{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T04:47:44Z","timestamp":1773204464461,"version":"3.50.1"},"reference-count":37,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,5,30]],"date-time":"2025-05-30T00:00:00Z","timestamp":1748563200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004242","name":"Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia","doi-asserted-by":"publisher","award":["(PNURSP2025R174)"],"award-info":[{"award-number":["(PNURSP2025R174)"]}],"id":[{"id":"10.13039\/501100004242","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper investigates the approximation of fixed points for mappings that satisfy the enriched (C) condition using a modified iterative process in a Banach space framework. We first establish a weak convergence result and then derive strong convergence theorems under suitable assumptions. To illustrate the applicability of our findings, we present a numerical example involving mappings that satisfy the enriched (C) condition but not the standard (C) condition. Additionally, numerical computations and graphical representations demonstrate that the proposed iterative process achieves a faster convergence rate compared to several existing methods. As a practical application, we introduce a projection based an iterative process for solving split feasibility problems (SFPs) in a Hilbert space setting. Our findings contribute to the ongoing development of iterative processes for solving optimization and feasibility problems in mathematical and applied sciences.<\/jats:p>","DOI":"10.3390\/axioms14060426","type":"journal-article","created":{"date-parts":[[2025,5,30]],"date-time":"2025-05-30T06:14:27Z","timestamp":1748585667000},"page":"426","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Fixed Point Approximation for Enriched Suzuki Nonexpansive Mappings in Banach Spaces"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2450-3043","authenticated-orcid":false,"given":"Doaa","family":"Filali","sequence":"first","affiliation":[{"name":"Department of Mathematical Science, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0006-9816-1719","authenticated-orcid":false,"given":"Fahad Maqbul","family":"Alamrani","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-0385-2557","authenticated-orcid":false,"given":"Esmail","family":"Alshaban","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-8154-0858","authenticated-orcid":false,"given":"Adel","family":"Alatawi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Amid Yousef","family":"Alanazi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8734-5516","authenticated-orcid":false,"given":"Faizan Ahmad","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1007\/BF01456931","article-title":"\u00dcber Abbildung von Mannigfaltigkeiten","volume":"71","author":"Brouwer","year":"1911","journal-title":"Math. Ann."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"819","DOI":"10.2307\/1968838","article-title":"On the fixed point formula","volume":"38","author":"Lefschetz","year":"1937","journal-title":"Ann. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1112\/plms\/s3-55.1.59","article-title":"The self-duality equations on a Riemann surface","volume":"55","author":"Hitchin","year":"1987","journal-title":"Proc. Lond. Math. Soc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1007\/s00222-003-0286-7","article-title":"Mirror symmetry, Langlands duality, and the Hitchin system","volume":"153","author":"Hausel","year":"2003","journal-title":"Invent. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2485","DOI":"10.2298\/FIL2108485A","article-title":"Common fixed point theorem for modified Kannan enriched contration pair in Banach spaces and its applications","volume":"35","author":"Anjum","year":"2021","journal-title":"Filomat"},{"key":"ref_6","first-page":"549","article-title":"On the strong and \u0394-convergence of S-iteration process for generalized nonexpansive mappings on CAT(0) space","volume":"12","author":"Basarir","year":"2014","journal-title":"Thai J. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"221","DOI":"10.37193\/CJM.2019.02.10","article-title":"Some new results of Miteration process in hyperbolic spaces","volume":"35","author":"Sahin","year":"2019","journal-title":"Carpathian J. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1041","DOI":"10.1073\/pnas.54.4.1041","article-title":"Nonexpansive nonlinear operators in a Banach space","volume":"54","author":"Browder","year":"1965","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1002\/mana.19650300312","article-title":"Zum Prinzip der Kontraktiven Abbildung","volume":"30","author":"Gohde","year":"1965","journal-title":"Math. Nachrichten"},{"key":"ref_10","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_11","doi-asserted-by":"crossref","first-page":"553","DOI":"10.1016\/j.topol.2013.01.006","article-title":"A Suzuki type fixed point theorem for a generalized multivalued mapping on partial Hausdorff metric spaces","volume":"160","author":"Abbas","year":"2013","journal-title":"Topol. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"293","DOI":"10.37193\/CJM.2019.03.04","article-title":"Approximating fixed points of enriched nonexpansive mappings by Krasnoselskij iteration in Hilbert spaces","volume":"35","author":"Berinde","year":"2019","journal-title":"Carpathian J. Math."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Ullah, K., Ahmad, J., Arshad, M., and Ma, Z. (2022). Approximation of fixed points for enriched Suzuki nonexpansive operators with an application in Hilbert spaces. Axioms, 11.","DOI":"10.3390\/axioms11010014"},{"key":"ref_14","first-page":"145","article-title":"M\u00e9moire sur la th\u00e9orie des \u00e9quations aux d\u00e9riv\u00e9es partielles et la m\u00e9thode des approximations successives","volume":"6","author":"Picard","year":"1890","journal-title":"J. Math. Pure Appl."},{"key":"ref_15","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":"Fundam. Math."},{"key":"ref_16","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":"Math. Vesnik"},{"key":"ref_17","first-page":"61","article-title":"Iterative construction of fixed points of nearly asymptotically non-expansive mappings","volume":"8","author":"Agarwal","year":"2007","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"693","DOI":"10.1007\/s00366-020-01172-y","article-title":"Stability and convergence of F iterative scheme with an application to the fractional differential equation","volume":"38","author":"Ali","year":"2020","journal-title":"Eng. Comput."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Alshehri, M.G., Khan, F.A., and Ali, F. (2022). An iterative algorithm to approximate fixed points of non-linear operators with an application. Mathematics, 10.","DOI":"10.3390\/math10071132"},{"key":"ref_20","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_21","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_22","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_23","doi-asserted-by":"crossref","first-page":"667","DOI":"10.1007\/s40314-020-01316-2","article-title":"Convergence, stability and data dependence of a new iterative algorithm with an application","volume":"39","author":"Ali","year":"2020","journal-title":"Comp. Appl. Math."},{"key":"ref_24","first-page":"147","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_25","doi-asserted-by":"crossref","first-page":"20247","DOI":"10.3934\/math.20221108","article-title":"On the convergence of an iterative process for enriched Suzuki nonexpansive mappings in Banach spaces","volume":"7","author":"Abdeljawad","year":"2022","journal-title":"AIMS Math."},{"key":"ref_26","unstructured":"Gursoy, F., and Karakaya, V. (2014). A Picard-S hybrid type iteration method for solving a differential equation with retarded argument. arXiv."},{"key":"ref_27","first-page":"2151","article-title":"A new iterative scheme for approximating fixed points with an application to delay differential equation","volume":"21","author":"Ali","year":"2020","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"187","DOI":"10.2298\/FIL1801187U","article-title":"Numerical reckoning fixed points for Suzuki\u2019s generalized nonexpansive mappings via new iteration process","volume":"32","author":"Ullah","year":"2018","journal-title":"Filomat"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Agarwal, R.P., O\u2019Regon, D., and Sahu, D.R. (2009). Fixed Point Theory for Lipschitzian-Type Mappings with Applications, Springer. Topological Fixed Point Theory and Its Applications.","DOI":"10.1155\/2009\/439176"},{"key":"ref_30","unstructured":"Takahashi, W. (2000). Nonlinear Functional Analysis, Yokohoma Publishers."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1090\/S0002-9904-1967-11761-0","article-title":"Weak and strong convergence of the sequence of successive approximations for nonexpansive mappings","volume":"73","author":"Opial","year":"1967","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1090\/S0002-9939-1974-0346608-8","article-title":"Approximating fixed points of non-expansive mappings","volume":"44","author":"Senter","year":"1974","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Ullah, K., Ahmad, J., and Sen, M.D. (2020). Some new results on a three-step iteration process. Axioms, 9.","DOI":"10.3390\/axioms9030110"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1017\/S0004972700028884","article-title":"Weak and strong convergence to fixed points of asymptotically nonexpansive mappings","volume":"43","author":"Schu","year":"1991","journal-title":"Bull. Aust. Math. Soc."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"396","DOI":"10.1090\/S0002-9947-1936-1501880-4","article-title":"Uniformly convex spaces","volume":"40","author":"Clarkson","year":"1936","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1007\/BF02142692","article-title":"A multiprojection algorithm using Bregman projections in a product space","volume":"8","author":"Censor","year":"1994","journal-title":"Numer. Algorithms"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1088\/0266-5611\/20\/1\/006","article-title":"A unified treatment of some iterative algorithms in signal processing and image reconstruction","volume":"20","author":"Byrne","year":"2004","journal-title":"Inverse Prob."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/6\/426\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:43:50Z","timestamp":1760031830000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/6\/426"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,30]]},"references-count":37,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2025,6]]}},"alternative-id":["axioms14060426"],"URL":"https:\/\/doi.org\/10.3390\/axioms14060426","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,5,30]]}}}