{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:55:11Z","timestamp":1760144111315,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,3,18]],"date-time":"2024-03-18T00:00:00Z","timestamp":1710720000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper aims to provide a comprehensive analysis of the advancements made in understanding Iterative Fixed-Point Schemes, which builds upon the concept of digital contraction mappings. Additionally, we introduce the notion of an Iterative Fixed-Point Schemes in digital metric spaces. In this study, we extend the idea of Iteration process Mann, Ishikawa, Agarwal, and Thakur based on the \u03dd-Stable Iterative Scheme in digital metric space. We also design some fractal images, which frame the compression of Fixed-Point Iterative Schemes and contractive mappings. Furthermore, we present a concrete example that exemplifies the motivation behind our investigations. Moreover, we provide an application of the proposed Fractal image and Sierpinski triangle that compress the works by storing images as a collection of digital contractions, which addresses the issue of storing images with less storage memory in this paper.<\/jats:p>","DOI":"10.3390\/sym16030368","type":"journal-article","created":{"date-parts":[[2024,3,18]],"date-time":"2024-03-18T10:54:38Z","timestamp":1710759278000},"page":"368","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Recent Developments in Iterative Algorithms for Digital Metrics"],"prefix":"10.3390","volume":"16","author":[{"given":"Aasma","family":"Shaheen","sequence":"first","affiliation":[{"name":"Department of Mathematics, Fatima Jinnah Women University, Islamabad 46000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4729-4785","authenticated-orcid":false,"given":"Afshan","family":"Batool","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Fatima Jinnah Women University, Islamabad 46000, Pakistan"}]},{"given":"Amjad","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada"},{"name":"Department of Mathematics & Statistics, International Islamic University, Islamabad 44000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5188-2830","authenticated-orcid":false,"given":"Hamed Al","family":"Sulami","sequence":"additional","affiliation":[{"name":"Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7742-5993","authenticated-orcid":false,"given":"Aftab","family":"Hussain","sequence":"additional","affiliation":[{"name":"Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"621","DOI":"10.1080\/00029890.1979.11994873","article-title":"Digital topology","volume":"86","author":"Rosenfeld","year":"1979","journal-title":"Am. Math. Mon."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"155","DOI":"10.4995\/agt.2019.10667","article-title":"Remarks on Fixed Point Assertions in Digital Topology, 2","volume":"20","author":"Boxer","year":"2019","journal-title":"Appl. Gen. Topol."},{"key":"ref_3","first-page":"198","article-title":"Some Results on Simplicial Homology Groups of 2D Digital Images","volume":"1","author":"Ege","year":"2012","journal-title":"Int. J. Inform. Comput. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1186\/1687-1812-2013-253","article-title":"Lefschetz Fixed Point Theorem for Digital Images","volume":"2013","author":"Ege","year":"2013","journal-title":"Fixed Point Theory Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"823","DOI":"10.36045\/bbms\/1420071856","article-title":"Applications of The Lefschetz Number to Digital Images","volume":"21","author":"Ege","year":"2014","journal-title":"Bull. Belg. Math. Soc. Simon Stevin"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"237","DOI":"10.22436\/jnsa.008.03.08","article-title":"Banach Fixed Point Theorem for Digital Images","volume":"8","author":"Ege","year":"2015","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_7","first-page":"V48P522","article-title":"Fixed Point Theorems for Digital Contractive Type Mappings in Digital Metric Spaces","volume":"48","author":"Sridevi","year":"2017","journal-title":"Int. J. Math. Trends Technol. (IJMTT)"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1023\/A:1008370600456","article-title":"A classical construction for the digital fundamental group","volume":"10","author":"Boxer","year":"1999","journal-title":"J. Math. Imaging Vis."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"895","DOI":"10.22436\/jnsa.009.03.19","article-title":"Banach fixed point theorem from the viewpoint of digital topology","volume":"9","author":"Han","year":"2015","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_10","first-page":"693","article-title":"Hicks, T.L. Stability results for fixed point iteration procedures","volume":"33","author":"Harder","year":"1988","journal-title":"Math. J."},{"key":"ref_11","first-page":"691","article-title":"Fixed point theorems and stability results for fixed point iteration procedures II","volume":"24","author":"Rhoades","year":"1993","journal-title":"Indian J. Pure Appl. Math."},{"key":"ref_12","first-page":"1229","article-title":"Short proofs of stability results for fixed point iteration procedures for a class of contractive-type mappings","volume":"30","author":"Osilike","year":"1999","journal-title":"Indian J. Pure Appl. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"292","DOI":"10.1007\/BF01304884","article-title":"Fix point theorems in metric spaces","volume":"23","author":"Zamfirescu","year":"1972","journal-title":"Arch. Math."},{"key":"ref_14","first-page":"7","article-title":"On the Stability of Some Fixed Point Procedures","volume":"XVIII","author":"Berinde","year":"2002","journal-title":"Bul. \u0219tiin\u021bific Univ. Baia Mare Ser. Mat. Inform."},{"key":"ref_15","first-page":"119","article-title":"On the convergence of the Ishikawa iteration in the class of quasi-contractive operators","volume":"LXXIII","author":"Berinde","year":"2004","journal-title":"Acta Math. Univ. Comen."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Berinde, V. (2007). Iterative Approximation of Fixed Points, Springer.","DOI":"10.1109\/SYNASC.2007.49"},{"key":"ref_17","first-page":"157","article-title":"Some remarks on a fixed point theorem for Ciric-type almost contractions","volume":"25","author":"Berinde","year":"2009","journal-title":"Carpathian J. Math."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Debnath, P., Torres, D.F.M., and Cho, Y.J. (2023). Advanced Mathematical Analysis and Its Applications, Chapman and Hall\/CRC.","DOI":"10.1201\/9781003388678"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"506","DOI":"10.1090\/S0002-9939-1953-0054846-3","article-title":"Mean value methods in iteration","volume":"44","author":"Mann","year":"1953","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1090\/S0002-9939-1974-0336469-5","article-title":"Fixed point by a new iteration method","volume":"44","author":"Ishikawa","year":"1974","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Agarwal, R.P., O\u2019Regan, D., and Sahu, D.R. (2009). It Fixed Point Theory for Lipschitzian Type Mappings with Applications, Topological Fixed Point Theory and Its Applications, Springer.","DOI":"10.1155\/2009\/439176"},{"key":"ref_22","first-page":"147","article-title":"A new iterative scheme for numerical reckoning fixed points of Suzuki\u2019s generalized nonexpansive mappings","volume":"275","author":"Thakur","year":"2016","journal-title":"Appl. Math. Comput."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"873","DOI":"10.3934\/math.2023042","article-title":"A novel scheme of k-step iterations in digital metric spaces","volume":"8","author":"Botmart","year":"2023","journal-title":"AIMS Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/3\/368\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:15:32Z","timestamp":1760105732000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/3\/368"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,18]]},"references-count":23,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,3]]}},"alternative-id":["sym16030368"],"URL":"https:\/\/doi.org\/10.3390\/sym16030368","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,3,18]]}}}