Some new generalized difference double sequence spaces via Orlicz functions.

[section]1. Introduction

Let [l.sub.[infinity]], c and [c.sub.o] be the Banach spaces of bounded, convergent and null sequences x = ([x.sub.k]) with the usual norm k[x.sub.k] = supk|[x.sub.k]|. Kizmaz[14]introduced the notion of difference sequence spaces as follows:

X ([DELTA]) = {x = ([x.sub.k]) : ([DELTA][x.sub.k]) [member of] X}

for X = [l.sub.[infinity]], c and [c.sub.o]. Later on, the notion was generalized by Et and Colak[15]as follows:

X ([[DELTA].sup.m]) = {x = ([x.sub.k]) : ([[DELTA].sup.m][x.sub.k]) [member of] X}

for X = [l.sub.[infinity]], c and [c.sub.o], where [[DELTA].sup.m]x = ([[DELTA].sup.m][x.sub.k]) =([[DELTA].sup.m- 1][x.sub.k] - [[DELTA].sup.m-1][x.sub.k+1]), [DELTA]0x = x and also this generalized difference notion has the following binomial representation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Subsequently, difference sequence spaces were studied by Esi[4], Esi and Tripathy[5], Tripathy et.al[13]and many others.

An Orlicz function M is a function M : [0,[infinity]) [right arrow] [0,[infinity]), which is continuous, convex, nondecreasing function define for x > 0 such that M(0) = 0, M(x) > 0 and M (x) [right arrow] [infinity] as x [right arrow] [infinity]. If convexity of Orlicz function is replaced by M(x + y) [less than or equal to] M (x) + M (y) then this function is called the modulus function and characterized by Ruckle[17]. An Orlicz function M is said to satisfy [[DELTA].sub.2]-condition for all values u, if there exists K > 0 such that M(2u) [less than or equal to] KM(u), u [greater than or equal to] 0.

Remark 1.1. An Orlicz function satisfies the inequality M ([lambda]x) [less than or equal to] [lambda]M (x) for all [lambda] with 0 < [lambda] < 1.

Lindenstrauss and Tzafriri [11]used the idea of Orlicz function to construct the sequence space

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is a Banach space normed by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The space [l.sub.M] is closely related to the space [l.sub.p], which is an Orlicz sequence space with M (x) = [|x|.sup.p], for 1 [less than or equal to] p < [infinity].

In the later stage different Orlicz sequence spaces were introduced and studied by Tripathy and Mahanta[12], Esi[1,2], Esi and Et[3], Parashar and Choudhary[16]and many others.

Let [w.sup.2] denote the set of all double sequences of complex numbers. By the convergence of a double sequence we mean the convergence on the Pringsheim sense that is, a double sequence x = ([x.sub.k,l]) has Pringsheim limit L (denoted by P - limx = L) provided that given [epsilon] > 0 there exists N [member of] N such that |[x.sub.k,l]- L| < [epsilon] whenever k,l > N [7]. We shall describe such an x = ([x.sub.k,l]) more briefly as "P-convergent". We shall denote the space of all P-convergent sequences by c2. The double sequence x = ([x.sub.k,l]) is bounded if and only if there exists a positive number M such that |[x.sub.k,l]| < M for all k and l. We shall denote all bounded double sequences by [l.sup.2.sub.[infinity]].

[section]2. Definitions and results

In this presentation our goal is to extend a few results known in the literature from ordinary (single) difference sequences to difference double sequences. Some studies on double sequence spaces can be found in 8-10.

Definition 2.1. Let M be an Orlicz function and p = ([p.sub.k,l]) be a factorable double sequence of strictly positive real numbers. Let X be a seminormed space over the complex field C with the seminorm q. We now define the following new generalized difference sequence spaces:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and also this generalized difference double notion has the following binomial representation [6]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Some double spaces are obtained by specializing M, p, q and r. Here are some examples:

(i) If M(x) = x, r = 0, [p.sub.k,l]= 1 for all k,l [member of] N, and q(x) = |x|, then we obtain ordinary double sequence spaces [w.sup.2], [w.sup.2.sub.o]and [w.sup.2.sub.[infinity]].

(ii) If M(x) = x, r = 0 and q(x) = |x|, then we obtain new double sequence spaces as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(iii) If r = 0 and q(x) = |x|, then we obtain new double sequence spaces as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where ([DELTA][x.sub.k,l]) = ([x.sub.k,l]- [x.sub.k,l+1] - [x.sub.k+1,l] + [x.sub.k+1,l+1]).

[section]3. Main results

Theorem 3.1. Let p = ([p.sub.k,l]) be bounded. The classes of [w.sup.2](M,[[DELTA].sup.r],p,q), [w.sup.2.sub.o](M,[[DELTA].sup.r],p,q) and [w.sup.2.sub.[infinity]](M,[[DELTA].sup.r],p,q) are linear spaces over the complex field C.

Proof. We give the proof only [w.sup.2.sub.[infinity]](M,[[DELTA].sup.r],p,q). The others can be treated similarly. Let x = ([x.sub.k,l]),y = ([y.sub.k,l]) [member of] [w.sup.2.sub.[infinity]](M,[[DELTA].sup.r],p,q). Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

Let [alpha],[beta] [member of] C be scalars and [rho] = max(2|[alpha]|[[rho].sub.1],2|[beta]|[rho]2). Since M is non- decreasing convex function, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where D = max(1,2H), H = su[p.sub.k,l][p.sub.k,l]< [infinity]. Now, from (1) and (2), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

Therefore [alpha]x + [beta]y [member of] [w.sup.2.sub.[infinity]](M,[[DELTA].sup.r],p,q). Hence [w.sup.2.sub.[infinity]](M,[[DELTA].sup.r],p,q) is a linear space.

Theorem 3.2. The double sequence spaces [w.sup.2](M,[[DELTA].sup.r],p,q), [w.sup.2.sub.o](M,[[DELTA].sup.r],p,q) and [w.sup.2.sub.[infinity]](M,[[DELTA].sup.r],p,q) are seminormed spaces, seminormed by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. Since q is a seminorm, so we have f (([x.sub.k,l])) [greater than or equal to] 0 for all x = ([x.sub.k,l]); f(- 2)= 0 and f (([lambda][x.sub.k,l])) = |[lambda]|f (([x.sub.k,l])) for all scalars [lambda].

Now, let x = ([x.sub.k,l]), y = ([y.sub.k,l]) [member of] [w.sup.2.sub.o](M,[[DELTA].sup.r],p,q). Then there exist [[rho].sub.1],[[rho].sub.2]> 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let [rho] = [[rho].sub.1]+ [[rho].sub.2]. Then we have,

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Since [[rho].sub.1], [[rho].sub.2]> 0, so we have,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore f is a seminorm.

Theorem 3.3. Let (X,q) be a complete seminormed space. Then the spaces [w.sup.2](M,[[DELTA].sup.r],p,q), [w.sup.2.sub.o](M,[[DELTA].sup.r],p,q) and [w.sup.2.sub.[infinity]](M,[[DELTA].sup.r],p,q) are complete seminormed spaces seminormed by f.

Proof. We prove the theorem for the space [w.sup.2.sub.o](M,[[DELTA].sup.r],p,q). The other cases can be establish following similar technique. Let [x.sup.i]= ([x.sup.i.sub.k,l]) be a Cauchy sequence in [w.sup.2.sub.o](M,[[DELTA].sup.r],p,q). Let [epsilon] > 0 be given and for r > 0, choose xo fixed such that M([rx.sub.o]/2) [greater than or equal to] 1 and there exists mo[member of] N such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By definition of seminorm, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This shows that q ([x.sup.i.sub.k,1]) and q([x.sup.j.sub.1,l]) (k,l [less than or equal to] r) are Cauchy sequences in (X,q). Since (X,q) is complete, so there exists [x.sub.k,1],[x.sub.1,l][member of] X such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now from (3), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

This implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So, q([[DELTA].sup.r]([x.sup.i.sub.k,l])) is a Cauchy sequence in (X,q). Since (X,q) is complete, there exists [x.sub.k,l][member of] X such that limi[[DELTA].sup.r]([x.sup.i.sub.k,l]) = [x.sub.k,l] for all k,l [member of] N. Since M is continuous, so for i [greater than or equal to] mo, on taking limit as j [right arrow] [infinity], we have from (4),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

On taking the infimum of such [rho]0s, we have

f(([x.sup.i.sub.k,l]- [x.sub.k,l]))< [epsilon], for all i [greater than or equal to] [m.sub.o].

Thus ([x.sup.i.sub.k,l]- [x.sub.k,l]) [member of] [w.sup.2.sub.o](M,[[DELTA].sup.r],p,q). By linearity of the space [w.sup.2.sub.o](M,[[DELTA].sup.r],p,q), we have for all i [greater than or equal to] [m.sub.o],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus [w.sup.2.sub.o](M,[[DELTA].sup.r],p,q) is a complete space.

Proposition 3.4. (a) [w.sup.2](M,[[DELTA].sup.r],p,q) [subset] [w.sup.2.sub.[infinity]](M,[[DELTA].sup.r],p,q), (b) [w.sup.2.sub.o](M,[[DELTA].sup.r],p,q) [subset] [w.sup.2.sub.[infinity]](M,[[DELTA].sup.r],p,q).

The inclusions are strict.

Proof. It is easy, so omitted.

To show that the inclusions are strict, consider the following example.

Example 3.5. Let M(x) = xp, p [greater than or equal to] 1, r = 1,q (x) = |x|, [p.sub.k,l]= 2 for all k, l [member of] N and consider the double sequence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here x = ([x.sub.k,l]) [member of] [w.sup.2.sub.[infinity]](M,[[DELTA].sup.r],p,q), but x = ([x.sub.k,l]) [not member of] [w.sup.2](M,[[DELTA].sup.r],p,q).

Theorem 3.6. The double spaces [w.sup.2](M,[[DELTA].sup.r],p,q) and [w.sup.2.sub.o](M,[[DELTA].sup.r],p,q) are nowhere dense subsets of [w.sup.2.sub.[infinity]](M,[[DELTA].sup.r],p,q).

Proof. The proof is obvious in view of Theorem 3.3 and Proposition 3.4.

Theorem 3.7. Let r [greater than or equal to] 1, then for all 0 < i [less than or equal to] r, [z.sup.2](M,[DELTA]i,p,q)[subset] [z.sup.2](M,[[DELTA].sup.r],p,q), where [z.sup.2]= [w.sup.2], [w.sup.2.sub.o]and [w.sup.2.sub.[infinity]]. The inclusions are strict.

Proof. We establish it for only [w.sup.2.sub.o] (M,[[DELTA].sup.r]-1,p,q) [subset] [w.sup.2.sub.o](M,[[DELTA].sup.r],p,q). Let x = ([x.sub.k,l]) [member of] [w.sup.2.sub.o] (M,[[DELTA].sup.r]-1,p,q). Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Thus from (5) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now for

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

from which it follows that x = ([x.sub.k,l]) [member of] [w.sup.2.sub.o](M,[[DELTA].sup.r],p,q) and hence [w.sup.2.sub.o] (M,[[DELTA].sup.r]-1,p,q) [subset] [w.sup.2.sub.o](M,[[DELTA].sup.r],p,q). On applying the principle of induction, it follows that [w.sup.2.sub.o] (M,[DELTA]i,p,q) [subset] [w.sup.2.sub.o](M,[[DELTA].sup.r],p,q) for i = 0,1,2,...,r - 1. The proof for the rest cases are similar. To show that the inclusions are strict, consider the following example.

Example 3.8. Let M(x) = xp, r = 1, q(x) = |x|, [p.sub.k,l] = 1 for all k odd and for all l [member of] N and [p.sub.k,l] = 2 otherwise. Consider the sequence x = ([x.sub.k,l]) defined by [x.sub.k,l] = k + l for all k,l [member of] N. We have [[DELTA].sup.r][x.sub.k,l] = 0 for all k,l [member of] N. Hence x = ([x.sub.k,l]) [member of] [w.sup.2.sub.o](M,[DELTA],p,q) but x = ([x.sub.k,l]) [not member of] [w.sup.2.sub.o](M,p,q).

Theorem 3.9. (a) If 0 < infk,l[p.sub.k,l][less than or equal to] [p.sub.k,l]< 1, then [z.sup.2](M,[[DELTA].sup.r],p,q) [subset] [z.sup.2](M,[[DELTA].sup.r],q),

(b) If 1 < [p.sub.k,l][less than or equal to] su[p.sub.k,l][p.sub.k,l]< [infinity], then [z.sup.2](M,[[DELTA].sup.r],q) [subset] [z.sup.2](M,[[DELTA].sup.r]-1,p,q), where Z2= [w.sup.2], [w.sup.2.sub.o]and [w.sup.2.sub.[infinity]].

Proof. The first part of the result follows from the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the second part of the result follows from the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Theorem 3.10. Let [M.sub.1] and [M.sub.2] be Orlicz functions satisfying [[DELTA].sub.2]-condition. If [beta] = [lim.sub.t[right arrow][infinity]] [M.sub.2](t)/t [greater than or equal to] 1, then [z.sup.2]([M.sub.1],[[DELTA].sup.r],p,q) = [z.sup.2]([M.sub.2]* [M.sub.1],[[DELTA].sup.r],p,q), where [z.sup.2]= [w.sup.2], [w.sup.2.sub.o]and [w.sup.2.sub.[infinity]].

Proof. We prove it for [z.sup.2]= [w.sup.2.sub.o]and the other cases will follows on applying similar techniques. Let x = ([x.sub.k]) [member of] [w.sup.2]([M.sub.1],[[DELTA].sup.r],p,q), then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let 0 < [epsilon] < 1 and [delta] with 0 < [delta] < 1 such that [M.sub.2](t) < [epsilon] for 0 [less than or equal to] t < [delta]. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where the first term is over [y.sub.k,l][less than or equal to] [delta] and the second is over [y.sub.k,l]> [delta]. From the first term in (6), using the Remark

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

On the other hand, we use the fact that

[y.sub.k,l]<[y.sub.k,l]/[delta] < 1 +[y.sub.k,l]/[delta].

Since [M.sub.2]is non-decreasing and convex, it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since [M.sub.2]satisfies [[DELTA].sub.2]-condition, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, from the second term in (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

By (7) and (8), taking limit in the Pringsheim sense, we have x = ([x.sub.k,l]) [member of] [w.sup.2.sub.o]([M.sub.2]* [M.sub.1],[[DELTA].sup.r],p,q). Observe that in this part of the proof we did not need [beta] [greater than or equal to] 1. Now, let [beta] [greater than or equal to] 1 and x = ([x.sub.k,l]) [member of] [w.sup.2.sub.o]([M.sub.1],[[DELTA].sup.r],p,q). Since [beta] [greater than or equal to] 1 we have [M.sub.2](t) [greater than or equal to] [beta]t for all t [greater than or equal to] 0. It follows that x = ([x.sub.k,l]) [member of] [w.sup.2.sub.o]([M.sub.2]* [M.sub.1],[[DELTA].sup.r],p,q) implies x = ([x.sub.k,l]) [member of] [w.sup.2.sub.o]([M.sub.1],[[DELTA].sup.r],p,q). This implies [w.sup.2.sub.o]([M.sub.2]* [M.sub.1],[[DELTA].sup.r],p,q) = [w.sup.2.sub.o]([M.sub.1],[[DELTA].sup.r],p,q).

Theorem 3.11. Let M, M1and [M.sub.2]be Orlicz functions, q, [q.sub.1]and [q.sub.2]be seminorms. Then

(i) [z.sup.2]([M.sub.1],[[DELTA].sup.r],p,q) [intersection] [z.sup.2]([M.sub.2],[[DELTA].sup.r],p,q) [subset] [z.sup.2]([M.sub.1]+ [M.sub.2],[[DELTA].sup.r],p,q).

(ii) [z.sup.2](M,[[DELTA].sup.r],p,[q.sub.1]) [intersection] [z.sup.2](M,[[DELTA].sup.r],p,q) [subset] [z.sup.2](M,[[DELTA].sup.r],p,[q.sub.1]+ [q.sub.2]).

(iii) If [q.sub.1]is stronger than [q.sub.2], then [z.sup.2](M,[[DELTA].sup.r],p,[q.sub.1]) [subset] [z.sup.2](M,[[DELTA].sup.r],p,[q.sub.2]), where [z.sup.2]= [w.sup.2], [w.sup.2.sub.o] and [w.sup.2.sub.[infinity]].

Proof. (i) We establish it for only [z.sup.2]= [w.sup.2.sub.o]. The rest cases are similar. Let x = ([x.sub.k,l]) [member of] [w.sup.2.sub.o]([M.sub.1],[[DELTA].sup.r],p,q) [intersection] [w.sup.2.sub.o]([M.sub.2],[[DELTA].sup.r],p,q). Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let [rho] = max([[rho].sub.1],[[rho].sub.2]). The result follows from the following inequality

The proofs of (ii) and (iii) follow obviously.

The proof of the following result is also routine work.

Proposition 3.12. For any modulus function, if [q.sub.1] u (equivalent to) [q.sub.2], then [z.sup.2](M,[[DELTA].sup.r],p,[q.sub.1]) = [z.sup.2](M,[[DELTA].sup.r],p,[q.sub.2]), where [z.sup.2]= [w.sup.2], [w.sup.2.sub.o]and [w.sup.2.sub.[infinity]].

References

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[2] A. Esi, Some new sequence spaces defined by Orlicz functions, Bull. Inst. Math. Acad. Sinica, No. 1, 27(1999), 776.

[3] A. Esi and M. Et, Some new sequence spaces defined by a sequence of Orlicz functions, Indian J. Pure Appl. Math., No. 8, 31(2000), 967-973.

[4] A. Esi, On some generalized difference sequence spaces of invariant means defined by a sequence of Orlicz functions, Journal of Computational Analysis and Applications, No. 3, 11(2009), 524-535.

[5] A. Esi and B. C. Tripathy, On some generalized new type difference sequence spaces defined by a modulus function in a seminormed space, Fasciculi Mathematici, 40(2008), 15-24.

[6] A. Esi, On some new generalized difference double sequence spaces defined by modulus functions, Journal of the Assam Academy of Mathematics, 2(2010), 109-118.

[7] A. Pringsheim, Zur theorie der zweifach unendlichen zahlenfolgen, Math. Ann. Soc., 53(1900), 289-321.

[8] A. Gokhan and R. Colak, The double sequence spaces [c.sup.2](p) and [c.sup.2.sub.o](p), Appl. Math. Comput., No. 2, 157(2004), 491-501.

[9] A. Gokhan and R. Colak, On double sequence spaces [c.sup.2.sub.o](p),[c.sup.2](p) and l2(p), Int. J. Pure Appl. Math., No. 3, 30(2006), 309-321.

[10] A. Gokhan and R. Colak, Double sequence space l2(p), Appl. Math. Comput., No. 1, 160(2005), 147-153.

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[12] B. C. Tripathy and S. Mahanta, On a class of generalized lacunary sequences defined by Orlicz functions, Acta Math. Appl. Sin. Eng. Ser., No. 2, 20(2004), 231-238.

[13] B. C. Tripathy, A. Esi and B. K. Tripathy, On a new type of generalized difference Cesaro sequence spaces, Soochow J. Math., No. 3, 31(2005), 333-340.

[14] H. Kizmaz, On certain sequence spaces, Canad. Math. Bull., No. 2, 24(1981), 169-176.

[15] M. Et and R. Colak, On generalized difference sequence spaces, Soochow J. Math., No. 4, 21(1995), 377-386.

[16] S. D. Parashar and B. Choudhary, Sequence spaces defined by Orlicz functions, Indian J. Pure Appl. Math., 25(1994), 419-428.

[17] W. H. Ruckle, FK spaces in which the sequence of coordinate vectors is bounded, Canad. J. Math., 25(1973), 973-978.

Ayhan Esi([dagger])and M. Necdet Catalbas([double dagger])

([dagger]) Adiyaman University, Science and Art Faculty, Department of Mathematics, 02040, Adiyaman, Turkey

([double dagger]) Firat University, Science and Art Faculty, Department of Mathematics, 23119, Elaz-g, Turkey E-mail: aesi23@hotmail.com ncatalbas@firat.edu.tr