{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,17]],"date-time":"2025-12-17T18:13:08Z","timestamp":1765995188580,"version":"build-2065373602"},"reference-count":44,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,7,11]],"date-time":"2023-07-11T00:00:00Z","timestamp":1689033600000},"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 study, we developed a new faster iterative scheme for approximate fixed points. This technique was applied to discuss some convergence and stability results for almost contraction mapping in a Banach space and for Suzuki generalized nonexpansive mapping in a uniformly convex Banach space. Moreover, some numerical experiments were investigated to illustrate the behavior and efficacy of our iterative scheme. The proposed method converges faster than symmetrical iterations of the S algorithm, Thakur algorithm and K* algorithm. Eventually, as an application, the nonlinear Volterra integral equation with delay was solved using the suggested method.<\/jats:p>","DOI":"10.3390\/sym15071400","type":"journal-article","created":{"date-parts":[[2023,7,12]],"date-time":"2023-07-12T00:50:41Z","timestamp":1689123041000},"page":"1400","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Fixed-Point Estimation by Iterative Strategies and Stability Analysis with Applications"],"prefix":"10.3390","volume":"15","author":[{"given":"Hasanen A.","family":"Hammad","sequence":"first","affiliation":[{"name":"Department of Mathematics, Unaizah College of Sciences and Arts, Qassim University, Buraydah 52571, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1213-5549","authenticated-orcid":false,"given":"Doha A.","family":"Kattan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Sciences and Art, King Abdulaziz University, Rabigh 25712, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,7,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Berinde, V. (2007). Iterative Approximation of Fixed Points, Springer.","DOI":"10.1109\/SYNASC.2007.49"},{"key":"ref_2","first-page":"1","article-title":"Applying new fixed point theorems on fractional and ordinary differential equations","volume":"2019","author":"Abdeljawad","year":"2019","journal-title":"Adv.Differ. Equ."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/1687-1812-2013-69","article-title":"A Picard-Mann hybrid iterative process","volume":"2013","author":"Khan","year":"2013","journal-title":"Fixed Point Theory Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"3006","DOI":"10.1016\/j.cam.2010.12.022","article-title":"On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval","volume":"235","author":"Phuengrattana","year":"2011","journal-title":"J. Comput. Appl. Math."},{"key":"ref_5","first-page":"510","article-title":"A general iterative method for approximation of fixed points and their applications","volume":"3","author":"Karahan","year":"2013","journal-title":"Adv. Fixed Point Theory"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"345","DOI":"10.4236\/ajcm.2012.24048","article-title":"Strong convergence of a new three step iterative scheme in Banach spaces","volume":"2","author":"Chugh","year":"2012","journal-title":"Am. J. Comp. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"292","DOI":"10.1007\/BF01304884","article-title":"Fixed point theorems in metric spaces","volume":"23","author":"Zamfirescu","year":"1972","journal-title":"Arch. Math."},{"key":"ref_8","first-page":"7","article-title":"On the approximation of fixed points of weak contractive mapping","volume":"19","author":"Berinde","year":"2003","journal-title":"Carpath. J. Math."},{"key":"ref_9","first-page":"155","article-title":"On the stability of Picard and Mann iteration processes","volume":"19","author":"Imoru","year":"2003","journal-title":"Carpath. J. Math."},{"key":"ref_10","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_11","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_12","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_13","first-page":"61","article-title":"Iterative construction of fixed points of nearly asymptotically nonexpansive mappings","volume":"8","author":"Agarwal","year":"2007","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_14","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. Vesn."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"333","DOI":"10.18576\/isl\/100214","article-title":"A novel four-step iterative scheme for approximating the fixed point with a supportive application","volume":"10","author":"Hammad","year":"2021","journal-title":"Inf. Sci. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Hammad, H.A., Rehman, H.U., and De la Sen, M. (2022). A New four-step iterative procedure for approximating fixed points with Application to 2D Volterra integral equations. Mathematics, 10.","DOI":"10.3390\/math10224257"},{"key":"ref_17","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_18","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_19","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_20","doi-asserted-by":"crossref","first-page":"625","DOI":"10.24193\/fpt-ro.2017.2.50","article-title":"Some fixed point results for a new three steps iteration process in Banach spaces","volume":"18","author":"Karakaya","year":"2017","journal-title":"Fixed Point Theory"},{"key":"ref_21","first-page":"45","article-title":"Application of fixed point method for solving Volterra-Hammerstein integral equation","volume":"74","author":"Maleknejad","year":"2012","journal-title":"U.P.B. Sci. Bull. Ser. A"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Hammad, H.A., Agarwal, P., Momani, S., and Alsharari, F. (2021). Solving a fractional-Order differential equation using rational symmetric contraction mappings. Fractal Fract., 5.","DOI":"10.3390\/fractalfract5040159"},{"key":"ref_23","first-page":"405","article-title":"A reliable treatment for mixed Volterra-Fredholm integral equations","volume":"127","author":"Wazwaz","year":"2002","journal-title":"Appl. Math. Comput."},{"key":"ref_24","first-page":"675","article-title":"Iterative solution of functional Volterra-Fredholm integral equation with deviating argument","volume":"18","author":"Atlan","year":"2017","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1186\/s13661-022-01684-0","article-title":"Solving systems of coupled nonlinear Atangana\u2013Baleanu-type fractional differential equations","volume":"2022","author":"Hammad","year":"2022","journal-title":"Bound Value Probl."},{"key":"ref_26","first-page":"79","article-title":"An efficient iterative method and its applications to a nonlinear integral equation and a delay differential equation in Banach spaces","volume":"4","author":"Ofem","year":"2020","journal-title":"Turk. J. Ineq."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1","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":"Comput. Appl. Math."},{"key":"ref_28","first-page":"111","article-title":"On modified Picard-S-AK hybrid iterative algorithm for approximating fixed point of Banach contraction map","volume":"4","author":"Hudson","year":"2019","journal-title":"MathLAB J."},{"key":"ref_29","first-page":"97","article-title":"Picard iteration converges faster than Mann iteration for a class of quasicontractive operators","volume":"2","author":"Berinde","year":"2004","journal-title":"Fixed Point Theory Appl."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1090\/S0002-9939-1974-0346608-8","article-title":"Approximating fixed points of nonexpansive mapping","volume":"44","author":"Senter","year":"1974","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_31","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. Math."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"242916","DOI":"10.1155\/2008\/242916","article-title":"Data dependence for Ishikawa iteration when dealing with contractive like operators","volume":"2008","author":"Soltuz","year":"2008","journal-title":"Fixed Point Theory Appl."},{"key":"ref_33","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_34","unstructured":"Harder, A.H. (1987). Fixed Point Theory and Stability Results for Fixed-Point Iteration Procedures. [Ph.D. Thesis, University of Missouri-Rolla]."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"8634050","DOI":"10.1155\/2020\/8634050","article-title":"Approximation of the fixed point of multivalued quasi-nonexpansive mappings via a faster iterative process with applications","volume":"2020","author":"Okeke","year":"2020","journal-title":"Discret. Dyn. Nat. Soc."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Konnov, I.V. (2001). Combined Relaxation Methods for Variational Inequalities, Springer.","DOI":"10.1007\/978-3-642-56886-2"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Facchinei, F., and Pang, J.S. (2003). Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer.","DOI":"10.1007\/b97544"},{"key":"ref_38","first-page":"747","article-title":"The extragradient method for finding saddle points and other problems","volume":"12","author":"Korpelevich","year":"1976","journal-title":"Ekon. Mat. Metod."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"2400","DOI":"10.1016\/j.na.2005.08.018","article-title":"Strong convergence of the CQ method for fixed point iteration processes","volume":"64","author":"Xu","year":"2006","journal-title":"Nonlinear Anal."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"7487383","DOI":"10.1155\/2020\/7487383","article-title":"Shrinking projection methods for accelerating relaxed inertial Tseng-type algorithm with applications","volume":"2020","author":"Hammad","year":"2020","journal-title":"Math. Probl. Eng."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"941","DOI":"10.18514\/MMN.2019.2954","article-title":"A modified shrinking projection methods for numerical reckoning fixed points of G-nonexpansive mappings in Hilbert spaces with graph","volume":"20","author":"Hammad","year":"2019","journal-title":"Miskolc Math. Notes"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1669","DOI":"10.1007\/s12215-020-00581-8","article-title":"Effect of shrinking projection and CQ-methods on two inertial forward-backward algorithms for solving variational inclusion problems","volume":"70","author":"Tuyen","year":"2021","journal-title":"Rend. Circ. Mat. Palermo II Ser."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Bauschke, H.H., and Combettes, P.L. (2011). Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer.","DOI":"10.1007\/978-1-4419-9467-7"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1137\/S0036144593251710","article-title":"On projection algorithms for solving convex feasibility problems","volume":"38","author":"Bauschke","year":"1996","journal-title":"SIAM Rev."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/7\/1400\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:10:33Z","timestamp":1760127033000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/7\/1400"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,11]]},"references-count":44,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2023,7]]}},"alternative-id":["sym15071400"],"URL":"https:\/\/doi.org\/10.3390\/sym15071400","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,7,11]]}}}