{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:32:53Z","timestamp":1760146373732,"version":"build-2065373602"},"reference-count":41,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2024,10,29]],"date-time":"2024-10-29T00:00:00Z","timestamp":1730160000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Fundamental Fund 2025, Chiang Mai University"},{"name":"Chiang Mai University"},{"name":"Faculty of Science, Chiang Mai University"},{"name":"Centre of Excellence in Mathematics, MHESI"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper introduces a novel parallel method for solving common variational inclusion and common fixed-point (CVI-CFP) problems. The proposed algorithm provides a strong convergence theorem established under specific conditions associated with the CVI-CFP problem. Numerical simulations demonstrate the algorithm\u2019s efficacy in the context of signal recovery problems involving various types of blurred filters. The results highlight the algorithm\u2019s potential for practical applications in image processing and other fields.<\/jats:p>","DOI":"10.3390\/axioms13110740","type":"journal-article","created":{"date-parts":[[2024,10,29]],"date-time":"2024-10-29T06:29:57Z","timestamp":1730183397000},"page":"740","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Refined Iterative Method for a Common Variational Inclusion and Common Fixed-Point Problem with Practical Applications"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9434-5373","authenticated-orcid":false,"given":"Chaiporn","family":"Thangthong","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0790-3811","authenticated-orcid":false,"given":"Raweerote","family":"Suparatulatorn","sequence":"additional","affiliation":[{"name":"Advanced Research Center for Computational Simulation, Chiang Mai University, Chiang Mai 50200, Thailand"},{"name":"Office of Research Administration, Chiang Mai University, Chiang Mai 50200, Thailand"},{"name":"Centre of Excellence in Mathematics, MHESI, Bangkok 10400, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3433-0003","authenticated-orcid":false,"given":"Tanadon","family":"Chaobankoh","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand"},{"name":"Advanced Research Center for Computational Simulation, Chiang Mai University, Chiang Mai 50200, Thailand"},{"name":"Centre of Excellence in Mathematics, MHESI, Bangkok 10400, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3246-673X","authenticated-orcid":false,"given":"Khuanchanok","family":"Chaichana","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand"},{"name":"Advanced Research Center for Computational Simulation, Chiang Mai University, Chiang Mai 50200, Thailand"},{"name":"Centre of Excellence in Mathematics, MHESI, Bangkok 10400, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1007\/s40314-021-01530-6","article-title":"A parallel monotone hybrid algorithm for a finite family of G-nonexpansive mappings in Hilbert spaces endowed with a graph applicable in signal recovery","volume":"40","author":"Suantai","year":"2021","journal-title":"Comput. Appl. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1016\/j.apnum.2021.12.006","article-title":"A strongly convergent algorithm for solving common variational inclusion with application to image recovery problems","volume":"173","author":"Suparatulatorn","year":"2022","journal-title":"Appl. Numer. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"189","DOI":"10.37193\/CJM.2023.01.12","article-title":"A parallel method for common variational inclusion and common fixed point problems with applications","volume":"39","author":"Mouktonglang","year":"2023","journal-title":"Carpathian J. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"964","DOI":"10.1137\/0716071","article-title":"Splitting algorithms for the sum of two nonlinear operators","volume":"16","author":"Lions","year":"1979","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1016\/0022-247X(79)90234-8","article-title":"Ergodic convergence to a zero of the sum of monotone operators in Hilbert space","volume":"72","author":"Passty","year":"1979","journal-title":"J. Math. Anal. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1016\/S0377-0427(02)00906-8","article-title":"Convergence of a splitting inertial proximal method for monotone operators","volume":"55","author":"Moudafi","year":"2003","journal-title":"J. Comput. Appl. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1137\/S0363012998338806","article-title":"A modified forward-backward splitting method for maximal monotone mappings","volume":"38","author":"Tseng","year":"2000","journal-title":"SIAM J. Control. Optim."},{"key":"ref_8","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_9","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1007\/s11784-018-0526-5","article-title":"An inertial forward-backward splitting method for solving inclusion problems in Hilbert spaces","volume":"30","author":"Cholamjiak","year":"2018","journal-title":"J. Fixed Point Theory Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1007\/s10092-018-0292-1","article-title":"Tseng type methods for solving inclusion problems and its applications","volume":"55","author":"Gibali","year":"2018","journal-title":"Calcolo"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"361","DOI":"10.37193\/CJM.2021.03.01","article-title":"An inertial iterative scheme for solving variational inclusion with application to Nash-Cournot equilibrium and image restoration problems","volume":"37","author":"Abubakar","year":"2021","journal-title":"Carpathian J. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1007\/s10915-021-01608-7","article-title":"Relaxed forward-backward splitting methods for solving variational inclusions and applications","volume":"88","author":"Cholamjiak","year":"2021","journal-title":"J. Sci. Comput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"492","DOI":"10.1186\/s13662-021-03647-8","article-title":"A parallel Tseng\u2019s splitting method for solving common variational inclusion applied to signal recovery problems","volume":"2021","author":"Suparatulatorn","year":"2021","journal-title":"Adv. Differ. Equ."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1007\/s13370-013-0210-2","article-title":"Strong convergence theorems of a finite family of quasi-nonexpansive and Lipschitz multi-valued mappings","volume":"26","author":"Suantai","year":"2015","journal-title":"Afr. Mat."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1007\/s12190-014-0801-6","article-title":"Parallel and sequential hybrid methods for a finite family of asymptotically quasi \u03d5-nonexpansive mappings","volume":"48","author":"Anh","year":"2015","journal-title":"J. Appl. Math. Comput."},{"key":"ref_16","first-page":"99","article-title":"Strong convergence analysis of common variational inclusion problems involving an inertial parallel monotone hybrid method for a novel application to image restoration. Revista de la Real Academia de Ciencias Exactas","volume":"114","author":"Cholamjiak","year":"2020","journal-title":"F\u00edsicas Nat. Ser. A Matem\u00e1t."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"4265","DOI":"10.1080\/02331934.2021.1939341","article-title":"Strong convergence theorem for common zero points of inverse strongly monotone mappings and common fixed points of generalized demimetric mappings","volume":"71","author":"Eslamian","year":"2021","journal-title":"Optimization"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"868","DOI":"10.3934\/jimo.2021210","article-title":"A novel algorithm for approximating common solution of a system of monotone inclusion problems and common fixed point problem","volume":"19","author":"Eslamian","year":"2021","journal-title":"J. Ind. Manag. Optim."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Wairojjana, N., Younis, M., Rehman, H.U., Pakkaranang, N., and Pholasa, N. (2020). Modified viscosity subgradient extragradient-like algorithms for solving monotone variational inequalities problems. Axioms, 9.","DOI":"10.3390\/axioms9040118"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1007\/s00009-023-02559-z","article-title":"Fixed points of automorphisms of the vector bundle moduli space over a compact Riemann surface","volume":"21","year":"2024","journal-title":"Mediterr. J. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1137\/080716542","article-title":"A fast iterative shrinkage-thresholding algorithm for linear inverse problems","volume":"2","author":"Beck","year":"2009","journal-title":"SIAM J. Imaging Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1007\/s10092-018-0300-5","article-title":"Iterative method with inertial for variational inequalities in Hilbert spaces","volume":"56","author":"Shehu","year":"2019","journal-title":"Calcolo"},{"key":"ref_23","first-page":"47","article-title":"Inertial hybrid projection methods with selection techniques for split common fixed point problems in Hilbert spaces","volume":"54","author":"Mouktonglang","year":"2022","journal-title":"Univ. Politeh. Buchar. Sci. Bull. Ser. A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"8147","DOI":"10.3934\/math.2022453","article-title":"A modified inertial proximal gradient method for minimization problems and applications","volume":"7","author":"Kesornprom","year":"2022","journal-title":"AIMS Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"112048","DOI":"10.1016\/j.chaos.2022.112048","article-title":"An inertial Mann forward-backward splitting algorithm of variational inclusion problems and its applications","volume":"158","author":"Peeyada","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1007\/s10614-020-10037-x","article-title":"A Generalized Time Iteration Method for Solving Dynamic Optimization Problems with Occasionally Binding Constraints","volume":"58","year":"2021","journal-title":"Comput. Econ."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"117451","DOI":"10.1016\/j.atmosenv.2020.117451","article-title":"Estimating ground-level PM2.5 using micro-satellite images by a convolutional neural network and random forest approach","volume":"230","author":"Zheng","year":"2020","journal-title":"Atmos. Environ."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2057","DOI":"10.1016\/j.procs.2020.04.221","article-title":"Forecasting Air Pollution Particulate Matter (PM2.5) Using Machine Learning Regression Models","volume":"171","author":"Doreswamy","year":"2020","journal-title":"Procedia Comput. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"102067","DOI":"10.1016\/j.ecoinf.2023.102067","article-title":"Spatio-temporal air quality analysis and PM2.5 prediction over Hyderabad City, India using artificial intelligence techniques","volume":"76","author":"Gokul","year":"2023","journal-title":"Ecol. Inform."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2243743","DOI":"10.1080\/23311916.2023.2243743","article-title":"Air quality analysis and PM2.5 modelling using machine learning techniques: A study of Hyderabad city in India","volume":"10","author":"Mathew","year":"2023","journal-title":"Cogent Eng."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1112\/S0024610702003332","article-title":"Iterative algorithms for nonlinear operators","volume":"66","author":"Xu","year":"2002","journal-title":"J. Lond. Math. Soc."},{"key":"ref_32","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_33","doi-asserted-by":"crossref","unstructured":"Goebel, K., and Kirk, W. (1990). Topics in Metric Fixed Point Theory, Cambridge University Press. Cambridge Studies in Advanced Mathematics.","DOI":"10.1017\/CBO9780511526152"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"336","DOI":"10.1016\/j.jmaa.2006.06.055","article-title":"Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces","volume":"329","author":"Marino","year":"2007","journal-title":"J. Math. Anal. Appl."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1007\/BF01418765","article-title":"Nonlinear maximal monotone operators in Banach space","volume":"175","author":"Browder","year":"1968","journal-title":"Math. Ann."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Bauschke, H.H., and Combettes, P.L. (2011). Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics.","DOI":"10.1007\/978-1-4419-9467-7"},{"key":"ref_37","unstructured":"Br\u00e9zis, H. (1973). Op\u00e9rateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, Mathematical Studies."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1387","DOI":"10.1007\/s12190-021-01576-z","article-title":"Viscosity-type inertial extragradient algorithms for solving variational inequality problems and fixed point problems","volume":"68","author":"Tan","year":"2022","journal-title":"J. Appl. Math. Comput."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1007\/s10589-020-00217-8","article-title":"Weak convergence of iterative methods for solving quasimonotone variational inequalities","volume":"77","author":"Liu","year":"2020","journal-title":"Comput. Optim. Appl."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1168","DOI":"10.1137\/050626090","article-title":"Signal recovery by proximal forward-backward splitting","volume":"4","author":"Combettes","year":"2005","journal-title":"Multiscale Model. Simul."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"600","DOI":"10.1109\/TIP.2003.819861","article-title":"Image quality assessment: From error visibility to structural similarity","volume":"13","author":"Wang","year":"2004","journal-title":"IEEE Trans. Image Process."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/11\/740\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:22:46Z","timestamp":1760113366000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/11\/740"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,29]]},"references-count":41,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2024,11]]}},"alternative-id":["axioms13110740"],"URL":"https:\/\/doi.org\/10.3390\/axioms13110740","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,10,29]]}}}