# zbMATH — the first resource for mathematics

Statistical convergence of double sequences in intuitionistic fuzzy normed spaces. (English) Zbl 1198.40007
Summary: Recently, the concept of intuitionistic fuzzy normed spaces was introduced by R. Saadati and J. H. Park [Chaos Solitons Fractals 27, No. 2, 331–344 (2006; Zbl 1083.54514)]. S. Karakus, K. Demirci and O. Duman [Chaos Solitons and Fractals 35, No. 4, 763–769 (2008; Zbl 1139.54006)] have quite recently studied the notion of statistical convergence for single sequences in intuitionistic fuzzy normed spaces. In this paper, we study the concept of statistically convergent and statistically Cauchy double sequences in intuitionistic fuzzy normed spaces. Furthermore, we construct an example of a double sequence to show that in IFNS statistical convergence does not imply convergence and our method of convergence even for double sequences is stronger than the usual convergence in intuitionistic fuzzy normed space.
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

##### MSC:
 40G15 Summability methods using statistical convergence 46S40 Fuzzy functional analysis
Full Text:
##### References:
 [1] Alaca, C.; Turkoglu, D.; Yildiz, C., Fixed points in intuitionistic fuzzy metric spaces, Chaos, solitons & fractals, 29, 1073-1078, (2006) · Zbl 1142.54362 [2] Alimohammady, M.; Roohi, M., Compactness in fuzzy minimal spaces, Chaos, solitons & fractals, 28, 906-912, (2006) · Zbl 1094.54501 [3] Barros, L.C.; Bassanezi, R.C.; Tonelli, P.A., Fuzzy modelling in population dynamics, Ecol model, 128, 27-33, (2000) [4] Christopher, J., The asymptotic density of some k-dimensional sets, Am math monthly, 63, 399-401, (1956) · Zbl 0070.04101 [5] El Naschie, M.S., On certainty of Cantorian geometry and two-slit experiment, Chaos, solitons & fractals, 9, 517-529, (1998) · Zbl 0935.81009 [6] El Naschie, M.S., A review of E-infinity theory and the mass spectrum of high energy particle physics, Chaos, solitons & fractals, 19, 209-236, (2004) · Zbl 1071.81501 [7] Erceg, M.A., Metric spaces in fuzzy set theory, J math anal appl, 69, 205-230, (1979) · Zbl 0409.54007 [8] Fast, H., Sur la convergence statistique, Colloq math, 2, 241-244, (1951) · Zbl 0044.33605 [9] Fradkov, A.L.; Evans, R.J., Control of chaos: methods and applications in engineering, Chaos, solitons & fractals, 29, 33-56, (2005) [10] Fridy, J.A., On statistical convergence, Analysis, 5, 301-313, (1985) · Zbl 0588.40001 [11] Giles, R., A computer program for fuzzy reasoning, Fuzzy sets syst, 4, 221-234, (1980) · Zbl 0445.03007 [12] Hong, L.; Sun, J.Q., Bifurcations of fuzzy nonlinear dynamical systems, Commun nonlinear sci numer simul, 1, 1-12, (2006) · Zbl 1078.37049 [13] Kaleva, O.; Seikkala, S., On fuzzy metric spaces, Fuzzy sets syst, 12, 215-229, (1984) · Zbl 0558.54003 [14] Karakus S, Demirci K. Statistical convergence of double sequences on probabilistic normed spaces. Int J Math Math Sci 2007; article ID 14737, 11 p. doi:10.1155/2007/14737. · Zbl 1147.54016 [15] Karakus, S.; Demirci, K.; Duman, O., Statistical convergence on intuitionistic fuzzy normed spaces, Chaos, solitons & fractals, 35, 763-769, (2008) · Zbl 1139.54006 [16] Madore, J., Fuzzy physics, Ann phys, 219, 187-198, (1992) [17] Mursaleen, M.; Edely Osama, H.H., Statistical convergence of double sequences, J math anal appl, 288, 223-231, (2003) · Zbl 1032.40001 [18] Park, J.H., Intuitionistic fuzzy metric spaces, Chaos, solitons & fractals, 22, 1039-1046, (2004) · Zbl 1060.54010 [19] Pringsheim, A., Zur theorie der Zweifach unendlichen zahlenfolgen, Math Z, 53, 289-2321, (1900) · JFM 31.0249.01 [20] Saadati, R.; Park, J.H., Intuitionistic fuzzy Euclidean normed spaces, Commun math anal, 12, 85-90, (2006) · Zbl 1140.54301 [21] Saadati, R.; Park, J.H., On the intuitionistic fuzzy topological spaces, Chaos, solitons & fractals, 27, 331-344, (2006) · Zbl 1083.54514 [22] Saadati, R.; Razani, A.; Abidi, H., A common fixed point theorem in $$\mathcal{L}$$-fuzzy metric spaces, Chaos, solitons & fractals, 33, 358-363, (2007) [23] Saadati, R., A note on some results on the IF-normed spaces, Chaos, solitons & fractals, 41, 1, 206-213, (2009) · Zbl 1198.54023 [24] Šalát, T., On statistically convergent sequences of real numbers, Math slovaca, 30, 139-150, (1980) · Zbl 0437.40003 [25] Savaş, E.; Mursaleen, M., On statistically convergent double sequences of fuzzy numbers, Inform sci, 162, 183-192, (2004) · Zbl 1057.40002 [26] Schweizer, B.; Sklar, A., Statistical metric spaces, Pacific J math, 10, 313-334, (1960) · Zbl 0091.29801 [27] Steinhaus, H., Sur la convergence ordinaire et la convergence asymptotique, Colloq math, 2, 73-74, (1951) [28] Zadeh, L.A., Fuzzy sets, Inform control, 8, 338-353, (1965) · Zbl 0139.24606
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.