{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T00:34:41Z","timestamp":1760402081934,"version":"build-2065373602"},"reference-count":9,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,4,14]],"date-time":"2020-04-14T00:00:00Z","timestamp":1586822400000},"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>In this paper, we study the stationary boundary value problem derived from the magnetic (non) insulated regime on a plane diode. Our main goal is to prove the existence of non-negative solutions for that nonlinear singular system of second-order ordinary differential equations. To attain such a goal, we reduce the boundary value problem to a singular system of coupled nonlinear Fredholm integral equations, then we analyze its solvability through the existence of fixed points for the related operators. This system of integral equations is studied by means of Leray-Schauder\u2019s topological degree theory.<\/jats:p>","DOI":"10.3390\/sym12040617","type":"journal-article","created":{"date-parts":[[2020,4,15]],"date-time":"2020-04-15T04:01:46Z","timestamp":1586923306000},"page":"617","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Boundary Value Problem for Noninsulated Magnetic Regime in a Vacuum Diode"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8593-0463","authenticated-orcid":false,"given":"Edixon M.","family":"Rojas","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Universidad Nacional de Colombia, sede Bogot\u00e1 2, Bogot\u00e1, Colombia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9331-1921","authenticated-orcid":false,"given":"Nikolai A.","family":"Sidorov","sequence":"additional","affiliation":[{"name":"Institute of Mathamatics and Information Technologies, Irkutsk State University, Irkutsk 664003, Russia"}]},{"given":"Aleksandr V.","family":"Sinitsyn","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Universidad Nacional de Colombia, sede Bogot\u00e1 2, Bogot\u00e1, Colombia"}]}],"member":"1968","published-online":{"date-parts":[[2020,4,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1103\/RevModPhys.3.191","article-title":"Electrical discharges in gases Part II. Fundamental phenomena in electrical discharges","volume":"3","author":"Langmuir","year":"1931","journal-title":"Rev. Mod. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1522","DOI":"10.1063\/1.872810","article-title":"Mathematical models of magnetic insulation","volume":"5","author":"Abdallah","year":"1998","journal-title":"Phys. Plasmas"},{"key":"ref_3","first-page":"115","article-title":"A numerical modelling of the limit problem for the magnetically noninsulated diode","volume":"162","author":"Dulov","year":"2005","journal-title":"Appl. Math. Comput."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1007\/BF02355910","article-title":"Analysis of bifurcation points and nontrivial branches of solutions to the stationary Vlasov\u2013Maxwell system","volume":"62","author":"Sidorov","year":"1997","journal-title":"Math. Notes"},{"key":"ref_5","unstructured":"Sidorov, N.A., and Sinitsyn, A.V. (1999). On bifurcation points of the stationary Vlasov-Maxwell system with bifurcation direction. ECMI Progress in Industrial Mathematics at ECMI 98 (European Consortium for Mathematics in Industry), Vieweg+Teubner Verlag."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Sidorov, N., Sidorov, D., and Sinitsyn, A. (2020). Towards General Theory of Differential-Operator and Kinetic Models, World Scientific.","DOI":"10.1142\/11651"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1070\/SM1995v186n02ABEH000017","article-title":"Explicit and implicit parametrizations in the construction of branching solutions by iterative methods","volume":"186","author":"Sidorov","year":"1995","journal-title":"Sb. Math."},{"key":"ref_8","first-page":"300","article-title":"A class of degenerate differential equations with convergence","volume":"35","author":"Sidorov","year":"1984","journal-title":"Math. Notes Acad. Sci. USSR"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Deimling, K. (1985). Nonlinear Functional Analysis, Springer.","DOI":"10.1007\/978-3-662-00547-7"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/4\/617\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:21:07Z","timestamp":1760361667000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/4\/617"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,14]]},"references-count":9,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,4]]}},"alternative-id":["sym12040617"],"URL":"https:\/\/doi.org\/10.3390\/sym12040617","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,4,14]]}}}