{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:10:16Z","timestamp":1760145016370,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,6,6]],"date-time":"2024-06-06T00:00:00Z","timestamp":1717632000000},"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 the present paper, under certain assumptions, we establish the convergence of iterates for self-mappings of complete metric spaces with graphs which are of a contractive type. The class of mappings considered in the paper contains the so-called cyclical mappings introduced by W. A. Kirk, P. S. Srinivasan and P. Veeramani in 2003 and the class of monotone nonexpansive operators. Our results hold in the case of a symmetric graph.<\/jats:p>","DOI":"10.3390\/sym16060705","type":"journal-article","created":{"date-parts":[[2024,6,6]],"date-time":"2024-06-06T06:40:26Z","timestamp":1717656026000},"page":"705","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Existence of a Fixed Point and Convergence of Iterates for Self-Mappings of Metric Spaces with Graphs"],"prefix":"10.3390","volume":"16","author":[{"given":"Alexander J.","family":"Zaslavski","sequence":"first","affiliation":[{"name":"Department of Mathematics, The Technion\u2014Israel Institute of Technology, Haifa 32000, Israel"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"133","DOI":"10.4064\/fm-3-1-133-181","article-title":"Sur les op\u00e9rations dans les ensembles abstraits et leur application aux \u00e9quations int\u00e9grales","volume":"3","author":"Banach","year":"1922","journal-title":"Fund. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"123482","DOI":"10.1016\/j.jmaa.2019.123482","article-title":"On the rate of convergence of iterated Bregman projections and of the alternating algorithm","volume":"481","author":"Bargetz","year":"2020","journal-title":"J. Math. Anal. Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1007\/s00574-018-0117-5","article-title":"Some generalizations of fixed point theorems of Caristi type and Mizoguchi\u2013Takahashi type under relaxed conditions","volume":"50","author":"Du","year":"2019","journal-title":"Bull. Braz. Math. Soc."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Goebel, K., and Kirk, W.A. (1990). Topics in Metric Fixed Point Theory, Cambridge University Press.","DOI":"10.1017\/CBO9780511526152"},{"key":"ref_5","unstructured":"Goebel, K., and Reich, S. (1984). Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1007\/s12215-020-00491-9","article-title":"On a theorem of Ciric in b-metric spaces","volume":"70","author":"Karapinar","year":"2021","journal-title":"Rend. Circ. Mat. Palermo"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Kozlowski, W.M. (2014). An Introduction to Fixed Point Theory in Modular Function Spaces, Springer.","DOI":"10.1007\/978-3-319-01586-6_5"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1090\/S0002-9939-1962-0148046-1","article-title":"A note on contractive mappings","volume":"13","author":"Rakotch","year":"1962","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_9","unstructured":"Reich, S., and Zaslavski, A.J. (2014). Developments in Mathematics, Springer."},{"key":"ref_10","unstructured":"Zaslavski, A.J. (2016). Springer Optimization and Its Applications, Springer."},{"key":"ref_11","unstructured":"Zaslavski, A.J. (2018). Springer Optimization and Its Applications, Springer."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"065008","DOI":"10.1088\/0266-5611\/26\/6\/065008","article-title":"Perturbation resilience and superiorization of iterative algorithms","volume":"26","author":"Censor","year":"2010","journal-title":"Inverse Probl."},{"key":"ref_13","first-page":"565","article-title":"Algorithms and convergence results of projection methods for inconsistent feasibility problems: A review","volume":"3","author":"Censor","year":"2018","journal-title":"Pure Appl. Func. Anal."},{"key":"ref_14","first-page":"243","article-title":"A new split inverse problem and an application to least intensity feasible solutions","volume":"2","author":"Gibali","year":"2017","journal-title":"Pure Appl. Funct. Anal."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1016\/j.topol.2011.10.013","article-title":"Some fixed point results on a metric space with a graph","volume":"159","author":"Aleomraninejad","year":"2012","journal-title":"Topol. Appl."},{"key":"ref_16","first-page":"85","article-title":"Fixed point of \u03d5-contraction in metric spaces endowed with a graph","volume":"37","author":"Bojor","year":"2010","journal-title":"An. Univ. Craiova Ser. Mat. Inform."},{"key":"ref_17","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."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1016\/j.jmaa.2009.03.023","article-title":"IFS on a metric space with a graph structure and extensions of the Kelisky-Rivlin theorem","volume":"356","author":"Jachymski","year":"2009","journal-title":"J. Math. Anal. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1359","DOI":"10.1090\/S0002-9939-07-09110-1","article-title":"The contraction principle for mappings on a metric space with a graph","volume":"136","author":"Jachymski","year":"2008","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Reich, S., and Zaslavski, A.J. (2021). Contractive mappings on metric spaces with graphs. Mathematics, 9.","DOI":"10.3390\/math9212774"},{"key":"ref_21","first-page":"141","article-title":"Some fixed point results on intuitionistic fuzzy metric spaces with a graph","volume":"13","author":"Samei","year":"2019","journal-title":"Sahand Commun. Math. Anal."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Suantai, S., Donganont, M., and Cholamjiak, W. (2019). Hybrid methods for a countable family of G-nonexpansive mappings in Hilbert spaces endowed with graphs. Mathematics, 7.","DOI":"10.3390\/math7100936"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1007\/s40314-021-01530-6","article-title":"A parallel monotone hybrid algorithm for a finite family of G-nonexpansive mappings in Hilbert spaces endowed with a graph applicable in signal recovery","volume":"40","author":"Suantai","year":"2021","journal-title":"Comp. Appl. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/s11075-017-0324-y","article-title":"A modified S-iteration process for G-nonexpansive mappings in Banach spaces with graphs","volume":"77","author":"Suparatulatorn","year":"2018","journal-title":"Numer. Algorithms"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/j.akcej.2017.01.001","article-title":"Hybrid methods for a finite family of G-nonexpansive mappings in Hilbert spaces endowed with graphs","volume":"14","author":"Suparatulatorn","year":"2017","journal-title":"AKCE Int. J. Graphs Comb."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1001","DOI":"10.1016\/j.jmaa.2005.10.081","article-title":"Existence and convergence of best proximity points","volume":"323","author":"Eldred","year":"2006","journal-title":"J. Math. Anal. Appl."},{"key":"ref_27","first-page":"79","article-title":"Fixed points for mappings satisfying cyclical contractive conditions","volume":"4","author":"Kirk","year":"2003","journal-title":"Fixed Point Theory"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"3332","DOI":"10.1016\/j.na.2008.04.037","article-title":"Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces","volume":"70","author":"Wlodarczyk","year":"2009","journal-title":"Nonlinear Anal."},{"key":"ref_29","first-page":"57","article-title":"Existence of a fixed point for a cyclical contractive mapping","volume":"32","author":"Zaslavski","year":"2022","journal-title":"Pan-Am. Math. J."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1186\/s13663-015-0437-8","article-title":"A note on fixed point theory for cyclic \u03d5-contractions","volume":"2015","author":"Radenovic","year":"2015","journal-title":"Fixed Point Theory Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1007\/s13370-015-0339-2","article-title":"Some remarks on mappings satisfying cyclical contractive conditions","volume":"27","author":"Radenovic","year":"2015","journal-title":"Afr. Mat."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1016\/j.amc.2015.09.089","article-title":"A note on some recent fixed point results for cyclic contractions in b-metric spaces and an application to integral equations","volume":"273","author":"Radenovic","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_33","first-page":"391","article-title":"Monotone contractive mappings","volume":"1","author":"Reich","year":"2017","journal-title":"J. Nonlinear Var. Anal."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/6\/705\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:54:41Z","timestamp":1760108081000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/6\/705"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,6]]},"references-count":33,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2024,6]]}},"alternative-id":["sym16060705"],"URL":"https:\/\/doi.org\/10.3390\/sym16060705","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,6,6]]}}}