IJANAL International Journal of Analysis 2314-4998 2314-498X Hindawi Publishing Corporation 684512 10.1155/2014/684512 684512 Research Article A New Double Sequence Space m 2 ( F , ϕ , p ) Defined by a Double Sequence of Modulus Functions Sağır Birsen Duyar Cenap Oğur Oğuz Qamar Shamsul Department of Mathematics Art and Science Faculty Ondokuz Mayıs University Kurupelit Campus, 55139 Samsun Turkey omu.edu.tr 2014 16 1 2014 2014 01 11 2013 08 01 2014 09 01 2014 11 2 2014 2014 Copyright © 2014 Birsen Sağır et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this work we introduce new spaces m 2 ( F , ϕ , p ) of double sequences defined by a double sequence of modulus functions, and we study some properties of this space.

1. Introduction

In this work, by w and w 2 , we denote the spaces of single complex sequences and double complex sequences, respectively. and denote the set of positive integers and complex numbers, respectively. If, for all ɛ > 0 , there is n ɛ such that x k , l - a X < ɛ where k > n ɛ and l > n ɛ , then a double sequence { x k , l } is said to be converge (in terms of Pringsheim) to a . A real double sequence { x k , l } is nondecreasing, if x k , l x p , q for ( k , l ) < ( p , q ) . A double series is infinity sum k , l = 1 x k , l and its convergence implies the convergence by | · | of partial sums sequence { S n , m } , where S n , m = k = 1 n l = 1 m x k , l (see ).

For 1 p < , p ( 2 ) denote the space of sequences x = { x k , l } such that (1) k , l = 1 | x k , l | p < . (see ).

A double sequence x = { x k , l } is said to be bounded if and only if sup k , l | x k , l | < . The space of all bounded double sequences is denoted by ( 2 ) . It is known that ( 2 ) is Banach space (see [5, 6]).

A double sequence space E is said to be normal if ( y k l ) E whenever | y k l | | x k l | for all k , l and ( x k l ) E .

The double sequence spaces in the various forms were introduced and studies by Khan and Tabassum in , by Khan in , and by Khan et al. in [16, 17].

Now let φ s be a family of subsets σ having most elements s in . Also φ s , t denote the class of subsets σ = σ 1 × σ 2 in × such that the elements of σ 1 and σ 2 are most s and t , respectively. Besides { ϕ k , l } is taken as a nondecreasing double sequence of the positive real numbers such that (2) k ϕ k + 1 , l ( k + 1 ) ϕ k , l , l ϕ k , l + 1 ( l + 1 ) ϕ k , l . (see ).

Let x = { x k , l } be a double sequence. A set S ( x ) is defined by (3) S ( x ) = { { x π 1 ( k ) , π 2 ( k ) } : π 1    and    π 2    are    permutations    of    } . A double sequence space E is said to be symmetric if u = ( u k l ) E and u = x whenever x = ( x k l ) E and u S ( x ) .

A BK-space is a Banach sequence space E in which the coordinate maps are continuous.

A function f : [ 0 , ) [ 0 , ) is said to be a modulus function if it satisfies the following:

f ( x ) = 0 if and only if x = 0 ;

f ( x + y ) f ( x ) + f ( y ) for all x , y [ 0 , ) ;

f is increasing;

f is continuous from right at 0 .

It follows that f is continuous on [ 0 , ) . The modulus function may be bounded or unbounded. For example, if we take f ( x ) = x / ( x + 1 ) , then f ( x ) is bounded. But, for 0 < p < 1 , f ( x ) = x p is not bounded.

The BK-spaces m ( ϕ ) , introduced by Sargent in , is in the form (4) m ( ϕ ) = { x = { x k } w : x m ( ϕ ) = sup s 1 , σ φ s 1 ϕ s k σ | x k | < } .

Sargent studied some properties of this space and examined relationship between this space and l p -space.

The space m ( ϕ ) was extended to m ( ϕ , p ) by Tripathy and Sen  as follows: (5) m ( ϕ , p ) = { ( k σ | x k | p ) 1 / p x = { x k } w : x m ( ϕ , p ) w = sup s 1 , σ φ s 1 ϕ s ( k σ | x k | p ) 1 / p < } . Recently, Raj et al.  introduced and studied the following sequence space m ( F , ϕ , p ) .

Let F = ( f k ) be a sequence of modulus functions. Then the space m ( F , ϕ , p ) is defined by (6) m ( F , ϕ , p ) = { x = { x k } w : sup s 1 , σ φ s 1 ϕ s ( k σ [ f k ( | x k | ρ ) ] p ) 1 / p w w e w w < , for    some    ρ > 0 ( k σ [ f k ( | x k | ρ ) ] p ) 1 / p } .

In this work we introduce sequence spaces m 2 ( F , ϕ , p ) defined by (7) m 2 ( F , ϕ , p ) = { { i σ 1 j σ 2 [ f i , j ( | x i , j | ρ ) ] p } x = ( x k l ) w 2 : ( ew ) sup ( s , t ) ( 1,1 ) sup σ 1 × σ 2 φ s t 1 ϕ s t { i σ 1 j σ 2 [ f i , j ( | x i , j | ρ ) ] p } 1 / p = ( e ) <    for    ρ > 0 { i σ 1 j σ 2 [ f i , j ( | x i , j | ρ ) ] p } } , where F = ( f i , j ) is a double sequence of modulus functions.

2. Main Results

The result stated in the first theorem is not hard. So, we give it without proof.

Theorem 1.

The sequence space m 2 ( F , ϕ , p ) is a linear space.

Theorem 2.

The sequence spaces m 2 ( F , ϕ , p ) are complete.

Proof.

Let { x ( i ) } be a double Cauchy sequence in m 2 ( F , ϕ , p ) such that x ( i ) = { x k , l ( i ) } k , l = 1 for all i . Then (8) sup ( s , t ) ( 1,1 ) sup σ 1 × σ 2 φ s t 1 ϕ s t { k σ 1 l σ 2 [ f k , l ( | x k , l ( i ) | ρ ) ] p } 1 / p < for some ρ > 0 and for all i . For each ɛ > 0 , there exists a positive integer n 0 such that (9) x ( i ) - x ( j ) m 2 ( F , ϕ , p ) < ɛ for all i , j n 0 . Hence (10) sup ( s , t ) ( 1,1 ) sup σ 1 × σ 2 φ s t 1 ϕ s t { k σ 1 l σ 2 [ f k , l ( | x k , l ( i ) - x k , l ( j ) | ρ ) ] p } 1 / p < ɛ for some ρ > 0 and for all i , j n 0 . This implies that (11) | x k , l ( i ) - x k , l ( j ) | < ε ϕ 1,1 for all i , j n 0 and for each fixed ( k , l ) × . Hence { x ( i ) } is a Cauchy sequence in .

Then, there exists x k , l such that x k , l ( i ) x k , l as i and let us define x = ( x k , l ) . From (10), for each fixed ( s , t ) , (12) { k σ 1 l σ 2 [ f k , l ( | x k , l ( i ) - x k , l ( j ) | ρ ) ] p } < ε p ϕ s t p for some ρ > 0 , for all i , j n 0 and σ 1 × σ 2 φ s t .

Letting j , we get (13) { k σ 1 l σ 2 [ f k , l ( | x k , l ( i ) - x k , l | ρ ) ] p } < ε p ϕ s t p for some ρ > 0 , for all i , j n 0 , and σ 1 × σ 2 φ s t . Thus we obtain (14) sup ( s , t ) ( 1,1 ) sup σ 1 × σ 2 φ s t 1 ϕ s t       { k σ 1 l σ 2 [ f k , l ( | x k , l ( i ) - x k , l | ρ ) ] p } 1 / p < ɛ for some ρ > 0 and for all i , j n 0 . This implies that x ( i ) - x m 2 ( F , ϕ , p ) for all i , j n 0 .

Hence x = x ( n 0 ) + x - x ( n 0 ) m 2 ( F , ϕ , p ) . By (14), (15) x ( i ) - x m 2 ( F , ϕ , p ) < ɛ for all i n 0 . This means that x ( i ) x as i . Hence m 2 ( F , ϕ , p ) is a Banach space.

Theorem 3.

The space m 2 ( F , ϕ , p ) is a BK-space.

Proof.

Suppose that { x ( i ) } m 2 ( F , ϕ , p ) with x ( i ) - x m 2 ( F , ϕ , p ) 0 as i . For each ɛ > 0 there exists n 0 N such that (16) x ( i ) - x m 2 ( F , ϕ , p ) < ɛ for all i n 0 . Thus (17) sup ( s , t ) ( 1,1 ) sup σ 1 × σ 2 φ s t 1 ϕ s t { k σ 1 l σ 2 [ f k , l ( | x k , l ( i ) - x k , l | ρ ) ] p } 1 / p < ɛ for some ρ > 0 and for all i n 0 . Hence we obtain (18) | x k , l ( i ) - x k , l | < ε ϕ 1,1 for all i n 0 and for all ( k , l ) × . This implies | x k , l ( i ) - x k , l | 0 as i . This completes the proof.

Corollary 4.

The space m 2 ( F , ϕ , p ) is a symmetric space.

Proof.

Let x = { x k , l } m 2 ( F , ϕ , p ) and let y = { y k , l } S ( x ) . Then we can write y k , l = x m k , m l . Thus we obtain (19) x m 2 ( F , ϕ , p ) = y m 2 ( F , ϕ , p ) .

Corollary 5.

The space m 2 ( F , ϕ , p ) is a normal space.

Proof.

It is obvious.

Theorem 6.

Consider (20) m 2 ( ϕ ) m 2 ( F , ϕ , p ) .

Proof.

Suppose that x m 2 ( ϕ ) . Then we have (21) x m 2 ( ϕ ) = sup ( s , t ) ( 1,1 ) sup σ 1 × σ 2 φ s t 1 ϕ s t { k σ 1 l σ 2 | x k , l | } x m 2 ( ϕ ) = K < . Thus for each fixed ( s , t ) and for σ 1 × σ 2 φ s t , (22) k σ 1 l σ 2 | x k , l | K ϕ s t for some ρ > 0 . Hence (23) sup ( s , t ) ( 1,1 ) sup σ 1 × σ 2 φ s t 1 ϕ s t { k σ 1 l σ 2 [ f k , l ( | x k , l | ρ ) ] p } 1 / p K for some ρ > 0 . This implies that x m 2 ( F , ϕ , p ) . Hence m 2 ( ϕ ) m 2 ( F , ϕ , p ) .

Theorem 7.

m 2 ( F , ϕ , p ) m 2 ( F , ψ , p ) if and only if sup ( s , t ) ( 1,1 ) ( ϕ s t / ψ s t ) < .

Proof.

Let K = sup ( s , t ) ( 1,1 ) ( ϕ s t / ψ s t ) < . Then ϕ s t K ψ s t for all ( s , t ) ( 1,1 ) . If x m 2 ( F , ϕ , p ) , then (24) sup ( s , t ) ( 1,1 ) sup σ 1 × σ 2 φ s t 1 ϕ s t { k σ 1 l σ 2 [ f k , l ( | x k , l | ρ ) ] p } 1 / p < for some ρ > 0 . Thus (25) sup ( s , t ) ( 1,1 ) sup σ 1 × σ 2 φ s t 1 K ψ s t { k σ 1 l σ 2 [ f k , l ( | x k , l | ρ ) ] p } 1 / p < for some ρ > 0 . Hence x m 2 ( F , ψ , p ) . This shows that m 2 ( F , ϕ , p ) m 2 ( F , ψ , p ) . Conversely, let m 2 ( F , ϕ , p ) m 2 ( F , ψ , p ) . We define γ s , t = ϕ s t / ψ s t . Let sup ( s , t ) ( 1,1 ) γ s , t = . Then there exists a subsequence { γ s i , t i } of { γ s , t } such that γ s i , t i as i . Then for x m 2 ( F , ϕ , p ) we have (26) sup ( s , t ) ( 1,1 ) sup σ 1 × σ 2 φ s t 1 ψ s t { k σ 1 l σ 2 [ f k , l ( | x k , l | ρ ) ] p } 1 / p sup ( s , t ) ( 1,1 ) sup σ 1 × σ 2 φ s t γ s , t ϕ s t { k σ 1 l σ 2 [ f k , l ( | x k , l | ρ ) ] p } 1 / p = for some ρ > 0 . This is a contradiction as x m 2 ( F , ψ , p ) and this completes the proof.

Proposition 8.

Consider (27) p ( 2 ) m 2 ( F , ϕ , p ) ( 2 ) .

Proof.

Clearly, p ( 2 ) = m 2 ( F , ψ , p ) , where ψ s t = 1 for s , t = 1,2 , when f k , l ( x ) = x and sup ( s , t ) ( 1,1 ) ( ψ s t / ϕ s t ) < by nondecreasing ( ϕ s t ) . Then, by Theorem 7, first inclusion is obtained. Suppose x m 2 ( F , ϕ , p ) . Then (28) sup ( s , t ) ( 1,1 ) sup σ 1 × σ 2 φ s t 1 ϕ s t { k σ 1 l σ 2 [ f k , l ( | x k , l | ρ ) ] p } 1 / p = K < for some ρ > 0 . Hence we obtain (29) | x k l | K ϕ 1,1 for all k , l . Thus x ( 2 ) and proof is completed.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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