SOME SETS OF DOUBLE LACUNARY INVARIANT SEQUENCES DEFINIED BY FOUR DIMENSIONAL SUMMABLE MATRICES AND ORLICZ FUNCTIONS

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1 iskolc athematical Notes HU e-issn Vol. 6 (5), No., pp. 5 6 OI.54/N.5.43 SOE SES OF OUBLE LCUNRY INVRIN SEQUENCES EFINIE BY FOUR IENSIONL SUBLE RICES N ORLICZ FUNCIONS YHN ESI N HEEN U Received November, 4 bstract. he paper introduces some sets of double lacunary invariant sequences defined by four dimensional RH -regular matrices Orlicz functions. First we study some basic properties of such sets by equipping them with linear topological structure. Further, we investigate some relationships among such sets under different conditions finally compare with a set of double lacunary statistically convergent sequences. athematics Subject Classification 45; 4B5 Keywords Orlicz function, double sequence, -convergence, invariant mean, double lacunary sequence. INROUCION n Orlicz function is a function W Œ;/ Œ;/ which is continuous, nondecreasing convex with./ ;.x/ > for x >.x/ as x n Orlicz function is said to satisfy the condition for all values of u, if there exists a constant K >, such that.u/ K.u/; u Note that, if < < ;then.x/.x/; for all x In 7 Lindenstrauss zafriri [6] used the idea of Orlicz function to construct the sequence space for single sequences as follows l ( x.x k / W jxk j < ; for some > k which is a Banach space normed by ( k.x k /k inf > W ) jxk j k ) ; c 5 iskolc University ress

2 6 YHN ESI N HEEN U he Orlicz sequence spaces was further investigated from sequence space point of view related summability theory by Esi [3], utta [], ripathy utta [5], ltin et.al. [] many others. Let be a one-to-one mapping from the set of natural numbers into itself. continuous linear functional on ` is said to be an invariant mean or a mean provided that.a/.x/ when the sequence x.x k / is such that x k for all k N;.b/.e/, wheree.;;;/;.c/.x/ x.k/ for all x.xk / ` For certain class of mapping every invariant mean ' extends the limit functional on space c, the space of all convergent sequences, in sense that '.x/ limx for all x.x k / c he space ŒV is of strongly convergent sequence space was introduced by ursaleen [] as follows sequence x.x k / is said to be strongly convergent if there exists a number L such that k k x i.m/ L as k, uniformly in m. i If.m/ mc, then ŒV Œbc ; the space of strongly almost convergent sequences, was introduced by addox in [7]. double sequence x x k;l has a ringsheim limit L [] (denoted by limx L) provided that given an > there exists an N N such that xk;l L < whenever k;l > N We shall describe such an x x k;l more briefly as converge nt. he four dimensional matrix is said to be RH regular if it maps every bounded converge nt sequence into a converge nt sequence with the same li mi t he assumption of boundedness was made because a double sequence which is convergent is not necessarily bounded. Using this definition Robison Hamilton, independently, both presented the following Silverman-oeplitz type characterization of RH regularity. roposition ([5, ]). he four dimensional matrix is RH regular if only if RH W lim m;n for each k l; RH W lim ; m;n k;l; I RH 3 W lim ; m;n am;n;k;l k;l; for each li RH 4 W lim ; m;n am;n;k;l k;l; ;for each ki RH 5 W ; am;n;k;l k;l; is convergenti RH 6 W here exist finite positive integers B such that am;n;k;l k;l>b <

3 SOE SES OF OUBLE LCUNRY INVRIN SEQUENCES 7 he double sequence r;s f.k r ;l s /g is called double lacunary sequence if there exist two increasing of integers such that k o ; h r k r k r as r l o ; h s l s l s as s Notations k r;s k r l s, h r h s r;s is determined by I r;s f.k;l/ W k r < k k r l s < l l s g; q r k r ;q s l s q r;s q r q s k r l s (see [7]). he set of all double lacunary sequences denoted by N r;s defined by Savaş atterson [3] as follows < N r;s x x = k;l W lim xk;l L ; for some L r;s ; For some relevant literature, we also refer to Savaş atterson [4]. Let r;s f.k r ;l s /g be a double lacunary sequence, be an Orlicz function, p p k;l be a factorable double sequence of strictly positive real numbers be a nonnegative RH regular summability matrix method. We now define the following new sets of double sequences ˆ ˆ ˆ w o.;;p/ r;s x x k;l W limr;s uniformly in.p; q/; for some > w.;;p/ r;s x x k;l W limr;s x x uniformly in.p;q/;for some > L w.;;p/ r;s x x k;l W for some > # pk;l ; L x # pk;l ; # pk;l < ; When.x/ x, for all x Œ;/ we have the following sets of double sequences w o.;p/ r;s >; ; >; ; >;

4 YHN ESI N HEEN U ( ( ) x x k;l W limr;s a x pk;l m;n;k;l ; ; uniformly in.p; q/ w.;p/ r;s x x k;l W limr;s x Lpk;l ; uniformly in.p; q/; for some L < x x k;l W w.;p/ r;s x h r;s = pk;l < ; When.p/ p C.q/ q C ; we have the following sets of double sequences bw o.;;p/ r;s < < < x x k;l W limr;s uniformly in.p; q/; for some > bw.;;p/ r;s x x k;l W limr;s jxkcp;lcqj jxkcp;lcq uniformly in.p;q/; for some > L bw.;;p/ r;s x x k;l W for some > jxkcp;lcqj pk;l ; ) pk;l Lj ; pk;l < ; When p k;l ; for all k;l N, we obtain the following sets of double sequences ˆ ˆ w o.; / r;s x x k;l W limr;s uniformly in.p; q/; for some > w.; / r;s x x k;l W limr;s x x uniformly in.p;q/;for some > L w.; / r;s # L # ; ; ; = ; ; = ; ; >; ; = ; >; ;

5 ˆ SOE SES OF OUBLE LCUNRY INVRIN SEQUENCES x x k;l W for some > x # < ; >;. IN RESULS In this section we shall investigate relevant properties of the sets defined in previous sections. heorem. Let p p k;l be bounded. he classes of sequences w o.;;p/ r;s ; w.;;p/ r;s w.;;p/ r;s are linear spaces. roof. It is easy, so we omit it. heorem. Let < h infp k;l p k;l H < ; let is a nonnegative RH regular summability matrix method. For any Orlicz function, if.t/ lim t t ; then w.;p/ r;s w.;;p/ r;s roof. Let < h infp k;l p k;l H < x x k;l w.;p/ r;s let < < ı with < ı < such that.t/ < for t < ı We can write for each m n # pk;l x L # pk;l x L C.k;l/Ir;s x Lı.k;l/Ir;s x L>ı x # pk;l L hen.k;l/ir;s x Lı h x L # pk;l (.)

6 YHN ESI N HEEN U On the other h, we use the fact that x x L < C # L where Œjtj denotes the integer part of t. Since is Orlicz function we have x L./ Now, let us consider the second part where the sum is taken over x L > ı hus # pk;l x L.k;l/Ir;s x L>ı.k;l/Ir;s x L>ı./ı H C x x L L pk;l ## pk;l his from (.) RH regularity of ; we are granted that x x k;l w.;;p/ r;s. Observe that in this part of the proof we did not use Let x x k;l w.;;p/ r;s. Since we have.t/ t for all t It follows that x x k;l w.;;p/ r;s implies x x k;l w.;p/ r;s this completes the proof. heorem 3. wo.;;p/ r;s, w.;;p/ r;s w.;;p/ r;s are complete linear topological spaces with the paranorm inf ˆ pk;l > g x k;l where max.;h /; H k;l p k;l < roof. Clearly g./ ; g. x/ g.x/ x # pk;l >;

7 SOE SES OF OUBLE LCUNRY INVRIN SEQUENCES Let x x k;l, y yk;l w.;;p/ r;s hen there exist some such # x Let C hen Œ C C By inkowsky s C Now C pk;l inf ˆ > g y # pk;l x C y # pk;l x C y # pk;l C C y x p k;l / x y # pk;l # pk;l p k;l x k;l C yk;l inff > W x C y # x # pk;l g >;

8 YHN ESI N HEEN U pk;l inf ˆ > C y # pk;l >; g x k;l C g yk;l Let C, then the continuity of the product follows from the following p k;l g x k;l inff > W # x pk;l ; > g inff.jjr/ p jjg x k;l ; > W # x pk;l ;r > g where r jj Now x s is a Cauchy sequence in w k;l.;;p/ r;s hen g x s k;l x t k;l as s;t For given > ; choose r > x o > be such that rx o > rx o Now g x s x t as s;t implies that there exists n k;l k;l o N such that g x s k;l x t k;l < for all s;t n o rx o his implies inff p k;l > W x s x t 3 5 p k;l C g < rx o (.)

9 SOE SES OF OUBLE LCUNRY INVRIN SEQUENCES 3 Now from (.) we have, x s x rxo x s x t ) rx o g x s x t k;l k;l ) x s x t < rx o rx o his implies that x s is a Cauchy sequence of real numbers. Let lim s x s x for all k;l N Using continuity of, we have x s x t t x s x Let s n o, then taking infimum of such s we have g x sk;l x k;l < hus x sk;l x k;l w.;;p/ r;s By linearity of the space w.;;p/ r;s we have x k;l w.;;p/ r;s Hence w.;;p/ r;s is a complete space. roposition. We have the following inclusions.a/ w.;;p/ r;s w.;;p/ r;s ;.b/ w o.;;p/ r;s w.;;p/ r;s roof. It is easy, so we omit it. heorem 4. he spaces w o.;;p/ r;s w.;;p/ r;s are nowhere dense subsets of w.;;p/ r;s roof. he proof is clear in view of heorem 3 roposition. heorem 5..a/ If < h infp k;l < p k;l ; then.b/ If p k;l p k;l < ; then w.;;p/ r;s w.; / r;s w.; / r;s w.;;p/ r;s

10 4 YHN ESI N HEEN U roof. (a) Let x x k;l w.;;p/ r;s ; since < h infp k;l < p k;l ; we obtain the following x L # pk;l x L thus x x k;l w.; / r;s.b/ Let p k;l for each k, l p k;l < let x x k;l w.; / r;s hen for each < < there exists a positive integer K such that x L < for all n;m K his implies that x x L # pk;l L hus x x k;l w.;;p/ r;s his completes the proof. 3. OUBLE SISICL CONVERGENCE he concept of statistical convergence for single sequences was introduced by Fast [4] in 5. Later, ursaleen Edely [] defined the statistical analogue for double sequence x x k;l as follows real double sequence x xk;l is said to be statistical convergence to L provided that for each > lim.k;l/ N N W k m;l ni xk;l L m;n mn where the vertical bars indicate the numbers of elements in the enclosed set. In this case, we write st lim k;l x k;l L we denote the set of all statistical convergent double sequences by st efinition ([]). Let r;s f.k r ;l s /g be a double lacunary sequence, a real double sequence x x k;l is said to be uniformly S converge nt or uniformly.;/ convergence to L provided that for each >.;/ lim max.k;l/ I r;s W x r;s p;q L

11 SOE SES OF OUBLE LCUNRY INVRIN SEQUENCES 5 In this case, we write S lim.;/ k;l x k;l L we denote the set of all statistical S convergent double sequences by S.;/.;/ In this section we give some relationship between the double sequence spaces S.;/ w. / r;s heorem 6. If be an Orlicz function, then w. / r;s S.;/ ; where w. / x x r;s k;l W x limr;s L ; ˆ uniformly in.p;q/;for some > L roof. Suppose that x x k;l w. / r;s > ; then we obtain the following for every p q x L x L x k.p/; l L.q/ >; Hence x x k;l S.;/./.k;l/ Ir;s W x L heorem 7. Let be an Orlicz function. hen S.;/ \ ` w. / r;s. roof. Let x x k;l S \` Since x `, we can find a positive number.;/ K such that.x/ K; for all x hen for each p q, we have x L x L C x k.p/; l L.q/ x k.p/; l L.q/ < K.k;l/ Ir;s W x L x L C./ thus the ringsheim s limit on r s grant us the result.

12 6 YHN ESI N HEEN U REFERENCES [] Y. ltin,. Et, B. ripathy, On sequence space Np.m;r;q;s/ on seminormed spaces, pp. ath. Comput., vol. 54, pp , 4, doi.6/s6-33(3)7-7. [] H. utta, characterization of the class of statistically pre-cauchy double sequences of fuzzy numbers, pp. ath. Inf. Sci., vol. 7, no. 4, pp , 3, doi.75/amis/743. [3]. Esi, Some new sequence spaces defined by Orlicz functions, Bull. Inst. ath., cademia Sinica, vol. 7, no., pp. 7 76,. [4] H. Fast, Sur la convergence statistique, Colloq. ath., vol., pp. 4 44, 5. [5] H. Hamilton, ransformations of multiple sequences, uke ath. J., vol., pp. 6, 36, doi.5/s [6] J. Lindenstrauss L. zafriri, On Orlicz sequence spaces, Israel J. ath., vol., pp. 37 3, 7, doi.7/bf [7] I. addox, Spaces of strongly summable sequences, Quart. J. ath., Oxford Ser., vol., no., pp , 67, doi.3/qmath/ []. ursaleen, atrix transformations between some new sequence spaces, Houston J. ath., vol., no. 4, pp. 55 5, 3. []. ursaleen O. Edely, Statistical convergence of double sequences, J. ath. nal. ppl., vol., no., pp. 3 3, 3, doi.6/j.jmaa []. ringsheim, Zur theorie der zweifach unendlichen zahlenfolgen, ath. nn., vol. 53, pp. 3,, doi.7/bf4477. [] G. Robison, ivergent double sequences series, mer. ath. Soc. rans., vol., pp. 5 73, 6, doi./s [] E. Savaş, On some new double lacunary sequence spaces via Orlicz function, J. Comput. nal. ppl., vol., no. 3, pp ,. [3] E. Savaş R. atterson, Lacunary statistical convergence of multiple sequences, ppl. ath. Lett., vol., pp , 6, doi.6/j.aml [4] E. Savaş R. atterson,. / -double sequence spaces via Orlicz functions double statistical convergence, Iran. J. Sci. echnol., rans., vol. 3, no. 4, pp , 7. [5] B. ripathy H. utta, On some lacunary difference sequence spaces defined by a sequence of Orlicz functions q-lacunary ım n statistical convergence, n. Şt. Univ. Ovidius Constanţa, vol., no., pp ,. uthors addresses yhan Esi diyaman University, Science rt Faculty, epartment of athematics, 4 diyaman, urkey address aesi3@adiyaman.edu.tr Hemen utta Gauhati University, athematics epartment, Guwahati, 74 ssam, India address hemen dutta@rediffmail.com