In this paper, the concepts of asymptotically I_σ-equivalence, σ-asymptotically equivalence, strongly σ-asymptotically equivalence and strongly σ-asymptotically p-equivalence for real number sequences are deﬁned. Also, we give relationships among these new type equivalence concepts and the concept of S_σ-asymptotically equivalence which is studied in [Savaş, E. and Patterson, R.F., σ-asymptotically lacunary statistical equivalent sequences, Cent. Eur. J. Math., 4 (2006), No. 4, 648–655

Ulusu, Uğur

English

In this study, the concepts of asymptotically I_σ-equivalence, σ-asymptotically equivalence, strongly σ-asymptotically equivalence and strongly σ-asymptotically p-equivalence for real number sequences are defined. Also, we give some relationships among this new type equivalence concepts and the concept of S_σ-asymptotically equivalence which is studied in this area befor

Afyon Kocatepe Üniversitesi Açık Erişim Sistemi

Asymptotically Ideal Invariant Equivalence  – Oral Presentation  / Asymptotically Ideal Invariant Equivalence    Department of Mathematics, Faculty of Science and Literature,  Afyon Kocatepe University, Afyonkarahisar-TURKEY  ulusu@aku.edu.tr   ABSTRACT  In this study, the concepts of asymptotically -equivalence, -asymptotically equivalence, strongly -asymptotically equivalence and strongly -asymptotically -equivalence for real number sequences are defined. Also, we give some relationships among this new type equivalence concepts and the concept of -asymptotically equivalence which is studied in this area before.  129

Asymptotically ideal invariant equivalence

Afyon Kocatepe Üniversitesi

CREAT. MATH. INFORM.27 (2018), No. 2, 215 - 220Online version at http://creative-mathematics.ubm.ro/Print Edition: ISSN 1584 - 286X Online Edition: ISSN 1843 - 441XAsymptotically ideal invariant equivalenceUǦUR ULUSUABSTRACT. In this paper, the concepts of asymptotically Iσ-equivalence, σ-asymptotically equivalence, st-rongly σ-asymptotically equivalence and strongly σ-asymptotically p-equivalence for real number sequences aredefined. Also, we give relationships among these new type equivalence concepts and the concept ofSσ-asymptotically equivalence which is studied in [Savaş, E. and Patterson, R. F., σ-asymptotically lacunary sta-tistical equivalent sequences, Cent. Eur. J. Math., 4 (2006), No. 4, 648–655]1. INTRODUCTION AND BACKGROUNDLet σ be a mapping of the positive integers into themselves. A continuous linear functi-onal φ on `∞, the space of real bounded sequences, is said to be an invariant mean or aσ-mean if it satisfies following conditions:(1) φ(x) ≥ 0, when the sequence x = (xn) has xn ≥ 0 for all n,(2) φ(e) = 1, where e = (1, 1, 1, ...) and(3) φ(xσ(n)) = φ(xn) for all x ∈ `∞.The mappings σ are assumed to be one-to-one and such that σm(n) 6= n for all positiveintegers n and m, where σm(n) denotes the m th iterate of the mapping σ at n. Thus,φ extends the limit functional on c, the space of convergent sequences, in the sense thatφ(x) = limx for all x ∈ c.In the case σ is translation mappings σ(n) = n+ 1, the σ-mean is often called a Banachlimit and the space Vσ , the set of bounded sequences all of whose invariant means areequal, is the set of almost convergent sequences ĉ.It can be shown thatVσ ={x = (xn) ∈ `∞ : limm→∞1mm∑k=1xσk(n) = L, uniformly in n}.Several authors have studied invariant convergent sequences (see, [5–9, 12–14, 16, 18]).The concept of strongly σ-convergence was defined by Mursaleen in [6] as follows:A bounded sequence x = (xk) is said to be strongly σ-convergent to L iflimm→∞1mm∑k=1|xσk(n) − L| = 0,uniformly in n. It is denoted by xk → L[Vσ].By [Vσ], we denote the set of all strongly σ-convergent sequences. In the caseσ(n) = n+ 1, the space [Vσ] is the set of strongly almost convergent sequences [ĉ].Received: 12.06.2017. In revised form: 13.01.2018. Accepted: 20.01.20182010 Mathematics Subject Classification. 34C41, 40A35.Key words and phrases. Asymptotically equivalence, statistical convergence, I-convergence, invariant convergence.215216 U. UlusuThe concept of strongly σ-convergence was generalized by Savaş [13] as below:[Vσ]p ={x = (xk) : limm→∞1mm∑k=1|xσk(n) − L|p = 0, uniformly in n},where 0 < p <∞.If p = 1, then [Vσ]p = [Vσ]. It is known that [Vσ]p ⊂ `∞.The idea of statistical convergence was introduced by Fast [1] and studied by manyauthors.A sequence x = (xk) is said to be statistically convergent to L if for every ε > 0limn→∞1n∣∣∣{k ≤ n : |xk − L| ≥ ε}∣∣∣ = 0,where the vertical bars indicate the number of elements in the enclosed set.The concept of σ-statistically convergent sequence was introduced by Savaş and Nurayin [16] as follows:A sequence x = (xk) is σ-statistically convergent to L if for every ε > 0limm→∞1m∣∣∣{k ≤ m : |xσk(n) − L| ≥ ε}∣∣∣ = 0,uniformly in n. It is denoted by Sσ − limx = L or xk → L(Sσ).The idea of I-convergence was introduced by Kostyrko et al. [3] as a generalization ofstatistical convergence which is based on the structure of the ideal I of subset of the set ofnatural numbers N. Similar concepts can be seen in [2, 9].A family of sets I ⊆ 2N is called an ideal if and only if (i) ∅ ∈ I, (ii) For each A,B ∈ Iwe have A ∪B ∈ I, (iii) For each A ∈ I and each B ⊆ A we have B ∈ I.An ideal is called non-trivial if N /∈ I and non-trivial ideal is called admissible if{n} ∈ I for each n ∈ N.All ideals in this paper are assumed to be admissible.A family of sets F ⊆ 2N is called a filter if and only if (i) ∅ /∈ F , (ii) For each A,B ∈ Fwe have A ∩B ∈ F , (iii) For each A ∈ F and each B ⊇ A we have B ∈ F .For any ideal there is a filter F(I) corresponding with I, given byF(I) ={M ⊂ N : (∃A ∈ I)(M = N\A)}.A sequence x = (xk) is said to be I-convergent to L if for every ε > 0, the setA(ε) ={k ∈ N : |xk − L| ≥ ε},belongs to I. If x = (xk) is I-convergent to L, then we write I − limx = L.Recently, the concepts of σ-uniform density of subset A of the set N of positive integersand corresponding Iσ-convergence for real number sequences was introduced by Nurayet al. [9].Asymptotically ideal invariant equivalence 217Let A ⊆ N andsm = minn∣∣∣A ∩ {σ(n), σ2(n), ..., σm(n)} ∣∣∣ and Sm = maxn∣∣∣A ∩ {σ(n), σ2(n), ..., σm(n)} ∣∣∣.If the following limits existsV (A) = limm→∞smm, V (A) = limm→∞Smm,then they are called a lower σ-uniform density and an upper σ-uniform density of the setA, respectively.If V (A) = V (A), then V (A) = V (A) = V (A) is called the σ-uniform density of A.Denote by Iσ the class of all A ⊆ N with V (A) = 0.Throughout the paper we take Iσ as an admissible ideal in N.A sequence x = (xk) is said to be Iσ-convergent to L if for every ε > 0, the setAε ={k : |xk − L| ≥ ε},belongs to Iσ ; i.e., V (Aε) = 0. It is denoted by Iσ − limxk = L.Marouf [4] presented definitions for asymptotically equivalent sequences and asymp-totic regular matrices. Then, the concept of asymptotically equivalence has been develo-ped by many other researchers (see, [10, 11, 15, 17]).Two nonnegative sequences x = (xk) and y = (yk) are said to be asymptotically equi-valent iflimkxkyk= 1.It is denoted by x ∼ y.Two nonnegative sequences x = (xk) and y = (yk) are Sσ-asymptotically equivalent ofmultiple L provided that for every ε > 0limn1n∣∣∣∣{k ≤ n : ∣∣∣∣xσk(m)yσk(m) − L∣∣∣∣ ≥ ε}∣∣∣∣ = 0,uniformly in m = 1, 2, ... , (denoted by x Sσ∼ y) and simply Sσ-asymptotically statisticalequivalent, if L = 1.2. ASYMPTOTICALLY Iσ -EQUIVALENCEIn this section, the concepts of asymptotically Iσ-equivalence, σ-asymptotically equiva-lence, strongly σ-asymptotically equivalence and strongly σ-asymptotically p-equivalencefor real number sequences are defined. Also, we examine relationships among these newtype equivalence concepts and the concept of Sσ-asymptotically equivalence which is stu-died in this area before.Definition 2.1. Two nonnegative sequences x = (xk) and y = (yk) are said to be asymp-totically Iσ-equivalent of multiple L if for every ε > 0Aε :={k ∈ N :∣∣∣∣xkyk − L∣∣∣∣ ≥ ε} ∈ Iσ;i.e., V (Aε) = 0. In this case, we write xILσ∼ y and simply asymptotically Iσ-equivalent, ifL = 1.The set of all asymptotically Iσ-equivalent of multiple L sequences will be denoted byILσ .218 U. UlusuDefinition 2.2. Two nonnegative sequence x = (xk) and y = (yk) are σ-asymptoticallyequivalent of multiple L iflimn→∞1nn∑k=1xσk(m)yσk(m)= L,uniformly in m. In this case, we write xV Lσ∼ y and simply σ-asymptotically equivalent, ifL = 1.Theorem 2.1. Suppose that x = (xk) and y = (yk) are bounded sequences. If x and y areasymptotically Iσ-equivalent of multiple L, then these sequences are σ-asymptotically equivalentof multiple L.Proof. Let m,n ∈ N be an arbitrary and ε > 0. Now, we calculatet(m,n) :=∣∣∣∣∣ 1nn∑k=1xσk(m)yσk(m)− L∣∣∣∣∣ .We havet(m,n) ≤ t(1)(m,n) + t(2)(m,n),wheret(1)(m,n) :=1nn∑k=1∣∣∣∣ xσk(m)yσk(m)−L∣∣∣∣≥ε∣∣∣∣xσk(m)yσk(m) − L∣∣∣∣ and t(2)(m,n) := 1nn∑k=1∣∣∣∣ xσk(m)yσk(m)−L∣∣∣∣<ε∣∣∣∣xσk(m)yσk(m) − L∣∣∣∣.We get t(2)(m,n) < ε, for everym = 1, 2, ... . The boundedness of x = (xk) and y = (yk)implies that there exists a M > 0 such that∣∣∣∣xσk(m)yσk(m) − L∣∣∣∣ ≤M,for k = 1, 2, ...; m = 1, 2, ... . Then, this implies thatt(1)(m,n) ≤ Mn∣∣∣∣{1 ≤ k ≤ n : ∣∣∣∣xσk(m)yσk(m) − L∣∣∣∣ ≥ ε}∣∣∣∣≤ Mmaxm∣∣∣{1 ≤ k ≤ n : ∣∣∣xσk(m)yσk(m)− L∣∣∣ ≥ ε}∣∣∣n=MSnn,hence x and y are σ-asymptotically equivalent to multiple L. The converse of Theorem 2.1 does not hold. For example, x = (xk) and y = (yk) are thesequences defined by following;xk :={2 , if k is an even integer0 , if k is an odd integer ; yk := 1When σ(m) = m+1, this sequence is σ-asymptotically equivalent but it is not asymptoti-cally Iσ-equivalent.Definition 2.3. Two nonnegative sequence x = (xk) and y = (yk) are strongly σ-asymptoticallyequivalent of multiple L iflimn→∞1nn∑k=1∣∣∣∣xσk(m)yσk(m) − L∣∣∣∣ = 0,Asymptotically ideal invariant equivalence 219uniformly in m. In this case, we write x[V Lσ ]∼ y and simply strongly σ-asymptoticallyequivalent, if L = 1.Definition 2.4. Let 0 < p < ∞. Two nonnegative sequence x = (xk) and y = (yk) arestrongly σ-asymptotically p-equivalent of multiple L iflimn→∞1nn∑k=1∣∣∣∣xσk(m)yσk(m) − L∣∣∣∣p = 0,uniformly in m. In this case, we write x[V Lσ ]p∼ y and simply strongly σ-asymptoticallyp-equivalent, if L = 1.The set of all strongly σ-asymptotically p-equivalent of multiple L sequences will bedenoted by [VLσ ]p.Theorem 2.2. Let 0 < p <∞. Then, x [VLσ ]p∼ y ⇒ x ILσ∼ y.Proof. Let x[V Lσ ]p∼ y and given ε > 0. Then, for every m ∈ N we haven∑k=1∣∣∣∣xσk(m)yσk(m) − L∣∣∣∣p ≥ n∑k=1∣∣∣∣ xσk(m)yσk(m)−L∣∣∣∣≥ε∣∣∣∣xσk(m)yσk(m) − L∣∣∣∣p ≥ εp · ∣∣∣∣{1 ≤ k ≤ n : ∣∣∣∣xσk(m)yσk(m) − L∣∣∣∣ ≥ ε}∣∣∣∣≥ εp ·maxm∣∣∣∣{1 ≤ k ≤ n : ∣∣∣∣xσk(m)yσk(m) − L∣∣∣∣ ≥ ε}∣∣∣∣and1nn∑k=1∣∣∣∣xσk(m)yσk(m) − L∣∣∣∣p ≥ εp · maxm∣∣∣{1 ≤ k ≤ n : ∣∣∣xσk(m)yσk(m)− L∣∣∣ ≥ ε}∣∣∣n= εp · Snn,for every m = 1, 2, ... . This implies limn→∞Snn= 0 and so xILσ∼ y. Theorem 2.3. Let 0 < p <∞ and x, y ∈ `∞. Then, xILσ∼ y ⇒ x [VLσ ]p∼ y .Proof. Suppose that x, y ∈ `∞ and xILσ∼ y. Let ε > 0. By assumption, we have V (Aε) = 0.The boundedness of x and y implies that there exists a M > 0 such that∣∣∣∣xσk(m)yσk(m) − L∣∣∣∣ ≤M,for k = 1, 2, ...; m = 1, 2, ... . Observe that, for every m ∈ N we have1nn∑k=1∣∣∣xσk(m)yσk(m)− L∣∣∣p = 1nn∑k=1∣∣∣∣ xσk(m)yσk(m)−L∣∣∣∣≥ε∣∣∣xσk(m)yσk(m)− L∣∣∣p + 1nn∑k=1∣∣∣∣ xσk(m)yσk(m)−L∣∣∣∣<ε∣∣∣xσk(m)yσk(m)− L∣∣∣p≤ Mmaxm∣∣∣{1 ≤ k ≤ n : ∣∣∣xσk(m)yσk(m)− L∣∣∣ ≥ ε}∣∣∣n+ εp≤ MSnn+ εp.220 U. UlusuHence, we obtainlimn→∞1nn∑k=1∣∣∣∣xσk(m)yσk(m) − L∣∣∣∣p = 0,uniformly in m.Theorem 2.4. Let 0 < p <∞. Then, ILσ ∩ `∞ = [VLσ ]p ∩ `∞.Proof. This is an immediate consequence of Theorem 2.2 and Theorem 2.3. Now we shall state a theorem that gives a relationship between asymptoticallyIσ-equivalence and Sσ-asymptotically equivalence.Theorem 2.5. The sequences x = (xk) and y = (yk) are asymptotically Iσ-equivalent to multipleL if and only if they are Sσ-asymptotically equivalent of multiple L.REFERENCES[1] Fast, H., Sur la convergence statistique, Colloq. Math., 2 (1951), 241–244[2] Kostyrko, P., Macaj, M., Šalát, T. and Sleziak, M., I-Convergence and Extermal I-limits points, Math. Slovaca,55 (2005), 443–464[3] Kostyrko, P., Šalát, T. and Wilczyński, W., I-Convergence, Real Anal. Exchange, 26 (2000), No. 2, 669–686[4] Marouf, M., Asymptotic equivalence and summability, Int. J. Math. Math. Sci., 16 (1993), No. 4, 755–762[5] Mursaleen, M., On finite matrices and invariant means, Indian J. Pure Appl. Math., 10 (1979), 457–460[6] Mursaleen, M., Matrix transformation between some new sequence spaces, Houston J. Math., 9 (1983), 505–509[7] Mursaleen, M. and Edely, O. H. H., On the invariant mean and statistical convergence, Appl. Math. Lett., 22(2009), No. 11, 1700–1704[8] Nuray, F. and Savaş, E., Invariant statistical convergence and A-invariant statistical convergence, Indian J. PureAppl. Math., 25 (1994), No. 3, 267–274[9] Nuray, F., Gök, H. and Ulusu, U., Iσ-convergence, Math. Commun., 16 (2011), 531–538[10] Patterson, R. F., On asymptotically statistically equivalent sequences, Demostratio Math., 36 (2003), No. 1,149–153[11] Patterson, R. F. and Savaş, E., On asymptotically lacunary statistically equivalent sequences, Thai J. Math., 4(2006), No. 2, 267–272[12] Raimi, R. A., Invariant means and invariant matrix methods of summability, Duke Math. J., 30 (1963), No. 1,81–94[13] Savaş, E., Some sequence spaces involving invariant means, Indian J. Math., 31 (1989), 1–8[14] Savaş, E., Strongly σ-convergent sequences, Bull. Calcutta Math., 81 (1989), 295–300[15] Savaş, E., On I-asymptotically lacunary statistical equivalent sequences, Adv. Difference Equ., 111 (2013),7 pages. doi:10.1186/1687-1847-2013-111[16] Savaş, E. and Nuray, F., On σ-statistically convergence and lacunary σ-statistically convergence, Math. Slovaca,43 (1993), No. 3, 309–315[17] Savaş, E. and Patterson, R. F., σ-asymptotically lacunary statistical equivalent sequences, Cent. Eur. J. Math., 4(2006), No. 4, 648–655[18] Schaefer, P., Infinite matrices and invariant means, Proc. Amer. Math. Soc., 36 (1972), 104–110DEPARTMENT OF MATHEMATICSAFYON KOCATEPE UNIVERSITY03200 AFYONKARAHISAR, TURKEYE-mail address: ulusu@aku.edu.tr

http://acikerisim.aku.edu.tr/xmlui/bitstream/11630/7625/1/Asymptotically%20Ideal%20Invariant%20Equivalence.pdf

Asymptotically ideal invariant equivalence

Abstract

Similar works

Full text

Available Versions

Afyon Kocatepe Üniversitesi Açık Erişim Sistemi

Afyon Kocatepe Üniversitesi

Afyon Kocatepe Üniversitesi

Afyon Kocatepe Üniversitesi Açık Erişim Sistemi