We define and study λ2-convergence of double sequences of interval numbers dened by Orlicz function and λ2-statistical convergence of double sequences of interval numbers. We also establish some inclusion relations between them

Esi, Ayhan

Acta et Commentationes Universitatis Tartuensis de Mathematica

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We define and study λ2-convergence of double sequences of interval numbers dened by Orlicz function and λ2-statistical convergence of double sequences of interval numbers. We also establish some inclusion relations between them.</jats:p

Ayhan Esi

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Double sequences of interval numbers defined by Orlicz functions

Journals from University of Tartu

ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICAVolume 17, Number 1, June 2013Available online at www.math.ut.ee/acta/Double sequences of interval numbers defined byOrlicz functionsAyhan EsiAbstract. We define and study λ2-convergence of double sequences ofinterval numbers defined by Orlicz function and λ2-statistical conver-gence of double sequences of interval numbers. We also establish someinclusion relations between them.1. IntroductionThe idea of statistical convergence for ordinary sequences was introducedby Fast [7] in 1951. Schoenberg [17] studied statistical convergence as asummability method and listed some elementary properties of statisticalconvergence. Both of these authors noted that if a bounded sequence isstatistically convergent, then it is Cesàro summable. Recently Mursaleen[15] defined and studied λ-statistical convergence for sequences as follows.Let λ = (λi) be a non-decreasing sequence of positive numbers tending toinfinity such that λi+1 ≤ λi + 1, λ1 = 1. Then a sequence x = (xk) is saidto be λ-statistically convergent to a number L if for every ε > 0,limi→∞1λi|{k ∈ Ii : d (xk, L) ≥ ε }| = 0,where Ii = [i− λi + 1, i].Recall (see [10]) that an Orlicz function M is a continuous, convex, non-decreasing function, defined for u ≥ 0, such that M(0) = 0 and M(u) > 0 ifu > 0. An Orlicz function M is said to satisfy ∆2-condition for all values ofu if there exists a number K > 0 such that M(2u) ≤ KM(u), u ≥ 0.Interval arithmetic was first suggested by Dwyer [2] in 1951. Developmentof interval arithmetic as a formal system and evidence of its value as acomputational device was provided by Moore [12] in 1959 and Moore andReceived February 17, 2012.2010 Mathematics Subject Classification. 40A05, 40A35, 40C05, 46A45.Key words and phrases. Double sequence space, interval numbers, Orlicz function, sta-tistical convergence.http://dx.doi.org/10.12697/ACUTM.2013.17.045758 AYHAN ESIYang [13] in 1962. Furthermore, Moore and others (see [3], [8], [12] and [11])have developed applications to differential equations.Chiao [1] introduced sequences of interval numbers and defined the usualconvergence of sequences of interval numbers. Şengönül and Eryılmaz [18]introduced and studied bounded and convergent sequence spaces of intervalnumbers and showed that these spaces are complete metric spaces. Recently,Esi in [4] and [5] defined and studied λ-statistical and lacunary statisticalconvergence of interval numbers, respectively.We denote the set of all real valued closed intervals by IR. Any element ofIR is called an interval number and is denoted by A = [xl, xr]. Let xl and xrbe the smallest and the greatest points of an interval number A, respectively.For interval numbers A1 = [x1l , x1r ], A2 = [x2l , x2r ] and a number α ∈ R wehaveA1 = A2 ⇐⇒ x1l = x2l , x1r = x2r ,A1 +A2 = {x ∈ R : x1l + x2l ≤ x ≤ x1r + x2r}andαA ={{x ∈ R : αx1l ≤ x ≤ αx1r} if α ≥ 0,{x ∈ R : αx1r ≤ x ≤ αx1l} if α < 0.The set of all interval numbers IR is a complete metric space with thedistanced(A1, A2)= max {|x1l − x2l | , |x1r − x2r |}(see [14]). In the special case A1 = [a, a] and A2 = [b, b] we obtain the usualmetric of R.Now we give the definition of a convergent sequence of interval numbers(see [1]).Definition 1.1. A sequence(Ak)of interval numbers is said to be conver-gent to an interval number A0 if for each ε > 0 there exists a positive integerk0 such that d(Ak, A0)< ε for all k ≥ k0. We denote it by limk Ak = A0.Thus, limk Ak = A0 if and only if limk xkl = x0l and limk xkr = x0r .In this paper, we introduce and study the concepts of λ2-summable andstatistically λ2-convergent double sequences of interval numbers, and rela-tions between them.2. DefinitionsRecall that a double sequence (ak,i) of real numbers is said to be convergentin the Pringsheim sense to a number L if for every ε > 0 there exists anindex n such that |ak,i − L| < ε whenever k, i > n (see [16]). We transferDOUBLE SEQUENCES OF INTERVAL NUMBERS 59this definition to the double sequences of interval numbers in the followingway.Definition 2.1. An interval valued double sequence(Ak,l)is said to beconvergent in the Pringsheim sense to an interval number A0 if for everyε > 0 there exists n ∈ N such thatd(Ak,l, A0)< ε for k, l > n.In this case we write P -limAk,l = A0.We denote by c2 the set of all convergent in the Pringsheim sense doublesequences of interval numbers.Definition 2.2. An interval valued double sequence(Ak,l)is said to bebounded if there exist an interval number A0 and a positive number B suchthat d(Ak,l, 0)≤ B for all k, l ∈ N.We will denote the set of all bounded double sequences of interval numberby the symbol `2∞. It should be noted that, similarly to the case of doublenumber sequences, c2 is not the subset of `2∞.Let a double sequence λ2 = (λi,j) of positive real numbers tend to infinityand satisfyλi+1,j ≤ λi,j + 1, λi,j+1 ≤ λi,j + 1,λi,j − λi+1,j ≤ λi,j+1 − λi+1,j+1, λ1,1 = 1.PutIi,j = {(k, l) : i− λi,j + 1 ≤ k ≤ i, j − λi,j + 1 ≤ l ≤ j} .Definition 2.3. An interval valued double sequence(Ak,l)is said to beλ2-summable if there exists an interval number A0 such thatP - limi,j1λi,j∑(k,l)∈Ii,jd(Ak,l, A0)= 0.Definition 2.4. Let M be an Orlicz function, let(Ak,l)be an intervalvalued double sequence, and let p = (pk,l) be a double sequence of positivereal numbers. Let λ2 = (λi,j) be the double sequence defined above. We60 AYHAN ESIdefine[V2λ2 ,M, p]=(Ak,l) : P - limi,j 1λi,j ∑(k,l)∈Ii,j[M(d(Ak,l, A0)r)]pk,l=0,for some r > 0 and A0 ∈ IR},[V2λ2 ,M, p]0=(Ak,l) : P - limi,j 1λi,j ∑(k,l)∈Ii,j[M(d(Ak,l, 0)r)]pk,l=0,for some r > 0}and[V2λ2 ,M, p]∞=(Ak,l) : supi,j 1λi,j ∑(k,l)∈Ii,j[M(d(Ak,l, 0)r)]pk,l<∞,for some r > 0},where 0 = [0, 0].If we consider various assignments of M , λ2 and p in Definition 2.4, thenwe obtain different special sets of sequences. For example, if pk,l = 1 forall k, l ∈ N, then these sets reduce to the sets denoted, respectively, by[V2λ2 ,M],[V2λ2 ,M]0and[V2λ2 ,M]∞. In the special case λi,j = ij (i, j ∈N), we write[c2,M]instead of[V2λ2 ,M].For M(x) = x we obtain[V2λ2 , p]=(Ak,l) : P - limi,j 1λi,j ∑(k,l)∈Ii,j[d(Ak,l, A0)]pk,l=0for some A0 ∈ IR},and similarly,[V2λ2 , p]0and[V2λ2 , p]∞.3. Main theoremsTheorem 3.1. If 0 < pk,l < qk,l and(qk,lpk,l)is bounded, then[V2λ2 ,M, p]⊂[V2λ2 ,M, q].DOUBLE SEQUENCES OF INTERVAL NUMBERS 61Proof. If we take[M(d(Ak,l,A0)r)]pk,l= wk,l for all k, l ∈ N, then usingthe same technique employed in the proof of Theorem 2.9 from [9] we getthe result. Corollary 3.2. The following statements are valid.(i) If 0 < infk,l pk,l ≤ 1 for all k, l ∈ N, then[V2λ2 ,M, p]⊂[V2λ2 ,M].(ii) If 1 ≤ pk,l ≤ supk,l pk,l = H < ∞ for all k, l ∈ N, then[V2λ2 ,M]⊂[V2λ2 ,M, p].Proof. (i) follows from Theorem 3.1 with qk,l = 1 for all k, l ∈ N and (ii)follows from Theorem 3.1 with pk,l = 1 for all k, l ∈ N. The proof of the following result is a routine work, so we omit it.Proposition 3.3. Let M be an Orlicz function such that ∆2-conditionis satisfied. Then we have[V2λ2 , p]0⊂[V2λ2 ,M, p]0,[V2λ2 , p]⊂[V2λ2 ,M, p]and[V2λ2 , p]∞⊂[V2λ2 ,M, p]∞.The following definition was presented by Esi [6] for a single sequenceof interval numbers. A sequence of interval numbers(Ak)is said to bestatistically λ-convergent to an interval number A0 if for every ε > 0,limn→∞1λn∣∣{k ∈ In : d (Ak, A0) ≥ ε }∣∣ = 0,where the vertical bars indicate the number of elements in the enclosed set.Now, we will give definitions of statistical convergence and statisticalλ2-convergence for double sequences of interval numbers.Definition 3.1. A double sequence(Ak,l)of interval numbers is said tobe statistically convergent to an interval number A0 provided that for eachε > 0,P - limi,j→∞1ij∣∣{(k, l) ∈ N× N; k ≤ i, l ≤ j : d (Ak,l, A0) ≥ ε }∣∣ = 0.We denote the set of all statistically convergent double sequences of intervalnumbers by s2.Definition 3.2. A double sequence(Ak,l)of interval numbers is said tobe statistically λ2-convergent to an interval number A0 if for each ε > 0,P - limi,j→∞1λi,j∣∣{(k, l) ∈ Ii,j : d (Ak,l, A0) ≥ ε }∣∣ = 0.We denote the set of all statistically λ2-convergent double sequences of in-terval numbers by s2λ2 .62 AYHAN ESITheorem 3.4. Let M be an Orlicz function. If 0 < h ≤ infk,l pk,l ≤supk,l pk,l = H <∞, then[V2λ2 ,M, p]⊂ s2λ2.Proof. Let(Ak,l)∈[V2λ2 ,M, p]. Then there exists r > 0 such that1λi,j∑(k,l)∈Ii,j[M(d(Ak,l, A0)r)]pk,l→ 0in the Pringsheim sense if i, j →∞. If ε > 0, then we obtain1λi,j∑(k,l)∈Ii,j[M(d(Ak,l, A0)r)]pk,l≥ 1λi,j∑(k,l)∈Ii,jd(Ak,l,A0)≥ε[M(d(Ak,l, A0)r)]pk,l≥ 1λi,j∑(k,l)∈Ii,jd(Ak,l,A0)≥ε[M(εr)]pk,l≥ 1λi,j∑(k,l)∈Ii,jd(Ak,l,A0)≥εmin{M(εr)h,M(εr)H}≥ 1λi,j∣∣{(k, l) ∈ Ii,j : d (Ak,l, A0) ≥ ε }∣∣min{M (εr)h ,M (εr)H}.Hence(Ak,l)∈ s2λ2 , which completes the proof. Theorem 3.5. Let M be an Orlicz function and let 0 < h ≤ infk,l pk,l ≤supk,l pk,l = H <∞. Then s2λ2 ∩ `2∞ ⊂[V2λ2 ,M, p].Proof. Let(Ak,l)∈ s2λ2 ∩ `2∞. Then there is a constant N > 0 such thatd(Ak,l, A0)≤ N for all k, l ∈ N. Given ε > 0, for an arbitrarily fixed r > 0DOUBLE SEQUENCES OF INTERVAL NUMBERS 63we have1λi,j∑(k,l)∈Ii,j[M(d(Ak,l, A0)r)]pk,l=1λi,j∑(k,l)∈Ii,jd(Ak,l,A0)<ε[M(d(Ak,l, A0)r)]pk,l+1λi,j∑(k,l)∈Ii,jd(Ak,l,A0)≥ε[M(d(Ak,l, A0)r)]pk,l≤ 1λi,j∑(k,l)∈Ii,jd(Ak,l,A0)<ε[M(εr)]pk,l+1λi,j∑(k,l)∈Ii,jd(Ak,l,A0)≥εmax{M(Nr)h,M(Nr)H}≤ max{M(εr)h,M(εr)H}+1λi,j∣∣{(k, l) ∈ Ii,j : d (Ak,l, A0) ≥ ε }∣∣max{M(Nr)h,M(Nr)H}.Hence(Ak,l)∈[V2λ2 ,M, p]. This completes the proof. The following corollary follows directly from Theorems 3.4 and 3.5.Corollary 3.6. If 0 < h ≤ infk,l pk,l ≤ supk,l pk,l = H < ∞, then s2λ2 ∩`2∞ =[V2λ2 ,M, p]∩ `2∞.If we take λi,j = ij and pk,l = 1 for all k, l ∈ N in Theorems 3.4, 3.5 andCorollary 3.6, then we getCorollary 3.7. Let M be an Orlicz function. Then the following state-ments hold.(i)[c2,M]⊂ s2.(ii) s2 ∩ `2∞ ⊂[c2,M].(iii)[c2,M]∩ `2∞ = s2 ∩ `2∞.AcknowledgementThe author is extremely grateful to the referee for his/her many valuablecomments and suggestions.64 AYHAN ESIReferences[1] K.-P. Chiao, Fundamental properties of interval vector max-norm, Tamsui Oxf. J.Math. Sci. 18 (2002), 219–233.[2] P. S. Dwyer, Linear Computation, Wiley, New York, 1951.[3] P. S. Dwyer, Errors of matrix computation. Simultaneous linear equations and thedetermination of eigenvalues, in: National Bureau of Standarts Applied MathematicsSeries, No. 29, U. S. Covernment Printing Office, Washington, 1953, pp. 49–58.[4] A. Esi, Strongly almost λ-convergence and statistically almost λ-convergence of inter-val numbers, Sci. Magna 7 (2011), 117–122.[5] A. Esi, Lacunary sequence spaces of interval numbers, Thai J. Math. 10 (2012),445–451.[6] A. Esi, λ-sequence spaces of interval numbers. (submitted)[7] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244.[8] P. S. Fischer, Automatic propagated and round-off error analysis, paper presented atthe 13th National Meeting of the Association of Computing Machinary, June 1958.[9] M. Güngör, M. Et, and Y. Altin, Strongly (Vσ, λ, q)-summable sequences defined byOrlicz functions, Appl. Math. Comput. 157 (2004), 561–571.[10] M. A. Krasnosel’skĭı and Ja. B. Rutickĭı, Convex Functions and Orlicz Spaces,P. Noordhoff Ltd., Groningen, 1961.[11] S. Markov, Quasilinear spaces and their relation to vector spaces, Electron. J. Math.Comput. 2 (2005), 1–21.[12] R. E. Moore, Automatic Error Analysis in Digital Computation, LSMD-48421,Lockheed Missiles and Space Company, 1959.[13] R. E. Moore and C. T. Yang, Theory of an Interval Algebra and its Application toNumeric Analysis, RAAG Memories II, Gaukutsu Bunken Fukeyu-kai, Tokyo, 1958.[14] R. E. Moore and C. T. Yang, Interval Analysis I, LMSD-285875, Lockheed Missilesand Space Company, 1962.[15] Mursaleen, λ-statistical convergence, Math. Slovaca 50 (2000), 111–115.[16] A. Pringsheim, Zur Theorie der zweifach unendlichen Zahlenfolgen, Math. Ann. 53(1900), 289–321.[17] I. J. Schoenberg, The integrability of certain functions and related summability meth-ods, Amer. Math. Monthly 66 (1959), 361–375.[18] M. Şengönül and A. Eryılmaz, On the sequence spaces of interval numbers, Thai J.Math. 8 (2010), 503–510.Adiyaman University, Science and Art Faculty, Department of Mathemat-ics, 02040, Adıyaman, TurkeyE-mail address: aesi23@hotmail.com.

We define and study Î»2-convergence of double sequences of interval numbers defined by Orlicz function and Î»2-statistical convergence of double sequences of interval numbers. We also establish some inclusion relations between them

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Double sequences of interval numbers defined by Orlicz Functions

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Double sequences of interval numbers defined by Orlicz functions

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