RIESZ LACUNARY ALMOST CONVERGENT DOUBLE SEQUENCE SPACES DEFINED BY SEQUENCE OF ORLICZ FUNCTIONS OVER N-NORMED SPACES

TWMS J. Pure Appl. Math., V.8, N., 207, pp.43-63 RIESZ LACUNARY ALMOST CONVERGENT DOUBLE SEQUENCE SPACES DEFINED BY SEQUENCE OF ORLICZ FUNCTIONS OVER N-NORMED SPACES M. MURSALEEN, S.. SHARMA 2 Abstract. In the present paper we introduce a new concept for strong almost convergence with respect to sequence of Orlicz function, difference sequence, Reisz mean for double sequence, double lacunary sequence and n-normed space. We examine some topological properties, inclusion relation between these newly defined sequence spaces and establish relation with Riesz lacunary almost statistically convergence sequence spaces. eywords: double sequence, difference sequence, Pringsheim convergence, Riesz convergence, statistical convergence, Orlicz function, n-normed space. AMS Subject Classification: 46A40, 40A35, 40G5, 46A45.. Introduction and preinaries The initial works on double sequences is found in Bromwich 5]. Later on, it was studied by Hardy 4], Moricz 23], Moricz and Rhoades 24], Mursaleen 25, 26], Başarir and Sonalcan 3], Altay and Basar 2], Başar and Sever 4], Mursaleen 30, 35], Shakhmurov 43] and many others. Mursaleen 27, 28] have recently introduced the statistical convergence and Cauchy convergence for double sequences and given the relation between statistical convergent and strongly Cesaro summable double sequences. Nextly, Mursaleen 25] and Mursaleen and Edely 29] have defined the almost strong regularity of matrices for double sequences and applied these matrices to establish a core theorem and introduced the M-core for double sequences and determined those four dimensional matrices transforming every bounded double sequences x x k,l ) into one whose core is a subset of the M-core of x. More recently, Altay and Başar 2] have defined the spaces BS, BSt), CS p, CS bp, CS r and BV of double sequences consisting of all double series whose sequence of partial sums are in the spaces M u, M u t), C p, C bp, C r and L u, respectively and also examined some properties of these sequence spaces and determined the α-duals of the spaces BS, BV, CS bp and the βv)-duals of the spaces CS bp and CS r of double series. Now, recently Başar and Sever 4] have introduced the Banach space L q of double sequences corresponding to the well known classical sequence space l q and examined some properties of the space L q. By the convergence of a double sequence we mean the convergence in the Pringsheim sense i.e. a double sequence x x k,l ) has Pringsheim it L denoted by P x L) provided that given ϵ > 0 there exists n N such that x k,l L < ϵ whenever k, l > n see 36]. We shall write more briefly as P -convergent. The double sequence x x k,l ) is bounded if there exists a positive number M such that x k,l < M for all k and l. For more details about sequence spaces see, 8, 9, 3-34, 37, 38, 44-46] and references therein. Department of Mathematics, Aligarh Mus university, Aligarh, India 2 Department of Mathematics, Model Institute of Engineering & Technology, ot Bhalwal, J&, India e-mail: murslaeenm@gmail.com, sunilksharma42@gmail.com Manuscript received July 206. 43

44 TWMS J. PURE APPL. MATH., V.8, N., 207 An Orlicz function M is a function, which is continuous, non-decreasing and convex with M0) 0, Mx) > 0 for x > 0 and Mx) as x. Lindenstrauss and Tzafriri 20] used the idea of Orlicz function to define the following sequence space. Let w be the space of all real or complex sequences x x k ), then l M x w : xk ) M < is called as an Orlicz sequence space. The space l M is a Banach space with the norm xk ) x inf > 0 : M. k k It is shown in 20] that every Orlicz sequence space l M contains a subspace isomorphic to l p p ). The 2 -condition is equivalent to MLx) klmx) for all values of x 0, and for L >. The notion of difference sequence spaces was introduced by ızmaz 6], who studied the difference sequence spaces l ), c ) and c 0 ). The notion was further generalized by Et and Çolak 7] by introducing the spaces l n ), c n ) and c 0 n ). Let n be a non-negative integer, then for Z c, c 0 and l, we have sequence spaces Z n ) x x k ) w : n x k ) Z, where n x n x k ) n x k n x k ) and 0 x k x k for all k N, which is equivalent to the following binomial representation n ) n x k ) v n x k+v. v v0 Taking n, we get the spaces l ), c ) and c 0 ) studied by Et and Çolak 7]. The concept of 2-normed spaces was initially developed by Gähler 0] in the mid of 960 s, while that of n-normed spaces one can see in Misiak 22]. Since then, many others have studied this concept and obtained various results, see Gunawan, 2] and Gunawan and Mashadi 3] and many others. Let n N and X be a linear space over the field, where is field of real or complex numbers of dimension d, where d n 2. A real valued function,, on X n satisfying the following four conditions: ) x, x 2,, x n 0 if and only if x, x 2,, x n are linearly dependent in X; 2) x, x 2,, x n is invariant under permutation; 3) αx, x 2,, x n α x, x 2,, x n for any α, and 4) x + x, x 2,, x n x, x 2,, x n + x, x 2,, x n is called a n-norm on X, and the pair X,,, ) is called a n-normed space over the field. For example, we may take X R n being equipped with the Euclidean n-norm x, x 2,, x n E the volume of the n-dimensional parallelopiped spanned by the vectors x, x 2,, x n which may be given explicitly by the formula x, x 2,, x n E detx ij ), where x i x i, x i2,, x in ) R n for each i, 2,, n. Let X,,, ) be an n-normed space of dimension d n 2 and a, a 2,, a n be linearly independent set in X. Then the following function,, on X n defined by x, x 2,, x n max x, x 2,, x n, a i : i, 2,, n

M. MURSALEEN, S.. SHARMA: RIESZ LACUNARY ALMOST CONVERGENT DOUBLE... 45 defines an n )-norm on X with respect to a, a 2,, a n. A sequence x k ) in a n-normed space X,,, ) is said to converge to some L X if x k L, z,, z n 0 for every z,, z n X. k A sequence x k ) in a n-normed space X,,, ) is said to be Cauchy if x k x i, z,, z n 0 for every z,, z n X. k,i If every Cauchy sequence in X converges to some L X, then X is said to be complete with respect to the n-norm. Any complete n-normed space is said to be n-banach space. Let A a mn jk ), j, k 0,,... be a doubly infinite matrix of real numbers for all m, n 0,,... Forming the sums y mn a mn jk x jk j0 k0 called the A- transform of the sequence x x j,k ), which yields a method of summability. More exactly, we say that a sequence x is A-summable to the it L if the A- transform Ax of x exists for all m, n 0,,... in the sense of Pringsheim s convergence: p q p,q j0 k0 a mn jk x jk y mn and y mn L. m,n We say that a triangular matrix A is bounded-regular or RH-regular if every bounded and convergent sequence x is A-summable to the same it and the A-means are also bounded. Necessary and sufficient conditions for A to be bounded-regular are ) m,n amn jk 2) 0 j, k 0,, 2,..) a mn jk m,n j0 k0 3) m,n j0 4) 5) m,n k0 j0 k0 a mn jk 0 k 0,,...) a mn jk 0 j 0,,...) a mn jk C < m, n 0,,...) These conditions were first established by Robison 39]. Actually ) is a consequence of each 3) and 4). We say that a matrix A is strongly regular if every almost convergent sequence x is A-summable to the same it, and the A-means are also bounded. Let n, m. to a it L if A double sequence x x k,l ) of real numbers is called almost P -convergent P sup n,m nm µ,η 0 µ+n kµ η+m lη x k,l L 0, i.e., the average value of x k,l ) taken over any rectangle k, l) : µ k µ + n, η l η + m

46 TWMS J. PURE APPL. MATH., V.8, N., 207 tends to L as both n and m tends to, and this convergence is uniform in µ and η. A double sequence x is called strongly almost P -convergent to a number L if P sup n,m µ,η 0 nm µ+n kµ η+m l η x k,l L 0. Let denote the set of sequences with this property as ĉ 2 ]. By ĉ 2 ], we denote the space of all almost convergent double sequences. It is easy to see that the inclusion c 2 ĉ 2 ] ĉ 2 l2 strictly hold, where l2 and c 2 denote the spaces of bounded and bounded convergent double sequences, respectively. As in the case of single sequences, every almost convergent double sequences is bounded. But a convergent double sequence need not be bounded. Thus a convergent double sequence need not be almost convergent. However, every bounded convergent double sequence is almost convergent. We use the following definition which may be called convergence in Pringsheim s sense as follows x k,l λ) 0), k, l ). Let p n ), p m ) be two sequences of positive numbers and Then the transformation given by P n p + p 2 +... + p n, Pm p + p 2 +... + p m. T n,m x) P n Pm n k l m x k,l is called the Riesz mean of double sequence x x k,l ). If P n,m T n,m L, L R, then the sequence x x k,l ) is said to be Riesz convergent to L. If x x k,l ) is Riesz convergent to L, then we write P R x L. The double sequence θ r,s k r, l s ) is called double lacunary if there exist two increasing sequences of integers such that k 0 0, h r k r k r as r and l 0 0, h s l s l s as s. Let r,s k r l s, h r,s h r hs and θ r,s is determined by I r,s k, l) : k r < k k r and; l s < l l s, qr k r k r, q s l s l s and q r,s q r q s. Let θ r,s k r, l s ) be a double lacunary sequence and p k ), p l ) be sequences of positive real numbers such that P kr p k, Pls p l and H r p k, Hs p l. k 0,k r] l 0,l s] k k r,k r] l l s,l s] Clearly, H r P kr P kr, Hs P ls P ls. If the Riesz transformation of double sequences is RH-regular, and H r P kr P kr as r, Hs P ls P ls, as s, then θ r,s P kr, P ls is a double lacunary sequence. Throughout the paper, we assume that P n p + p 2 +... + p n as n, Pm p + p 2 +... p m, as m, such that H r P kr P kr as r and H s P ls P ls as s. Let P kr,s P kr Pls, H r Hs, I r,s k, l) : P kr < k P kr and Pls < l P ls, θ r P kr P kr, θs P ls P ls and θ r,s θ r θs. If we take p k, p l for all k and l, then, P kr,s, θ r,s and I r,s reduce to h r,s, k r,s, q r,s and I r,s. Let P kr,s P kr Pls, H r Hs, I r,s k, l) : P kr < k P kr and P lr < l P ls, Q r P kr P kr, Qs P l s P ls & Q r,s Q r Qs. Let M ) be a sequence of Orlicz function and u u k,l ) be any factorable double sequence of strictly positive real numbers. In this section we define the following sequence

M. MURSALEEN, S.. SHARMA: RIESZ LACUNARY ALMOST CONVERGENT DOUBLE... 47 spaces over n-normed spaces: R2, θ r,s, M, p, v, u,,, ] x x k,l ) : P v x k+m,l+n L, z,, z n r,s k,l) I r,s 0, uniformly in m and n for some L and > 0 R2, θ r,s, M, p, v, u,,, ] 0 x x k,l ) : P v x )] k+m,l+n, z,, z n r,s k,l) I r,s 0, uniformly in m and n for some > 0. If we take Mx) x, we have R2, θ r,s, p, v, u,,, ] x x k,l ) : P v x k+m,l+n L ), z,, z n r,s k,l) I r,s 0, uniformly in m and n for some L and > 0 R2, θ r,s, p, v, u,,, ] 0 x x k,l ) : P v x ) k+m,l+n, z,, z n r,s k,l) I r,s 0, uniformly in m and n for some > 0. If we take u u k,l ), we have R2, θ r,s, M, p, v,,, ] x x k,l ) : P v x k+m,l+n L )], z,, z n r,s k,l) I r,s 0, uniformly in m and n for some L and > 0 R2, θ r,s, M, p, v,,, ] 0 x x k,l ) : P v x )] k+m,l+n, z,, z n r,s k,l) I r,s 0, uniformly in m and n for some > 0. If we take p k, p l for] all k and l, then we obtain the following ] sequence spaces AC θr,s, M, p, v,,, and AC θr,s, M, p, v,,, which can be seen in 34]. The following inequality will be used throughout the paper. If 0 p k,l sup p k,l H, D max, 2 H ) then a k,l + b k,l p k,l D a k,l p k,l + b k,l p k,l ) 0

48 TWMS J. PURE APPL. MATH., V.8, N., 207 for all k, l and a k,l, b k,l C. Also a p k,l max, a H ) for all a C. The R2 main aim of this paper is to introduce the new type of sequence spaces, θ r,s, M, p, v, u,, ] R2 and, θ r,s, M, p, v, u,, ] in the first section. In the second section of this paper we prove some topological properties and in the third section we establish inclusion relation between above defined sequence spaces and the sequence spaces which we defined in the third section of the paper. In the last section of paper we make an effort to study statistical convergence. 0 2. Some topological properties Theorem 2. Let M ) be a sequence of Orlicz functions and u u k,l ) be a factorable double sequence of positive real numbers. Then the spaces R2, θ r,s, M, p, v, u,,, ] and R2, θ r,s, M, p, v, u,,, ] are linear spaces over the field C of complex numbers. 0 ] Proof. Let x x k,l ), y y k,l ) R2, θ r,s, M, p, v, u,, 0 and α, β C. Then there exist positive numbers and 2 such that v x )] k+m,l+n, z,, z n 0, uniformly in m and n r,s k,l) I r,s for some > 0, and r,s k,l) I r,s v x )] k+m,l+n, z,, z n 0, uniformly in m and n 2 for some 2 > 0. Let 3 max2 α, 2 β 2 ). Since M ) is a sequence of non-decreasing convex functions, by using inequality.), we have v αx k+m,l+n + βy k+m,l+n ), z,, z n 3 k,l) I r,s D + D D + D 2 u p k p k,l l v x k+m,l+n ), z,, z n k,l) I r,s 2 u p k p k,l l v y k+m,l+n ), z,, z n 2 k,l) I r,s k,l) I r,s k,l) I r,s v x k+m,l+n ), z,, z n v y k+m,l+n ), z,, z n 2 0 as r, s, uniformly in m and n. R2 Thus, we have αx+βy, θ r,s, M, p, v, u,,, ] R2 ]. Hence, θ r,s, M, p, v, u,,, R2 0 is a linear space. Similarly, we can prove that, θ r,s, M, p, v, u,,, ] is a linear space. 0

M. MURSALEEN, S.. SHARMA: RIESZ LACUNARY ALMOST CONVERGENT DOUBLE... 49 Theorem 2.2 For any sequence of Orlicz functions M ) and u u k,l ) be a factorable double sequence of positive real numbers, the space R2, θ r,s, M, p, v, u,,, ] is a topological linear space paranormed by gx) inf ur,s : k,l I r,s where max, sup k,l u k,l < ). v x k+m,l+n ), z,, z n 0, r, s N, R2 Proof. Clearly gx) 0 for x x k,l ), θ r,s, M, p, v, u,, ]. Since 0) 0, we 0 get g0) 0. Again, if gx) 0, then inf u r,s : k,l) I r,s v x k+m,l+n ), z,, z n This implies that for a given ϵ > 0, there exists some ϵ 0 < ϵ < ϵ) such that Thus k,l) I r,s, k,l I r,s v x k+m,l+n ϵ v x )] k+m,l+n ), z,, z n ϵ k,l) I r,s ), z,, z n, r, s N 0.. v x )] k+m,l+n ), z,, z n ϵ for each r, s, m and n. Suppose that x k,l 0 for each k, l N. This implies ) that v x k+m,l+n 0, for each k, l, m, n N. Let ϵ 0, then v x k+m,l+n ϵ, z,, z n. It follows that k,l) I r,s v x k+m,l+n ϵ ), z,, z n, which is a contradiction. Therefore, v x k+m,l+n 0 for each k, l, m and n and thus x k,l 0 for each k, l N. Let > 0 and 2 > 0 be such that k,l) I r,s v x )] k+m,l+n ), z,, z n and k,l) I r,s v y )] k+m,l+n ), z,, z n 2

50 TWMS J. PURE APPL. MATH., V.8, N., 207 for each r, s, m and n. Let + 2. Then, by Minkowski s inequality, we have v x k+m,l+n + y k+m,l+n ) ), z,, z n k,l) I r,s ) v x k+m,l+n ) )), z,, z n + 2 + +. k,l) I r,s ) 2 + 2 ) + 2 ) 2 + 2 v y k+m,l+n ), z,, z n 2 k,l) I r,s k,l) I r,s Since s are non-negative, so we have gx + y) inf ur,s : k,l) I r,s, r, s N, inf + inf ur,s : ur,s 2 : k,l) I r,s, r, s N k,l) I r,s, r, s N. ))] pk,l ) v x k+m,l+n ), z,, z n ) v y k+m,l+n ) ), z,, z n 2 v x k+m,l+n + y k+m,l+n ) ), z,, z n v x k+m,l+n ) ), z,, z n v y k+m,l+n ) ), z,, z n 2 Therefore, gx + y) gx) + gy). Finally, we prove that the scalar multiplication is continuous. Let λ be any complex number. By definition, Then gλx) inf gλx) inf u r,s : λ t) ur,s : k,l) I r,s, r, s N. k,l) I r,s, r, s N, v λx k+m,l+n v x k+m,l+n t ), z,, z n ), z,, z n

M. MURSALEEN, S.. SHARMA: RIESZ LACUNARY ALMOST CONVERGENT DOUBLE... 5 where t λ. Since λ ur,s max, λ sup ur,s ), we have gλx) max, λ sup u r,s ) inf t ur,s : k,l) I r,s v x k+m,l+n t ), z,, z n, r, s N. So, the fact that scalar multiplication is continuous follows from the above inequality. completes the proof of the theorem. To prove the next theorem we need the following lemma. This Lemma 2.. Let M be an Orlicz function which satisfies 2 -condition and let 0 < δ <. Then for each x δ we have Mx) < δ M2) for some constant > 0. Theorem 2.3 For a sequence of Orlicz functions M ) which satisfies 2 -condition, we have R2, θ r,s, p, v,,, ] R2, θ r,s, M, p, v,,, ]. R2 Proof. Let x x k,l ), θ r,s, p, v,,, ] so that for each m and n, we have D r,s x x k,l ) : P r,s v x k+m,l+n L k,l) I r,s uniformly in m and n for some L., z,, z n 0, Let ϵ > 0 and choose δ with 0 < δ < such that t) < ϵ for every t with 0 t < δ. Now, we have v x k+m,l+n L, z,, z n r,s k,l) I r,s r,s + r,s ϵ) + r,s < k,l) Ir,s v x k+m,l+n L,z,,z n <δ k,l) Ir,s v x k+m,l+n L,z,,z n δ k,l) Ir,s v x k+m,l+n L,z,,z n >δ ϵ) + δ ) Mk,l 2)Hr,s D r,s. v x k+m,l+n L v x k+m,l+n L v x k+m,l+n L )), z,, z n )), z,, z n )), z,, z n Therefore by above Lemma as R2 r and s goes to infinity in the Pringsheim sense, for each m and n, we have x x k,l ), θ r,s, M, p, v,,, ]. This completes the proof of the theorem. Theorem 2.4 Let 0 < inf u k,l h u k,l sup u k,l H < and M ), M M k,l ) be two sequences of Orlicz functions which satisfying 2 -condition, we have 0

52 TWMS J. PURE APPL. MATH., V.8, N., 207 i) R2, θ r,s, M, p, v, u,,, ] R2, θ r,s, M M, p, v, u,,, ] and ] ] ii) R2, θ r,s, M, p, v, u,,, R2, θ r,s, M M, p, v, u,,,. R2 0 0 Proof. Let x x k,l ), θ r,s, M, p, v, u,,, ]. Then we have r,s v x k+m,l+n L, z,, z n )] 0, k,l) I r,s uniformly in m and n for some L and > 0. Let ϵ > 0 and choose δ with 0 < δ < such that t) < ϵ for 0 t δ. Let y k,l v x k+m,l+n L )), z,, z n for all k, l N. We can write k,l) Ir,s y k,l δ y k,l h r,s k,l) I r,s + Since M ) satisfies 2 -condition, we have y k,l )] u k,l )] H For y k,l > δ 2)] H k,l) Ir,s y k,l δ k,l) Ir,s y k,l>δ k,l) Ir,s y k,l δ k,l) Ir,s y k,l δ u y k,l)] k,l u y k,l)] k,l. y k,l )] u k,l y k,l < y k,l < + y k,l δ δ. Since M ) is non-decreasing and convex, it follows that Mk,l y k,l ) ) ) < Mk,l + y ) k,l < δ 2 p k p l Mk,l 2)) + 2 Also ) satisfies 2 -condition, we can write ) y k,l ) Hence, k,l) Ir,s y k,l>δ u y k,l)] k,l y k,l )] p k,l 2.) 2yk,l δ )). < 2 T y k,l δ Mk,l 2) ) + 2 T y k,l δ p k p Mk,l 2) ) T y k,l δ Mk,l 2) ). T pk p l Mk,l 2) ) ) H ) max, δ k,l) Ir,s y k,l >δ by the inequalities 2.) and 2.2), we have x x k,l ) y k,l )] u k,l 2.2) R2, θ r,s, M M, p, v, u,,, ]. This completes the proof of i). Similarly, we can prove that R2, θ r,s, M, p, v, u,,, ] 0 R2, θ r,s, M M, p, v, u,,, ] 0.

M. MURSALEEN, S.. SHARMA: RIESZ LACUNARY ALMOST CONVERGENT DOUBLE... 53 3. Inclusion relations Let M ) be a sequence of Orlicz function, u u k,l ) be any factorable double sequence of strictly positive real numbers and p µ ), p η ) be sequences of positive numbers and P µ p + p 2 +... + p µ, P η p + p 2 +... + p η. In this section we define the following sequence spaces: R2, M, p, v, u,,, ] x x k,l ) : P µ,η and R2, M, p, v, u,,, ] x x k,l ) : P µ,η If we take Mx) x, we have R2, p, v, u,,, ] x x k,l ) : P and R2, p, v, u,,, ] µ η, z,, z n )] 0, P µ Pη k l uniformly in m and n, for some > 0 0 µ η v x k+m,l+n, z,, z n )] 0, P µ Pη k l uniformly in m and n, for some > 0. µ η, z,, z n ) 0, P µ Pη k l uniformly in m and n, for some > 0 µ,η 0 x x k,l ) : P µ η v x k+m,l+n, z,, z n ) 0, P µ Pη k l uniformly in m and n, for some > 0. µ,η If we take u k,l, for all k, l N, then we get R2, M, p, v,,, ] x x k,l ) : P µ,η and R2, M, p, v,,, ] x x k,l ) : P 0 µ,η µ η )], z,, z n 0, P µ Pη k l uniformly in m and n, for some > 0 µ η v x )] k+m,l+n, z,, z n 0, P µ Pη k l uniformly in m and n, for some > 0.

54 TWMS J. PURE APPL. MATH., V.8, N., 207 In R2 this section of the paper we study inclusion relations between the spaces, θ r,s, M, p, v, u,,, ] R2,, θ r,s, M, p, v, u,,, ] R2 ],, M, p, v, u,,, R2 0 and, M, p, v, u,,, ] 0. Theorem 3. Let M ) be a sequence of Orlicz functions, θ r,s k r, l s ) be a double lacunary sequence and p k ), p l ) be the sequences of positive numbers. If inf Q r > and r inf Q s >, then R2, M, p, v, u,,, ] R2, θ r,s, M, p, v, u,, ]. s Proof. Assume that inf Q r > and inf Q s >, then there exists δ > 0 such that Q r > +δ r s and Q s > +δ. This implies H r P k δ r +δ and H s P δ R2 ] l +δ. Then for x, M, p, v, u,,,, s we can write for each m and n A r,s, z,, z n k,l) I r,s k r l s k l k r l s k l k r kk r + l k r l s l s k ll s + P k r Pls P k Pls P k r Pls k r H r kk r + Hs l s ll s + k r l s k l P kr Pls k r l s k l l, z,, z n, z,, z n l P ls s H s P ls kl, z,, z n, z,, z n ), z,, z n ), z,, z n, z,, z n k P kr r., z,, z n )] H r P kr R2 Since x, M, p, v, u,,, ] the last two terms tend to zero uniformly in m and n in the Pringsheim sense, thus for each m and n, we have A r,s P k r Pls s P k r Pls k r l s k l P kr Pls k r l s k l ), z,, z n ), z,, z n + 0).

M. MURSALEEN, S.. SHARMA: RIESZ LACUNARY ALMOST CONVERGENT DOUBLE... 55 Since P kr P ls P kr P ls, for each m and n we have the following: P kr P ls + δ δ and P k r P ls δ. The terms and P kr Pls P kr Pls k r l s k l k r k l l s, z,, z n, z,, z n are both Pringsheim null sequences R2 for all m and n. Thus A r,s is a Pringsheim null sequence for each m and n. Therefore x, θ r,s, M, p, v, u,, ]. This completes the proof of the theorem. Theorem 3.2 Let M ) be a sequence of Orlicz functions, θ r,s k r, l s ) be a double lacunary sequence and p k ), p l ) be sequences of positive numbers. If sup Q r < and r sup Q s <, then R2, θ r,s, M, p, v, u,, ] R2, M, p, v, u,,, ]. s Proof. Since sup Q r < and sup Q s <, there exists H > 0 such that Q r < H and r r Q R2 s < H for all r and s. Let x, θ r,s, M, p, v, u,, ] and ϵ > 0. Then there exist r 0 > 0 and s 0 > 0 such that for every i r 0 and j s 0 and for all m and n, A i,j H i,j v x k+m,l+n, z,, z n )] < ϵ. k,l) I i,j Let M max A i,j : i r 0 and j s 0 and n and m be such that kr < n k r and l s < m l s. Thus we obtain the following: P n Pm n m k l, z,, z n k r l s P kr Pls k l r,s P kr Pls t,u, r 0,s 0 P kr Pls t,u, k,l) I t,u H t,u A t,u + P kr Pls r 0 <t r) s 0 <u s), z,, z n ), z,, z n H t,u A t,u

56 TWMS J. PURE APPL. MATH., V.8, N., 207 M P kr0 ) Pls0 + sup A t,u P kr Pls t r 0 u s 0 M P kr0 Pls0 P kr Pls + M P kr0 Pls0 P kr Pls + P k r P kr ϵ H t,u P kr Pls r 0 <t r) s 0 <u s) H t,u P kr Pls r 0 <t r) s 0 <u s) Pls P ls ϵ M P kr0 Pls0 P kr Pls + Q r Qs ϵ M P kr0 Pls0 P kr Pls + ϵh 2. Since P k,l and P ls as r, s, it follows that n m, z,, z n )] 0, P n Pm k l R2 uniformly in m and n. Therefore x, M, p, v, u,,, ]. Theorem 3.3 Let M ) be a sequence of Orlicz functions, θ r,s k r, l s ) be a double lacunary sequence and p k ), p l ) be the sequences of positive numbers. If sup Q r < and r sup Q s <, then R2, θ r,s, M, p, v, u,,, ] R2, M, p, v, u,,, ]. s Proof. It is easy to prove by using Theorem 3. and Theorem 3.2. Theorem 3.4 Let M ) be a sequence of Orlicz functions. Then the following statements are true: a) If p k < and p l < for all k, l N, then ] AC θr,s, M, v, u,,, R2, θ r,s, M, p, v, u,,, ] with P x R2, θ r,s, M, p, v, u,,, ] P x L. b) If p k > and p l > for all k, l N, then ] R2, θ r,s, M, p, v, u,,, ] AC θr,s, M, v, u,,, with ] R2, θ r,s, M, p, v, u,,, ] P x AC θr,s, M, v, u,,, ] AC θr,s, M, v, u,,, P x L. Proof. a) If p k < and p l < for all k, l N, then H r < h r for all r N and H s < h s for all s N, respectively. Then there exist two constants M and N such that 0 < M Hr h r < for all r N and 0 < N H s < for all s N. Let x x h s k,l ) be a double sequence with ] P x L in AC θr,s, M, v, u,,,, then for each m and n k,l) I r,s < H r Hs k,l) I r,s Mh r N h s MN k,l) I r,s h r,s k,l) I r,s, z,, z n, z,, z n v x k+m,l+n., z, z n )]

M. MURSALEEN, S.. SHARMA: RIESZ LACUNARY ALMOST CONVERGENT DOUBLE... 57 Hence, we obtain the result by taking the P -it as r, s. b) Let Hr h r and H s be upper bounded and p h s k > for all k N and p l > for all l N. Then H r > h r for all r N and H s > h s for all s N. Let there exists two constants M and N such that < H r h r N < for all s N. Suppose that the M < for all r N and < H s h s double sequence x x k,l ) converges to the P L R2, θ r,s, M, p, v, u,, ], with R2, θ r,s, M, p, v, u,, ] P x L, then for each m and n we have )] h r,s k,l) I r,s v x k+m,l+n L h r hs < M H r N H s MN k,l) I r,s, z,, z n, z,, z n k,l) I r,s v x k+m,l+n., z, z n )] k,l) I r,s Hence, the result is obtained by taking the P as r, s. 4. Statistical convergence The notion of statistical convergence was introduced by Fast 8] and Schoenberg 42] independently. Over the years and under different names, statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory and number theory. Later on, it was further investigated from the sequence space point of view and linked with summability theory by Fridy 9], Connor 6], Salat 40], Mursaleen and Edely 29], Mursaleen and Mohiuddine 30], Isık 5], Savas 4], olk 7], Maddox 2] and many others. In recent years, generalizations of statistical convergence have appeared in the study of strong integral summability and the structure of ideals of bounded continuous functions on locally compact spaces. Statistical convergence and its generalizations are also connected with subsets of the Stone-Cech compactification of natural numbers. Moreover, statistical convergence is closely related to the concept of convergence in probability. The notion depends on the density of subsets of the set N of natural numbers. A subset E of N is said to have the natural density δe) if the following it exists: δe) n n n k χ Ek), where χ E is the characteristic function of E. It is clear that any finite subset of N has zero natural density and δe c ) δe). A sequence x x k,l ) is said to be lacunary v -statistically convergent to L, if for every ϵ > 0 k, l) I r,s : v x k,l L), z,, z n ϵ 0. r,s In this case we write x k,l L S θr,s v )). The set of all lacunary v -statistically convergent sequences is denoted by S θr,s v ). Let θ r,s k r, l s ) be a double lacunary sequence. The double number sequence x is said to be S R 2,θ r,s, v P -convergent to L provided that for every ϵ > 0, P sup k, l) I r,s r,s : v x k+m,l+n L ϵ 0. m,n

58 TWMS J. PURE APPL. MATH., V.8, N., 207 In this case we write S R 2,θ r,s, v P x L. A double sequence x x k,l ) is said to be Riesz lacunary almost P -convergent to L if P ω mn r,s rs x) L, uniformly in m and n, where ωrs mn ω mn rs x) k,l) I r,s v x k+m,l+n ). A double sequence x x k,l ) is said to be Riesz lacunary almost statistically summable to L if for every ϵ > 0 the set ϵ r, s) N N : ω mn rs L ϵ has double natural density zero, i.e., δ 2 ϵ ) 0. In this case, we write R, θ, v ) st2 P x L. That is, for every ϵ > 0, P m,n r m, s n : ω mn rs L ϵ 0, uniformly in m and n. Hence, a double sequence x x k,l ) is Riesz lacunary almost statistically summable to L if and only if the double sequence ωrs mn x)) is almost statistically P -convergent to L. Note that since a convergent double sequence is also statistically convergent to the same value, a Riesz lacunary almost convergent double sequence is also Riesz lacunary almost statistically summable with the same P -it. A double sequence x x k,l ) is said to be strongly R2, θ r,s, p, v ] q -almost convergent 0 < q < ) to the number L if P rs v x L q ) 0, uniformly in m and n. In this R2 ] ) R2 ] case, we write x k,l L, θ r,s, p, v and L is called, θ r,s, p, v P of x. R2 q ] q Also, we denote the set of all strongly, θ r,s, p, v -almost P -convergent double sequences R2 ] q by, θ r,s, p, v q. r,s ω mn Theorem 4. Let θ r,s k r, l s ) be a double lacunary sequence. If I r,s I r,s, then R2, θ r,s, p, v] S R 2,θ r,s, v ). Proof. Pr,s ϵ) k, l) I r,s : v x k+m,l+n L ϵ. 2) Suppose that x R2, θ r,s, p, v]. Then for each m and n. Since P r,s k,l) I r,s v x k+m,l+n L 0. v x k+m,l+n L k,l) I r,s k,l) Ir,s k,l) Pr,sϵ) v x k+m,l+n L + k,l) Ir,s k,l) P r,sϵ) k,l) I r,s k,l) Ir,s k,l)/ Pr,sϵ) v x k+m,l+n L P r,s ϵ) v x k+m,l+n L v x k+m,l+n L m and n we get P r,s Pr,sϵ) 0 for each m and n. This implies that x S R 2,θ r,s, v ).

M. MURSALEEN, S.. SHARMA: RIESZ LACUNARY ALMOST CONVERGENT DOUBLE... 59 Theorem 4.2 Let M be a constant such that v x k+m,l+n L M, k, l N and for all m and n. If I r,s I r,s, then S R 2,θ r,s, v ) R2, θ r,s, p, v] with R2, θ r,s, p, v] P x S R 2,θ r,s, v ) P x L. Proof. Suppose that θ r,s k r, l s ) be a double lacunary sequence v x k+m,l+n L M, k, l N and for all m and n. Let I r,s I r,s and Pr,s be defined in eqn. 4.). Since x S R 2,θ r,s, v ) with S R 2,θ r,s, v ) P x L, then P and for all m and n we have v x k+m,l+n L k,l) I r,s k,l) I r,s k,l) P r,sϵ) v x k+m,l+n L + M P r,s + ϵ. k,l) I r,s k,l) I r,s k,l)/ P r,sϵ) 0. For a given ϵ > 0 r,s Pr,s Since ϵ is arbitrary, we get x R2, θ r,s, p, v] with the same P. v x k+m,l+n L v x k+m,l+n L Theorem 4.3 If p k, for all k N and p l for all l N, then S θr,s, v ) S R 2,θ r,s, v ) with S θr,s, v ) P x S R 2,θ r,s, v ) P x L. Proof. If p k for all k N and p l for all l N, then H r h r for all r N and H s h s for all s N. So, there exist constants M and N such that 0 < M Hr h r for all r N and 0 < N < H s for all s N. Let x x h s k,l ) be a double sequence which converges to the P -it L in S θr,s, v ), then for an arbitrary ϵ > 0, and for all m and n, we have k, l) I r,s : υ x x+m,l+n L ε H r H s Pkr < k P kr and P ls < l P ls : υ x k+m,l+n L ε Pkr k r < k P kr k r and MN h r h s P ls < l s < l P ls l s : υ x k+m,l+n L ε k r < k k r MN h r.s and l s < l l s : υ x x+m,l+n L ε k, l) I r,s : υ x x+m,l+n L ε. MN h r.s Hence, we obtain the result by taking the P -it as r, s. Theorem 4.4 If p k, for all k N and p l for all l N and Hr h r, H s are upper bounded, h s then S R 2,θ r,s, v ) S θr,s, v ) with S R 2,θ r,s, v ) P x S θr,s, v ) P x L. Proof. If p k for all k N and p l for all l N and H r h r, H s are upper bounded, then h s H r h r for all r N and H s h s for all s N. So, there exist constants M and N such that Hr h r M for all r N and H s h s N for all s N. Let x x k,l ) be a double sequence

60 TWMS J. PURE APPL. MATH., V.8, N., 207 which converges to the P -it L in S R 2,θ r,s, v ) with S R 2,θ r,s, v ) P x L, then for an arbitrary ϵ > 0, and for all m and n, we have h r hs k, l) I r,s : υ x k+m,l+n L ε H r Hs Pkr < k P kr and P ls < l P ls : υ x k+m,l+n L ε Pkr k r < k P kr k r MN h r hs and P ls l s < l P ls l s : υ x k+m,l+n L ε k r < k k r MN h r,s and l s < l l s : υ x k+m,l+n L ε k, l) I r,s : υ x k+m,l+n L ε. MN h r,s Hence, we obtain the result by taking the P -it as r, s. Theorem 4.5 Let I r,s I r,s and v x k+m,l+n L M for all k, l N and for all m and n. If the following hold a) 0 < q < and v x k+m,l+n L < b) q < and 0 v x k+m,l+n L <, then S R 2,θ r,s, v ) R 2, θ r,s, v, p] q and S R 2,θ r,s, v ) P x R 2, θ r,s, v, p] q P x L. Proof. Let x x k,l ) S R 2,θ r,s, v ) with P Pr,s 0, where Pr,s ϵ) was defined r,s in eqn. 4.). Since v x k+m,l+n L M for all k, l N and for all m and n. If I r,s I r,s, then for a given ϵ > 0 and for all m and n, we have where and k, l) I r,s : υ x k+m,l+n L ε h r hs + k,l) I r,s k,l)/ P r,s ε) k,l) I r,s k,l)/ P r,s ε) υ x k+m,l+n L q + k,l) I r,s υ x k+m,l+n L q A r,s + B r,s A r,s B r,s k,l) Ir,s k,l)/ P r,s ϵ) k,l) Ir,s k,l)/ P r,s ϵ) v x k+m,l+n L q v x k+m,l+n L q. υ x k+m,l+n L q

M. MURSALEEN, S.. SHARMA: RIESZ LACUNARY ALMOST CONVERGENT DOUBLE... 6 For k, l) / Pr,s ϵ), we have A r,s k,l) Ir,s k,l)/ Pr,s ϵ) If k, l) P Pr,s ϵ), then v x k+m,l+n L q B r,s k,l) Ir,s k,l)/ P r,s ϵ) k,l) I r,s k,l) P r,s ϵ) k,l) Ir,s k,l)/ Pr,s ϵ) v x k+m,l+n L q v x k+m,l+n L M Pr,s ϵ). v x k+m,l+n L ϵ. Hence v x k+m,l+n L q 0, uniformly in m and n. r,s k,l) I r,s This completes the proof. Theorem 4.6 Let I r,s I r,s and. If the following hold a) 0 < q < and 0 v x k+m,l+n L < b) q < and v x k+m,l+n L <, then R 2, θ r,s, v, p] q S R 2,θ r,s, v ) and R 2, θ r,s, v, p] q P x S R 2,θ r,s, v ) P x L. Proof. Let x x k,l ) R 2, θ r,s, v, p] q -almost P -convergent to the it L. Since v x k+m,l+n L q v x k+m,l+n L for case a) and b), then for all m and n, we have v x k+m,l+n L q v x k+m,l+n L k,l) I r,s k,l) I r,s k,l) P r,s ϵ) k,l) I r,s v x k+m,l+n L q ϵ Pr,s ϵ) where Pr,s was defined in eqn.4.). Taking it as r, s in both sides of the above inequality, we conclude that S R 2,θ r,s, v P x L. This completes the proof of the theorem. References ] Alotaibi, A., Mursaleen, M., Sharma, S.., 204), Double sequence spaces over n-normed spaces defined by a sequence of Orlicz functions, J. Inequal. Appl., 26, 2p. 2] Altay, B., Başar, F., 2005), Some new spaces of double sequences, J. Math. Anal. Appl., 309, pp.70-90. 3] Başarır, M., Sonalcan, O., 999), On some double sequence spaces, J. Indian Acad. Math., 2, pp.93-200. 4] Başar, F., Sever, Y., 2009), The space L p of double sequences, Math. J. Okayama Univ., 5, pp.49-57. 5] Bromwich, T.J., 965), An Introduction to the Theory of Infinite Series, Macmillan, New York. 6] Connor, J.S., 988), The statistical and strong p-cesaro convergence of sequeces, Analysis Munich), 8, pp.47-63. 7] Et, M., Çolak, R., 995), On generalized difference sequence spaces, Soochow J. Math., 2, pp.377-386. 8] Fast, H., 95), Sur la convergence statistique, Colloq. Math., 2, pp.24-244. 9] Fridy, J.A., 985), On the statistical convergence, Analysis Munich), 5, pp.30-303. 0] Gahler, S., 965), Linear 2-normietre Rume, Math. Nachr., 28, pp.-43. ] Gunawan, H., 200), On n-inner Product, n-norms, and the Cauchy-Schwartz Inequality, Sci. Math. Jpn., 5, pp.47-54.

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M. MURSALEEN, S.. SHARMA: RIESZ LACUNARY ALMOST CONVERGENT DOUBLE... 63 45] Yeşilkayagil, M.F., Başar, F., 206), Some topological properties of the spaces of almost null and almost convergent double sequences, Turkish J. Math., 403), pp.624-630. 46] Yeşilkayagil, M.F., Başar, F., 206), Mercerian theorem for four dimensional matrices, Commun. Fac. Sci. Univ. Ankara Ser. A, 65), pp.47-55. M. Mursaleen is a full Professor & Chairman of Department of Mathematics, Aligarh Mus University, India. His main research interests are sequence spaces, summability, approximation theory and operator theory. Sunil. Sharma is an Assistant Professor in Department of Mathematics, Model Institute of Engineering and technology, ot Bhalwal, Jammu. His area of research interest is functional analysis and sequence spaces.