On Ideal Convergence for Triple Sequences on L-Fuzzy Normed Space

Year 2025, Volume: 21 Issue: 2, 64 - 71, 27.06.2025

Reha Yapalı

https://doi.org/10.18466/cbayarfbe.1539485

https://izlik.org/JA42LK73JN

Abstract

In this paper, we delve into the exploration of ideal convergence within the framework of triple sequences on L -fuzzy normed spaces. Our primary focus is to establish a comprehensive characterization of ideal convergence for these triple sequences, particularly in relation to their convergence in the classical sense. Through rigorous analysis, we demonstrate that the notion of ideal convergence, as developed in this context, exhibits a weaker form of convergence compared to the traditional convergence criteria applied to triple sequences in L - fuzzy normed spaces. This weaker form of convergence, while more generalized, retains significant applicability and provides a broader understanding of the behavior of sequences within these structured spaces. The results presented herein offer new insights into the subtleties of sequence convergence in fuzzy normed spaces, paving the way for further advancements in this area of mathematical analysis.

Keywords

L− fuzzy normed space , Ideal convergence , Triple sequence

References

• [1]. A.A. Nabiev, E. Savaş, M. Gürdal, Statistically localized sequences in metric spaces, J. Appl. Anal. Comput., 9(2) (2019), 739-746.
• [2]. C. Alaca, D. Turkoglu, and C. Yildiz. "Fixed points in intuitionistic fuzzy metric spaces." Chaos, Solitons & Fractals 29.5 (2006): 1073-1078.
• [3]. C. Alaca, D. Turkoglu, and C. Yildiz. "Common Fixed Points of Compatible Maps in Intuitionistic Fuzzy Metric Spaces." Southeast Asian Bull. Math. 32.1 (2008).
• [4]. K. Atanassov. ”Intuitionistic fuzzy sets.” Fuzzy Sets Syst. 1986; 20:87–96
• [5]. A. Nabiev, S. Pehlivan, M. Gürdal, On I-Cauchy sequences. Taiwanese J. Math., 11(2) (2007), 569-576.
• [6]. A. Şahiner, M. Gürdal, T. Yiğit, Ideal convergence characterization of the completion of linear n-normed spaces, Comput. Math. Appl., 61(3) (2011), 683-689.
• [7]. A. Şahiner, M. Gürdal, S. Saltan, H. Gunawan, Ideal convergence in 2-normed spaces. Taiwanese J. Math., 11(5) (2007), 1477-1484.
• [8]. N. Eghbali, M. Ganji. Generalized Statistical Convergence in the Non-Archimedean L-fuzzy Normed Spaces. Azerb. J. Math., 2016; 6(1), 15-22.
• [9]. E. Dündar, M. R. Türkmen, On I_2-Convergence and I_2^* -Convergence of Double Sequences in Fuzzy Normed Spaces, Konuralp J. Math. 7(2) (2019), 405–409.
• [10]. E. Dündar, M. R. Türkmen, On I_2-Cauchy Double Sequences in Fuzzy Normed Spaces, Commun. Adv. Math. Sci., 2(2) (2019), 154–160.
• [11]. M. Gürdal, M.B. Huban, On I-convergence of double sequences in the topology induced by random 2-norms, Mat. Vesnik, 66(1) (2014), 73-83.
• [12]. M. R. Türkmen, E. Dündar, On Lacunary Statistical Convergence of Double Sequences and Some Properties in Fuzzy Normed Spaces, J. Intell. Fuzzy Syst., 36(2) (2019), 1683–1690, DOI:10.3233/JIFS-181796
• [13]. E. Dündar, B. Altay I_2-uniform convergence of double sequences of functions. Filomat 30(5) (2016) 1273–1281.
• [14]. E. Dündar. Ö. Talo, I_2-convergence of double sequences of fuzzy numbers, Iran. J. Fuzzy Syst., 10(3) (2013), 37–50.
• [15]. E. Dündar, B. Altay, I2-convergence and I_2-Cauchy of double sequences, Acta Math. Sci., 34B(2) (2014), 343–353.
• [16]. E. Dündar, B. Altay, On some properties of I_2-convergence and I_2-Cauchy of double sequences, Gen. Math. Notes, 7(1)(2011), 1–12.
• [17]. E. Dündar, Ö. Talo, I_2-Cauchy Double Sequences of Fuzzy Numbers, Gen. Math. Notes, 16(2) (2013), 103–114.
• [18]. E. Dündar, Ö. Talo, F. Başar, Regularly (I_2,I)-Convergence and Regularly (I_2,I)-Cauchy Double Sequences of Fuzzy Numbers, Int. J. Anal., vol. 2013, pp. 1–7, Apr. 2013.
• [19]. E. Dündar, M. R. Türkmen and N. Pancaroglu Akın, Regularly ideal convergence of double sequences in fuzzy normed spaces,Bull. Math. Anal. Appl., 12(2) (2020), 12–26.
• [20]. J.A. Goguen. L-fuzzy Sets. J. Math. Anal. Appl. 18, 145–174 (1967)
• [21]. M. Grabiec Fixed points in fuzzy metric spaces. Fuzzy Sets Syst. 1988; 27:385–389
• [22]. V. Gregori, JJ. Miñana, S. Morillas, A. Sapena. Cauchyness and convergence in fuzzy metric spaces. RACSAM 2017; 111(1): 25–37
• [23]. M. Gürdal and E. Savaş. "An investigation on the triple ideal convergent sequences in fuzzy metric spaces."Commun. Fac. Sci. Univ. Ankara Ser. A1 Math. Stat. 71.1 (2022): 13-24.
• [24]. S. Karakuş, K. Demirci, O. Duman.. Statistical convergence on intuitionistic fuzzy normed spaces. Chaos, Solitons & Fractals 2008; 35:763–769
• [25]. P. Kostyrko, S. Tibor, and W. Władysław. "I-convergence. "Real Anal. Exchange (2000): 669-685.
• [26]. P. Kostyrko, et al. "I-convergence and extremal I-limit points." Math. Slovaca 55.4 (2005): 443-464.
• [27]. V. Kumar and K. Kuldeep. "On the ideal convergence of sequences of fuzzy numbers."Inf. Sci. 178.24 (2008): 4670-4678.
• [28]. SA. Mohiuddine and QM. Danish Lohani. On generalized statistical convergence in intuitionistic fuzzy normed space. Chaos, Solitons & Fractals 2009; 42: 1731–1737
• [29]. M. Mursaleen and SA. Mohiuddine. On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space. J. Comput. Appl. Math. 2009; 233:142–149
• [30]. M. Mursaleen, SA. Mohiuddine and H.E. Edely. "On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces." Comput. Math. Appl. 59.2 (2010): 603-611.
• [31]. JH. Park Intuitionistic fuzzy metric spaces. Chaos, Solitons & Fractals 2004; 22:1039–1046
• [32]. S. Pandit, A. Ahmad, and A. Esi. "On intuitionistic fuzzy metric space and ideal convergence of triple sequence space."Sahand Commun. Math. Anal. 20.1 (2023): 35-44.
• [33]. D. Rath, B. Tripathy. 1994. On statistically convergent and statistically Cauchy sequences, Indian J. Pure Appl. Math., 25, 381-386.
• [34]. R. Saadati, JH. Park. On the intuitionistic fuzzy topological spaces. Chaos, Solitons & Fractals 2006; 27:331–344
• [35]. E. Savaş and M. Gürdal. "Ideal convergent function sequences in random 2-normed spaces." Filomat 30.3 (2016): 557-567.
• [36]. E. Savas and D. Pratulananda. "A generalized statistical convergence via ideals." Appl. Math. Lett. 24.6 (2011): 826-830.
• [37]. E. Savaş, S. Debnath, and D. Rakshit. "On I-statistically rough convergence."Publ. Inst. Math. (Beograd) 105.119 (2019): 145-150.
• [38]. E. Savas and M. Gürdal. "A generalized statistical convergence in intuitionistic fuzzy normed spaces." Sci. Asia 41 (2015): 289-294.
• [39]. S. Shakeri, R. Saadati, C. Park Stability of the quadratic functional equation in non-Archimedean L-fuzzy normed spaces.Int. J. Nonlinear Anal. Appl., 2010, 1.2: 72-83.
• [40]. H. Steinhaus. 1951. Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math, 2, 73-74.
• [41]. S. Kastyro, T. Salat, W. Wilczynski, I-Convergence, Real Anal. Exchange 26(2000) 669-686.2001.
• [42]. A. Şahiner, M. Gürdal, F.K. Düden, Triple sequences and their statistical convergence, Selcuk J. Appl. Math., 8(2) (2007), 49-55.
• [43]. U. Yamancı, M. Gürdal, On lacunary ideal convergence in random nnormed space, J. Math., 2013; 868457, 1-8.
• [44]. R. Yapalı and U. Gürdal. "Pringsheim and tatistical convergence for double sequences on L- fuzzy normed space." AIMS Math. 6.12 (2021): 13726-13733.
• [45]. R. Yapalı and Ö. Talo. "Tauberian conditions for double sequences which are statistically summable (C, 1, 1) in fuzzy number space." J. Intell. Fuzzy Syst. 33.2 (2017): 947-956.
• [46]. R. Yapalı, H. Çoşkun, and U. Gürdal. "Statistical convergence on L- fuzzy normed space." Filomat 37.7 (2023): 2077-2085.
• [47]. R. Yapalı and H. Coşkun. "Lacunary Statistical Convergence for Double Sequences on L- Fuzzy Normed Space." J. Math. Sci. Model. 6.1 (2023): 24-31.
• [48]. R. Yapali et al. "Lacunary statistical convergence on L-fuzzy normed space." J. Intell. Fuzzy Syst.46.1 (2024): 1985-1993.
• [49]. LA. Zadeh Fuzzy sets. Inf. Control1965; 8:338–353