{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T15:04:59Z","timestamp":1773414299700,"version":"3.50.1"},"reference-count":40,"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":"Scientific Research Deanship at University of Ha\u2019il, Saudi Arabia","award":["RG-24 161"],"award-info":[{"award-number":["RG-24 161"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Neutral transmission line models are essential for analyzing stability and periodicity in systems influenced by nonlinear and delayed dynamics, particularly in modern smart grids. This study utilizes Krasnoselskii\u2019s fixed-point theorem to establish sufficient conditions for the existence and asymptotic stability of periodic solutions, eliminating the need for differentiability in delay terms and coefficients. The results extend existing findings and are validated through a single test example, demonstrating the theoretical applicability of the proposed approach. These findings provide a mathematical framework for understanding the behavior of power distribution systems under nonlinear and delayed influences.<\/jats:p>","DOI":"10.3390\/axioms14040269","type":"journal-article","created":{"date-parts":[[2025,4,2]],"date-time":"2025-04-02T16:19:16Z","timestamp":1743610756000},"page":"269","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Qualitative Properties of Nonlinear Neutral Transmission Line Models and Their Applications"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3963-6503","authenticated-orcid":false,"given":"Mouataz Billah","family":"Mesmouli","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, University of Ha\u2019il, Ha\u2019il 2440, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0216-1265","authenticated-orcid":false,"given":"Abdelouaheb","family":"Ardjouni","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Souk Ahras, P.O. Box 1553, Souk Ahras 41000, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8042-1806","authenticated-orcid":false,"given":"Ioan-Lucian","family":"Popa","sequence":"additional","affiliation":[{"name":"Department of Computing, Mathematics and Electronics, 1 Decembrie 1918 University of Alba Iulia, 510009 Alba Iulia, Romania"},{"name":"Faculty of Mathematics and Computer Science, Transilvania University of Brasov, Iuliu Maniu Street 50, 500091 Brasov, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3453-9277","authenticated-orcid":false,"given":"Hicham","family":"Saber","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, University of Ha\u2019il, Ha\u2019il 2440, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-4773-7046","authenticated-orcid":false,"given":"Faten H.","family":"Damag","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, University of Ha\u2019il, Ha\u2019il 2440, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-9006-8626","authenticated-orcid":false,"given":"Yasir A.","family":"Madani","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, University of Ha\u2019il, Ha\u2019il 2440, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2907-3353","authenticated-orcid":false,"given":"Taher S.","family":"Hassan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, University of Ha\u2019il, Ha\u2019il 2440, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,2]]},"reference":[{"key":"ref_1","unstructured":"Burton, T.A. (2006). Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications."},{"key":"ref_2","unstructured":"Kuang, Y. (1993). Delay Differential Equations with Application in Population Dynamics, Academic Press."},{"key":"ref_3","first-page":"349","article-title":"Periodic solutions in totally nonlinear dynamic equations with functional delay on a time scale","volume":"68","author":"Ardjouni","year":"2010","journal-title":"Rend. Del Semin. Mat. Della Univ. Torino"},{"key":"ref_4","first-page":"181","article-title":"Liapunov functionals, fixed points and stability by Krasnoselskii\u2019s theorem","volume":"9","author":"Burton","year":"2002","journal-title":"Nonlinear Stud."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1002\/mma.3981","article-title":"Existence of positive periodic solutions of first-order neutral differential equations","volume":"40","author":"Candan","year":"2017","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"142","DOI":"10.1016\/j.aml.2015.08.014","article-title":"Existence of positive periodic solutions of first-order neutral differential equations with variable coefficients","volume":"52","author":"Candan","year":"2016","journal-title":"Appl. Math. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2281","DOI":"10.1002\/mma.2975","article-title":"Existence of positive periodic solution for variable-coefficient third-order differential equation with singularity","volume":"37","author":"Cheng","year":"2014","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"527162","DOI":"10.1155\/2014\/527162","article-title":"Multiplicity Results for variable-coefficient singular third-order differential equation with a parameter","volume":"2014","author":"Cheng","year":"2014","journal-title":"Abstr. Appl. Anal."},{"key":"ref_9","first-page":"1279","article-title":"Positive periodic solutions of neutral Lotka-Volterra system with feedback control","volume":"162","author":"Chen","year":"2005","journal-title":"Appl. Math. Comput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1007\/s10255-005-0214-2","article-title":"Periodicity in a nonlinear predator-prey system with state dependent delays","volume":"21","author":"Chen","year":"2005","journal-title":"Acta Math. Appl. Sin. Engl. Ser."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"801","DOI":"10.1007\/s10114-003-0311-1","article-title":"Periodicity and stability in periodic n-species Lotka-Volterra competition system with feedback controls and deviating arguments","volume":"19","author":"Fan","year":"2003","journal-title":"Acta Math. Appl. Sin. Engl. Ser."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"689","DOI":"10.1137\/0523035","article-title":"Periodic solutions of single-species models with periodic delay","volume":"23","author":"Freedman","year":"1992","journal-title":"Siam J. Math. Anal."},{"key":"ref_13","first-page":"235","article-title":"Positive periodic solutions of nonlinear Duffing equations with delay and variable coefficients","volume":"20","author":"Liu","year":"2005","journal-title":"Tamsui Oxf. J. Math. Sci."},{"key":"ref_14","first-page":"1591","article-title":"An constructive proof of the existence theorem for periodic solutions of Duffing equations","volume":"42","author":"Li","year":"1997","journal-title":"Chin. Sci. Bull."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1216\/jie.2022.34.349","article-title":"Stability conditions for a mixed linear Levin\u2013Nohel integrodifferential system","volume":"34","author":"Mesmouli","year":"2022","journal-title":"J. Integral Equations Appl."},{"key":"ref_16","first-page":"6319312","article-title":"Dichotomy Condition and Periodic Solutions for Two Nonlinear Neutral Systems","volume":"2022","author":"Mesmouli","year":"2022","journal-title":"J. Funct. Spaces"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Mesmouli, M.B., Ardjouni, A., and Saber, H. (2023). Asymptotic Behavior of Solutions in Nonlinear Neutral System with Two Volterra Terms. Mathematics, 11.","DOI":"10.3390\/math11122676"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"14274","DOI":"10.3934\/math.2024693","article-title":"Effectiveness of matrix measure in finding periodic solutions for nonlinear systems of differential and integro-differential equations with delays","volume":"9","author":"Mesmouli","year":"2024","journal-title":"Aims Math."},{"key":"ref_19","first-page":"609","article-title":"A study of the stability in neutral nonlinear differential equations with functional delay via fixed points, Facta Universitatis","volume":"31","author":"Mesmouli","year":"2016","journal-title":"Ser. Math. Inform."},{"key":"ref_20","first-page":"789","article-title":"Positive periodic solutions of neutral delay equations","volume":"6","author":"Wang","year":"1996","journal-title":"Acta Math. Sin. (N.S.)"},{"key":"ref_21","first-page":"110","article-title":"Periodic solutions for a second order nonlinear functional differential equation","volume":"20","author":"Wang","year":"2007","journal-title":"Appl. Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"175479","DOI":"10.1155\/2013\/175479","article-title":"On the existence and stability of periodic solutions for a nonlinear neutral functional differential equation","volume":"2013","author":"Yuan","year":"2013","journal-title":"Abstr. Appl. Anal."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1007\/BF02560018","article-title":"Almost periodic solutions for nonlinear Duffing equations","volume":"13","author":"Zeng","year":"1997","journal-title":"Acta Math. Sin. (N.S.)"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1155\/S1085337502000878","article-title":"Positive periodic solutions of non autonomous functional differential equations depending on a parameter","volume":"7","author":"Zhang","year":"2002","journal-title":"Abstr. Appl. Anal."},{"key":"ref_25","unstructured":"Volterra, V. (2005). Theory of Functionals and of Integral and Integro-Differential Equations, Dover Publications."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1007\/BF02418820","article-title":"Sur les \u00e9quations int\u00e9gro-diff\u00e9rentielles et leurs applications","volume":"35","author":"Volterra","year":"1912","journal-title":"Acta Math."},{"key":"ref_27","first-page":"440","article-title":"On the qualitative behaviors of Volterra-Fredholm integro differential equations with multiple time\u2013Varying delays","volume":"31","author":"Tunc","year":"2024","journal-title":"Arab. J. Basic Appl. Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1186\/s13660-024-03085-6","article-title":"On the qualitative behaviors of stochastic delay integro-differential equations of second order","volume":"2024","author":"Mahmoud","year":"2024","journal-title":"J. Inequalities Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"26260","DOI":"10.3934\/math.20231339","article-title":"Boundary value problems of hybrid fractional integro-differential systems involving the conformable fractional derivative","volume":"8","author":"Taier","year":"2023","journal-title":"Aims Math."},{"key":"ref_30","first-page":"191","article-title":"Existence of periodic solutions for a second-order nonlinear neutral differential equation with variable delay","volume":"3","author":"Ardjouni","year":"2014","journal-title":"Palest. J. Math."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1220","DOI":"10.1016\/j.nonrwa.2009.02.013","article-title":"Periodicity and stability in neutral equations by Krasnoselskii\u2019s fixed point theorem","volume":"11","author":"Ding","year":"2010","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"15","DOI":"10.4067\/S0719-06462017000300015","article-title":"Periodicity and stability in neutral nonlinear differential equations by Krasnoselskii\u2019s fixed point theorem","volume":"19","author":"Mansouri","year":"2017","journal-title":"Cubo Math. J."},{"key":"ref_33","first-page":"608","article-title":"Priority-and delay-aware medium access for wireless sensor networks in the smart grid","volume":"8","author":"Mouftah","year":"2013","journal-title":"IEEE Syst. J."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"262","DOI":"10.1016\/j.solener.2017.05.063","article-title":"Modeling and analysis of a smart grid monitoring system for renewable energy sources","volume":"153","author":"Kabalci","year":"2017","journal-title":"Sol. Energy"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"770","DOI":"10.1016\/j.rser.2015.07.140","article-title":"Challenges of integrating renewable energy sources to smart grids: A review","volume":"52","author":"Eltigani","year":"2015","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_36","unstructured":"Kundur, P. (1994). Power System Stability and Control, McGraw-Hill."},{"key":"ref_37","unstructured":"Lund, H. (2014). Renewable Energy Systems: A Smart Energy Systems Approach to the Choice and Modeling of 100% Renewable Solutions, Academic Press."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1170","DOI":"10.1109\/TSG.2012.2203341","article-title":"Advanced demand side management for the future smart grid using mechanism design","volume":"3","author":"Samadi","year":"2012","journal-title":"IEEE Trans. Smart Grid"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"2370","DOI":"10.1109\/TPWRS.2014.2306495","article-title":"Distributed Optimal power Flow Using ADMM","volume":"29","author":"Erseghe","year":"2014","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_40","unstructured":"Smart, D.R. (1980). Fixed Points Theorems, Cambridge University Press."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/4\/269\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:08:50Z","timestamp":1760029730000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/4\/269"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,2]]},"references-count":40,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["axioms14040269"],"URL":"https:\/\/doi.org\/10.3390\/axioms14040269","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,4,2]]}}}