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Article Content:-
Abstract
In this paper, we present the extension of ∆-statistically convergent and ∆-statistically Cauchy sequences via neutrosophic normed space (NNS) to double sequences. The study in analogy also define and introduce for which lim or where denote set of all statistically convergent sequences. Furthermore, we present their feature utilizing double density and establish some inclusion relations between these concepts and prove some essentials analogous properties for double sequences.
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References
Aiyub (2015). Strongly (V^λ,A,Δ_((vm))^n,p,q)-summable sequence spaces defined by modulus function and statistical convergence. Proyecciones Journal of Mathematics, 34(2), 191-203.
Aizpuru, Listan-Garcia and Rambla-Barreno, F. (2014). Density by moduli and Statistical convergence. Quaestiones Mathematicae, 37(4),
-530.
Alotaibi and Alroqi (2012). Statistical convergence in a paranormed space. Journal of Inequalities and Applications, 39.
Altinok and Kasap (2016). f−Statistical Convergence of order β for Sequences of Fuzzy Numbers. Math, F. A. 1-10.
Altinok Altin and Isik (2018). Statistical convergence of order βfor double sequences of fuzzy numbers defined by a modulus function. AIP Conference Proceedings, 1-6.
Belen and Yildirim (2015). Generalized statistical convergence and some sequence spaces in 2-normed spaces. Hacettepe Journal of Mathematics and Statistics Volume 44 (3), 513 – 519.
Bhardwaj and Dhawan (2017). Density by moduli and Wijsman lacunary statistical convergence of sequences of sets. Journal of Inequalities and Applications, 25, 1-20.
Brono and Ali (2016). λ_2-Statistical Convergence in 2n-normed Spaces. Far East Journal of Mathematical Science, 99(10), 1551-1569.
Brono and Ali (2016). A Matrix Characterisation of Cesa ́ro C_1.1- Statistically Convergent Double Sequences.Far East Journal of Mathematical Sciences, 99(11), 1693-1702.
Brono and Ali (2016). I-Cesaro Statistical Core of Double Sequences. IOSR Journal of Mathematics. IOSR Journal of Mathematics, 12(2), 102-108.
Brono and Ali (2016). On Statistical Convergence of Double Sequences and Statistical Monotonicity. IOSR Journal of Mathematics, 12(1), 45-51
Burgin (2000). Theory of fuzzy limits, fuzzy sets and systems 115, 433-443
Bera and Mahapatra (2017). Neutrosophic soft normed linear spaces. Neutrosophic Sets and Systems, 23(2018), 52–71.
Bera and Mahapatra (2017). Neutrosophic soft linear spaces. Fuzzy Information and Engineering, 9, 299–324.
Barros Bassanezi and Tonelli (2000). Fuzzy modelling in population dynamics, Ecol Model, 128, 27–33.
Cakalli and Kaplan (2016). A variation on strongly lacunary ward continuity. Journal of Mathematical Analysis, 7(3), 13-20.
Cakalli and Kaplan (2017). A variation on lacunary statistical quasi Cauchy sequences, Cummun. Fac. Sci. Univ. Ank. Series A1, 66(2), 71-79.
Cakan and Altin (2015). Some classes of statistically convergent sequences of fuzzy numbers generated by a modulus function. Iranian Journal of Fuzzy Systems, 12(3), 47-55.
Connor and Kline (1996). On statistical limit points and the consistency of statistical convergence, Journal of Mathematical Anal Application.197, 393-399.
Connor and Swardson (1993). Strong integral summability and stone-chech compactification of the half-line. Pacific Journal of Mathematics.157, 201-224.
Connor and Ganichev and Kadets (2000). A characterization of Banach spaces with separable duals via week statistical convergence, Journal of Mathematical Analysis and Application. 244(1), 251-261.
Granados and Dhital (2021). Statistical Convergence of Double Sequences in Neutrosophic Normed Spaces. Neutrosophic Sets and Systems, Vol. 42, 344
Das and Savas (2014). On I-statistical and I-lacunary statistical convergence of order α. Bulletin of Iranian Mathematical Society, 40(2), 459-472.
Das and Savas, E. (2014). On I-statistically pre-Cauchy sequences. Taiwanese J. Math., 18(1), 115-126.
Deepmala and Mishra (2016). The ∫Γ^3λIstatistical Convergence of pre-Cauchy over the p-Metric Space Defined by Musielak Orlicz Function.
Duman, Khan and Orhan (2003). A-Statistical convergence of approximating operators, Math. Inequal. Appl. 6(4), 689-699.
Dutta (2013). A New Class of Strongly Summable and Statistical Convergence Sequences of Fuzzy Numbers An International Journal of Applied Mathematics and Information Science,7(6), 2369-2372.
Erkus and Dumam (2003). A-Statistical extension of the Korovkin type approximation Theorem Proc. Indian Academic Science (Mathematical Sciences) 115(4), 499–507.
Fast (1951). Sur la convergence statistique. Colloquium Mathematicum, 2 (3–4), 241–244.
Gumus (2015). Lacunary Weak I-Statistical Convergence. Gen. Math. Notes, 28(1), 50-58.
Gurdal and Ozgur (2015). A generalized statistical convergence via moduli. Electronic Journal of Mathematical Analysis and Applications,3(2),173-178.
George and Veeramani (1994). On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64, 395–399.
George and Veeramani (1997). On some results of analysis for fuzzy metric spaces, Fuzzy Sets and Systems, 90, 365–368.
Giles (1980). Computer program for fuzzy reasoning, Fuzzy Sets Syst., 4, 221–234.
Hazarika and Savas (2013). λ-statistical convergence in n-normed spaces.An. St. Univ. Ovidius Constanta. 21(2): 141-153.
Hong and Sun (2006). Bifurcations of fuzzy nonlinear dynamical systems, Commun Nonlinear Sci Numer Simul 1, 1–12.
Indu (2011). On Weak Statistical Convergence of Sequence of Functionals. International Journal of Pure and Applied Mathematics.70 (5): 647-653
Ilkhan and Kara (2018). A new type of Statistical Cauchy sequence and its relation to Bourkaki completeness. Cogent Mathematics & Statistics, 5, 1-9.
Kaya, Kucukaslan and Wagner (2013). On Statistical Convergence and Statistical Monotonicity.Annales Univ. Sci. Budapest., Sect. Comp. 39: 257-270.
Khan, Shafiq and Rababah (2015). On I-convergent sequence spaces defined by a compact operator and a modulus function. Cogent Mathematics2, 1-13.
Kukul (2014). αβ-Statistical convergence. Unpublished M. Sc. Dissertation, Eastern Mediterrenean University, Gazimagusa, North Cyprus.
Kaleva and Seikkala (1984). On fuzzy metric spaces, Fuzzy Sets and Systems, 12, 215– 229.
Kirisci and Simsek (2020). Neutrosophic normed spaces and statistical convergence. The Journal of Analysis, 28, 1059–1073.
Mursaleen and Edely (2003). Statistical convergence of double sequences, J. Math. Anal. Appl. 288 223–231.
Maddox (1988). Statistical convergence in a locally convex space, Math. Proc. Cambridge Phil. Soc. 104, 141-145.
Malik and Ghosh (2017). On I-statistical cluster point of double sequences. Functional Analysis, 205-212.
Miller (1995). A measure theoretical subsequence characterization of statistical convergence, Trans, Amer. Math. Soc. 347, 1811-1819.
Madore (1992). Fuzzy physics, Ann Phys 219, 187–98.
Menger (1942). Statistical metrics. Proceedings of the National Academy of Sciences 28: 535– 537.
Nuray (2000). Generalized Statistical Convergence and Convergence Free Spaces. Journal of Mathematical Analysis and Applications 245,513-527.
Niven, Zuckerman and Montgomery (1991). An introduction to the theory of numbers (5th ed.). New York, NY: Wiley.
Nazmiye (2022). Hybrid ∆- statistical convergence for neutrosophic normed space. Hindawi Journal of Mathematics, Volume 22, 10 Pages
Pancaroglu and Nuray (2014). Invariant Statistical Convergence of Sequences of Sets with respect to a Modulus Function. Hindawi Journal of Abstract and Applied Analysis.
Park (2004). Intuitionistic fuzzy metric spaces, Chaos, Solitons and Fractals, 22, 1039–1046.
Raj and Sharma (2014). Some spaces of double difference sequences of fuzzy numbers. Matematiqki Vesnik, 66(1), 91–100.
Schoenberg (1959). The integrability of certain functions and related summability methods. American Mathematics. Monthly 66:361-375.
Steinhauss (1951). Sur la convergence orndinaireet la convergence asymptotique, Colloq. Mathematics, 2:72-73.
Steinhauss (1951). Sur la convergence ordinate et al convergence asymptotique. Colloquium Mathematicum, 2, 73–84.
Smarandache (2005). Neutrosophic set, a Generalisation of the Intuitionistic Fuzzy Sets, International Journal of Pure and Applied Mathematics, 24, 287–297.
Smarandache (1998). Neutrosophy, Neutrosophic Probability, Set, and Logic, ProQuest Information and Learning. Ann Arbor, Michigan, USA.
Smarandache (2016). Degree of Dependence and Independence of the (sub) Components of Fuzzy Set and Neutrosophic Set. Neutrosophic Sets and Systems, Vol. 11, pp. 95-97
Sariana and Dauda (2017). A Review on Neutrosophic Set and Its Development. Menemui Matematik (Discovering Mathematics) Vol. 39, No. 2: 61-69
Vakeel et al (2021). Space of Neutrosophic λ- Statistical Convergence Sequences and their Properties. Journal of Mathematics and Computer Science, 23, 1-9
Vinod and Rani (2012). Weak ideal convergence in lp Spaces. International Journal of Pure and Applied Mathematics. 75(2), 247-256.
Yamanci and Gurdal (2014). I-statistically pre-Cauchy double sequences. Global Journal of Mathematical Analysis, 2(4), 297-303.
Zygmund (1979). Trigonometric series, second edition. Cambridge University Press, Londo
Zadeh (1965) Fuzzy sets, Information and Control, 8(3), 338–353.