{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,20]],"date-time":"2026-05-20T14:59:50Z","timestamp":1779289190580,"version":"3.51.4"},"reference-count":44,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2022,1,30]],"date-time":"2022-01-30T00:00:00Z","timestamp":1643500800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,1,30]],"date-time":"2022-01-30T00:00:00Z","timestamp":1643500800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Soft Comput"],"published-print":{"date-parts":[[2022,3]]},"DOI":"10.1007\/s00500-021-06649-6","type":"journal-article","created":{"date-parts":[[2022,1,30]],"date-time":"2022-01-30T00:03:06Z","timestamp":1643500986000},"page":"2625-2634","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["On deferred statistical convergence of order $$\\beta $$ for fuzzy fractional difference sequence and applications"],"prefix":"10.1007","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5618-0413","authenticated-orcid":false,"given":"P.","family":"Baliarsingh","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"L.","family":"Nayak","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,1,30]]},"reference":[{"key":"6649_CR1","doi-asserted-by":"publisher","first-page":"413","DOI":"10.2307\/1968524","volume":"33","author":"RP Agnew","year":"1932","unstructured":"Agnew RP (1932) On deferred Ces\u00e0ro mean. Annal Math 33:413\u2013421","journal-title":"Annal Math"},{"key":"6649_CR2","doi-asserted-by":"publisher","first-page":"174","DOI":"10.1016\/j.cam.2013.05.012","volume":"259","author":"H Aktu\u01e7lu","year":"2014","unstructured":"Aktu\u01e7lu H (2014) Korovkin type approximation theorems proved via $$A$$-statistical convergence. J Comput Appl Math 259:174\u2013181","journal-title":"J Comput Appl Math"},{"issue":"6","key":"6649_CR3","doi-asserted-by":"publisher","first-page":"379","DOI":"10.3233\/IFS-2010-0458","volume":"21","author":"Y Altin","year":"2010","unstructured":"Altin Y, Mursaleen M, Altinok H (2010) Statistical summability $$(C, 1)$$ for sequences of fuzzy real numbers and a Tauberian theorem. J Intell Fuzzy Syst 21(6):379\u2013384","journal-title":"J Intell Fuzzy Syst"},{"key":"6649_CR4","doi-asserted-by":"publisher","first-page":"847","DOI":"10.3233\/IFS-130776","volume":"26","author":"H Altinok","year":"2014","unstructured":"Altinok H (2014) Statistical convergence of order $$\\beta $$ for generalized difference sequences of fuzzy numbers. J Intell Fuzzy Syst 26:847\u2013856","journal-title":"J Intell Fuzzy Syst"},{"issue":"21","key":"6649_CR5","doi-asserted-by":"publisher","first-page":"3128","DOI":"10.1016\/j.fss.2009.06.002","volume":"60","author":"H Altinok","year":"2009","unstructured":"Altinok H, Colak R, Et M (2009) $$\\;\\lambda -$$ difference sequences spaces of fuzzy numbers. Fuzzy Sets Syst 60(21):3128\u20133139","journal-title":"Fuzzy Sets Syst"},{"key":"6649_CR6","doi-asserted-by":"publisher","first-page":"6017","DOI":"10.1007\/s00500-018-3569-z","volume":"23","author":"H Altinok","year":"2019","unstructured":"Altinok H, Et M (2019) Statistical convergence of order $$(\\beta , \\gamma )$$ for sequence of fuzzy numbers. Soft Comput 23:6017\u20136022","journal-title":"Soft Comput"},{"key":"6649_CR7","first-page":"63","volume":"10","author":"GA Anastassiou","year":"2007","unstructured":"Anastassiou GA (2007) Fuzzy random Korovkin theory and inequalities. Math Inequal Appl 10:63\u201394","journal-title":"Math Inequal Appl"},{"key":"6649_CR8","doi-asserted-by":"publisher","first-page":"573","DOI":"10.1016\/j.camwa.2007.05.007","volume":"55","author":"GA Anastassiou","year":"2008","unstructured":"Anastassiou GA, Duman O (2008) Statistical fuzzy approximation by fuzzy positive linear operators. Comput Math Appl 55:573\u2013580","journal-title":"Comput Math Appl"},{"issue":"3","key":"6649_CR9","first-page":"677","volume":"157","author":"C Ayd\u0131n","year":"2004","unstructured":"Ayd\u0131n C, Ba\u015far F (2004) Some new difference sequence spaces. Appl Math Comput 157(3):677\u2013693","journal-title":"Appl Math Comput"},{"issue":"18","key":"6649_CR10","first-page":"9737","volume":"219","author":"P Baliarsingh","year":"2013","unstructured":"Baliarsingh P (2013) Some new difference sequence spaces of fractional order and their dual spaces. Appl Math Comput 219(18):9737\u20139742","journal-title":"Appl Math Comput"},{"key":"6649_CR11","first-page":"665","volume":"250","author":"P Baliarsingh","year":"2015","unstructured":"Baliarsingh P, Dutta S (2015) On the classes of fractional order difference sequence spaces and their matrix transformations. Appl Math Comput 250:665\u2013674","journal-title":"Appl Math Comput"},{"issue":"1","key":"6649_CR12","doi-asserted-by":"publisher","first-page":"49","DOI":"10.5269\/bspm.v33i1.19884","volume":"33","author":"P Baliarsingh","year":"2015","unstructured":"Baliarsingh P, Dutta S (2015) A unifying approach to the difference operators and their applications. Boletim da Sociedade Paranaense de Matematica 33(1):49\u201357","journal-title":"Boletim da Sociedade Paranaense de Matematica"},{"issue":"2","key":"6649_CR13","doi-asserted-by":"publisher","first-page":"1811","DOI":"10.1016\/j.aej.2016.03.037","volume":"55","author":"P Baliarsingh","year":"2016","unstructured":"Baliarsingh P (2016) On a fractional difference operator. Alex Eng J 55(2):1811\u20131816","journal-title":"Alex Eng J"},{"issue":"2","key":"6649_CR14","doi-asserted-by":"publisher","first-page":"1051","DOI":"10.1016\/j.aej.2017.02.022","volume":"57","author":"P Baliarsingh","year":"2018","unstructured":"Baliarsingh P, Nayak L (2018) A note fractional difference operators. Alex Eng J 57(2):1051\u20131054","journal-title":"Alex Eng J"},{"key":"6649_CR15","doi-asserted-by":"publisher","DOI":"10.1002\/mma.6417","author":"P Baliarsingh","year":"2020","unstructured":"Baliarsingh P (2020) On certain dynamic properties of difference sequences and the fractional derivatives. Math Methods tghe Appl Sci. https:\/\/doi.org\/10.1002\/mma.6417","journal-title":"Math Methods tghe Appl Sci"},{"issue":"8","key":"6649_CR16","doi-asserted-by":"publisher","first-page":"1117","DOI":"10.2989\/16073606.2017.1420705","volume":"41","author":"P Baliarsingh","year":"2018","unstructured":"Baliarsingh P, Kadak U, Mursaleen M (2018) On statistical convergence of difference sequences of fractional order and related Korovkin type approximation theorems. Quaestiones Mathematicae 41(8):1117\u20131133","journal-title":"Quaestiones Mathematicae"},{"key":"6649_CR17","first-page":"162","volume":"228","author":"NL Braha","year":"2014","unstructured":"Braha NL, Srivastava HM, Mohiuddine SA (2014) A Korovkin\u2019s type approximation theorem for periodic functions via the statistical summability of the generalized De La Vallee Poussin mean. Appl Math Comput 228:162\u2013169","journal-title":"Appl Math Comput"},{"key":"6649_CR18","first-page":"121","volume-title":"Statistic al convergence of order $$\\alpha ,\\;$$ Modern Methods in Analysis and its Applications","author":"R Colak","year":"2010","unstructured":"Colak R (2010) Statistic al convergence of order $$\\alpha ,\\;$$ Modern Methods in Analysis and its Applications. Anamaya Publ., New Delhi, India, pp 121\u2013129"},{"key":"6649_CR19","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1524\/anly.1988.8.12.47","volume":"8","author":"JS Connor","year":"1988","unstructured":"Connor JS (1988) The statistical and strong $$p$$- Ces\u00e0ro convergence of sequence. Analysis 8:47\u201363","journal-title":"Analysis"},{"issue":"4","key":"6649_CR20","first-page":"377","volume":"21","author":"M Et","year":"1995","unstructured":"Et M, Colak R (1995) On some generalized difference sequence spaces. Soochow J Math 21(4):377\u2013386","journal-title":"Soochow J Math"},{"key":"6649_CR21","doi-asserted-by":"crossref","unstructured":"Et M, Mohiuddine S. A, Alotaibi A (2013) On $$\\lambda $$-statistical convergence and strongly $$\\lambda $$summable functions of order $$\\alpha ,\\;$$ J Inequal Appl, 1 , 1-8","DOI":"10.1186\/1029-242X-2013-469"},{"key":"6649_CR22","doi-asserted-by":"publisher","first-page":"241","DOI":"10.4064\/cm-2-3-4-241-244","volume":"2","author":"H Fast","year":"1951","unstructured":"Fast H (1951) Sur la convergence statistique. Colloquium Mathematicum 2:241\u2013244","journal-title":"Colloquium Mathematicum"},{"issue":"4","key":"6649_CR23","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1524\/anly.1985.5.4.301","volume":"5","author":"JA Fridy","year":"1985","unstructured":"Fridy JA (1985) On statistical convergence. Analysis 5(4):301\u2013313","journal-title":"Analysis"},{"issue":"1","key":"6649_CR24","first-page":"129","volume":"32","author":"AD Gadjiev","year":"1980","unstructured":"Gadjiev AD, Orhan C (1980) Some approximation theorems via statistical convergence. Rocky Mt J Math 32(1):129\u2013138","journal-title":"Rocky Mt J Math"},{"key":"6649_CR25","doi-asserted-by":"publisher","first-page":"2986","DOI":"10.1016\/j.mcm.2011.07.026","volume":"54","author":"B Hazarika","year":"2011","unstructured":"Hazarika B, Savas E (2011) Some $$I$$-convergent $$\\lambda -$$summable difference sequences spaces of fuzzy real numbers defined by a sequence of Orlicz functions. Math Comput Model 54:2986\u20132998","journal-title":"Math Comput Model"},{"issue":"2","key":"6649_CR26","doi-asserted-by":"publisher","first-page":"169","DOI":"10.4153\/CMB-1981-027-5","volume":"24","author":"H K\u0131zmaz","year":"1981","unstructured":"K\u0131zmaz H (1981) On Certain Sequence spaces. Can Math Bull 24(2):169\u2013176","journal-title":"Can Math Bull"},{"key":"6649_CR27","unstructured":"Korovkin PP (1960) Linear operators and approximation theory. Delhi 1960 Hindustan Publishing Corporation"},{"key":"6649_CR28","doi-asserted-by":"publisher","first-page":"357","DOI":"10.5666\/KMJ.2016.56.2.357","volume":"56","author":"M K\u00fc\u00e7\u00fckaslan","year":"2016","unstructured":"K\u00fc\u00e7\u00fckaslan M, Yilmazt\u00fcrk M (2016) On deferred statistical convergence of sequences. Kyungpook Math J 56:357\u2013366","journal-title":"Kyungpook Math J"},{"key":"6649_CR29","first-page":"28","volume":"28","author":"M Matloka","year":"1986","unstructured":"Matloka M (1986) Sequence of fuzzy number. Busefal 28:28\u201337","journal-title":"Busefal"},{"issue":"5","key":"6649_CR30","doi-asserted-by":"publisher","first-page":"492","DOI":"10.1080\/03081079.2019.1608985","volume":"48","author":"SA Mohiuddine","year":"2019","unstructured":"Mohiuddine SA, Asiri A, Hazarika B (2019) Weighted statistical convergence through difference operator of sequences of fuzzy numbers with application to fuzzy approximation theorems. Int J Gen Syst 48(5):492\u2013506","journal-title":"Int J Gen Syst"},{"key":"6649_CR31","doi-asserted-by":"publisher","unstructured":"Mohiuddine SA, Alamri BAS (2018) Generalization of equi-statistical convergence via weighted lacunary sequence with associated Korovkin and Voronovskaya type approximation theorems, Revista de la Real Academia de Ciencias Exactas. F\u00edsicasy Naturales Serie A Matem\u00e1ticas. https:\/\/doi.org\/10.1007\/s13398-018-0591-z","DOI":"10.1007\/s13398-018-0591-z"},{"issue":"5","key":"6649_CR32","doi-asserted-by":"publisher","first-page":"2414","DOI":"10.1016\/j.chaos.2008.09.018","volume":"41","author":"M Mursaleen","year":"2009","unstructured":"Mursaleen M, Mohiuddine SA (2009) Statistical convergence of double sequences in intuitionistic fuzzy normed spaces. Chaos Solitons Fractals 41(5):2414\u20132421","journal-title":"Chaos Solitons Fractals"},{"key":"6649_CR33","first-page":"9132","volume":"218","author":"M Mursaleen","year":"2012","unstructured":"Mursaleen M, Karakaya V, Erturk M, Gursoy F (2012) Weighted statistical convergence and its application to Korovkin type approximation theorem. Appl Math Comput 218:9132\u20139137","journal-title":"Appl Math Comput"},{"key":"6649_CR34","doi-asserted-by":"publisher","first-page":"603","DOI":"10.1016\/j.mcm.2010.04.006","volume":"52","author":"M Mursaleen","year":"2010","unstructured":"Mursaleen M, Noman AK (2010) On some new difference sequence spaces of non-absolute type. Math Comput Model 52:603\u2013617","journal-title":"Math Comput Model"},{"issue":"1","key":"6649_CR35","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1016\/j.laa.2011.06.014","volume":"436","author":"M Mursaleen","year":"2012","unstructured":"Mursaleen M, Noman AK (2012) Compactness of matrix operators on some new difference sequence spaces. Linear Algebra Appl 436(1):41\u201352","journal-title":"Linear Algebra Appl"},{"key":"6649_CR36","doi-asserted-by":"publisher","first-page":"814","DOI":"10.1016\/j.camwa.2011.06.011","volume":"62","author":"M Mursaleen","year":"2011","unstructured":"Mursaleen M, Karakaya V, Polat H, Simsek N (2011) Measure of noncompactness of matrix operators on some difference sequence spaces of weighted means. Comput Math Appl 62:814\u2013820","journal-title":"Comput Math Appl"},{"key":"6649_CR37","doi-asserted-by":"publisher","first-page":"563","DOI":"10.1016\/j.jmaa.2014.06.001","volume":"420","author":"L Nayak","year":"2014","unstructured":"Nayak L, Das G, Ray BK (2014) An estimate of the rate of convergence of Fourier series inthegeneralized H\u00f6lder metric by deferred ces\u00e0ro mean. Journal of Mathematical Analysis and Applications 420:563\u2013575","journal-title":"Journal of Mathematical Analysis and Applications"},{"key":"6649_CR38","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1016\/0165-0114(89)90222-4","volume":"33","author":"S Nanda","year":"1989","unstructured":"Nanda S (1989) On sequences of fuzzy number. Fuzzy Sets Syst 33:123\u2013126","journal-title":"Fuzzy Sets Syst"},{"issue":"3","key":"6649_CR39","first-page":"269","volume":"45","author":"F Nuray","year":"1995","unstructured":"Nuray F, Savas E (1995) Statistical convergence of sequences of fuzzy real numbers. Mathematica Slovaca 45(3):269\u2013273","journal-title":"Mathematica Slovaca"},{"issue":"3","key":"6649_CR40","first-page":"269","volume":"45","author":"T Salat","year":"1980","unstructured":"Salat T (1980) On statistical convergent sequence of fuzzy real numbers. Mathematica Slovaca 45(3):269\u2013273","journal-title":"Mathematica Slovaca"},{"key":"6649_CR41","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1016\/j.ins.2003.09.005","volume":"162","author":"E Savas","year":"2004","unstructured":"Savas E, Mursaleen M (2004) On statistically convergent double sequence of fuzzy numbers. Inf Sci 162:183\u2013192","journal-title":"Inf Sci"},{"key":"6649_CR42","doi-asserted-by":"publisher","first-page":"361","DOI":"10.1080\/00029890.1959.11989303","volume":"66","author":"IJ Schoenberg","year":"1959","unstructured":"Schoenberg IJ (1959) The integrability of certain function and related summability methods. Am Math Mon 66:361\u2013375","journal-title":"Am Math Mon"},{"key":"6649_CR43","first-page":"883","volume":"8","author":"BK Tripathy","year":"2000","unstructured":"Tripathy BK, Nanda S (2000) Absolute value of fuzzy real number and fuzzy sequence spaces. J Fuzzy Math 8:883\u2013892","journal-title":"J Fuzzy Math"},{"key":"6649_CR44","doi-asserted-by":"publisher","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","volume":"8","author":"LA Zadeh","year":"1965","unstructured":"Zadeh LA (1965) Fuzzy set. Inf Control 8:338\u2013353","journal-title":"Inf Control"}],"container-title":["Soft Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00500-021-06649-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00500-021-06649-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00500-021-06649-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,3,1]],"date-time":"2022-03-01T12:14:26Z","timestamp":1646136866000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00500-021-06649-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,30]]},"references-count":44,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2022,3]]}},"alternative-id":["6649"],"URL":"https:\/\/doi.org\/10.1007\/s00500-021-06649-6","relation":{},"ISSN":["1432-7643","1433-7479"],"issn-type":[{"value":"1432-7643","type":"print"},{"value":"1433-7479","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,1,30]]},"assertion":[{"value":"2 December 2021","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 January 2022","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The first author declares that he has no conflict of interest. The second author also declares that she has no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"This article does not contain any studies with human participants or animals performed by any of the authors.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval"}}]}}