{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T01:15:19Z","timestamp":1767143719424,"version":"build-2238731810"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T00:00:00Z","timestamp":1743638400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"},{"start":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T00:00:00Z","timestamp":1743638400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Complex Intell. Syst."],"published-print":{"date-parts":[[2025,5]]},"DOI":"10.1007\/s40747-025-01849-1","type":"journal-article","created":{"date-parts":[[2025,4,4]],"date-time":"2025-04-04T15:38:25Z","timestamp":1743781105000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Fractals in $$S_b$$-metric spaces"],"prefix":"10.1007","volume":"11","author":[{"given":"Fahim","family":"Ud\u00a0Din","sequence":"first","affiliation":[]},{"given":"Sheeza","family":"Nawaz","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6747-425X","authenticated-orcid":false,"given":"Adil","family":"Jhangeer","sequence":"additional","affiliation":[]},{"given":"Fairouz","family":"Tchier","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,4,3]]},"reference":[{"issue":"5","key":"1849_CR1","doi-asserted-by":"publisher","first-page":"713","DOI":"10.1512\/iumj.1981.30.30055","volume":"30","author":"JE Hutchinson","year":"1981","unstructured":"Hutchinson JE (1981) Fractals and self similarity. Indiana Univ Math J 30(5):713\u2013747","journal-title":"Indiana Univ Math J"},{"key":"1849_CR2","volume-title":"Fractals everywhere","author":"MF Barnsley","year":"1993","unstructured":"Barnsley MF, Rising H (1993) Fractals everywhere. Morgan Kaufmann, Los Altos"},{"key":"1849_CR3","doi-asserted-by":"publisher","first-page":"299","DOI":"10.1007\/s13373-013-0041-3","volume":"3","author":"M Barnsley","year":"2013","unstructured":"Barnsley M, Vince A (2013) Developments in fractal geometry. Bull Math Sci 3:299\u2013348","journal-title":"Bull Math Sci"},{"issue":"1","key":"1849_CR4","doi-asserted-by":"publisher","first-page":"133","DOI":"10.4064\/fm-3-1-133-181","volume":"3","author":"S Banach","year":"1922","unstructured":"Banach S (1922) Sur les op\u00e9rations dans les ensembles abstraits et leur application aux \u00e9quations int\u00e9grales. Fundam Math 3(1):133\u2013181","journal-title":"Fundam Math"},{"issue":"1","key":"1849_CR5","doi-asserted-by":"publisher","first-page":"86","DOI":"10.3390\/sym11010086","volume":"11","author":"E Ameer","year":"2019","unstructured":"Ameer E, Aydi H, Arshad M, Alsamir H, Noorani MS (2019) Hybrid multivalued type contraction mappings in K-complete S b-metric spaces and applications. Symmetry 11(1):86","journal-title":"Symmetry"},{"issue":"1","key":"1849_CR6","doi-asserted-by":"publisher","first-page":"111","DOI":"10.18514\/MMN.2018.1652","volume":"19","author":"H Aydi","year":"2018","unstructured":"Aydi H, Felhi A, Karapinar E, Sahmim S (2018) A Nadler-type fixed point theorem in dislocated spaces and applications. Miskolc Math Notes 19(1):111\u2013124","journal-title":"Miskolc Math Notes"},{"key":"1849_CR7","first-page":"2364","volume-title":"Metric fixed point theory: applications in science, engineering and behavioural sciences","year":"2021","unstructured":"Debnath P, Konwar N, Radenovi\u0107 S (eds) (2021) Metric fixed point theory: applications in science, engineering and behavioural sciences. Springer, Singapore, pp 2364\u20136756"},{"key":"1849_CR8","first-page":"1","volume":"1","author":"HA Hammad","year":"2020","unstructured":"Hammad HA, La Sen MD (2020) A technique of tripled coincidence points for solving a system of nonlinear integral equations in POCML spaces. J Inequal Appl 1:1\u201327","journal-title":"J Inequal Appl"},{"issue":"16","key":"1849_CR9","doi-asserted-by":"publisher","first-page":"2012","DOI":"10.3390\/math9162012","volume":"9","author":"HA Hammad","year":"2021","unstructured":"Hammad HA, Agarwal P, Guirao JL (2021) Applications to boundary value problems and homotopy theory via tripled fixed point techniques in Sly metric spaces. Mathematics 9(16):2012","journal-title":"Mathematics"},{"key":"1849_CR10","first-page":"1","volume":"2021","author":"K Javed","year":"2021","unstructured":"Javed K, Aydi H, Uddin F, Arshad M (2021) On orthogonal S b-metric spaces with an application. J Math 2021:1\u20137","journal-title":"J Math"},{"issue":"11","key":"1849_CR11","doi-asserted-by":"publisher","first-page":"256","DOI":"10.3390\/math6110256","volume":"6","author":"E Karapinar","year":"2018","unstructured":"Karapinar E, Agarwal R, Aydi H (2018) Interpolative Reich\u2013Rus\u2013\u0106iri\u0107 type contractions on S metric spaces. Mathematics 6(11):256","journal-title":"Mathematics"},{"key":"1849_CR12","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1016\/j.cam.2018.02.008","volume":"340","author":"G Lin","year":"2018","unstructured":"Lin G, Cheng X, Zhang Y (2018) A parametric level set based collage method for an inverse problem in elliptic S differential equations. J Comput Appl Math 340:101\u2013121","journal-title":"J Comput Appl Math"},{"key":"1849_CR13","doi-asserted-by":"crossref","unstructured":"Zhang Y, Hofmann B (2020) Two new non-negativity preserving iterative regularization methods for ill-posed inverse problems. arXiv preprint arXiv:2002.04474","DOI":"10.3934\/ipi.2020062"},{"key":"1849_CR14","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-22591-9","volume-title":"Harmonic quasiconformal mappings and hyperbolic type metrics","author":"V Todor\u010devi\u0107","year":"2019","unstructured":"Todor\u010devi\u0107 V (2019) Harmonic quasiconformal mappings and hyperbolic type metrics. Springer, Berlin"},{"issue":"1","key":"1849_CR15","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1007\/s00009-011-0116-x","volume":"9","author":"NA Secelean","year":"2012","unstructured":"Secelean NA (2012) The existence of the attractor of countable iterated function systems. Mediterr J Math 9(1):61\u201379","journal-title":"Mediterr J Math"},{"key":"1849_CR16","doi-asserted-by":"crossref","unstructured":"Nadler Jr SB (1969) Multi-valued contraction mappings","DOI":"10.1007\/BFb0069723"},{"key":"1849_CR17","first-page":"26","volume":"30","author":"IA Bakhtin","year":"1989","unstructured":"Bakhtin IA (1989) The contraction mapping in almost metric spaces. Funct Funct Ana Gos Ped Inst Unianowsk 30:26\u201337","journal-title":"Funct Funct Ana Gos Ped Inst Unianowsk"},{"issue":"1","key":"1849_CR18","first-page":"5","volume":"1","author":"S Czerwik","year":"1993","unstructured":"Czerwik S (1993) Contraction mappings in $$ b $$-metric spaces. Acta Math Inf Univ Ostrav 1(1):5\u201311","journal-title":"Acta Math Inf Univ Ostrav"},{"key":"1849_CR19","first-page":"329","volume":"84","author":"BC Dhage","year":"1992","unstructured":"Dhage BC (1992) Generalized metric spaces mappings with fixed point. Bull Calcutta Mat Soc 84:329\u2013326","journal-title":"Bull Calcutta Mat Soc"},{"issue":"2","key":"1849_CR20","first-page":"289","volume":"7","author":"Z Mustafa","year":"2006","unstructured":"Mustafa Z, Sims B (2006) A new approach to generalized metric spaces. J Nonlinear Convex Anal 7(2):289","journal-title":"J Nonlinear Convex Anal"},{"key":"1849_CR21","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1155\/2007\/27906","volume":"2007","author":"S Sedghi","year":"2007","unstructured":"Sedghi S, Shobe N, Zhou H (2007) A common fixed point theorem in-metric spaces. Fixed point theory and Applications 2007:1\u201313","journal-title":"Fixed point theory and Applications"},{"key":"1849_CR22","unstructured":"Sedghi S, Shobe N, Aliouche A (2012) A generalization of fixed point theorems in S-metric spaces. Mate Vesn 64(249):258\u2013266"},{"issue":"315","key":"1849_CR23","doi-asserted-by":"publisher","first-page":"131","DOI":"10.22436\/jmcs.016.02.01","volume":"16","author":"N Souayah","year":"2016","unstructured":"Souayah N, Mlaiki N (2016) A fixed point theorem in Sb-metric spaces. J Math Comput Sci 16(315):131\u2013139","journal-title":"J Math Comput Sci"},{"key":"1849_CR24","unstructured":"Nazir T, Silvestrov S, Qi X (2016) Fractals of generalized-Hutchinson operator in-metric spaces. J Oper 2016:5250394"},{"key":"1849_CR25","doi-asserted-by":"publisher","first-page":"2287","DOI":"10.1049\/cth2.12569","volume":"18","author":"Q Shi","year":"2023","unstructured":"Shi Q, He S, Wang H, Stojanovic V, Shi K, Lv W (2023) Extended state observer based fractional order sliding mode control for steer-by-wire systems. IET Control Theory Appl 18:2287\u20132295","journal-title":"IET Control Theory Appl"},{"issue":"13","key":"1849_CR26","doi-asserted-by":"publisher","first-page":"12181","DOI":"10.1007\/s11071-023-08456-0","volume":"111","author":"X Song","year":"2023","unstructured":"Song X, Sun P, Song S, Stojanovic V (2023) Finite-time adaptive neural resilient DSC for fractional-order nonlinear large-scale systems against sensor-actuator faults. Nonlinear Dyn 111(13):12181\u201312196","journal-title":"Nonlinear Dyn"},{"issue":"1","key":"1849_CR27","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1111\/j.1749-6632.1994.tb44144.x","volume":"728","author":"SG Matthews","year":"1994","unstructured":"Matthews SG (1994) Partial metric topology. Ann N Y Acad Sci 728(1):183\u2013197","journal-title":"Ann N Y Acad Sci"},{"issue":"1","key":"1849_CR28","first-page":"24","volume":"9","author":"TA\u015e Nihal","year":"2016","unstructured":"Nihal TA\u015e, \u00d6zg\u00fcr N (2016) New generalized fixed point results on $$ S_ b $$-metric spaces. Konuralp J Math 9(1):24\u201332","journal-title":"Konuralp J Math"},{"key":"1849_CR29","doi-asserted-by":"crossref","unstructured":"Sarkar J, Pandey M, Som T, Choudhury BS (2023) Generalized Hausdorff metric on $$ S_ {b} $$-metric space and some fixed point results. arXiv preprint arXiv:2303.02619","DOI":"10.1007\/s41478-024-00842-0"},{"issue":"1","key":"1849_CR30","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1186\/1687-1812-2012-94","volume":"2012","author":"D Wardowski","year":"2012","unstructured":"Wardowski D (2012) Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl 2012(1):1\u20136","journal-title":"Fixed Point Theory Appl"},{"key":"1849_CR31","first-page":"1","volume":"1","author":"D Klim","year":"2015","unstructured":"Klim D, Wardowski D (2015) Fixed points of dynamic processes of set-valued F-contractions and application to functional equations. Fixed Point Theory Appl 1:1\u20139","journal-title":"Fixed Point Theory Appl"},{"key":"1849_CR32","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1186\/s13663-015-0317-2","volume":"2015","author":"M Cosentino","year":"2015","unstructured":"Cosentino M, Jleli M, Samet B, Vetro C (2015) Solvability of integrodifferential problems via fixed point theory in b-metric spaces. Fixed Point Theory Appl 2015:1\u201315","journal-title":"Fixed Point Theory Appl"},{"key":"1849_CR33","doi-asserted-by":"crossref","unstructured":"Goyal K, Prasad B (2021) Generalized iterated function systems in multi-valued mapping. In: AIP conference proceedings, vol 2316(1). AIP Publishing LLC, p 040001","DOI":"10.1063\/5.0036921"},{"key":"1849_CR34","doi-asserted-by":"crossref","unstructured":"Jha S, Roychowdhury MK, Verma S (Eds) (2024) Recent developments in fractal geometry and dynamical systems","DOI":"10.1090\/conm\/797"},{"issue":"21","key":"1849_CR35","doi-asserted-by":"publisher","first-page":"3947","DOI":"10.1140\/epjs\/s11734-021-00336-1","volume":"230","author":"Q Rong","year":"2021","unstructured":"Rong Q, Thangaraj C, Easwaramoorthy D, He S (2021) Multifractal based image processing for estimating the complexity of COVID-19 dynamics. Eur Phy J Spec Top 230(21):3947\u20133954","journal-title":"Eur Phy J Spec Top"},{"issue":"4","key":"1849_CR36","first-page":"1402","volume":"13","author":"B Rama","year":"2018","unstructured":"Rama B, Mishra J (2018) Developing fractals using iterated function systems. ARPN J Eng Appl Sci 13(4):1402\u20131409","journal-title":"ARPN J Eng Appl Sci"}],"updated-by":[{"DOI":"10.1007\/s40747-025-01972-z","type":"correction","label":"Correction","source":"publisher","updated":{"date-parts":[[2025,6,4]],"date-time":"2025-06-04T00:00:00Z","timestamp":1748995200000}}],"container-title":["Complex &amp; Intelligent Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40747-025-01849-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40747-025-01849-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40747-025-01849-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,7]],"date-time":"2025-06-07T02:07:53Z","timestamp":1749262073000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40747-025-01849-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,3]]},"references-count":36,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2025,5]]}},"alternative-id":["1849"],"URL":"https:\/\/doi.org\/10.1007\/s40747-025-01849-1","relation":{},"ISSN":["2199-4536","2198-6053"],"issn-type":[{"value":"2199-4536","type":"print"},{"value":"2198-6053","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,4,3]]},"assertion":[{"value":"27 March 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 February 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 April 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 June 2025","order":4,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Correction","order":5,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"A Correction to this paper has been published:","order":6,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"https:\/\/doi.org\/10.1007\/s40747-025-01972-z","URL":"https:\/\/doi.org\/10.1007\/s40747-025-01972-z","order":7,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"233"}}