Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 482392, 8 pages doi:10.1155/2010/482392 Research Article On Some New Sequence Spaces in 2-Normed Spaces Using Ideal Convergence and an Orlicz Function E. Savas¸ Department of Mathematics, Istanbul Ticaret University, ¨ Usk ¨ udar, 34672 Istanbul, Turkey Correspondence should be addressed to E. Savas¸, ekremsavas@yahoo.com Received 25 July 2010; Accepted 17 August 2010 Academic Editor: Radu Precup Copyright q 2010 E. Savas¸. 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 purpose of this paper is to introduce certain new sequence spaces using ideal convergence and an Orlicz function in 2-normed spaces and examine some of their properties. 1. Introduction The notion of ideal convergence was introduced first by Kostyrko et al. 1 as a generalization of statistical convergence which was further studied in topological spaces 2.More applications of ideals can be seen in 3, 4. The concept of 2-normed space was initially introduced by G ¨ ahler 5 as an interesting nonlinear generalization of a normed linear space which was subsequently studied by many authors see, 6, 7. Recently, a lot of activities have started t o study summability, sequence spaces and related topics in these nonlinear spaces see, 8–10. Recall in 11 that an Orlicz function M : 0, ∞ → 0, ∞ is continuous, convex, nondecreasing function such that M00andMx > 0forx>0, and Mx →∞as x →∞. Subsequently Orlicz function was used to define sequence spaces by Parashar and Choudhary 12 and others. If convexity of Orlicz function, M is replaced by Mx  y ≤ MxMy, then this function is called Modulus function, which was presented and discussed by Ruckle 13 and Maddox 14. Note that if M is an Orlicz function then Mλx ≤ λMx for all λ with 0 <λ<1. 2 Journal of Inequalities and Applications Let X, · be a normed space. Recall that a sequence x n  n∈N of elements of X is called to be statistically convergent to x ∈ X if the set Aε{n ∈ N : x n − x≥ε} has natural density zero for each ε>0. A family I⊂2 Y of subsets a nonempty set Y is said to be an ideal in Y if i ∅∈I; ii A, B ∈Iimply A ∪ B ∈I; iii A ∈I,B ⊂ A imply B ∈I, while an admissible ideal I of Y further satisfies {x}∈Ifor each x ∈ Y , 9, 10. Given I⊂2 N is a nontrivial ideal in N. The sequence x n  n∈N in X is said to be I- convergent to x ∈ X, if for each ε>0thesetAε{n ∈ N : x n − x≥ε} belongs to I, 1, 3. Let X be a real vector space of dimension d, where 2 ≤ d<∞. A2-normonX is a function ·, · : X × X → R which satisfies i x, y  0 if and only if x and y are linearly dependent, ii x, y  y, x, iii αx, y  |α|x, y,α∈ R,andiv x, y  z≤x, y  x, z. The pair X, ·, · is then called a 2-normed space 6 . Recall that X, ·, · is a 2-Banach space if every Cauchy sequence in X is convergent to some x in X. Quite recently Savas¸ 15 defined some sequence spaces by using Orlicz function and ideals in 2-normed spaces. In this paper, we continue to study certain new sequence spaces by using Orlicz function and ideals in 2-normed spaces. In this context it should be noted that though sequence spaces have been studied before they have not been studied in nonlinear structures like 2-normed spaces and their ideals were not used. 2. Main Results Let Λλ n  be a nondecreasing sequence of positive numbers tending to ∞ such that λ n1 ≥ λ n  1,λ 1  0andletI be an admissible ideal of N,letM be an Orlicz function, and let X, ·, · be a 2-normed space. Further, let p p k  be a bounded sequence of positive real numbers. By S2 − X we denote the space of all sequences defined over X, ·, ·.Now,we define the following sequence spaces: W I  λ, M, p,  , ·,     x ∈ S  2 − X  : ∀ε>0  n ∈ N : 1 λ n  k∈I n  M      x k − L ρ ,z      p k ≥ ε  ∈ I for some ρ>0,L∈ X and each z ∈ X  , W I 0  λ, M, p,  , ·,     x ∈ S  2 − X  : ∀ε>0  n ∈ N : 1 λ n  k∈I n  M      x k ρ ,z      p k ≥ ε  ∈ I for some ρ>0, and each z ∈ X  , Journal of Inequalities and Applications 3 W ∞  λ, M, p,  , ·,     x ∈ S  2 − X  : ∃K>0s.t. sup n∈N 1 λ n  k∈I n  M      x k ρ ,z      p k ≤ K for some ρ>0, and each z ∈ X  , W I ∞  λ, M, p,  , ·,     x ∈ S  2 − X  : ∃K>0s.t.  n ∈ N : 1 λ n  k∈I n  M      x k ρ ,z      p k ≥ K  ∈ I for some ρ>0, and each z ∈ X  , 2.1 where I n n − λ n  1,n. The following well-known inequality 16, page 190 will be used in the study. If 0 ≤ p k ≤ sup p k  H, D  max  1, 2 H−1  2.2 then | a k  b k | p k ≤ D  | a k | p k  | b k | p k  2.3 for all k and a k ,b k ∈ C.Also|a| p k ≤ max1, |a| H  for all a ∈ C. Theorem 2.1. W I λ, M, p, , ·, ,W I 0 λ, M, p, , ·, , and W I ∞ λ, M, p, , ·,  are linear spaces. Proof. We will prove the assertion for W I 0 λ, M, p, , ·,  only and the others can be proved similarly. Assume that x, y ∈ W I 0 λ, M, , ·,  and α, β ∈ R,so  n ∈ N : 1 λ n  k∈I n  M      x k ρ 1 ,z      p k ≥ ε  ∈ I for some ρ 1 > 0,  n ∈ N : 1 λ n  k∈I r  M      x k ρ 2 ,z      p k ≥ ε  ∈ I for some ρ 2 > 0. 2.4 4 Journal of Inequalities and Applications Since , ·,  is a 2-norm, and M is an Orlicz function the following inequality holds: 1 λ n  k∈I n  M        αx k  βy k   | α | ρ 1    β   ρ 2  ,z        p k ≤ D 1 λ n  k∈I n  | α |  | α | ρ 1    β   ρ 2  M      x k ρ 1 ,z       p k  D 1 λ n  k∈I n    β    | α | ρ 1    β   ρ 2  M      y k ρ 2 ,z       p k ≤ DF 1 λ n  k∈I n  M      x k ρ 1 ,z      p k  DF 1 λ n  k∈I n  M      y k ρ 2 ,z      p k , 2.5 where F  max ⎡ ⎣ 1,  | α |  | α | ρ 1    β   ρ 2   H ,    β    | α | ρ 1    β   ρ 2   H ⎤ ⎦ . 2.6 From the above inequality, we get  n ∈ N : 1 λ n  k∈I n  M        αx k  βy k   | α | ρ 1    β   ρ 2  ,z        p k ≥ ε  ⊆  n ∈ N : DF 1 λ n  k∈I n  M      x k ρ 1 ,z      p k ≥ ε 2  ∪  n ∈ N : DF 1 λ n  k∈I n  M      y k ρ 2 ,z      p k ≥ ε 2  . 2.7 Two sets on the right hand side belong to I and this completes the proof. It is also easy to see that the space W ∞ λ, M, p, , ·,  is also a linear space and we now have the following. Theorem 2.2. For any fixed n ∈ N, W ∞ λ, M, p, , ·,  is paranormed space with respect to the paranorm defined by g n  x   inf ⎧ ⎨ ⎩ ρ p n /H : ρ>0 s.t.  sup n 1 λ n  k∈I n  M      x k ρ ,z      p k  1/H ≤ 1, ∀z ∈ X ⎫ ⎬ ⎭ . 2.8 Proof. That g n θ0andg n −xgx are easy t o prove. So we omit them. Journal of Inequalities and Applications 5 iii Let us take x x k  and y y k  in W ∞ λ, M, p, , ·, .Let A  x    ρ>0:sup n 1 λ n  k∈I n  M      x k ρ ,z      p k ≤ 1, ∀z ∈ X  , A  y    ρ>0:sup n 1 λ n  k∈I n  M      y k ρ ,z      p k ≤ 1, ∀z ∈ X  . 2.9 Let ρ 1 ∈ Ax and ρ 2 ∈ Ay, then if ρ  ρ 1  ρ 2 , then, we have sup n 1 λ n  n∈I n M        x k  y k  ρ ,z       ≤ ρ 1 ρ 1  ρ 2 sup n 1 λ n  k∈I n M      x k ρ 1 ,z       ρ 2 ρ 1  ρ 2 sup n 1 λ n  k∈I n M      y k ρ 2 ,z      . 2.10 Thus, sup n 1/λ n   n∈I n Mx k  y k /ρ 1  ρ 2 ,z p k ≤ 1and g n  x  y  ≤ inf   ρ 1  ρ 2  p n /H : ρ 1 ∈ A  x  ,ρ 2 ∈ A  y   ≤ inf  ρ p n /H 1 : ρ 1 ∈ A  x    inf  ρ p n /H 2 : ρ 2 ∈ A  y    g n  x   g n  y  . 2.11 iv Finally using the same technique of Theorem 2 of Savas¸ 15 it can be easily seen that scalar multiplication is continuous. This completes the proof. Corollary 2.3. It should be noted that for a fixed F ∈ I the space W ∞  F   λ, M, p,  , ·,     x ∈ S  2 − X  : ∃K>0 s.t. sup n∈N−F 1 λ n  k∈I n  M      x k ρ ,z      p k ≤ K for some ρ>0, and each z ∈ X  , 2.12 which is a subspace of the space W I ∞ λ, M, p, , ·,  is a paranormed space with the paranorms g n for n / ∈ F and g F  inf n∈N−F g n . Theorem 2.4. Let M,M 1 ,M 2 , be Orlicz functions. Then we have i W I 0 λ, M 1 ,p,, ·,  ⊆ W I 0 λ, M ◦ M 1 ,p,, ·,  provided p k  is such that H 0  inf p k > 0. ii W I 0 λ, M 1 ,p, , ·,  ∩ W I 0 λ, M 2 ,p,, ·,  ⊆ W I 0 λ, M 1  M 2 ,p,, ·, . 6 Journal of Inequalities and Applications Proof. i For given ε>0, first choose ε 0 > 0 such that max{ε H 0 ,ε H 0 0 } <ε.Now using the continuity of M choose 0 <δ<1 such that 0 <t<δ⇒ Mt <ε 0 .Letx k  ∈ W 0 λ, M 1 ,p,, ·, . Now from the definition A  δ    n ∈ N : 1 λ n  n∈I n  M 1      x k ρ ,z      p k ≥ δ H  ∈ I. 2.13 Thus if n / ∈ Aδ then 1 λ n  n∈I n  M 1      x k ρ ,z      p k <δ H , 2.14 that is,  n∈I n  M 1      x k ρ ,z      p k <λ n δ H , 2.15 that is,  M 1      x k ρ ,z      p k <δ H , ∀k ∈ I n , 2.16 that is, M 1      x k ρ ,z      <δ, ∀k ∈ I n . 2.17 Hence from above using the continuity of M we must have M  M 1      x k ρ ,z      <ε 0 , ∀k ∈ I n , 2.18 which consequently implies that  k∈I n  M  M 1      x k ρ ,z      p k <λ n max  ε H 0 ,ε H 0 0  <λ n ε, 2.19 that is, 1 λ n  k∈I n  M  M 1      x k ρ ,z      p k <ε. 2.20 Journal of Inequalities and Applications 7 This shows that  n ∈ N : 1 λ n  k∈I n  M  M 1      x k ρ ,z      p k ≥ ε  ⊂ A  δ  2.21 and so belongs to I. This proves the result. ii Let x k  ∈ W I 0 M 1 ,p,, ·,  ∩ W I 0 M 2 ,p,, ·, , then the fact 1 λ n  M 1  M 2      x k ρ ,z      p k ≤ D 1 λ n  M 1      x k ρ ,z      p k  D 1 λ n  M 2      x k ρ ,z      p k 2.22 gives us the result. Definition 2.5. Let X be a sequence space. Then X is called solid if α k x k  ∈ X whenever x k  ∈ X for all sequences α k  of scalars with |α k |≤1 for all k ∈ N. Theorem 2.6. The sequence spaces W I 0 λ, M, p, , ·, ,W I ∞ λ, M, p, , ·,  are solid. Proof. We give the proof for W I 0 λ, M, p, , ·,  only. Let x k  ∈ W I 0 λ, M, p, , ·,  and let α k  be a sequence of scalars such that |α k |≤1 for all k ∈ N. Then we have  n ∈ N : 1 λ n  k∈I n  M      α k x k  ρ ,z      p k ≥ ε  ⊆  n ∈ N : C λ n  k∈I n  M      x k ρ ,z      p k ≥ ε  ∈ I, 2.23 where C  max k {1, |α k | H }. Hence α k x k  ∈ W I 0 λ, M, p, , ·,  for all sequences of scalars α k  with |α k |≤1 for all k ∈ N whenever x k  ∈ W I 0 λ, M, p, , ·, . References 1 P. Kostyrko, T. ˇ Sal ´ at, and W. Wilczy ´ nski, “I-convergence,” Real Analysis Exchange, vol. 26, no. 2, pp. 669–686, 2000. 2 B. K. Lahiri and P. Das, “I and I ∗ -convergence in topological spaces,” Mathematica Bohemica, vol. 130, no. 2, pp. 153–160, 2005. 3 P. Kostyrko, M. Ma ˇ caj, T. ˇ Sal ´ at, and M. Sleziak, “I-convergence and extremal I-limit points,” Mathematica Slovaca, vol. 55, no. 4, pp. 443–464, 2005. 4 P. Das and P. Malik, “On the statistical and I- variation of double sequences,” Real Analysis Exchange, vol. 33, no. 2, pp. 351–364, 2008. 5 S. G ¨ ahler, “2-metrische R ¨ aume und ihre topologische Struktur,” Mathematische Nachrichten, vol. 26, pp. 115–148, 1963. 6 H. Gunawan and Mashadi, “On finite-dimensional 2-normed spaces,” Soochow Journal of Mathematics, vol. 27, no. 3, pp. 321–329, 2001. 7 R. W. Freese and Y. J. Cho, Geometry of Linear 2-Normed Spaces, Nova Science, Hauppauge, NY, U SA, 2001. 8 A. S¸ahiner, M. G ¨ urdal, S. Saltan, and H. Gunawan, “Ideal convergence in 2-normed spaces,” Taiwanese Journal of Mathematics, vol. 11, no. 5, pp. 1477–1484, 2007. 9 M. G ¨ urdal and S. Pehlivan, “Statistical convergence in 2-normed spaces,” Southeast Asian Bulletin of Mathematics, vol. 33, no. 2, pp. 257–264, 2009. 8 Journal of Inequalities and Applications 10 M. G ¨ urdal, A. S¸ahiner, and I. Ac¸ık, “Approximation theory in 2-Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 5-6, pp. 1654–1661, 2009. 11 M. A. Krasnoselskii and Y. B. Rutisky, Convex Function and Orlicz Spaces, Noordhoff, Groningen, The Netherlands, 1961. 12 S. D. Parashar and B. Choudhary, “Sequence spaces defined by Orlicz functions,” Indian Journal of Pure and Applied Mathematics, vol. 25, no. 4, pp. 419–428, 1994. 13 W. H. Ruckle, “FK spaces in which the sequence of coordinate vectors is bounded,” Canadian Journal of Mathematics, vol. 25, pp. 973–978, 1973. 14 I. J. Maddox, “Sequence spaces defined by a modulus,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 100, no. 1, pp. 161–166, 1986. 15 E. Savas¸, “Δ m -strongly summable sequences spaces in 2-normed spaces defined by ideal convergence and an Orlicz function,” Applied Mathematics and Computation, vol. 217, no. 1, pp. 271–276, 2010. 16 I. J. Maddox, Elements of Functional Analysis, Cambridge University Press, London, UK, 1970. . is to introduce certain new sequence spaces using ideal convergence and an Orlicz function in 2-normed spaces and examine some of their properties. 1. Introduction The notion of ideal convergence. in X. Quite recently Savas¸ 15 defined some sequence spaces by using Orlicz function and ideals in 2-normed spaces. In this paper, we continue to study certain new sequence spaces by using Orlicz function. summability, sequence spaces and related topics in these nonlinear spaces see, 8–10. Recall in 11 that an Orlicz function M : 0, ∞ → 0, ∞ is continuous, convex, nondecreasing function such