{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:58:50Z","timestamp":1760230730590,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,8,11]],"date-time":"2022-08-11T00:00:00Z","timestamp":1660176000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["11872043","2021ZYD0017","2020YGJC03","cx2021150"],"award-info":[{"award-number":["11872043","2021ZYD0017","2020YGJC03","cx2021150"]}]},{"name":"Central Government Funds of Guiding Local Scientific and Technological Development for Sichuan Province","award":["11872043","2021ZYD0017","2020YGJC03","cx2021150"],"award-info":[{"award-number":["11872043","2021ZYD0017","2020YGJC03","cx2021150"]}]},{"name":"Zigong Science and Technology Program","award":["11872043","2021ZYD0017","2020YGJC03","cx2021150"],"award-info":[{"award-number":["11872043","2021ZYD0017","2020YGJC03","cx2021150"]}]},{"name":"Sichuan University of Science and Engineering","award":["11872043","2021ZYD0017","2020YGJC03","cx2021150"],"award-info":[{"award-number":["11872043","2021ZYD0017","2020YGJC03","cx2021150"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The purpose of this paper is to develop coupled fixed point theorems by C-class functions with mixed monotonicity, discuss the existence of the coupled fixed point of two mappings and obtain a coupled coincidence point theorem via inverse Ck-class functions in partially ordered H-metric spaces. This improves the existing results. In addition, some instances and an application are given to illustrate the usability and validity of the results.<\/jats:p>","DOI":"10.3390\/sym14081663","type":"journal-article","created":{"date-parts":[[2022,8,11]],"date-time":"2022-08-11T23:05:49Z","timestamp":1660259149000},"page":"1663","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Coupled Fixed Point Theorems with Rational Type Contractive Condition via C-Class Functions and Inverse Ck-Class Functions"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8904-6782","authenticated-orcid":false,"given":"Xiaolan","family":"Liu","sequence":"first","affiliation":[{"name":"College of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 643000, China"},{"name":"South Sichuan Center for Applied Mathematics, Zigong 643000, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6496-4824","authenticated-orcid":false,"given":"Mi","family":"Zhou","sequence":"additional","affiliation":[{"name":"School of Science and Techology, University of Sanya, Sanya 572022, China"},{"name":"Center for Mathematical Reaserch, University of Sanya, Sanya 572022, China"},{"name":"Academician Guoliang Chen Team Innovation Center, University of Sanya, Sanya 572022, China"},{"name":"Academician Chunming Rong Workstation, University of Sanya, Sanya 572022, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5527-1284","authenticated-orcid":false,"given":"Arslan H.","family":"Ansari","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj 6915136111, Iran"},{"name":"Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, Pretoria 0204, South Africa"}]},{"given":"Kalyan","family":"Chakrabarti","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Scottish Church College, Urquhart Square, Kolkata 700006, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5528-1207","authenticated-orcid":false,"given":"Mujahid","family":"Abbas","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Applied Mathematics, University of Pretoria, Lynnwood Road, Pretoria 0002, South Africa"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2659-7568","authenticated-orcid":false,"given":"Laxmi","family":"Rathour","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak 484887, India"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1435","DOI":"10.1090\/S0002-9939-03-07220-4","article-title":"A fixed point theorem in partially ordered sets and some applications to matrix equations","volume":"132","author":"Ran","year":"2004","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1379","DOI":"10.1016\/j.na.2005.10.017","article-title":"Fixed point theorems in partially ordered metric spaces and applications","volume":"65","author":"Bhaskar","year":"2006","journal-title":"Nonl. Anal."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"4341","DOI":"10.1016\/j.na.2008.09.020","article-title":"Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces","volume":"70","author":"Lakshmikantham","year":"2009","journal-title":"Nonl. Anal."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1188","DOI":"10.1016\/j.na.2009.08.003","article-title":"Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations","volume":"72","author":"Harjani","year":"2010","journal-title":"Nonl. Anal."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"945915","DOI":"10.1155\/2012\/945915","article-title":"Common fixed-point theorems in complete generalized metric spaces","volume":"2012","author":"Chen","year":"2012","journal-title":"J. Appl. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"14303-01e","DOI":"10.18540\/jcecvl8iss3pp14303-01e","article-title":"Expansive fixed point theorems for tri-simulation functions","volume":"8","author":"Shahi","year":"2022","journal-title":"J. Eng. Exact Sci."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Sanatee, A.G., Rathour, L., Mishra, V.N., and Dewangan, V. (2022). Some fixed point theorems in regular modular metric spaces and application to Caratheodory\u2019s type anti-periodic boundary value problem. J. Anal., 1\u201314.","DOI":"10.1007\/s41478-022-00469-z"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Pathak, V.K., and Mishra, L.N. (2022). Application of fixed point theorem to solvability for non-linear fractional hadamard functional integral equations. Mathematics, 10.","DOI":"10.3390\/math10142400"},{"key":"ref_9","first-page":"600","article-title":"Unique fixed point theorems for generalized contractive mappings in partial metric spaces","volume":"2015","author":"Mishra","year":"2015","journal-title":"J. Funct. Spaces"},{"key":"ref_10","first-page":"600","article-title":"Fixed point theorems for generalized weakly S-contractive mappings in partial metric spaces","volume":"5","author":"Mishra","year":"2015","journal-title":"J. Appl. Anal. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Mishra, V.N., S\u00e1nchez Ruiz, L.M., Gautam, P., and Verma, S. (2020). Interpolative Reich-Rus-\u0106iri\u0107 and Hardy-Rogers contraction on quasi-partial b-metric space and related fixed point results. Mathematics, 8.","DOI":"10.3390\/math8091598"},{"key":"ref_12","first-page":"1","article-title":"Coupled fixed point theorems for mappings satisfying a contractive condition of rational type on a partially ordered metric space","volume":"2","author":"Olatinwo","year":"2012","journal-title":"Adv. Fixed Point Theory"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"7347","DOI":"10.1016\/j.na.2011.07.053","article-title":"Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces","volume":"74","author":"Berinde","year":"2011","journal-title":"Non. Anal."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/j.mcm.2011.01.036","article-title":"Coupled fixed point results in generalized metric spaces","volume":"54","author":"Choudhury","year":"2011","journal-title":"Math. Comput. Model."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"206","DOI":"10.1186\/1687-1812-2013-206","article-title":"Coupled coincidence point results for (\u03c8,\u03d5)-weakly contractive mappings in partially ordered Gb-metric spaces","volume":"2013","author":"Mustafa","year":"2013","journal-title":"Fixed Point Theory A."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"568718","DOI":"10.1155\/2014\/568718","article-title":"Coupled and tripled Coincidence Point Results with Application to fredholm integral equations","volume":"2014","author":"Kutbi","year":"2014","journal-title":"Abstr. Appl. Anal."},{"key":"ref_17","first-page":"289","article-title":"A new approach to generalized metric spaces","volume":"7","author":"Mustafa","year":"2006","journal-title":"J. Nonlinear Convex A."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"797","DOI":"10.1016\/j.mcm.2010.05.009","article-title":"Fixed point theorems in generalized partially ordered G-metric spaces","volume":"52","author":"Saadati","year":"2010","journal-title":"Math. Comput. Model."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"29","DOI":"10.12785\/msl\/020104","article-title":"Fixed point theorems in G-metric spaces with W maps","volume":"2","author":"Chakrabarti","year":"2013","journal-title":"Math. Sci. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1186\/1687-1812-2013-154","article-title":"Further fixed point results on G-metric spaces","volume":"2013","author":"Agarwal","year":"2013","journal-title":"Fixed Point Theory A."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1186\/1029-242X-2013-453","article-title":"Existence of tripled coincidence point in ordered Gb-metric spaces and applications to system of integral equations","volume":"2013","author":"Mustafa","year":"2013","journal-title":"J. Inequal. Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1186\/2251-7456-7-18","article-title":"Common fixed points of six mappings in partially ordered G-metric spaces","volume":"7","author":"Parvaneh","year":"2013","journal-title":"Math. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"701910","DOI":"10.1155\/2012\/701910","article-title":"On generalized weakly G-contractive mappings in partially ordered G-metric spaces","volume":"2012","author":"Razani","year":"2012","journal-title":"Abstr. Appl. Anal."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"326","DOI":"10.1186\/1687-1812-2013-326","article-title":"Some coincidence point results for generalized (\u03c8,\u03d5)-weakly contractive mappings in ordered G-metric spaces","volume":"2013","author":"Mustafa","year":"2013","journal-title":"Fixed Point Theory A"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"585964","DOI":"10.1155\/2014\/585964","article-title":"Fixed point results for G-\u03b1-contractive maps with application to boundary value problems","volume":"2014","author":"Hussain","year":"2014","journal-title":"Sci. World J."},{"key":"ref_26","first-page":"180698","article-title":"A unification of G-metric, Partial metric and b-metric spaces","volume":"2014","author":"Latif","year":"2014","journal-title":"Abstr. Appl. Anal."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1186\/1687-1812-2013-317","article-title":"Some fixed point results in ordered Gp-metric spaces","volume":"2013","author":"Alsulami","year":"2013","journal-title":"Fixed Point Theory A"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Harjani, J., L\u00f3pez, B., and Sadarangani, K. (2010). A fixed point theorem for mappings satisfying a contractive condition of rational type on a partially ordered metric space. Abstr. Appl. Anal., 2010.","DOI":"10.1155\/2010\/190701"},{"key":"ref_29","first-page":"223","article-title":"Some unique fixed point theorems","volume":"8","author":"Jaggi","year":"1977","journal-title":"Indian J. Pure. Ap. Mat."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"785357","DOI":"10.1155\/2014\/785357","article-title":"Coupled fixed point theorems with rational type contractive condition in a partially ordered G-metric space","volume":"2014","author":"Chakrabarti","year":"2014","journal-title":"J. Math."},{"key":"ref_31","unstructured":"Ansari, A.H. (, January September). Note on \u201c\u03c6-\u03c8-contractive type mappings and related fixed point\u201d. Proceedings of the the 2nd Regional Conference on Mathematics and Applications, Payame Noor University, Tehran, Iran."},{"key":"ref_32","first-page":"23","article-title":"Some fixed point theorems of inverse C-class funtion under weak semi compatibility","volume":"2018","author":"Saleem","year":"2018","journal-title":"J. Fixed Point Theory"},{"key":"ref_33","first-page":"503","article-title":"On various types of compatible maps and common fixed point theorems for non-continuous maps","volume":"40","author":"Singh","year":"2011","journal-title":"Hacettepe J. Math. Stat."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/8\/1663\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:07:11Z","timestamp":1760141231000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/8\/1663"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,11]]},"references-count":33,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2022,8]]}},"alternative-id":["sym14081663"],"URL":"https:\/\/doi.org\/10.3390\/sym14081663","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,8,11]]}}}