This paper is a continuation of the research on selection properties of certain classes of double sequences of positive real numbers that was began in [6].Publishe

CASERTA, Agata

MİTROVİĆ, Melanija

Đurčić, Dragan

Hacettepe Journal of Mathematics and Statistics

SCIDAR - Дигитална архива Универзитета у Крагујевцу

Hacettepe Journal of Mathematics and StatisticsVolume 43 (5) (2014), 725 – 730Selection principles and double sequences IIAgata Caserta ∗ , Dragan Djurčić † and Melanija Mitrović ‡Received 23 : 05 : 2013 : Accepted 12 : 08 : 2013Dedicated to Prof. Ljubiša Kočinac on the occasion of his 65th birthdayAbstractThis paper is a continuation of the research on selection properties ofcertain classes of double sequences of positive real numbers that wasbegan in [6].2000 AMS Classification: 40A05, 54A20.Keywords: Pringsheim convergence, selection principles, translationally rapidly vary-ing sequence.1. IntroductionIn recent years a number of papers concerning relations between selection principlestheory and the theory of convergence/divergence of sequences of positive real numbersappeared in the literature [1, 2, 3, 4, 7]. Special attention have been paid to connectionsbetween αi selection principles [12] and classes of sequences important in Karamata’stheory of regular variation (see the papers [1, 3] and references therein, and also the papers[13, 17] for important applications). On the other hand, in [6] the authors introducedmodified αi selection properties for double real sequences and gave their relations withPringsheim’s convergence of double sequences (see [14] and also [9, 10, 15]).In this note we continue investigation began in [6] and extend results from this paperconsidering the class of translationally rapidly varying double sequences following someideas from [3, 16].∗Department of Mathematics, Second University of NaplesCaserta, ItalyEmail: agata.caserta@unina2.it†University of Kragujevac, Faculty of Technical Sciences in ČačakSvetog Save 65, 32000 Čačak, SerbiaEmail:dragandj@tfc.kg.ac.rs Corresponding author.‡University of Niš, Mechanical Engineering Faculty, 18000 Niš, SerbiaEmail:meli@masfak.ni.ac.rsSupported by the Ministry of Education and Science, Serbia, Grants 174026 and 174032.Recall definitions of two selection principles that we consider in this note. If A and Bare families of subsets of an infinite set X, then:(1) S1(A,B) denotes the selection hypothesis that for each sequence (An : n ∈ N) inA there is a sequence (bn : n ∈ N) such that for each n, bn ∈ An and {bn : n ∈ N} ∈ B.(2) α2(A,B) denotes the selection hypothesis that for each sequence (An : n ∈ N) inA there is an element B ∈ B such that for each n, B ∩An is infinite (see [12]).For more details on selection principles see [11].2. ResultsGiven a ∈ R, by ca2 we denote the set of double sequences of real numbers whichconverge to a in the sense of Pringsheim [14]. Letca2,+ := {x = (xm,n)m,n∈N ∈ ca2 : xm,n > 0 for all m,n ∈ N}.We say that a positive double sequence x = (xm,n)m,n∈N belongs to the class Tr(R−∞,s2)of translationally rapidly varying double sequences iflimmin{m,n}→∞x[m+α],[n+β]xm,n= 0for each α ≥ 0 and each β ≥ 0 such that max{α, β} ≥ 1. Here [x] denotes the integerpart of x ∈ R.Notice that the class Tr(R−∞,s2) is nonempty, because it contains the double sequence(xm,n) defined byxm,n =1(m+ n)!, m ∈ N, n ∈ N.2.1. Theorem. Tr(R−∞,s2) & c02,+.Proof. Let x = (xm,n)m,n∈N ∈ Tr(R−∞,s2). Let ε = 12 and α = β = 1. There isN0 = N0(1/2, 1, 1) ∈ N such thatxm+1,n+1xm,n≤ 12for all m,n ≥ N0. Therefore, for m = n ≥ N0 we have xn+1,n+1 ≤ 12xn,n, and itfollows that limn→∞ xn,n = 0. Similarly, for ε = 12 and α = 1, β = 0, there is N1 =N1(1/2, 1, 0) ∈ N such that xm+1,n ≤ 12xm,n for all m,n ≥ N1. It implies that for n ≥ N1we have limm→∞ xm,n = 0. Finally for ε = 12 , α = 0, β = 1 there is N2 = N2(1/2, 0, 1) ∈N such that xm,n+1 ≤ 12xm,n for all m,n ≥ N2. From here we obtain limn→∞ xm,n = 0,for each m ≥ N2.Let ε > 0 be arbitrary (and fixed). Then there is nε ∈ N such that xn,n ≤ ε for eachn ≥ nε. Set n∗ = max{nε, N1, N2}. Then xm,n ≤ ε for each m,n ≥ n∗, which meansthat x ∈ c02,+.The double sequence x = (xm,n)m,n∈N defined byxm,n ={1/m for m ∈ N, n ∈ {1, 2, · · · ,m},1/n for n ∈ N, m ∈ {1, 2, · · · , n}.evidently belongs to the class c02,+, but it does not belong to Tr(R−∞,s2) because forα = β = 1 and m = n we havelimn→∞xm+1,n+1xm,n= limn→∞nn+ 1= 1.In what follows we need two definitions from [6].Let A and B be as above. Then:(a) S(d)1 (A,B) denotes the selection hypothesis: for each double sequence (Am,n :m,n ∈ N) of elements of A there are elements am,n ∈ Am,n, m,n ∈ N, such that thedouble sequence (am,n)m,n∈N belongs to B.(b) α(d)2 (A,B) denotes the selection hypothesis: for each double sequence (Am,n :m,n ∈ N) of elements of A there is an element B in B such that B ∩Am,n is infinite forall (m,n) ∈ N× N.2.2. Theorem. The selection principle S(d)1 (c02,+,Tr(R−∞,s2)) is satisfied.Proof. Let (xm,n,j,k) be a double sequence of double sequences such that for a fixed(j0, k0) ∈ N×N, (xm,n,j0,k0) ∈ c02,+. We create a new double sequence y = (yj,k)j,k∈N inthe following way.10. y1,1 = xm,n,1,1 for an arbitrary fixed (m,n) ∈ N× N;20. y1,2 = xm,n,1,2 such that y1,2 < 12y1,1, y2,1 = xm,n,2,1 such that y2,1 <12y1,1, andy2,2 = xm,n,2,2 such that y2,2 < 12 min{y1,2, y2,1}.p0, p ≥ 3. Choose yp,1 = xm,n,p,1 so that yp,1 <(12)pyp−1,1. For ` ∈ {2, 3, · · · , p− 1}pick yp,` = xm,n,p,` such that yp,` <(12)pyp,`−1 and yp,` <(12)pyp−1,`. Similarly, lety1,p = xm,n,1,p be such that y1,p <(12)py1,p−1. Choose also y`,p = xm,n,`,p such thaty`,p <(12)py`−1,p and y`,p <(12)py`,p−1. Finally, take yp,p to be some xm,n,p,p such thatyp,p <(12)pmin{yp,p−1, yp−1,p}.We prove that y ∈ Tr(R−∞,s2). Let ε > 0 and α, β ≥ 0 with max{α, β} ≥ 1 be given.Set h = h(α, β) = [α] + [β]. There is s0 ∈ N such that ( 12 )s ≤ ε for each s ≥ s0. Forj ≥ s0, k ≥ s0 we haveyj+1,kyj,k≤(12)s0+1andyj,k+1yj,k≤(12)s0+1,and thus we havey[j+α],[k+β]yj,k=yj+[α],k+[β]yj,k≤(12)(s0+1)h≤(12)s0≤ ε,which means that y ∈ Tr(R−∞,s2). 2.3. Theorem. The selection principle α(d)2 (c02,+,Tr(R−∞,s2)) is satisfied.Proof. Let (xm,n,j,k) be a double sequence of double sequences such that for a fixed(j0, k0) ∈ N× N, (xm,n,j0,k0) ∈ c02,+. Form a double sequence y = (yp,t)p,t∈N as follows.Step 1. Using some standard method arrange the given double sequence of doublesequences in a sequence (xn,m,r) of double sequences, where for each r0 ∈ N the doublesequence (xm,n,r0) belongs to c02,+.Step 2. Consider the sequence of sequences (xn,n,r), r ∈ N. Observe that for eachr0 ∈ N it holds (xn,n,r0) ∈ S0, where S0 denotes the set of all sequences of positive realnumbers converging to 0 (see, for instance, [7]). Let J = {(an)n∈N ∈ S : a1 > 0, an+1 ≤ann+1}, where S is the set of all sequences of positive real numbers. It holds J & S0 andthe selection principle S1(S0, J) is satisfied.Step 3. (In this part of the proof we use some techniques from [2]) Take an increasingsequence (pi)i∈N of prime numbers, (p1 = 2), and a fixed r ∈ N. Consider subsequences(xpni ,pni ,r), i ∈ N, of the sequence (xn,n,r). These subsequences are in the class S0.Varying i and r in N, arrange those subsequences in a sequence of sequences of S0.Applying S1(S0, J) one finds a sequence (zj) ∈ J such that (zj) has infinitely manyelements with the sequence (xn,n,r) for each r ∈ N. In other words, we conclude that theselection principle α2(S0, J) is true.Let now yj,j = zj , j ∈ N. For j ≥ 2 we choose ys,j =√s+ 1 · ys+1,j for s ∈{1, 2, · · · , j − 1}, and yj,s =√s+ 1 · yj,s+1. It is easy to see that the double sequencey = (yp,t) obtained in this way has infinitely many common elements with each doublesequence (xm,n,j.k) for arbitrary and fixed (j, k) ∈ N× N.It remains to prove y ∈ Tr(R−∞,s2). Let ε > 0 and α ≥ 0, β ≥ 0 with max{α, β} ≥ 1,be given. Set h = [α] + [β]. There is N0 ∈ N such that(1√N+1)h≤ ε for each N ∈ Nwith N ≥ N0 (N0 ≥ ε−(2/h) − 1). For p, t ≥ N0 we haveyp+1,typ,t≤ 1√N0 + 1andyp,t+1yp,t≤ 1√N0 + 1.So we havey[p+α],[t+β]yp,t=yp+[α],t+[β]yp,t(1√N0 + 1)h≤ ε,i.e. y ∈ Tr(R−∞,s2).2.4. Remark. (1) The selection principles α(d)i (c02,+,Tr(R−∞,s2)), i = 3, 4, are alsosatisfied; see the papers [8, 6] in connection with these selection principles.(2) From the proof of Theorem 2.3 it follows that selection principles αi(c02,+,Tr(R−∞,s2)),i ∈ {2, 3, 4}, are true; see [12] for these selection properties.(3) From the proof of Theorem 2.2 one concludes that this theorem remains true ifthe first coordinate c02,+ in it is replaced by the class of double sequences of positivereal numbers which possesses at least one Pringsheim’s limit point equal to 0 (see, forinstance, [6]).(4) Similarly, Theorem 2.3 remains true if the first coordinate c02,+ is replaced by theclass of double sequences (xm,n) having property that the sequence (xn,n) contains asubsequence converging to 0.For a double sequence x = (xm,n) we defineωn(x) := sup{|xj,k − xr,s| : j ≥ n, k ≥ n, r ≥ n, s ≥ n}, n ∈ N.The sequence (ωn(x)) is called the Landau-Hurwicz sequence of x (compare with [4]).2.5. Proposition. A double sequence x = (xm,n) belongs to the class ca2 , a ∈ R, if andonly if limn→∞ ωn(x) = 0.Proof. (⇒) Assume that x = (xm,n) is a double sequence from ca2 for some arbitrary andfixed a ∈ R. Let ε > 0 be given. There is n0 = n0(ε) ∈ N such that |xj,k − a| ≤ ε/2 foreach j ≥ n0 and each k ≥ n0. Therefore we have|xj,k − xr,s| = |xj,k − a+ a− xr,s| ≤ |xj,k − a|+ |xr,s − a| ≤ ε/2 + ε/2for all j, k, r, s ≥ n0. This implies that for each n ≥ n0 we have0 ≤ ωn(x) ≤ sup{|xj,k − xr,s| : j ≥ n0, k ≥ n0, r ≥ n0, s ≥ n0} ≤ ε,i.e. limn→∞ ωn(x) = 0.(⇐) Let x = (xm,n) be a double sequence with limn→∞ ωn(x) = 0. For a given ε > 0,there is n1 = n1(ε) ∈ N such that 0 ≤ |xj,k − xr,s| ≤ ε/2 for j ≥ n1, k ≥ n1, r ≥ n1,s ≥ n1, because0 ≤ ωn(x) = sup{|xj,k − xr,s| : j ≥ n1, k ≥ n1, r ≥ n1, s ≥ n1} ≤ ε/2for n ≥ n1. Since for all j, r ≥ n1 it holds |xj,j − xr,r| ≤ ε/2, it follows that the sequence(xt,t) is convergent (as a Cauchy sequence), i.e. there is A ∈ R such that limt→∞ xt,t = A.This implies there is n2 = n2(ε) ∈ N such that |xt,t−A| ≤ ε/2 for each t ≥ n2. Therefore,for n0 = max{n1, n2} and all j, k ≥ n0 we have|xj,k −A| ≤ |xj,k − xj,j |+ |xj,j −A| ≤ ε.For a ∈ R we definecaTr(R−∞,s),2 := {x ∈ ca2 : (ωn(x)) ∈ Tr(R−∞,s)}.(For the definition of Tr(R−∞,s) see [3].)2.6. Example. Given a ∈ R, consider the double sequence x = (xj,k) defined byxj,k ={a for j 6= k,a+ 1/j for j = k.It is clear that x ∈ ca2 . However, x /∈ caTr(R−∞,s),2 because ωn(x) = 1/n for each n ∈ Nandlimn→∞ωn+1(x)ωn(x)= 1.2.7. Theorem. The following selection principles are satisfied:(1) S(d)1 (c02,+, c0Tr(R−∞,s),2,+);(2) α(d)2 (c02,+, c0Tr(R−∞,s),2,+).Proof. (1) Consider the double sequence y = (yj,k) which was the selector in the proofof Theorem 2.2. We haveωn(y) = sup{|yj,k − yr,s| : j ≥ n, k ≥ n, r ≥ n, s ≥ n} = yn,n, n ∈ N,which implieslimn→∞ωn+1(y)ωn(y)= limn→∞yn+1,n+1yn,n≤ limn→∞(12)n+1= 0,i.e. (1) is true.(2) Consider the double sequence y = (yj,k) which was the selector in the proof ofTheorem 2.3. For this double sequence we have ωn(y) = yn,n, n ∈ N. Sincelimn→∞ωn+1(y)ωn(y)= limn→∞yn+1,n+1yn,n≤ limn→∞1n+ 1= 0,one concludes that (2) is satisfied. We recall a definition from [6]. Let A and B be as in Introduction. Then Sϕ1 (A,B)denotes the selection hypothesis: for each sequence (At) of elements from A there is anelement B = (bj,k) ∈ B such that bj,k ∈ At for t = ϕ(j, k), where ϕ : N × N → N is agiven bijection.2.8. Theorem. The selection principle Sϕ1 (c02,+,Tr(R−∞,s2)) is satisfied.Proof. Suppose that (At) is a sequence of double sequences At = (xm,n,t) in c02,+. Letus consider the double sequence of double sequences (xm,n,j,k) (constructed from thesequence (At)), where (j, k) = (j(t), k(t)) = ϕ−1(t), t ∈ N. To this double sequenceof double sequences apply the procedure from the proof of Theorem 2.2 to obtain thedouble sequence y = (yj,k) which will witness that the theorem is true. From the proof of Theorem 2.8 and Theorem 2.7(1) we have the following corollary.2.9. Corollary. The selection principle Sϕ1 (c02,+, c0Tr(R−∞,s),2,+) is satisfied.References[1] D. Djurčić, Lj.D.R. Kočinac, M.R. Žižović, Some properties of rapidly varying sequences, J.Math. Anal. Appl. 327 (2007), 1297–1306.[2] D. Djurčić, Lj.D.R. Kočinac, M.R. Žižović, Relations between sequences and selection prop-erties, Abst. Appl. Anal. 2007 (2007), Article ID 43081, 8 pages.[3] D. Djurčić, Lj.D.R. Kočinac, M.R. Žižović, Classes of sequences of real numbers, games andselection properties, Topology Appl. 156 (2008), 46–55.[4] D. Djurčić, Lj.D.R. Kočinac, M.R. Žižović, Rapidly varying sequences and rapid conver-gence, Topology Appl. 155 (2008), 2143–2149.[5] D. Djurčić, Lj.D.R. Kočinac, M.R. 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Řehák, Rapidly varying decreasing solutions of half-linear difference equa-tions, Math. Comp. Modelling 49 (2009), 1692–1699.[14] A. Pringsheim, Zur Theorie der zweifach unendlichen Zahlenfolgen, Math. Ann. 53 (1900),289–321.[15] G.M. Robison, Divergent double sequences and series, Trans. Amer. Math. Soc. 28 (1926),50–73.[16] M. Tasković, Fundamental facts on translationally O-regularly varying functions, Math.Morav. 7 (2003), 107–152.[17] J. Vítovec, Theory of rapid variation on time-scales with applications to dynamic equations,Arch. Math. (Brno) 46 (2010), 263–284.

Selection principles and double sequences II

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