Koncaand Başarır Journal of Inequalities and Applications 204 204:8 http://www.journalofinequalitiesandapplications.com/content/204//8 R E S E A R C H Open Access On some spaces of almost lacunary convergent sequences derived by Riesz mean and weighted almost lacunary statistical convergence in a real n-normed space Şükran Konca * and Metin Başarır * Correspondence: skonca@sakarya.edu.tr Department of Mathematics Sakarya University Sakarya 5487 Turkey Abstract In this paper we introduce some new spaces of almost convergent sequences derived by Riesz mean and the lacunary sequence in a real n-normed space. By combining the definitions of lacunary sequence and Riesz mean we obtain a new concept of statistical convergence which will be called weighted almost lacunary statistical convergence in a real n-normed space. We examine some connections between this notion with the concept of almost lacunary statistical convergence and weighted almost statistical convergence where the base space is a real n-normed space. MSC: Primary 40C05; secondary 40A35; 46A45; 40A05; 40F05 Keywords: Riesz mean; weighted lacunary statistical convergence; almost convergence; lacunary sequence; n-norm Introduction The concept of 2-normed space has been initially introduced by Gähler []. Later this concept was generalized to the concept of n-normedspacesbymisiak[2]. Since then many others have studied these concepts and obtained various results [3 0]. The idea of statistical convergence was given by Zygmund [] in 935 in order to extend the convergence of sequences. The concept was formally introduced by Fast [2] and Steinhaus [3] and later on by Schoenberg [4] and also independently by Buck [5]. Many years later it has been discussed in the theory of Fourier analysis ergodic theory and number theory under different names. In 993 Fridy and Orhan [6]introducedthe concept of lacunary statistical convergence. Statistical convergence has been generalized to the concept of a 2-normed space by Gürdal and Pehlivan [3] and to the concept of an n-normed space by Reddy [9]. Moricz and Orhan [7] have defined the concept of statistical summability (R p r ). Later on Karakaya and Chishti [8] haveused(r p r )-summability to generalize the concept of statistical convergence and have called this new method weighted statistical convergence. Mursaleen et al. [9] have altered the definition of weighted statistical convergence and have found its relation with the concept of statistical (R p r )-summability. In general the statistical convergence of weighted mean is studied as a regular matrix transformation. In 204 Konca and Başarır; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0) which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited.
Koncaand Başarır Journal of Inequalities and Applications 204 204:8 Page 2 of http://www.journalofinequalitiesandapplications.com/content/204//8 [8]and[9] the concept of statistical convergence is generalized by using a Riesz summability method and it is called weighted statistical convergence. For more details related to this topic we may refer to [5 20 23]. In this paper we introduce some new spaces of almost convergent sequences derived by Riesz mean and lacunarysequence in a real n-normed space. By combining the definitions of lacunary sequence and Riesz mean we obtain a new concept of statistical convergence which will be called weighted almost lacunary statistical convergence in a real n-normed space. We examine some connections between this notion with the concept of almost lacunary statistical convergence and weighted almost statistical convergence where the base space is a real n-normed space. 2 Definitions and preliminaries Let K be a subset of natural numbers N and we denote the set K n = j K : j n. The cardinality of K n is denoted by K n. The natural density of K is given by δ(k):=lim r r K r if it exists. The sequence x =(x j ) is statistically convergent to ξ provided that for every ε >0thesetK = K(ε):=j N : x j ξ ε has natural density zero. Let ( ) be a sequence of non-negative real numbers and P r = p + p 2 + + p r for r N. Then the Riesz transformation of x =(x k )isdefinedas t r := P r r x k. (2.) k= If the sequence t r has a finite limit ξ then the sequence x is said to be (R p r )-convergent to ξ. Let us note that if P r as r then the Riesz transformation is a regular summability method that is it transforms every convergent sequence to convergent sequence and preserves the limit. If =forallk N in (2.) then the Riesz mean reduces to the Cesaro mean C of order one. By a lacunary sequence θ =(k r ) where k 0 = 0 we will mean an increasing sequence of non-negative integers with k r k r as r.theintervalsdeterminedbyθ will be denoted by I r =(k r k r ]. We write = k r k r.theratio k r k r will be denoted by q r. Throughoutthepaperwewillusethefollowingnotationswhichhavebeendefinedin [24]. Let θ =(k r )bealacunarysequence( ) be a sequence of positive real numbers such that := P kr := k (0k r ] P kr := k (0k r ] Q r := P kr P P 0 = 0 and the intervals kr determined by θ and ( )aredenotedbyi r =(P k r P kr ] = P kr P kr.if =forall k Nthen P kr P kr Q r and I r reduce to k r k r q r and I r respectively. If θ =(k r )isalacunarysequenceandp r as r thenθ =(P kr )isalacunary sequence that is P 0 =00<P kr < P kr and = P kr P kr as r. Throughout the paper we will take P r as r unlessotherwisestated. Lorentz [25] hasprovedthatasequencex is almost convergent to a number ξ if and only if t km (x) ξ as k uniformlyinmwhere t km (x)= x m + x m+ + + x m+k k N m 0. (2.2) k
Koncaand Başarır Journal of Inequalities and Applications 204 204:8 Page 3 of http://www.journalofinequalitiesandapplications.com/content/204//8 We write f lim x = ξ if x is almost convergent to ξ. Maddox [26] hasdefinedx =(x j )to be strongly almost convergent to a number ξ if and only if t km ( x ξe ) 0ask uniformly in mwherex ξe =(x j ξ) for all j and e =(...). Let n N and X be a real vector space of dimension d n 2. A real-valued function... : X n R satisfying the following conditions is called an n-norm on X and the pair (X... ) is called a linear n-normed space: () x...x n =0if and only if x...x n are linearly dependent (2) x...x n is invariant under permutation (3) αx...x n x n = α x...x n x n for any α R (4) x...x n y + z x...x n y + x...x n z for all y z x...x n X. Asequencex =(x j )inann-normed space (X... ) is said to be convergent to some ξ X in the n-norm if for each ε > 0 there exists a positive integer j 0 = j 0 (ε) suchthat x j ξ z...z n < ε for all j j 0 and for every nonzero z...z n X. Asequencex =(x j ) is said to be statistically convergent to ξ if for every ε >0thesetK := j N : x j ξ z...z n ε has natural density zero for every nonzero z...z n X in other words x =(x j ) is statistically convergent to ξ in n-normed space (X... ) if lim j j j N : x j ξ z...z n ε = 0 for every nonzero z...z n X.Forξ =0 we say this is statistically null. 3 Main results Throughout the paper w(x) l (X) denote the spaces of all and bounded X valued sequence spaces respectively where (X... )isarealn-normed space. The set of all almost convergent sequences and strongly almost convergent sequences witespect to the n-norm are denoted by F and [F] respectively as follows: x l (X):lim k t km (x ξe) z...z n =0 uniformlyinm F = for every nonzero z...z n X and x l (X):lim k t km ( x ξe z...z n )=0 uniformlyinm [F]= for every nonzero z...z n X where t km (x) isdefinedasin(2.2). We write F lim x = ξ if x is almost convergent to ξ witespect to the n-norm and [F] lim x = ξ if x is strongly almost convergent to ξ with respect to the n-norm. It is easy to see that the inclusions [F] F l (X)hold. Now we define some new sequence spaces in a real n-normed space as follows: [ R Hr k I p r θ] n = r t km (x ξe) z...z n =0 uniformlyinm for some ξ and for every nonzero z...z n X ( R p r θ) n = t km (x ξe) z...z n =0 uniformlyinm for some ξ and for every nonzero z...z n X R p r θ n = t km ( x ξe z...z n )=0 uniformlyinm. for some ξ and for every nonzero z...z n X The following results are obtained for some special cases:
Koncaand Başarır Journal of Inequalities and Applications 204 204:8 Page 4 of http://www.journalofinequalitiesandapplications.com/content/204//8 () If we take m =0then the sequence spaces above are reduced to the sequence spaces [C θ] n (C θ) n C θ n respectively as follows: [C θ] n = t k0 (x ξe) z...z n =0 for some ξ and for every nonzero z...z n X (C θ) n = t k0 (x ξe) z...z n =0 for some ξ and for every nonzero z...z n X C θ n = t k0 x ξe z...z n =0. for some ξ and for every nonzero z...z n X (2) If we take =for all k N then the sequence spaces above are reduced to the following spaces: hr k I [w θ ] n = r t km (x ξe) z...z n =0 uniformly in m for some ξ and for every nonzero z...z n X (w θ ) n = t km (x ξe) z...z n =0 uniformly in m for some ξ and for every nonzero z...z n X w θ n = t km ( x ξe z...z n )=0 uniformly in m. for some ξ and for every nonzero z...z n X (3) Let us choose θ =(k r )=2 r for r >0 then these sequence spaces above are reduced to the following spaces: [ R r p r ] n = P r k= t km (x ξe)z...z n =0 uniformly in m for some ξ and for every nonzero z...z n X x : lim r r ( R p r ) n = P r k= t km (x ξe) z...z n =0 uniformly in m for some ξ and for every nonzero z...z n X x : lim r r R p r n = P r k= t km ( x ξe z...z n )=0 uniformly. in m for some ξ and for every nonzero z...z n X (4) If we select θ =(k r )=2 r for r >0and the base space as (X ) then these sequence spaces above are reduced to the sequence spaces which can be seen in [5]. (5) If we choose =for all k N and θ =(k r )=2 r for r >0 then these sequence spaces above are reduced to the sequence spaces [C ] n (C ) n C n respectively. Now we give the following theorem to demonstrate some inclusion relations among the sequence spaces R p r θ n ( R p r θ) n [ R p r θ] n C θ n (C θ) n [C θ] n with the spaces F and [F]. Theorem 3. The following statements are true: () [F] F ( R p r θ) n [ R p r θ] n [C θ] n. (2) [F] R p r θ n ( R p r θ) n [ R p r θ] n [C θ] n. (3) [F] R p r θ n C θ (C θ) n [C θ] n.
Koncaand Başarır Journal of Inequalities and Applications 204 204:8 Page 5 of http://www.journalofinequalitiesandapplications.com/content/204//8 Proof We give the proof only for (2). The proofs of () and (3) can be done similarly. So we omit them. Let x [F] and[f] lim x = ξ. Thent km ( x ξe z...z n ) 0as k uniformlyinm for every nonzero z...z n X.Since as r then its weighted lacunary mean also converges to ξ as r uniformly in m. This proves that x R p r θ n and [F] lim x = R p r θ n lim x = ξ.alsosince t km (x ξe) z...z n H r tkm (x ξe) z...z n ( t km x ξe z...z n ) then it follows that [F] R p r θ n ( R p r θ) n [ R p r θ] n and [F] lim x = R p r θ n lim x =( R p r θ) n lim x =[ R p r θ] n lim x = ξ. Since uniform convergence of t km (x ξe) z...z n witespect to m asr impliesconvergencefor m = 0 and for every nonzero z...z n X. It follows that [ R p r θ] n [C θ] n and [ R p r θ] n lim x =[C θ] n lim x = ξ. This completes the proof. Theorem 3.2 Let θ =(k r ) be a lacunary sequence and lim inf r Q r >.Then ( R p r ) n ( R p r θ) n with ( R p r ) n lim x =( R p r θ) n lim x = ξ. Proof Suppose that lim inf r Q r > then there exists a δ >0suchthatQ r +δ for sufficiently large values of r which implies that δ P kr +δ.ifx ( R p r ) n with ( R p r ) n lim x = ξ then for sufficiently large values of rwehave P kr k r k= = P kr t km (x ξe) z...z n ( kr t km (x ξe) z...z n + k= k r k=k r + H ( r ) tkm (x ξe) z...z n P kr δ +δ. tkm (x ξe) z...z n t km (x ξe) z...z n ) for each m 0 and for every nonzero z...z n X. Then it follows that x ( R p r θ) n with ( R p r θ) n lim x = ξ by taking the limit as r. This completes the proof. Theorem 3.3 Let θ =(k r ) be a lacunary sequence with lim sup r Q r <. Then ( R p r θ) n ( R p r ) n with ( R p r θ) n lim x =( R p r ) n lim x = ξ. Proof Let x ( R p r θ) n with ( R p r θ) n lim x = ξ.thenforε >0thereexistsq 0 such that for every q > q 0 L q = H q k I q tkm (x ξe) z...z n < ε (3.)
Koncaand Başarır Journal of Inequalities and Applications 204 204:8 Page 6 of http://www.journalofinequalitiesandapplications.com/content/204//8 for each m 0 and for every nonzero z...z n X thatiswecanfindsomepositive constant M such that L q M for all q. (3.2) lim sup r Q r < implies that there exists some positive number K such that Q r K for all r. (3.3) Therefore for k r < r k r wehaveby(3.) (3.2) and (3.3) P r r p tkm k (x ξe) z...z n k= P kr = P kr + k r k= ( k I q0 tkm (x ξe) z...z n k I t km (x ξe) z...z n + k I2 t km (x ξe) z...z n + tkm (x ξe) z...z n + + = P kr (L H + L 2 H 2 + + L q0 H q0 + L q0 +H q0 + + + L r ) M (H + H 2 + + H q0 )+ ε (H q0 + + + ) P kr P kr ) tkm (x ξe) z...z n = M (P k P k0 + + P kq0 P P )+ ε kq0 (P kq0 P kr P + + P kq0 k r P kr ) kr = M P k q0 P kr + ε P k r P kq0 P kr M P k q0 P kr + εk for each m 0 and for every nonzero z...z n X. SinceP kr as r weget x ( R p r ) n with ( R p r ) n lim x = ξ. This completes the proof. Corollary 3.4 Let <lim inf r Q r lim sup r Q r <. Then ( R p r θ) n =( R p r ) n ( R p r θ) n lim x =( R p r ) n lim x = ξ. Proof It follows from Theorem 3.2 and Theorem 3.3. and In the following theorem we give the relations between the sequence spaces (w θ ) n and ( R p r ) n. Theorem 3.5 () If <for all k N then (w θ ) n ( R p r ) n and (w θ ) n lim x =( R p r ) n lim x = ξ. (2) If >for all k N and ( ) is upper-bounded then ( R p r ) n (w θ ) n and ( R p r ) n lim x =(w θ ) n lim x = ξ.
Koncaand Başarır Journal of Inequalities and Applications 204 204:8 Page 7 of http://www.journalofinequalitiesandapplications.com/content/204//8 Proof () If <for all k Nthen < for all r N. So there exists an M aconstant such that 0<M <for all r N.Letx (w θ ) n with (w θ ) n lim x = ξthenfor an arbitrary ε >0we have tkm (x ξe) z...z n M tkm (x ξe) z...z n for each m 0 and for every nonzero z...z n X. Therefore we get the result by taking the limit as r. (2) Let >for all k Nthen > for all r N.Supposethat( ) is upper-bounded then there exists an M 2 aconstantsuchthat< M 2 < for all r N.Letx ( R p r ) n and ( R p r ) n lim x = ξ. So the result is obtained by taking the limit as r for each m 0 and for every nonzero z...z n Xfromthe following inequality: t km (x ξe) z...z n M 2 t km (x ξe) z...z n. Nowwe define a new concept of statistical convergence in n-normed space which will be called weighted almost lacunary statistical convergence: Definition 3.6 The weighted almost lacunary density of K N is denoted by δ ( Rθ) (K)= lim r K r (ε) if the limit exists. We say that the sequence x =(x j )isweightedalmost lacunary statistically convergent to ξ if for every ε >0thesetK r (ε) =k I r : t km (x ξe) z...z n ε has weighted lacunary density zero i.e. lim r k I r : tkm (x ξe) z...z n ε =0 (3.4) uniformly in m for every nonzero z...z n X.Inthiscasewewrite(S ( Rθ) n) lim k x k = ξ. By(S ( Rθ) n) we denote the set of all weighted almost lacunary statistically convergent sequences in n-normed space. () If we take =for all k N in (3.4) then we obtain the definition of almost lacunary statistical convergence in n normed space that is x is called almost lacunary statistically convergent to ξ if for every ε >0theset K θ (ε)=k I r : t km (x ξe) z...z n ε has lacunary density zero i.e. lim r k Ir : tkm (x ξe) z...z n ε =0 (3.5) uniformly in m for every nonzero z...z n X.Inthiscasewewrite (S θ n) lim j x j = ξ.by(s θ n) we denote the set of all weighted almost lacunary statistically convergent sequencesin n-normed space. (2) Let us choose θ =(k r ) for r >0then the definition of weighted almost lacunary statistical convergence which is given in (3.4) is reduced to the definition of weighted almost statistically convergence that is x is called weighted almost
Koncaand Başarır Journal of Inequalities and Applications 204 204:8 Page 8 of http://www.journalofinequalitiesandapplications.com/content/204//8 statistically convergent to ξ if for every ε >0theset K Pr (ε)=k P r : t km (x ξe) z...z n ε has weighted density zero i.e. lim r P r k Pr : t km (x ξe) z...z n ε =0 (3.6) uniformly in m for every nonzero z...z n X.Inthiscasewewrite (S R n) lim j x j = ξ.by(s R n) we denote the set of all weighted almost lacunary statistically convergent sequencesin n-normed space. (3) Let us choose θ =(k r ) for r >0and =for all k N then the definition of weighted almost lacunary statistical convergence which is given in (3.4) is reduced to the definition of almost statistical convergence. Theorem 3.7 If the sequence x is ( R p r θ) n -convergent to ξ then the sequence x is weighted almost lacunary statistically convergent to ξ. Proof Let the sequence x be ( R p r θ) n -convergent to ξ and K rm (ε) =k I r : t km (x ξe) z...z n ε.thenforagivenε >0wehave tkm (x ξe) z...z n k K rm (ε) ε K rm (ε) tkm (x ξe) z...z n for each m 0 and for every nonzero z X. Hence we see that the sequence x is weighted almost statistically convergent to ξ by taking the limit as r. Theorem 3.8 Let t km (x ξe) z...z n Mforallk N for each m 0 and for every nonzero z...z n X. Then (S ( Rθ) n) ( R p r θ) n with (S ( Rθ) n) lim x =( R p r θ) n lim x = ξ. Proof Let x be convergent to ξ in (S ( Rθ) n)andletustake K rm (ε)= k I r : p k tkm (x ξe) z...z n ε. Since t km (x ξe) z...z n M for all k N for each m 0 for every nonzero z...z n X and as r thenforagivenε >0wehave tkm (x ξe) z...z n = k K rm(ε) + k / K rm(ε) tkm (x ξe) z...z n tkm (x ξe) z...z n M K rm(ε) + ε M K rm(ε) + ε
Koncaand Başarır Journal of Inequalities and Applications 204 204:8 Page 9 of http://www.journalofinequalitiesandapplications.com/content/204//8 for each m 0 and for every nonzero z...z n X. Sinceε is arbitrary we have x ( R p r θ) n by taking the limit as r. Theorem 3.9 The following statements are true. () If for all k N then (S θ n) (S ( Rθ) n). (2) Let for all k N and ( ) be upper-bounded then (S ( Rθ) n) (S θ n). Proof () If for all k Nthen for all r N.SothereexistM and M 2 constants such that 0<M M 2 for all r N.Letx (S θ n) with (S θ n) lim x = ξ then for an arbitrary ε >0we have k I r : t km (x ξe) z...z n ε = Pkr < k P kr : tkm (x ξe) z...z n ε M Pkr k r < k P kr k r : tkm (x ξe) z...z n ε = M kr < k k r : t km (x ξe) z...z n ε = M k Ir : tkm (x ξe) z...z n ε for each m 0 and for every nonzero z...z n X. Hence we obtain the result by taking the limit as r. (2) Let ( ) be upper-bounded then there exist M and M 2 constants such that M M 2 < for all r N.Supposethat for all k Nthen for all r N.Letx ( R p r ) n and ( R p r ) n lim x = ξ then for an arbitrary ε >0we have k Ir : tkm (x ξe) z...z n ε = kr < k k r : t km (x ξe) z...z n ε M 2 kr P kr < k k r P kr : tkm (x ξe) z...z n ε = M 2 Pkr < k P kr : t km (x ξe) z...z n ε = M 2 k I r : tkm (x ξe) z...z n ε for each m 0 and for every nonzero z...z n X. Hencetheresultisobtainedby taking the limit as r. Theorem 3.0 For any lacunary sequence θ if lim inf r Q r >then (S R n) (S ( Rθ) n) and (S R n) lim x =(S ( Rθ) n) lim x = ξ. Proof Suppose that lim inf r Q r > then there exists a δ >0suchthatQ r +δ for sufficiently large values of r which implies that δ P kr +δ.ifx (S R n) with(s R n) lim x = ξ
Koncaand Başarır Journal of Inequalities and Applications 204 204:8 http://www.journalofinequalitiesandapplications.com/content/204//8 Page 0of then for every ε > 0 and for sufficiently large values of rwehave P kr k Pkr : t km (x ξe) z...z n ε Pkr < k P kr : t km (x ξe) z...z n ε P kr = H ( r Pkr < k P kr : t km (x ξe) z...z n ε ) P kr δ +δ ( k I r : t km (x ξe) z...z n ε for each m 0 and for every nonzero z...z n X. Hencewegettheresultbytaking the limit as r. Theorem 3. Let θ =(k r ) be a lacunary sequence with lim sup r Q r < then (S ( Rθ) n) (S R n) and (S R n) lim x =(S ( Rθ) n) lim x = ξ. Proof If lim sup r Q r < then there is a K >0suchthatQ r K for all r N.Supposethat x (S ( Rθ) n)with(s ( Rθ) n) lim x = ξ and let N r := k I r : p kt km (x ξe) z...z n ε. (3.7) ) By (3.7) given ε >0thereisar 0 N such that N r < ε for all r > r 0.NowletM := maxn r : r r 0 and let r be any integer satisfying k r < r k r thenforeachm 0 and for every nonzero z...z n X we can write P r k Pr : t km (x ξe) z...z n ε P kr Pkr < k P kr : tkm (x ξe) z...z n ε = P kr (N + N 2 + + N r0 + N r0 + + + N r ) M.r 0 P kr + P kr ε(0 + + + ) = M.r 0 P kr + ε (P k r P kr0 ) P kr M.r 0 + εq r M.r 0 + εk P kr P kr which completes the proof by taking the limit as r. Corollary 3.2 Let <lim inf r Q r lim sup r Q r <. Then (S ( Rθ) n)=(s R n) and (S R n) lim x =(S ( Rθ) n) lim x = ξ. Proof It follows from Theorem 3.0 and Theorem 3..
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