IJRRAS 7 (2) ● May 2011
www.arpapress.com/Volumes/Vol7Issue2/IJRRAS_7_2_01.pdf
ON SOME GENERALIZED DIFFERENCE SEQUENCE SPACES DEFINED BY A SEQUENCE OF MODULI Vakeel A. Khan1,* & Ayaz Ahmad2 Department of Mathematics, A.M.U. Aligarh (INDIA) 2 National Institute of Technology , Patna ,Bihar (INDIA) * Email :
[email protected] 1
ABSTRACT In this paper we define some generalized sequenc spaces defined by a sequence of moduli. The results here in proved are analogous to those by ASMA BEKTAS Cigdem (2006)[Journal of Zhejiang University Science A (2006),7(12) 2093-2096] . Keywords: Difference sequence space , Sequence of moduli, Strongly almost convergent . 1.
INTRODUCTION Let
null sequences
0
be the set of all complex sequences and
l , c and c0 be the sets of all bounded, convergent and
x = ( xk ) with complex terms, respectively , normed by || x || = sup | xk |, where k I
N = {1,2,}.
k
The idea of difference sequence space was introduced by Kizmaz (1981). In 1981, Kizmaz defined the sequence spaces : 1
l () = {x = ( xk ) 0 : (xk ) l }, c() = {x = ( xk ) 0 : (xk ) c}, c0 () = {x = ( xk ) 0 : (xk ) c0 },
and where
x = ( xk xk 1 ) . These are Banach spaces with the norm
|| x || =| x1 | || x || . After then R. Colak and M. Et (1995) defined the sequence spaces :
l ( m ) = {x = ( xk ) 0 : ( m xk ) l }, c( m ) = {x = ( xk ) 0 : ( m xk ) c}, c0 ( m ) = {x = ( xk ) 0 : ( m xk ) c0 },
and where m I
N, 0 x = ( xk ),
x = ( xk xk 1 ), m x = ( m1 xk m1 xk 1 ), and so that m xk = (1) i xk i . i =0 and show that these are Banach spaces with the norm m
m
i
m
|| x || = | xi | || m x || . i =1
1
AMS Subject Classification 2000: 40C05, 46A45.
106
IJRRAS 7 (2) ● May 2011
Khan & Ahmad ● Generalized Difference Sequence Spaces
Definition 1.1. A function f : [0, ) [0, ) is called a modulus if 1.
f (t ) = 0 if and only if t = 0, f (t u) f (t ) f (u) for all t , u 0,
2. 3. f is increasing, and 4. f is continuous from the right of 0. Let X be a sequence space. Then the sequence space X ( f ) is defined as
X ( f ) = {x = ( xk ) 0 : ( f (| xk |)) X } for a modulus f(Maddox 1986 and Ruckle 1973). Kolk (1993, 1994) gave an extension of X ( f ) by considering a sequence of moduli
F = ( f k ) i.e.
X ( F ) = {x = ( xk ) : ( f k (| xk |)) X }. 0
A sequence x l is said to be almost convergent (Lorentz, 1984) if all Banach limits of Lorentz(1984) proved that
x coincide.
1 n xk s , uniformly in s}. n n k =1 Maddox (1967; 1978) has defined x to be strongly almost convergent to L if 1 n lim | xk s L |= 0, uniformly in s, for some L > 0. n n k =1 Let p = ( pk ) be a sequence of strictly positive real numbers . Nanda (1984) defined cˆ:= {x = ( xk ) 0 : lim
[cˆ, p] := {x = ( xk ) 0 : lim n
1 n p | xk s L | k = 0, uniformly in s, for some L > 0}, n k =1
1 n p | xk s | k = 0, uniformly in s,}, n n k =1 1 n p [cˆ, p] := {x = ( xk ) 0 : sup | xk s L | k = 0, uniformly in s,}. n n k =1 [cˆ, p]0 := {x = ( xk ) 0 : lim
2.
MAIN RESULTS Let F = ( f k ) be a sequence of moduli,
u = (uk ) be any sequence such that uk = 0 for all k and
p = ( pk ) be any sequence space of strictly positive real numbers then we define the following sequence spaces : [cˆ, F , p]( um ) := {x = ( xk ) 0 : lim n
If
1 n p [ f k (| uk m xk s L |)] k = 0, uniformly in s n k =1
, for some L> 0}, 1 n p [cˆ, F , p]0 ( um ) := {x = ( xk ) 0 : lim [ f k (| uk m xk s |)] k = 0, uniformly in s}, n n k =1 1 n p [cˆ, F , p] ( um ) := {x = ( xk ) 0 : sup [ f k (| uk m xk s |)] k < , uniformly in s}, s , n n k =1 f k ( x) = x for every k , then [cˆ, F , p](um ) = [cˆ, p] , [cˆ, F , p]0 (um ) = [cˆ, p]0 (um )
and
[cˆ, F , p] ( ) = [cˆ, p] ( ) .We denote [cˆ, F , p]( ) , [cˆ, F , p]0 ( ) and [cˆ, F , p] ( ) by m u
m u
m u
[cˆ, F ](um ) , [cˆ, F ]0 (um ) and [cˆ, F ] (um ) , when pk = 1 for all k .
107
m u
m u
IJRRAS 7 (2) ● May 2011
Khan & Ahmad ● Generalized Difference Sequence Spaces
Theorem 2.1. For a sequence
F = ( f k ) of moduli , the following statements are equivalent:
1. [cˆ, p] ( ) [cˆ, F , p] (u ), m u
m
2. [cˆ, p]0 (u ) [cˆ, F , p]0 (u ), m
3. sup n
m
1 n p [ f k (t )] k < n k =1
(t > 0).
Proof. (i) (ii) is clear. (ii)
(iii): Let [cˆ, p]0 (um ) [cˆ, F , p]0 (um ) . Suppose that (iii) is not true. Then for some t 1 n p sup [ f k (t )] k = n n k =1
and there exists a sequence
(ni ) of positive integers such that 1 ni
Now we define
.
ni
f
k
k =1
1 ( ) > i for i = 1,2,(1) i
x = {xk } by 1 xk = i 0
, if
1 k ni ,
i = 1,2, ,
, (k > ni ).
Then x [cˆ, p]0 ( ) but by Eqn (1), x [cˆ, F , p] (u ) which contradicts (ii). m u
m
Hence (iii) is true. (iii) (i) : Let (iii) is true and x [cˆ, p] (u ). If we suppose that x [cˆ, F , p] (u ) . Then m
m
1 n p sup [ f k (| uk m xk s |)] k = .(2) s , n n k =1 m If we take t =| uk xk s | for each k and fixed s , then by Eqn(2) 1 n p [ f k (t )] k = , n n k =1 m m which contradicts (iii). Hence [cˆ, p] (u ) [cˆ, F , p] (u ) . sup
Theorem 2.2. Let
1 pk sup pk < . For a sequence of moduli F = ( f k ) the following statements are k
equivalent: 1. [cˆ, F , p]0 (u ) [cˆ, p]0 (u ), m
m
2. [cˆ, F , p]0 (u ) [cˆ, p] (u ), m
3. inf n
m
1 n p [ f k (t )] k > 0 n k =1
(t > 0).
Proof. (i) (ii) is clear. (ii)
(iii): Let [cˆ, F , p]0 (um ) [cˆ, p] (um ). Suppose that (iii) does not hold. Then 1 n p inf [ f k (t )] k = 0 (t > 0), (3) n n k =1
and there exists a sequence
(ni ) of positive integers such that 108
IJRRAS 7 (2) ● May 2011
Khan & Ahmad ● Generalized Difference Sequence Spaces
1 ni Now define the sequence
ni
[ f
k
(i)]
pk
k =1
1 < , for i = 1,2,. i
x = {xk } by
i , if 1 k ni , for i = 1,2,, xk = 0 , k > ni . m m By Eqn.(3), x [cˆ, F , p]0 (u ) but x [cˆ, p] (u ), which contradicts (ii). Hence (iii) is true. (iii) (i) : Let (iii) is true and x [cˆ, F , p]0 (u ) i.e. m
1 n p [ f k (| uk m xk s |)] k = 0, uniformly in s}.(4) n n k =1 m Suppose that x [cˆ, p]0 ( u ) .Then for some number 0 > 0 and positive integer n0 we have lim
| uk m xk s | 0
for some
s s and 1 k n0 . Therefore [ f k ( 0 )]
n
and hence lim n
[ f
k
( 0 )]
pk
pk
[ f k (| uk m xk s |)]
pk
= 0 , which contradicts (iii). Thus [cˆ, F , p]0 (um ) [cˆ, p]0 (um ).
k =1
Theorem 2.3. Let
1 pk sup pk < . The inclusion [cˆ, F , p] (um ) [cˆ, p]0 (um ), holds if and only if k
1 n p lim [ f k (t )] k = for t > 0.(5) n n k =1 m m Proof. Let [cˆ, F , p] (u ) [cˆ, p]0 (u ), . Suppose Eqn(5) does not hold. Then there exists a number t0 > 0 and a sequence
(ni ) of positive integers such that n
1 i p [ f k (t0 )] k M < , i = 1,2,.(6) ni k =1 Noe we define the sequence x = ( xk ) by
t0 , if 1 k ni for i = 1,2,, xk = 0 , k > ni . m m Thus by Eqn (6), x [cˆ, F , p] (u ) but x [cˆ, p]0 ( u ) .So that Eqn (5) must hold. Conversely let Eqn (5) hold. If x [cˆ, F , p] (u ) , then for each m
s and n
1 n p [ f k (| uk m xk |)] k M < .(7) n k =1 m Suppose that x [cˆ, p]0 ( u ) . Then for some number 0 > 0 and positive integer s0 and index n0 we have | uk m xk s | 0
for
s s0 . Therefore
[ f k ( 0 )] and hence for each
pk
[ f k (| uk m xk s |)] k , p
k and s we get 109
IJRRAS 7 (2) ● May 2011
Khan & Ahmad ● Generalized Difference Sequence Spaces
1 n p [ f k ( 0 )] k M < , n k =1
M > 0 , by Eqn (7) which contradicts Eqn (5). Hence [cˆ, F , p] (um ) [cˆ, p]0 (um )
for some
. Theorem 2.4. Let
1 pk sup pk < . The inclusion [cˆ, p] (um ) [cˆ, F , p]0 (um ), holds if and only if k
1 n p [ f k (t0 )] k = 0 for t > 0.(8) n n k =1 m m Proof. Suppose that [cˆ, p] (u ) [cˆ, F , p]0 (u ), but (8) does not hold. Then for some t0 > 0 lim
1 n p lim [ f k (t0 )] k = L = 0.(9) n n k =1 Define the sequence
x = ( xk ) by
m k 1 xk = t0 (1) m k =0 m for k = 1,2,. Then x c0 ( F , p, v ) , by Eqn (6).Hence Eqn (5) must hold. k
Conversely let x [cˆ, p] ( u ) , and suppose that Eqn (8) holds. Then for every m
k and s
| uk m xk s | M < . Therefore
[ f k (| uk m xk s |)]
pk
[ f k (M )] k , p
and
1 n 1 n pk p m [ f (| u x |)] ) lim k k lim [ f k ( M )] k = 0, by Eqn(8). k s n n k =1 n n k =1 m Hence x [cˆ, F , p]0 (u ) . 3. REFERENCES [1]. Çi g dem . A. B., 2006. On some difference sequence spaces defined by a sequence of orlicz functions,Journal of Zhejiang University Science A .,7(12) :2093-2096. [2]. Colak, R., and M. Et,1995. On some generalized difference sequence spaces , Soochow J. Math., 21(4): 377-386. [3]. Kizmaz,H., 1981. on certain sequence spaces Candian Math. Bull.,24(2): 169-176. [4]. Kolk,E., 1993. on strong boundedness and summmability with respect to a sequence of moduli, Acta Comment. Univ. Tartu,960 :41-50 [5]. Kolk,E., 1994. Inclusion theorems for some sequence spaces defined by a sequence of moduli, Acta Comment. Univ. Tartu,970 :65-72 [6]. Maddox, I.J.,1978. A new type convergence, Math.Proc.Camp.Phil.Soc., 83: 61-64. [7]. Maddox, I.J.,1986. Sequence spaces defined by a modulus, Math. Camb. Phil. Soc., 100, 161-166. [8]. Maddox,I.J.,1967. Spaces of strongly summable sequences. Quart. J. Math., 18 : 61 - 64. [9]. Ruckle,W.H., 1973. FK spaces in which the sequence of coordinate vectors is bounded , Canad. J.Math.,25,973-975. [10]. Lorentz, G.G., 1984. A contribution to the theory of divergent sequences. Acta Mathematica, 80 (1) : 167 - 190. [11]. Nanda , S., 1984. Strongly almost convergent sequences. Bull. Cal Math.Soc., 76 : 236 - 240.
110
