{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T18:18:42Z","timestamp":1769192322788,"version":"3.49.0"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,1,5]],"date-time":"2025-01-05T00:00:00Z","timestamp":1736035200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,1,5]],"date-time":"2025-01-05T00:00:00Z","timestamp":1736035200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/100014718","name":"Innovative Research Group Project of the National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11901007"],"award-info":[{"award-number":["11901007"]}],"id":[{"id":"10.13039\/100014718","id-type":"DOI","asserted-by":"publisher"}]},{"name":"National research support project","award":["1100513623XN-283-23"],"award-info":[{"award-number":["1100513623XN-283-23"]}]},{"name":"North China University of Technology organized scientific research project","award":["2023YZZKY19"],"award-info":[{"award-number":["2023YZZKY19"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2025,4]]},"DOI":"10.1007\/s40314-024-03064-z","type":"journal-article","created":{"date-parts":[[2025,1,5]],"date-time":"2025-01-05T07:35:40Z","timestamp":1736062540000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A novel definition of fuzzy derivatives based on non-increasing and left-continuous fuzzy real numbers"],"prefix":"10.1007","volume":"44","author":[{"given":"Yu","family":"Zhong","sequence":"first","affiliation":[]},{"given":"Xin","family":"Wu","sequence":"additional","affiliation":[]},{"given":"Meng Zhen","family":"Wang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,1,5]]},"reference":[{"key":"3064_CR1","doi-asserted-by":"publisher","first-page":"421","DOI":"10.1016\/S0377-0427(00)00342-3","volume":"121","author":"G Alefeld","year":"2000","unstructured":"Alefeld G, Mayer G (2000) Interval analysis: theory and applications. J Comput Appl Math 121:421\u2013464","journal-title":"J Comput Appl Math"},{"key":"3064_CR2","first-page":"27","volume":"3","author":"A Armand","year":"2018","unstructured":"Armand A, Allahviranloo T, Gouyandeh Z (2018) Some fundamental results on fuzzy calculus. Iran J Fuzzy Syst 3:27\u201346","journal-title":"Iran J Fuzzy Syst"},{"key":"3064_CR3","doi-asserted-by":"publisher","first-page":"581","DOI":"10.1016\/j.fss.2004.08.001","volume":"151","author":"B Bede","year":"2005","unstructured":"Bede B, Gal SG (2005) Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Sets Syst 151:581\u2013599","journal-title":"Fuzzy Sets Syst"},{"key":"3064_CR4","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1016\/j.fss.2012.10.003","volume":"230","author":"B Bede","year":"2013","unstructured":"Bede B, Stefanini L (2013) Generalized differentiability of fuzzy-valued functions. Fuzzy Sets Syst 230:119\u2013141","journal-title":"Fuzzy Sets Syst"},{"key":"3064_CR5","doi-asserted-by":"publisher","first-page":"433","DOI":"10.1016\/S0165-0114(98)00338-8","volume":"115","author":"M Burgin","year":"2000","unstructured":"Burgin M (2000) Theory of fuzzy limits. Fuzzy Sets Syst 115:433\u2013443","journal-title":"Fuzzy Sets Syst"},{"key":"3064_CR6","first-page":"9","volume":"1","author":"D Chakraborty","year":"2010","unstructured":"Chakraborty D, Guha D (2010) Addition of two generalized fuzzy numbers. Int J Ind Math 1:9\u201320","journal-title":"Int J Ind Math"},{"key":"3064_CR7","unstructured":"Chalco-Cano Y, Rom\u00e1n-Flores H, Jim\u00e9nez-Gamero MD (2009) Fuzzy differential equation with $$\\pi $$-derivative. In: Proceedings IFSA-EUSFLAT, Lisbon, Portugal, pp 703\u2013706"},{"key":"3064_CR8","doi-asserted-by":"publisher","first-page":"2177","DOI":"10.1016\/j.ins.2011.01.023","volume":"181","author":"Y Chalco-Cano","year":"2011","unstructured":"Chalco-Cano Y, Rom\u00e1n-Flores H, Jim\u00e9nez-Gamero MD (2011) Generalized derivative and $$\\pi $$-derivative for set-valued functions. Inf Sci 181:2177\u20132188","journal-title":"Inf Sci"},{"key":"3064_CR9","doi-asserted-by":"publisher","first-page":"30","DOI":"10.1109\/TSMC.1972.5408553","volume":"1","author":"SL Chang","year":"1972","unstructured":"Chang SL, Zadeh LA (1972) On fuzzy mapping and control. IEEE Trans Syst Man Cybern 1:30\u201334","journal-title":"IEEE Trans Syst Man Cybern"},{"key":"3064_CR10","doi-asserted-by":"publisher","first-page":"583","DOI":"10.1109\/91.873581","volume":"8","author":"P Diamond","year":"2000","unstructured":"Diamond P (2000) Stability and periodicity in fuzzy differential equations. IEEE Trans Fuzzy Syst 8:583\u2013590","journal-title":"IEEE Trans Fuzzy Syst"},{"key":"3064_CR11","doi-asserted-by":"publisher","DOI":"10.1142\/2326","volume-title":"Metric spaces of fuzzy sets: theory and application","author":"P Diamond","year":"1994","unstructured":"Diamond P, Kloeden P (1994) Metric spaces of fuzzy sets: theory and application. World Scientific, Singapore"},{"key":"3064_CR12","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1016\/S0165-0114(82)80001-8","volume":"8","author":"D Dubois","year":"1982","unstructured":"Dubois D, Prade H (1982) Towards fuzzy differential calculus. Part 3: differentiation. Fuzzy Sets Syst 8:225\u2013233","journal-title":"Fuzzy Sets Syst"},{"key":"3064_CR13","first-page":"57","volume":"8","author":"M Friedman","year":"1996","unstructured":"Friedman M, Ming M, Kandel A (1996) Fuzzy derivatives and fuzzy Chauchy problems using $$LP$$ metric. Fuzzy Logic Found Ind Appl 8:57\u201372","journal-title":"Fuzzy Logic Found Ind Appl"},{"key":"3064_CR14","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1016\/S0165-0114(03)00180-5","volume":"143","author":"R Full\u00e9r","year":"2004","unstructured":"Full\u00e9r R, Majlender P (2004) On interactive fuzzy numbers. Fuzzy Sets Syst 143:355\u2013369","journal-title":"Fuzzy Sets Syst"},{"key":"3064_CR15","volume-title":"Fuzzy sets and fuzzy logic theory and application","author":"JK George","year":"1995","unstructured":"George JK, Gazdik BY (1995) Fuzzy sets and fuzzy logic theory and application. Prentice Hall PTR, Englewood Cliffs"},{"key":"3064_CR16","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1016\/0165-0114(86)90026-6","volume":"18","author":"R Goetschel","year":"1986","unstructured":"Goetschel R, Voxman W (1986) Elementary fuzzy calculus. Fuzzy Sets Syst 18:31\u201343","journal-title":"Fuzzy Sets Syst"},{"key":"3064_CR17","doi-asserted-by":"publisher","first-page":"142","DOI":"10.1016\/j.fss.2015.02.015","volume":"280","author":"LT Gomes","year":"2015","unstructured":"Gomes LT, Barros LC (2015) A note on the generalized difference and the generalized differentiability. Fuzzy Sets Syst 280:142\u2013145","journal-title":"Fuzzy Sets Syst"},{"key":"3064_CR18","doi-asserted-by":"publisher","first-page":"1141","DOI":"10.1016\/j.ins.2007.10.001","volume":"178","author":"HL Huang","year":"2008","unstructured":"Huang HL, Shi FG (2008) $$L$$-fuzzy numbers and their properties. Inf Sci 178:1141\u20131151","journal-title":"Inf Sci"},{"key":"3064_CR19","first-page":"205","volume":"10","author":"PM Hukuhara","year":"1967","unstructured":"Hukuhara PM (1967) Int\u00e9gration des applications measurables dont la valeur est un compact convexe (in French). Funkcialaj Ekvacioj 10:205\u2013223","journal-title":"Funkcialaj Ekvacioj"},{"key":"3064_CR20","doi-asserted-by":"publisher","first-page":"279","DOI":"10.1016\/j.aml.2011.08.027","volume":"25","author":"H Lian","year":"2012","unstructured":"Lian H, Xin X (2012) The nullities for $$M$$-fuzzifying matroids. Appl Math Lett 25:279\u2013286","journal-title":"Appl Math Lett"},{"key":"3064_CR21","doi-asserted-by":"publisher","first-page":"310","DOI":"10.1109\/TFUZZ.2017.2659731","volume":"26","author":"M Mazandarani","year":"2018","unstructured":"Mazandarani M, Pariz N, Kamyad AV (2018) Granular differentiability of fuzzy-number-valued function. IEEE Trans Fuzzy Syst 26:310\u2013323","journal-title":"IEEE Trans Fuzzy Syst"},{"key":"3064_CR22","volume-title":"Interval analysis","author":"R Moore","year":"1966","unstructured":"Moore R (1966) Interval analysis. Prentice-Hall, Englewood Cliffs"},{"key":"3064_CR23","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898717716","volume-title":"Introduction to interval analysis","author":"R Moore","year":"2009","unstructured":"Moore R, Kearfott RB, Cloud MJ (2009) Introduction to interval analysis. SIAM, Philadelphia"},{"key":"3064_CR24","first-page":"197","volume":"1","author":"AV Plotnikov","year":"2009","unstructured":"Plotnikov AV, Komleva TA, Arsiry AV (2009) Necessary and sufficient optimality conditions for a control fuzzy linear problem. Int J Ind Math 1:197\u2013207","journal-title":"Int J Ind Math"},{"key":"3064_CR25","doi-asserted-by":"publisher","first-page":"552","DOI":"10.1016\/0022-247X(83)90169-5","volume":"91","author":"ML Puri","year":"1983","unstructured":"Puri ML, Ralescu DA (1983) Differentials of fuzzy functions. J Math Anal Appl 91:552\u2013558","journal-title":"J Math Anal Appl"},{"key":"3064_CR26","doi-asserted-by":"publisher","first-page":"158","DOI":"10.1016\/j.fss.2020.07.012","volume":"419","author":"D Qiu","year":"2021","unstructured":"Qiu D (2021) The generalized Hukuhara differentiability of interval-valued function is not fully equivalent to the one-sided differentiability of its endpoint functions. Fuzzy Sets Syst 419:158\u2013168","journal-title":"Fuzzy Sets Syst"},{"key":"3064_CR27","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.fss.2021.05.006","volume":"434","author":"D Qiu","year":"2022","unstructured":"Qiu D, Ouyang C (2022) Optimality conditions for fuzzy optimization in several variables under generalized differentiability. Fuzzy Sets Syst 434:1\u201319","journal-title":"Fuzzy Sets Syst"},{"key":"3064_CR28","first-page":"258","volume":"5","author":"H Rouhparvar","year":"2011","unstructured":"Rouhparvar H, Panahi A, Zanjani AN (2011) Fuzzy distance measure for fuzzy numbers. Aust J Basic Appl Sci 5:258\u2013265","journal-title":"Aust J Basic Appl Sci"},{"key":"3064_CR29","first-page":"209","volume":"1","author":"R Saneifard","year":"2009","unstructured":"Saneifard R (2009) A method for defuzzification by weighted distance. Int J Ind Math 1:209\u2013217","journal-title":"Int J Ind Math"},{"key":"3064_CR30","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1016\/0165-0114(87)90030-3","volume":"24","author":"S Seikkala","year":"1987","unstructured":"Seikkala S (1987) On the fuzzy initial value problem. Fuzzy Sets Syst 24:319\u2013330","journal-title":"Fuzzy Sets Syst"},{"key":"3064_CR31","doi-asserted-by":"publisher","first-page":"6027","DOI":"10.3233\/JIFS-212908","volume":"42","author":"YH Shen","year":"2022","unstructured":"Shen YH (2022) A novel difference and derivative of linearly correlated fuzzy number-valued functions. J Intell Fuzzy Syst 42:6027\u20136043","journal-title":"J Intell Fuzzy Syst"},{"key":"3064_CR32","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1016\/j.fss.2021.02.017","volume":"429","author":"YH Shen","year":"2022","unstructured":"Shen YH (2022) Calculus for linearly correlated fuzzy number-valued functions. Fuzzy Sets Syst 429:101\u2013135","journal-title":"Fuzzy Sets Syst"},{"key":"3064_CR33","doi-asserted-by":"publisher","first-page":"696","DOI":"10.1016\/j.fss.2008.05.007","volume":"160","author":"FG Shi","year":"2009","unstructured":"Shi FG (2009) A new approach to the fuzzification of matroids. Fuzzy Sets Syst 160:696\u2013705","journal-title":"Fuzzy Sets Syst"},{"key":"3064_CR34","doi-asserted-by":"publisher","first-page":"1564","DOI":"10.1016\/j.fss.2009.06.009","volume":"161","author":"L Stefanini","year":"2010","unstructured":"Stefanini L (2010) A generalization of Hukuhara difference and division for interval and fuzzy arithmetic. Fuzzy Sets Syst 161:1564\u20131584","journal-title":"Fuzzy Sets Syst"},{"key":"3064_CR35","doi-asserted-by":"publisher","first-page":"1311","DOI":"10.1016\/j.na.2008.12.005","volume":"71","author":"L Stefanini","year":"2009","unstructured":"Stefanini L, Bede B (2009) Generalized Hukuhara differentiability of interval-valued functions and interval differential equations. Nonlinear Anal 71:1311\u20131328","journal-title":"Nonlinear Anal"},{"key":"3064_CR36","doi-asserted-by":"publisher","first-page":"353","DOI":"10.1016\/S0165-0114(97)00090-0","volume":"100","author":"W Voxman","year":"1998","unstructured":"Voxman W (1998) Some remarks on distances between fuzzy numbers. Fuzzy Sets Syst 100:353\u2013365","journal-title":"Fuzzy Sets Syst"},{"key":"3064_CR37","first-page":"31","volume":"39","author":"X Xin","year":"2010","unstructured":"Xin X, Shi FG (2010) Rank functions for closed and perfect $$[0, 1]$$-matroids. Hacettepe J Math Stat 39:31\u201339","journal-title":"Hacettepe J Math Stat"},{"key":"3064_CR38","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 sets. Inf Control 8:338\u2013353","journal-title":"Inf Control"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-024-03064-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-024-03064-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-024-03064-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,14]],"date-time":"2025-03-14T06:07:02Z","timestamp":1741932422000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-024-03064-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,5]]},"references-count":38,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,4]]}},"alternative-id":["3064"],"URL":"https:\/\/doi.org\/10.1007\/s40314-024-03064-z","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1,5]]},"assertion":[{"value":"31 October 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 November 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 December 2024","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 January 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"103"}}