{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:13:02Z","timestamp":1760145182693,"version":"build-2065373602"},"reference-count":47,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2024,6,27]],"date-time":"2024-06-27T00:00:00Z","timestamp":1719446400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"NNSFs of China","award":["11761052","11862019","2020ZD01","2023KFZD01","NMGIRT2317","JC2019003"],"award-info":[{"award-number":["11761052","11862019","2020ZD01","2023KFZD01","NMGIRT2317","JC2019003"]}]},{"name":"NSF of Inner Mongolia","award":["11761052","11862019","2020ZD01","2023KFZD01","NMGIRT2317","JC2019003"],"award-info":[{"award-number":["11761052","11862019","2020ZD01","2023KFZD01","NMGIRT2317","JC2019003"]}]},{"name":"Key Laboratory of Infinite-Dimensional Hamilton System and Its Algorithm Application (Inner Mongolia Normal University), Ministry of Education","award":["11761052","11862019","2020ZD01","2023KFZD01","NMGIRT2317","JC2019003"],"award-info":[{"award-number":["11761052","11862019","2020ZD01","2023KFZD01","NMGIRT2317","JC2019003"]}]},{"name":"Inner Mongolia Autonomous Region University Innovation Team Development Plan","award":["11761052","11862019","2020ZD01","2023KFZD01","NMGIRT2317","JC2019003"],"award-info":[{"award-number":["11761052","11862019","2020ZD01","2023KFZD01","NMGIRT2317","JC2019003"]}]},{"name":"Inner Mongolia Agricultural University Basic Discipline Research Launch Fund Project","award":["11761052","11862019","2020ZD01","2023KFZD01","NMGIRT2317","JC2019003"],"award-info":[{"award-number":["11761052","11862019","2020ZD01","2023KFZD01","NMGIRT2317","JC2019003"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we give the necessary and sufficient conditions for the existence of general solutions, self-adjoint solutions, and positive solutions to the system of AiXBi+CiYDi+EiZFi=Gi(i=1,2) under additional conditions. In addition, we derive the representation of general solutions to the system of AiXBi+CiYDi+EiZFi=Gi(i=1,2), and provide the matrix representation of the self-adjoint solutions and the positive solutions in the sense of the star order.<\/jats:p>","DOI":"10.3390\/axioms13070435","type":"journal-article","created":{"date-parts":[[2024,6,27]],"date-time":"2024-06-27T13:26:43Z","timestamp":1719494803000},"page":"435","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Existence and Representation of the Solutions to the System of Operator Equations AiXBi + CiYDi + EiZFi = Gi(i = 1, 2)"],"prefix":"10.3390","volume":"13","author":[{"given":"Gen","family":"Che","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China"},{"name":"College of Science, Inner Mongolia Agricultural University, Hohhot 010018, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guojun","family":"Hai","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jiarui","family":"Mei","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiang","family":"Cao","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"581","DOI":"10.1137\/0612044","article-title":"The matrix equation AXB + CYD = E over a principal ideal domain","volume":"12","year":"1991","journal-title":"SIAM J. 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