Research Article On Some Lacunary Almost Convergent Double Sequence Spaces and Banach Limits

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1 Abstract and Applied Analysis Volume 202, Article ID , 7 pages doi:0.55/202/ Research Article On Some Lacunary Almost Convergent Double Sequence Spaces and Banach Limits Metin Başarır and Şükran Konca Department of Mathematics, Sakarya University, 5487 Sakarya, Turkey Correspondence should be addressed to Metin Başarır, basarir@sakarya.edu.tr Received 3 June 202; Accepted 9 July 202 Academic Editor: Chaitan Gupta Copyright q 202 M. Başarır andş. Konca. 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. The object of this paper is to introduce some new sequence spaces related with the concept of lacunary strong almost convergence for double sequences and also to characterize these spaces through sublinear functionals that both dominate and generate Banach limits and to establish some inclusion relations.. Introduction and Preliminaries Let w 2 be the set of all real or complex double sequences. We mean the convergence in the Pringsheim sense, that is, a double sequence x x i,j has a Pringsheim limit λ denoted i,j0 by P lim x λ provided that given ε>0and there exists N N such that x i,j λ <ε whenever i, j N. We denote by c 2, the space of P-convergent sequences. A double sequence x x i,j is bounded if x i,j 0 x i,j <. Letl 2 and c 2 be the set of all real or complex bounded double sequences and the set of bounded and convergent double sequences, respectively. Moricz and Rhoades 2 defined the almost convergence of double sequences that x x i,j is said to be almost convergent to a number λ if mp lim nq ) ) x i,j λ m,n 0 p q 0,. im jn that is, the average value of x i,j taken over any rectangle D { i, j ) : m i m p, n j n q },.2

2 2 Abstract and Applied Analysis tends to λ as both p and q tend to and this convergence is uniform in m and n. We denote the space of almost convergent double sequences by f 2,as f 2 { x x i,j ) : lim k,l t klpq x λ } 0, uniformly in p, q,.3 where pk ql t klpq x x i,j. k l ip jq.4 The notion of almost convergence for single sequences was introduced by Lorentz 3 and for double sequences by Moricz and Rhoades 2 and some further studies are in 4 4. A double sequence x is called strongly almost convergent to a number λ if lim k,l pk ql xi,j λe 0, uniformly in p, q..5 k l ip jq By f 2, we denote the space of all strongly almost convergent double sequences. It is easy to see that the inclusions c 2 f 2 f 2 l 2 strictly hold. As in the case of single sequences, every almost convergent double sequence 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. The notion of strong almost convergence for single sequences has been introduced by Maddox 5, 6 and for double sequences by Başarir 7. A linear functional L on l 2 is said to be Banach limit if it has the following properties 7, Lx 0ifx 0 i.e., x i,j 0 for all i, j, 2 Le, where e e i,j with e i,j for all i, j and 3 Lx LS 0 xls 0 xls x where the shift operators S 0 x, S 0 x, S x are defined by S 0 x x i,j,s 0 x x i,j,s x x i,j. Let B 2 be the set of all Banach limits on l 2. A double sequence x x i,j is said to be almost convergent to a number λ if Lx λ for all L B 2.Ifϕis any sublinear functional on l 2, then we write {l 2,ϕ} to denote the set of all linear functionals F on l 2, such that F ϕ, that is, Fx ϕx, x l 2. A sublinear functional ϕ is said to generate Banach limits if F {l 2,ϕ} implies that F is a Banach limit; ϕ is said to dominate Banach limits if F is a Banach limit implies that F {l 2,ϕ}. Then if ϕ both generates and dominates Banach limits, then {l 2,ϕ} is the set of all Banach limits. Using the notations for single sequences, we present the notations for double-lacunary sequences that can be seen in 0. The double sequence r,s {k r,l s } is called a doublelacunary if there exist two increasing sequences of nonnegative 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.letk r,s k r l s, h r h s, r,s is determined by I r,s {i, j : k r <i k r and l s <j l s }.Also k r l s k r l s and r,s is determined by I r,s {i, j : k r <i k r l s <j l s }/I I 2, where

3 Abstract and Applied Analysis 3 I {i, j : k r <i k r and l s <j<.} and I 2 {i, j : l s <j l s and k r <i<.} with q r k r /k r,q s l s /l s and q r,s q r q s. Das and Mishra 8 introduced the space of lacunary almost convergent sequences by combining the space of lacunary convergent sequences and the space of almost convergent sequences. Savaş and Patterson 0 extended the notions of lacunary almost convergence and lacunary strongly almost convergence to double-lacunary P-convergence and doublelacunary strongly almost P-convergence. They also established multidimensional analogues of Das and Patel s results. We will use the following definition which may be called convergence in Pringsheim s sense with a bound: xi,j L ) O, i, j ),.6 and also we will use the following definition which may be called convergence in Pringsheim s sense as follows: xi,j L ) o, i, j )..7 The following sequence spaces were introduced and examined by Başarir 9: { w x : lim r { w x : lim r { w x : lim r i i } t ki x s 0, for some s, h r k I r } t ki x s 0, for some s, h r k I r } t ki x s 0, for some s, h r k I r i.8 with respect to sublinear functionals on l the set of all real or complex bounded single sequences by φ x lim r ψ x lim r ζ x lim r i i i t ki x, h r k I r t ki x, h r k I r t ki x, h r k I r.9 where t ki x /k ik ji x j and x x j j. It can be easily seen that each of the above functionals are finite, well defined, and sublinear on l. There is a very close connection among these sequence spaces with the sublinear functionals which were given by Başarir 9. Recently Mursaleen and

4 4 Abstract and Applied Analysis Mohiuddine 7 generalized the sequence spaces which were studied by Das and Sahoo 20 for single sequences, to the double sequences as follows: { w 2 x ) m n x i,j : t klpq x λe 0, m n k0 l0 } as m, n, uniformly in p, q, for some, { w 2 x ) m n x i,j : t klpq x λe 0, m n k0 l0 as m, n, uniformly in p, q, for some λ, }.0 { w 2 x ) m n x i,j : t klpq x λe 0, m n k0 l0 as m, n, uniformly in p, q, for some λ } by using.4. The object of the present paper is to determine some new sublinear functionals involving double-lacunary sequence that both dominates and generates Banach limits. We also extend the sequence spaces which were introduced for single sequences by Başarir 9 to the double sequences with respect to these sublinear functionals. Furthermore, we present some inclusion relations with these new sequence spaces between the sequence spaces which were introduced by Mursaleen and Mohiuddine 7, earlier. 2. Sublinear Functionals and Double-Lacunary Sequence Spaces In this section, we introduce the following sequence spaces: w 2 x x i,j ) : P lim [ ] w 2 x ) x i,j : P lim w 2 x x i,j ) : P lim r,s r,s r,s W, 2 x ) x i,j : P lim r,s t klpq x λe 0, for some λ, t klpq x λe 0, for some λ, t klpq x λe 0, for some λ, t kl00 x λe 0, for some λ,

5 Abstract and Applied Analysis 5 W, 2 x ) x i,j : P lim t kl00 x λe 0, for some λ r,s, W, 2 x ) x i,j : P lim t kl00 x λe 0, for some λ r,s. 2. It may be noted that almost convergent double sequences are necessarily bounded but the sequence spaces w 2 and w2 may contain unbounded sequences. Now we define the following functionals on l 2 for a double-lacunary sequence r,s by, φ 2 x lim ψ 2 x lim ϕ 2 x lim r,s r,s r,s t klpq x, tklpq x, t klpq x, 2.2 ζ 2 x lim η 2 x lim k,l k,l t klpq x, t klpq x. It is easy to see that each of the above functionals are finite, well defined, and sublinear on l 2. Throughout the paper we will write lim r,s for P lim r,s and by this notation we shall mean the convergence in the Pringsheim sense. In the following theorem, we demonstrate that {l 2,φ2 } is the set of all Banach limits on l 2 and characterize the space w2 l 2 in terms of the sublinear functional φ 2. Theorem 2.. One has the following. ) The sublinear functional φ 2 both dominates and generates Banach limits, that is, φ2 x ζ 2 x, for all x x i,j l 2. 2) { f 2 x ) } x i,j l 2 : φ2 x φ2 x x ) x i,j l 2 : t klpq x λ, as r, s, uniformly in p, q. 2.3 w 2 l 2.

6 6 Abstract and Applied Analysis Proof. From the definition of ζ 2, for given ε>0 there exist k 0,l 0 such that t klpq x <ζ 2 x ε, 2.4 for k k 0, l l 0 and for all p, q. This implies that φ 2 x <ζ 2x ε, 2.5 for all x x i,j l 2. Since ε is arbitrary, so that φ2 x ζ 2x, for all x x i,j l 2 and hence { } l 2,φ2 { l 2,ζ 2} B2, 2.6 that is, φ 2 generates Banach limits. Conversely, pose that L B 2.AsL is the shift invariant, that is, LS xlx LS 0 xls 0 x and using the properties of L B 2,weobtain Lx L pk ql x i,j L t klpq x ) k l ip jq L t klpq x t klpq x. 2.7 It follows from the definition of φ 2, that for given ε>0 there exist r 0,s 0 such that t klpq x <φ 2 x ε, 2.8 for r r 0, s s 0 and for all p, q. Hence by 2.8 and properties and 2 of Banach limits, we have L ) ) t klpq x <L φ 2 x ε e φ 2 x ε, 2.9 for r r 0, s s 0 and for all p, q; where e e i,j with e i,j for all i, j. Since ε is arbitrary, it follows from 2.7 and 2.9 that Lx φ 2 x, x x i,j ) l Hence B 2 { } l 2,φ2. 2.

7 Abstract and Applied Analysis 7 That is, φ 2 dominates Banach limits. Combining 2.6 and 2.,weget { l 2,ζ } { } 2 l 2,φ2, 2.2 this implies that φ 2 dominates and generates Banach limits and φ2 x ζ 2x for all x l 2. 2 As a consequence of Hahn-Banach theorem, {l 2,φ2 } is non empty and a linear functional F {l 2,φ2 } is not necessarily uniquely defined at any particular value of x. This is evident in the manner the linear functionals are constructed. But in order that all the functionals {l 2,φ2 } coincide at x x i,j, it is necessary and sufficient that φ 2 x φ2 x, 2.3 we have lim r,s t klpq x lim inf r,s inf t klpq x. 2.4 But 2.4 holds if and only if t klpq x λ, as r, s, uniformly in p, q. 2.5 Hence, x x i,j w 2 l 2.But2.3 is equivalent to ζ 2x ζ 2 x, this holds if and only if x x i,j f 2. This completes the proof of the theorem. If Fx λe 0 for all F {l 2,ψ2 }, then we say that x x i,j is ψ 2 -convergent to λ. Similarly we define the ϕ 2 -convergent sequences. In the following theorem we characterize the spaces w 2 l 2 and w2 l 2 in terms of the sublinear functionals. Theorem 2.2. One has the following: w 2 l 2 {x x i,j : ψ 2 x λe 0, for some λ} = {x x i,j : Fx λe 0, for all F {l 2,ψ2 }, for some λ} 2 w 2 l 2 {x x i,j : ϕ 2 x λe 0, for some λ} = {x x i,j : Fx λe 0, for all F {l 2,ϕ2 }, for some λ}. Proof. It can be easily verified that x x i,j w 2 l 2 if and only if ψ 2 x λe ψ2 λe x. 2.6 Since ψ 2 x ψ2 x then 2.6 reduces to ψ 2 x λe

8 8 Abstract and Applied Analysis Now if F {l 2,ψ2 } then from 2.7 and linearity of F, we have Fx λe Conversely, pose that Fx λe 0 for all F {l 2,ψ2 } and hence by Hahn-Banach theorem, there exists F 0 {l 2,ψ2 } such that F 0x ψ 2 x. Hence 0 F 0 x λe ψ 2 x λe The proof is similar to the proof of, above. 3. Inclusion Relations We establish here some inclusion relations between the sequence spaces defined in Section 2. Theorem 3.. We have the following proper inclusions and the limit is preserved in each case. f 2 w 2 w2 w2 W, 2. 2 w 2 w2 W, 2 W, 2. 3 w 2 W, 2 W, 2 W, 2. Proof. Let x f 2 with f 2 lim x λ. Then t klpq x λe 0, as k, l, uniformly in p, q. 3. This implies that t klpq x λe 0, as k, l, uniformly in p, q. 3.2 This proves that x w 2 and f 2 lim x w 2 lim x λ. Since t klpq x λe tklpq x λe t klpq x λe, 3.3 this implies that w 2 w2 w2 and w2 lim x w2 lim x w2 lim x λ. Since t klpq x λe 3.4

9 Abstract and Applied Analysis 9 converges uniformly in p, q as r, s, implies the convergence for p 0 q. It follows that w 2 W, 2 and w2 lim x W, 2 lim x λ. This completes the proof of. It is easy to see the proof of 2 and 3. So we omit them. Theorem 3.2. One has the following proper inclusions; [ ] ) [ ] ) f2 w 2 l 2 w 2 l 2 f Proof. The proof of the theorem is similar as in 7, Theorem 4.2. So we omit it. Prior to giving Lemmas 3.3 and 3.5, we need the following notations used in 0: I C { i, j ) : q j q n, p m<i< }, I 2 C { i, j ) : q n<j<, p i p m }, C m,n { ) } i, j :p i p m or ) \ I q j q n C I2 C, I D { i, j ) : q y ) h s <j<, p xh r i p x h r }, I 2 D { i, j ) : q yh s j q y ) h s, p x h r <i< }, D x,y { ) } i, j :p xhr i p x h r or q yh s j q y ) ) \ I h s D I2 D. 3.6 Lemma 3.3. Suppose ε>0 there exist m 0,n 0,p 0, and q 0 such that C m,n k,l C m,n C pk,ql xi,j λe <ε, 3.7 for m m 0,n n 0 and p p 0,q q 0.Thenx w 2. Proof. Let ε>0 be given. Choose m 0,n 0, p 0 and q 0 such that C m,n k,l C m,n C pk,ql xi,j λe < ε for m m 0,n n 0,p p 0,q q 0. We need only to show that given ε>0there exist m 2 0 and n 2 0 such that C m,n k,l C m,n C pk,ql x i,j λe <ε, 3.9

10 0 Abstract and Applied Analysis for m m 2 0,n n2 0 and 0 p p 0, 0 q q 0. If we take m 0 max{m 0,m2 0 } and n 0 max{n 0,n2 0 }, then 3.9 holds for m m 0,n n 0 and for all p and q, which gives the result. Once p 0 and q 0 have been chosen, they are fixed, so k,l C p 0,q 0 C pk,ql i,j C p 0,q 0 xi,j λe M, 3.0 is finite. Now taking 0 p p 0,0 q q 0 and m p 0, n q 0 we have from 3.8 and 3.0 C m,n k,l C m,n C m,n C m,n C m,n C m,n C pk,ql k,l C p 0,q 0 k,l C p 0,q 0 k,l C m,n p 0,q 0 i,j C m,n p 0,q 0 C pk,ql C pk,ql x i,j λe i,j C p 0,q 0 C pk,ql p 0,q 0 C pk,ql x i,j λe i,j C p 0,q 0 p 0,q 0 xi,j λe x i,j λe xi,j λe 3. M 3 ε 6 M ε C m,n 2. C m,n Therefore taking m and n sufficiently large, we can make M/ C m,n ε/2 <εwhich gives 3.9 and hence the result. Theorem 3.4. We have w 2 w 2 for every r,s. Proof. Let x w 2 ; then given ε>0 there exist r 0,s 0 and λ such that hr,hs k,l C t klpq x λe <ε, 3.2

11 Abstract and Applied Analysis for r r 0,s s 0, p k r α and q l s β where α, β 0. Let m h r such that m δ h r where δ is an integer. Also let n h s such that n δ 2 h s 2 where δ 2 is an integer and h r, 2 h s. Since m h r for δ andn h s for δ 2 we have, C m,n k,l C m,n C m,n t klpq x λe k,l C δ hr,δ 2 hs δ,δ 2 C m,n x,y k,l D x,y t klpq x λe t klpq x λe 3.3 δ,δ 2 C m,n x,y ε ) δ δ 2 C m,n ) ε o, which gives the result. Therefore by Lemma 3.3, w 2 w 2. It is clear that w 2 w2 for every r,s. This completes the proof. Lemma 3.5. Suppose, for a given ε>0, thereexistm 0,n 0,p 0, and q 0 such that C m,n k,l C m,n C pk,ql xi,j λe ) <ε, 3.4 for all m m 0,n n 0 and p p 0,q q 0.Thenx w 2. Proof. Let ε>0 be given and choose m 0,n 0,p 0 and q 0 such that C m,n k,l C m,n C pk,ql xi,j λe ) < ε for all m m 0,n n 0,p p 0,andq q 0.AsinLemma 3.3, it is enough to show that there exist m 0 and n 0 such that for m m 0,n n 0 implies C m,n k,l C m,n C pk,ql xi,j λe ) <ε, 3.6

12 2 Abstract and Applied Analysis for all p and q with 0 p p 0 and 0 q q 0. Since p 0 and q 0 are fixed, k,l C p 0,q 0 C pk,ql i,j C p 0,q 0 xi,j λe M. 3.7 Now, let 0 p p 0, 0 q q 0,andm p 0,n q 0 and consider the following; C m,n k,l C m,n C m,n C m,n C pk,ql k,l C p 0,q 0 k,l C p 0,q 0 C pk,ql C pk,ql xi,j λe ) i,j C p 0,q 0 C m,n k,l C m,n p 0,q 0 C pk,ql M C m,n C m,n C m,n k,l C p 0,q 0 k,l C m,n p 0,q 0 p 0,q 0 C pk,ql xi,j λe tklpq x λe. xi,j λe ) xi,j λe ) p 0,q 0 xi,j λe ) 3.8 Let k p 0 m 0, then k p p 0 m 0 for 0 p p 0.Alsoifweletl q 0 n 0, then l q q 0 n 0 for 0 q q 0. Therefore from 3.5 C m,n k,l C p 0,q 0 < C p 0,q 0 C pk,ql k,l C p 0,q 0 p 0,q 0 xi,j λe ) C p 0pk p 0,q 0 ql q 0 p 0,q 0 i,j C p 0 pk p 0,q 0 ql q 0 p 0,q 0 xi,j λe ) 3.9 < ε 4.

13 Abstract and Applied Analysis 3 From 3.8 and 3.9 C m,n k,l C m,n C pk,ql xi,j λe ) M 2 ε C m,n 4 <ε, 3.20 for sufficiently large values of m and n. Hence the result. Theorem 3.6. For every r,s, One has w 2 l 2 w 2 l 2. Proof. Let x w 2 l 2. For ε>0, there exist r 0,s 0,p 0 and q 0 such that hr,hs k,l C tklpq x λe < ε for r r 0,s s 0,p p 0 and q q 0 with p k r α where α 0, q l s β and β 0. Let m h r and n h s where m andn. Then C m,n k,l C m,n C pk,ql C m,n C m,n C m,n k,l C δ hr,δ 2 hs δ,δ 2 x, y0, 0 k,l D x,y k,l C m,n δ hr,δ 2 hs xi,j λe ) tklpq x λe tklpq x λe C pk,ql C m,n k,l C m,n δ hr,δ 2 hs xi,j λe. tklpq x λe 3.22 Since x i,j l 2 for all i and j, there exists M such that x i,j λe M.From3.2 and 3.22, we have the following: C m,n k,l C m,n C pk,ql xi,j λe ) δ δ 2 C m,n ε ) 2 Mhr,s C m,n Thus for m and n sufficiently large, we have the following: C m,n k,l C m,n C pk,ql xi,j λe ) <ε, 3.24

14 4 Abstract and Applied Analysis for r r 0,s s 0 and p p 0,q q 0.ThusbyLemma 3.5, we have w 2 l 2 clear that w 2 l 2 w2 l 2. This completes the proof of the theorem. w 2 l 2.Itis Corollary 3.7. f 2 w 2 w2 l 2 w2 l 2 f 2. Proof. It is easy to see by combining Theorem 3.4, Theorem 3.6 with 7, Theorems 4. and 3.ii. So we omit it. A paranormed space X, g is a topological linear space with the topology given by the paranorm g. It may be recalled that a paranorm g is a real subadditive function on X such that g 0, gx g x and scalar multiplication is continuous, that is, μ n μ, x n x imply that μ n x n μx where μ n,μare scalars and x n,x X. Let u u k,l be a bounded double sequence of positive real numbers, that is, u k,l > 0 for all k, l with k,l u k,l H<.Let ] [w 2 u x ) x i,j : lim r,s tklpq x λe u k,l 0 for some λ If u u k,l is constant we write w 2 u in place of w2 u. If we take u u k,l with u k,l for all k and l, then w 2 u is reduced to w2 which is defined in Section 2. Theorem 3.8. Let u u k,l be a bounded sequence of positive real numbers with k,l u k,l H<.Thenw 2 u is a complete linear topological space paranormed by gx r,s, t klpq x u k,l /M, 3.26 where M max,h. In the case u is constant, w 2 is a Banach space if u and is a p-normed u space if 0 <u<. Proof. It is easy to see that w 2 u is a linear space with coordinatewise addition and scalar multiplication. Clearly g 0, gx g xand g is subadditive. To prove the continuity of multiplication, assume that x w 2 u. Since u u k,l is bounded and positive there exists a constant δ>0such that u k,l δ for all k, l. Nowfor μ, μ u k,l μ δ and hence gμx μ δ/m gx. This proves the fact that g is a paranorm on w 2 u. To prove that w 2 u is complete, assume that xm is a Cauchy sequence in w 2 u, that is, gx m x n 0asm, n. Since tklpq x m x n u k,l [ gx m x n ] M, 3.27 it follows that t klpq x m x n u k,l 0 as m, n for each k, l, p,andq. In particular t 00pq x m x n x m x n 0 as m, n, for each fixed p and q. 3.28

15 Abstract and Applied Analysis 5 Hence, x m is a Cauchy sequence in R or C. Since R or C is complete, there exists x R or C such that x m x coordinate wise as m. It follows from 3.27 that given ε>0, there exists m 0 N such that t klpq x m x n u k,l /M <ε, 3.29 for m, n>m 0. Now making n and then taking remum with respect to p and q in 3.29 we obtain gx m x ε for m>m 0. This proves that x m x and x w 2 u. Hence u is complete. When u is constant, it is easy to derive the rest of the theorem. w 2 Theorem 3.9. Let 0 <ρ k,l σ k,l < for each k and l.thenw 2 ρ w2 σ. Proof. Let x w 2 ρ. By the definition of w2 ρ, that for given ε>0 there exist r 0,s 0 such that tklpq x λe ρ k,l <ε, 3.30 for r>r 0,s>s 0 and for all p, q. Since / 0asr, s, then tklpq x λe ρ k,l <, 3.3 for r>r 0,s>s 0 and for all p, q. This implies that t klpq x λe <, 3.32 for sufficiently large values of k, l and for all p, q. Then we get, tklpq x λe σ k,l tklpq x λe ρ k,l o, 3.33 as r, s and for all p, q. Hence, lim r,s t klpq x λe ρ k,l 0, 3.34 is obtained and consequently we have x w 2 ρ. This completes the proof.

16 6 Abstract and Applied Analysis Theorem 3.0. One has the following. Let 0 < inf k,l u k,l u k,l for each k and l. Thenw 2 u w2. 2 Let u k,l k,l u k,l < for each k and l.thenw 2 w2 u. Proof. It is clear from the above theorem. If we take ρ k,l u k,l and σ k,l for each k and l, then we have w 2 u w2. 2 From the above theorem, if we take ρ k,l andσ k,l u k,l for each k and l, then we have w 2 w2 u. This completes the proof. Acknowledgment The authors are grateful to anonymous referees for their careful reading of the paper which improved it greatly. References A. Pringsheim, Zur Theorie der zweifach unendlichen Zahlenfolgen, Mathematische Annalen, vol. 53, no. 3, pp , F. Moricz and B. E. Rhoades, Almost convergence of double sequences and strong regularity of summability matrices, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 04, no. 2, pp , G. G. Lorentz, A contribution to the theory of divergent sequences, Acta Mathematica, vol. 80, pp , M. Başarır and O. Sonalcan, On some double sequence spaces, Indian Academy of Mathematics. Journal, vol. 2, no. 2, pp , M. Mursaleen, Almost strongly regular matrices and a core theorem for double sequences, Journal of Mathematical Analysis and Applications, vol. 293, no. 2, pp , M. Mursaleen and O. H. H. Edely, Almost convergence and a core theorem for double sequences, Journal of Mathematical Analysis and Applications, vol. 293, no. 2, pp , M. Mursaleen and S. A. Mohiuddine, Banach limit and some new spaces of double sequences, Turkish Journal of Mathematics, vol. 36, no., pp. 2 30, M. Mursaleen and E. Savaş, Almost regular matrices for double sequences, Studia Scientiarum Mathematicarum Hungarica. A Quarterly of the Hungarian Academy of Sciences, vol. 40, no. -2, pp , E. Savaş and R. F. Patterson, On some double almost lacunary sequence spaces defined by Orlicz functions, Filomat, Faculty of Sciences and Mathematics. University of Niš, no. 9, pp , E. Savaş and R. F. Patterson, Double sequence spaces characterized by lacunary sequences, Applied Mathematics Letters, vol. 20, no. 9, pp , A. Gokhan, M. Et, and M. Mursaleen, Almost lacunary statistical and strongly almost lacunary convergence of sequences of fuzzy numbers, Mathematical and Computer Modelling, vol. 49, no. 3-4, pp , M. Mursaleen and S. A. Mohiuddine, On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space, Journal of Computational and Applied Mathematics, vol. 233, no. 2, pp , M. Zeltser, M. Mursaleen, and S. A. Mohiuddine, On almost conservative matrix methods for double sequence spaces, Publicationes Mathematicae Debrecen, vol. 75, no. 3-4, pp , G. A. Anastassiou, M. Mursaleen, and S. A. Mohiuddine, Some approximation theorems for functions of two variables through almost convergence of double sequences, Journal of Computational Analysis and Applications, vol. 3, no., pp , I. J. Maddox, A new type of convergence, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 83, no., pp. 6 64, 978.

17 Abstract and Applied Analysis 7 6 I. J. Maddox, On strong almost convergence, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 85, no. 2, pp , M. Başarır, On the strong almost convergence of double sequences, Periodica Mathematica Hungarica, vol. 30, no. 3, pp. 77 8, G. Das and S. K. Mishra, Banach limits and lacunary strong almost convergence, Journal of Orissa Mathematical Society, vol. 2, no. 2, pp. 6 70, M. Başarır, On some new sequence spaces, Rivista di Matematica della UniversitàdiParma, vol. 5, no., pp , G. Das and S. K. Sahoo, On some sequence spaces, Journal of Mathematical Analysis and Applications, vol. 64, no. 2, pp , 992.

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