Received May 27, 2009; accepted January 14, 2011

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1 MATHEMATICAL COMMUNICATIONS 53 Math. Coun. 6(20), I σ -Convergence Fatih Nuray,, Hafize Gök and Uǧur Ulusu Departent of Matheatics, Afyon Kocatepe University, Afyonkarahisar, Turkey Received May 27, 2009; accepted January 4, 20 Abstract. In this paper, the concepts of σ-unifor density of subsets A of the set N of positive integers and corresponding I σ -convergence were introduced. Furtherore, inclusion relations between I σ-convergence and invariant convergence also I σ-convergence and [V σ ] p -convergence were given. AMS subject classifications: 40A05, 40D25 Key words: statistical convergence, I-convergence, invariant convergence, strongly σ- convergence, σ-unifor density, I σ -convergence. Introduction and background A sequence x = (x k ) is said to be strongly Cesaro suable to the nuber L if n n n x k L = 0. k= A continuous linear functional φ on l, the space of real bounded sequences, is said to be a Banach it if (a) φ(x) 0, when the sequence x = (x n ) has x n 0 for all n, (b) φ(e) =, where e = (,,,...), and (c) φ(x n+ ) = φ(x n ) for all x l. A sequence x l is said to be alost convergent to the value L if all of its Banach its are equal to L. Lorentz [4] has given the following characterization. A bounded sequence (x n ) is said to be alost convergent to L if and only if x n+k = L uniforly in n. ĉ denotes the set of all alost convergent sequences. k= Corresponding author. Eail addresses: fnuray@aku.edu.tr (F. Nuray), hgok@aku.edu.tr (H. Gök), ulusu@aku.edu.tr (U. Ulusu) c 20 Departent of Matheatics, University of Osijek

2 532 F. Nuray, H. Gök and U. Ulusu Maddox [5] has defined a strongly alost convergent sequence as follows: A bounded sequence (x n ) is said to be strongly alost convergent to L if and only if x n+k L = 0 k= uniforly in n. [ĉ] denotes the set of all strongly alost convergent sequences. Let σ be a apping of the positive integers into theselves. A continuous linear functional φ on l is said to be an invariant ean or a σ-ean if it satisfies conditions (a) and (b) stated above and (d) φ(x σ(n) ) = φ(x n ) for all x l. The appings σ are assued to be one-to-one and such that σ (n) n for all positive integers n and, where σ (n) denotes the th iterate of the apping σ at n. Thus φ extends the it functional on c, the space of convergent sequences, in the sense that φ(x) = x for all x c. Consequently, c V σ. In the case σ is the translation apping σ(n) = n +, the σ-ean is often called a Banach it and V σ, the set of bounded sequences all of whose invariant eans are equal, is the set of alost convergent sequences ĉ. It can be shown that V σ = {x = (x n ) l : x σ k (n) = L uniforly in n}, where l denotes the set of all bounded sequences. The set of all such σ appings will be denoted by M. Raii [] proved that {Vσ : σ M} = l k= and {Vσ : σ M} = c, where c denotes the set of all convergent sequences. The following inclusion relation between ĉ and V σ can be written: {ĉ} {V σ : σ M}. Several authors including Raii [], Schaefer [4], Mursaleen [8], Savaş [2] and others have studied invariant convergent sequences. The concept of strongly σ-convergence was defined by Mursaleen in [7]: A bounded sequence x = (x k ) is said to be strongly σ convergent to L if x σ k (n) L = 0 k= uniforly in n. In this case we will write x k L[V σ ]. By [V σ ], we denote the set of all strongly σ-convergent sequences. In the case σ(n) = n+, the space [V σ ] is the set of strongly alost convergent sequences [ĉ].

3 I σ -Convergence 533 Recently, the concept of strong σ-convergence was generalized by Savaş [2] as below [V σ ] p := {x = (x k ) : x σk (n) L p = 0 uniforly in n}, k= where 0 < p <. If p =, then [V σ ] p = [V σ ]. It is known that [V σ ] p l. A sequence x = (x k ) is said to be statistically convergent to the nuber L if for every ɛ > 0, n n {k n : x k L ɛ} = 0, where the vertical bars indicate the nuber of eleents in the enclosed set. The idea of statistical convergence was introduced by Fast [3] and studied by any authors. There is a natural relationship between statistical convergence and strong Cesaro suability [2]. The concept of a σ-statistically convergent sequence was introduced by Nuray and Savaş in [0] as follows: A sequence x = (x k ) is σ-statistically convergent to L if for everyɛ > 0, {k : x σ k (n) L ɛ} = 0 uniforly in n. In this case we write S σ x = L or x k L(S σ ) and define 2. I σ -convergence Definition. Let A N and If the following its exist S σ := {x = (x k ) : S σ x = L, for soe L}. s := in n A {σ(n), σ 2 (n),..., σ (n)} S := ax n A {σ(n), σ 2 (n),..., σ (n)}. s V (A) :=, S V (A) := then they are called a lower and an upper σ-unifor density of the set A, respectively. If V (A) = V (A), then V (A) = V (A) = V (A) is called the σ-unifor density of A. In the case σ(n) = n +, this definition gives a definition of unifor density u in []. A non-epty subset of I of P (N) is called an ideal on N if (i) B I whenever B A for soe A I, (ii) A B I whenever A, B I.

4 534 F. Nuray, H. Gök and U. Ulusu An ideal I is called proper if N / I. An ideal I is called adissible if it is proper and contains all finite subsets. For any ideal I there is a filter F (I) corresponding to I, given by F (I) = {K N : N \ K I}. Let I P (N) be a proper ideal in N. The sequence x = (x k ) is said to be I-convergent to L, if for ɛ > 0 the set A ɛ := {k : x k L ɛ} belongs to I. If x = (x k ) is I-convergent to L, then we write I x = L. A sequence x = (x k ) is said to be I -convergent to the nuber L if there exists a set M = { < 2 <...} F (I) such that k x k = L. In this case we write I x k = L (see [3]). Denote by I σ the class of all A N with V (A) = 0. Definition 2. A sequence x = (x k ) is said to be I σ -convergent to the nuber L if for every ɛ > 0 A ɛ := {k : x k L ɛ} belongs to I σ ; i.e., V (A ɛ ) = 0. In this case we write I σ x k = L. The set of all I σ -convergent sequences will be denoted by I σ. In the case σ(n) = n +, I σ -convergence coincides with I u - convergence which was defined in []. We can also write {I u } {I σ : σ M}, where I u denotes the set of all I u -convergent sequences. We can easily verify that if I σ x n = L and I σ y n = L 2, then I σ (x n + y n ) = L + L 2 and if a is a constant, then I σ ax n = al. Theore. Suppose x = (x k ) is a bounded sequence. If x is I σ -convergent to L, then x is invariant convergent to L. Proof. Let, n N be arbitrary and ɛ > 0. We estiate t(n, ) = x σ(n) + x σ 2 (n) x σ (n) L. We have t(n, ) t () (n, ) + t (2) (n, ), where and t () (n, ) = t (2) (n, ) = j ; j ; x σ j (n) L x σ j (n) L ɛ x σ j (n) L. x σ j (n) L <ɛ

5 I σ -Convergence 535 We have t (2) (n, ) < ɛ, for every n =, 2,... The boundedness of x = (x k ) iplies that there exist K > 0 such that x σ j (n) L K, (j =, 2,...; n =, 2,...), then this iplies that t () (n, ) K { j : x σ j (n) L ɛ} hence x is invariant convergent to L. K ax n { j : x σj (n) L ɛ} = K S, The converse of the previous theore does not hold. For exaple, x = (x k ) is the sequence defined by x k = if k is even and x k = 0 if k is odd. When σ(n) = n +, this sequence is invariant convergent to 2 but it is not I σ-convergent. In [2], Connor gave soe inclusion relations between strong p-cesaro convergence and statistical convergence and showed that these are equivalent for bounded sequences. Now we shall give an analogous theore which states inclusion relations between [V σ ] p -convergence and I σ -convergence and show that these are equivalent for bounded sequences. Theore 2. (a) If 0 < p < and x k L([V σ ] p ), then x = (x n ) is I σ -convergent to L. (b) If x = (x n ) l and I σ -converges to L, then x k L([V σ ] p ). (c) If x = (x n ) l, then x = (x n ) is I σ -convergent to L if and only if x k L([V σ ] p ) (0 < p < ). Proof. (a) Let x k ([V σ ] p ), 0 < p <. Suppose ɛ > 0. Then for every n N, we have and x σ j (n) L p j ; x σ j (n) L ɛ x σ j (n) L p ɛ p { j : x σ j (n) L ɛ} ɛ p ax n { j : x σ j (n) L ɛ} x σ j (n) L p ɛ p ax n { j : x σ j (n) L ɛ} = ɛ p S for every n =, 2, 3,.... This iplies S = 0 and so I σ x k = L. (b) Now suppose that x l and I σ -convergent to L. Let 0 < p < and ɛ > 0. By assuption, we have V (A ɛ ) = 0. The boundedness of x = (x k ) iplies that

6 536 F. Nuray, H. Gök and U. Ulusu there exist M > 0 such that x σ j (n) L M, that for every n N we have that x σ j (n) L p = j= Hence, we obtain + j ; x σ j (n) L ɛ j ; x σ j (n) L <ɛ (j =, 2,...; n =, 2,...). Observe x σ j (n) L p x σj (n) L p M ax n { j : x σ j (n) L ɛ} M S + ɛp. x σ j (n) L p = 0 j= uniforly in n. (c) This is a corollary of (a) and (b). In the case σ(n) = n+ in the above theores, we have Theore and Theore 2 in []. Definition 3. A sequence x = (x k ) is said to be Iσ-convergent to the nuber L if there exists a set M = { < 2 <...} F (I σ ) such that k x k = L. In this case we write Iσ x k = L. I σ- convergence is better applicable in soe situations. Theore 3. Let I σ be an adissible ideal. If a sequence x = (x k ) is I σ-convergent to L, then this sequence is I σ -convergent to L. Proof. By assuption, there exists a set H I σ such that for M = N \ H = { < 2 <... < k <...} we have + ɛ p x k = L. () k Let ɛ > 0. By (), there exists k 0 N such that x k L < ɛ for each k > k 0. Then obviously {k N : x k l ɛ} H { < 2 <... < k0 }.(2) (2) The set on the right-hand side of (2) belongs to I σ (since I σ is adissible). So x = (x k ) is I σ -convergent to L. The converse of Theore 3 holds if I σ has property (AP). Definition 4 (see [3]). An adissible ideal I is said to satisfy the condition (AP) if for every countable faily of utually disjoint sets {A, A 2,...} belonging to I there exits a countable faily of sets {B, B 2,...} such that the syetric difference A j B j is a finite set for j N and B = ( j= B j) I.

7 I σ -Convergence 537 Theore 4. Let I σ be an adissible ideal and let it have property (AP). If x is I σ -convergent to L, then x is I σ-convergent to L. Proof. Suppose that I σ satisfies condition (AP). Let I σ x k = L. Then for ɛ > 0, {k : x k L ɛ} belongs to I σ. Put A = {k : x k L } and A n = {k : n x k L < n } for n 2, n N. Obviously, A i B j = for i j. By condition (AP) there exits a sequence of {B n } n N such that A j B j are finite sets for j N and B = ( j= B j) I σ. It is sufficient to prove that for M = N \ B we have Let λ > 0. Choose n N such that n+ < λ. Then x k = L. (3) k M;k {k : x k L λ} Since A j B j, j =, 2,..., n + are finite sets, there exists k 0 N such that n+ ( j= n+ B j ) {k : k > k 0 } = ( j= n+ j= A j. A j ) {k : k > k 0 } (4) If k > k 0 and k / B, then k / n+ j= B j and by (4), k / n+ j= A j. But then x k L < n + < λ so (3) holds and hence we have I σ x k = L. Now we shall state a theore that gives a relation between S σ -convergence and I σ -convergence. Theore 5. A sequence x = (x k ) is S σ -convergent to L if and only if it is I σ - convergent to L. References [] V. Baláž, T. Šalát, Unifor density u and corresponding I u-convergence, Math. Coun. (2006), 7. [2] J.S. Connor, The statistical and strong p - Cesaro convergence of sequences, Analysis 8(988), [3] H. Fast, Sur la convergence statistique, Cooloq. Math. 2(95), [4] P. Kostryko,T. Šalát, W. Wilczinski, I-convergence, Real. Anal. Exchange 26( ), [5] G. Lorentz, A contribution to the theory of divergent sequences, Acta Math. 80(948), [6] I. J. Maddox, A new type of convergence, Math. Proc. Cabridge Phil. Soc. 83(978), 6 64.

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