{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T15:00:59Z","timestamp":1772550059139,"version":"3.50.1"},"reference-count":36,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2020,10,14]],"date-time":"2020-10-14T00:00:00Z","timestamp":1602633600000},"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>A recently introduced nonexpansive-type condition is subjected to an in-depth analysis. New examples are provided to highlight the relationship with Suzuki-type mappings. Furthermore, a convergence survey is conducted based on the iteration procedure Sn. Issues related to data dependence and the stability of this iterative process are also being studied. Our study is performed in the framework of Banach spaces, in which the symmetry of the associated metric is a fundamental axiom and plays a key role while proving many results of this paper.<\/jats:p>","DOI":"10.3390\/sym12101676","type":"journal-article","created":{"date-parts":[[2020,10,17]],"date-time":"2020-10-17T07:23:22Z","timestamp":1602919402000},"page":"1676","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["On Iteration Sn for Operators with Condition (D)"],"prefix":"10.3390","volume":"12","author":[{"given":"Cristian","family":"Ciobanescu","sequence":"first","affiliation":[{"name":"Department of Mathematics and Informatics, University Politehnica of Bucharest, 060042 Bucharest, Romania"}]},{"given":"Teodor","family":"Turcanu","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Informatics, University Politehnica of Bucharest, 060042 Bucharest, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2020,10,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1088","DOI":"10.1016\/j.jmaa.2007.09.023","article-title":"Fixed point theorems and convergence theorems for some generalized nonexpansive mappings","volume":"340","author":"Suzuki","year":"2008","journal-title":"J. Math. Anal. Appl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1742","DOI":"10.1080\/01630563.2018.1497650","article-title":"Fixed point theorems and convergence theorems for a new generalized nonexpansive mapping","volume":"39","author":"Donghan","year":"2018","journal-title":"Numer. Funct. Anal. Optim."},{"key":"ref_3","first-page":"827","article-title":"A unifying approach for some nonexpansiveness conditions on modular vector spaces","volume":"25","author":"Bejenaru","year":"2020","journal-title":"Nonlinear Anal. Model. Control."},{"key":"ref_4","first-page":"123","article-title":"Two remarks on the method of succesive approximations","volume":"10","year":"1955","journal-title":"Uspekhi Math. Nauk"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"506","DOI":"10.1090\/S0002-9939-1953-0054846-3","article-title":"Mean value methods in iteration","volume":"4","author":"Mann","year":"1953","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1090\/S0002-9939-1974-0336469-5","article-title":"Fixed points by a new iteration method","volume":"44","author":"Ishikawa","year":"1974","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1006\/jmaa.2000.7042","article-title":"New approximation schemes for general variational inequalities","volume":"251","author":"Noor","year":"2000","journal-title":"J. Math. Anal. Appl."},{"key":"ref_8","first-page":"223","article-title":"A new faster iteration process applied to constrained minimization and feasibility problems","volume":"66","author":"Abbas","year":"2014","journal-title":"Mat. Vesn."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/j.amc.2015.11.065","article-title":"A new iterative scheme for numerical reckoning fixed points of Suzuki\u2019s generalized nonexpansive mappings","volume":"275","author":"Thakur","year":"2016","journal-title":"Appl. Math. Comput."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Usurelu, G.I., and Postolache, M. (2019). Convergence analysis for a three-step Thakur iteration for Suzuki-type nonexpansive mappings with visualization. Symmetry, 11.","DOI":"10.3390\/sym11121441"},{"key":"ref_11","first-page":"1","article-title":"On a three-step iteration process for Suzuki mappings with qualitative study","volume":"1","author":"Maniu","year":"2020","journal-title":"Numer. Funct. Anal. Optim."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2553","DOI":"10.22436\/jnsa.009.05.53","article-title":"On a new iteration scheme for numerical reckoning fixed points of Berinde mappings with convergence analysis","volume":"9","author":"Sintunavarat","year":"2016","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_13","first-page":"97","article-title":"Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators","volume":"2","author":"Berinde","year":"2004","journal-title":"Fixed Point Theory Appl."},{"key":"ref_14","first-page":"105","article-title":"A strong convergence theorem for the split common fixed-point problem of demicontractive mappings","volume":"2","author":"Wang","year":"2019","journal-title":"Appl. Set-Valued Anal. Optim."},{"key":"ref_15","first-page":"1","article-title":"Strong convergence of a self-adaptive method for the split feasibility problem","volume":"201","author":"Yao","year":"2013","journal-title":"Fixed Point Theory Appl."},{"key":"ref_16","first-page":"875","article-title":"Mann-type iteration method for solving the split common fixed point problem","volume":"5","author":"Yao","year":"2017","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1080\/02331934.2019.1602772","article-title":"Gradient methods with selection technique for the multiple-sets split feasibility problem","volume":"2","author":"Yao","year":"2020","journal-title":"Optimization"},{"key":"ref_18","first-page":"37","article-title":"Iterative algorithms for the proximal split feasibility problem","volume":"80","author":"Yao","year":"2018","journal-title":"U. Politeh. Buch. Ser. A"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Wang, Y., Chen, J., and Pitea, A. (2020). The split equality fixed point problem of demicontractive operators with numerical example and application. Symmetry, 6.","DOI":"10.3390\/sym12060902"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1007\/s40065-018-0236-2","article-title":"Forward-backward splitting algorithm for fixed point problems and zeros of the sum of monotone operators","volume":"1","author":"Dadashi","year":"2020","journal-title":"Arab. J. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1007\/s11075-019-00688-9","article-title":"A new iteration technique for nonlinear operators as concerns convex programming and feasibility problems","volume":"83","author":"Sahu","year":"2020","journal-title":"Numer. Algorithms"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Goebel, K., and Kirk, W.A. (1990). Topic in Metric Fixed Point Theory, Cambridge University Press.","DOI":"10.1017\/CBO9780511526152"},{"key":"ref_23","unstructured":"Goebel, K., and Reich, S. (1984). Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"595","DOI":"10.1090\/S0002-9904-1967-11761-0","article-title":"Weak convergence of the sequence of succesive approximations for nonexpansive mappings","volume":"73","author":"Opial","year":"1967","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1017\/S0004972700028884","article-title":"Weak and strong convergence of fixed points of asymptotically nonexpansive mappings","volume":"43","author":"Schu","year":"1991","journal-title":"Bull. Austral. Math. Soc."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1090\/S0002-9939-1974-0358451-4","article-title":"Fixed point theorems in uniformly convex Banach spaces","volume":"44","author":"Edelstein","year":"1974","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1090\/S0002-9939-1974-0346608-8","article-title":"Approximating fixed points of nonexpansive mappings","volume":"44","author":"Senter","year":"1974","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_28","first-page":"687","article-title":"A stable iteration procedure for nonexpansive mappings","volume":"33","author":"Harder","year":"1988","journal-title":"Math. Japon"},{"key":"ref_29","unstructured":"Berinde, V. (2002). Iterative Approximation of Fixed Points, Springer."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"6727","DOI":"10.1016\/j.amc.2011.12.038","article-title":"Stability results for Jungck-type iterative processes in convex metric spaces","volume":"218","author":"Olatinwo","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"727","DOI":"10.1090\/S0002-9939-1991-1086345-8","article-title":"Fixed point iteration for local strictly pseudocontractive mapping","volume":"113","author":"Weng","year":"1991","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_32","first-page":"79","article-title":"Data dependency of fixed points set of weakly Picard operators","volume":"43","author":"Rus","year":"1998","journal-title":"Stud. Univ. Babe\u015f-Bolyai Math."},{"key":"ref_33","first-page":"7","article-title":"On the approximation of fixed points of weak contractive mappings","volume":"19","author":"Berinde","year":"2003","journal-title":"Carpathian J. Math."},{"key":"ref_34","first-page":"151","article-title":"Some results on the continuous dependence of the fixed points in normed linear space","volume":"10","author":"Olatinwo","year":"2009","journal-title":"Fixed Point Theory Appl."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Usurelu, G.I., Bejenaru, A., and Postolache, M. (2020). Newton-like methods and polynomiographic visualization of modified Thakur processes. Int. J. Comput. Math.","DOI":"10.1080\/00207160.2020.1802017"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"242916","DOI":"10.1155\/2008\/242916","article-title":"Data dependence for Ishikawa iteration when dealing with contrative like operators","volume":"2008","author":"Soltuz","year":"2008","journal-title":"Fixed Point Theory Appl."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/10\/1676\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:20:52Z","timestamp":1760178052000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/10\/1676"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,14]]},"references-count":36,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2020,10]]}},"alternative-id":["sym12101676"],"URL":"https:\/\/doi.org\/10.3390\/sym12101676","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,10,14]]}}}