ON SOME GENERALIZED DEFERRED STATISTICAL CONVERGENCE OF ORDER αβ FOR FUZZY VARIABLE SEQUENCES IN CREDIBILITY SPACE

Ekrem Savaş, Ömer Kişi, Mehmet Gürdal

DOI Number
https://doi.org/10.22190/FUMI230220038S
First page
599
Last page
620

Abstract


In this paper, we investigate the concepts of deferred statistical convergence of order αβ and strongly s-deferred Cesaro summability of order αβ for fuzzy variable sequences in credibility space. Furthermore, the conditions of deferred statistical convergence almost surely of order αβ, deferred statistical convergence in credibility of order αβ, deferred statistical convergence in mean of order αβ, deferred statistical convergence in distribution of order αβ, and deferred statistical convergence uniformly almost surely of order αβ of fuzzy variable sequences have been examined. We have proved relations between these notions.


Keywords

Statistical convergence, Deferred statistical convergence, fuzzy variable sequences.

Full Text:

PDF

References


bibitem{agnew}

{sc R. P. Agnew}: textit{On deferred Ces`{a}ro means}, Ann. Math. {bf 33} (3) (1932), 413--421.

bibitem{B2012}

{sc F. Bac{s}ar}: textit{Summability Theory and its

Applications}, Second edition, CRC Press/Taylor & Francis Group, 2022.

bibitem{Moh1}

{sc C. Belen {rm and} S. A. Mohiuddine}: textit{Generalized weighted

statistical convergence and application}, Appl. Math. Comput. {bf 219} (2013), 9821--9826.

bibitem{BDP}

{sc S. Bhunia, P. Das {rm and} S. K. Pal}: textit{Restricting statistical convergence}, Acta Math. Hungar. {bf 134} (1-2) (2012), 153--161.

bibitem{CNW2016}

{sc X. Chen, Y. Ning {rm and} X. Wang}: textit{Convergence of complex

uncertain sequence}, J. Intell. Fuzzy Systems {bf 30} (6) (2016), 3357--3366.

bibitem{C1}

{sc R. c{C}olak}: textit{Statistical convergence of order $alpha$}, In:

Modern Methods in Analysis and Its Applications, Anamaya

Publisher, New Delhi, India, (2010), 121--129.

bibitem{CB}

{sc R. c{C}olak {rm and} C. A. Bektaş}: textit{$lambda$-Statistical convergence of

order $alpha$}, Acta Math. Sin. Engl. Ser. {bf 31} (3) (2011), 953--959.

bibitem{DP}

{sc D. Dubois {rm and} H. Prade}: textit{Possibility theory: An approach

to computerized processing of uncertainty}, New York, Plenum, 1998.

bibitem{E3}

{sc M. Et, H. S. Kandemir {rm and} M. Cinar}: textit{Generalized deferred

statistical convergence}, In: Approximation Theory, Sequence Spaces and

Applications, Springer, (2022), 35--52.

bibitem{E1}

{sc M. Et {rm and} H. Şengül}: textit{Some Ces`{a}ro-type summability spaces of

order $alpha$ and lacunary statistical convergence of order $alpha$}, Filomat {bf 28} (8) (2014), 1593--1602.

bibitem{HF}

{sc H. Fast}: textit{Sur la convergence statistique}, Colloq. Math. {bf 2} (1951), 241--244.

bibitem{Fri}

{sc J. A. Fridy}: textit{On statistical convergence}, Analysis {bf 5} (1985), 301--313.

bibitem{GO}

{sc A. D. Gadjiev {rm and} C. Orhan}: textit{Some approximation theorems via

statistical convergence}, Rocky Mountain J. Math. {bf 32} (1) (2002), 129--138.

bibitem{G1}

{sc M. Gürdal, A. Şahiner {rm and} I. Açık}: textit{Approximation theory in 2-Banach spaces}, Nonlinear Anal. {bf 71} (5-6) (2009), 1654--1661.

bibitem{Moh5}

{sc B. Hazarika, A. Alotaibi {rm and} S. A. Mohiuddine}: textit{Statistical convergence in measure for double sequences of

fuzzy-valued functions}, Soft Comput. {bf 24} (2020), 6613--6622.

bibitem{Jiang}

{sc Q. Jiang}: textit{Some remarks on convergence in cedibility

distribution of fuzzy variable}, In: Proceedings of a Conference on Intelligence

Science and Information Engineering, Wuhan, China, 2011, pp. 446--449.

bibitem{UKFB}

{sc U. Kadak {rm and} F. Bac{s}ar}: textit{Power series with real or

fuzzy coefficients}, Filomat {bf 25} (3)

(2012), 519--528.

bibitem{Moh2}

{sc U. Kadak {rm and} S. A. Mohiuddine}: textit{Generalized

statistically almost convergence based on the difference operator which

includes the (p,q)-Gamma function and related approximation theorems}, Results

Math. {bf 73} (9) (2018), 1--31.

bibitem{Kaufmann}

{sc A. Kaufmann}: textit{Introduction to the theory of fuzzy

subsets}, New York: Academic Press, 1975.

bibitem{kisi1}

{sc "{O}. Kic{s}i, M. G"{u}rdal {rm and} E. Savac{s}}: textit{On lacunary convergence in credibility space}, Facta Univ. Math. Inform. {bf 37} (4) (2022), 683--708.

bibitem{kisi2}

{sc "{O}. Kic{s}i, M. G"{u}rdal {rm and} E. Savac{s}}: textit{On deferred statistical convergence of fuzzy variables}, Appl. Appl. Math. {bf 17} (2) (2022), 366--385.

bibitem{k1}

{sc M. Küçükaslan {rm and} M. Yilmaztürk}: textit{On deferred statistical

convergence of sequences}, Kyungpook Math. J. {bf 56} (2) (2016), 357--366.

bibitem{K1978}

{sc H. Kwakernaak}: textit{Fuzzy random variables-I: definition

and theorem}, Inf. Sci. {bf 15} (1) (1978), 1--29.

bibitem{X2006}

{sc X. Li {rm and} B. Liu}: textit{A sufficient and necessary condition

for credibility measures}, Internat. J. Uncertain. Fuzziness Knowledge-Based

Systems {bf 14} (2006), 527--535.

bibitem{Lin}

{sc X. Lin}: textit{Characteristics of convex function}, Journal

of Guangxi University for Nationalities (Natural Science Edition) {bf 6} (4) (2000), 250--253.

bibitem{BLiu2002}

{sc B. Liu {rm and} Y. K. Liu}: textit{Expected value of fuzzy

variable and fuzzy expected value models}, IEEE Trans. Fuzzy Syst. {bf 10} (4) (2002), 445--450.

bibitem{Liu2003}

{sc B. Liu}: textit{Inequalities and convergence concepts of fuzzy and rough variables}, Fuzzy

Optim. Decis. Mak. {bf 2} (2) (2003), 87--100.

bibitem{Liu2004}

{sc B. Liu}: textit{Uncertainty theory}, Springer, Berlin, Heidelberg, Second Edition, 2007.

bibitem{Liu2006}

{sc B. Liu}: textit{A survey of credibility theory}, Fuzzy

Optim. Decis. Mak. {bf 5} (4) (2006), 387--408.

bibitem{LW}

{sc Y. K. Liu {rm and} S. M. Wang}: textit{Theory of fuzzy random optimization,

China Agricultural University Press}, Beijing, p. 280, 2006.

bibitem{Ma}

{sc S. Ma}: textit{The convergences properties of the credibility

distribution sequence of fuzzy variables}, J. Modern Math. Frontier. {bf 3} (1) (2014), 24--27.

bibitem{Moh0}

{sc S. A. Mohiuddine, H. c{S}evli {rm and} M. Cancan}: textit{Statistical convergence in fuzzy 2-normed space}, J. Comput. Anal.

Appl. {bf 12} (4) (2010), 787--798.

bibitem{Moh3}

{sc S. A. Mohiuddine, A. Asiri {rm and} B. Hazarika}: textit{Weighted

statistical convergence through difference operator of sequences of fuzzy

numbers with application to fuzzy approximation theorems}, Int. J. Gen. Syst. {bf 48} (5) (2019), 492--506.

bibitem{Moh4}

{sc S. A. Mohiuddine, B. Hazarika {rm and} A. Alotaibi}: textit{On

statistical convergence of double sequences of fuzzy valued functions}, J.

Intell. Fuzzy Syst. {bf 32} (2017), 4331--4342.

bibitem{Moh6}

{sc S. A. Mohiuddine {rm and} B. A. S. Alamri}: textit{Generalization of

equi-statistical convergence via weighted lacunary sequence with associated

Korovkin and Voronovskaya type approximation theorems}, Rev. Real Acad. Cienc.

Exactas Fis. Nat. Ser. A-Mat. RACSAM {bf 113} (3) (2019), 1955--1973.

bibitem{MB2020}

{sc M. Mursaleen {rm and} F. Bac{s}ar}: textit{Sequence Spaces:

Topics in Modern Summability Theory}, CRC Press, Taylor & Francis Group,

Series: Mathematics and Its Applications, Boca Raton, London, New York, 2020.

bibitem{Nahmias}

{sc S. Nahmias}: textit{Fuzzy variables}, Fuzzy Sets Syst. {bf 1} (1978), 97--110.

bibitem{NS95}

{sc F. Nuray {rm and} E. Savac{s}}: textit{Statistical convergence of

sequences of fuzzy numbers}, Math. Slovaca {bf 45} (1995), 269--273.

bibitem{Rath}

{sc D. Rath {rm and} B. C. Tripathy}: textit{On statistically convergent

and statistical Cauchy sequences}, Indian J. Pure Appl. Math. {bf 25} (4) (1994), 381--386.

bibitem{Rath1}

{sc D. Rath {rm and} B. C. Tripathy}: textit{Matrix maps on sequence spaces

associated with sets of integers}, Indian J. Pure Appl. Math. {bf 27} (2) (1996), 197--206.

bibitem{Salat}

{sc T. v{S}al'{a}t}: textit{On statistically convergent

sequences of real numbers}, Math. Slovaca {bf 30} (2) (1980), 139--150.

bibitem{G2}

{sc E. Savac{s} {rm and} M. G"{u}rdal}: textit{A generalized statistical convergence in intuitionistic fuzzy normed spaces}, Scienceasia {bf 41} (4) (2015), 289--294

bibitem{kisi3}

{sc E. Savac{s}, "{O}. Kic{s}i {rm and} M. G"{u}rdal}: textit{On statistical convergence in credibility space}, Numer. Funct. Anal. Optim. {bf 43} (8) (2022), 987--1008.

bibitem{E2}

{sc H. Şengül {rm and} M. Et}: textit{On lacunary statistical convergence of order

$alpha$}, Acta Math. Sci. Ser. B Engl. Ed. {bf 34} (2) (2014), 473--482.

bibitem{Steinhaus}

{sc H. Steinhaus}: textit{Sur la convergence ordinarie et la

convergence asymptotique}, Colloq. Math. {bf 2} (1951), 73--74.

bibitem{OTFB3}

{sc "{O}. Talo {rm and} F. Bac{s}ar}: textit{Quasilinearity of the

classical sets of sequences of fuzzy numbers and some related results}, Taiwanese J. Math. {bf 14} (5) (2010), 1799--1819.

bibitem{t}

{sc F. Temizsu, M. Et, {rm and} M. Cinar}: textit{$Delta^{m}$-deferred statistical

convergence of order $alpha$}, Filomat {bf 30} (3) (2016), 667--673.

bibitem{tri}

{sc B. C. Tripathy}: textit{Matrix transformations between some classes of

sequences}, J. Math. Anal. {bf 206} (1997), 448--450.

bibitem{tri1}

{sc B. C. Tripathy {rm and} M. Sen}: textit{On generalized statistically

convergent sequences}, Indian J. Pure Appl. Math. {bf 32} (11) (2001), 1689--1694.

bibitem{WL2003}

{sc G. Wang {rm and} B. Liu}: textit{New theorems for fuzzy sequence

convergence}, In: Proceedings of a Conference on Information and Management Science, Chengdu, China, 2003, pp. 100--105.

bibitem{X2011}

{sc Y. Xia}: textit{Convergence of uncertain sequences}, M.S.

Thesis, Suzhou University of Science and Technology, 2011.

bibitem{Y2009}

{sc C. You}: textit{On the convergence of uncertain sequences}, Math. Comput. Model. {bf 49} (3-4) (2009), 482--487.

bibitem{k2}

{sc M. Yilmaztürk {rm and} M. Küçükaslan}: textit{On strongly deferred Ces`{a}ro

summability and deferred statistical convergence of the sequences}, Bitlis Eren

Univ. J. Sci. Technol. {bf 3} (2011), 22--25.

bibitem{Y2019}

{sc C. You, R. Zhang {rm and} K. Su}: textit{On the convergence of

fuzzy variables}, J. Intell. Fuzzy Systems {bf 36} (2) (2019), 1663--1670.

bibitem{Zadeh65}

{sc L. A. Zadeh}: textit{Fuzzy set}, Inf. Control. {bf 8} (3) (1965), 338--353.




DOI: https://doi.org/10.22190/FUMI230220038S

Refbacks

  • There are currently no refbacks.




© University of Niš | Created on November, 2013
ISSN 0352-9665 (Print)
ISSN 2406-047X (Online)