A Alotaibi

Abdullah Alotaibi

C Belen

E Kolk

E Malkowsky

E Savaş

F Moricz

H Fast

JA Fridy

JS Connor

M Mursaleen

Mohammad Mursaleen

OHH Edely

R Colak

SA Mohiuddine

Advances in Difference Equations

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Alotaibi and Mursaleen Advances in Diﬀerence Equations 2013, 2013:212http://www.advancesindifferenceequations.com/content/2013/1/212RESEARCH Open AccessGeneralized statistical convergence ofdifference sequencesAbdullah Alotaibi1 and Mohammad Mursaleen2**Correspondence:mursaleenm@gmail.com2Department of Mathematics,Aligarh Muslim University, Aligarh,202002, IndiaFull list of author information isavailable at the end of the articleAbstractIn this paper we deﬁne the λ(u)-statistical convergence that generalizes, in a certainsense, the notion of λ-statistical convergence. We ﬁnd some relations with sets ofsequences which are related to the notion of strong convergence.MSC: 40A05; 40H05Keywords: statistical convergence; λ-statistical convergence; diﬀerence sequences1 Introduction and preliminariesThe notion of statistical convergence (see Fast []) has been studied in various setups, andits various generalizations, extensions and variants have been studied by various authorsso far. For example, lacunary statistical convergence [], A-statistical convergence [, ],statistical summability (C, ) [, ], statistical λ-summability [], statistical σ -convergence[], statisticalA-summability [], λ-statistical convergence with order α [], lacunary andλ-statistical convergence in a solid Riesz space [, ], lacunary statistical convergenceand ideal convergence in random -normed spaces [, ], generalized weighted statisti-cal convergence [] etc. In this paper we deﬁne the notion of λ-statistical convergence as amatrix domain of a diﬀerence operator [], which is obtained by replacing the sequence xby ux, wherex = (xk – xk+)∞k= and u = (uk)∞k= is another sequence with uk =  for all k.We ﬁnd some relations with sets of sequences which are related to the notion of strongconvergence [].Let K be a subset of the set of natural numbers N. Then the asymptotic density of Kdenoted by δ(K) is deﬁned as δ(K) = limn n |{k ≤ n : k ∈ K}|, where the vertical bars denotethe cardinality of the enclosed set.A number sequence x = (xk) is said to be statistically convergent to the number L if foreach  > , the set K() = {k ≤ n : |xk – L| > } has asymptotic density zero, i.e.,limnn∣∣{k ≤ n : |xk – L|}∣∣ = .In this case, we write S- limx = L.Let λ = (λn) be a non-decreasing sequence of positive numbers tending to ∞ such thatλn+ ≤ λn + , λ = .© 2013 Alotaibi and Mursaleen; licensee Springer. This is an Open Access article distributed under the terms of the Creative Com-mons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and repro-duction in any medium, provided the original work is properly cited.Alotaibi and Mursaleen Advances in Diﬀerence Equations 2013, 2013:212 Page 2 of 5http://www.advancesindifferenceequations.com/content/2013/1/212The generalized de la Vallée-Poussin mean is deﬁned bytn(x) =:λn∑j∈Inxj,where In = [n – λn + ,n].A sequence x = (xj) is said to be (V ,λ)-summable to a number L iftn(x)→ L as n→ ∞.In this case, L is called the λ-limit of x.Let K ⊆N. Then the λ-density of K is deﬁned byδλ(K) = limnλn∣∣{n – λn + ≤ j≤ n : j ∈ K}∣∣.In case λn = n, λ-density reduces to the asymptotic density. Also, since (λn/n)≤ , δ(K)≤δλ(K) for every K ⊆N.A sequence x = (xk) is said to be λ-statistically convergent (see []) to L if for every  > the set K := {k ∈N : |xk – L| ≥ } has λ-density zero, i.e., δλ(K) = . That is,limnλn∣∣{k ∈ In : |xk – L| ≥ }∣∣ = .In this case, we write Sλ- limx = L and we denote the set of all λ-statistically convergentsequences by Sλ.2 λ(u)-Statistical convergenceWeconsider the inﬁnitematrix of ﬁrst diﬀerence = (anm)n,m≥ deﬁned by ann = , an,n+ =– and anm =  otherwise. LetDu be the diagonalmatrix deﬁned by [Du]nn = un for all n andconsider the set U of all sequences such that un =  for all n. Then we write (u) = Dufor u ∈U .From the generalized de la Vallée-Poussin mean deﬁned bytn(x) =λn∑k∈Inxk for x = (xk)k ,we are led to deﬁne the following sets:[V ,λ]((u))={x = (xk) : limn→∞λn∑k∈In∣∣(u)xk∣∣ = }={x = (xk) : limn→∞λn∑k∈In∣∣uk(xk – xk+)∣∣ = },[V ,λ]∞((u))={x = (xk) : supnλn∑k∈In∣∣(u)xk∣∣ <∞}={x = (xk) : supnλn∑k∈In∣∣uk(xk – xk+)∣∣ = }.Alotaibi and Mursaleen Advances in Diﬀerence Equations 2013, 2013:212 Page 3 of 5http://www.advancesindifferenceequations.com/content/2013/1/212In the case when λn = n, we write the previous sets [V ]((u)) and [V ]∞((u)), respec-tively. Now we can state the deﬁnition of λ(u)-statistical convergence to zero.A sequence x = (xk)k≥ is said to be λ(u)-statistically convergent to zero if for every ε > ,limn→∞λn∣∣{k ∈ In : ∣∣(u)xk∣∣≥ ε}∣∣ = .In this case, we write xk → Sλ((u)). If λn = n for all n, we then write xk → S((u)).3 Main resultsWe are ready to prove the following result.Theorem  Let u ∈U . Then(a) [V ,λ]((u))⊂ Sλ((u)) and the inclusion is proper,(b) if x ∈ l∞ and xk → Sλ((u)), then x ∈ [V ,λ]((u)),(c) Sλ((u))∩ l∞ = [V ,λ]((u))∩ l∞.Proof (a) Let ε >  be given and x ∈ [V ,λ]((u)). Then we haveλn∑k∈In∣∣(u)xk∣∣≥ λn∑k∈In|xk–L|≥ε∣∣(u)xk∣∣≥ ελn∣∣{k ∈ In : ∣∣(u)xk∣∣≥ ε}∣∣.Therefore x ∈ Sλ((u)). The following example shows that the inclusion is proper: Letx = (xk) be deﬁned byxk ={∑∞j=k j, for n – [√λn] + ≤ k ≤ n,, otherwise.Then x /∈ l∞ and for  < ε ≤ ,λn∣∣{k ∈ In : ∣∣(u)xk∣∣≥ ε}∣∣ = [√λn]λn→  (n→ ∞),i.e., x ∈ Sλ((u)). Butλn∑k∈In∣∣(u)xk∣∣  (n→ ∞),i.e., x /∈ [V ,λ]((u)).(b) Let x ∈ l∞ and xk → Sλ((u)). Then |(u)xk| ≤M for all k, whereM > . For ε > ,we haveλn∑k∈In∣∣(u)xk∣∣ = λn∑k∈In|xk–L|≥∣∣(u)xk∣∣ + λn∑k∈In|xk–L|<∣∣(u)xk∣∣≤ Mλn∣∣{k ∈ In : ∣∣(u)xk∣∣≥ ε}∣∣ + ε.Hence, x ∈ [V ,λ]((u)).Alotaibi and Mursaleen Advances in Diﬀerence Equations 2013, 2013:212 Page 4 of 5http://www.advancesindifferenceequations.com/content/2013/1/212(c) This immediately follows from (a) and (b).This completes the proof of the theorem. Theorem  S((u))⊆ Sλ((u)) if and only iflim infn→∞λnn > , (∗)where by x ∈ S((u)) (or x ∈ Sλ((u))) we mean xk → S((u)) (or xk → Sλ((u))).Proof For ε > , we have{k ∈ In :∣∣(u)xk∣∣≥ ε}⊂ {k ≤ n : ∣∣(u)xk∣∣≥ ε}.Thereforen∣∣{k ≤ n : ∣∣(u)xk∣∣≥ ε}∣∣ ≥ n∣∣{k ∈ In : ∣∣(u)xk∣∣≥ ε}∣∣≥ λnn ·λn∣∣{k ∈ In : ∣∣(u)xk∣∣≥ ε}∣∣.Taking the limit as n→ ∞ and using (∗), we get the inclusion.Conversely, suppose thatlim infn→∞λnn = .Choose a subsequence (n(j))j≥ such thatλn(j)n(j) <j . Deﬁne a sequence x = (xk)k≥ suchthatxk ={, for k ∈ In(j), j = , , , . . . ,, otherwise.Then x ∈ [C, ] and hence, by Theorem . of [], x ∈ S((u)). On the other hand,x /∈ [V ,λ]((u)) and Theorem (b) implies that x /∈ Sλ((u)). Hence, (∗) is neces-sary.This completes the proof of the theorem. Presently, for the reverse inclusion, we have only one way condition.Theorem  If lim supn(n – λn) <∞, then Sλ((u))⊆ S((u)).Proof Let lim supn(n – λn) < ∞. Then there exists M >  such that n – λn ≤ M for all n.Since n ≤ λn and { ≤ k ≤ n : |(u)xk| ≥ ε} ⊆ {k ∈ In : |(u)xk| ≥ ε} ∪ { ≤ k ≤ n – λn :|(u)xk| ≥ ε}, we haven∣∣{≤ k ≤ n : ∣∣(u)xk∣∣≥ ε}∣∣≤ λn∣∣{≤ k ≤ n : ∣∣(u)xk∣∣≥ ε}∣∣Alotaibi and Mursaleen Advances in Diﬀerence Equations 2013, 2013:212 Page 5 of 5http://www.advancesindifferenceequations.com/content/2013/1/212≤ λn∣∣{k ∈ In : ∣∣(u)xk∣∣≥ ε}∣∣ + λn∣∣{k ≤ n – λn : ∣∣(u)xk∣∣≥ ε}∣∣≤ λn∣∣{k ∈ In : ∣∣(u)xk∣∣≥ ε}∣∣ + Mλn.Now, taking the limit as n→ ∞, we get Sλ((u))⊆ S((u)).This completes the proof of the theorem. Competing interestsThe authors declare that they have no competing interests.Authors’ contributionsBoth authors contributed equally and signiﬁcantly in writing this paper. Both authors read and approved the ﬁnalmanuscript.Author details1Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia. 2Department ofMathematics, Aligarh Muslim University, Aligarh, 202002, India.AcknowledgementsThis project was funded by the Deanship of Scientiﬁc Research (DSR), King Abdulaziz University, Jeddah under GrantNo. (174/130/1433). The authors, therefore, acknowledge with thanks DSR technical and ﬁnancial support.Received: 16 March 2013 Accepted: 26 June 2013 Published: 13 July 2013References1. Fast, H: Sur la convergence statistique. Colloq. Math. 2, 241-244 (1951)2. Fridy, JA, Orhan, C: Lacunary statistical convergence. Pac. J. Math. 160, 43-51 (1993)3. Alotaibi, A, Mursaleen, M: A-Statistical summability of Fourier series and Walsh-Fourier series. Appl. Math. Inf. Sci. 6(3),535-538 (2012)4. Kolk, E: The statistical convergence in Banach spaces. Tartu Ülik. Toim. 928, 41-52 (1991)5. Moricz, F: Tauberian conditions under which statistical convergence follows from statistical summability (C, 1).J. Math. Anal. Appl. 275, 277-287 (2002)6. Mohiuddine, SA, Alotaibi, A: Korovkin second theorem via statistical summability (C, 1). J. Inequal. Appl. 2013, 149(2013)7. Mursaleen, M, Alotaibi, A: Statistical summability and approximation by de la Vallée-Poussin mean. Appl. Math. Lett.24, 320-324 (2011) (Erratum: Appl. Math. Lett. 25, 665 (2012))8. Mursaleen, M, Edely, OHH: On the invariant mean and statistical convergence. Appl. Math. Lett. 22, 1700-1704 (2009)9. Edely, OHH, Mursaleen, M: On statistical A-summability. Math. Comput. Model. 49, 672-680 (2009)10. Colak, R, Bektas, CA: λ-Statistical convergence of order α . Acta Math. Sci. 31(3), 953-959 (2011)11. Mohiuddine, SA, Alghamdi, MA: Statistical summability through a lacunary sequence in locally solid Riesz spaces.J. Inequal. Appl. 2012, 225 (2012)12. Mohiuddine, SA, Alotaibi, A, Mursaleen, M: Statistical convergence through de la Vallée-Poussin mean in locally solidRiesz spaces. Adv. Diﬀer. Equ. 2013, 66 (2013)13. Mohiuddine, SA, Aiyub, M: Lacunary statistical convergence in random 2-normed spaces. Appl. Math. Inf. Sci. 6(3),581-585 (2012)14. Mohiuddine, SA, Alotaibi, A, Alsulami, SM: Ideal convergence of double sequences in random 2-normed spaces. Adv.Diﬀer. Equ. 2012, 149 (2012)15. Belen, C, Mohiuddine, SA: Generalized weighted statistical convergence and application. Appl. Math. Comput. 219,9821-9826 (2013)16. Malkowsky, E, Mursaleen, M, Suantai, S: The dual spaces of sets of diﬀerence sequences of orderm and matrixtransformations. Acta Math. Sin. Engl. Ser. 23, 521-532 (2007)17. Savas¸, E, Kiliçman, A: A note on some strongly sequence spaces. Abstr. Appl. Anal. 2011, Article ID 598393 (2011)18. Connor, JS: The statistical and strong p-Cesàro convergence of sequences. Analysis 8, 47-63 (1988)doi:10.1186/1687-1847-2013-212Cite this article as: Alotaibi and Mursaleen: Generalized statistical convergence of difference sequences. Advances inDiﬀerence Equations 2013 2013:212.

Generalized statistical convergence of difference sequences

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Generalized statistical convergence of difference sequences

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