Generalized I of strongly Lacunary of χ 2 over p metric spaces defined by Musielak Orlicz function

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1 Available at htt://vamuedu/aam Al Al Math ISSN: Vol 11, Issue 2 (December 2016), Alications and Alied Mathematics: An International Journal (AAM) Generalized I of strongly Lacunary of χ 2 over metric saces defined by Musielak Orlicz function 1 Deemala, 2 N Subramanian & 3 Lakshmi N Mishra 1 SQC and OR Unit, Indian Statistical Institute, 203 B T Road Kolkata West Bengal, India dmrai23@gmailcom; deemaladm23@gmailcom 2 Deartment of Mathematics, SASTRA University Thanjavur , India nsmaths@yahoocom 3 Deartment of Mathematics, National Institute of Technology Silchar Assam, India lakshminarayanmishra04@gmailcom Received: February 27, 2016; Acceted: May 31, 2016 Abstract In this aer, we introduce generalized difference sequence saces via ideal convergence, lacunary of χ 2 sequence saces over metric saces defined by Musielak function, and examine the Musielak-Orlicz function which satisfies uniform 2 condition, and we also discuss some toological roerties of the resulting saces of χ 2 with resect to ideal structures which is solid and monotone Hence, given an examle of the sace χ 2, this is not solid and not monotone This theory is very useful for statistical convergence and also is alicable to rough convergence Keywords: Analytic sequence; double sequences; χ 2 sace; difference sequence sace; Musielak Orlicz function; metric sace; Lacunary sequence; ideal 888

2 AAM: Intern J, Vol 11, Issue 2 (December 2016) 889 MSC 2010 No: 40A05, 40C05, 40D05 1 Introduction Throughout w, χ and Λ denote the classes of all gai and analytic scalar valued single sequences, resectively We write w 2 for the set of all comlex sequences (x mn ), where m, n N, the set of ositive integers Then, w 2 is a linear sace under coordinatewise addition and scalar multilication Some initial work on double sequence saces is found in Bromwich (1965) Later on, they were investigated by Hardy (1917), Moricz (1991), Moricz and Rhoades (1988), Basarir and Solankan (1999), Triathy (2003), Turkmenoglu (1999), Mishra et al (2007, 2012) and many others We rocure the following sets of double sequences: M u (t) := (x mn ) w 2 : su m,n N x mn tmn <, C (t) := (x mn ) w 2 : lim m,n x mn l tmn = 1 for some l C, C 0 (t) := (x mn ) w 2 : lim m,n x mn tmn = 1, L u (t) := (x mn ) w 2 : m=1 n=1 x mn tmn <, C b (t) := C (t) M u (t) and C 0b (t) = C 0 (t) M u (t), where t = (t mn ) is the sequence of strictly ositive reals t mn for all m, n N and lim m,n denotes the limit in the Pringsheim s sense In the case t mn = 1 for all m, n N; M u (t), C (t), C 0 (t), L u (t), C b (t) and C 0b (t) reduce to the sets M u, C, C 0, L u, C b and C 0b, resectively Now, we may summarize the knowledge given in some documents related to the double sequence saces Gökhan and Colak (2004, 2005) have roved that M u (t) and C (t), C b (t) are comlete aranormed saces of double sequences and gave the α, β, γ duals of the saces M u (t) and C b (t) Quite recently, in her PhD thesis, Zelter (2001) has essentially studied both the theory of toological double sequence saces and the theory of summability of double sequences Mursaleen and Edely (2003) and Triathy (2003) have indeendently introduced the statistical convergence and Cauchy for double sequences and given the relation between statistical convergent and strongly Cesàro summable double sequences Altay and Basar (2005) have defined the saces BS, BS (t), CS, CS b, CS r and BV of double sequences consisting of all double series whose sequence of artial sums are in the saces M u, M u (t), C, C b, C r and L u, resectively, and also examined some roerties of those sequence saces and determined the α duals of the saces BS, BV, CS b and the β (ϑ) duals of the saces CS b and CS r of double series Basar and Sever (2009) have introduced the Banach sace L q of double sequences corresonding to the well-known sace l q of single sequences and examined some roerties of the sace L q Quite recently Subramanian and Mishra (2010) have studied the sace χ 2 M (, q, u) of double sequences and gave some inclusion relations

3 890 Deemala et al The class of sequences which are strongly Cesàro summable with resect to a modulus was introduced by Maddox (1986) as an extension of the definition of strongly Cesàro summable sequences Connor (1989) further extended this definition to a definition of strong A summability with resect to a modulus where A = (a n,k ) is a nonnegative regular matrix and established some connections between strong A summability, strong A summability with resect to a modulus, and A statistical convergence In Pringsheim (1900) the notion of convergence of double sequences was resented by A Pringsheim Also, in Hamilton (1936, 1938), the four dimensional matrix transformation (Ax) k,l = m=1 Robison (1926) and Hamilton (1939) The double series m,n=1 convergent, where s mn = n=1 amn kl x mn was studied extensively by x mn is called convergent if and only if the double sequence (s mn ) is m,n i,j=1 x ij (m, n N) A sequence x = (x mn ) is said to be double analytic if su mn x mn 1/m+n < The vector sace of all double analytic sequences will be denoted by Λ 2 A sequence x = (x mn ) is called a double gai sequence if ((m + n)! x mn ) 1/m+n 0 as m, n The double gai sequences will be denoted by χ 2 Let φ = all finite sequences Consider a double sequence x = (x ij ) The (m, n) th section x m,n] of the sequence is defined by m,n x m,n] = x ij I ij for all m, n N, where I ij denotes the double sequence whose only non i,j=0 zero term is 1 (i+j)! in the (i, j) th lace for each i, j N An FK-sace (or a metric sace) X is said to have AK roerty if (I mn ) is a Schauder basis for X, or equivalently x m,n] x An FDK-sace is a double sequence sace endowed with a comlete metrizable locally convex toology under which the coordinate maings x = (x k ) (x mn )(m, n N) are also continuous Let M and Φ be mutually comlementary modulus functions Then, we have: (i) For all u, y 0, uy M (u) + Φ (y), (Young s inequality) (see Kamthan et al (1981)), (1) (ii) for all u 0, uη (u) = M (u) + Φ (η (u)), (2) and

4 AAM: Intern J, Vol 11, Issue 2 (December 2016) 891 (iii) for all u 0 and 0 < λ < 1, M (λu) λm (u) (3) Lindenstrauss and Tzafriri (1971) used the idea of Orlicz function to construct the Orlicz sequence sace l M = x w : ( ) k=1 M xk <, for some ρ > 0 ρ The sace l M with the norm x = inf ρ > 0 : ( ) k=1 M xk 1, ρ becomes a Banach sace which is called an Orlicz sequence sace For M (t) = t (1 < ), the saces l M coincide with the classical sequence sace l A sequence f = (f mn ) of modulus function is called a Musielak-modulus function A sequence g = (g mn ) defined by g mn (v) = su v u (f mn ) (u) : u 0, m, n = 1, 2, is called the comlementary function of a Musielak-modulus function f For a given Musielak modulus function f, the Musielak-modulus sequence sace t f and its subsace h f are defined as follows: t f = x w 2 : I f ( x mn ) 1/m+n 0 as m, n, h f = where I f is a convex modular defined by x w 2 : I f ( x mn ) 1/m+n 0 as m, n, I f (x) = m=1 n=1 f mn ( x mn ) 1/m+n, x = (x mn ) t f We consider t f equied with the Luxemburg metric d (x, y) = su mn inf ( m=1 n=1 f mn If X is a sequence sace, we give the following definitions: ( xmn 1/m+n mn )) 1 (1) X = the continuous dual of X, (2) X α = a = (a mn ) : m,n=1 a mn x mn <, for each x X, (3) X β = a = (a mn ) : m,n=1 a mn x mn is convegent, for each x X, (4) X γ = a = (a mn ) : su mn 1 M,N m,n=1 a mnx mn <, for each x X, (5) let X be an FK-sace φ; then X f = f(i mn ) : f X,

5 892 Deemala et al (6) X δ = a = (a mn ) : su mn a mn x mn 1/m+n <, for each x X, where X α, X β and X γ are called α (or Köthe-Toelitz) dual of X, β (or generalized-köthe- Toelitz) dual of X, γ dual of X, and δ dual of X, resectively X α is defined by Guta and Kamtan (1981) It is clear that X α X β and X α X γ, but X β X γ does not hold, since the sequence of artial sums of a double convergent series need not to be bounded The notion of difference sequence saces (for single sequences) was introduced by Kizmaz as follows Z ( ) = x = (x k ) w : ( x k ) Z, for Z = c, c 0 and l, where x k = x k x k+1, for all k N Here, c, c 0 and l denote the classes of convergent, null,and bounded scalar valued single sequences, resectively The difference sequence sace bv of the classical sace l is introduced and studied in the case 1 by Başar and Altay and in the case 0 < < 1 by Altay and Başar (2005) The saces c ( ), c 0 ( ), l ( ) and bv are Banach saces normed by x = x 1 + su k 1 x k and x bv = ( k=1 x k ) 1/, (1 < ) Later on the notion was further investigated by many others We now introduce the following difference double sequence saces defined by where Z = Λ 2, χ 2, and Z ( ) = x = (x mn ) w 2 : ( x mn ) Z, x mn = (x mn x mn+1 ) (x m+1n x m+1n+1 ) = x mn x mn+1 x m+1n + x m+1n+1, m, n N The generalized difference double notion has the following reresentation: m x mn = m 1 x mn m 1 x mn+1 m 1 x m+1n + m 1 x m+1n+1, and also this generalized difference double notion has the following binomial reresentation: m m ( )( ) m m m x mn = ( 1) i+j x m+i,n+j i j i=0 j=0 2 Definition and Preliminaries Let n N and X be a real vector sace of dimension w, where n w A real valued function d (x 1,, x n ) = (d 1 (x 1 ),, d n (x n )) on X satisfes the following four conditions: (1) (d 1 (x 1 ),, d n (x n )) = 0 if and and only if d 1 (x 1 ),, d n (x n ) are linearly deendent,

6 AAM: Intern J, Vol 11, Issue 2 (December 2016) 893 (2) (d 1 (x 1 ),, d n (x n )) is invariant under ermutation, (3) (αd 1 (x 1 ),, d n (x n )) = α (d 1 (x 1 ),, d n (x n )), α R, (4) d ((x 1, y 1 ), (x 2, y 2 ) (x n, y n )) = (d X (x 1, x 2, x n ) + d Y (y 1, y 2, y n ) ) 1/, for 1 <, or (5) d ((x 1, y 1 ), (x 2, y 2 ), (x n, y n )) := su d X (x 1, x 2, x n ), d Y (y 1, y 2, y n ), for x 1, x 2, x n X, y 1, y 2, y n Y is called the roduct metric A trivial examle of the roduct metric of n metric sace is the norm sace X = R equied with the following Euclidean metric in the roduct sace with the norm: (d 1 (x 1 ),, d n (x n )) E = su ( det(d mn (x mn )) ) d 11 (x 11 ) d 12 (x 12 ) d 1n (x 1n ) d 21 (x 21 ) d 22 (x 22 ) d 2n (x 1n ) = su, d n1 (x n1 ) d n2 (x n2 ) d nn (x nn ) where x i = (x i1, x in ) R n for each i = 1, 2, n If every Cauchy sequence in X converges to some L X, then X is said to be comlete with resect to the metric Any comlete metric sace is said to be a Banach metric sace Let X be a linear metric sace A function w : X R is called aranorm, if (1) w (x) 0, for all x X, (2) w ( x) = w (x), for all x X, (3) w (x + y) w (x) + w (y), for all x, y X, and (4) If (σ mn ) is a sequence of scalars with σ mn σ as m, n and (x mn ) is a sequence of vectors with w (x mn x) 0 as m, n, then w (σ mn x mn σx) 0 as m, n A aranorm w for which w (x) = 0 imlies x = 0 is called total aranorm and the air (X, w) is called a total aranormed sace It is well known that the metric of any linear metric sace is given by some total aranorm (see Wilansky (1984), Theorem 1042, 183) Let X be a non-emty set Then a family of sets I 2 X (the class of all subsets of X) is called an ideal if and only if for each A, B I, we have A B I and for each A I and each B A, we have B I A non-emty family of sets F 2 X is a filter on X if and only if φ / F, for each A, B F, we have A B F and each A F and A B, we have B F An ideal I is called non-trivial ideal if I φ and X I Clearly I 2 X is a non-trivial ideal if and only F = F (I) = X/A : A I is a filter on X A non-trivial ideal I 2 X is called admissible if and only if x x X I A sequence (x mn ) m,n N in X is said to be I convergent to 0 X, if for each ɛ > 0 the set A (ɛ) = m, n N : (d 1 (x 1 ),, d n (x n )) 0 ɛ belongs to I Further details on ideals of 2 X can be found in Kostyrko et al (2001) The notion was further investigated by Salat et al (2004) and others

7 894 Deemala et al By the convergence of a double sequence we mean the convergence on the Pringsheim sense, that is, a double sequence x = (x mn ) has Prinsheim limit L (denoted by P lim x = L) rovided that given ɛ > 0 there exists n N such that x mn L < ɛ whenever m, n > n We shall write this more briefly as P convergent The double sequence = (m r, n s ) is called a double lacunary sequence if there exist two increasing integers such that m 0 = 0, ϕ r = m r m r 1 as r and n 0 = 0, ϕ s = n s n s 1 as s Notations: m rs = m r n s, h rs = ϕ r ϕ s, are determined by I rs = (m, n) : m r 1 < m m r and n s 1 < n n s, q r = kr k r 1, q s = ns n s 1 and q rs = q r q s The notion of λ double gai and double analytic sequences are as follows Let λ = (λ mn ) m,n=0 be a strictly increasing sequence of ositive real numbers tending to infinity, that is, 0 < λ 00 < λ 11 < and λ mn as m, n and said that a sequence x = (x mn ) w 2 is λ convergent to 0 is called λ limit of x, if where µ m n (x) = 1 ϕ rs m I rs µ m n (x) 0 as m, n, n I rs ( m 1 λ m,n m 1 λ m,n+1 m 1 λ m+1,n + m 1 λ m+1,n+1 ) x mn 1/m+n The sequence x = (x mn ) w 2 is λ double analytic if su uv µ mn (x) < If lim mn x mn = 0 in the ordinary sense of convergence, then This imlies that 1 lim mn ϕ rs m I rs lim µ mn (x) 0 mn n I rs ( m 1 λ m,n m 1 λ m,n+1 m 1 λ m+1,n 1 = lim mn ϕ rs = 0, + m 1 λ m+1,n+1 ) ((m + n)! x mn 0 ) 1/m+n = 0 m I rs n I rs ( m 1 λ m,n m 1 λ m,n+1 m 1 λ m+1,n + m 1 λ m+1,n+1 ) ((m + n)! x mn 0 ) 1/m+n

8 AAM: Intern J, Vol 11, Issue 2 (December 2016) 895 which yields that lim uv µ mn (x) = 0 and hence x = (x mn ) w 2 is λ convergent to 0 Let I 2 be an admissible ideal of 2 N N, be a double lacunary sequence, ) f = (f mn ) be a Musielak-modulus function, and (X, (d (x 1 ), d (x 2 ),, d (x n 1 )) be a metric sace, q = (q mn ) be double analytic sequence of strictly ositive real numbers By w 2 () X) we denote the sace of all sequences defined over (X, (d (x 1 ), d (x 2 ),, d (x n 1 )) The following inequality will be used throughout the aer If 0 q mn su q mn = H, K = max ( 1, 2 H 1), then for all m, n and a mn, b mn C ( Also a qmn max 1, a H) for all a C a mn + b mn qmn K a mn qmn + b mn qmn, (4) In the resent aer we define the following sequence saces: and fµ, (d (x 1), d (x 2 ),, d (x n 1 )) ϕ 2 θ rs = r, s I rs : f mn ( µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ɛ I 2, fµ, (d (x 1), d (x 2 ),, d (x n 1 )) ϕ 2 θ rs = r, s I rs : f mn ( µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) K I 2 If we take f mn (x) = x, we get and fµ, (d (x 1), d (x 2 ),, d (x n 1 )) ϕ 2 θ rs = r, s I rs : ( µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ɛ I 2, fµ, (d (x 1), d (x 2 ),, d (x n 1 )) ϕ 2 θ rs = r, s I rs : ( µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) K I 2 If we take q = (q mn ) = 1, we get χ 2 fµ, (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ θ rs )] = r, s I rs : f mn ( µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ɛ I 2,

9 896 Deemala et al and Λ 2 fµ, (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ 2 θ rs )] = r, s I rs : f mn ( µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) K I 2 In the resent aer we lan to study some toological roerties and inclusion relations between the above defined sequence saces, 2 2 fµ, (d (x 1), d (x 2 ),, d (x n 1 )) ϕ and fµ, (d (x 1), d (x 2 ),, d (x n 1 )) ϕ, which we shall discuss in this aer 3 Main Results Theorem 1 Let f = (f mn ) be a Musielak-Orlicz function and q = (q mn ) be a double analytic sequence of strictly ositive real numbers The sequence saces fµ, (d (x 1), d (x 2 ),, d (x n 1 )) ϕ 2 and 2 fµ, (d (x 1), d (x 2 ),, d (x n 1 )) ϕ are linear saces It is routine verification Therefore the roof is omitted Theorem 2 Let f = (f mn ) be a Musielak-Orlicz function and q = (q mn ) be a double analytic sequence of strictly ositive real numbers The sequence sace fµ, (d (x 1), d (x 2 ),, d (x n 1 )) ϕ 2 is a aranormed sace with resect to the aranorm defined by g (x) = inf f mn ( µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) 1 Clearly g (x) 0 for x = (x mn ) Since f mn (0) = 0, we get g (0) = 0 Conversely, suose that g (x) = 0 fµ, (d (x 1), d (x 2 ),, d (x n 1 )) ϕ 2

10 AAM: Intern J, Vol 11, Issue 2 (December 2016) 897 Then, inf f mn ( µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) 1 Suose that µ mn (x) 0 for each m, n N Then, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ It follows that (f mn ( µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ) 1/H, which is a contradiction Therefore, µ mn (x) = 0 ( ) 1/H Let (f mn µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) 1 and (f mn ( µ mn (y), (d (x 1 ), d (x 2 ),, d (x n 1 )) ) 1/H 1 Then, by using Minkowski s inequality, we have (f mn ( µ mn (x + y), (d (x 1 ), d (x 2 ),, d (x n 1 )) ) 1/H ) 1/H (f mn ( µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ( ) 1/H + (f mn µ mn (y), (d (x 1 ), d (x 2 ),, d (x n 1 )) So we have g(x + y) = inf f mn ( µ mn (x + y), (d (x 1 ), d (x 2 ),, d (x n 1 )) 1 inf f mn ( µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) 1 + inf f mn ( µ mn (y), (d (x 1 ), d (x 2 ),, d (x n 1 )) 1 Therefore, g (x + y) g (x) + g (y) Finally, to rove that the scalar multilication is continuous, let λ be any comlex number By definition, g (λx) = inf f mn ( µ mn (λx), (d (x 1 ), d (x 2 ),, d (x n 1 )) 1 Then, g (λx) = inf (( λ t) qmn/h : f mn ( µ mn (λx), (d (x 1 ), d (x 2 ),, d (x n 1 )) 1, where t = 1 Since max (1, λ su λ λ qmn mn ), we have )] g (λx) max (1, λ su qmn mn ) inf t qmn/h : f mn ( µ mn (λx), (d (x 1 ), d (x 2 ),, d (x n 1 )) 1

11 898 Deemala et al This comletes the roof Theorem 3 (i) If the sequence (f mn ) satisfies uniform 2 condition, then α = µ g, µ uv (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ (ii) If the sequence (g mn ) satisfies uniform 2 condition, then µ g, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ 2α = fµ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ Let the sequence (f mn ) satisfy uniform 2 condition We get µ g, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ 2 fµ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ To rove the inclusion α let a (x mn ) µ g fµ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ fµ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ m=1 n=1, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ 2α θrs 2 Then for all x mn with we have 2 2 2α 2, x mn a mn < (5) Since the sequence (f mn ) satisfies uniform 2 condition, then (y mn ) and we get ϕ m=1 n=1 rsy mna mn m λ mn(m+n)! < by (5) Thus (ϕ rs a mn ) fµ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ = µ g, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ, 2 2

12 AAM: Intern J, Vol 11, Issue 2 (December 2016) 899 and hence (a mn ) µ g, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ 2 This gives that α From above we have, µ g α (ii) Similarly, one can rove that µ g = µ g, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ 2α if the sequence (g mn ) satisfies uniform 2 condition Proosition 1 If 0 < q mn < mn < for each m and m, then, fµ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ Λ 2 fµ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ The roof is standard, so we omit it Proosition 2 (i) If 0 < inf q mn q mn < 1, then, Λ 2 fµ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ

13 900 Deemala et al (ii) If 1 q mn su q mn <, then, Λ 2 fµ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ 2 fµ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ The roof is standard, so we omit it Proosition 3 Let f = ( ) f mn and f = ( f mn) are sequences of Musielak functions, we have, µ f µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ 2 f µ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ The roof is easy so we omit it Proosition 4 2 f +f µ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ 2 For any sequence of Musielak functions f = (f mn ) and q = (q mn ) be double analytic sequence of strictly ositive real numbers Then, fµ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ The roof is easy so we omit it Proosition 5 The sequence sace Let x = (x mn ), ie is solid su mn fµ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ 2 < 2 2

14 AAM: Intern J, Vol 11, Issue 2 (December 2016) 901 Let (α mn ) be double sequence of scalars such that α mn 1 for all m, n N N Then we get su fµ, µ mn (αx), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ 2 mn su mn This comletes the roof Proosition 6 The sequence sace The roof follows from Proosition 5 Proosition 7 fµ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ If f = (f mn ) be any Musielak function Then, 2 is monotone 2 fµ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ fµ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ if and only if, su r,s 1 ϕ rs ϕ rs Let x we get < 2 fµ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ fµ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ rs Thus x = N 2 and N = su r,s 1 ϕ rs ϕ rs fµ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ rs 2, < Then 2 = 0 2 fµ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ Conversely, suose that fµ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ N θ u fµ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ and x 2 fµ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ Then θ ] rs I 2 fµ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ < ɛ, for every ɛ > 0 2

15 902 Deemala et al Suose that su r,s 1 ϕ rs ϕ rs lim j,k ϕ jk ϕ jk = Hence, we have =, then there exists a sequence of members (rs jk ) such that 2 fµ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ rs = Therefore x / 2 fµ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ, which is a contradiction This comletes the roof Proosition 8 If f = (f mn ) be any Musielak function Then 2 fµ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ = 2 fµ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ, if and only if su r,s 1 ϕ rs ϕ rs It is easy to rove so we omit Proosition 9 The sequence sace <, su r,s 1 ϕ rs ϕ rs > The result follows from the following examle Examle Consider x = (x mn ) = Let is not solid 1 m+n 1 m+n 1 m+n 1 m+n 1 m+n 1 m+n α mn =, for all m, n N 1 m+n 1 m+n 1 m+n

16 AAM: Intern J, Vol 11, Issue 2 (December 2016) 903 Then α mn x mn / fµ, µ mn (x), (d (x 1 ), d (x 2 ),, d (x n 1 )) ϕ is not solid Proosition 10 The sequence sace The roof follows from Proosition 9 2 Hence is not monotone 4 Conclusion We introduce generalized difference sequence saces via ideal convergence, lacunary of χ 2 sequence saces over metric saces defined by Musielak-Orlicz function and also discuss some toological roerties of our roved results on these saces The growing interest in this field is strongly stimulated by the treatment of recent roblems in elasticity, fluid dynamics, calculus of variations, and differential equations One can extend our results for more general saces Acknowledgments The authors are extremely grateful to the anonymous learned referee(s) for their keen reading, valuable suggestion and constructive comments for the imrovement of the manuscrit The authors are thankful to the editor(s) and reviewers of Alications and Alied Mathematics, and also the second author wishes to thank the Deartment of Science and Technology, Government of India for the financial sanction towards this work under FIST rogram SR/FST/MSI-107/2015 REFERENCES Altay, B and BaŞar, F (2005) Some new saces of double sequences, J Math Anal Al, Vol 309, No 1, BaŞar, F and Sever, Y (2009) The sace L of double sequences, Math J Okayama Univ, Vol 51, Basarir, M and Solancan, O (1999) On some double sequence saces, J Indian Acad Math, Vol 21, No 2, Bromwich, TJI A (1965) An introduction to the theory of infinite series, Macmillan and Co Ltd, New York Hardy, GH (1917) On the convergence of certain multile series, Proc Camb Phil Soc, Vol 19, Chandra, P and Triathy, BC (2002) On generalized Kothe-Toelitz duals of some sequence saces, Indian Journal of Pure and Alied Mathematics, Vol 33, No 8, Cannor, J (1989) On strong matrix summability with resect to a modulus and statistical

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