Generalized Statistical Convergence of Double Sequences in Paranormed Spaces
Main Article Content
Abstract
We introduce the notion of (λ, µ)-statistical convergence of double sequences in a setting of paranormed space and prove that every convergent sequence is (λ, µ)-statistically convergent but not conversely by supporting an illustrative example. We also define the notions of (λ, µ)-statistical Cauchy and strongly (λ, µ)p-summable double sequences in a paranormed space and obtain their relationship with (λ, µ)- statistical convergence.
Article Details
References
- T. Acar and S. A. Mohiuddine, Statistical (C, 1)(E, 1) summability and Korovkin's theorem, Filomat 30(2) (2016), 387-393.
- A. Alotaibi and A. M. Alroqi, Statistical convergence in a paranormed space, J. Inequal. Appl. 2012 (2012), 39.
- A. Alotaibi, M. Mursaleen, S. A. Mohiuddine, Korovkin type approximation theorems for σ-convergence of double sequences, J. Nonlinear Convex Anal. 16(1) (2015), 183-192.
- M. A. Alghamdi and M. Mursaleen, λ-Statistical convergence in paranormed space, Abstr. Appl. Anal. 2013 (2013), Article ID 264520.
- F. A. Arani, M. E. Gordji and S. Talebi, Statistical convergence of double sequence in paranormed spaces, J. Math. Comput. Sci. 10 (2014), 47-53.
- C. Belen and S. A. Mohiuddine, Generalized weighted statistical convergence and application, Appl. Math. Comput. 219 (2013), 9821-9826.
- N. L. Braha, H. M. Srivastava and S. A. Mohiuddine, A Korovkin's type approximation theorem for periodic functions via the statistical summability of the generalized de la Vall ´ee Poussin mean, Appl. Math. Comput. 228 (2014), 162-169.
- H. C ¸ akalli, On statistical convergence in topological groups, Pure Appl. Math. Sci. 43 (1996), 27-31.
- J. S. Connor, The statistical and strong p-Cesaro convergence of sequences, Analysis 8 (1988), 47-63.
- O. H. H. Edely, S. A. Mohiuddine and A. K. Noman, Korovkin type approximation theorems obtained through generalized statistical convergence, Appl. Math. Lett. 23 (2010), 1382-1387.
- H. Fast, Sur la convergence statistique, Coll. Math. 2 (1951), 241-244.
- J. A. Fridy, On statistical convergence, Analysis 5(4) (1985), 301-313.
- B. Hazarika and V. Kumar, On asymptotically double lacunary statistical equivalent sequences in ideal context, J. Inequal. Appl. 2013 (2013), 543.
- U. Kadak and S. A. Mohiuddine, Generalized statistically almost convergence based on the difference operator which includes the (p, q)-Gamma function and related approximation theorems, Results Math. 73 (2018), 9.
- E. Kolk, The statistical convergence in Banach spaces, Tartu Ul. Toime. 928 (1991), 41-52.
- I. J. Maddox, Statistical convergence in a locally convex space, Math. Camb. Phil. Soc. 104 (1988), 141-145.
- S. A. Mohiuddine, Statistical weighted A-summability with application to Korovkin's type approximation theorem, J. Inequal. Appl. 2016 (2016), 101.
- S. A. Mohiuddine and B. A. S. Alamri, Generalization of equi-statistical convergence via weighted lacunary sequence with associated Korovkin and Voronovskaya type approximation theorems, Rev. R. Acad. Cienc. Exactas Fs. Nat., Ser. A Mat., RACSAM 113 (3) (2019), 1955-1973.
- S. A. Mohiuddine and M. Aiyub, Lacunary statistical convergence in random 2-normed spaces, Appl. Math. Inf. Sci. 6(3) (2012), 581-585.
- S. A. Mohiuddine, A. Alotaibi and M. Mursaleen, Statistical convergence of double sequences in locally solid Riesz spaces, Abstr. Appl. Anal. 2012 (2012), 719729.
- S. A. Mohiuddine, A. Asiri and B. Hazarika, Weighted statistical convergence through difference operator of sequences of fuzzy numbers with application to fuzzy approximation theorems, Int. J. Gen. Syst. 48(5) (2019), 492-506.
- S. A. Mohiuddine and B. Hazarika, Some classes of ideal convergent sequences and generalized difference matrix operator, Filomat 31(6) (2017), 1827-1834
- S. A. Mohiuddine, B. Hazarika and A. Alotaibi, On statistical convergence of double sequences of fuzzy valued functions, J. Intell. Fuzzy Syst. 32 (2017), 4331-4342.
- M. Mursaleen, λ-Statistical convergence, Math. Slovaca 50 (2000), 111-115.
- M. Mursaleen, C. Cakan, S. A. Mohiuddine and E. Sava ¸s, Generalized statistical convergence and statistical core of double sequences, Acta Math. Sin. 26(11) (2010), 2131-2144.
- M. Mursaleen and O. H. H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl. 288 (2003), 223-231.
- M. Mursaleen and S. A. Mohiuddine, Statistical convergence of double sequences in intuitionistic fuzzy normed spaces, Chaos Solitons Fractals 41(5) (2009), 2414-2421.
- M. Mursaleen and S. A. Mohiuddine, On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space, J. Comput. Appl. Math. 233 (2009), 142-149.
- A. Pringsheim, Zur Ttheorie der zweifach unendlichen Zahlenfolgen, Math. Ann. 53 (1900), 289-321.
- I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959), 361-375.
- T. Salat, On statistically convergent sequences of real numbers, Math. Slovaca 30 (1980), 139-150.
- A. Zygmund, Trigonometrical Series, vol. 5(1935) of Monografyas de Matematicas, Warszawa-Lwow.