This lemma implies that F l is a filter. If M N is a finite set, then it is clear that M C F l, i.e. F l satisfies the condition v. Then it is absolutely clear that l(m = 0. Let us show that F l is a monotone close filter. Let K K 2... (l(k n =, n N l(k c n = 0, n N. Therefore {n k } k N N, n < n 2 <... : k= χ K c m (k k < m, n n m. Similarly to the previous example, let N 0 = Ñ0 I n, where Ñ 0 {k N : n m k n m+ (k K c m }. It is clear that l(n 0 = l (Ñ0. Let n N m N : n m < n n m+. As before, we assume that n > n. It is clear that ( is true, i.e.. ( I n Ñ0 (I n Km. c Hence k= χñ0 (k k k= χ K c m (k k < m, n n m. Consequently, l (Ñ0 = 0 l(n 0 = 0 l(n c 0 = Nc 0 F l. It is obvious that N c 0 {k N : n m < k n m+ (k K m }. It directly follows that F l is a right filter. Thus, we have proved Statement 2. Filter F l, generated by logarithmic density, is a monotone close and right filter. Note that, if δ(m l(m l(m = δ(m. The converse iot generally true. IV. Uniform convergence. Let M N (t Z + ;s N. Assume M (t+;t+s = n M : t+ n t+s. Put β s (M = liminf t M (t+;t+s, β s (M = limsupm (t+;t+s. t

8 If lim β s(m s s = lim β s (M s s = β(m, then the quantity β(m is called the uniform density of the set M. Let F β {M N : β(m = }. Let us show that F β is a filter. It is clear that M (t+;t+s+m c (t+;t+s = [t+,t+s] = s. Hence it directly follows that β(m = β(m c = 0. I β {M N : β(m = 0} is a non-trivial ideal [23]. Therefore, F β is a filter. Itis clear that F β satisfies thecondition v. Let us show that F β is a monotone close filter. Let K K 2... (β(k n =, n N β(k c n = 0, n N {n k} k N N, n < n 2 <... : 9 β s (K c m s < m, s n m. As before, we set N 0 = Ñ0 I n, where Ñ0 {k N : n m k n m+ (k Km}. c It is clear that β(n 0 = β (Ñ0. Let n > n be an arbitrary integer. Then m N : n m < n n m+. It is obvious that the inclusion ( I n Ñ0 (I n Km c, is true in this case, too. From the arbitrariness of n we have (Ñ0 [t+;t+s] (K c m [t+;t+s]. Consequently and as a result Ñ 0 (t+;t+s K c m (t+;t+s, β s( Ñ 0 β s (K c m. Thus β (Ñ0 s βs (Km c < s s m, s n m. From this relation it directly follows β (Ñ0 = 0 β(n 0 = 0 β(n c 0 = Nc 0 F β, where N c 0 {k N : n m < k n m+ (k K m }, i.e. F β is a monotone close filter. As a result, we obtain the validity of the following Statement 3. Filter F β, generated by the uniform convergence, is a monotone close and right filter. Following [23], number of such examples can be extended. Remark. Similar results can be obtained with respect to concepts of I-convergence and I -convergence.

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