{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:34:47Z","timestamp":1760236487712,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2021,11,25]],"date-time":"2021-11-25T00:00:00Z","timestamp":1637798400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Thailand Science Research and Innovation Fund, King Mongkut\u2019s University of Technology North Bangkok","award":["KMUTNB-FF-65-13"],"award-info":[{"award-number":["KMUTNB-FF-65-13"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The goal of this study was to show how a modified variational inclusion problem can be solved based on Tseng\u2019s method. In this study, we propose a modified Tseng\u2019s method and increase the reliability of the proposed method. This method is to modify the relaxed inertial Tseng\u2019s method by using certain conditions and the parallel technique. We also prove a weak convergence theorem under appropriate assumptions and some symmetry properties and then provide numerical experiments to demonstrate the convergence behavior of the proposed method. Moreover, the proposed method is used for image restoration technology, which takes a corrupt\/noisy image and estimates the clean, original image. Finally, we show the signal-to-noise ratio (SNR) to guarantee image quality.<\/jats:p>","DOI":"10.3390\/sym13122250","type":"journal-article","created":{"date-parts":[[2021,12,1]],"date-time":"2021-12-01T03:12:40Z","timestamp":1638328360000},"page":"2250","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["A Modified Tseng\u2019s Method for Solving the Modified Variational Inclusion Problems and Its Applications"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1240-1181","authenticated-orcid":false,"given":"Thidaporn","family":"Seangwattana","sequence":"first","affiliation":[{"name":"Faculty of Science Energy and Environment, King Mongkut\u2019s University of Technology North Bangkok, Rayong Campus (KMUTNB), Rayong 21120, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8775-4790","authenticated-orcid":false,"given":"Kamonrat","family":"Sombut","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi (RMUTT), Pathum Thani 12110, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9569-4057","authenticated-orcid":false,"given":"Areerat","family":"Arunchai","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Faculty of Science and Technology, Nakhon Sawan Rajabhat University, Nakhon Sawan 60000, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8496-7803","authenticated-orcid":false,"given":"Kanokwan","family":"Sitthithakerngkiet","sequence":"additional","affiliation":[{"name":"Intelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut\u2019s University of Technology North Bangkok (KMUTNB), Wongsawang, Bangsue, Bangkok 10587, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,25]]},"reference":[{"unstructured":"Baiocchi, C. (1984). Variational and Quasivariational Inequalities. Applications to Free-Boundary Problems, Springer.","key":"ref_1"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1088\/0266-5611\/20\/1\/006","article-title":"A unified treatment of some iterative algorithms in signal processing and image reconstruction","volume":"20","author":"Byrne","year":"2004","journal-title":"Inverse Probl."},{"key":"ref_3","first-page":"258","article-title":"Application of Khobotov\u2019s algorithm to varaitional inequalities and network equilibrium problems","volume":"29","author":"Marcotte","year":"1991","journal-title":"INFOR Inf. Syst. Oper. Res."},{"doi-asserted-by":"crossref","unstructured":"Hanjing, A., and Suantai, S. (2020). A fast image restoration algorithm based on a fixed point and optimization method. Mathematics, 8.","key":"ref_4","DOI":"10.3390\/math8030378"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1007\/s10092-018-0292-1","article-title":"Tseng type method for solving inclusion problems and its applications","volume":"55","author":"Gibali","year":"2018","journal-title":"Calcolo"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1007\/s40314-019-0855-z","article-title":"Strong convergence of a forward-backward splitting method with a new step size for solving monotone inclusions","volume":"38","author":"Thong","year":"2019","journal-title":"Comput. Appl. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1016\/0041-5553(87)90058-9","article-title":"Modification of the extra-gradient method for solving variational inequalities and certain optimization problems","volume":"27","author":"Khobotov","year":"1987","journal-title":"USSR Comput. Math. Math. Phys."},{"key":"ref_8","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":"1998","journal-title":"SIAM J. Control Optim."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"964","DOI":"10.1137\/0716071","article-title":"Splitting algorithms for the sum of two nonliner operators","volume":"16","author":"Lions","year":"1979","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_10","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."},{"doi-asserted-by":"crossref","unstructured":"Bauschke, H.H., and Combettes, P.L. (2011). Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer.","key":"ref_11","DOI":"10.1007\/978-1-4419-9467-7"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1007\/BF01581204","article-title":"On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators","volume":"55","author":"Eckstein","year":"1992","journal-title":"Math. Program."},{"key":"ref_13","first-page":"543","article-title":"A method for unconstrained convex minimization problem with the rate of convergence O(1k2)","volume":"269","author":"Nesterov","year":"1983","journal-title":"Doklady Ussr."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1102","DOI":"10.1137\/S0363012998335802","article-title":"On the minimizing property of a second order dissipative system in Hilbert spaces","volume":"38","author":"Alvarez","year":"2000","journal-title":"SIAM J. Control Optim."},{"key":"ref_15","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."},{"doi-asserted-by":"crossref","unstructured":"Abubarkar, J., Kumam, P., Rehman, H., and Ibrahim, A.H. (2020). Inertial iterative schemes with variable step sizes for variational inequality problem involving pseudo monotone operator. Mathematics, 8.","key":"ref_16","DOI":"10.3390\/math8040609"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"93","DOI":"10.24193\/fpt-ro.2020.1.07","article-title":"A Modified inertial subgradient extragradient method for solving pseudomonotone variational inequalities and common fixed point problems","volume":"21","author":"Ceng","year":"2020","journal-title":"Fixed Point Theory"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1337","DOI":"10.1080\/02331934.2020.1858832","article-title":"Two inertial subgradient extragradient algorithms for variational inequalities with fixed-point constraints","volume":"70","author":"Ceng","year":"2021","journal-title":"Optimization"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1080\/01630563.2020.1867866","article-title":"Quasi-inertial Tseng\u2019s extragradient algorithms for pseudomonotone variational inequalities and fixed point problems of quasi-nonexpansive operators","volume":"42","author":"Zhao","year":"2020","journal-title":"Numer. Funct. Anal. Optim."},{"key":"ref_20","first-page":"1","article-title":"Strong convergence for monotone bilevel equilibria with constraints of variational inequalities and fixed points using subgradient extragradient implicit rule","volume":"146","author":"He","year":"2021","journal-title":"J. Inequal. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1007\/s10107-019-01412-0","article-title":"Convergence of a relaxed inertial proximal algorithm for maximally monotone operators","volume":"184","author":"Attouch","year":"2020","journal-title":"Math. Program."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1281","DOI":"10.1080\/02331934.2019.1696337","article-title":"Convergence rate of a relaxed inertial proximal algorithm for convex minimization","volume":"69","author":"Attouch","year":"2019","journal-title":"Optimization"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"547","DOI":"10.1007\/s00245-019-09584-z","article-title":"Convergence of a relaxed inertial forward-backward algorithm for structured monotone inclusions","volume":"80","author":"Attouch","year":"2019","journal-title":"Appl. Math. Optim."},{"key":"ref_24","first-page":"472","article-title":"Inertial Douglas-Rachford splitting for monotone inclusion problems","volume":"256","author":"Bot","year":"2015","journal-title":"Appl. Math. Comput."},{"doi-asserted-by":"crossref","unstructured":"Oyewole, O.K., Abass, H.A., Mebawondu, A.A., and Aremu, K.O. (2021). A Tseng extragradient method for solving variational inequality problems in Banach spaces. Numer. Algor., 1\u201321.","key":"ref_25","DOI":"10.1007\/s11075-021-01133-6"},{"doi-asserted-by":"crossref","unstructured":"Abubakar, A., Kumam, P., Ibrahim, A.H., and Padcharoen, A. (2020). Relaxed inertial Tseng\u2019s type method for solving the inclusion problem with application to image restoration. Mathematics, 8.","key":"ref_26","DOI":"10.3390\/math8050818"},{"doi-asserted-by":"crossref","unstructured":"Khuangsatung, W., and Kangtunyakarn, A. (2014). Algorithm of a new variational inclusion problem and strictly pseudononspreading mapping with application. Fixed Point Theory Appl., 209.","key":"ref_27","DOI":"10.1186\/1687-1812-2014-209"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1172","DOI":"10.1080\/00036811.2017.1307965","article-title":"A theorem of variational inclusion problems and various nonlinear mappings","volume":"97","author":"Khuangsatung","year":"2018","journal-title":"Appl. Anal."},{"key":"ref_29","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_30","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1007\/s10013-015-0129-z","article-title":"Parallel hybrid iterative methods for variational inequalities, equilibrium problems and common fixed point problems","volume":"44","author":"Anh","year":"2016","journal-title":"Vietnam J. Math."},{"key":"ref_31","first-page":"351","article-title":"Strong convergence analysis of common variational inclusion problems involving an inertial parallel monotone hybrid method for a novel application to image restoration","volume":"114","author":"Cholamjiak","year":"2020","journal-title":"Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. Mat."},{"unstructured":"Takahashi, W. (2000). Nonlinear Function Analysis, Yokohama Publishers.","key":"ref_32"},{"key":"ref_33","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_34","doi-asserted-by":"crossref","first-page":"722","DOI":"10.1016\/j.jmaa.2005.08.076","article-title":"Strong convergence theorem for uniformly L-Lipschitzian asymptotically pseudo contractive mapping in real Banach space","volume":"321","author":"Ofoedu","year":"2006","journal-title":"J. Math. Anal. Appl."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/12\/2250\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:35:45Z","timestamp":1760168145000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/12\/2250"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,25]]},"references-count":34,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2021,12]]}},"alternative-id":["sym13122250"],"URL":"https:\/\/doi.org\/10.3390\/sym13122250","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,11,25]]}}}