{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:06:07Z","timestamp":1760058367309,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,2]],"date-time":"2025-04-02T00:00:00Z","timestamp":1743552000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Umm Al-Qura University, Saudi Arabia","award":["25UQU4331214GSSR01"],"award-info":[{"award-number":["25UQU4331214GSSR01"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The aims of this paper are (a) to introduce the concept of the 0-complete m-metric spaces, (b) to obtain the results for mw-Caristi mapping using Kirk\u2019s approach, (c) to investigate the problem of non-cooperative equilibrium (abbreviated as NCE) in two- and three-person games in the structure of game theory and find the solution by employing coupled and tripled fixed-point results within the framework of 0-complete m-metric spaces (m-metric spaces, respectively), and (d) to establish some coupled fixed-point results which extend the scope of metric fixed point theory. We provide some examples to support the concepts and results presented in this paper. As an application of our results in this paper, we obtain the existence of a solution for a nonlinear integral equation.<\/jats:p>","DOI":"10.3390\/sym17040544","type":"journal-article","created":{"date-parts":[[2025,4,2]],"date-time":"2025-04-02T16:19:16Z","timestamp":1743610756000},"page":"544","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Solution to the Non-Cooperative Equilibrium Problem for Two and Three Players Using the Fixed-Point Technique"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0009-0005-0738-0186","authenticated-orcid":false,"given":"Muhammad","family":"Tariq","sequence":"first","affiliation":[{"name":"Department of Mathematics, MY University, Islamabad 44000, Pakistan"}]},{"given":"Sabeur","family":"Mansour","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Sciences, Umm Al-Qura University, Mecca 21955, Saudi Arabia"}]},{"given":"Mujahid","family":"Abbas","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering Sciences, Faculty of Engineering and the Built Environment, Doornfontein Campus, University of Johannesburg, Johannesburg 2006, South Africa"}]},{"given":"Abdullah","family":"Assiry","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Sciences, Umm Al-Qura University, Mecca 21955, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"133","DOI":"10.4064\/fm-3-1-133-181","article-title":"Sur les operations dans les ensembles abstraits et leurs applications aux equations integrales","volume":"3","author":"Banach","year":"1922","journal-title":"Fund. 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