On some generalized statistically convergent sequence spaces

Transcription

1 Kuwait J. Sci. 42 (3) pp , 215 KULDIP RAJ AND SEEMA JAMWAL School of Mathematics, Shri Mata Vaishno Devi University Katra , J&K, INDIA Corresponding author uldipraj68@gmail.com ABSTRACT In this paper we construct some generalized new difference statistically convergent sequence spaces defined by a Musiela-Orlicz function over n normed spaces. We also study several properties relevant to topological structures and inclusion relations between these spaces. Keywords: Generalized difference sequence space; Musiela-Orlicz function; normed space; paranorm; statistical convergence. INTRODUCTION AND PRELIMINARIES n The notion of difference sequence spaces was introduced by Kızmaz(1981) who studied the difference sequence spaces and c () Δ. The notion was further generalized by Et & Çola(1995) in which they introduced the spaces and. Later the concept has been studied by Betaş et al. (24); Et & Esi (2). Another type of generalization of the difference sequence spaces is due to Tripathy & Esi (26) who studied the spaces and. Recently, Esi et al. (27); Tripathy et al. (25) have introduced a new type of generalized difference operators and unified those as follows. Let m, v be non-negative integers, then for Z a given sequence space, we have For Z = c, c and where and Δ m x = x all N, which is equivalent to the following binomial representation for Taing m = 1, we get the spaces and studied by Et & Çola (1995). Taing m = v = 1, we get the spaces and c() Δ introduced and studied by Kızmaz (1981). For more details about sequence spaces see (Mursaleen, 1996; Raj et al., 211; Raj & Sharma, 213a; Tripathy et al., 28; Tripathy et al. 212b) and references therein.

2 87 Başar & Altay (23) introduced the generalized difference matrix 1 which is a generalization of () difference operator as follows: Δ 1 for all, v N, r, s R \ {}. Recently, Başarir & Kayıçı (29) have defined the v generalized difference matrix B of order v which reduced the difference operator 1 Δ () 1 in case r = 1, s = 1 the binomial representation of this operator is where r, s R \ {} and v N. Thus for any sequence space Z, the space is more general and more comprehensive than the corresponding space For details one may refer to (Başarir & Nuray, 1991; Kayıçı & Başarir, 21) and references therein. The idea of statistical convergence was given by Zygmund (211). The concept was further studied by Fast (1951) ; Schoenberg (1959) independently for the real sequences. Later on, it was further investigated from sequence point of view and lined with the summability theory by Fridy (1985). The idea is based on the notion of natural density of subsets of N, the set of positive integers, which is defined as follows. The natural density of a subset E of N is denoted by where the vertical bar denotes the cardinality of the enclosed set. An Orlicz function is a function, which is continuous, non-decreasing and convex for x > and as x. Lindenstrauss & Tzafriri (1971) used the idea of Orlicz function to define the following sequence space: which is called as an Orlicz sequence space. The space l norm is a Banach space with the

3 Kuldip Raj and Seema Jamwal 88 It is shown by Lindenstrauss & Tzafriri (1971) that every Orlicz sequence space l contains a subspace isomorphic to l p ( p 1). In the later stage different Orlicz sequence spaces were introduced and studied see (Parashar & Choudhary, 1994; Esi & Et, 2; Tripathy & Mahanta 23; Mursaleen, 1996; Sen & Roy, 213; Esi, 213) and many others. The Δ 2 condition is equivalent to for all values of x and for L > 1. A sequence M = of Orlicz functions is called a Musiela-Orlicz function N defined by (Maligranda, 1989; Musiela, 1983). A sequence N ( ) is called the complementary function of a Musiela-Orlicz function M. For a given Musiela-Orlicz function the Musiela-Orlicz sequence space t M and its subspace h M are defined as follows: where I M is a convex modular defined by We consider t M equipped with the Luxemburg norm or equipped with the Orlicz norm For more details about Orlicz functions see (Et et al., 26; Tripathy & Dutta, 212) Let X be a linear metric space. A function (1) for all x X, (2) for all x X, (3) for all x, y X, p : X R is called paranorm, if (4) if is a sequence of scalars with λ n λ as n and is a sequence of vectors with p( x n x) as n, then p( λ x λ x) as n. A paranorm p for which implies x = is called total paranorm and the pair n n

4 89 ( p) X, is called a total paranormed space. It is well nown that the metric of any linear metric space is given by some total paranorm see (Wilansy, 1984) Theorem 1.4.2, pp By w we denote the set of all real or complex sequences. Let c, c, c, c, l, m and m denote the sets of all convergent, null, statistically convergent, statistically null, bounded, bounded statistically convergent and bounded statistically null sequences, respectively. A sequence space E is said to be solid (or normal) if implies ; for all sequences of scalars with for all N. The concept of 2-normed spaces was initially developed by Gӓhler (1965) in the mid of 196 s, while that of n normed spaces was introduced by Misia (1989). Since then, many others have studied these concepts and obtained various results, see (Gunawan, 21a; Gunawan, 21b; Gunawan & Mashadi, 21). Let n N and X be a linear space over the field K, where K is the 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: (1) f and only if ( x x,..., ), 2 (2) is invariant under permutation, x n 1 are linearly dependent in X, (3) for any α K, and (4) is called an n norm on X, and the pair is called an n normed space over the field K. Example 1.1. We may tae X = R n equipped with the n norm the volume of the which may be given explicitly by the formula n dimensional parallelopiped spanned by the vectors ( x x,..., ) 1, 2 x n where R n for each i 1,2,..., n. Let be an n normed space of dimension d n 2 and be linearly independent set in X. n 1 Then the following function on X defined by defines an ( n 1) norm on X with respect to.

5 Kuldip Raj and Seema Jamwal 9 A sequence in an n normed space is said to converge to some L X if lim x L, z1,..., z n 1 = for every z1,..., z n 1 X. A sequence in an n normed space is said to be Cauchy if lim x x p, z1,..., z n 1 = for every z1,..., z n 1 X., p 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. For more details about n normed space see ( Altin et al., 24; Raj & Sharma, 212; Raj et al., 21; Tripathy et al., 212a; Tripathy & Borgogain, 213) and references therein. A sequence is said to be statistically convergent to L if for every ε >, the set N: has natural density zero for each nonzero z1,..., z n 1 X, in other words statistically converges to L in n normed space if N: for each nonzero. For L > we say this is statistically null. First we give the following lemma, which we need to establish our main results. Lemma 1.2. Every closed linear subspace F of an arbitrary linear normed space E (different from E is a nowhere dense set in E (Tripathy & Dutta, 21). Throughout the paper and ( X ) denote the spaces of all, convergent, null, statistically convergent, statistically null, bounded, bounded statistically convergent and bounded statistically null X valued sequence spaces respectively, where is an n-normed space. By we mean the zero element of X. By x stat we mean that x is statistically convergent to zero. Let m, v be non-negative integers, M be a Musiela-Orlicz function, be a bounded sequence of positive real numbers and be a sequence of positive real numbers. Let be an n normed space. In this paper we define the following sequence spaces: m

6 91 c( = for every non zero and some c( = for every non zero and some = c( = for every non zero and some c( = for every non zero and some for every non zero and some

7 Kuldip Raj and Seema Jamwal 92 W( = for every non zero and some m( = c( B( v m ), u,.,...,. ) and m( B( v m ), u,.,...,. ) = c( B( v m ), u,.,...,. ), where and B ( m) x = x for all N, which is equivalent to the binomial representation as follows: v In this representation, we obtain the matrix B() 1 defined in Başarir & Kayıçı (29) for v > 1 and in Başar & Altay (23) for v =1. 1) If we tae v = then the above sequence spaces are reduced to c( c( c( c W( m( and m ( ( respectively. 2) If we tae r = 1 and s = 1 then the above sequence spaces are reduced to ( ( c ( c c( ( ( c W m and m ( respectively. 3) By taing and for all then these sequence spaces are reduces to c( c( c( c( W( m( and m( respectively. 4) If we tae M = I where I is the Identity ma then we get the sequence spaces c c ( c ( (

8 93 ( ( c W ( m ( and m respectively. 5) If we replace the base space X which is a linear n normed space by C, complete normed linear space and tae m = 1 and tae then the above spaces reduces to c( c( c( c( W( m( and m( respectively. 6) If we replace the base space X which is an n normed space by 2-normed space, M = I, for all and we obtained the sequence spaces introduced by (Başarir et al., 213). 7) Moreover if we tae X = C, M = I, and for all we get the spaces and m, respectively. The following inequality will be used throughout the paper. Let be a sequence of positive real numbers with < p sup p = H and let Then, for the sequences and in the complex plane, we have and for λ C, where h = inf p. The main purpose of this paper is to introduce and study the above defined sequence spaces. We mae an effort to study some topological and algebraic properties of these spaces. Further we show that the sequence spaces m( B( v m), u,.,...,. ) and m( B( v m), u,.,...,. ) are complete paranormed spaces, when the base space is n Banach space. We have also studied some inclusion relations between these spaces. MAIN RESULTS Theorem 2.1. Let be a M = Musiela-Orlicz function, be a bounded sequence of positive real numbers and be a sequence of positive real numbers. Then the sequence spaces Z( B( v m ), u,.,...,. ) are linear spaces over the field of complex number C, where Z = c c, l, W, m,., m (1)

9 Kuldip Raj and Seema Jamwal 94 Proof. We prove the theorem only for the space c( B( v m ), u,.,...,. ) and for the other spaces it will follow on applying similar argument. Let c( B( v m ), u,.,...,. ) and C. Then there exists L, J X and positive real numbers such that for every and Let Since are non-decreasing convex so by using the inequality (1) we have stat as. Thus, c( B( v m ), u,.,...,. ). Hence c( B( v m ), u,.,...,. ) is a linear space over the field of complex number C. Theorem 2.2. Let M = be a Musiela-Orlicz function, be a bounded sequence of positive real numbers and be a sequence of strictly positive real numbers. Let be a n Banach space. Then the spaces m( and m( are complete paranormed sequence spaces, paranormed by

10 95 where, H = sup p and h = inf p. Proof. We shall prove the theorem for the space m(. Clearly,, where is the zero sequence and. Let m(. Then there exists positive real numbers and such that and for some for some Let, thus This implies that To prove the continuity of scalar multiplication, assume that be any sequence of points in m( such that as v and be a sequence of scalars such that λ λ v. Since the inequality holds by subadditivity of is bounded. Thus,

11 Kuldip Raj and Seema Jamwal 96 Thus, is a paranorm. To prove the completeness of space m( assume that be a Cauchy sequence in m(. Then for a given there exists a positive integer N such that for all i, j N. This implies that for all i, j N. It follows that for every nonzero for each 1 and for all i, j N. Therefore, is a Cauchy sequence in X for all N. Since X is n Banach space, is convergent in X for all N. We write as i. Now we have for all i, j N.

12 97 It follows that ( a linear space, so we have ( can prove for the space m( Theorem 2.3. The space Z( Z = c, c, m, m. m and m( is m. Similarly we. This completes the proof. is not solid in general, where Proof. To show the space is not solid in general, consider the following examples. Example 2.4. Let m = 3, v = 1, r = 1, s = 1 and consider the n normed space as defined in example (1.1). Let p = 5, u = 1 and, (the identity map); for all N. Consider the sequence, where is defined by for each fixed N. Then x Z( for Z = c, m. Let, then Z(.Thus Z( for Z = c, m is not solid in general. Example 2.5. Let m = 3, v = 1, r = 1, s = 1 and consider the n normed space as defined in example (1.1). Let p =1, for all odd and p = 2 for all even, u = 1and, the identity ma for all N. Consider the sequence, where is defined by for each fixed N. Then x Z( for Z = c m. Let, then Z(. Thus Z (, for Z = c, m is not solid in general. Theorem 2.6. The spaces m( and m( are nowhere dense subsets of. Proof. From Theorem 2.2, it follows that m( and m( are closed subspace of. Since the inclusion relations and are strict, the spaces m( and m( nowhere dense subsets of by lemma 1.2. Theorem 2.7. Let are be a non-negative bounded sequence of positive real

13 Kuldip Raj and Seema Jamwal 98 numbers such that inf p >. Then Proof. Let ( x ) W(. Then for a given we have If we tae the limit for n. It follows that ( x ) W( from the above inequality. Since ( x ) W(. we have the result. Theorem 2.8. If p < q <, for each. Then < Proof. Let then their exists some such that This implies that for sufficiently large value of. Since are non-decreasing, we get

14 99 <. Thus,. This completes the proof. Theorem 2.9. (i) If inf p p < 1, then < (ii) If 1 p sup p <, then Proof. (i) Let. Since inf p 1, we have < Hence (ii) Let p 1 for each N and sup p <. Let Then for each <ε <1, there exists a positive integer N such that for all N. This implies that Hence

15 Kuldip Raj and Seema Jamwal 1 Theorem 2.1. Let M' = and M'' = be two Musiela-Orlicz functions. Then we have Proof. Let Then and Adding the above inequalities from = 1 to, we have. Thus, we get This completes the proof.

16 11 CONCLUSION In this paper we have introduced some new statistically convergent sequence spaces defined by a Musiela-Orlicz function over n normed spaces. We have studied some topological properties and interesting inclusion relations between these sequence spaces. There are many applications of sequence spaces in Science and Engineering see (Raj & Sharma, 213b). The solutions obtained here are potentially significant and important for the explanation of some practical physical problems. REFERENCES Altin, Y., Et, M. & Tripathy, B. C. 24. The sequence space on seminormed spaces, Applied Mathematics and Computation, 154: Betaş, Ç. A., Et, M. & Çola, R. 24. Generalized difference sequence spaces and their dual spaces, Journal of Mathematical Analysis and Applications, 292: Başarir, M., Konca, S. & Kara, E. E Some generalized difference statistically convergent sequence spaces in 2-normed space, Journal of Inequalities and Applications, 213: 177. Başarir, M. & Nuray, F Paranormed difference sequence spaces generated by infinite matrices, Pure & Applied Mathematia Sciences, 34: m Başarir, M. & Kayıçı, M. 29. On the generalized B Riesz difference sequence space and β property, Journal of Inequalities and Applications, Article ID Başar, F. & Altay, B. 23. On the space of sequences of p bounded variation and related matrix mappings, Urainian Mathematical Journal, 55: Esi, A Lacunary strong A convergent sequence spaces defined by sequences of modulus, Kuwait Journal of Science, 4: Esi, A., Tripathy, B. C. & Sarma, B. 27. On some new type generalized difference sequence spaces, Mathematica Slovaca, 57: Esi, A. & Et, M. 2. Some new sequence spaces defined by a sequence of Orlicz functions, Indian Journal of Pure and Applied Mathematics, 31: Et, M. & Çola, R On generalized difference sequence spaces, Soochow Journal of Mathematics, 21: Et, M. & Esi, A. 2. On Köthe - Toeplitz duals of generalized difference sequence spaces, Bulletin of the Malaysian Mathematical Sciences Society, 23: Et, M., Altin, Y., Choudhary, B. & Tripathy, B. C. 26. On some classes of sequences defined by sequences Orlicz functions, Mathematical Inequalities and Applications, 9: Fast, H Sur la convergence statistique, Colloquium Mathematicum, 2: Fridy, J. A On statistical convergence, Analysis, 5: Gӓhler, S Linear 2-normietre Rume, Mathematische Nachrichten, 28: Gunawan, H. 21a. On n inner product, n norms, and the Cauchy-Schwartz inequality, Scientiae Mathematicae Japonicae, 5: Gunawan, H. 21b. The space of p summable sequence and its natural n norm, Bulletin of the Australian Mathematical Society, 64:

17 Kuldip Raj and Seema Jamwal 12 Gunawan, H. & Mashadi, M. 21. On n normed spaces, International Journal of Mathematics and Mathematical Sciences, 27: Kizmaz, H On certain sequence spaces, Canadian Mathematical Bulletin, 24: Kirişçi, M. & Başar, F. 21. Some new sequence spaces derived by the domain of generalized diffrence matrix, Computers & Mathematics with Applications, 6: Lindenstrauss, J. & Tzafriri, L On Orlicz sequence spaces, Israel Journal of Mathematics, 1: Misia, A n Inner product spaces, Mathematische Nachrichten, 14: Mursaleen, M Generalized spaces of difference sequences, Journal of Mathematical Analysis and Applications, 23: Maligranda, L Orlicz spaces and interpolation, Seminars in Mathematics 5, Polish Academy of Science. Musiela, J Orlicz spaces and modular spaces, Lecture Notes in Mathematics, 134. Parasher, S. D. & Choudhary, B Sequence spaces defined by Orlicz function, Indian Journal of Pure and Applied Mathematics, 25: Raj, K., Sharma, A. K. & Sharma, S. K A sequence space defined by Musiela-Orlicz functions, International Journal of Pure and Applied Mathematics, 67: Raj, K. & Sharma, S. K A new sequence space defined by a sequence of Orlicz functions over n normed spaces, Acta Universitatis Palacianae Olomucensis, Facultas Rerum Naturalium, Mathematica, 51: Raj, K., Sharma, S. K. & Sharma, A. K. 21. Some difference sequence spaces in n-normed spaces defined by Musiela-Orlicz function, Armenian journal of Mathematics, 3: Raj, K. & Sharma, S. K. 213a. Some difference sequence spaces defined by Musiela-Orlicz functions, Mathematica Pannonica, 24: Raj, K. & Sharma, S. K. 213b. Some new sequence spaces, Applications and Applied Mathematics, 8: Schoenberg, I. J The integrability of certain functions and related summability methods, American Mathematical Monthly, 66: Sen, M. & Roy, S On paranormed type fuzzy I-convergent double multiplier sequences, Kuwait Journal of Science, 4: Tripathy, B. C. & Dutta, H. 21. On some new paranormed difference sequence spaces defined by Orlicz functions, Kyungpoo Mathematical Journal, 5: Tripathy, B. C., Esi, A. & Tripathy, B. 25. On a new type of generalized difference Ces`aro sequence spaces, Soochow Journal of Mathematics, 31: Tripathy, B. C. & Esi, A. 26. A new type of difference sequence spaces, International Journal of Science and Technology, 1: Tripathy, B. C. & Mahanta, S. 23. On a class of sequences related to the lp space defined by Orlicz functions, Soochow Journal of Mathematics, 29: Tripathy, B. C., Altin, Y. & Et, M. 28. Generalized difference sequence spaces on seminormed spaces defined by Orlicz functions, Mathematica Slovaca, 58: Tripathy, B. C. & Dutta, H On some lacunary difference sequence spaces defined by sequence of Orlicz functions and q lacunary Δ n m Statistical convergence, Analele Stiintifice ale Universitatii Ovidius, Seria Matematica, 2:

18 13 Tripathy, B. C., Sen, M. & Nath, S. 212a. I convergence in probabilistic n normed space, Soft Computing, 16: , DOI 1.17/s Tripathy, B. C., Baruah, A., Et, M. & Gungor, M. 212b. On almost statistical convergence of new type of generalized difference sequence of fuzzy numbers, Iranian Journal of Science and Technology Transaction A: Science, 36: Tripathy, B. C. & Borgogain, S On a class of n normed sequences related to the l p space, Boletim da Sociedade Paranaense de Matemàtica, 31: Wilansy, A Summability through functional analysis, North-Holland Mathematics Studies Zygmund, A Trigonometric Series, Cambridge University Press, Cambridge 51: Submitted : 6/2/214 Revised : 2/4/214 Accepted : 3/4/214

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