{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:25:53Z","timestamp":1760149553777,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2023,8,16]],"date-time":"2023-08-16T00:00:00Z","timestamp":1692144000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this article, we present the use of a unique and common fixed point for a pair of mappings that satisfy certain rational-type inequalities in complex-valued b-metric spaces. We also provide applications related to authenticity concerns in integral equations. Our results combine well-known contractions, such as the \u0106iri\u0107 contraction and almost contractions.<\/jats:p>","DOI":"10.3390\/axioms12080794","type":"journal-article","created":{"date-parts":[[2023,8,16]],"date-time":"2023-08-16T10:09:33Z","timestamp":1692180573000},"page":"794","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Almost \u0106iri\u0107 Type Contractions and Their Applications in Complex Valued b-Metric Spaces"],"prefix":"10.3390","volume":"12","author":[{"given":"Muhammad","family":"Sarwar","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Malakand, Chakdara 18800, Pakistan"}]},{"given":"Syed Khayyam","family":"Shah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Malakand, Chakdara 18800, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9993-9082","authenticated-orcid":false,"given":"Zoran D.","family":"Mitrovi\u0107","sequence":"additional","affiliation":[{"name":"Faculty of Electrical Engineering, University of Banja Luka, Patre 5, 78000 Banja Luka, Bosnia and Herzegovina"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8798-3297","authenticated-orcid":false,"given":"Aiman","family":"Mukheimer","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7986-886X","authenticated-orcid":false,"given":"Nabil","family":"Mlaiki","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"133","DOI":"10.4064\/fm-3-1-133-181","article-title":"Sur les oprations dans les ensembles abstraits et leurs applications aux equatins integrales","volume":"3","author":"Banach","year":"1922","journal-title":"Fundam. Math."},{"key":"ref_2","first-page":"7167","article-title":"Boyd-Wong-type common fixed point results in cone metric spaces","volume":"217","author":"Kadelburg","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_3","first-page":"1385","article-title":"Common fixed point theorem for weakly compatible maps in complex valued metric spaces","volume":"1","author":"Bhatt","year":"2011","journal-title":"Int. J. Math. Sci. Appl."},{"key":"ref_4","first-page":"123","article-title":"Fixed points of generalized multi-valued contractive mappings in metric type spaces","volume":"6","author":"Latif","year":"2022","journal-title":"J. Nonlinear Var. Anal."},{"key":"ref_5","first-page":"45","article-title":"Some fixed point theorems with rational type contraction in controlled metric spaces","volume":"16","author":"Xu","year":"2023","journal-title":"J. Adv. Math. Stud."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"458","DOI":"10.1090\/S0002-9939-1969-0239559-9","article-title":"On nonlinear contractions","volume":"20","author":"Boyd","year":"1969","journal-title":"Proc. Am. Math. Soc."},{"unstructured":"Dugundji, J., and Granas, A. (1982). Fixed Point Theory, Panstwowe Wydawnictwo Naukowe (PWN\u2014Polish Scientific Publishers). Monografie Matematyczne, Tom 61.","key":"ref_7"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1090\/S0002-9947-1974-0362283-5","article-title":"An approach to fixed-point theorems on uniform spaces","volume":"191","author":"Tarafdar","year":"1974","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"9314","DOI":"10.3934\/math.2023468","article-title":"New fixed point results in controlled metric type spaces based on new contractive conditions","volume":"8","author":"Shatanawi","year":"2023","journal-title":"AIMS Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"7225","DOI":"10.3934\/math.2023363","article-title":"Common fixed point results via A\u03bd-\u03b1-contractions with a pair and two pairs of self-mappings in the frame of an extended quasi b-metric space","volume":"8","author":"Rezazgui","year":"2023","journal-title":"AIMS Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"4407","DOI":"10.3934\/math.2023220","article-title":"On fixed points, their geometry and application to satellite web coupling problem in S-metric spaces","volume":"8","author":"Joshi","year":"2023","journal-title":"AIMS Math."},{"key":"ref_12","first-page":"19","article-title":"Generalized contraction and fixed point theorems","volume":"12","year":"1971","journal-title":"Publ. Inst. Math. Nouv. S\u00e9r."},{"key":"ref_13","first-page":"43","article-title":"Approximation fixed points of weak contractions using the Picard iteration","volume":"9","author":"Berinde","year":"2004","journal-title":"Nonlinear Anal. Forum"},{"unstructured":"Banach, S. (1932). Th\u00e9orie des Op\u00e9rations Lin\u00e9aires, Monografie Matematyczne.","key":"ref_14"},{"key":"ref_15","first-page":"727","article-title":"Fixed-point theorems","volume":"25","author":"Chatterjea","year":"1972","journal-title":"Comptes Rendus Acad. Bulg. Sci."},{"key":"ref_16","first-page":"71","article-title":"Some resulta on fixed points","volume":"10","author":"Kannan","year":"1968","journal-title":"Bull. Calcutta Math. Soc."},{"key":"ref_17","first-page":"6","article-title":"The contraction mapping principle in quasimetric spaces","volume":"30","author":"Bakhtin","year":"1989","journal-title":"Funct. Anal."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1014","DOI":"10.22436\/jnsa.008.06.12","article-title":"Complex valued rectangular b-metric spacesand an application to linear equations","volume":"8","author":"Ege","year":"2015","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1186\/1687-1812-2013-130","article-title":"Existance of tripled coincidence points in order b-matric spaces and an application to system of integral equations","volume":"2013","author":"Parvaneh","year":"2013","journal-title":"Fixed Point Theory Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1186\/1687-1812-2014-135","article-title":"Contration conditions usingcomparison functionson b-metric spaces","volume":"2014","author":"Shatanawi","year":"2014","journal-title":"Fixed Point Theory Appl."},{"key":"ref_21","first-page":"1","article-title":"Fixed and common fixed for mapping satisfying some nonlinear contraction in b-metric spaces","volume":"7","author":"Shatanawi","year":"2016","journal-title":"J. Math. Anal."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1080\/01630563.2011.533046","article-title":"Common fixed point theorems in complex valued metric space","volume":"32","author":"Azam","year":"2011","journal-title":"Numer. Funct. Anal. Optim."},{"key":"ref_23","first-page":"1","article-title":"A common fixed point theorem in complex valued b-metric spaces","volume":"1","author":"Rao","year":"2013","journal-title":"Bull. Math. Stat. Res."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1019","DOI":"10.3934\/math.2019.3.1019","article-title":"Applications and theorem on common fixed point in complex valued b-metric space","volume":"4","author":"Berrah","year":"2019","journal-title":"AIMS Math."},{"key":"ref_25","first-page":"347","article-title":"Common Fixed Point Theorems Using Property (E.A) in Complex-Valued Metric Spaces","volume":"11","author":"Verma","year":"2013","journal-title":"Thai J. Math."},{"key":"ref_26","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_27","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 Appl."},{"key":"ref_28","first-page":"4","article-title":"A note on the results of Suzuki, Miculescu and Mihail","volume":"21","year":"2019","journal-title":"J. Fixed Point Theory Appl."},{"doi-asserted-by":"crossref","unstructured":"Aslam, M.S., Guran, L., and Saleem, N. (2021). Common Fixed Point Technique for Existence of a Solution of Urysohn Type Integral Equations System in Complex Valued b-Metric Spaces. Mathematics, 9.","key":"ref_29","DOI":"10.3390\/math9040400"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1186\/1687-1847-2013-49","article-title":"Urysohn integral equations approach by common fixed points in complex-valued metric spaces","volume":"2013","author":"Sintunavarat","year":"2013","journal-title":"Adv. Differ. Equ."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/8\/794\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:35:01Z","timestamp":1760128501000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/8\/794"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,16]]},"references-count":30,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2023,8]]}},"alternative-id":["axioms12080794"],"URL":"https:\/\/doi.org\/10.3390\/axioms12080794","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2023,8,16]]}}}