{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,1]],"date-time":"2026-02-01T12:56:48Z","timestamp":1769950608304,"version":"3.49.0"},"reference-count":23,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2020,7,1]],"date-time":"2020-07-01T00:00:00Z","timestamp":1593561600000},"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 metric function generalizes the concept of distance between two points and hence includes the symmetric property. The aim of this article is to introduce a new and proper extension of Kannan\u2019s fixed point theorem to the case of multivalued maps using Wardowski\u2019s F-contraction. We show that our result is applicable to a class of mappings where neither the multivalued version of Kannan\u2019s theorem nor that of Wardowski\u2019s can be applied to determine the existence of fixed points. Application of our result to the solution of integral equations has been provided. A multivalued Reich type generalized version of the result is also established.<\/jats:p>","DOI":"10.3390\/sym12071090","type":"journal-article","created":{"date-parts":[[2020,7,2]],"date-time":"2020-07-02T05:00:08Z","timestamp":1593666008000},"page":"1090","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":33,"title":["New Extensions of Kannan\u2019s and Reich\u2019s Fixed Point Theorems for Multivalued Maps Using Wardowski\u2019s Technique with Application to Integral Equations"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0097-0819","authenticated-orcid":false,"given":"Pradip","family":"Debnath","sequence":"first","affiliation":[{"name":"Department of Applied Science and Humanities, Assam University, Silchar, Cachar, Assam 788011, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9277-8092","authenticated-orcid":false,"given":"Hari Mohan","family":"Srivastava","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada"},{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"},{"name":"Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, Baku AZ1007, Azerbaijan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,7,1]]},"reference":[{"key":"ref_1","first-page":"71","article-title":"Some results on fixed points","volume":"60","author":"Kannan","year":"1968","journal-title":"Bull. Calcutta Math. Soc."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1007\/BF01472580","article-title":"Completeness and fixed points","volume":"80","author":"Subrahmanyam","year":"1975","journal-title":"Monatsh. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"121","DOI":"10.4153\/CMB-1971-024-9","article-title":"Some remarks concerning contraction mappings","volume":"14","author":"Reich","year":"1971","journal-title":"Can. Math. Bull."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1186\/1687-1812-2012-94","article-title":"Fixed points of a new type of contractive mappings in complete metric space","volume":"2012","author":"Wardowski","year":"2012","journal-title":"Fixed Point Theory Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"475","DOI":"10.2140\/pjm.1969.30.475","article-title":"Multi-valued contraction mappings","volume":"30","author":"Nadler","year":"1969","journal-title":"Pac. J. Math."},{"key":"ref_6","first-page":"65","article-title":"Fixed point theorems for some multivalued generalized contraction in b-metric spaces","volume":"6","author":"Boriceanu","year":"2010","journal-title":"Int. J. Math. Stat."},{"key":"ref_7","first-page":"263","article-title":"Nonlinear set-valued contraction mappings in b-metric spaces","volume":"46","author":"Czerwik","year":"1998","journal-title":"Atti Sem. Mat. Univ. Modena"},{"key":"ref_8","first-page":"26","article-title":"Fixed points of contractive functions","volume":"4","author":"Reich","year":"1972","journal-title":"Boll. Unione Mat. Ital."},{"key":"ref_9","first-page":"659","article-title":"Multivalued F-contractions on complete metric spaces","volume":"16","author":"Altun","year":"2015","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Aydi, H., Chen, C.M., and Karapinar, E. (2019). Interpolative \u0106iri\u0107-Reich-Rus type contractions via the Branciari distance. Mathematics, 7.","DOI":"10.3390\/math7010084"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Aydi, H., Karapinar, E., and Hierro, A.F.R. (2019). \u03c9-Interpolative \u0106iri\u0107-Reich-Rus-type contractions. Mathematics, 7.","DOI":"10.3390\/math7010057"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3895","DOI":"10.1016\/j.na.2012.02.009","article-title":"Fixed point theorems for Reich type contractions on metric spaces with a graph","volume":"75","author":"Bojor","year":"2012","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"ref_13","first-page":"31","article-title":"Fixed points of Kannan mappings in metric spaces endowed with a graph","volume":"20","author":"Bojor","year":"2012","journal-title":"An. Univ. Ovidius Constanta Ser. Mat."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Choudhury, B.S., and Kundu, A. (2013). A Kannan-like contraction in partially ordered spaces. Demonstr. Math., XLVI.","DOI":"10.1515\/dema-2013-0462"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Debnath, P., and de La Sen, M. (2019). Set-valued interpolative Hardy-Rogers and set-valued Reich-Rus-\u0106iri\u0107-type contractions in b-metric spaces. Mathematics, 7.","DOI":"10.3390\/math7090849"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Debnath, P., and de La Sen, M. (2020). Fixed points of interpolative \u0106iri\u0107-Reich-Rus-type contractions in b-metric spaces. Symmetry, 12.","DOI":"10.3390\/sym12010012"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Debnath, P., Mitrovi\u0107, Z., and Radenovi\u0107, S. (2019). Interpolative Hardy-Rogers and Reich-Rus-\u0106iri\u0107 type contractions in b-metric spaces and rectangular b-metric spaces. Math. Vesnik, in Press.","DOI":"10.3390\/math7090849"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Debnath, P., Neog, M., and Radenovi\u0107, S. (2019). Set valued Reich type G-contractions in a complete metric space with graph. Rend. Circ. Mat. Palermo II Ser., in Press.","DOI":"10.1007\/s12215-019-00446-9"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2145","DOI":"10.1007\/s11784-017-0402-8","article-title":"Fixed point theorems for Kannan type mappings","volume":"19","author":"Gornicki","year":"2017","journal-title":"J. Fixed Point Theory Appl."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Karapinar, E., Agarwal, R.P., and Aydi, H. (2018). Interpolative Reich-Rus-\u0106iri\u0107 type contractions on partial metric spaces. Mathematics, 6.","DOI":"10.3390\/math6110256"},{"key":"ref_21","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_22","doi-asserted-by":"crossref","first-page":"1259","DOI":"10.2298\/FIL1307259S","article-title":"Multi-valued F-contractions and the solution of certain functional and integral equations","volume":"27","author":"Sgroi","year":"2013","journal-title":"Filomat"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"3779","DOI":"10.2298\/FIL1614779A","article-title":"Multi-valued F-contractions and related fixed point theorems with an application","volume":"30","author":"Ali","year":"2016","journal-title":"Filomat"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/7\/1090\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:45:55Z","timestamp":1760175955000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/7\/1090"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,1]]},"references-count":23,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2020,7]]}},"alternative-id":["sym12071090"],"URL":"https:\/\/doi.org\/10.3390\/sym12071090","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,7,1]]}}}