{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,19]],"date-time":"2025-12-19T09:50:57Z","timestamp":1766137857824,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2021,12,3]],"date-time":"2021-12-03T00:00:00Z","timestamp":1638489600000},"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>Symmetry plays an important role in solving practical problems of applied science, especially in algorithm innovation. In this paper, we propose what we call the self-adaptive inertial-like proximal point algorithms for solving the split common null point problem, which use a new inertial structure to avoid the traditional convergence condition in general inertial methods and avoid computing the norm of the difference between xn and xn\u22121 before choosing the inertial parameter. In addition, the selection of the step-sizes in the inertial-like proximal point algorithms does not need prior knowledge of operator norms. Numerical experiments are presented to illustrate the performance of the algorithms. The proposed algorithms provide enlightenment for the further development of applied science in order to dig deep into symmetry under the background of technological innovation.<\/jats:p>","DOI":"10.3390\/sym13122316","type":"journal-article","created":{"date-parts":[[2021,12,7]],"date-time":"2021-12-07T02:48:13Z","timestamp":1638845293000},"page":"2316","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["New Self-Adaptive Inertial-like Proximal Point Methods for the Split Common Null Point Problem"],"prefix":"10.3390","volume":"13","author":[{"given":"Yan","family":"Tang","sequence":"first","affiliation":[{"name":"College of Mathematics, Sichuan University, Chengdu 600014, China"}]},{"given":"Yeyu","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Mathematics, Sichuan University, Chengdu 600014, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2150-553X","authenticated-orcid":false,"given":"Aviv","family":"Gibali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, ORT Braude College of Engineering, Karmiel 2161002, Israel"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1007\/BF02142692","article-title":"A multiprojection algorithm using Bregman projections in product space","volume":"8","author":"Censor","year":"1994","journal-title":"Numer. Algorithms"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1007\/s10957-011-9814-6","article-title":"Split monotone variational inclusions","volume":"150","author":"Moudafi","year":"2011","journal-title":"J. Optim. Theory Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1088\/0266-5611\/18\/2\/310","article-title":"Iterative oblique projection onto convex sets and the split feasibility problem","volume":"18","author":"Byrne","year":"2002","journal-title":"Inverse Probl."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Moudafi, A., and Thakur, B.S. (2013). Solving proximal split feasibilty problem without prior knowledge of matrix norms. Optim. Lett., 8.","DOI":"10.1007\/s11590-013-0708-4"},{"key":"ref_5","first-page":"759","article-title":"Weak and strong convergence of algorithms for the split common null point problem","volume":"13","author":"Byrne","year":"2012","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1023\/A:1011253113155","article-title":"An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping","volume":"9","author":"Alvarez","year":"2001","journal-title":"Set-Valued Anal."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"773","DOI":"10.1137\/S1052623403427859","article-title":"Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert spaces","volume":"14","author":"Alvarez","year":"2004","journal-title":"SIAM J. Optim."},{"key":"ref_8","unstructured":"Attouch, H., and Chbani, Z. (2016). Fast inertial dynamics and FISTA algorithms in convex optimization, perturbation aspects. arXiv."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1007\/s10107-016-0992-8","article-title":"Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity","volume":"168","author":"Attouch","year":"2018","journal-title":"Math. Program."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"232","DOI":"10.1137\/130910294","article-title":"A dynamical approach to an inertial forward-backward algorithm for convex minimization","volume":"24","author":"Attouch","year":"2014","journal-title":"SIAM J. Optim."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"478","DOI":"10.1016\/j.chaos.2018.07.032","article-title":"A novel method for a fractional derivative with non-local and non-singular kernel","volume":"114","year":"2018","journal-title":"Chaos Solitons Fractals"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"226","DOI":"10.3934\/math.2020014","article-title":"The existence and uniqueness of solution for linear system of mixed Volterra-Fredholm integral equations in Banach space","volume":"5","author":"Hasan","year":"2020","journal-title":"AIMS Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1140\/epjp\/i2019-12665-3","article-title":"An optimized Steffensen-type iterative method with memory associated with annuity calculation","volume":"134","author":"Khdhr","year":"2019","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1388","DOI":"10.1137\/130942954","article-title":"iPiano: Inertial proximal algorithm for non-convex optimization","volume":"7","author":"Ochs","year":"2014","journal-title":"SIAM J. Imaging Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1007\/s10851-015-0565-0","article-title":"iPiasco: Inertial proximal algorithm for strongly convex optimization","volume":"53","author":"Ochs","year":"2015","journal-title":"J. Math. Imaging Vis."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1383","DOI":"10.3934\/jimo.2016078","article-title":"Inertial accelerated algorithms for solving a split feasibility problem","volume":"13","author":"Dang","year":"2017","journal-title":"J. Ind. Manag. Optim."},{"key":"ref_17","first-page":"889","article-title":"European option valuation under the Bates PIDE in finance: A numerical implementation of the Gaussian scheme","volume":"13","author":"Soleymani","year":"2018","journal-title":"Discret. Contin. Dyn. Syst.-S"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3345","DOI":"10.4995\/agt.2017.7257","article-title":"Convergence theorems for finding the split common null point in Banach spaces","volume":"18","author":"Suantai","year":"2017","journal-title":"Appl. Gen. Topol."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1595","DOI":"10.3934\/jimo.2018023","article-title":"The modified inertial relaxed CQ algorithm for solving the split feasibility problems","volume":"14","author":"Suantai","year":"2017","journal-title":"J. Ind. Manag. Optim."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"687","DOI":"10.1007\/s10898-017-0506-0","article-title":"Inertial projection and contraction algorithms for variational inequalities","volume":"70","author":"Dong","year":"2018","journal-title":"J. Glob. Optim."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"801","DOI":"10.1007\/s11075-017-0462-2","article-title":"Convergence analysis of a general iterative algorithm for finding a common solution of split variational inclusion and optimization problems","volume":"79","author":"Sitthithakerngkiet","year":"2018","journal-title":"Numer Algorithms"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1113","DOI":"10.1007\/s11590-013-0629-2","article-title":"An iterative method for split variational inclusion problem and fixed point problem for a nonexpansive mapping","volume":"8","author":"Kazmi","year":"2014","journal-title":"Optim. Lett."},{"key":"ref_23","first-page":"27","article-title":"Viscosity approximation method for split common null point problems between Banach spaces and Hilbert spaces","volume":"9","author":"Promluang","year":"2017","journal-title":"J. Inform. Math. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1007\/s00009-017-0922-x","article-title":"Split common null point and common fixed point problems between Banach spaces and Hilbert spaces","volume":"14","author":"Ealamian","year":"2017","journal-title":"Mediterr. J. Math."},{"key":"ref_25","first-page":"131","article-title":"Three generalizations of firmly nonexpansive mappings: Their relations and continuity properties","volume":"10","author":"Aoyama","year":"2009","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1112\/S0024610702003332","article-title":"Iterative algorithms for nonliear operators","volume":"66","author":"Xu","year":"2002","journal-title":"J. Lond. Math. Soc."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1016\/j.jmaa.2005.12.066","article-title":"Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces","volume":"325","year":"2007","journal-title":"J. Math. Anal. Appl."},{"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 nonexpansivemappings","volume":"73","author":"Opial","year":"1967","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"899","DOI":"10.1007\/s11228-008-0102-z","article-title":"Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization","volume":"16","year":"2008","journal-title":"Set-Valued Anal."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1007\/s11075-011-9490-5","article-title":"Algorithms for the split variational inequality problem","volume":"59","author":"Censor","year":"2012","journal-title":"Numer. Algorithms"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"192795","DOI":"10.1155\/2013\/192795","article-title":"Deterministic sensing matrices in compressive sensing: A survey","volume":"2013","author":"Nguyen","year":"2013","journal-title":"Sci. World J."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"817","DOI":"10.1007\/s11590-017-1148-3","article-title":"Note on the modified relaxation CQ algorithm for the split feasibility problem","volume":"12","author":"Gibali","year":"2018","journal-title":"Optim. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"739","DOI":"10.1007\/s11075-017-0398-6","article-title":"l1 \u2212 l2 Regularization of split feasibility problems","volume":"78","author":"Moudafi","year":"2018","journal-title":"Numer. Algorithms"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1186\/s13660-019-1971-4","article-title":"New inertial algorithm for solving split common null point problem in Banach spaces","volume":"2019","author":"Tang","year":"2019","journal-title":"J. Inequal. Appl."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/12\/2316\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:39:30Z","timestamp":1760168370000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/12\/2316"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,3]]},"references-count":34,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2021,12]]}},"alternative-id":["sym13122316"],"URL":"https:\/\/doi.org\/10.3390\/sym13122316","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,12,3]]}}}