{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T11:35:38Z","timestamp":1771673738306,"version":"3.50.1"},"reference-count":15,"publisher":"Wiley","license":[{"start":{"date-parts":[[2022,4,22]],"date-time":"2022-04-22T00:00:00Z","timestamp":1650585600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100009694","name":"Bu-Ali Sina University","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100009694","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Problems in Engineering"],"published-print":{"date-parts":[[2022,4,22]]},"abstract":"<jats:p>This article studies the existence theory of an innovation type of generalized proportional fractional differential equations with the assistance of the technique of Kuratowski measure on noncompactness combined with the fixed-point theorem of M\u00f6nch. Also, we use Lebesgue\u2019s dominated convergence theorem and Arzel\u00e1\u2013Ascoli fixed point theorem on existence and uniqueness results. An application is also presented by employing two illustrative iks, which enrich our outcomes.<\/jats:p>","DOI":"10.1155\/2022\/7018170","type":"journal-article","created":{"date-parts":[[2022,4,22]],"date-time":"2022-04-22T23:05:10Z","timestamp":1650668710000},"page":"1-21","source":"Crossref","is-referenced-by-count":10,"title":["Investigation of the Generalized Proportional Langevin and Sturm\u2013Liouville Fractional Differential Equations via Variable Coefficients and Antiperiodic Boundary Conditions with a Control Theory Application Arising from Complex Networks"],"prefix":"10.1155","volume":"2022","author":[{"given":"Abdellatif","family":"Boutiara","sequence":"first","affiliation":[{"name":"Laboratory of Mathematics and Applied Sciences, University of Ghardaia, Bounoura 47000, Algeria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2260-0341","authenticated-orcid":true,"given":"Mohammed K. 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