{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:02:55Z","timestamp":1760144575535,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2024,5,2]],"date-time":"2024-05-02T00:00:00Z","timestamp":1714608000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Faisal University, Saudi Arabia","award":["6187"],"award-info":[{"award-number":["6187"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The concept of symmetry has played a major role in Hilbert space setting owing to the structure of a complete inner product space. Subsequently, different studies pertaining to symmetry, including symmetric operators, have investigated real Hilbert spaces. In this paper, we study the solutions to multiple-set split feasibility problems for a pair of finite families of \u03b2-enriched, strictly pseudocontractive mappings in the setup of a real Hilbert space. In view of this, we constructed an iterative scheme that properly included these two mappings into the formula. Under this iterative scheme, an appropriate condition for the existence of solutions and strong and weak convergent results are presented. No sum condition is imposed on the countably finite family of the iteration parameters in obtaining our results unlike for several other results in this direction. In addition, we prove that a slight modification of our iterative scheme could be applied in studying hierarchical variational inequality problems in a real Hilbert space. Our results improve, extend and generalize several results currently existing in the literature.<\/jats:p>","DOI":"10.3390\/sym16050546","type":"journal-article","created":{"date-parts":[[2024,5,2]],"date-time":"2024-05-02T07:04:14Z","timestamp":1714633454000},"page":"546","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Strong and Weak Convergence Theorems for the Split Feasibility Problem of (\u03b2,k)-Enriched Strict Pseudocontractive Mappings with an Application in Hilbert Spaces"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7579-8972","authenticated-orcid":false,"given":"Asima","family":"Razzaque","sequence":"first","affiliation":[{"name":"Department of Basic Sciences, Preparatory Year, King Faisal University, Al-Ahsa 31982, Saudi Arabia"},{"name":"Department of Mathematics, College of Science, King Faisal University, Al-Ahsa 31982, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1485-6163","authenticated-orcid":false,"given":"Naeem","family":"Saleem","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Management and Technology, Lahore 54770, Pakistan"},{"name":"Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Pretoria 0204, South Africa"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4535-1974","authenticated-orcid":false,"given":"Imo Kalu","family":"Agwu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Michael Okpara University of Agriculture, Umudike 440101, Abia State, Nigeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5228-1073","authenticated-orcid":false,"given":"Umar","family":"Ishtiaq","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Quaid-i-Azam University, Islamabad 45320, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9640-7846","authenticated-orcid":false,"given":"Maggie","family":"Aphane","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Pretoria 0204, South Africa"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Saleem, N., Agwu, I.K., Ishtiaq, U., and Radenovi\u0107, S. (2022). Strong Convergence Theorems for a Finite Family of Enriched Strictly Pseudocontractive Mappings and \u03a6T-Enriched Lipschitizian Mappings Using a New Modified Mixed-Type Ishikawa Iteration Scheme with Error. Symmetry, 14.","DOI":"10.3390\/sym14051032"},{"key":"ref_2","first-page":"137","article-title":"Remarks on asymptotically regularity and fixed points","volume":"4","author":"Gormicki","year":"2003","journal-title":"J. Fixed Point Theory Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1090\/S0002-9939-1976-0412909-X","article-title":"Fixed point iteration of nonexpansive mapping in Banach space","volume":"59","author":"Ishikawa","year":"1976","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"45","DOI":"10.28924\/ada\/ma.1.45","article-title":"Convergence of a three-step iteration scheme to the common fixed points of mixed-type total asymptotically nonexpansive mappings","volume":"1","author":"Igbokwe","year":"2021","journal-title":"Eur. J. Math. Anal."},{"key":"ref_5","first-page":"129","article-title":"Weak and strong convergence theorems for approximating fixed points of nonexpansive mappings using composite hybrid iteration method","volume":"33","author":"Igbokwe","year":"2014","journal-title":"J. Niger. Math. Soc."},{"key":"ref_6","first-page":"137","article-title":"Construction of fixed points of strictly pseudocontractive mappings of Brouwder-Petryshyn-type in arbitrary Banach space","volume":"4","author":"Igbokwe","year":"2003","journal-title":"Adv. Fixed Point Theory Appl."},{"key":"ref_7","first-page":"120","article-title":"Hybrid-type iteration scheme for approximating fixed points of Lipschitz \u03b1-hemicontractive mappings","volume":"5","author":"Agwu","year":"2015","journal-title":"Adv. Fixed Point Theory"},{"key":"ref_8","first-page":"277","article-title":"Approximating fixed points of enriched nonexpansive mappings by Krasnolselkii iteration in Hilbert spaces","volume":"3","author":"Berinde","year":"2019","journal-title":"Carpathian J. Math."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Berinde, V. (2019). Approximating fixed points of enriched quasi nonexpansive mappings and applications. arXiv.","DOI":"10.1007\/s11784-020-0769-9"},{"key":"ref_10","first-page":"13","article-title":"Weak and strong convergence theorems for the Kreasnolselkij iteration algorithm in the class of enriched strictly pseudocontractive operators","volume":"2","author":"Berinde","year":"2018","journal-title":"Analele Univ. Vest Timisoara Ser. Math. Inform."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Jeribi, A., and Krichen, B. (2016). Functional Analysis in Banach Spaces and Banach Algebras Construction of Fixed Points of Strictly Pseudocontractive Mappings of Brouwder-Petryshyn-Type in Arbitrary Banach Space, CRC Press-Taylor and Francis Group.","DOI":"10.1201\/b18790"},{"key":"ref_12","first-page":"67","article-title":"Weak and srtong convergence theorems for the iterative approximation of fixed points of strictly pseudocontractive maps in arbitrary Banach spaces","volume":"5","author":"Igbokwe","year":"2002","journal-title":"J. Inequal. Pure Appl. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2258","DOI":"10.1016\/j.na.2006.08.036","article-title":"Iteration methods for strict pseudocontractions in Hilbert space","volume":"67","author":"Acedo","year":"2007","journal-title":"Nonlinear Anal."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1363","DOI":"10.1090\/S0002-9939-97-03858-6","article-title":"Approximation of fixed points of a strictly pseudocontrative mapping","volume":"125","author":"Liu","year":"1997","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"249","DOI":"10.24193\/fpt-ro.2024.1.16","article-title":"Convergence analysis of a new relaxed algorith with inertial for solving split feaseability problems","volume":"25","author":"Ogbuisi","year":"2024","journal-title":"Fixed Point Theory"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1016\/0022-247X(67)90085-6","article-title":"Construction of fixed points of nonlinear mappings in Hilbert space","volume":"20","author":"Browder","year":"1967","journal-title":"J. Math. Anal. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"10416","DOI":"10.1016\/j.amc.2013.04.020","article-title":"Split feasibility problem for Quasi-nonexpansive multi-valued mappings and total asymptotically strict pseudo contractive mappings","volume":"219","author":"Chang","year":"2013","journal-title":"Appl. Math. Comp."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1186\/1687-1812-2011-77","article-title":"Multiple-set split feasibility problems for total asymptotically strict pseudo contractions mappings","volume":"2011","author":"Yang","year":"2011","journal-title":"Fixed Point Theory Appl."},{"key":"ref_19","first-page":"25212538","article-title":"Strong convergence theorems by hybrid methods for the split feasibility problem in Banach spaces","volume":"16","author":"Alsulami","year":"2015","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_20","first-page":"759775","article-title":"The split common null point problem","volume":"13","author":"Byrne","year":"2012","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_21","first-page":"367371","article-title":"A note on the multiple-set split feasibility problem in Hilbert space","volume":"7","author":"Masad","year":"2008","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_22","first-page":"587600","article-title":"The split common fixed point problem for directed operators","volume":"16","author":"Censor","year":"2009","journal-title":"J. Convex Anal."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Moudafi, A., and Thera, M. (1999). Proximal and Dynamical Approaches to Equilibrium Problems, Springer.","DOI":"10.1007\/978-3-642-45780-7_12"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"055007","DOI":"10.1088\/0266-5611\/26\/5\/055007","article-title":"The split common fixed point problem for demicontractive mappings","volume":"26","author":"Moudafi","year":"2010","journal-title":"J. Inverse Probl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1186\/s13663-015-0324-3","article-title":"Strong convergence theorems by hybrid methods for the split common null point problem in Banach spaces","volume":"205","author":"Takahashi","year":"2015","journal-title":"Fixed Point Theory Appl."},{"key":"ref_26","first-page":"205221","article-title":"On the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings in Banach spaces","volume":"23","author":"Tang","year":"2015","journal-title":"J. Inequal. Appl."},{"key":"ref_27","first-page":"6489","article-title":"A modified halpern-type iteration algorithm for totally quasi-f-asymptotically nonexpansive mappings with applications","volume":"218","author":"Chang","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1186\/1687-1812-2014-11","article-title":"Weak- and strong-convergence theorems of solutions to split feasibility problem for nonspreading type mapping in Hilbert spaces","volume":"2014","author":"Chang","year":"2014","journal-title":"Fixed Point Theory Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"315835","DOI":"10.1155\/2012\/315835","article-title":"Computing the Fixed points of strictly pseudocontractive mappings by the implicit and explicit iterations","volume":"2012","author":"Lion","year":"2012","journal-title":"Abstr. Appl. Anal."},{"key":"ref_30","first-page":"3287968","article-title":"New iteration scheme for approximating a common fixed point od finite family of mappings","volume":"2020","author":"Isogugu","year":"2020","journal-title":"Hindawi J. Math."},{"key":"ref_31","first-page":"3287968","article-title":"On some convergent properties of the modified Ishikawa scheme for asymptotically demicontractive mappings with metricial parameterizing sequences","volume":"2018","year":"2018","journal-title":"J. Math."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"274","DOI":"10.1016\/0022-247X(66)90027-8","article-title":"Construction of fixed points for demicompact mappings in Hilbert space","volume":"14","author":"Petryshyn","year":"1966","journal-title":"J. Math. Anal. Appl."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"126393","DOI":"10.1016\/j.jmaa.2022.126393","article-title":"Shift-like operators on Lp(X)","volume":"515","author":"Darji","year":"2022","journal-title":"J. Math. Anal. Appl."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1007\/s43037-022-00196-4","article-title":"Expansivity and strong structural stability for composition operators on spaces","volume":"16","author":"Maiuriello","year":"2022","journal-title":"Banach J. Math. Anal."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/5\/546\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:38:38Z","timestamp":1760107118000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/5\/546"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,2]]},"references-count":34,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2024,5]]}},"alternative-id":["sym16050546"],"URL":"https:\/\/doi.org\/10.3390\/sym16050546","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,5,2]]}}}