{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,10]],"date-time":"2026-07-10T20:53:35Z","timestamp":1783716815241,"version":"3.55.0"},"reference-count":38,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,7,3]],"date-time":"2021-07-03T00:00:00Z","timestamp":1625270400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this work, we introduce a new inertial accelerated Mann algorithm for finding a point in the set of fixed points of asymptotically nonexpansive mapping in a real uniformly convex Banach space. We also establish weak and strong convergence theorems of the scheme. Finally, we give a numerical experiment to validate the performance of our algorithm and compare with some existing methods. Our results generalize and improve some recent results in the literature.<\/jats:p>","DOI":"10.3390\/axioms10030147","type":"journal-article","created":{"date-parts":[[2021,7,4]],"date-time":"2021-07-04T22:34:36Z","timestamp":1625438076000},"page":"147","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Inertial Accelerated Algorithm for Fixed Point of Asymptotically Nonexpansive Mapping in Real Uniformly Convex Banach Spaces"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6298-9171","authenticated-orcid":false,"given":"Murtala Haruna","family":"Harbau","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, Bayero University, Kano, Nigeria"},{"name":"Department of Science and Technology Education, Bayero University, Kano, Nigeria"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Godwin Chidi","family":"Ugwunnadi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science and Engineering, University of Eswatini, Private Bag 4, Kwaluseni M201, Eswatini"},{"name":"Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, P.O. Box 94, Medunsa, Pretoria 0204, South Africa"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4838-7465","authenticated-orcid":false,"given":"Lateef Olakunle","family":"Jolaoso","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, P.O. Box 94, Medunsa, Pretoria 0204, South Africa"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ahmad","family":"Abdulwahab","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Federal College of Education, Katsina, Nigeria"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1090\/S0002-9939-1972-0298500-3","article-title":"A fixed point theorem for asymptotically nonexpansive mappings","volume":"35","author":"Goebel","year":"1972","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Ali, B., and Harbau, M.H. (2019). Covergence theorems for pseudomonotone equilibrium problems, split feasibility problems and multivalued strictly pseudocontractive mappings. Numer. Funct. Anal. Opt., 40.","DOI":"10.1080\/01630563.2019.1599014"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1407","DOI":"10.1109\/TSP.2018.2890065","article-title":"Rational optimization for nonlinear reconstruction with approximate \u21130 penalization","volume":"67","author":"Castella","year":"2019","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"645","DOI":"10.1007\/s10107-016-1044-0","article-title":"Asynchronous block-iterative primal-dual decomposition methods for monotone inclusions","volume":"B168","author":"Combettes","year":"2018","journal-title":"Math. Program."},{"key":"ref_5","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_6","doi-asserted-by":"crossref","first-page":"2021","DOI":"10.1088\/0266-5611\/22\/6\/007","article-title":"A variable Krasnosel\u2019skii-Mann algorithm and the multiple-set split feasibility problem","volume":"22","author":"Xu","year":"2006","journal-title":"Inverse Probl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"869","DOI":"10.1007\/s11075-016-0121-z","article-title":"Iterative algorithms for solving variational inequalities and fixed point problems for asymptotically nonexpansive mappings in Banach spaces","volume":"73","author":"Cai","year":"2016","journal-title":"Numer. Algorithms"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"957","DOI":"10.1090\/S0002-9904-1967-11864-0","article-title":"Fixed points of nonexpansive maps","volume":"73","author":"Halpern","year":"1967","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_9","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_10","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. Pures Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"733","DOI":"10.1090\/S0002-9939-1994-1203993-5","article-title":"Fixed Point Iteration Proccesses for Asymtotically Nonnexpansive Mappings","volume":"122","author":"Tan","year":"1994","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1090\/S0002-9939-1978-0493543-4","article-title":"Weak convergence to the fixed point of an asymptotically nonexpansive map","volume":"68","author":"Bose","year":"1978","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_13","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_14","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1016\/0022-247X(91)90245-U","article-title":"Iterative construction of fixed points of asymptotically nonexpansive mappings","volume":"158","author":"Schu","year":"1991","journal-title":"J. Math. Anal. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1181","DOI":"10.1016\/S0895-7177(00)00199-0","article-title":"Weak and Strong Convergence Theorems for Fixed Points of Asymtotically Nonexpansive Mappings","volume":"32","author":"Osilike","year":"2000","journal-title":"Math. Comput. Model."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Dong, Q.L., and Yuan, H.B. (2015). Accelerated Mann and CQ algorithms for finding a fixed point of a nonexpansive mapping. Fixed Point Theory Appl., 125.","DOI":"10.1186\/s13663-015-0374-6"},{"key":"ref_17","unstructured":"Nocedal, J., and Wright, S.J. (2006). Numerical Optimization, Springer. [2nd ed.]."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"232","DOI":"10.1137\/130910294","article-title":"A dynamical approach to an inertial forward\u2013backward algorithm for convex minimization","volume":"24","author":"Attouch","year":"2014","journal-title":"SIAM J. Optim."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1142\/S0219199700000025","article-title":"The heavy ball with friction. I. The continuous dynamical system","volume":"2","author":"Attouch","year":"2000","journal-title":"Commun. Contemp. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1824","DOI":"10.1137\/15M1046095","article-title":"The rate of convergence of Nesterov\u2019s accelarated forward-backward method is actually faster than 1k2","volume":"26","author":"Attouch","year":"2016","journal-title":"SIAM J. Optim."},{"key":"ref_21","first-page":"472","article-title":"Inertial Douglas-Rachford splitting for monotone inclusion problems","volume":"256","author":"Bot","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1007\/s11590-016-1102-9","article-title":"Modified inertial Mann algorithm and inertial CQ-algorithm for nonexpansive mappings","volume":"12","author":"Dong","year":"2016","journal-title":"Optim. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1007\/s11784-019-0699-6","article-title":"Inertial Krasnol\u2019skii-Mann type hybrid algorithms for solving hierarchical fixed point problems","volume":"21","author":"Dong","year":"2019","journal-title":"J. Fixed Point Theory Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1007\/s10851-014-0523-2","article-title":"An inertial forward-backward algorithm for monotone inclusions","volume":"51","author":"Lorenz","year":"2015","journal-title":"J. Math. Imaging Vis."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0041-5553(64)90137-5","article-title":"Some methods of speeding up the convergence of iteration methods","volume":"4","author":"Polyak","year":"1964","journal-title":"USSR Comput. Math. Math. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Shehu, Y., and Gibali, A. (2020). Inertial Krasnol\u2019skii-Mann method in Banach spaces. Mathematics, 8.","DOI":"10.3390\/math8040638"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1007\/BF01109805","article-title":"Convergence theorems for sequence of nonlinear mappings in Hilbert spaces","volume":"100","author":"Browder","year":"1967","journal-title":"Math. Z."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1090\/S0002-9904-1967-11761-0","article-title":"Weak convergence of the sequence of successive approximations for nonexpansive mappings","volume":"73","author":"Opial","year":"1967","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1007\/BF00282247","article-title":"Fixed point theorems for nonlinear semicontractive mappings in Banach spaces","volume":"21","author":"Browder","year":"1966","journal-title":"Arch. Ration. Mech. Anal."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"565","DOI":"10.2140\/pjm.1972.40.565","article-title":"Some geometric properties related to the fixed point theory for nonexpansive mappings","volume":"40","author":"Gossez","year":"1972","journal-title":"Pac. J. Math."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1127","DOI":"10.1016\/0362-546X(91)90200-K","article-title":"Inequalities in Banach Spaces with Applications","volume":"16","author":"Xu","year":"1991","journal-title":"Nonlinear Anal."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1137\/080716542","article-title":"A fast iterative shrinkage thresholding algorithm for linear inverse problem","volume":"2","author":"Beck","year":"2009","journal-title":"SIAM J. Imaging Sci."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Agarwal, R.P., Regan, D.O., and Sahu, D.R. (2009). Fixed Point Theory for Lipschitzian-Type Mappings with Applications, Springer.","DOI":"10.1155\/2009\/439176"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Pan, C., and Wang, Y. (2019). Generalized viscosity implicit iterative process for asymptotically non-expansive mappings in Banach spaces. Mathematics, 7.","DOI":"10.3390\/math7050379"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1217","DOI":"10.1007\/s11075-019-00798-4","article-title":"Generalized viscosity implicit schemewith Meir-Keeler contraction for asymptotically nonexpansive mapping in Banach spaces","volume":"84","author":"Vaish","year":"2020","journal-title":"Numer. Algorithms"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"586","DOI":"10.1109\/JSTSP.2007.910281","article-title":"Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems","volume":"1","author":"Figueiredo","year":"2007","journal-title":"IEEE J. Sel. Top. Signal Process."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11784-019-0684-0","article-title":"A self-adaptive projection method with an inertial technique for split feasibility problems in Banach spaces with applications to image restoration problems","volume":"21","author":"Shehu","year":"2019","journal-title":"J. Fixed Point Theory Appl."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1321","DOI":"10.1007\/s11075-019-00727-5","article-title":"Iterative method with inertial terms for nonexpansive mappings, Applications to compressed sensing","volume":"83","author":"Shehu","year":"2020","journal-title":"Numer. Algorithms"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/3\/147\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:25:29Z","timestamp":1760163929000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/3\/147"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,3]]},"references-count":38,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2021,9]]}},"alternative-id":["axioms10030147"],"URL":"https:\/\/doi.org\/10.3390\/axioms10030147","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,7,3]]}}}