On Generalized I Convergent Paranormed Spaces

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1 Math Sci Lett 4, No 2, (2015) 165 Mathematical Sciences Letters An International Journal On Generalized I Convergent Paranormed Spaces Kuldip Raj and Seema Jamwal School of Mathematics, Shri Mata Vaishno Devi University Katra , J&K, India Received: 2 Feb 2014, Revised: 5 Apr 2014, Accepted: 9 Apr 2014 Published online: 1 May 2015 Abstract: In the present paper we introduce some generalized I convergent sequence spaces and study some topological and algebraic properties of these spaces We also make an effort to study some inclusion relations between these spaces Keywords: double sequence, σ mean, σ bounded variation, ideal convergence, paranorm 1 Introduction and Preliminaries Let w denote the space of all real or complex sequences A double sequence of complex numbers is defined as a function x : N N C We denote a double sequence as (x ) where the two subscripts run through the sequence of natural numbers independent of each other A number a C is called a double limit of a double sequence(x ) if for every > 0 there exists some N = N() N such that (x ) a <, i, j N The study of double sequence spaces was initiated by Bromwich [2] and further generalized and studied by Hardy [6], Moricz [15], Moricz and Rhoades [16], Tripathy ([27], [28]), Başarir and Sonalcan [4] and many others Quite recently, Zeltser [1] in her PhD thesis has essentially studied both the theory of topological double sequence spaces and the theory of summability of double sequences For more details about double sequence spaces (see [20], [17],[18]) and references therein Let l and c denote the Banach spaces of bounded and convergent sequences, respectively, with norm x = sup x k Let V denote the space of sequences of k bounded variation that is, V = x=(x k ) : where V is a Banach space normed by x = x k x k 1 <, x 1 = 0, x k x k 1, (see [19]) Let σ be a mapping of the set of the positive integers into itself having no finite orbits A continuous linear functional φ on l is said to be an invariant mean or σ-mean if and only if (i) φ(x) 0 when the sequence x=(x k ) has x k 0 for all k; (ii) φ(e)=1, where e=1,1,1,; (ii) φ(x σ(n) )=φ(x) for all x l In case σ is the translation mapping n n + 1, a σ mean is often called a Banach limit (see []) and V σ the set of bounded sequences all of whose invariant means are equal, is the set of almost convergent sequences (see [14]) If x = (x k ), then Tx=(T x k )=(x σ(n) ) It can be shown that V σ = x=(x k ) : m=1 where m 0, k>0 Consider t m,k (x)=l uniformally in k L=σ limx t m,k (x)= x k+ x σ(k) + x σ 2 (k) ++x σ m (k), t 1,k = 0, m+1 where σ m (k) denote the mth iterate of σ(k) at k The special case of (1) in which σ(n) = n+1 was given by Lorentz [[14], Theorem 1], and that the general result can be proved in a similar way It is familiar that a Banach limit extends the limit functional on c A σ-mean extends the limit functional on c in the sense that φ(x)=limx for all x cif and only if σ has no finite orbits that is to say, if and only if, for all k 0, j 1, (see [19]) σ j (k) k (1) Corresponding author kuldipraj68@gmailcom

2 166 K Raj, S Jamwal: On Generalized I Convergent Paranormed Spaces Put φ m,k (x)= t m,k (x) t m 1,k (x), assuming that t 1,k = 0 A straight forward calculation shows (see [21]) that φ m,k (x)= x k, 1 m(m+1) m j=1 J(x σ j (k) x σ j 1 (k) ), For any sequence x, y and scalar λ, we have φ m,k (x+y)= φ m,k (x)+φ m,k (y), φ m,k (λ x)=λ φ m,k (x) (m 1), (m=0) A sequence x l is of σ-bounded variations if and only if (i) φ m,k (x) converges uniformly in m; (ii) lim t m,k(x), which must exist, should take the same m value for all k We denote by BV σ, the space of all sequences of σ-bounded variations (see [8]): BV σ = x l : φ m,k (x) <, uniformaly in k m BV σ is a Banach space normed by x =sup k φ m,k (x) (see [22]) Subsequently, invariant mean have been studied by Ahmad and Mursaleen [1], Mursaleen et al ([19],[21]), Raimi [2], Vakeel et al ([9], [10], [11]), and many others For the first time, I convergence was studied by Kostyrko et al [1] Later on, it was studied by Salat et al [26], Tripathy and Hazarika [29] and many others The notion of difference sequence spaces was introduced by Kızmaz [7], who defined the sequence spaces Z( )=x=(x k ) w:( x k ) Z for Z = c,c 0 and l where x = ( x k ) = (x k x k+1 ) The notion was further generalized by Et and Çolak [5] by introducing the spaces Let r be a non-negative integer, then Z( r )=x=(x k ) w:( r x k ) Z for Z = c,c 0 and l where r x=( r x k )=( r 1 x k r 1 x k+1 ) and 0 x k = x k for all k N The generalized difference sequence has the following binomial representation r r ( ) x k = ( 1) v r x v k+v v=0 Let N be a non empty set Then a family of sets I 2 N (Power set of N) is said to be an ideal if I is additive ie A,B I A B I and A B A B I A non empty family of sets (I) 2 N is said to be filter on N if and only if Φ / (I) for A,B (I) we have A B (I) and for each A (I) and A B implies B (I) An ideal I 2 N is called non trivial if I 2 N A non trivial ideal I 2 N is called admissible if x : x N I A non-trivial ideal is maximal if there cannot exist any non trivial ideal J I containing I as a subset For each ideal I, there exist a filter (I) corresponding to I ie (I) = K N : K c where K c =N\K Definition 11 A double sequence (x ) w is said to be I-convergent to a number L if for every > 0, the set i, j N : x L I In this case we write I limx = L Definition 12 A double sequence (x ) w is said to be I-null if L=0 In this case we write I limx = 0 Definition 1 A double sequence (x ) w is said to be I-Cauchy if for every > 0, there exist a number a=a() and b=b() such that i, j N: x x ab I Definition 14 A double sequence (x ) w is said to be I-bounded if there exist M > 0 such that i, j N: x >M I Definition 15 A double-sequence space E is said to be solid or normal if (x ) E implies (α x ) E for all sequence of scalars(α ) with α <1 for all i, j N Definition 16 Let X be a linear metric space A function p : X R is called paranorm, if 1p(x) 0 for all x X; 2p( x)= p(x) for all x X; p(x+y) p(x)+ p(y) for all x,y X; 4if (λ n ) is a sequence of scalars with λ n λ as n and (x n ) is a sequence of vectors with p(x n x) 0 as n, then p(λ n x n λ x) 0 as n A paranorm p for which p(x) = 0 implies x = 0 is called total paranorm and the pair (X, p) is called a total paranormed space It is well known that the metric of any linear metric space is given by some total paranorm ([Theorem 1042, pp 18] see [0]) For more details about sequence spaces see ([24], [25]) and references therein Let p=(p ) be any double bounded sequence of positive real numbers and u = (u ) be a double sequence of strictly positive real numbers In this paper we define the following sequence space: 2BVσ I (u, p, r ) = x=(x ) w: i, j N: φ mn, (u r x) L p for some L C If we take u = (u ) = 1, p = (p ) = 1, for all i, j and r = 0 then we get the sequence space defined by Vakeel and Nazneen [12] The main purpose of this paper is to introduce the sequence space 2 BV I σ(u, p, r ) We have also make an

3 Math Sci Lett 4, No 2, (2015) / wwwnaturalspublishingcom/journalsasp 167 attempt to study some topological, algebraic properties and inclusion relations between the sequence spaces 2BV I σ(u, p, r ) 2 Main Results Theorem 21 Let p = (p ) be a double bounded sequence of positive real numbers and u = (u ) be a double sequence of strictly positive real numbers Then the space 2 BV I σ (u, p, r ) is a linear space over the complex field C Proof Let x=(x ), y=(y ) 2 BVσ I(u, p, r ) and α, β C Then for a given > 0, we have i, j N: φ mn, (u r x) L 1 p 2 i, j N: φ mn, (u r y) L 2 p 2 for somel 1 C, I, for somel 2 C Now let A 1 = i, j N: φ mn, (u r x) L 1 p 2 for somel 1 C, A 2 = i, j N: φ mn, (u r y) L 2 p 2 for somel 2 C be such that A c 1,Ac 2 I Now consider φ mn, (u r (αx+β y)) (αl 1 + β L 2 ) p = φ mn, (αu r x)+φ mn, (β u r y) αl 1 β L 2 p = φ mn, (αu r x) αl 1 + φ mn, (β u r y) β L 2 p φ mn, (αu r x) αl 1 p + φ mn, (β u r y) β L 2 p = α φ mn, (u r x) L 1 p + β φ mn, (u r y) L 2 p α 2 + β 2 = ( α + β ) 2 (say) This implies that the sequence space A = i, j N : φ mn, (u r (αx+β y)) (αl 1 + β L 2 ) p < for some L 1, L 2 C Hence (αx+β y) 2 BVσ(u, I p, r ) Therefore 2 BVσ(u, I p, r ) is a linear space over the complex field C This completes the proof Theorem 22 Let p = (p ) be a double bounded sequence of positive real numbers and u = (u ) be a double sequence of strictly positive real numbers Then the space 2BV I σ(u, p, r ) is a paranormed space, paranormed by g(x )=sup φ mn, (u r x) p Proof For x = (x ) = 0, g(x ) = 0 is trivial For x=(x ) 0, g(x ) 0, we have (i) g(x) = sup φ mn, (u r x) p 0, for all x 2 BVσ I(u, p, r ) (ii) g( x) = sup φ mn, ( u r x) p = sup φ mn, (u r x) p = sup for all x 2 BVσ I(u, p, r ) (iii) sup g(x + y) = sup φ mn, (u r x) p + sup φ mn, (u r x) p = g(x), φ mn, (u r x + u r y) p φ mn, (u r y) p = g(x)+g(y) (iv) Let λ be a sequence of scalars with λ λ as ( ) and x 2 BV I σ (u, p, r ) such that in the sense that φ mn, (u r x) L as ( ) g(φ mn, (u r x) L) p 0 as ( ) Therefore g(λ φ mn, (u r x) λ L) p g(λ φ mn, (u r x)) p g(λ L) p = λ g(φ mn, (u r x)) p λ g(l) p 0 as Hence 2 BV I σ(u, p, r ) is a paranormed space This completes the proof Theorem 2 The space 2 BV I σ(u, p, r ) is solid and monotone Proof Let x = (x ) 2 BV I σ (u, p, r ) and (α ) be a sequence of scalars with α 1, for all i, j N Then we have α φ mn, (u r x) p α φ mn, (u r x) p φ mn, (u r x) p, i, j N The space 2 BVσ I(u, p, r ) is solid follows from the following inclusion relation: i, j N: φ mn, (u r x) p i, j N: α φ mn, (u r x) p Also a sequence is solid implies monotone Hence the space 2BVσ(u, I p, r ) is monotone This completes the proof Theorem 24 2BVσ(u, I p, r ) is a closed subspace of 2l(u, I p, r ) Proof Let(x (bd) ) be a Cauchy sequence in 2 BVσ I(u, p, r ) such that x (bd) x We show that x 2 BVσ I(u, p, r ) Since (x (bd) ) 2 BVσ I(u, p, r ), then there exist a bd such that i, j N: φ mn, (u r x (bd) ) a bd p I We need to show that (i)(a bd ) converges to a

4 168 K Raj, S Jamwal: On Generalized I Convergent Paranormed Spaces (ii) If U = i, j N: x a <, then U c I Since (x (bd) ) is a Cauchy sequence in 2 BV I σ (u, p, r ) Then for a given > 0, their exists k 0 N such that (e f) sup φ mn, (u r x (bd) ) φ mn, (u r x b,d,e, f k 0 For a given > 0, we have B bde f = i, j N: φ mn, (u r x (bd) ) φ mn, (u r (e f) x ) p < B bd = i, j N: φ mn, (u r x (bd) B e f = i, j N: φ mn, (u r x ) p <,, ) a bd p < (e f) ) a e f p < Then B c bde f, Bc bd and Bc e f I Let Bc = B c bde f Bc bd Bc e f, where B = i, j N : a bd a e f < Then B c I We choose k 0 B c, then for each b,d,e, f k 0, we have i, j N : a bd a e f < φ mn, (u r x (bd) ) a bd p < i, j N: φ mn, (u r x (bd), i, j N : ) φ mn, (u r (e f) x ) p < (e f) i, j N: φ mn, (u r x ) a e f p < Then (a bd ) is a Cauchy sequence of scalars in N, so their exists a scalar a C such that(a bd ) aas b,d For the next step, let 0<δ < 1 be given Then, we show that if U = i, j N : φ mn, (u r x) a p < δ, then U c I Since φ mn, (u r x (bd) ) φ mn, (u r x), then their exist a scalar b 0 d 0 N such that P= i, j N: φ mn, (u r x (b 0d 0 ) ) φ mn, (u r x) p < δ (2) which implies that P c I The number b 0 d 0 can be so chosen together with, we have Q= i, j N: a b0 d 0 a p < δ such that Q c I Since i, j N : φ mn, (u r x (b 0d 0 ) ) a b0 d 0 p δ then we have a subset S of N such that S c where S= i, j N: φ mn, (u r x (b 0d 0 ) ) a b0 d 0 p < δ Let U c = P c Q c S c, where U = i, j N: φ mn, (u r x) a p < δ, therefore for each i, j U c, we have i, j N: φ mn, (u r x) a p < δ i, j N: φ mn, (u r x (b 0d 0 ) ) φ mn, (u r x) p < δ i, j N: φ mn, (u r x (b 0d 0 ) ) a b0 d 0 p < δ i, j N: a b0 d 0 a p < δ Hence the result 2 BVσ I(u, p, r,) 2 l I (u, p, r ) follows This completes the proof Theorem 25 The space 2 BV I σ(u, p, r ) is nowhere dense subset of 2 l I (u, p, r ) Proof Proof of the result follows from the previous theorem Theorem 26 The inclusions 2C0 I(u, p, r ) 2 BVσ(u, I p, r,) 2 l(u, I p, r ) are proper Proof Let x=(x ) 2 C0 I(u, p, r ) Then, we have i, j N : u r x p I Since 2C 0 2 BV σ, x=(x ) 2 BV I σ (u, p, r ) implies Now let i, j N: φ mn, (u r x) p I A 1 = i, j N: u r x p <, A 2 = i, j N: φ mn, (u r x) p < be such that A c 1, Ac 2 I As 2l(u, I p, r ) = u r x p < x = (x ) : sup taking supremum over i, j we get A c 1 Ac 2 Hence 2C I 0(u, p, r ) 2 BV I σ(u, p, r ) 2 l I (u, p, r ) Next we show that the inclusion is proper First for 2C I 0 (u, p, r ) 2BV I σ (u, p, r ) Consider x 2 BV I σ(u, p, r ), then by the definition 2BVσ I (u, p, r ) = x=(x ) w: i, j N: φ mn, (u r x) L p for some L C, we have φ mn, (u r x)= t mn, (u r x) t (m 1)(n 1), (u r x), where t mn, (u r x)= u r x + u r x σ() + u r x σ 2 () ++u r x σ mn () mn Therefore

5 Math Sci Lett 4, No 2, (2015) / wwwnaturalspublishingcom/journalsasp 169 t mn, (u r x) t (m 1)(n 1), (u r x) = u r x + u r x σ() + u r x σ 2 () mn ++ u r x σ mn () mn u r x + u r x σ() + u r x σ 2 () (m 1)(n 1) ++ u r x σ (m 1)(n 1) () (m 1)(n 1) = (m 1)(n 1)(u r x + u r x σ() + u r x σ 2 () ++ u r x σ mn () mn(u r x + u r x σ() + u r x σ 2 () ++ u r x σ (m 1)(n 1) () On solving we get φ mn, (u r x)= mnu r x σ mn () + (1 m n)(u r x + u r x σ() + u r x σ 2 () ++ u r x σ mn () As σ is a translation map, that is σ(n) = n + 1, we have φ mn, (u r x)= mnu r x (i+m)( j+n) + (1 m n)(u r x + u r x (i+1)( j+1) ++u r x (i+m)( j+n) taking limit i, j, we have lim φ mn,(u r x) (i, j) [ = lim (mnu r (i, j) x (i+m)( j+n) +(1 m n)(u r x + u r x (i+1)( j+1) ++u r x (i+m)( j+n) ) () 1], [ L()= lim mnu r x (i+m)( j+n) + (i, j) (1 m n)(u r x ] + u r x (i+1)( j+1) ++ u r x (i+m)( j+n) ) Since m,n,l 0, therefore lim φ mn,(u r x) 0 (i, j) which implies that x / 2 C0 I(u, p, r ) Hence we get that the inclusion is proper For 2 BVσ I(u, p, r ) 2 l I (u, p, r ), the result of this part follows from the proof of the Theorem (24) This completes the proof Theorem 27 The inclusions 2C I (u, p, r ) 2 BVσ(u, I p, r ) 2 l(u, I p, r ) are proper Proof Let x = (x ) 2 C I (u, p, r ) Then, we have i, j N : u r x L p I Since 2C0 I(u, p, r ) 2 BVσ I(u, p, r ) 2 l I (u, p, r ), which implies x=(x ) 2 BVσ(u, I p, r ) then i, j N: u r φ mn, (x) L p I Now let B 1 = i, j N: u r x L p <, B 2 = i, j N: φ mn, (u r x) L p < be such that B c 1, Bc 2 I As 2l(u, I p, r ) = x = (x ) : sup u r x p < taking lim sup over i, j we get B c 1 Bc 2 Hence 2C I (u, p, r ) 2 BVσ I(u, p, r ) 2 l I (u, p, r ) Next we show that the inclusion is proper First for 2C I (u, p, r ) 2BVσ(u, I p, r ) Let x=(x ) 2 BVσ I(u, p, r ), then by the definition 2BVσ(u, I p, r ) = x=(x ) w: i, j N: φ mn, (u r x) L p for some L C We have φ mn, (u r x) L p We say that the I lim (φ mn, (u r x))=l Now considering the case when φ mn, (u r x) L p < Then t mn, (u r x) t (m 1)(n 1), (u r x) L p < when m,n=0, then we have Therefore, we get Hence, φ mn, (u r x)=t (u r x)=u r x x / 2 C I (u, p, r )= u r x L p <, i, j N i, j N: u r x L p I Hence, the inclusion is proper For 2BV I σ (u, p, r ) 2 l I (u, p, r ), the result of this part follows from the proof of the Theorem (24) This completes the proof

6 170 K Raj, S Jamwal: On Generalized I Convergent Paranormed Spaces Acknowledgement The authors are grateful to the anonymous referee for a careful checking of the details and for helpful comments that improved this paper References [1] Z U Ahmad and M Mursaleen, Proc Amer Math Soc 10, (1988) [2] T J Bromwich, Macmillan and Co Ltd New York 1965 [] S Banach, Theorie des Operations Lineaires, Warszawa, Poland, (192) [4] M Başarir and O Sonalcan, J Indian Acad Math 21, (1999) [5] M Et and R Çolak, Soochow J Math (1995) [6] G H Hardy, Proc Camb Phil Soc 19, (1917) [7] H Kizmaz, Canad Math Bull 24, (1981) [8] V A Khan, Comm Fac Sci Univ Ankara Sér A, 57, 25 (2008) [9] V A Khan, K Ebadullah and S Suantai, Analysis, 2, (2012) [10] V A Khan and K Ebadullah, Theory and Applications of Mathematics and Computer Science, 1, 22-0 (2011) [11] V A Khan and K Ebadullah, J Math Comput Sci 2, (2012) [12] V A Khan and Nazneen Khan, Int J Anal (201) 7 pages [1] P Kostyrko, T Salat and W Wilczynski, Real Analysis Exchange, 26, (2000) [14] G G Lorentz, Acta Mathematica, 80, (1948) [15] F Moricz, Acta Math Hungarica, 57, (1991) [16] F Moricz and B E Rhoades, Math Proc Camb Phil Soc 104, (1988) [17] M Mursaleen, J Math Anal Appl 29, (2004) [18] M Mursaleen and O H H Edely, J Math Anal Appl 29, (2004) [19] M Mursaleen, Quarterly J Math 4, (198) [20] M Mursaleen and O H H Edely, J Math Anal Appl 288, (200) [21] M Mursaleen, Houston J Math 9, (198) [22] M Mursaleen and S A Mohiuddine, Glas Mat 45, (2010) [2] R A Raimi, Duke Math J (196) [24] K Raj, A K Sharma and S K Sharma, Int J Pure Appl Math 67, (2011) [25] K Raj and S K Sharma, Acta Univ Palacki Olomuc Fac rer nat 51, (2012) [26] T Salat, B C Tripathy and M Ziman, Tatra Mt Math Publ 28, (2004) [27] B C Tripathy, Soochow J Math 0, (2004) [28] B C Tripathy, Tamkang J Math 4, (200) [29] B C Tripathy and B Hazarika, 59, (2009) [0] A Wilansky, Summability through Functional Analysis, North- Holland Math Stud (1984) [1] M Zeltser, Diss Math Univ Tartu, Tartu University Press, Univ of Tartu, Faculty of Mathematics and Computer Science, Tartu 25 (2001) Kuldip Raj got his BSc(1990), MSc(1992) and PhD(1999) from University of Jammu, Jammu, India His research interests are in the areas of Functional Analysis, Operator Theory, Sequences, Series and Summability Theory In 1998, Kuldip Raj had been appointed in Education Department of Jammu and Kashmir Govt In 2007, Kuldip Raj appointed as Assistant Professor in the School of Mathematics, Shri Mata Vaishno Devi University, Katra J&K India He has been Published more than 120 papers in reputed International Journals of Mathematics Presently he has been working in school of Mathematics Shri Mata Vaishno Devi University Katra J&K India Seema Jamwal is a Research Scholar in the School of Mathematics Shri Mata Vaishno Devi University Katra J&K India