{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:46:26Z","timestamp":1760150786340,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,12,29]],"date-time":"2023-12-29T00:00:00Z","timestamp":1703808000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["11861003","2023AAC03301","2021BEG03049","2022RKX03","NXYLXK2017B09"],"award-info":[{"award-number":["11861003","2023AAC03301","2021BEG03049","2022RKX03","NXYLXK2017B09"]}]},{"name":"Natural Science Foundation of Ningxia province","award":["11861003","2023AAC03301","2021BEG03049","2022RKX03","NXYLXK2017B09"],"award-info":[{"award-number":["11861003","2023AAC03301","2021BEG03049","2022RKX03","NXYLXK2017B09"]}]},{"name":"Major Research Projects of NingXia","award":["11861003","2023AAC03301","2021BEG03049","2022RKX03","NXYLXK2017B09"],"award-info":[{"award-number":["11861003","2023AAC03301","2021BEG03049","2022RKX03","NXYLXK2017B09"]}]},{"name":"Major Scientific and Technological Innovation Projects of YinChuan","award":["11861003","2023AAC03301","2021BEG03049","2022RKX03","NXYLXK2017B09"],"award-info":[{"award-number":["11861003","2023AAC03301","2021BEG03049","2022RKX03","NXYLXK2017B09"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In the context of a complete metric space, the most important fixed-point result for multi-valued operators was given in 1969. Many extensions of this fixed-point principle for multi-valued operators were proved by different authors. Based on some of the above-mentioned results, we introduce the notion of the multi-valued Feng\u2013Liu-type operator and we construct an abstract fixed-point theory for this general class of multi-valued operators. Our results extend and complement some theorems in metric fixed-point theory for multi-valued operators. An application to a Cauchy problem related to a first-order differential inclusion is also given. In this case, our theorem improves several previous theorems on this subject by relaxing the contraction-type condition (with respect to its second argument) on the multi-valued right-hand side.<\/jats:p>","DOI":"10.3390\/axioms13010024","type":"journal-article","created":{"date-parts":[[2023,12,29]],"date-time":"2023-12-29T06:18:13Z","timestamp":1703830693000},"page":"24","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On Some Properties of Multi-Valued Feng\u2013Liu-Type Operators in Metric Spaces"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5629-5667","authenticated-orcid":false,"given":"Adrian","family":"Petru\u015fel","sequence":"first","affiliation":[{"name":"Department of Mathematics, Babe\u015f-Bolyai University, Kog\u0103lniceanu Street No. 1, 400084 Cluj-Napoca, Romania"},{"name":"Academy of Romanian Scientists, Ilfov Street No. 3, 50044 Bucharest, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8405-1977","authenticated-orcid":false,"given":"Gabriela","family":"Petru\u015fel","sequence":"additional","affiliation":[{"name":"Department of Business, Babe\u015f-Bolyai University Cluj-Napoca, Horea Street No. 7, 400174 Cluj-Napoca, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8883-5034","authenticated-orcid":false,"given":"Lijun","family":"Zhu","sequence":"additional","affiliation":[{"name":"The Key Laboratory of Intelligent Information and Big Data Processing of NingXia Province, Health Big Data Research Institute, North Minzu University, Yinchuan 750021, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,12,29]]},"reference":[{"key":"ref_1","unstructured":"Rus, I.A., Petru\u015fel, A., and Petru\u015fel, G. (2008). Fixed Point Theory, Cluj University Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"772","DOI":"10.1016\/j.jmaa.2006.03.016","article-title":"On a general class of multi-valued weakly Picard mappings","volume":"326","author":"Berinde","year":"2007","journal-title":"J. Math. Anal. Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1007\/BF02771543","article-title":"Multi-valued contraction mappings in generalized metric spaces","volume":"8","author":"Covitz","year":"1970","journal-title":"Isr. J. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/j.jmaa.2005.12.004","article-title":"Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings","volume":"317","author":"Feng","year":"2006","journal-title":"J. Math. Anal. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1016\/0022-247X(85)90306-3","article-title":"On fixed point stability for set-valued contractive mappings with applications to generalized differential equations","volume":"110","author":"Lim","year":"1985","journal-title":"J. Math. Anal. Appl."},{"key":"ref_6","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_7","doi-asserted-by":"crossref","first-page":"2195","DOI":"10.11650\/twjm\/1500406430","article-title":"Ulam-Hyers stability for operatorial equations and inclusions via nonself operators","volume":"15","author":"Petru","year":"2011","journal-title":"Taiwan J. 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Nonlinear Convex Anal."},{"key":"ref_12","first-page":"2187","article-title":"Some variants of the contraction principle for multi-valued operators, generalizations and applications","volume":"20","year":"2019","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_13","first-page":"393","article-title":"Well-posedness of fixed point problems","volume":"46","author":"Reich","year":"2011","journal-title":"Far East J. Math. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"157","DOI":"10.24193\/fpt-ro.2021.1.11","article-title":"Fixed and periodic point results in cone b-metric spaces over Banach algebras; a survey","volume":"22","author":"Rahimi","year":"2021","journal-title":"Fixed Point Theory"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Lotfali Ghasab, E., Majani, H., De la Sen, M., and Soleimani Rad, G. (2021). e-distance in Menger PGM Spaces with an application. Axioms, 10.","DOI":"10.3390\/axioms10010003"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Aubin, J.P., and Frankowska, H. (2009). Set-Valued Analysis, Birkh\u00e4user.","DOI":"10.1007\/978-0-8176-4848-0"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Deimling, K. (1992). Multivalued Differential Equations, Walter de Gruyter.","DOI":"10.1515\/9783110874228"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"187","DOI":"10.4064\/fm-125-3-187-193","article-title":"On Carath\u00e9odory type selections","volume":"125","author":"Rybinski","year":"1985","journal-title":"Fund. Math."},{"key":"ref_19","first-page":"159","article-title":"Lipschitzian generalized differential equations","volume":"48","author":"Himmelberg","year":"1972","journal-title":"Rend. Semin. Mat. Univ. Padova"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/1\/24\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:44:09Z","timestamp":1760132649000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/1\/24"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,12,29]]},"references-count":19,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2024,1]]}},"alternative-id":["axioms13010024"],"URL":"https:\/\/doi.org\/10.3390\/axioms13010024","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2023,12,29]]}}}