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On Fibonacci Ideal Convergence of Double Sequences in Intuitionistic Fuzzy Normed Linear Spaces

Year 2019, Volume: 11, 46 - 55, 30.12.2019

Ömer Kişi , Erhan Güler

Abstract

The aim of this article is to introduce and study the notion of Fibonacci $% \mathcal{I}_{2}$-convergence on intuitionistic fuzzy normed linear space. We define the Fibonacci $\mathcal{I}_{2}$-Cauchy sequences and the Fibonacci $% \mathcal{I}_{2}$ completeness with respect to an intuitionistic fuzzy normed linear space.

Keywords

Fibonacci I₂-convergence Fibonacci I₂-Cauchy sequence intuitionistic fuzzy normed linear space

References

  • Anastassiou, G.A., {\em Fuzzy approximation by fuzzy convolution type operators}, Comput. Math. Appl., \textbf{48}(2004), 1369--1386.
  • Arslan M., D\"{u}ndar E., \textit{$\mathcal{I}$-convergence and $% \mathcal{I}$-Cauchy sequence of functions in 2-normed spaces}, Konuralp J. Math., \textbf{6:1}(2018), 57-- 62.
  • Atanassov, K.T., \textit{Intuitionistic fuzzy sets}, Fuzzy Sets Syst., \textbf{20}(1986), 87--96.
  • Atanassov, K., Pasi, G., Yager, R., Intuitionistic fuzzy interpretations of multi-person multicriteria decision making, in: Proceedings of 2002 First International IEEE Symposium Intelligent Systems, \textbf{1}(2002), 115-119.
  • Bakery, A.A.,\textit{ Operator ideal of Cesaro type sequence spaces involving lacunary sequence,} Abst. Appl. Anal, \textbf{2014}(2014), 6 pp., Article ID 419560, http://dx.doi.org/10.1155/2014/419560.
  • Bakery, A.A., Mohammed, M.M., \textit{On lacunary mean ideal convergence in generalized random $n$-normed spaces}, Abstract Appl. Anal., \textbf{2014}(2014), 11 pp., Article ID:101782, http://dx.doi.org/10.1155/2014/101782.
  • Barros, L.C., Bassanezi, R.C., Tonelli, P.A., \textit{Fuzzy modelling in population dynamics}, Ecol Model, \textbf{128}(2000), 27--33.
  • Debnath, P., \textit{Lacunary ideal convergence in intuitionistic fuzzy normed linear spaces}, Comput. Math. Appl., \textbf{63}(2012), 708--715.
  • D\"{u}ndar E., Altay B., \textit{$\mathcal{I}_{2}$-convergence and $% \mathcal{I}_{2}$-Cauchy of double sequences}, Acta Math. Sci., \textbf{34B(2)}(2014), 343--353.
  • El Naschie, M.S.,\textit{ On the unification of heterotic strings, $% m$-theory and $\epsilon ^{\infty }$-theory}, Chaos Solitons Fractals, \textbf{11}(2000), 2397--2408.
  • Erceg, M.A., \textit{Metric spaces in fuzzy set theory}, J. Math. Anal. Appl., \textbf{69}(1979), 205--230.
  • Fast, H., \textit{Sur la convergence statistique}, Colloq. Math., \textbf{2}(1951), 241--244.
  • Fradkov, A.L., Evans, R.J., \textit{Control of chaos: methods and applications in engineering}, Chaos Solitons Fractals, \textbf{29}(1) (2005), 33--56.
  • Fridy, J.A., Orhan, C., \textit{Lacunary statistical convergence}, Pacific J. Math., \textbf{160(1)}(1993), 43--51.
  • Fridy, J.A., Orhan, C., \textit{Lacunary statistical summability}, J. Math. Anal Appl., \textbf{173}(1993), 497--504.
  • George, A., Veeramani P., \textit{On some results in fuzzy metric spaces}, Fuzzy Sets Syst., \textbf{64(3)}(1994), 395--399.
  • Giles, R., \textit{A computer program for fuzzy reasoning}, Fuzzy Sets Syst., \textbf{4}(1980), 221--234.
  • Goonatilake, S., Toward a Global Science, p. 126, Indiana University Press, (1998).
  • Hazarika, B., \textit{ Lacunary $\mathcal{I}$-convergent sequence of fuzzy real numbers}, Pacific J. Sci. Technol., \textbf{10(2)}(2009), 203--206.
  • Hazarika, B., \textit{Fuzzy real valued lacunary $\mathcal{I}$% -convergent sequences}, Appl. Math. Lett., \textbf{25}(2012), 466--470.
  • Hazarika, B., \textit{Lacunary ideal convergent double sequences of fuzzy real numbers}, J. Intell. Fuzzy Syst. \textbf{27(1)}(2014), 495--504.
  • Hazarika, B., \textit{Lacunary ideal convergence of multiple sequences}, J. Egyptian Math. Soc., \textbf{24(1)}(2016), 54--59.
  • Hong, L., Sun, J.Q., \textit{Bifurcations of fuzzy nonlinear dynamical systems}, Commun. Nonlinear Sci. Numer. Simul., \textbf{1}(2006), 1--12.
  • Hosseini, S.B., O'Regan, D., Saadati, R., \textit{Some results of intuitionistic fuzzy spaces}, Iranian J. Fuzzy Syst., \textbf{4(1)}(2007), 53--64.
  • J\"{a}ger, G., \textit{Fuzzy uniform convergence and equicontinuity}, Fuzzy Sets Syst., \textbf{109}(2000), 187--198.
  • Kaleva, O., Seikkala, S., \textit{On fuzzy metric spaces}, Fuzzy Sets Syst., \textbf{12}(1984), 215--229.
  • Kara, E.E., Ba\c{s}ar\i r, M., \textit{An application of Fibonacci numbers into infinite Toeplitz matrices}, Caspian J. Math. Sci., \textbf{1(1)}(2012), 43--47.
  • Kara, EE., \textit{Some topological and geometrical properties of new Banach sequence spaces}, J. Inequal. Appl., \textbf{38}(2013), doi:10.1186/1029-242X-2013-38.
  • Karakus, S., Demirci, K., Duman, O., \textit{Statistical convergence on intuitionistic fuzzy normed spaces}, Chaos Solitons Fractals, \textbf{35}(2008), 763--769.
  • Kiri\c{s}\c{c}i, M, \textit{Fibonacci statistical convergence on intuitionistic fuzzy normed spaces}, J. Intell Fuzzy Syst, \textbf{36(6)}(2019), 5597--5604.
  • Kiri\c{s}\c{c}i, M, Karaisa, A., \textit{Fibonacci statistical convergence and Korovkin type approximation theorems}, J. Inequal. Appl., \textbf{229}(2017), doi: 10.1186/s13660-017-1503-z.
  • Kocinac, Lj.D.R., Khan, V.A., Alshlool, K.M.A.S., Altaf, H., \textit{On some topological properties of intuitionistic 2-fuzzy n-normed linear spaces}, Hacet. J. Math. Stat., in press.
  • Kocinac, Lj.D.R., Rashid, M.H.M., \textit{On ideal convergence of double sequences in the topology induced by a fuzzy 2-norm}, TWMS J. Pure Appl. Math., \textbf{8:1}(2017), 97--111.
  • Koshy, T., Fibonacci and Lucas Numbers with Applications, Wiley, New York. 2001.
  • Kostyrko, P., Salat, T., Wilczynsski, W., \textit{$\mathcal{I}$% -convergence}, Real Anal. Exchange, \textbf{26(2)}(2000-2001), 669--686.
  • Li, J., \textit{Lacunary statistical convergence and inclusion properties between lacunary methods}, Int. J Math. Math. Sci., \textbf{23(3)}(2000), 175--180.
  • Madore, J., \textit{Fuzzy physics}, Ann. Phys., \textbf{219}(1992), 187--198.
  • Mohiuddine, S.A., Alghamdi, M.A., \textit{Statistical summability through lacunary sequence in locally solid Riesz spaces}, J. Ineq Appl., \textbf{225}(2012), 1--9.
  • Mursaleen, M., Mohiuddine, S.A., \textit{Statistical convergence of double sequences in intuitionistic fuzzy normed space}, Chaos Solitons Fractals, \textbf{41}(2009), 2414--2421.
  • Mursaleen, M., Mohiuddine, S.A., \textit{On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space}, J. Comput. Appl. Math., \textbf{233(2)}(2009), 142--149.
  • Mursaleen, M., Mohiuddine, S.A., Edely, H.H., \textit{On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces}, Comput. Math. Appl., \textbf{59}(2010), 603--611.
  • Narayanan, A., Vijayabalaji, S., Thillaigovindan, N. \textit{Intuitionistic fuzzy bounded linear operators}, Iranian J. Fuzzy Syst., \textbf{4}(2007), 89--101.
  • Park, J.H., \textit{Intuitionistic fuzzy metric spaces}, Chaos Solitons Fractals, \textbf{22}(2004), 1039--1046.
  • Saadati, R., Park, J.H., \textit{On the intuitionistic fuzzy topological spaces}, Chaos Solitons Fractals, \textbf{27}(2006), 331--344.
  • Salat, T., Tripathy, B.C., Ziman, M., \textit{On some properties of $% \mathcal{I}$-convergence}, Tatra Mt. Math Publ., \textbf{28}(2004), 279--286.
  • Schoenberg, I.J., \textit{The integrability of certain functions and related summability methods}, Am. Math. Mon., \textbf{66}(1959), 361--375.
  • Sen, M., Debnath P., \textit{Lacunary statistical convergence in intuitionistic fuzzy $n$-normed linear spaces}, Math. Comput. Modelling, \textbf{54}(2011), 2978--2985.
  • Tripathy, B.C., Hazarika, B., \textit{$\mathcal{I}$% -monotonic and $\mathcal{I}$-convergent sequences}, Kyungpook Math. J., \textbf{51}(2011), 233--239.
  • Tripathy, B.C., Hazarika, B., Choudhary, B., \textit{Lacunary $\mathcal{I}$-convergent sequences}, Kyungpook Math. J., \textbf{52(4)}(2012), 473--482.
  • Wu, K., \textit{Convergences of fuzzy sets based on decomposition theory and fuzzy polynomial function}, Fuzzy Sets Syst., \textbf{109}(2000), 173--185.
  • Zadeh, L.A.,\textit{ Fuzzy sets}, Inf. Control., \textbf{8}(1965), 338--353.

Year 2019, Volume: 11, 46 - 55, 30.12.2019

Ömer Kişi , Erhan Güler

Abstract

References

  • Anastassiou, G.A., {\em Fuzzy approximation by fuzzy convolution type operators}, Comput. Math. Appl., \textbf{48}(2004), 1369--1386.
  • Arslan M., D\"{u}ndar E., \textit{$\mathcal{I}$-convergence and $% \mathcal{I}$-Cauchy sequence of functions in 2-normed spaces}, Konuralp J. Math., \textbf{6:1}(2018), 57-- 62.
  • Atanassov, K.T., \textit{Intuitionistic fuzzy sets}, Fuzzy Sets Syst., \textbf{20}(1986), 87--96.
  • Atanassov, K., Pasi, G., Yager, R., Intuitionistic fuzzy interpretations of multi-person multicriteria decision making, in: Proceedings of 2002 First International IEEE Symposium Intelligent Systems, \textbf{1}(2002), 115-119.
  • Bakery, A.A.,\textit{ Operator ideal of Cesaro type sequence spaces involving lacunary sequence,} Abst. Appl. Anal, \textbf{2014}(2014), 6 pp., Article ID 419560, http://dx.doi.org/10.1155/2014/419560.
  • Bakery, A.A., Mohammed, M.M., \textit{On lacunary mean ideal convergence in generalized random $n$-normed spaces}, Abstract Appl. Anal., \textbf{2014}(2014), 11 pp., Article ID:101782, http://dx.doi.org/10.1155/2014/101782.
  • Barros, L.C., Bassanezi, R.C., Tonelli, P.A., \textit{Fuzzy modelling in population dynamics}, Ecol Model, \textbf{128}(2000), 27--33.
  • Debnath, P., \textit{Lacunary ideal convergence in intuitionistic fuzzy normed linear spaces}, Comput. Math. Appl., \textbf{63}(2012), 708--715.
  • D\"{u}ndar E., Altay B., \textit{$\mathcal{I}_{2}$-convergence and $% \mathcal{I}_{2}$-Cauchy of double sequences}, Acta Math. Sci., \textbf{34B(2)}(2014), 343--353.
  • El Naschie, M.S.,\textit{ On the unification of heterotic strings, $% m$-theory and $\epsilon ^{\infty }$-theory}, Chaos Solitons Fractals, \textbf{11}(2000), 2397--2408.
  • Erceg, M.A., \textit{Metric spaces in fuzzy set theory}, J. Math. Anal. Appl., \textbf{69}(1979), 205--230.
  • Fast, H., \textit{Sur la convergence statistique}, Colloq. Math., \textbf{2}(1951), 241--244.
  • Fradkov, A.L., Evans, R.J., \textit{Control of chaos: methods and applications in engineering}, Chaos Solitons Fractals, \textbf{29}(1) (2005), 33--56.
  • Fridy, J.A., Orhan, C., \textit{Lacunary statistical convergence}, Pacific J. Math., \textbf{160(1)}(1993), 43--51.
  • Fridy, J.A., Orhan, C., \textit{Lacunary statistical summability}, J. Math. Anal Appl., \textbf{173}(1993), 497--504.
  • George, A., Veeramani P., \textit{On some results in fuzzy metric spaces}, Fuzzy Sets Syst., \textbf{64(3)}(1994), 395--399.
  • Giles, R., \textit{A computer program for fuzzy reasoning}, Fuzzy Sets Syst., \textbf{4}(1980), 221--234.
  • Goonatilake, S., Toward a Global Science, p. 126, Indiana University Press, (1998).
  • Hazarika, B., \textit{ Lacunary $\mathcal{I}$-convergent sequence of fuzzy real numbers}, Pacific J. Sci. Technol., \textbf{10(2)}(2009), 203--206.
  • Hazarika, B., \textit{Fuzzy real valued lacunary $\mathcal{I}$% -convergent sequences}, Appl. Math. Lett., \textbf{25}(2012), 466--470.
  • Hazarika, B., \textit{Lacunary ideal convergent double sequences of fuzzy real numbers}, J. Intell. Fuzzy Syst. \textbf{27(1)}(2014), 495--504.
  • Hazarika, B., \textit{Lacunary ideal convergence of multiple sequences}, J. Egyptian Math. Soc., \textbf{24(1)}(2016), 54--59.
  • Hong, L., Sun, J.Q., \textit{Bifurcations of fuzzy nonlinear dynamical systems}, Commun. Nonlinear Sci. Numer. Simul., \textbf{1}(2006), 1--12.
  • Hosseini, S.B., O'Regan, D., Saadati, R., \textit{Some results of intuitionistic fuzzy spaces}, Iranian J. Fuzzy Syst., \textbf{4(1)}(2007), 53--64.
  • J\"{a}ger, G., \textit{Fuzzy uniform convergence and equicontinuity}, Fuzzy Sets Syst., \textbf{109}(2000), 187--198.
  • Kaleva, O., Seikkala, S., \textit{On fuzzy metric spaces}, Fuzzy Sets Syst., \textbf{12}(1984), 215--229.
  • Kara, E.E., Ba\c{s}ar\i r, M., \textit{An application of Fibonacci numbers into infinite Toeplitz matrices}, Caspian J. Math. Sci., \textbf{1(1)}(2012), 43--47.
  • Kara, EE., \textit{Some topological and geometrical properties of new Banach sequence spaces}, J. Inequal. Appl., \textbf{38}(2013), doi:10.1186/1029-242X-2013-38.
  • Karakus, S., Demirci, K., Duman, O., \textit{Statistical convergence on intuitionistic fuzzy normed spaces}, Chaos Solitons Fractals, \textbf{35}(2008), 763--769.
  • Kiri\c{s}\c{c}i, M, \textit{Fibonacci statistical convergence on intuitionistic fuzzy normed spaces}, J. Intell Fuzzy Syst, \textbf{36(6)}(2019), 5597--5604.
  • Kiri\c{s}\c{c}i, M, Karaisa, A., \textit{Fibonacci statistical convergence and Korovkin type approximation theorems}, J. Inequal. Appl., \textbf{229}(2017), doi: 10.1186/s13660-017-1503-z.
  • Kocinac, Lj.D.R., Khan, V.A., Alshlool, K.M.A.S., Altaf, H., \textit{On some topological properties of intuitionistic 2-fuzzy n-normed linear spaces}, Hacet. J. Math. Stat., in press.
  • Kocinac, Lj.D.R., Rashid, M.H.M., \textit{On ideal convergence of double sequences in the topology induced by a fuzzy 2-norm}, TWMS J. Pure Appl. Math., \textbf{8:1}(2017), 97--111.
  • Koshy, T., Fibonacci and Lucas Numbers with Applications, Wiley, New York. 2001.
  • Kostyrko, P., Salat, T., Wilczynsski, W., \textit{$\mathcal{I}$% -convergence}, Real Anal. Exchange, \textbf{26(2)}(2000-2001), 669--686.
  • Li, J., \textit{Lacunary statistical convergence and inclusion properties between lacunary methods}, Int. J Math. Math. Sci., \textbf{23(3)}(2000), 175--180.
  • Madore, J., \textit{Fuzzy physics}, Ann. Phys., \textbf{219}(1992), 187--198.
  • Mohiuddine, S.A., Alghamdi, M.A., \textit{Statistical summability through lacunary sequence in locally solid Riesz spaces}, J. Ineq Appl., \textbf{225}(2012), 1--9.
  • Mursaleen, M., Mohiuddine, S.A., \textit{Statistical convergence of double sequences in intuitionistic fuzzy normed space}, Chaos Solitons Fractals, \textbf{41}(2009), 2414--2421.
  • Mursaleen, M., Mohiuddine, S.A., \textit{On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space}, J. Comput. Appl. Math., \textbf{233(2)}(2009), 142--149.
  • Mursaleen, M., Mohiuddine, S.A., Edely, H.H., \textit{On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces}, Comput. Math. Appl., \textbf{59}(2010), 603--611.
  • Narayanan, A., Vijayabalaji, S., Thillaigovindan, N. \textit{Intuitionistic fuzzy bounded linear operators}, Iranian J. Fuzzy Syst., \textbf{4}(2007), 89--101.
  • Park, J.H., \textit{Intuitionistic fuzzy metric spaces}, Chaos Solitons Fractals, \textbf{22}(2004), 1039--1046.
  • Saadati, R., Park, J.H., \textit{On the intuitionistic fuzzy topological spaces}, Chaos Solitons Fractals, \textbf{27}(2006), 331--344.
  • Salat, T., Tripathy, B.C., Ziman, M., \textit{On some properties of $% \mathcal{I}$-convergence}, Tatra Mt. Math Publ., \textbf{28}(2004), 279--286.
  • Schoenberg, I.J., \textit{The integrability of certain functions and related summability methods}, Am. Math. Mon., \textbf{66}(1959), 361--375.
  • Sen, M., Debnath P., \textit{Lacunary statistical convergence in intuitionistic fuzzy $n$-normed linear spaces}, Math. Comput. Modelling, \textbf{54}(2011), 2978--2985.
  • Tripathy, B.C., Hazarika, B., \textit{$\mathcal{I}$% -monotonic and $\mathcal{I}$-convergent sequences}, Kyungpook Math. J., \textbf{51}(2011), 233--239.
  • Tripathy, B.C., Hazarika, B., Choudhary, B., \textit{Lacunary $\mathcal{I}$-convergent sequences}, Kyungpook Math. J., \textbf{52(4)}(2012), 473--482.
  • Wu, K., \textit{Convergences of fuzzy sets based on decomposition theory and fuzzy polynomial function}, Fuzzy Sets Syst., \textbf{109}(2000), 173--185.
  • Zadeh, L.A.,\textit{ Fuzzy sets}, Inf. Control., \textbf{8}(1965), 338--353.
There are 51 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Ömer Kişi 0000-0001-6844-3092

Erhan Güler 0000-0003-3264-6239

Publication Date December 30, 2019
Published in Issue Year 2019 Volume: 11

Cite

APA Kişi, Ö., & Güler, E. (2019). On Fibonacci Ideal Convergence of Double Sequences in Intuitionistic Fuzzy Normed Linear Spaces. Turkish Journal of Mathematics and Computer Science, 11, 46-55.
AMA Kişi Ö, Güler E. On Fibonacci Ideal Convergence of Double Sequences in Intuitionistic Fuzzy Normed Linear Spaces. TJMCS. December 2019;11:46-55.
Chicago Kişi, Ömer, and Erhan Güler. “On Fibonacci Ideal Convergence of Double Sequences in Intuitionistic Fuzzy Normed Linear Spaces”. Turkish Journal of Mathematics and Computer Science 11, December (December 2019): 46-55.
EndNote Kişi Ö, Güler E (December 1, 2019) On Fibonacci Ideal Convergence of Double Sequences in Intuitionistic Fuzzy Normed Linear Spaces. Turkish Journal of Mathematics and Computer Science 11 46–55.
IEEE Ö. Kişi and E. Güler, “On Fibonacci Ideal Convergence of Double Sequences in Intuitionistic Fuzzy Normed Linear Spaces”, TJMCS, vol. 11, pp. 46–55, 2019.
ISNAD Kişi, Ömer - Güler, Erhan. “On Fibonacci Ideal Convergence of Double Sequences in Intuitionistic Fuzzy Normed Linear Spaces”. Turkish Journal of Mathematics and Computer Science 11 (December 2019), 46-55.
JAMA Kişi Ö, Güler E. On Fibonacci Ideal Convergence of Double Sequences in Intuitionistic Fuzzy Normed Linear Spaces. TJMCS. 2019;11:46–55.
MLA Kişi, Ömer and Erhan Güler. “On Fibonacci Ideal Convergence of Double Sequences in Intuitionistic Fuzzy Normed Linear Spaces”. Turkish Journal of Mathematics and Computer Science, vol. 11, 2019, pp. 46-55.
Vancouver Kişi Ö, Güler E. On Fibonacci Ideal Convergence of Double Sequences in Intuitionistic Fuzzy Normed Linear Spaces. TJMCS. 2019;11:46-55.
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