{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,29]],"date-time":"2026-01-29T14:14:27Z","timestamp":1769696067139,"version":"3.49.0"},"reference-count":19,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2023,8,1]],"date-time":"2023-08-01T00:00:00Z","timestamp":1690848000000},"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>The focus of this paper is to conduct a comprehensive analysis of the advancements made in the understanding of Interpolative contraction, building upon the ideas initially introduced by Karapinar in 2018. In this paper, we develop the notion of Interpolative contraction mappings to the case of non-linear Kannan Interpolative, Riech Rus \u0106iri\u0107 interpolative and Hardy\u2013Roger Interpolative contraction mappings based on controlled function, and prove some fixed point results in the context of controlled metric space, thereby enhancing the current understanding of this particular analysis. Furthermore, we provide a concrete example that illustrates the underlying drive for the investigations presented in this context. An application of the proposed non-linear Interpolative-contractions to the Liouville\u2013Caputo fractional derivatives and fractional differential equations is provided in this paper.<\/jats:p>","DOI":"10.3390\/sym15081515","type":"journal-article","created":{"date-parts":[[2023,8,1]],"date-time":"2023-08-01T09:06:44Z","timestamp":1690880804000},"page":"1515","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Recent Advancements in KRH-Interpolative-Type Contractions"],"prefix":"10.3390","volume":"15","author":[{"given":"Ansar","family":"Abbas","sequence":"first","affiliation":[{"name":"Department of Mathematics & Statistics, International Islamic University, Islamabad 44000, Pakistan"}]},{"given":"Amjad","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Mathematics & Statistics, International Islamic University, Islamabad 44000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5188-2830","authenticated-orcid":false,"given":"Hamed","family":"Al 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":[[2023,8,1]]},"reference":[{"key":"ref_1","unstructured":"Czerwik, S. (1993). Acta mathematica et Informatica Universitatis Ostraviensis, University of Ostrava."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Chifu, L.C., and Karapinar, E. (2019). Admissible hybrid Z-contractions in b-metric spaces. Axioms, 9.","DOI":"10.3390\/axioms9010002"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"321","DOI":"10.24193\/fpt-ro.2018.1.25","article-title":"Kaj\u00e1nt\u00f3, A. Fixed point theorems for various types of F-contractions in complete b-metric spaces","volume":"19","year":"2018","journal-title":"Fixed Point Theory"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2153","DOI":"10.1007\/s11784-016-0400-2","article-title":"New fixed point theorems for set-valued contractions in b-metric spaces","volume":"19","author":"Miculescu","year":"2017","journal-title":"J. Fixed Point Theory"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Kamran, T., Samreen, M., and UL Ain, Q. (2017). A generalization of b-metric space and some fixed point theorems. Mathematics, 5.","DOI":"10.3390\/math5020019"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1186\/s13660-018-1745-4","article-title":"Common fixed point results on an extended b-metric space","volume":"1","author":"Alqahtani","year":"2018","journal-title":"J. Inequal. Appl."},{"key":"ref_7","first-page":"473","article-title":"The new results in extended $ b $-metric spaces and applications","volume":"11","author":"Mitrovic","year":"2020","journal-title":"Int. J. Nonlinear Anal. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Aydi, H., Karapinar, E., and Rold\u00e1n L\u00f3pez de Hierro, A.F. (2019). \u03c9-Interpolative \u0106iri\u0107\u2013Reich\u2013Rus-type contractions. Mathematics, 7.","DOI":"10.3390\/math7010057"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"386","DOI":"10.1515\/math-2020-0139","article-title":"On multivalued \u03b8-contraction maps and related applications","volume":"18","author":"Ali","year":"2020","journal-title":"Open Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"122669","DOI":"10.1016\/j.physa.2019.122669","article-title":"Hybrid fixed point results via generalized dynamic process for F-HRS type contractions with application","volume":"538","author":"Ali","year":"2020","journal-title":"Phys. A"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Gaba, Y.U., and Karapinar, E. (2019). A new approach to the interpolative contractions. Axioms, 8.","DOI":"10.3390\/axioms8040110"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Mlaiki, N., Aydi, H., Souayah, N., and Abdeljawad, T. (2018). Controlled metric type spaces and the related contraction principle. Mathematics, 6.","DOI":"10.3390\/math6100194"},{"key":"ref_13","first-page":"85","article-title":"Revisiting the Kannan type contractions via interpolation","volume":"2","author":"Karpinar","year":"2018","journal-title":"Adv. Theory Nonlinear Anal. Its Appl."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Karapinar, E., Agarwal, R., and Aydi, H. (2018). Interpolative Reich\u2013Rus\u2013Ciric Type Contractions on Partial Metric Spaces. Mathematics, 6.","DOI":"10.3390\/math6110256"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Karapinar, E., Alqahtani, O., and Aydi, H. (2019). On interpolative Hardy\u2013Rogers-type contractions. Symmetry, 11.","DOI":"10.3390\/sym11010008"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"290","DOI":"10.1186\/s13660-019-2227-z","article-title":"On extended interpolative \u0106iri\u0107-Reich-Rus type F-contractions and an application","volume":"2019","author":"Mohammadi","year":"2019","journal-title":"J. Inequal. Appl."},{"key":"ref_17","first-page":"2108167","article-title":"On fixed point results in controlled metric spaces","volume":"2020","author":"Ahmad","year":"2020","journal-title":"J. Funct. Spaces"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1007\/s11784-015-0275-7","article-title":"New fixed point results for mappings of contractive type with an application to nonlinear fractional differential equations","volume":"18","author":"Lakzian","year":"2016","journal-title":"J. Fixed Point Theory A"},{"key":"ref_19","first-page":"17","article-title":"The existence of solution for k-dimensional system of Langevin Hadamard-type fractional differential inclusions with 2k different fractional orders","volume":"50","author":"Samei","year":"2020","journal-title":"Mediterr. J. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/8\/1515\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:23:37Z","timestamp":1760127817000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/8\/1515"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,1]]},"references-count":19,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2023,8]]}},"alternative-id":["sym15081515"],"URL":"https:\/\/doi.org\/10.3390\/sym15081515","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,8,1]]}}}