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SOOCHOW JOURNAL OF MATHEMATICS

Volume 32, No. 2, pp. 211-221, April 2006

STATISTICALLY CONVERGENT DOUBLE SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS BY BINOD CHANDRA TRIPATHY AND BIPUL SARMA Abstract. In this article we define different type of statistically convergent, statistically null and statistically bounded double sequence spaces on a seminormed space by Orlicz functions. We study their different properties like solidness, denseness, symmetricity, completeness etc. We obtain some inclusion relations.

1. Introduction The notion of statistical convergence was introduced by Fast [7] and Schoenberg [17] independently. It is also found in Zygmund [21]. Later on it was studied by Fridy and Orhan [8], Maddox [10], Salat [16], Rath and Tripathy [15], Tripathy [19, 20] and many others. Throughout the article χE denotes the characteristic function of E. The notion of statistical convergence depends on the density of subsets of N . A subset E of N is said to have density δ(E) if ∞ 1X χE (k) exists. n→∞ n k=1

δ(E) = lim

A sequence (xn ) is said to be statistically convergent to L if for every > 0 δ({k ∈ N : |xk − L| ≥ }) = 0. Throughout a double sequence will be denoted by A =< a nk > i.e. a double infinite array of elements ank , for all n, k ∈ N . The notion of statistical Received August 14, 2004; revised August 22, 2005. AMS Subject Classification. 40A05, 40B05, 46E30. Key words. Orlicz functions, ∆2 -condition, statistical convergence, solid space, complete space. 211

212

BINOD CHANDRA TRIPATHY AND BIPUL SARMA

convergence for double sequences was introduced by Tripathy [20]. For this he introduced the notion of density of subsets of N × N as follows: A subset E of N × N is said to have density ρ(E) if 1 X X χE (n, k) exists. p,q→∞ pq n≤p k≤q

ρ(E) = lim

Throughout (X, q) will represent a seminormed space seminormed by q. An Orlicz function M is a mapping M : [0, ∞) → [0, ∞) such that it is continuous, non-decreasing and convex with M (0) = 0, M (x) > 0, for x > 0 and M (x) → ∞, as x → ∞. An Orlicz function M is said to satisfy ∆ 2 - condition if there exists a constant K > 0 such that M (2u) ≤ KM (u) for all values of u ≥ 0. 2. Definitions and Preliminaries A double sequence A =< ank > is said to converge in Pringsheim’s sense to L if lim ank = L, where n and k tend to ∞, independent of each other.

n,k→∞

A double sequence < ank > is said to converge regularly if it converges in Pringsheim’s sense and in addition the following limits exist. lim ank = Lk

n→∞

(k = 1, 2, 3, . . .)

and lim ank = Jn (n = 1, 2, 3 . . .).

k→∞

Throughout the article 2 w(q), 2 `∞ (q), 2 c(q), 2 c0 (q), 2 cR (q), 2 cR 0 (q) will denote the spaces of all, bounded, convergent in Pringsheim’s sense, null in Pringsheim’s sense, regularly convergent, regularly null X-valued double sequence spaces respectively. An X-valued double sequence < ank > is said to be statistically convergent to L if for every > 0, ρ({(n, k) ∈ N × N : q(a nk − L) ≥ }) = 0.

STATISTICALLY CONVERGENT DOUBLE SEQUENCE SPACES

213

An X-valued double sequence A is said to be statistically regularly convergent if it converges in Pringsheim’s sense and the following statistical limits exist. stat-lim ank = Lk n→∞

(k = 1, 2, 3, . . .)

and stat-lim ank = Jn (n = 1, 2, 3 . . .). k→∞

An X-valued double sequence A is said to be statistically bounded if there exists G > 0 such that ρ({(n, k) : q(ank ) > G}) = 0. For M an Orlicz function, we now introduce the following double sequence spaces: 2 `∞ (M, q)

ank )) < ∞, for some r > 0}. r n,k ank − L >∈ 2 w(q) : stat-lim M (q( )) = 0, for some r > 0 n,k→∞ r and L ∈ X}. ank )) = 0, for some r > 0}. >∈ 2 w(q) : stat-lim M (q( n,k→∞ r

= {< ank >∈ 2 w(q) : sup M (q(

¯(M, q) 2c

= {< ank

2 c¯0 (M, q)

= {< ank

A sequence < ank >∈ (2 c¯)R (M, q), if < ank >∈ 2 c¯(M, q) and the following statistical limits exist ank − Ln )) = 0, for n = 1, 2, 3, . . . , r1 ank − Jk stat-lim M (q( )) = 0, for k = 1, 2, 3, . . . . n→∞ r2

stat-lim M (q( k→∞

(1) (2)

A sequence < ank >∈ (2 c¯0 )R (M, q), if < ank >∈ 2 c¯0 (M, q) and (1) and (2) hold with Ln = Jk = θ, the zero element of X, for all n, k ∈ N . A double sequence space E is said to be solid if < α nk ank >∈ E whenever < ank >∈ E for all sequences < αnk > of scalars with |αnk | ≤ 1 for all n, k ∈ N. A double sequence space E is said to be symmetric if < a nk >∈ E implies < aπ(n)π(k) >∈ E, where π is a permutations on N . A double sequence space E is said to be a sequence algebra if < a nk bnk >∈ E, whenever < ank >, < bnk >∈ E. A double sequence space E is said to be monotone if it contains the canonical preimages of all its step spaces.

214


A double sequence space E is said to be convergence free if < b nk >∈ E, whenever < ank >∈ E and ank = θ implies bnk = θ. Remark 1. A sequence space E is solid implies E is monotone. The zero single sequence will be denoted by θ¯ = (θ, θ, θ, θ, . . .) and the zero double sequence will be denoted by 2 θ¯ =< θ >. Throughout e = (1, 1, 1, . . .) and ek = (0, 0, . . . , 1, 0, 0, . . .), where the only 1 appears at the k-th place. Throughout the article (2 c¯)R (M, q), (2 c¯0 )R (M, q),

¯(M, q), 2 c¯0 (M, q), (2 c¯)B (M, q), 2c (2 c¯)BR (M, q), (2 c¯0 )BR (M, q) denote

(2 c¯0 )B (M, q), the spaces of

statistically convergent in Pringsheim’s sense, statistically null in Pringsheim’s sense, bounded statistically convergent in Pringsheim’s sense, bounded statistically null in Pringsheim’s sense, regularly statistically convergent, regularly statistically null, bounded regularly convergent, bounded regularly null X-valued double sequences defined by Orlicz function respectively. 3. Main Results The proof of the following result is a routine verification in view of the existing technique. Theorem 1. The classes Z(M, q), where Z = (2 c¯0 )R , (2 c¯)BR , (2 c¯0 )BR , 2 `¯∞ are linear spaces.

¯, 2 c¯0 , (2 c¯) 2c

B,

(2 c¯0 )B , (2 c¯)R ,

Theorem 2. The spaces Z(M, q), where Z = ( 2 c¯)B , (2 c¯0 )B , (2 c¯)BR , (2 c¯0 )BR and 2 `∞ are seminormed spaces, seminormed by f (< ank >) = inf{ρ > 0 : sup M (q( n,k

ank )) ≤ 1}. ρ

¯ = 0 and Proof. Since q is a seminorm, so we have f (A) ≥ 0 for all A; f ( 2 θ) f (λA) = |λ|f (A) for all scalars λ. Let < ank > and < bnk >∈ (2 c¯)B (M, q). There exist ρ1 , ρ2 > 0 such that sup M (q( n,k

and

ank )) ≤ 1 ρ1


sup M (q( n,k

215

bnk )) ≤ 1. ρ2

Let ρ = ρ1 + ρ2 . Then we have, sup M (q( n,k

ρ1 ank ρ2 bnk ank + bnk )) ≤ ( ) sup M (q( )) + ( ) sup M (q( )) ≤ 1. ρ ρ1 + ρ2 n,k ρ1 ρ1 + ρ2 n,k ρ2

Since ρ1 , ρ2 > 0, so we have, f (< ank > + < bnk >) = inf{ρ = ρ1 + ρ2 > 0 : sup M (q( n,k

≤ inf{ρ1 > 0 : sup M (q( n,k

ank + bnk )) ≤ 1} ρ

ank bnk )) ≤ 1} + inf{ρ2 > 0 : sup M (q( )) ≤ 1} ρ1 ρ2 n,k

= f (< ank >) + f (< bnk >). Hence f is a seminorm. Theorem 3. Let (X, q) be a complete seminormed space. Then the spaces Z(M, q), where Z = (2 c¯)B , (2 c¯0 )B , (2 c¯)BR , (2 c¯0 )BR and 2 `∞ are complete seminormed spaces seminormed by f . Proof. We prove the theorem for the space ( 2 c¯)B (M, q) and the proof for the other cases can be established following similar technique. Let A i =< aink > be a Cauchy sequence in (2 c¯)B (M, q). We have to show the following: (i) aink → ank , as i → ∞, for each (n, k) ∈ N × N . (ii) ai → a, as i → ∞, where stat-lim aink = ai , for each i ∈ N. (iii) ank stat −→ a (relative to M ). Let > 0 be given. For a fixed x0 > 0, choose r > 0 such that M ( rx30 ) ≥ 1 and m0 ∈ N be such that f (< aink − ajnk >)
) = , for all i, j ≥ m0 . 3 rx0 3

(3)

216


Hence < aink > is a Cauchy sequence in X for all (n, k) ∈ N × N. Since X is complete, so there exists ank ∈ X, such that aink → ank , as i → ∞, for each (n, k) ∈ N × N. (ii) We have stat-lim aink = ai for each i ∈ N . Thus there exists a subset Ei ⊂ N × N such that ρ(Ei ) = 1, for each i ∈ N and for a given > 0, aink −ai )) ≤ M ( ), for all (n, k) ∈ Ei , for each i ∈ N and some r > 0. r 3r i ⇒ q(ank −ai ) < , for all (n, k) ∈ Ei , for each i ∈ N and by continuity of M . (4) 3 M (q(

Let i, j ≥ m0 and (n, k) ∈ Ei ∩ Ej . Then we have q(ai − aj ) ≤ q(aink − ai ) + q(aink − ajnk ) + q(ajnk − aj ) < + + = , by (3) and (4). 3 3 3 Hence < ai > is a Cauchy sequence in X, which is complete. Thus < a i > converges in X and let limi→∞ ai = a. (iii) For 1 > 0 given, let i ≥ m0 and r > 0 be so chosen that M ( r ) < 1 and the following hold. From (ii) we have a subset E ⊂ N × N such that q(aink − ai ) < . 3 By (i) we have q(ank − aink ) < 3 , for all i ≥ m0 . By (ii) we have q(ai − a) < 3 , for all i ≥ m0 . Hence for all i ≥ m0 and for all (n, k) ∈ E with ρ(E) = 1, we have q(ank − a) ≤ q(ank − aink ) + q(aink − ai ) + q(ai − a)
0 and all (n, k) ∈ E with ρ(E) = 1. r r ⇒ stat-lim ank = a. ⇒ M (q(

Hence < ank >∈ (2 c¯)B (M, q). Thus (2 c¯)B (M, q) is a complete seminormed space. Proposition 4. The spaces 2 c¯0 (M, q), (2 c¯0 )BR (M, q), (2 c¯0 )B (M, q), (2 c¯0 )R (M, q), 2 `∞ (M, q) and 2 `¯∞ (M, q) are solid and hence are monotone. Proof. Let < αnk > be a double sequence of scalars such that |α nk | ≤ 1, for all n, k ∈ N .


217

Then the proof for 2 c¯0 (M, q), (2 c¯0 )BR (M, q), (2 c¯0 )B (M, q), (2 c¯0 )R (M, q), ¯ 2 `∞ (M, q) and 2 `∞ (M, q) are obvious in view of the following inequality, M (q(

αnk ank ank )) ≤ M (q( )), for all (n, k) ∈ K. ρ ρ

The rest of the proof of first part follows from Remark 1. Proposition 5. The spaces Z(M, q), where Z = 2 c¯, 2 c¯0 , (2 c¯)B , (2 c¯0 )B , (2 c¯)R , (2 c¯0 )R , (2 c¯)BR , (2 c¯0 )BR , 2 `¯∞ are not symmetric. The above result follows from the following examples:

xp ,

Example 1. Let X = c, the class of convergent single sequences and M (x) = p ≥ 1 and q(x) = supi |xi |, for x = (xi ) ∈ c. Define < ank > by ank =

  e, 

θ,

for all n = k ∈ N, otherwise.

Let < bnk > be a rearrangement of < ank >, defined as bnk =

  e, 

θ,

for all k even and all n ∈ N, otherwise.

Then < bnk >∈ / Z(M, q), but < ank >∈ Z(M, q), for Z =

¯, 2 c¯0 , 2c

(2 c¯)R ,

(2 c¯0 )R . Example 2. Let X = `∞ , M (x) = xp , p ≥ 1 and q(x) = supi |xi |, for x = (xi ) ∈ `∞ . Define the sequence < ank > by ank =

  ie, 

θ,

for k = i2 , i ∈ N and n even, otherwise.

Let < bnk > be a rearrangement of < ank >, defined as follows bnk =

  k+1 e, 2

 θ,

if k is odd, and for all n ∈ N, otherwise.

/ 2 `¯∞ (M, q). Then < ank >∈ 2 `¯∞ (M, q), but < bnk >∈ Hence 2 `¯∞ (M, q) is not symmetric.

218


Theorem 6. Let M and M1 be two Orlicz functions, then Z(M1 , q) ⊆ Z(M ◦ M1 , q), for Z = 2 c¯, 2 c¯0 , (2 c¯)B , (2 c¯0 )B , (2 c¯)R , (2 c¯0 )R , (2 c¯)BR , (2 c¯0 )BR and ¯ 2 `∞ (M, q). Proof. We prove it for the case Z = 2 c¯0 , the other cases can be proved following similar technique. Let > 0 be given. Since M is continuous, so there exists η > 0 such that M (η) = . Let < ank >∈ 2 c¯0 (M1 , q). Then there exists a subset K ⊂ N × N with ρ(K) = 1 such that ank )) < η, for all (n, k) ∈ K. r ank )) < . ⇒ M ◦ M1 (q( r M1 (q(

Hence < ank >∈ 2 c¯0 (M ◦ M1 , q). Thus 2 c¯0 (M1 , q) ⊂ 2 c¯0 (M ◦ M1 , q). Theorem 7. If M and M1 are two Orlicz functions then Z(M1 , q) ∩ Z(M2 , q) ⊆ Z(M1 + M2 , q), for Z = 2 c¯, 2 c¯0 , (2 c¯)B , (2 c¯0 )B , (2 c¯)R , (2 c¯0 )R , (2 c¯)BR , (2 c¯0 )BR , 2 `¯∞ . Proof. We prove the result for 2 c¯(M, q). The other cases can be established following similar technique. Let < ank >∈ 2 c¯(M1 , q) ∩ 2 c¯(M2 , q). Let > 0 be given. Then there exist subsets K and D of N × N such that ρ(K) = ρ(D) = 1 and M1 (q(

ank − L )) < , for all (n, k) ∈ K, for some r1 > 0, r1 2

M2 (q(

ank − L )) < , for all (n, k) ∈ D, for some r2 > 0. r2 2

and

Let r = max{r1 , r2 }. Then for all (n, k) ∈ K ∩ D we have (M1 + M2 )(q(

ank − L ank − L ank − L )) ≤ M1 (q( )) + M2 (q( )) < . r r1 r2

Hence < ank >∈ 2 c¯(M1 + M2 , q). This completes the proof. The proof of the following result is a consequence of Theorem 6.


219

Corollary 8. Let M be an Orlicz function then we have Z(q) ⊂ Z(M, q), for Z = 2 c¯, 2 c¯0 , (2 c¯)B , (2 c¯0 )B , (2 c¯)R , (2 c¯0 )R , (2 c¯)BR , (2 c¯0 )BR and 2 `¯∞ . The proof of the following result is a routine work. Theorem 9. Let M be an Orlicz function, q 1 and q2 be seminorms. Then (i) Z(M, q1 ) ∩ Z(M, q2 ) ⊆ Z(M, q1 + q2 ). (ii) If q1 is stronger than q2 , then Z(M, q1 ) ⊂ Z(M, q2 ), for Z = 2 c¯, 2 c¯0 , (2 c¯)B , (2 c¯0 )B , (2 c¯)R , (2 c¯0 )R , (2 c¯)BR , (2 c¯0 )BR and 2 `¯∞ . The following result can be proved by using standard technique. Theorem 10. Let M be an Orlicz function. Then Z(M, q) ⊂ for Z = (2 c¯)B , (2 c¯0 )B , (2 c¯)BR , (2 c¯0 )BR and the inclusions are strict.

2 `∞ (M, q),

The following result is a consequence of the above Theorem and Theorem 3. Corollary 11. The spaces Z(M, q) for Z = ( 2 c¯)B , (2 c¯0 )B , (2 c¯)BR , (2 c¯0 )BR are nowhere dense subset of 2 `∞ . Theorem 12. The spaces Z(M, q) where Z = 2 c¯, 2 c¯0 , (2 c¯)B , (2 c¯0 )B , (2 c¯)R , (2 c¯0 )R , (2 c¯)BR , (2 c¯0 )BR and 2 `¯∞ are not convergence free. The above result is clear from the following example. Example 3. Let M (x) = xp , for some 1 ≤ p < ∞, X = `∞ , q(x) = supi |xi |, for x = (xi ) ∈ `∞ . Then the double sequence < ank > defined as ank = (k −1 , k −1 , k −1 , . . .), for all n, k ∈ N belongs to all the spaces. Consider the sequence < bnk > defined as bnk = (k, k, k, . . .), for all n, k ∈ N . Then < b nk > does not belong to any of these spaces. Hence none of the spaces is convergence free. Proposition 13. The spaces Z(M, q), for Z = 2 c¯, (2 c¯)B , (2 c¯)R , (2 c¯)BR are not monotone and hence are not solid. Proof. The first part follows from the following example. The second part follows from Remark 1. Example 4. Let M (x) = x, X = C and q(x) = |x|. Then the double

220


sequence < ank >, defined by ank = 1, for all n, k ∈ N belong to Z(M, q), for Z = 2 c¯, (2 c¯)B , (2 c¯)R , (2 c¯)BR . Consider its pre-image < bnk > defined as bnk =

  0, 

1,

for k even, for all n ∈ N, for k odd, for all n ∈ N.

Then < bnk > does not belong to any of these spaces. Remark 2. Let (X, k.k) be a normed linear space. Then the spaces Z(M, k.k), for Z = (2 c¯)B , (2 c¯0 )B , (2 c¯)BR , (2 c¯0 )BR and 2 `∞ will be normed linear spaces normed by f (< ank >) = inf{ρ > 0 : sup M (k( n,k

ank )k) ≤ 1}. ρ

Remark 3. If X is a Banach space then it is clear that the spaces Z(M, k.k), for Z = (2 c¯)B , (2 c¯0 )B , (2 c¯)BR , (2 c¯0 )BR and 2 `∞ are Banach spaces under the norm f. References [1] M. Basarir and O. Solankan, On some double sequence spaces, J Indian Acad. Math., 21:2 (1999), 193-200. [2] V. K. Bhardwaj and N. Singh, Some sequences defined by Orlicz functions, Demonstratio Math., 33:3(2000), 571-582. [3] T. J. IA Bromwich, An Introduction to the Theory of Infinite Series, Macmillan and Co. Ltd., New york, 1965. [4] A. Esi, Some new sequence spaces defined by Orlicz functions, Bull. Inst. Math. Acad. Sinica., 27:1(1999), 71-76. [5] A. Esi and M. Et, Some new sequence spaces defined by a sequence of Orlicz functions, Indian J. Pure Appl. Math, 31:8(2000), 967-972. [6] M. Et, On some new Orlicz sequence spaces, J. Analysis, 9(2001), 21-28. [7] H. Fast, Sur la convergence statistique, Colloq. Math., 2(1951), 241-244. [8] J. A. Fridy and C. Orhan, Statistical limit superior and limit inferior, Proc. Amer. Math. Soc., 125:12(1997), 3625-3631. [9] G. H. Hardy, On the convergence of certain multiple series, Proc. Camb. Phil. Soc., 19 (1917), 86-95. [10] I. J. Maddox, A Tauberion condition for statistical convergence, Math. Proc. Camb. Phil. Soc., (1989), 272-280. [11] F. Moricz, Extension of the spaces c and c0 from single to double sequences, Acta. Math. Hung., 57:1-2(1991), 129-136.


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[12] F. Morciz and B. E. Rhoades, Almost convergence of double sequences and strong regularity of summability matrices, Math. Proc. Camb. Phil. Soc., 104(1988), 283-294. [13] S. D. Parashar and B. Choudhary, Sequence spaces defined by Orlicz functions, Indian J. Pure Appl. Math., 25:4(1994), 419-428. [14] R. F. Patterson, Analogues of some fundamental theorems of summability theory, Internet. J. Math. & Math. Sci., 23:1(2000), 1-9. [15] D. Rath and B. C. Tripathy, Matrix maps on sequence spaces associated with sets of integers, Indian J. Pure Appl. Math., 27:2(1996), 197-206. [16] T. Salat, On statistically convergent sequences of real numbers, Math. Slovaca, 30:2(1980), 139-150. [17] I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 (1959), 361-375. [18] B. C. Tripathy, Matrix transformation between some class of sequences, J. Math. Anal. Appl., 206:2(1997), 448-450. [19] B. C. Tripathy, On statistical convergence, Proc. Estonian Acad. Sci., Phy. Math., 47:4 (1998), 299-303. [20] B. C. Tripathy, Statistically convergent double sequences, Tamkang J. Math., 34:3(2003), 231-237. [21] A. Zygumd, Trigonometric Series, vol II, Cambridge, 1993. Mathematical Sciences Division, Institute of Advanced Study in Science and Technology, Paschim Boragaon, GARCHUK, GUWAHATI -781 035, India. E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]

statistically convergent double sequence spaces ... - Semantic Scholar

statistically convergent double sequence spaces ... - Semantic Scholar

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