In this paper we deﬁne the WS_θ-analog of the Cauchy criterion for convergence and show that it is equivalent to Wijsman lacunary statistical convergence. Also, Wijsman lacunary statistical convergence is compared to other summability methods which are deﬁned in this paper. After giving new deﬁnitions for convergence, we prove a result comparing them. In addition, we give the relationship between Wijsman lacunary statistical convergence and Hausdorf lacunary statistical convergenc

Nuray, Fatih

Ulusu, Uğur

English

Afyon Kocatepe Üniversitesi Açık Erişim Sistemi

Konuralp Journal of MathematicsVolume 3 No. 2 pp. 176–184 (2015) c©KJMLACUNARY STATISTICAL SUMMABILITY OF SEQUENCES OFSETSUĞUR ULUSU AND FATİH NURAYAbstract. In this paper we define the WSθ−analog of the Cauchy criterionfor convergence and show that it is equivalent to Wijsman lacunary statisticalconvergence. Also, Wijsman lacunary statistical convergence is compared toother summability methods which are defined in this paper. After giving newdefinitions for convergence, we prove a result comparing them. In addition, wegive the relationship between Wijsman lacunary statistical convergence andHausdorf lacunary statistical convergence.1. INTRODUCTION AND BACKGROUNDThe concept of convergence of a sequence of real numbers has been extended tostatistical convergence independently by Fast [5] and Schoenberg [11]. The conceptof lacunary statistical convergence and summability were defined by Fridy andOrhan in [7, 8].The concept of convergence of sequences of numbers has been extended byseveral authors to convergence of sequences of sets. The one of these such ex-tensions considered in this paper is the concept of Wijsman convergence (see,[1],[2],[3],[4],[9],[12],[13],[14]). Nuray and Rhoades [9] extended the notion of con-vergence of set sequences to statistical convergence and gave some basic theorems.Ulusu and Nuray [12] defined the Wijsman lacunary statistical convergence of se-quence of sets and considered its relation with Wijsman statistical convergence,which was defined by Nuray and Rhoades.In this paper, we shall define the concept of Wijsman lacunary statistical Cauchysequences for sequences of sets and show that this concept is equivalent to the con-cept of Wijsman lacunary statistically convergence. Also, Wijsman lacunary statis-tical convergence will be compared to newly defined Wijsman lacunary summabilitymethods. Further, the definition of Wijsman lacunary almost convergence for se-quences of sets is introduced and some comparison theorems are given.Date: April 30, 2015 and, in revised form, July 1, 2015.2010 Mathematics Subject Classification. 40A35, 40G15.Key words and phrases. Statistical convergence, lacunary sequence, sequence of sets, Wijsmanconvergence, Hausdorff convergence, Cesàro summability, almost convergence.176LACUNARY STATISTICAL SUMMABILITY OF SEQUENCES OF SETS 1772. DEFINITIONS AND NOTATIONSNow, we recall the concept of statistical, lacunary statistical, Wijsman, Haus-dorff, Wijsman statistiscal, Hausdorff statistical, Wijsman strongly almost, Wijs-man almost statistical, Wijsman lacunary statistical convergence, Wijsman lacu-nary summability, Wijsman strongly lacunary summability and Wijsman Cesàrosummability of the sequences of sets (see, [2],[6],[7],[9],[12])Definition 2.1. A sequence x = (xk) is said to be statistically convergent to thenumber L if for every ε > 0,limn→∞1n|{k ≤ n : |xk − L| ≥ ε}| = 0.In this case, we write st− limxk = L.By a lacunary sequence we mean an increasing integer sequence θ = {kr} suchthat k0 = 0 and hr = kr − kr−1 → ∞ as r → ∞. Throughout this paper theintervals determined by θ will be denoted by Ir = (kr−1, kr].Definition 2.2. A sequence x = (xk) is said to be lacunary statistically convergentto the number L if for every ε > 0,limr1hr|{k ∈ Ir : |xk − L| ≥ ε}| = 0.In this case, we write Sθ − limxk = L or xk → L(Sθ).Let (X, ρ) be a metric space. For any point x ∈ X and any non-empty subset Aof X, we define the distance from x to A byd(x,A) = infa∈Aρ(x, a).Definition 2.3. Let (X, ρ) be a metric space. For any non-empty closed subsetsA,Ak ⊆ X, we say that the sequence {Ak} is Wijsman convergent to A iflimk→∞d(x,Ak) = d(x,A),for each x ∈ X. In this case, we write W − limAk = A.As an example, consider the following sequence of circles in the (x, y)-plane:Ak = {(x, y) : x2 + y2 + 2kx = 0}.As k →∞ the sequence is Wijsman convergent to the y-axis A = {(x, y) : x = 0}.Definition 2.4. Let (X, ρ) be a metric space. For any non-empty closed subsetsA,Ak ⊆ X, we say that the sequence {Ak} is Hausdorff convergent to A iflimk→∞supx∈X|d(x,Ak)− d(x,A)| = 0.In this case, we write H − limAk = A.The concepts of Wijsman statistical convergence and Hausdorff statistical con-vergence were given by Nuray and Rhoades [9] as follows:178 UĞUR ULUSU AND FATİH NURAYDefinition 2.5. Let (X, ρ) be a metric space. For any non-empty closed subsetsA,Ak ⊆ X, we say that the sequence {Ak} is Wijsman statistical convergent to Aif {d(x,Ak)} is statistically convergent to d(x,A); i.e., for every ε > 0 and for eachx ∈ X,limn→∞1n|{k ≤ n : |d(x,Ak)− d(x,A)| ≥ ε}| = 0.In this case, we write st− limW Ak = A or Ak → A(WS).Definition 2.6. Let (X, ρ) be a metric space. For any non-empty closed subsetsA,Ak ⊆ X, we say that the sequence {Ak} is Hausdorff statistical convergent to Aif for each ε > 0,limn→∞1n|{k ≤ n : supx∈X|d(x,Ak)− d(x,A)| ≥ ε}| = 0.In this case, we write st− limH Ak = A or Ak → A(HS).Let (X, ρ) be a metric space. For any non-empty closed subsets Ak of X, we saythat the sequence {Ak} is bounded if supk d(x,Ak) <∞, for each x ∈ X.Also, the concepts of Wijsman Cesàro Summability, Wijsman strongly almostconvergence and Wijsman almost statistical convergence for sequences of sets weregiven by Nuray and Rhoades [9] as follows:Definition 2.7. Let (X, ρ) be a metric space. For any non-empty closed subsetsA, Ak ⊆ X, we say that {Ak} is Wijsman Cesàro summable to A if {d(x,Ak)} isCesàro summable to d(x,A); i.e., for each x ∈ X,limn→∞1nn∑k=1d(x,Ak) = d(x,A).Definition 2.8. Let (X, ρ) be a metric space. For any non-empty closed subsetsA, Ak ⊆ X, we say that {Ak} is Wijsman strongly almost convergent to A if foreach x ∈ X,limn→∞1nn∑k=1|d(x,Ak+i)− d(x,A)| = 0,uniformly in i.Definition 2.9. Let (X, ρ) be a metric space. For any non-empty closed subsets A,Ak ⊆ X, we say that the sequence {Ak} is Wijsman almost statistically convergentto A if for each ε > 0 and for each x ∈ X,limn→∞1n|{k ≤ n : |d(x,Ak+i)− d(x,A)| ≥ ε}| = 0,uniformly in i.The concepts of Wijsman lacunary summability, Wijsman strongly lacunarySummability and Wijsman lacunary statistical convergence of sequences of setswere given by Ulusu and Nuray [12] as follows:Definition 2.10. Let (X, ρ) be a metric space and θ = {kr} be a lacunary sequence.For any non-empty closed subsetsA, Ak ⊆ X, we say that {Ak} is Wijsman lacunarysummable to A, if {d(x,Ak)} is lacunary summable to d(x,A); i.e., for each x ∈ X,limr→∞1hr∑Ird(x,Ak) = d(x,A).LACUNARY STATISTICAL SUMMABILITY OF SEQUENCES OF SETS 179Definition 2.11. Let (X, ρ) be a metric space and θ = {kr} be a lacunary sequence.For any non-empty closed subsets A, Ak ⊆ X, we say that {Ak} is Wijsman stronglylacunary Summable to A, if {d(x,Ak)} is strongly lacunary summable to d(x,A);i.e., for each x ∈ X,limr→∞1hr∑Ir|d(x,Ak)− d(x,A)| = 0.Definition 2.12. Let (X, ρ) be a metric space and θ = {kr} be a lacunary sequence.For any non-empty closed subsets A,Ak ⊆ X, we say that the sequence {Ak} isWijsman lacunary statistical convergent to A, if {d(x,Ak)} is lacunary statisticallyconvergent to d(x,A); i.e., for every ε > 0 and for each x ∈ X,limr1hr|k ∈ Ir : |d(x,Ak)− d(x,A)| ≥ ε| = 0.In this case, we write Sθ − limW Ak = A or Ak → A(WSθ).Example 2.1. Let X = R and we define a sequence {Ak} as follows:Ak :={x ∈ R : 2 ≤ x ≤ k} , if k ≥ 2, kr−1 < k ≤ krand k is a square integer,{1} , otherwise.As k → ∞ this sequence is Wijsman lacunary statistical converget to the setA = {1}.3. MAIN RESULTSDefinition 3.1. Let (X, ρ) be a metric space and θ = {kr} be a lacunary sequence.For any non-empty closed subsets A, Ak ⊆ X, we say that the sequence {Ak} issaid to be a Wijsman lacunary statistical Cauchy sequence if there is a subsequence{Ak′(r)} of {Ak} such that k′(r) ∈ Ir for each r, W − limr Ak′(r) = A, and for everyε > 0 and x ∈ X,(3.1) limr→∞1hr|{k ∈ Ir : |d(x,Ak)− d(x,Ak′(r))| ≥ ε}| = 0.Theorem 3.1. Let (X, ρ) be a metric space and θ = {kr} be a lacunary sequence.The sequence {Ak} is Wijsman lacunary statistical convergent if and only if {Ak}is a Wijsman lacunary statistical Cauchy sequence.Proof. (⇒) Let Ak → A(WSθ) and writeK(j) :={k ∈ N : |d(x,Ak)− d(x,A)| <1j},for each x ∈ X and each j ∈ N. Hence, for each j,K(j) ⊇ K(j+1) andlimr→∞|K(j) ∩ Ir|hr= 1.Choose m(1) such that r ≥ m(1) implies |K(1) ∩ Ir|hr> 0, i.e., K(1) ∩ Ir 6= ∅.Next choose m(2) > m(1) so that r ≥ m(2) implies K(2) ∩ Ir 6= ∅. Then, foreach r satisfying m(1) ≤ r < m(2), choose k′(r) ∈ Ir such that k′(r) ∈ Ir ∩K(1),i.e., |d(x,Ak′(r)) − d(x,A)| < 1. In general, choose m(p + 1) > m(p) such that180 UĞUR ULUSU AND FATİH NURAYr > m(p+1) implies Ir∩K(p+1) 6= ∅. Then, for all r satisfying m(p) ≤ r < m(p+1),choose k′(r) ∈ Ir ∩K(p), i.e.,(3.2) |d(x,Ak′(r))− d(A, x)| <1p.Hence, we get k′(r) ∈ Ir for every r and (3.2) implies thatW − limrd(x,Ak′(r)) = d(x,A).Furthermore, for every ε > 0 we have,1hr|{k ∈ Ir : |d(x,Ak)− d(x,Ak′(r))| ≥ ε}|≤ 1hr|{k ∈ Ir : |d(x,Ak)− d(x,A)| ≥ε2}|+1hr|{k ∈ Ir : |d(x,Ak′(r))− d(x,A)| ≥ε2}|.Using the assumptions that Ak → A(WSθ) and W − limr d(x,Ak′(r)) = d(x,A), weinfer (3.1), whence Ak is a Wijsman lacunary statistical Cauchy sequence.(⇐) Conversely, suppose that {Ak} is a Wijsman lacunary statistical Cauchysequence. For every ε > 0, we have|{k ∈ Ir : |d(x,Ak)− d(x,A)| ≥ ε}| ≤∣∣∣{k ∈ Ir : |d(x,Ak)− d(x,Ak′(r))| ≥ ε2}∣∣∣+∣∣∣{k ∈ Ir : |d(x,Ak′(r))− d(x,A)| ≥ ε2}∣∣∣from which it follows that Ak → A(WSθ). Now we give following theorem where ∆ denotes the forward difference operatordefined by ∆d(x,Ai) = d(x,Ai)− d(x,Ai+1).Theorem 3.2. Let (X, ρ) be a metric space and θ = {kr} be a lacunary sequence.If Ak → A(WSθ) and for each x ∈ Xmax{|∆d(x,Ai)| : i ∈ Ir} = o(1hr) as r →∞,then W − limAk = A.Proof. Assume that Ak → A(WSθ) and by Theorem (3.1), choose a subsequence{Ak′(r)} of {Ak} as in Definition (3.1). Since k′(r) ∈ Ir, for each x ∈ X we have|d(x,Ak)− d(x,Ak′(r))| ≤k′(r)−1∑i=k|∆d(x,Ai)|≤ hr. (maxi∈Ir{|∆d(x,Ai)| : i ∈ Ir})= o(1)and therefore Ak′(r) → A(WS) implies that Ak → A(WS). Theorem 3.3. Let (X, ρ) be a metric space and θ = {kr} be a lacunary sequence.If {Ak} is a bounded sequence and Ak → A(WSθ), then {Ak} is Wijsman Cesàrosummable to A.LACUNARY STATISTICAL SUMMABILITY OF SEQUENCES OF SETS 181Proof. Let (X, ρ) be a metric space, θ = {kr} be a lacunary sequence and let n bea positive integer with n ∈ Ir; then1nn∑k=1(d(x,Ak)− d(x,A))=1nr−1∑p=1∑k∈Ip(d(x,Ak)− d(x,A))(3.3)+1nn∑k=1+kr−1(d(x,Ak)− d(x,A)).Consider the first term on the right in (3.3),1nr−1∑p=1∑k∈Ip(d(x,Ak)− d(x,A))≤ 1kr−1r−1∑p=1∑k∈Ip∣∣∣∣d(x,Ak)− d(x,A)∣∣∣∣(3.4) =1kr−1r−1∑p=1hp.tp = (Ht)r,say, wheretp =1hp∑k∈Ip|d(x,Ak)− d(x,A)| .Since {Ak} is bounded and Ak → A(WSθ), it follows from Theorem 1 (ii) of [12]that tp → 0. Moreoverkr−1 =r−1∑p=1hp →∞ as r →∞,because θ is a lacunary sequence, which implies that (3.4) is a regular weightedmean matrix transform of t in [10]; hence,(3.5) (Ht)r → 0.Now consider the second term on the right in (3.3). Since {Ak} is bounded,there is a constant M > 0 such that |d(x,Ak)− d(x,A)| ≤M , for all k. Therefore,for every ε > 0 we have,(3.6)∣∣∣∣∣ 1n n∑k=1+kr−1(d(x,Ak)− d(x,A))∣∣∣∣∣ ≤ 1n ∑kr−1<k≤n|d(x,Ak)−d(x,A)|≥ε|d(x,Ak)− d(x,A)|+1n∑kr−1<k≤n|d(x,Ak)−d(x,A)|<ε|d(x,Ak)− d(x,A)|≤ Mhr|{k ∈ Ir : |d(x,Ak)− d(x,A)| ≥ ε}|+ ε.Since Ak → A(WSθ) and ε is an arbitrary, the expression on the left side of (3.6)tends to zero as r →∞. Hence, (3.3), (3.5) and (3.6) imply that {Ak} is WijsmanCesàro summable to A. 182 UĞUR ULUSU AND FATİH NURAYDefinition 3.2. Let (X, ρ) be a metric space and θ = {kr} be a lacunary se-quence. For any non-empty closed subsets A, Ak ⊆ X, we say that {Ak} is Wijsmanstrongly p−lacunary summable to A if {d(x,Ak)} is strongly p−lacunary summableto d(x,A); i.e., for each p positive real number and for each x ∈ Xlimr→∞1hr∑Ir|d(x,Ak)− d(x,A)|p = 0.Theorem 3.4. Let (X, ρ) be a metric space, θ = {kr} be a lacunary sequence andlet p positive real number. Then, for any non-empty closed subsets A, Ak ⊆ X;(i) {Ak} is Wijsman lacunary statistical convergent to A if it is Wijsmanstrongly p−lacunary summable to A.(ii) If {Ak} is bounded and Wijsman lacunary statistical convergent to A thenit is Wijsman strongly p−lacunary summable to A.Proof. (i) For any {Ak}, fix an ε > 0. Then∑Ir|d(x,Ak)− d(x,A)|p ≥ εp|{k ∈ Ir : |d(x,Ak)− d(x,A)| ≥ ε}|,and it follows that if {Ak} is Wijsman strongly p−lacunary summable to A then{Ak} is Wijsman lacunary statistical convergent to A.(ii) Let {Ak} be bounded and Wijsman lacunary statistical convergent to A.Since {Ak} is bounded setsupk{d(x,Ak)}+ d(x,A) = M.Since {Ak} is Wijsman lacunary statistically convergent to A, for given ε > 0 wecan select Nε such that for each x ∈ X1hr∣∣∣∣{k ∈ Ir : |d(x,Ak)− d(x,A)| ≥ (ε2) 1p}∣∣∣∣ < ε2Mp ,for all r > Nε and we let the setLr ={k ∈ Ir : |d(x,Ak)− d(x,A)| ≥(ε2) 1p}.Then, for each x ∈ X1hr∑Ir|d(x,Ak)− d(x,A)|p =1hr ∑k∈Irk∈Lr|d(x,Ak)− d(x,A)|p+∑k∈Irk/∈Lr|d(x,Ak)− d(x,A)|p<1hr.hr.ε2MpMp +1hr.hr.ε2=ε2+ε2= ε.Hence, {Ak} is Wijsman strongly p−lacunary summable to A. LACUNARY STATISTICAL SUMMABILITY OF SEQUENCES OF SETS 183Definition 3.3. Let (X, ρ) be a metric space and θ = {kr} be a lacunary sequence.For any non-empty closed subsets A, Ak ⊆ X, we say that {Ak} is Wijsmanlacunary almost convergent to A, if for each ε > 0 and for each x ∈ X,limr→∞1hr∑Ird(x,Ak+i) = d(x,A),uniformly in i.Definition 3.4. Let (X, ρ) be a metric space and θ = {kr} be a lacunary sequence.For any non-empty closed subsets A, Ak ⊆ X, we say that {Ak} is Wijsmanlacunary strongly almost convergent to A, if for each ε > 0 and for each x ∈ X,limr→∞1hr∑Ir|d(x,Ak+i)− d(x,A)| = 0,uniformly in i.Example 3.1. Let X = R2 and we define a sequence {Ak} as follows:Ak :={(x, y) ∈ R2 : (x− 1)2 + (y + 1)2 = 1k}, if kr−1 < k < kr−1 + [√hr],{(1, 0)} , otherwise.As k → ∞ this sequence is Wijsman lacunary strongly almost convergent to theset A = {(1, 0)}.Definition 3.5. Let (X, ρ) be a metric space and θ = {kr} be a lacunary sequence.For any non-empty closed subsets A, Ak ⊆ X, we say that {Ak} is Wijsmanlacunary strongly p−almost convergent to A, if for each ε > 0 and for each x ∈ X,limr→∞1hr∑Ir|d(x,Ak+i)− d(x,A)|p = 0,uniformly in i, where p is a positive real number.Definition 3.6. Let (X, ρ) be a metric space and θ = {kr} be a lacunary sequence.For any non-empty closed subsets A, Ak ⊆ X, we say that the sequence {Ak} isWijsman lacunary almost statistically convergent to A, if for each ε > 0 and foreach x ∈ X,limr→∞1hr|{k ∈ Ir : |d(x,Ak+i)− d(x,A)| ≥ ε| = 0,uniformly in i.Let L∞, C, (WAC)θ and |WAC|θ, respectively, denote the sets of the all bounded,Wijsman convergent, Wijsman lacunary almost convergent and Wijsman lacunarystrongly almost convergent sequences of sets. It is easy to see thatC ⊂ (WAC)θ ⊂ |WAC|θ ⊂ L∞.Theorem 3.5. Let (X, ρ) be a metric space, θ = {kr} be a lacunary sequence andp be a positive number. Then, for any non-empty closed subsets A, Ak ⊆ X,(i) {Ak} is Wijsman lacunary almost statistically convergent to A, if it is Wijs-man lacunary strongly p−almost converget to A,184 UĞUR ULUSU AND FATİH NURAY(ii) If {Ak} is bounded and Wijsman lacunary almost statistically convergentto A, then it is Wijsman lacunary strongly p−almost convergent to A.Proof. The proof is similar to the proof of Theorem (3.4). Definition 3.7. Let (X, ρ) be a metric space and θ = {kr} be a lacunary sequence.For any non-empty closed subsets A, Ak ⊆ X, we say that the sequence {Ak} isHausdorff lacunary statistically convergent to A, if for each ε > 0,limr→∞1hr∣∣∣∣{k ∈ Ir : supx∈X|d(x,Ak)− d(x,A)| ≥ ε}∣∣∣∣ = 0i.e.,supx∈X|d(x,Ak)− d(x,A)| < ε a.a.k.in this case, we write HSθ − limAk = A, Sθ − limH Ak = A, Ak → A(HSθ).Theorem 3.6. Let (X, ρ) be a metric space, θ = {kr} be a lacunary and {Ak} be asequence of non-empty closed subsets of X. If {Ak} is Hausdorff lacunary statisticalconverget, then {Ak} is Wijsman lacunary statistical convergent.Proof. For any sequence {Ak} and for every ε > 0, since|{k ∈ Ir : |d(x,Ak)− d(x,A)| ≥ ε}| ≤∣∣∣∣{k ∈ Ir : supx∈X|d(x,Ak)− d(x,A)| ≥ ε}∣∣∣∣we get the result. References[1] J.-P. Aubin and H. Frankowska, Set-valued analysis, Birkhauser, Boston, 1990.[2] M. Baronti and P. Papini, Convergence of sequences of sets. In: Methods of FunctionalAnalysis in Approximation Theory, ISNM 76, Birkhauser, Basel, 133-155, 1986.[3] G. Beer, On convergence of closed sets in a metric space and distance functions, Bull. Austral.Math. Soc., 31 (1985), 421-432.[4] G. Beer, Wijsman convergence: A survey. Set-Valued Var. Anal., 2 (1994), 77-94.[5] H. Fast, Sur la convergence statistique, Collog. Math., 2 (1951), 241-244.[6] J.A. Fridy, On statistical convergence, Analysis, 5 (1985), 301-313.[7] J.A. Fridy and C. Orhan, Lacunary statistical convergence, Pacific J. Math., 160(1) (1993),43-51.[8] J.A. Fridy and C. Orhan, Lacunary statistical summability, J. Math. Anal. Appl., 173 (1993),497-504.[9] F. Nuray and B.E. Rhoades, Statistical convergence of sequences of sets, Fasc. Math., 49(2012), 87-99.[10] R.E. Powel and S.M. Shah, Summability theory and its applications, Van Nostrand-Rheinhold, London, 1972.[11] I.J. Schoenberg, The integrability of certain functions and related summability methods,Amer. Math. Monthly, 66 (1959), 361-375.[12] U. Ulusu and F. Nuray, Lacunary statistical convergence of sequences of sets, Progress inApplied Mathematics, 4 (2012), 99-109.[13] R.A. Wijsman, Convergence of sequences of convex sets, cones and functions, Bull. Amer.Math. Soc., 70 (1964), 186-188.[14] R. A. Wijsman, Convergence of sequences of Convex sets, Cones and Functions II, Trans.Amer. Math. Soc., 123 (1) (1966), 32-45.DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE AND LITERATURE,AFYON KOCATEPE UNIVERSITY, AFYONKARAHİSAR, TURKEYE-mail address: ulusu@aku.edu.tr, fnuray@aku.edu.tr

Lacunary statistical summability of sequences of sets

Afyon Kocatepe Üniversitesi

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Lacunary statistical summability of sequences of sets

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