SOME GENERALIZED SEQUENCE SPACES OF INVARIANT MEANS DEFINED BY IDEAL AND MODULUS FUNCTIONS IN N-NORMED SPACES

ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS N. 39 208 579 595) 579 SOME GENERALIZED SEQUENCE SPACES OF INVARIANT MEANS DEFINED BY IDEAL AND MODULUS FUNCTIONS IN N-NORMED SPACES Kuldip Raj School of Mathematics Shri Mata Vaishno Devi University Katra-82320, J&K India kuldipraj68@gmail.com S.A. Mohiuddine Operator Theory and Applications Research Group Department of Mathematics Faculty of Science King Abdulaziz University P.O. Box 80203 Jeddah 2589 Saudi Arabia mohiuddine@gmail.com M. Ayman Mursaleen Department of Mathematics Aligarh Muslilm University Aligarh 202002 India mohdaymanm@gmail.com Abstract. In the present paper we introduce and study some generalized difference sequence spaces of invariant means defined by ideal and a sequence of modulus functions over n-normed space. We study some topological properties and prove some inclusion results between these spaces. Further, we also study some results on statistical convergence. Keywords: ideal, Difference sequence space, modulus function, Lacunary sequence, Invariant mean, Statistical convergence.. Introduction and preliminaries Let σ be the mapping of the set of positive integers into itself. A continuous linear functional φ on l is said to be an invariant mean or σ-mean c.f. 5, 34, 35) if and only if. φx) 0 when the sequence x = x k ) has x k 0, for all k,. Corresponding author

580 KULDIP RAJ, S.A. MOHIUDDINE and M. AYMAN MURSALEEN 2. φe) =, where e =,,, ) and 3. φx σk) ) = φx), for all x l. If x = x n ), write T x = T x n = x σn) ). It can be shown in 4 that V σ = x l : lim t kn x) = l, uniformly in n, l = σ lim x, k where t kn x) = x n + x σ n +... + x σ k n. k + In the case σ is the translation mapping n n +, σ-mean is often called a Banach limit and V σ, the set of bounded sequences all of whose invariant means are equal, is the set of almost convergent sequences 24 also see 27). By using the concept of invariant means Mursaleen et al. 36 introduced the following sequence spaces: n w σ = x : lim t km x k l) 0, uniformally in m, n n + k=0 n w σ = x : lim t km x k l) 0, uniformally in m, n n + k=0 n w σ = x : lim t km x k l ) 0, uniformally in m, n n + k=0 and investigate some of its properties. The notion of statistical convergence has been introduced by Fast in 95 and later developed by Fridy 2, Salát 40, Mohiuddine and Belen 3 and many others. Furthermore, Kostyrko et al. 2 presented a very interesting generalization of statistical convergence called as I-convergence. The detailed history and development in this regard can be found by Connor 6, Maddox 25 and many others. By a lacunary sequence θ = k r ) where k 0 = 0, we shall mean an increasing sequence of non-negative integers with k r k r as r. The intervals determined by θ will be denoted by I r = k r, k r. We write = k r k r. k r k r The ratio will be denoted by q r. The space of lacunary strongly convergent sequence was defined by Freedman et al. 5 as follows: N θ = x = x k ) : lim x k L = 0 for some L. r Fridy and Orhan 3 generalized the concept of statistical convergence by using lacunary sequence which is called lacunary statistical convergence. Further, lacunary sequences have been studied by Fridy and Orhan 4. Quite recently, Karakaya 22 combined the approach of lacunary sequence with invariant means and introduced the notion of strong σ lacunary statistically convergence as follows:

SOME GENERALIZED SEQUENCE SPACES OF INVARIANT MEANS... 58 Definition.. 22 Let θ = k r ) be a lacunary sequence. A sequence x = x k ) is said to be lacunary strong σ lacunary statistically convergent if for every ε > 0, lim r k I r : t km x L) ε = 0 uniformaly in m where St θσ denotes the set of all lacunary strong σ lacunary statistically convergent sequences. 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 i.e A, B I A B I and A I, 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 i.e I) = K N : K c I, where K c = N \ K. Recently, Das et al. 7 unified the idea of lacunary statistical convergence with ideal convergence and presented the following interesting generalization of statistical convergence. Definition.2. 7 Let θ = k r ) be a lacunary sequence. A sequence x = x k ) is said to be I lacunary statistical convergent or S θ I) convergent to L, if for every ε > 0 and δ > 0, k I r : x k L ε δ I. In this case x k LS θ I)) or S θ I) lim k x k = L. The set of all I-lacunary statistically convergent sequences will be denoted by S θ I). Definition.3. 7 Let θ = k r ) be a lacunary sequence. A sequence x = x k ) is said to be N θ I) convergent to L, if for every ε > 0 we have x k L ε I. In this case x k LN θ I)). The concept of 2-normed spaces was initially developed by Gähler 6 in the mid of 960 s, while that of n-normed spaces one can see in Misiak 26. Since then, many others have studied this concept and obtained various results, see Gunawan 7,8) and Gunawan and Mashadi 9. Let n N and X be a linear space over the field R of reals 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;

582 KULDIP RAJ, S.A. MOHIUDDINE and M. AYMAN MURSALEEN 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 R, 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 R. 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 a 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 defines an n )-norm on X witespect 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 lim 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 lim x k x p, z,, z n = 0 for every z,, z n X. k,p If every Cauchy sequence in X converges to some L X, then X is said to be complete witespect to the n-norm. Any complete n-normed space is said to be n-banach space. Definition.4. Let I 2 N. A sequence x = x k ) in a n-normed space X,,, ) is said to be I-convergent to a number L if for every ϵ > 0, the set Aε) = k N : x k L, z,, z n ϵ I. In this case we write I lim k x k, z,, z n = L, z,, z n. Definition.5. A sequence x = x k ) in a n-normed space X,,, ) is said to be statistical convergent to some L X if for each ε > 0, the set Aε) = k N : x k L, z,, z n ε having its natural density zero. The notion of difference sequence spaces was introduced by Kızmaz 20 who studied the difference sequence spaces l ), c ) and c 0 ). The notion was further generalized by Et. and Çolak 9 by introducing the spaces l n ),

SOME GENERALIZED SEQUENCE SPACES OF INVARIANT MEANS... 583 c n ) and c 0 n ). Let w be the space of all complex or real sequences x = x k ) and let m, v be non-negative integers, then for Z = l, c, c 0 we have sequence spaces Z m v ) = x = x k ) w : m v x k ) Z, where m v x = m v x k ) = x k m v x k+ ) and 0 x k = x k, for all k N, which is equivalent to the following binomial representation m ) m v x k = ) s m x s k+vs. m v s=0 Taking v =, we get the spaces which were introduced and studied by Et. and Çolak 9. Taking m = v =, we get the spaces which were studied by Kızmaz 20. For more details about sequence spaces see 23, 32, 33, 36, 37, 39) and reference therein. A modulus function is a function f : 0, ) 0, ) such that. fx) = 0 if and only if x = 0, 2. fx + y) fx) + fy), for all x, y 0, 3. f is increasing, 4. f is continuous from the right at 0. It follows that f must be continuous everywhere on 0, ). The modulus x x+, function may be bounded or unbounded. For example, if we take fx) = then fx) is bounded. If fx) = x p, 0 < p < then the modulus function fx) is unbounded. Subsequently, modulus function has been discussed in 2, 38) and references therein. Lemma.6. Let F = ) be a sequence of modulus functions and 0 < δ <. Then for each x > δ we have x) 2)x δ. Let F = ) be a sequence of modulus functions, X,,, ) be a n- normed space, p = p k ) be a bounded sequence of strictly positive real numbers and u = ) be any sequence of positive real numbers. By Sn X) we denote the space of all sequences defined over X,,, ). In this paper we define the following sequence spaces: F, I, u, p,,, w 0 σ = x Sn X) : r N: for some > 0, w σ F, I, u, p,,, = x Sn X): r N : for some l and > 0, m v x k ) m v x k l) ) pk ε, z,, z n I, ) pk ε, z,, z n I,

584 KULDIP RAJ, S.A. MOHIUDDINE and M. AYMAN MURSALEEN and wσ F, I, u, p,,, = x Sn X) : K > 0, r N: m v x k ) ) pk K, z,, z n I, for some > 0, uniformly in n. If we take Fx) = x, we get the spaces wσ 0 I, u, p,,, = x Sn X): r N : for some > 0, w σ I, u, p,,, = x Sn X): r N: for some l and > 0, and w σ = I, u, p,,, x Sn X) : K > 0, m v x k ) m v x k ) m v x k l) If we take p = p k ) =, we get the spaces wσ 0 F, I, u,,, = x Sn X) : m v x k ) ), z,, z n w σ F, I, u,,, = x Sn X) : m v x k l) ) pk ε, z,, z n I, ) pk ε, z,, z n I, ) pk, z,, z n K I, for some > 0. ), z,, z n ε I, for some > 0, ε I, for some l and > 0,

SOME GENERALIZED SEQUENCE SPACES OF INVARIANT MEANS... 585 and w σ = F, I, u,,, x Sn X) : K > 0, m v x k ) ), z,, z n K I, for some > 0. The following inequality will be used throughout the paper. If 0 p k sup p k = H, K = max, 2 H ) then.) a k + b k p k D a k p k + b k p k for all k and a k, b k R. Also a p k max, a H ), for all a R. The main aim of the present paper is to study some topological properties and prove some inclusion relations between above defined sequence spaces. 2. Main results Theorem 2.. Let F= ) be a sequence of modulus functions, p=p k ) be a bounded sequence of strictly positive real numbers and u= ) be any sequence of positive real numbers. Then the classes of sequences w 0 σf, I, u, p,,, θ m v ), w σ F, I, u, p,,, θ m v ) and w σ F, I, u, p,,, θ m v ) are linear spaces over the real field R. Proof. We shall prove the assertion for w 0 σf, I, u, p,,, θ m v ) only and the others can be proved similarly. Let x = x k ), y = y k ) w 0 σf, I, u, p,,, θ m v ) and α, β R. Then there exist positive real numbers and 2 such that for every ε > 0 and z,, z n X, we have and ε ) A θ = 2 ε ) B θ = 2 m v x k ) m v y k ) 2 ) pk ε, z,, z n 2 ) pk ε, z,, z n 2

586 KULDIP RAJ, S.A. MOHIUDDINE and M. AYMAN MURSALEEN belongs to I. Let 3 = max2 α, 2 β 2 ). Since,, is a n-norm on X and s are modulus functions so by using inequality.), we have + α m v x k + β m v y k ) 3 D. 2 p k + D. 2 p k α t k n m v x k ) 3 β t k n m v y k ) 3 m v x k ) m v y k ) 2 ) pk, z,, z n ) pk, z,, z n ) pk, z,, z n ) pk, z,, z n ) pk., z,, z n For given ε > 0 and for all z,, z n X, we have the following containment α m v x k + β m v y k ) 3 m v x k ) m v y k ) 2 ) pk, z,, z n ε ) pk ε, z,, z n 2D ) pk ε, z,, z n 2D By using the property of an ideal the set on the left hand side in the above expression belongs to I. Thus, αx + βy w 0 σf, I, u, p,,, θ m v ). This completes the proof. Theorem 2.2. Let p = p k ) be a bounded sequence of strictly positive real numbers and u = ) be a sequence of positive real numbers. Then for m, we have: i) w 0 σf, I, u, p,,, θ m v ii) w σ F, I, u, p,,, θ m v iii) w σ F, I, u, p,,, θ m v ) wσf, 0 u, p,,, θ m v ) is strict. ) w σ F, u, p,,, θ m v ) is strict. ) wσ F, u, p,,, θ m v ) is strict. Proof. We shall prove the result for wσf, 0 I, u, p,,, θ m v ) only. The others can be proved similarly. Suppose x wσf, 0 I, u, p,,, θ m v ), by definition for every ε > 0 and z,, z n X, we have 2.) m v x k ) ) pk, z,, z n ε I,.

SOME GENERALIZED SEQUENCE SPACES OF INVARIANT MEANS... 587 uniformly in n. Since F = ) is a sequence of modulus functions, we have the following inequality: m v x k ) m + m D + D uniformly in n. Now for given ε > 0, we have ) pk, z,, z n v x k ) v x k+ ) m ), z,, z n ) p k, z,, z n v x k ) m m v x k ) m m v x k+ ) ) pk, z,, z n ) pk, z,, z n ) pk, z,, z n ε v x k ) v x k+ ) ) pk ε, z,, z n 2D ) pk ε, z,, z n 2D uniformly in n. Both the sets on the right hand side in the above containment belong to I by 2.). It follows that x wσf, 0 I, u, p,,, θ m v ). Clearly the inclusion is strict because if we take x = x k ) = k m, x) = x, p k =, =, t 0n x) = x n ) and θ = 2 r ) for all k N, then x k wσf, 0 u, p,,, θ m v ) but x k / wσf, 0 I, u, p,,, θ m v ). Theorem 2.3. Let F = f k ) and F = f k ) be two sequences of modulus functions. If lim sup t) t t) = P > 0, then w0 σf, u, p,,, θ m v ) wσf 0, I, u, p,,, θ m v ). Proof. Let lim sup t) t = P, then there exists a positive number K > 0 t) such that f k t) Kf k t), for all t 0. Therefore, for each z,, z n X,,

588 KULDIP RAJ, S.A. MOHIUDDINE and M. AYMAN MURSALEEN we have K) H m v x k ) f k ) pk, z,, z n m v x k ), z,, z n ) pk, uniformly in n. Then for every ε > 0 and z,, z n X, we have following relationship f k m v x k ) m v x k ) ) pk, z,, z n ε ) pk, z,, z n εk) H uniformly in n. Therefore, the above containment gives the result. Theorem 2.4. Suppose F = ), F = f k ) and F = f k ) are sequences of modulus functions, then i) w σ F, I, u, p,,, θ m v ) w σ F F, I, u, p,,, θ m v ). ii) w σ F, I, u, p,,, θ m v ) w σ F, I, u, p,,, θ m v ) w σ F + F, I, u, p,,, θ m v ). Proof. i) Let x = x k ) w σ F, I, u, p,,, θ m v ), then for every ε > 0 choose δ 0, ) such that t) < ε, for all 0 < t < δ, we have 2.2) A δ = m v x k l) ) pk, z,, z n δ I, uniformly in n. On the other hand, we have h ) m v x k l), z,, z n r = & f k t kn m v x k l),z,,z n ) p k <δ ) m v x k l), z,, z n + & t kn m v x k l),z,,z n ) p k δ ) m v x k l), z,, z n ε) H + max, 2. ) ) H ) δ ) pk ) pk ) pk m v x k l) ) pk, z,, z n,

SOME GENERALIZED SEQUENCE SPACES OF INVARIANT MEANS... 589 uniformly in n by lemma.6). Then for any η > 0, h ) m v x k l) ) pk, z,, z n η r m v x k l) ) pk η ε, z,, z n, K where K = max, 2. ) δ ) H ). By using 2.2), we obtain x w σ F F, I, u, p,,, θ m v ). ii) This part of the theorem proved by using the following inequality h + ) r D + D f k m v x k l) m v x k l) m v x k l) where sup k p k = H and D = max, 2 H ). ) pk, z,, z n ) pk, z,, z n, z,, z n ) pk, Theorem 2.5. Let F = ) be a sequence of modulus functions and p = p k ) be a bounded sequence of strictly positive real numbers, then w σ I, u, p,,, θ m v ) w σ F, I, u, p,,, θ m v ). Proof. This can be proved by using the techniques similar to those used in Theorem 2.4 i). Theorem 2.6. Let F = ) be a sequence of modulus functions and p = p k ) be f a bounded sequence of strictly positive real numbers, if lim sup k t) t t = Q > 0, then w σ F, I, u, p,,, w σ u, p,,,. Proof. Suppose x = x k ) w σ F, I, u, p,,, θ m f v ) and lim sup k t) t t = Q > 0, then there exists a constant R > 0 such that t) Rt, for all t 0. Thus we have R) H m v x k l) ) pk, z,, z n m v x k l) pk,, z,, z n ) uniformly in n and for each z,, z n X. Which gives the result.

590 KULDIP RAJ, S.A. MOHIUDDINE and M. AYMAN MURSALEEN Theorem 2.7. If 0 < p k q k and q k p k ) be bounded, then w σ F, I, u, q,,, θ m v ) w σ F, I, u, p,,, θ m v ) Proof. The proof is easy so omitted. 3. Statistical convergence The concept of convergence of a sequence of real numbers had been extended to statistical convergence by Fast and later studied by many authors. We refer to the recent work in, 3, 4, 8, 28, 29, 30 for some applications of statistical summability to approximation theorems. Here, we define the notion of S m v I, u-convergence with the help of an ideal and invariant means. We also made an effort to establish a strong connection between the spaces S m v I, u,,, and w σ F, I, u, p,,, θ m v ). Definition 3.. Let I P N) be a non-trivial ideal. A sequence x = x k ) X is said to be S m v I, u-convergent to a number l provided that for every ε > 0, δ > 0 and z,, z n X, the set k I r : m v x k l), z,, z n ε δ I. uniformly in n. In this case, we write S m v I, u lim k x k = l. Let S m v I, u,,,, denotes the set of all S m v I, u-convergent sequences in X. Theorem 3.2. Let F = ) be a sequence of modulus functions and 0 < in p k = h p k sup k p k = H <. Then w σ F, I, u, p,,, θ m v ) I, u,,. S m v Proof. Suppose x = x k ) w σ F, I, u, p,,, θ m v ) and ε > 0 be given. Then for each z,, z n X, we obtain t kn m v x k l), z,, z n ) p k = m v x k l) & t kn m v x k l),z,,z n ) ε + & t kn m v x k l),z,,z n )<ε & t kn m v x k l),z,,z n ) ε m v x k l) m v x k l) ) pk, z,, z n ) pk, z,, z n ) pk, z,, z n

SOME GENERALIZED SEQUENCE SPACES OF INVARIANT MEANS... 59 ε) pk min ε) h, ε) H ) K k I r : m v x k l),, z,, z n ε where K = min ε) h, ε) H ). Then for every δ > 0 and z,, z n X, we have k I r : m v x k l) m v x k l) Since x k w σ F, I, u, p,,, θ m v ) so that x S m v, z,, z n ε δ ) pk, z,, z n Kδ. I, u,,,. Theorem 3.3. Let F = ) be a sequence of modulus functions and p = p k ) be a bounded sequence of strictly positive real numbers. If 0 < in p k = h p k sup k p k = H < then S m v I, u,,, w σ F, I, u, p,,, θ m v ). Proof. Using the same technique of 0, Theorem 3.5), it is easy to prove. Theorem 3.4. Let F = ) be a bounded sequence of modulus functions and p = p k ) be a bounded sequence of strictly positive real numbers. If 0 < in p k = h p k sup k p k = H < then S m v I, u,,, = w σ F, I, u, p,,, θ m v ) if and only if F = ) is bounded. Proof. Direct part can be obtained by combining Theorems 3.2. and 3.3. Conversely suppose F = ) be unbounded defined by k) = k, for all k N. Let θ = 2 r ) be a lacunary sequence. We take a fixed set B I, where I be an admissible ideal and define x = x k ) as follows k m+, if r / B, 2 r + k 2 r +, x k = k m+, if r B, 2 r < k 2 r +, 0, otherwise. where I r = 2 r, 2 r and = 2 r 2 r. For given ε > 0 and for each z,, z n X we have, lim r k Ir : m v x k 0), z,, z n ε < 0,

592 KULDIP RAJ, S.A. MOHIUDDINE and M. AYMAN MURSALEEN for all r / B, uniformly in n. Hence for δ > 0, there exists a positive integer r 0 such that k Ir : t kn m v x k 0), z,, z n ε < δ for r / B and r r 0. Now we have k I r : m v x k 0), z,, z n ε δ B, 2,..., r 0 ). Since I be an admissible ideal. It follows that S m v I, u lim k x k, z,, z n 0 for each z,, z n X. On the other hand, if we take p = p k ) =, for all k N then x k / w σ F, I, u, p,,, θ m v ). This contradicts the fact S m v I, u,,, = w σ F, I, u, p,,, θ m v ), so our supposition is wrong. Acknowledgments The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia. References T. Acar, S. A. Mohiuddine, Statistical C, )E, ) summability and Korovkin s theorem, Filomat, 302) 206), 387-393. 2 Y. Altin, M. Et, Generalized difference sequence spaces defined by a modulus function in a locally convex space, Soochow. J. Math., 3 2005), 233-243. 3 C. Belen, S. A. Mohiuddine, Generalized weighted statistical convergence and application, Appl. Math. Comput., 29 203), 982-9826. 4 N. L. Braha, H. M. Srivastava, S. A. Mohiuddine, A Korovkin s type approximation theorem for periodic functions via the statistical summability of the generalized de la Vallee Poussin mean, Appl. Math. Comput., 228 204), 62-69. 5 C. Cakan, B. Altay, M. Mursaleen, The σ-convergence and σ-core of double sequences, Appl. Math. Lett., 9 2006), 22-28. 6 J. S. Connor, The statistical and strong p Cesàro convergence of sequences, International Mathematical Journal of Analysis and its Applications, 8 988), 47-63. 7 P. Das, E. Savas, S. K. Ghosal, On generalizations of certain summability methods using ideals, App. Math. Lett., 24 20), 509-54.

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