On Regularly Generated Double Sequences

Transcription

1 Filomat 3:3 207, DOI 02298/FIL703809T Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: On Regularly Generated Double Sequences Ümit Totur a, İbrahim Çanak b a Department of Mathematics, Adnan Menderes University, Aydin, Turkey b Department of Mathematics, Ege University, Izmir, Turkey Abstract In this paper, we introduce regularly generated sequences for double sequence of real numbers, and obtain some Tauberian theorems for C,, summability method using the concept of regularly generated sequence Introduction and Definitions A double sequence u u mn is called Pringsheim convergent or P- convergent [] to l if for a given ε > 0 there exists a positive integer N 0 such that u mn l < ε for all nonnegative integers m, n N 0 The C,, means of u mn are defined by σ mn u m n for nonnegative integers m, n see [2] The sequence u mn is said to be C,, summable to a finite number l if lim σ mn u l Every convergent double sequence in Pringsheim s sense need not be C,, summable For example, the sequence u mn defined by n, if m 0; n 0,, 2, u mn 0, otherwise, is convergent to 0 But, the limit lim m n u ij lim does not tend to a finite limit Therefore, u mn is not C,, summable The C,, 0 and C, 0, means of u mn are defined respectively by σ 0 mn u m u ij n 2 n 2n m u in and σ 0 mn u n u mj 200 Mathematics Subject Classification Primary 40E05; Secondary 40G05 Keywords Tauberian theorems, double sequence, regularly generated sequence, one-sided condition, slowly oscillating sequence Received: 23 January 205; Revised 29 March 205; Accepted: 04 April 205 Communicated by Ljubiša D R Kočinac addresses: utotur@aduedutr Ümit Totur, ibrahimcanak@egeedutr İbrahim Çanak

2 Ü Totur, İ Çanak / Filomat 3:3 207, for nonnegative integers m, n The sequence u mn is said to be C,, 0 summable to a finite number l if lim σ 0 mn u l In the light of above discussion, the C, 0, summability is defined analogously A double sequence u mn is said to be bounded if there exists a real number C > 0 such that u mn C for all nonnegative integers m, n Note that every P-convergent double sequence need not be bounded For example, the double sequence u mn is P-convergent to 0, but it is not bounded A double sequence u mn is said to be one-sided bounded if there exists a real number C > 0 such that u mn C for all nonnegative integers m, n Let N, B, and B > denote the space of all double sequences which is P-converging to 0, bounded, one-sided bounded, respectively For a double sequence u mn, we define n u mn u mn u m,n, m u mn u mn u m,n, and m,n u mn m n u mn m n u mn n m u mn for all integers m, n We define de la Vallée Poussin means of the double sequence u mn as follows: If λ > τ > mnu and if 0 < λ < τ < mnu [λm] m[λn] n m [λm]n [λn] jm kn u jk, j[λm] k[λn] for sufficiently large nonnegative integers m, n Now, give the concept of slow oscillation in different senses for a double sequence Definition A double sequence u mn is said to be slowly oscillating in sense, if lim λ max u jk, m j [λm] n k [λn] u mn is said to be slowly oscillating in sense, 0 if lim λ u mn is said to be slowly oscillating in sense 0, if lim λ max rm sn m j [λm] rm max n k [λn] sn r,s u rs 0, r u rn 0, s u ms 0 S, S 0, and S 0 denote the classes of all slowly oscillating sequences in sense,,, 0, and 0,, respectively Notice that every P-convergent sequence is slowly oscillating in senses,,, 0, and 0, However the converse may not be true The following example provides slowly oscillating sequences in senses,,, 0, and 0,, but they are not P-convergent

3 Example 2 u mn log m log n S Indeed, since we have Therefore, Ü Totur, İ Çanak / Filomat 3:3 207, r,s log r log s log r log s logr log s log r logs logr logs, rm sn From this, we obtain r,s log r log s log r log r,s log r log s max m j [λm] n k [λn] rm sn rm sn s s log logr log r r log s s [λm] r,s log r log s log m s s log log [λn] After taking of both sides as m, n, we obtain max r,s log r log s log2 λ m j [λm] n k [λn] rm sn Finally, taking the limit of both sides as λ, we get lim max λ ii i u mn log m S 0 iii u mn log n S 0 m j [λm] n k [λn] The C,, means of mn m,n u mn is defined by rm sn V mn m,n u : m n Moreover, the C,, 0 means of m m u mn is defined by V 0 mn m u : m and the C, 0, means of n n u mn is defined by V 0 mn n u : n n j log m r,s u rs lim λ log2 λ 0 ij i,j u ij in i,n u in, j m u mj The Kronecker identity for single sequences takes the following form for double sequences see [3] For all nonnegative integers m, n, u mn σ 0 mn u σ 0 mn u σ mn u V mn m,n u k n

4 Ü Totur, İ Çanak / Filomat 3:3 207, We write the following identities similar to the Kronecker identity for single sequences u mn σ 0 mn u V 0 mn m u, 2 u mn σ 0 mn u V 0 mn n u The following lemma shows the relationships between Cesàro means σ mn u, σ 0 mn u, σ 0 mn u and V mn m,n u, V 0 mn m u, V 0 mn n u, respectively Lemma 3 For a double sequence u mn of real numbers, 3 mn m,n σ mn u V mn m,n u, m m σ 0 mn u V 0 mn m u, n n σ 0 mn u V 0 mn m u, for all nonnegative integers m, n Proof First, we prove the identity 4 We have m,n σ mn u σ mn u σ u σ u σ m,n m,n m n mn m mnm n m nm From these lines we deduce that mn m,n σ mn u We finally obtain mn m,n σ mn u u ij m n m,n u n u ij u ij m n u ij mn n mn u ij mn m n u ij m n u ij m n m n mn m u ij mn m n n mnu mn mn u ij mn u ij n u mj mn m n m m n m n u ij m n u in m n u ij m n u mn u in u mj u ij m n m n V mn m,n u i ij ij u ij j u mj u mj

5 Now, we prove the identity 5 We have m σ 0 mn u σ 0 mn u σ 0 m Ü Totur, İ Çanak / Filomat 3:3 207, m,n u u in m m mm From these lines we obtain m m σ 0 mn u m m m m m u in m u in m u in m m u in m u in u in m mu mn u in The identity 6 can be similarly showed m i i u in V 0 mn m u u mn u in 2 Regularly Generated Double Sequences The idea of regularly generated sequence for single sequences has been introduced by Dik et al [4] Using the concept of regularly generated sequence, some Tauberian theorems for Abel summability methods have been obtained by many authors see [5 7] In the light of this information, we introduce the concept of regularly generated sequence for double sequences Let L be any linear space of real double sequences and A, B, C be subclasses of L If u mn ξ mn ν mn i j η ij ij η mn, for some ξ mn, ν mn, η mn A, we say that the double sequence u mn is regularly generated by the double sequences ξ mn, ν mn, η mn and the double sequences ξ mn, ν mn, η mn are called the generators of u mn The classes of all sequences regularly generated by ξ ξ mn, ν ν mn, η η mn are denoted by U ξ, ν, η If ξ in u mn ξ mn, i for some ξ mn B, we say that the double sequence u mn is regularly generated by the double sequence ξ mn and the double sequence ξ mn is called a generator of u mn The classes of all sequences regularly generated by ξ ξ mn are denoted by U 2 ξ If ν mj u mn ν mn, j for some ν mn C, we say that the double sequence u mn is regularly generated by the double sequence ν mn and the double sequence ν mn is called a generator of u mn The classes of all sequences regularly generated by ν ν mn are denoted by U 3 ν i j

6 Example 2 Ü Totur, İ Çanak / Filomat 3:3 207, a If S is the class of slowly oscillating sequences in sense,, then U S, S, B is the class of sequence b If SB is the class of all bounded and slowly oscillating sequences in sense, 0, then U 2 SB is the class of all slowly oscillating sequences in sense, 0 For a double sequence u mn of real numbers, σ mn u n n V ij i,j u i j ij, σ 0 mn u m V 0 ij i u i i σ 0 mn u n V 0 ij j u j j by Lemma 3 Since u mn can be expressed as u mn V 0 mn m u V 0 mn n u i j V ij i,j u ij V mn m,n u, the sequences V 0 mn m u, V 0 mn n u, and V mn m,n u are generators of u mn In addition, the sequence u mn can also be represented as or u mn V 0 mn m u u mn V 0 mn n u i j V 0 in V 0 iu, i mj j u j We can say that both V 0 mn m u and V 0 mn n u are generators of u mn Lemma 22 Let u mn L and B, C L i If u mn U 2 B, then V 0 mn m u B ii If u mn U 3 C, then V 0 mn n u C Proof i Since u mn U 2 B, then u mn ξ mn for some ξ mn B Hence, we have m u mn m ξ mn ξ mn m, and m mu mn m m ξ mn ξ mn Therefore, taking C,, 0 means of both sides, we get V 0 mn m u V 0 mn m ξ σ 0 mn ξ It follows from the Kronecker identity that V 0 mn m u ξ mn This completes the proof ii The proof of ii is similar to that of i Lemma 23 [8] Let u mn be a double sequence of real numbers For sufficiently large integers m, n: i If λ > u mn σ mn u [λm] [λn] [λm] m[λn] n [λm] [λm] m i ξ in i σ u σ u σ u σ [λm],[λn] [λm],n m,[λn] mn u σ u σ [λm],n m,nu [λn] [λn] n [λm] m[λn] n jm kn u jk u mn, σ u σ m,[λn] m,nu and

7 Ü Totur, İ Çanak / Filomat 3:3 207, ii if 0 < λ < u mn σ mn u [λm] [λn] m [λm]n [λn] [λm] m [λm] σ mn u σ u σ u σ [λm],n m,[λn] σ mn u σ [λm],n u [λn] n [λn] m [λm]n [λn] j[λm] k[λn] u mn u jk, where [λn] and [λm] denote the integer part of λn and λm, respectively [λm],[λn] u σ mn u σ m,[λn] u Remark 24 In analogy to Lemma 23, we have the following identities i For λ >, u mn σ 0 mn u [λm] 0 σ u σ m [λm] [λm],n m,nu [λm] [λm] m jm u jn u mn ii For 0 < λ <, u mn σ 0 mn u [λm] σ [λm] m m,nu σ 0 [λm],n u [λm] m [λm] j[λm] u mn u jn We can show the identities as in the proof of the corresponding lemma for single sequence in [9] We do not give details Moreover, we note that we can similarly represent the difference u mn σ 0 mn u in two different ways as in Remark 24 Lemma 25 i If u mn is C,, 0 summable to l, and the condition m m u mn C is satisfied for some C > 0 and large enough m, n, then u mn is P-convergent to l ii If u mn is C, 0, summable to l, and the condition n n u mn C is satisfied for some C > 0 and large enough m, n, then u mn is P-convergent to l Proof The proof of lemma is done step by step using the identities in Remark 24 as in the proof of the one-sidedly Tauberian theorem for single sequence Lemma 26 i If u mn is C,, 0 summable to l, and u mn is slowly oscillating in sense, 0, then u mn is P-convergent to l ii If u mn is C, 0, summable to l, and u mn is slowly oscillating in sense 0,, then u mn is P-convergent to l Proof The proof of lemma is done step by step using the identities in Remark 24 as in the proof of the generalized littlewood theorem for single sequence

8 Ü Totur, İ Çanak / Filomat 3:3 207, Some Tauberian Theorems for Regularly Generated Double Sequences If a double sequence is P-convergent to l, then it is C,, summable to l provided that it is bounded [0] However the converse is not necessarily true Namely, a double sequence which is bounded and C,, summable may not be P-convergent We can recover P-convergence of a double sequence from its C,, summability under some suitable conditions Such a condition is called a Tauberian condition and the resulting theorem is called a Tauberian theorem Now, let us give some classical type Tauberian theorems, which are called Landau s theorem and generalized Littlewood theorem for C,, summability method of a double sequence, respectively see [2] Theorem 3 If u mn is C,, summable to l, and mn m,n u mn B >, m m u mn B >, and n n u mn B >, 7 then u mn is P-convergent to l Note that Stadtmüller [] indicated that the condition mn m,n u mn B > in the Theorem 3 is superfluous Theorem 32 If u mn is C,, summable to l, and u mn S, u mn S 0, u mn S 0 8 then u mn is P-convergent to l Note that Stadtmüller [] indicated that the condition u mn S in the Theorem 32 is superfluous Now, we should mention the main goal of the present paper Certain conditions on the double sequence u mn or the sequence V mn m,n u in a class of sequence which is regularly generated sequences are sufficient conditions for C,, summable sequence to be P-convergent Furthermore, we extended some classical type Tauberian theorems for C,, summability method Theorem 33 If u mn is C,, summable to l, and u mn U N, N, N, u mn U 2 N, and u mn U 3 N, 9 then u mn is P-convergent to l Proof Since u mn U 2 N, then V 0 mn m u N, 0 by Lemma 22 i On the other hand, since u mn U 3 N, then V 0 mn n u N, from Lemma 22 ii By the hypothesis u mn U N, N, N, it follows u mn ξ mn ν mn n n η ij i j ij η mn, where ξ mn N, ν mn N, and η mn N From this, we get and m,n u mn m,n ξ mn m,n ν mn η mn mn m,nη mn, mn m,n u mn mn m,n ξ mn mn m,n ν mn η mn mn m,n η mn

9 Ü Totur, İ Çanak / Filomat 3:3 207, Therefore, taking C,, means of both sides of the last identity, we get V mn m,n u V mn m,n ξ V mn m,n ν σ mn η V mn m,n η 2 Applying identities, 2, 3 to sequences ξ mn, ν mn, and η mn, respectively, we obtain V mn m,n ξ N, V mn m,n ν N, V mn m,n η N, and σ mn η N Therefore, we have V mn m,n u N 3 By the identity, the proof is completed Remark 34 If the double sequence u mn is in B, then the condition u mn U N, N, N is omitted Indeed, it follows from the identity and u mn B that Therefore, we obtain V mn m,n u N V 0 mn m u σ 0 mn V 0 mn m u V mn m,n u, V 0 mn m u N σ 0 mn V 0 mn m u N Theorem 35 Let the double sequence u mn be bounded If u mn is C,, summable to l, and V 0 mn n u U 2 S 0, V 0 mn m u U 3 S 0, 4 then u mn is P-convergent to l Proof Since V 0 mn n u U 2 S 0 and V 0 mn m u U 3 S 0, then V mn m,n u S 0, V mn m,n u S 0, 5 6 by Lemma 22 On the other hand, since u mn is bounded and C,, summable to l, σ mn u is P-convergent to l We know that the C,,, C,, 0 and C, 0, summability methods are regular, so σ mn u is C,, summable to l, σ 0 mn u is C,, summable to l and σ 0 mn u is C,, summable to l It follows from the identity that V mn m,n u is C,, summable to 0 If we replace u mn by V mn m,n u in Lemma 23 i, we obtain V mn m,n u σ mn V [λm] [λn] m,n u σ [λm] m[λn] n [λm],[λn] V m,n u σ [λm],n V m,n u σ m,[λn] V m,n u σ mn V m,n u [λm] σ [λm] m [λm],n V m,n u σ m,nv m,n u [λn] σ [λn] n m,[λn] V m,n u σ m,nv m,n u [λm] m[λn] n jm kn V jk j,k u V mn m,n u

10 Ü Totur, İ Çanak / Filomat 3:3 207, for λ > From this, we get V mn m,n u σ mn V m,n u [λm] [λn] [λm] m[λn] n σ [λm],[λn] V m,n u σ [λm],n V m,n u σ m,[λn] V m,n u σ mn V m,n u [λm] σ [λm],n V m,n u σ m,nv m,n u [λm] m [λn] [λn] n [λm] m[λn] n σ m,[λn] V m,n u σ m,nv m,n u jm kn jk j,k u V mn m,n u V 7 From the last term on the right-hand side of the inequality 7, we have [λm] m[λn] n and then [λm] m[λn] n jm kn V [λm] m[λn] n jm kn jk j,k u V mn m,n u r V rk r,k u V jm kn rm jk j,k u V mn m,n u max m j [λm] r V rk r,k u rm Taking of both sides of the inequality 7 as m, n, then we have V mn m,n u σ mn V m,n u σ max s V ms m,s u, sn n k [λn] sn s V ms m,s u λ 2 [λm],[λn] V m,n u λ λ lim inf 2 λ σ [λm],n V m,n u λ λ lim inf 2 λ σ m,[λn] V m,n u λ 2λ σ 2 λ m,nv m,n u max m j [λm] r V rk r,k u rm max n k [λn] s V ms m,s u Since the sequence σ mn V m,n u is P-convergent, then the terms on the right-hand side of the last inequality vanish Therefore, taking the limit of both sides as λ, we obtain sn

11 Ü Totur, İ Çanak / Filomat 3:3 207, V mn m,n u σ mn V m,n u lim λ max m j [λm] rm max n k [λn] sn Since V mn m,n u is slowly oscillating in senses, 0, and 0,, we get by 5 and 6 Hence, we obtain V mn m,n u σ mn V m,n u 0 r V rk r,k u s V ms m,s u V mn m,n u o 8 On the other hand, since u mn is C,, summable to l, then σ 0 mn u is C,, 0 summable to l Moreover, σ 0 mn u is C, 0, summable to l Therefore, we get σ 0 mn V 0 n u is C,, 0 summable to 0 by the identity 2, and σ 0 mn V 0 m u is C, 0, summable to 0 by the identity 3 P-convergence of the sequence V mn m,n u implies the slow oscillation in sense, 0 of V 0 mn n u by Lemma 22 Therefore, we obtain V 0 mn n u o, 9 by Lemma 26 i Similarly, since the sequence V mn m,n u is P-convergent, then V 0 mn m u is slowly oscillating in sense 0, by Lemma 22 Hence, we obtain V 0 mn m u o 20 by Lemma 26ii Taking 8, 9, and 20 into consideration completes the proof by identity 2 Theorem 36 Let the double sequence u mn be bounded If u mn is C,, summable to l, and m m u mn U 2 B >, and n n u mn U 3 B >, 2 n n V 0 mn n u U 2 B >, and m m V 0 mn m u U 3 B >, 22 then u mn is P-convergent to l Proof Since n n V 0 mn n u U 2 B > and m m V 0 mn m u U 3 B >, then n n V mn m,n u B >, m m V mn m,n u B >, by Lemma 22 Since u mn is bounded and C,, summable to l, then it can be satisfied exactly in the same way as in Theorem 36 in order to prove the C,, summability of V mn m,n u to 0

12 Ü Totur, İ Çanak / Filomat 3:3 207, For λ >, if we replace u mn by V mn m,n u in Lemma 23 i, we have V mn m,n u σ mn V [λm] [λn] m,n u σ [λm] m[λn] n [λm],[λn] V m,n u σ [λm],n V m,n u σ m,[λn] V m,n u σ mn V m,n u [λm] σ [λm] m [λm],n V m,n u σ m,nv m,n u [λn] σ [λn] n m,[λn] V m,n u σ m,nv m,n u [λm] m[λn] n jm kn V Taking of both sides of the previous equation as m, n, we get jk j,k u V mn m,n u V mn m,n u σ mn V m,n u λ λ λ λ λ 2 σ [λm],[λn] V m,n u σ [λm],n V m,n u σ m,[λn] V m,n u σ m,nv m,n u σ [λm],n V m,n u σ m,nv m,n u σ m,[λn] V m,n u σ m,nv m,n u [λm] m[λn] n jm kn V jk j,k u V mn m,n u From this, we have V mn m,n u σ mn V m,n u σ λ 2 [λm],[λn] V m,n u λ λ lim inf 2 λ σ [λm],n V m,n u λ λ lim inf 2 λ σ m,[λn] V m,n u λ 2λ σ 2 λ m,nv m,n u [λm] m[λn] n rm r V rk r,k u jm kn s V ms m,s u sn Since the sequence σ mn V m,n u is P-convergent, then the terms on the right-hand side of the last

13 Ü Totur, İ Çanak / Filomat 3:3 207, inequality vanish Hence, we obtain by the conditions 23 and 24 V mn m,n u σ for some C, C 2 > 0 Therefore, we get mn V m,n u [λm] m[λn] n C r rm C [λm] m[λn] n j log m C log V mn m,n u σ mn V m,n u C 3 log λ, for some C 3 > 0 Taking the limit of both sides as λ, we have [λm] m C 2 log jm kn C s sn jm kn k log n [λn], V mn m,n u σ mn V m,n u 0 25 For 0 < λ <, in a similar way using Lemma 23 ii we have lim inf V mn m,n u σ mn V m,n u 0 26 By the inequalities 25 and 26, we obtain n V mn m,n u o 27 On the other hand, by hypothesis, since m m u mn U 2 B > and n n u mn U 3 B >, then m m V 0 mn m u B >, n n V 0 mn n u B >, by Lemma 22 Since u mn is C,, summable to l, then σ 0 mn u is C,, 0 summable to l Moreover, σ 0 mn u is C, 0, summable to l As a result, we get σ 0 mn V 0 m u is C,, 0 summable to 0 by the identity 3, and σ 0 mnv 0 m u is C, 0, integrable to 0 by the identity 2 Using the identity 2, we have m m V 0 mn n u m m σ 0 mn V 0 n u m m V mn m,n u By 27 and Lemma 3, it follows that m m V 0 mn n u C, 30 for some C > 0 Moreover, m m σ 0 mn V 0 n u C, for some C > 0 Since the sequence σ 0 mn V 0 u is C,, 0 summable to 0, then we get σ 0 mn V 0 u is P-convergent to 0 from Lemma 25i Therefore, we obtain that V 0 mn u is C, 0, summable to 0 By the condition 3 and Lemma 25ii, we have V 0 mn n u o 3

14 Similarly, from 3, 27, and Lemma 3, we obtain Ü Totur, İ Çanak / Filomat 3:3 207, n n V 0 mn n u C, 32 for some C > 0 Moreover, n n σ 0 mn V 0 m u C, for some C > 0 Since the sequence σ 0 mn V 0 m u is C,, 0 summable to 0, then we have σ 0 mn V 0 m u is P-convergent to 0 by Lemma 25 ii Hence, we deduce that V 0 mn m u is C,, 0 summable to 0 By the condition 3 and Lemma 25 i, we have V 0 mn m u o 33 The proof is completed by using 27, 3, and 33 in the identity References [] A Pringsheim, Zur Theorie der zweifach unendlichen Zahlenfolgen, Math Ann [2] F Móricz, Tauberian theorems for Cesàro summable double sequences, Stud Math [3] K Knopp, Limitierungs-Umkehrsätze für Doppelfolgen, Math Z [4] M Dik, F Dik, İ Çanak, Classical and neoclassical Tauberian theorems for regularly generated sequences, Far East J Math Sci [5] İ Çanak, Ü Totur, A note on Tauberian theorems for regularly generated sequences, Tamkang J Math [6] İ Çanak, Ü Totur, M Dik, Some conditions under which subsequential convergence follows from A, m summability, Filomat [7] İ Çanak, F Hasekiler, D Kebapcı, Some Tauberian theorems for regularly generated sequences, Comput Math Appl [8] Ü Totur, Classical Tauberian theorems for C,, summability method, Analele Ştiintifice ale Universităţii Al I Cuza din Iaşi, Vol 6, No: 2, 205, [9] Č V Stanojević, Analysis of Divergence: Control and Management of Divergent Process, Graduate Research Seminar Lecture Notes, edited by İ Çanak, University of Missouri - Rolla, 998 [0] O H H Edely, M Mursaleen, Tauberian theorems for statistically convergent double sequences, Inform Sci [] U Stadtmüller, Tauberian theorems for weighted means of double sequences, Anal Math