Annals of Functional Analysis

The λ+r(μ)-statistical convergence

B. de Malafosse, M. Mursaleen, and V. Rakočević

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Let λ=(λn)n1 be a nondecreasing sequence of positive numbers tending to infinity such that λ1=1 and λn+1λn+1 for all n, and let In=[nλn+1,n] for n=1,2,. Then for any given nonzero sequence μ, we define by Δ+(μ) the operator that generalizes the operator of the first difference and is defined by Δ+(μ)xk=μk(xkxk+1). In this article, for any given integer r1, we deal with the λ+r(μ) -statistical convergence that generalizes in a certain sense the well-known λEr-statistical convergence. The main results consist in determining sets of sequences χ and χ of the form sξ0 satisfying χ[V,λ]0(Δ+r(μ))χ and sets κ and κ of the form sξ satisfying κ[V,λ](λ+r(μ))κ. This study is justified since the infinite matrix associated with the operator Δ+r(μ) cannot be explicitly calculated for all r.

Article information

Source
Ann. Funct. Anal. Volume 8, Number 1 (2017), 1-15.

Dates
Received: 16 December 2015
Accepted: 14 May 2016
First available in Project Euclid: 14 October 2016

Permanent link to this document
https://projecteuclid.org/euclid.afa/1476450343

Digital Object Identifier
doi:10.1215/20088752-3720471

Mathematical Reviews number (MathSciNet)
MR3558300

Subjects
Primary: 46A15
Secondary: 40C05: Matrix methods 40J05: Summability in abstract structures [See also 43A55, 46A35, 46B15] (should also be assigned at least one other classification number in this section)

Keywords
matrix transformations operator of first-difference statistical convergence $BK$ space

Citation

de Malafosse, B.; Mursaleen, M.; Rakočević, V. The $\lambda ^{+r}(\mu )$ -statistical convergence. Ann. Funct. Anal. 8 (2017), no. 1, 1--15. doi:10.1215/20088752-3720471. https://projecteuclid.org/euclid.afa/1476450343


Export citation

References

  • [1] A. Alotaibi, B. Hazarika, and S. A. Mohiuddine, On lacunary statistical convergence of double sequences in locally solid Riesz spaces, J. Comput. Anal. Appl. 17 (2014), no. 1, 156–165.
  • [2] A. Alotaibi and M. Mursaleen, Generalized statistical convergence of difference sequences, Adv. Difference Equ. 2013 (2013), 5 pp.
  • [3] R. Çolak and Ç. Bektaş, $\lambda $-statistical convergence of order $\alpha$, Acta Math. Sci. Ser. B Engl. Ed. 31 (2011), no. 3, 953–959.
  • [4] J. S. Connor, The statistical and strong p-Cesàro convergence of sequences, Analysis (Berlin) 8 (1988), no. 1–2, 47–63.
  • [5] M. Et, Spaces of Cesàro difference sequences of order $r$ defined by a modulus function in a locally convex space, Taiwanese J. Math. 10 (2006), no. 4, 865–879.
  • [6] M. Et, Y. Altin, and H. Altinok, On some generalized difference sequence spaces defined by a modulus function, Filomat 17 (2003), 23–33.
  • [7] M. Et and F. Nuray, $\Delta ^{m}$-statistical convergence, Indian J. Pure Appl. Math. 32 (2001), no. 6, 961–969.
  • [8] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244.
  • [9] J. A. Fridy, On statistical convergence, Analysis (Berlin) 5 (1985), 301–313.
  • [10] J. A. Fridy, Statistical limit points, Proc. Amer. Math. Soc. 118 (1993), 1187–1192.
  • [11] J. A. Fridy and C. Orhan, Lacunary statistical convergence, Pacific J. Math. 160 (1993), no. 1, 43–51.
  • [12] J. A. Fridy and C. Orhan, Statistical core theorems, J. Math. Anal. Appl. 208 (1997), no. 2, 520–527.
  • [13] I. J. Maddox, Infinite Matrices of Operators, Lecture Notes in Math. 786, Springer, Berlin, 1980.
  • [14] B. de Malafosse, On some BK space, Int. J. Math. Math. Sci. 28 (2003), no. 28, 1783–1801.
  • [15] B. de Malafosse, On the sets of sequences that are strongly $\alpha$-bounded and $\alpha$-convergent to naught with index p, Rend. Semin. Mat. Univ. Politec. Torino 61 (2003), no. 1, 13–32.
  • [16] B. de Malafosse, Calculations on some sequence spaces, Int. J. of Math. and Math. Sci. 31 (2004), no. 29–32, 1653–1670.
  • [17] B. de Malafosse and E. Malkowsky, Sequence spaces and inverse of an infinite matrix, Rend. Circ. Mat. Palermo (2) 51 (2002), no. 2, 277–294.
  • [18] B. de Malafosse and E. Malkowsky, Matrix transformations in the sets $\chi (\overline{N}_{p}\overline{N}_{q})$ where $\chi $ is in the form s$_{\xi }$, or s$_{\xi}^{{{}^{\circ }}}$, or s$_{\xi }^{(c)}$, Filomat 17 (2003), 85–106.
  • [19] S. A. Mohiuddine and A. Alotaibi, Statistical summability of double sequences through de la Vallée-Poussin mean in probabilistic normed spaces, Abst. Appl. Anal. 2013, art. ID 215612.
  • [20] A. Wilansky, Summability through Functional Analysis, North-Holland Math. Stud. 85, North-Holland, Amsterdam, 1984.