In this paper we study the concepts of Wijsman almost asymptotically statistical equivalent, Wijsman almost asymptotically lacunary statistical equivalent and Wijsman strongly almost asymptotically lacunary equivalent
sequences of sets and investigate the relationship between the

Ulusu, Uğur

English

Afyon Kocatepe Üniversitesi Açık Erişim Sistemi

Electronic Journal of Mathematical Analysis and Applications,Vol. 2(2) July 2014, pp. 56-66.ISSN: 2090-792X (online)http://fcag-egypt.com/Journals/EJMAA/ON ALMOST ASYMPTOTICALLY LACUNARY STATISTICALEQUIVALENCE OF SEQUENCES OF SETSUUR ULUSUAbstract. In this paper we study the concepts of Wijsman almost asymptoti-cally statistical equivalent, Wijsman almost asymptotically lacunary statisticalequivalent and Wijsman strongly almost asymptotically lacunary equivalentsequences of sets and investigate the relationship between them.1. Introduction and BackgroundThe concept of convergence of a sequence of real numbers has been extended tostatistical convergence independently by Fast [6] and Schoenberg [16].Denition 1.1. (Fridy, [7]) The sequence x = (xk) is said to be statisticallyconvergent to the number L if for every ε > 0,limn1n|{k ≤ n : |xk − L| ≥ ε}| = 0,(denoted by st− limxk = L).The concept of convergence of sequences of numbers has been extended by severalauthors to convergence of sequences of sets. The one of these such extensionsconsidered in this paper is the concept of Wijsman convergence (see, [2, 4, 12, 17,19]).Let (X, ρ) be a metric space. For any point x ∈ X and any non-empty subset Aof X, we dene the distance from x to A byd(x,A) = infa∈Aρ(x,A).Denition 1.2. (Baronti & Papini, [2]) Let (X, ρ) be a metric space. For anynon-empty closed subsets A,Ak ⊆ X, we say that the sequence {Ak} is Wijsmanconvergent to A iflimk→∞d(x,Ak) = d(x,A)for each x ∈ X. In this case we write W − limAk = A.2010 Mathematics Subject Classication. 40A05, 40A35.Key words and phrases. Asymptotic equivalence, statistical convergence, lacunary sequence,almost convergence, Wijsman convergence, sequences of sets.Submitted Nov. 20, 2013.56EJMAA-2014/2(2) ON ALMOST ASYMP. LACUNARY STATIS. EQUIVALENCE OF 57Let (X, ρ) a metric space. For any non-empty closed subsets Ak of X, thesequence {Ak} is said to be bounded if supk d(x,Ak) < ∞ for each x ∈ X.Nuray and Rhoades [12] extended the notion of convergence of set sequences tostatistical convergence, and gave some basic theorems. Also the concept of almoststatistical convergence for sequences of sets was given by Nuray and Rhoades in[12].Denition 1.3. (Nuray & Rhoades, [12]) Let (X, ρ) be a metric space. For anynon-empty closed subsets A,Ak ⊆ X, we say that the sequence {Ak} is Wijsmanstatistical convergent to A if {d(x,Ak)} is statistically convergent to d(x,A); thatis, for ε > 0 and for each x ∈ X,limn→∞1n|{k ≤ n : |d(x,Ak)− d(x,A)| ≥ ε}| = 0.Denition 1.4. (Nuray & Rhoades, [12]) Let (X, ρ) be a metric space. For anynon-empty closed subsets A,Ak ⊆ X, we say that the sequence {Ak} is Wijsmanalmost statistical convergent to A if for ε > 0 and for each x ∈ X,limn→∞1n|{k ≤ n : |d(x,Ak+i)− d(x,A)| ≥ ε}| = 0,uniformly in i.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], and ratiokrkr−1will beabbreviated by qr.Ulusu and Nuray [17] dened the Wijsman lacunary statistical convergence ofsequences of sets, and considered its relation with Wijsman statistical convergence,which was dened by Nuray and Rhoades. Also, the concept of Wijsman lacunaryalmost statistical convergence and Wijsman lacunary strongly almost convergencewere given by Ulusu and Nuray in [17].Denition 1.5. (Ulusu & Nuray, [17]) Let (X, ρ) a metric space and θ = {kr}be a lacunary sequence. For any non-empty closed subsets A,Ak ⊆ X, we say thatthe sequence {Ak} is Wijsman lacunary statistical convergent to A if {d(x,Ak)} islacunary statistically convergent to d(x,A); that is, for 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 = A or Ak → A(WSθ).Denition 1.6. (Ulusu and Nuray, [17]) Let (X, ρ) a metric space and θ = {kr}be a lacunary sequence. For any non-empty closed subsets A,Ak ⊆ X, we say thatthe sequence {Ak} is Wijsman lacunary almost statistical convergent to A if foreach ε > 0 and for each x ∈ X,limr→∞1hr| {k ∈ Ir : |d(x,Ak+i)− d(x,A)| ≥ ε} | = 0,uniformly in i.58 UUR ULUSU EJMAA-2014/2(2)Denition 1.7. (Ulusu and Nuray, [17]) Let (X, ρ) a metric space and θ = {kr}be a lacunary sequence. For any non-empty closed subsets A,Ak ⊆ X, we say thatthe sequence {Ak} is Wijsman lacunary strongly almost convergent to A if for eachx ∈ X,limr→∞1hr∑k∈Ir|d(x,Ak+i)− d(x,A)| = 0,uniformly in i.Marouf [11] presented denitions for asymptotically equivalent sequences andasymptotic regular matrices. Patterson [13] dened asymptotically statistical equiv-alent sequences by using the denition of statistical convergence.Denition 1.8. (Marouf, [11]) Two nonnegative sequences x = (xk) and y = (yk)are said to be asymptotically equivalent iflimkxkyk= 1,(denoted by x ∼ y).Denition 1.9. (Patterson, [13]) Two nonnegative sequences x = (xk) and y =(yk) are said to be asymptotically statistical equivalent of multiple L provided thatfor every ε > 0,limn1n{the number of k < n :∣∣∣∣xkyk − L∣∣∣∣ ≥ ε} = 0,(denoted by xSL∼ y), and simply asymptotically statistical equivalent if L = 1.Patterson and Sava³ [14] extended the denitions presented in [13] to lacunarysequences. In addition to these denitions, natural inclusion theorems were pre-sented.Denition 1.10. (Patterson and Sava³, [14]) Let θ = {kr} be a lacunary sequence,two nonnegative sequences [x] and [y] are said to be asymptotically lacunary statis-tical equivalent of multiple L provided that for every ε > 0,limr1hr∣∣∣∣{k ∈ Ir : ∣∣∣∣xkyk − L∣∣∣∣ ≥ ε}∣∣∣∣ = 0,(denoted by xSLθ∼ y) and simply asymptotically lacunary statistical equivalent ifL = 1. Furthermore, let SLθ denote the set of x and y such that xSLθ∼ y.Denition 1.11. (Patterson and Sava³, [14]) Let θ = {kr} be a lacunary sequence,two number sequences x = (xk) and y = (yk) are said to be strong asymptoticallylacunary equivalent of multiple L provided that,limr1hr∑k∈Ir∣∣∣∣xkyk − L∣∣∣∣ = 0,(denoted by xNLθ∼ y) and strong simply asymptotically lacunary equivalent if L = 1.In addition, let NLθ denote the set of x and y such that xNLθ∼ y.EJMAA-2014/2(2) ON ALMOST ASYMP. LACUNARY STATIS. EQUIVALENCE OF 59Ulusu and Nuray [18] extended the denitions presented in [14] to sequencesof sets in the Wijsman sense. In addition to these denitions, natural inclusiontheorems are presented.Denition 1.12. (Ulusu & Nuray, [18]) Let (X, ρ) be a metric space. For anynon-empty closed subsets Ak, Bk ⊆ X such that d(x,Ak) > 0 and d(x,Bk) > 0 foreach x ∈ X. We say that the sequences {Ak} and {Bk}are Wijsman asymptoticallystatistical equivalent of multiple L if for every ε > 0 and for each x ∈ X,limn1n∣∣∣∣{k ≤ n : ∣∣∣∣d(x,Ak)d(x,Bk) − L∣∣∣∣ ≥ ε}∣∣∣∣ = 0,(denoted by {Ak}WSL∼ {Bk}) and simply Wijsman asymptotically statistical equiv-alent if L = 1.Denition 1.13. (Ulusu & Nuray,[18]) Let (X, ρ) be a metric space and θ bea lacunary sequence. For any non-empty closed subsets Ak, Bk ⊆ X such thatd(x,Ak) > 0 and d(x,Bk) > 0 for each x ∈ X. We say that the sequences {Ak}and {Bk} are Wijsman asymptotically lacunary statistical equivalent of multiple Lif for every ε > 0 and each x ∈ X,limr1hr∣∣∣∣{k ∈ Ir : ∣∣∣∣d(x,Ak)d(x,Bk) − L∣∣∣∣ ≥ ε}∣∣∣∣ = 0,(denoted by {Ak}WSLθ∼ {Bk}) and simply Wijsman asymptotically lacunary statis-tical equivalent if L = 1.2. Main ResultsIn this section we shall give some new denitions and new theorems.Denition 2.1. Let (X, ρ) be a metric space. For any non-empty closed subsetsAk, Bk ⊆ X. We say that the sequences {Ak} and {Bk} are Wijsman almostasymptotically statistical equivalent of multiple L if for every ε > 0 and for eachx ∈ X,limn1n|{k ≤ n : |d(x;Ak+i, Bk+i)− L| ≥ ε}| = 0,uniformly in i whered(x;Ak, Bk) =d(x,Ak)d(x,Bk), x ̸∈ Ak ∪BkL , x ∈ Ak ∪Bk.In this case we write {Ak}(ŴS)L∼ {Bk} and simply Wijsman almost asymptoticallystatistical equivalent if L = 1. Furthermore, let(ŴS)Ldenote the set of {Ak} and{Bk} such that {Ak}(ŴS)L∼ {Bk}.Example 2.2. Consider the following sequences;Ak ={{(x, y) : x2 + y2 − 2kx = 0} , if k is a square integer,{(1, 1)} , otherwise.60 UUR ULUSU EJMAA-2014/2(2)andBk ={{(x, y) : x2 + y2 + 2kx = 0} , if k is a square integer,{(1, 1)} , otherwise.Sincelimn1n|{k ≤ n : |d(x;Ak+i, Bk+i)− 1| ≥ ε}| = 0,uniformly in i, the sequences {Ak} and {Bk} is Wijsman almost asymptoticallystatistical equivalent. That is {Ak}(ŴS)1∼ {Bk}.Denition 2.3. Let (X, ρ) be a metric space and θ be a lacunary sequence. Forany non-empty closed subsets Ak, Bk ⊆ X. We say that the sequences {Ak} and{Bk} are Wijsman almost asymptotically lacunary statistical equivalent of multipleL if for every ε > 0 and for each x ∈ X,limr1hr|{k ∈ Ir : |d(x;Ak+i, Bk+i)− L| ≥ ε}| = 0,uniformly in i. In this case we write {Ak}(ŴSθ)L∼ {Bk} and simply Wijsman almostasymptotically lacunary statistical equivalent if L = 1. In addition, let(ŴSθ)Ldenote the set of {Ak} and {Bk} such that {Ak}(ŴSθ)L∼ {Bk}.Example 2.4. Consider the following sequences;Ak :={(x, y) ∈ R2 : (x+ 1)2 + y2 = 1k},if kr−1 < k < kr−1 + [√hr] andk is a square integer,{(0, 0)} , otherwise.andBk :={(x, y) ∈ R2 : (x− 1)2 + y2 = 1k},if kr−1 < k < kr−1 + [√hr] andk is a square integer,{(0, 0)} , otherwise.Sincelimr1hr|{k ∈ Ir : |d(x;Ak+i, Bk+i)− 1| ≥ ε}| = 0,uniformly in i, the sequences {Ak} and {Bk} is Wijsman almost asymptoticallylacunary statistical equivalent. That is {Ak}(ŴSθ)1∼ {Bk}.Denition 2.5. Let (X, ρ) be a metric space and θ be a lacunary sequence. Forany non-empty closed subsets Ak, Bk ⊆ X. We say that the sequences {Ak} and{Bk} are Wijsman strongly almost asymptotically lacunary equivalent of multipleL if for each x ∈ X,limr1hr∑k∈Ir|d(x;Ak+i, Bk+i)− L| = 0,uniformly in i. In this case we write {Ak}[ŴNθ]L∼ {Bk} and simply Wijsmanstrongly almost asymptotically lacunary equivalent if L = 1. In addition, let(ŴNθ)Ldenote the set of {Ak} and {Bk} such that {Ak}(ŴNθ)L∼ {Bk}.EJMAA-2014/2(2) ON ALMOST ASYMP. LACUNARY STATIS. EQUIVALENCE OF 61Example 2.6. Consider the following sequences;Ak :={(x, y) ∈ R2 : (x+√k)2k+y22k= 1}, if kr−1 < k < kr−1 + [√hr]{(1, 1)} , otherwise.andBk :={(x, y) ∈ R2 : (x−√k)2k+y22k= 1}, if kr−1 < k < kr−1 + [√hr]{(1, 1)} , otherwise.Sincelimr1hr∑k∈Ir|d(x;Ak+i, Bk+i)− 1| = 0,uniformly in i, the sequences {Ak} and {Bk} is Wijsman strongly almost asymp-totically lacunary equivalent. That is {Ak}[ŴNθ]1∼ {Bk}.Theorem 2.7. Let (X, ρ) be a metric space, θ = {kr} be a lacunary sequence andAk, Bk be non-empty closed subsets of X;(i) (a) {Ak}[ŴNθ]L∼ {Bk} ⇒ {Ak}(ŴSθ)L∼ {Bk},(b)[ŴNθ]Lis a proper subset of(ŴSθ)L;(ii) d (x,Ak) = O (d(x,Bk)) and {Ak}(ŴSθ)L∼ {Bk} ⇒ {Ak}[ŴNθ]L∼ {Bk}.Proof. (i)− (a). Let ε > 0 and {Ak}[WN ]Lθ∼ {Bk}. For each x ∈ X, we can write∑k∈Ir|d(x;Ak+i, Bk+i)− L| ≥∑k∈Ir|d(x;Ak+i,Bk+i)−L|≥ε|d(x;Ak+i, Bk+i)− L|≥ ε · |{k ∈ Ir : |d(x;Ak+i, Bk+i)− L| ≥ ε}|uniformly in i which yields the result.(i) − (b). Suppose that[ŴNθ]L⊂(ŴSθ)L. Let {Ak} and {Bk} be followingsequences;Ak ={{k} , if kr−1 < k ≤ kr−1 +[√hr]r = 1, 2, · · ·{0} , otherwise.andBk = {0} for all k.Note that {Ak} is not bounded. For every ε > 0 and for each x ∈ X, we have1hr|{k ∈ Ir : |d(x;Ak+i, Bk+i)− 1| ≥ ε}| ≤[√hr]hr→ 0, as r → ∞,uniformly in i, that is, {Ak}(ŴSθ)1∼ {Bk}. On the other hand, there exists x in X62 UUR ULUSU EJMAA-2014/2(2)1hr∑k∈Ir|d(x;Ak+i, Bk+i)− L| ̸→ 0, as r → ∞,uniformly in i. Hence {Ak}[ŴNθ]L̸∼ {Bk}.(ii) Suppose that d (x,Ak) = O (d(x,Bk)) and {Ak}(ŴSθ)L∼ {Bk}. Then we canassume that|d(x;Ak+i, Bk+i)− L| ≤ Mfor each x ∈ X, for all k and uniformly in i.Given ε > 0 and for each x ∈ X, we get1hr∑k∈Ir|d(x;Ak+i, Bk+i)− L| =1hr∑k∈Ir|d(x;Ak+i,Bk+i)−L|≥ε|d(x;Ak+i, Bk+i)− L|+1hr∑k∈Ir|d(x;Ak+i,Bk+i)−L|<ε|d(x;Ak+i, Bk+i)− L|≤ Mhr|{k ∈ Ir : |d(x;Ak+i, Bk+i)− L| ≥ ε}|+ ε,uniformly in i. Therefore {Ak}[ŴNθ]L∼ {Bk}. Lemma 2.8. Suppose that for given ε1 > 0 and every ε > 0 there exist i0 and j0such that1j|{0 ≤ k ≤ j − 1 : |d(x;Ak+i, Bk+i)− L| ≥ ε}| < ε1,for each x ∈ X, for all j ≥ j0 and i ≥ i0, then {Ak}(ŴS)L∼ {Bk}.Proof. Let ε1 > 0 be given. For every ε > 0 and for each x ∈ X, choose j′0 and i0such that1j|{0 ≤ k ≤ j − 1 : |d(x;Ak+i, Bk+i)− L| ≥ ε}| <ε12,for all j ≥ j′0 and i ≥ i0. It is enough to prove that there exists j′′0 such that1j|{0 ≤ k ≤ j − 1 : |d(x;Ak+i, Bk+i)− L| ≥ ε}| < ε1, (2.1)for each x ∈ X, for j ≥ j′′0 and 0 ≤ i ≤ i0. If we let j0 = max{j′0, j′′0 }, (2.1) will betrue for j > j0 and for all i. Once i0 has been chosen, i0 is xed, so|{0 ≤ k ≤ i0 − 1 : |d(x;Ak+i, Bk+i)− L| ≥ ε}| = M,EJMAA-2014/2(2) ON ALMOST ASYMP. LACUNARY STATIS. EQUIVALENCE OF 63for each x ∈ X. Now, taking 0 ≤ i ≤ i0, and j > i0, for each x ∈ X we have1j|{0 ≤ k ≤ j − 1 : |d(x;Ak+i, Bk+i)− L| ≥ ε}|≤ 1j|{0 ≤ k ≤ i0 − 1 : |d(x;Ak+i, Bk+i)− L| ≥ ε}|+ |{i0 ≤ k ≤ j − 1 : |d(x;Ak+i, Bk+i)− L| ≥ ε}|≤ Mj+1j|{i0 ≤ k ≤ j − 1 : |d(x;Ak+i, Bk+i)− L| ≥ ε}|≤ Mj+ε12.Thus, for j suciently large and for each x ∈ X1j|{0 ≤ k ≤ j − 1 : |d(x;Ak+i, Bk+i)− L| ≥ ε}| ≤Mj+ε12< ε1,uniformly in i which gives (2.1) and this step concludes the proof. Theorem 2.9. For every lacunary sequence θ = {kr},(ŴSθ)L=(ŴS)L.Proof. Let {Ak} ∈(ŴSθ)L, then from Denition (2.3) assures us that, givenε1 > 0 there exist ε > 0, ∃r0 and L such that1hr|{k ∈ Ir : |d(x;Ak+i, Bk+i)− L| ≥ ε}| < ε1,for each x ∈ X, for r ≥ r0 and i = kr−1 + 1 + v where v ≥ 0.Let j ≥ hr and write j = m · hr + t where 0 ≤ t ≤ hr and m is an integer.Since j ≥ hr and m ≥ 0, we obtain the following, for each x ∈ X,1j|{0 ≤ k ≤ j − 1 : |d(x;Ak+i, Bk+i)− L| ≥ ε}|≤ 1j|{0 ≤ k ≤ (m+ 1)hr − 1 : |d(x;Ak+i, Bk+i)− L| ≥ ε}|=1jm∑n=0|{n · hr ≤ k ≤ (n+ 1)hr − 1 : |d(x;Ak+i, Bk+i)− L| ≥ ε}|≤ (m+ 1)jhr · ε1≤ 2m · hr · ε1jfor m ≥ 1,uniformly in i.Forhrj≤ 1, since m · hrj≤ 1 we have, for each x ∈ X,1j|{0 ≤ k ≤ j − 1 : |d(x;Ak+i, Bk+i)− L| ≥ ε}| ≤ 2ε1,64 UUR ULUSU EJMAA-2014/2(2)uniformly in i. Then, Lemma 2.8 implies(ŴSθ)L⊆(ŴS)L.It is also clear that (ŴS)L⊆(ŴSθ)Lfor every lacunay sequence θ. Hence, we have the result. Theorem 2.10. Let (X, ρ) be a metric space and Ak, Bk be non-empty closedsubsets of X. If θ = {kr} be a lacunary sequence with lim infr qr > 1, then{Ak}(ŴS)L∼ {Bk} ⇒ {Ak}(ŴSθ)L∼ {Bk}.Proof. Suppose rst that lim infr qr > 1, then there exists a λ > 0 such thatqr ≥ 1 + λ for suciently large r, which implies thatλ1 + λ≤ hrkr.If {Ak}(ŴS)L∼ {Bk}, then for every ε > 0, for each x ∈ X and for suciently larger, we have1kr|{k ≤ kr : |d(x;Ak+i, Bk+i)− L| ≥ ε}| ≥1kr|{k ∈ Ir : |d(x;Ak+i, Bk+i)− L| ≥ ε}|≥ λ1 + λ.(1hr|{k ∈ Ir : |d(x;Ak+i, Bk+i)− L| ≥ ε}|),uniformly in i. This completes the proof. Theorem 2.11. Let (X, ρ) be a metric space and Ak, Bk be non-empty closedsubsets of X. If θ = {kr} be a lacunary sequence with lim supr qr < ∞, then{Ak}(ŴSθ)L∼ {Bk} ⇒ {Ak}(ŴS)L∼ {Bk}.Proof. Let θ = {kr} be a lacunary sequence with lim supr qr < ∞. Then there isan M > 0 such that qr < M for all r ≥ 1. Let {Ak}(ŴSθ)L∼ {Bk}, and ε1 > 0.There exists R > 0 and ε > 0 such that for every j ≥ R and for each x ∈ X,Aj =1hj|{k ∈ Ij : |d(x;Ak+i, Bk+i)− L| ≥ ε}| < ε1,uniformly in i. We can also nd H > 0 such that Aj < H for all j = 1, 2, ... . Nowlet t be any integer with satisfying kr−1 < t ≤ kr, where r > R. Then we can write,EJMAA-2014/2(2) ON ALMOST ASYMP. LACUNARY STATIS. EQUIVALENCE OF 65for each x ∈ X,1t|{k ≤ t : |d(x;Ak+i, Bk+i)− L| ≥ ε}| ≤1kr−1|{k ≤ kr : |d(x;Ak+i, Bk+i)− L| ≥ ε}|=1kr−1{|{k ∈ I1 : |d(x;Ak+i, Bk+i)− L| ≥ ε}|}+1kr−1{|{k ∈ I2 : |d(x;Ak+i, Bk+i)− L| ≥ ε}|}+ · · ·+ 1kr−1{|{k ∈ Ir : |d(x;Ak+i, Bk+i)− L| ≥ ε}|}=k1kr−1 · k1|{k ∈ I1 : |d(x;Ak+i, Bk+i)− L| ≥ ε}|+k2 − k1kr−1(k2 − k1)|{k ∈ I2 : |d(x;Ak+i, Bk+i)− L| ≥ ε}|+ · · ·+ kR − kR−1kr−1(kR − kR−1)|{k ∈ IR : |d(x;Ak+i, Bk+i)− L| ≥ ε}|+ · · ·+ kr − kr−1kr−1(kr − kr−1)|{k ∈ Ir : |d(x;Ak+i, Bk+i)− L| ≥ ε}|=k1kr−1A1 +k2 − k1kr−1A2 + · · ·+kR − kR−1kr−1AR+kR+1 − kRkr−1AR+1 + · · ·+kr − kr−1kr−1Ar≤{supj≥1 Aj} kRkr−1+{supj≥R Aj} kr − kRkr−1≤ H kRkr−1+ εM,uniformly in i. This completes the proof. Combining Theorem (2.10) and (2.11) we have following Theorem.Theorem 2.12. Let (X, ρ) be a metric space and Ak, Bk be non-empty closedsubsets of X. If θ = {kr} be a lacunary sequence with 1 < lim infr qr ≤ lim supr qr <∞, then{Ak}(ŴSθ)L∼ {Bk} ⇔ {Ak}(ŴS)L∼ {Bk}.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 functional analysisin approximation theory, ISNM 76, 133-155, Birkhauser, Basel, 1986.66 UUR ULUSU EJMAA-2014/2(2)[3] G. Beer, On convergence of closed sets in a metric space and distance functions, Bull. Austral.Math. Soc., 31, 421-432, 1985.[4] G. Beer, Wijsman convergence: A survey. Set-Valued Var. Anal., 2, 77-94, 1994.[5] G. Das and B.K. Patel, Lacunary distribution of sequences, Indian J. Pure Appl. Math., 20,64-74, 1989.[6] H. Fast, Sur la convergence statistique. Colloq. Math., 2, 241-244, 1951.[7] J. A. Fridy, On statistical convergence, Analysis, 5, 301-313, 1985.[8] J. A. Fridy and C. Orhan, Lacunary statistical convergence, Pacic J. Math., 160, no.1, 43-51,1993.[9] G. G. Lorentz, A contribution to the theory of divergent sequences, Acta Mathematica, 80,167-190, 1948.[10] I. J. Maddox, A new type of convergence, Math. Proc. Cambridge Philos. Soc., 83, 61-64,1978.[11] M. Marouf, Asymptotic equivalence and summability. Internat. J. Math. Sci., 16, no.4, 755-762, 1993.[12] F. Nuray and B. E. Rhoades, Statistical convergence of sequences of sets, Fasc. Math., 49,87-99, 2012.[13] R. F. Patterson, On asymptotically statistically equivalent sequences, Demostratio Math.,36, no.1, 149-153, 2003.[14] R. F. Patterson and E. Sava³, On asymptotically lacunary statistically equivalent sequences,Thai J. Math., 4, 267-272, 2006.[15] R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer-Verlag, Berlin, 2004.[16] I. J. Schoenberg, The integrability of certain functions and related summability methods,Amer. Math. Monthly, 66, 361-375, 1959.[17] U. Ulusu and F. Nuray, Lacunary statistical convergence of sequence of sets, Progress inApplied Mathematics, 4, no.2, 99-109, 2012.[18] U. Ulusu and F. Nuray, On asymptotically lacunary statistical equivalent set sequences,Journal of Mathematics, Vol. 2013, Article ID 310438, 5 pages.[19] R. A. Wijsman, Convergence of sequences of convex sets, cones and functions, Bull. Amer.Math. Soc., 70, 186-188, 1964.[20] R. A. Wijsman, Convergence of sequences of convex sets, cones and functions II, Trans. Amer.Math. Soc., 123, no.1, 32-45, 1966.U§ur Ulusu, Faculty of Science and Literature, Afyon Kocatepe University, Af-yonkarahisar, TurkeyE-mail address: ulusu@aku.edu.tr

On almost asymptotically lacunary statistical equivalence of sequences of sets

Afyon Kocatepe Üniversitesi

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On almost asymptotically lacunary statistical equivalence of sequences of sets

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