{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T11:27:34Z","timestamp":1771673254837,"version":"3.50.1"},"reference-count":36,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2019,5,14]],"date-time":"2019-05-14T00:00:00Z","timestamp":1557792000000},"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>In this article, a general contractive mapping is presented and some fixed point results in complete b-metric-like spaces are studied. The results obtained here extend and improve some related results in the literature. Also, new common fixed point results for a graphic contraction mappings are proved. Some comparative examples are given to support the obtained results. Moreover, an analytical solution of an integral equation has been presented as an application.<\/jats:p>","DOI":"10.3390\/sym11050667","type":"journal-article","created":{"date-parts":[[2019,5,14]],"date-time":"2019-05-14T10:42:33Z","timestamp":1557830553000},"page":"667","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["Generalized Contractive Mappings and Related Results in b-Metric Like Spaces with an Application"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8724-9367","authenticated-orcid":false,"given":"Hasanen A.","family":"Hammad","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9320-9433","authenticated-orcid":false,"given":"Manuel","family":"De la Sen","sequence":"additional","affiliation":[{"name":"Institute of Research and Development of Processes, University of the Basque Country, 48940 Leioa (Bizkaia), Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,5,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1016\/j.amc.2014.09.096","article-title":"Generalized Krasnoselskii fixed point theorem involving auxiliary functions in bimetric spaces and application to two-point boundary value problem","volume":"248","author":"Berzig","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"955","DOI":"10.1016\/j.amc.2014.01.135","article-title":"Some new fixed point theorems in manger PM-spaces with application to Volterra type integral equation","volume":"232","author":"Gopal","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_3","first-page":"1","article-title":"Fixed point theorems for mappings satisfying contractive condition of integral type and applications","volume":"64","author":"Liu","year":"2011","journal-title":"Fixed Point Theory Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"961","DOI":"10.1016\/j.camwa.2006.08.046","article-title":"A common fixed point theorem and its application to nonlinear integral equations","volume":"53","author":"Pathak","year":"2007","journal-title":"Comput. Math. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1292","DOI":"10.1016\/j.amc.2015.06.074","article-title":"A fixed point theorem in partial quasi-metric spaces and an application to software engineering","volume":"268","author":"Shahzad","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_6","first-page":"1","article-title":"Krasnoselskii-type fixed point theorems with applications to Volterra integral equations","volume":"2013","author":"Hussain","year":"2013","journal-title":"Fixed Point Theory Appl."},{"key":"ref_7","first-page":"26","article-title":"The contraction mapping principle in quasi-metric spaces","volume":"30","author":"Bakhtin","year":"1989","journal-title":"Funct. Anal. Ulianowsk Gos. Ped. Inst."},{"key":"ref_8","first-page":"5","article-title":"Contraction mappings in b-metric spaces","volume":"1","author":"Czerwik","year":"1993","journal-title":"Acta Math. Inform. Univ. Ostrav."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"128","DOI":"10.1515\/math-2016-0010","article-title":"Existence of a common solution for a system of nonlinear integral equations via fixed point methods in b-metric spaces","volume":"14","author":"Yamaod","year":"2017","journal-title":"Open Math."},{"key":"ref_10","first-page":"2417","article-title":"\u03b1-implicit contractive pair of mappings on quasi b-metric spaces and an application to integral equations","volume":"17","author":"Aydi","year":"2016","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/1029-242X-2013-562","article-title":"Some common fixed point results in ordered b-metric spaces","volume":"2013","author":"Mustafa","year":"2013","journal-title":"J. Inequal. Appl."},{"key":"ref_12","first-page":"70","article-title":"Fixed point theorems for Ciric\u2019s and generalized contractions in b-metric spaces","volume":"7","author":"Sarwar","year":"2015","journal-title":"Int. J. Anal. Appl."},{"key":"ref_13","first-page":"125","article-title":"Sequences of almost contractions and fixed points in b-metric spaces","volume":"48","author":"Pacurar","year":"2010","journal-title":"Anal. Univ. Vest Timis. Ser. Mat-Inform."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"725","DOI":"10.1016\/j.amc.2013.10.043","article-title":"Some coincidence point results in ordered b-metric spaces and applications in a system of integral equations","volume":"262","author":"Roshan","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_15","first-page":"1","article-title":"Common fixed points of almost generalized (\u03c8,\u03d5)-contractive mappings in ordered b-metric spaces","volume":"2013","author":"Roshan","year":"2013","journal-title":"Fixed Point Theory Appl."},{"key":"ref_16","first-page":"1","article-title":"Metric-like spaces, partial metric spaces and fixed points","volume":"2012","author":"Harandi","year":"2012","journal-title":"Fixed Point Theory Appl."},{"key":"ref_17","first-page":"1","article-title":"Fixed points of contractive mappings in b-metric-like spaces","volume":"2014","author":"Hussain","year":"2014","journal-title":"Sci. World J."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1524","DOI":"10.22436\/jnsa.010.04.20","article-title":"Common fixed points via implicit contractions on b-metric-like spaces","volume":"10","author":"Aydi","year":"2017","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3057","DOI":"10.2298\/FIL1711057N","article-title":"Existence of solutions of Cantilever Beam Problem via \u03b1-\u03b2-FG-contractions in b-metric-like spaces","volume":"31","author":"Nashine","year":"2017","journal-title":"Filomat"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"3727","DOI":"10.2298\/FIL1712727A","article-title":"\u0106iri\u0107-Berinde fixed point theorems for multi-valued mappings on \u03b1 complete metric-like spaces","volume":"31","author":"Aydi","year":"2017","journal-title":"Filomat"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"780","DOI":"10.22436\/jnsa.010.02.38","article-title":"New Geraghty type contractions on metric-like spaces","volume":"10","author":"Aydi","year":"2017","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_22","first-page":"1","article-title":"Fixed point results for generalized \u03b1-\u03c8-contractions in metric-like spaces and applications","volume":"133","author":"Aydi","year":"2015","journal-title":"Electron. J. Differ. Equ."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1515\/math-2016-0097","article-title":"An Ulam stability result on quasi-b-metric-like spaces","volume":"14","author":"Alsulami","year":"2016","journal-title":"Open Math."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Hammad, H.A., and De la Sen, M. (2019). Solution of nonlinear integral equation via fixed point of cyclic \u03b1L\u03c8-rational contraction mappings in metric-like spaces. Bull. Braz. Math. Soc. New Ser.","DOI":"10.1007\/s00574-019-00144-1"},{"key":"ref_25","first-page":"1","article-title":"Fixed point and coupled fixed point theorems on b-metric-like spaces","volume":"402","author":"Alghmandi","year":"2013","journal-title":"J. Inequal. Appl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2492","DOI":"10.22436\/jnsa.009.05.48","article-title":"On common fixed points for (\u03b1,\u03c8)-contractions and generalized cyclic contractions in b-metric-like spaces and consequences","volume":"9","author":"Aydi","year":"2016","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1515\/math-2018-0024","article-title":"On some fixed point results for (s,p,\u03b1)-contractive mappings in b-metric-like spaces and applications to integral equations","volume":"16","author":"Zoto","year":"2018","journal-title":"Open Math."},{"key":"ref_28","first-page":"199","article-title":"Common fixed points for noncontinuous nonself maps on nonmetric spaces","volume":"4","author":"Jungck","year":"1996","journal-title":"Far East J. Math. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"416","DOI":"10.1016\/j.jmaa.2007.09.070","article-title":"Common fixed point results for noncommuting mappings without continuity in cone metric spaces","volume":"341","author":"Abbas","year":"2008","journal-title":"J. Math. Anal. Appl."},{"key":"ref_30","first-page":"1359","article-title":"The contraction principle for mappings on a metric space with a graph","volume":"1","author":"Jachymski","year":"2008","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"4838","DOI":"10.22436\/jnsa.009.06.119","article-title":"Some common fixed point results of graphs on b-metric space","volume":"9","author":"Mustafa","year":"2016","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Nash, J.F., and Rassias, M.T. (2016). Open Problems in Mathematics, Springer.","DOI":"10.1007\/978-3-319-32162-2"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"713","DOI":"10.1512\/iumj.1981.30.30055","article-title":"Fractals and self similarity","volume":"30","author":"Hutchinson","year":"1981","journal-title":"Indiana Univ. Math. J."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Guariglia, E. (2019). Primality, Fractality and image analysis. Entropy, 21.","DOI":"10.3390\/e21030304"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/j.indag.2015.07.006","article-title":"A new fixed point theorem in the fractal space","volume":"27","author":"Ri","year":"2016","journal-title":"Indag. Math.-New Ser."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1016\/j.indag.2015.01.003","article-title":"On contractive and convergence properties of a class of truncated operators represented through Schauder bases with applications","volume":"26","year":"2015","journal-title":"Indag. Math.-New Ser."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/5\/667\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:51:43Z","timestamp":1760187103000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/5\/667"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,14]]},"references-count":36,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2019,5]]}},"alternative-id":["sym11050667"],"URL":"https:\/\/doi.org\/10.3390\/sym11050667","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,5,14]]}}}