{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:33:27Z","timestamp":1760240007662,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2019,2,16]],"date-time":"2019-02-16T00:00:00Z","timestamp":1550275200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The results presented in this paper deal with the existence of solutions of a first order fully coupled system of three equations, and they are split in two parts: 1. Case with coupled functional boundary conditions, and 2. Case with periodic boundary conditions. Functional boundary conditions, which are becoming increasingly popular in the literature, as they generalize most of the classical cases and in addition can be used to tackle global conditions, such as maximum or minimum conditions. The arguments used are based on the Arz\u00e8la Ascoli theorem and Schauder\u2019s fixed point theorem. The existence results are directly applied to an epidemic SIRS (Susceptible-Infectious-Recovered-Susceptible) model, with global boundary conditions.<\/jats:p>","DOI":"10.3390\/axioms8010023","type":"journal-article","created":{"date-parts":[[2019,2,17]],"date-time":"2019-02-17T22:11:50Z","timestamp":1550441510000},"page":"23","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["First Order Coupled Systems With Functional and Periodic Boundary Conditions: Existence Results and Application to an SIRS Model"],"prefix":"10.3390","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9048-6793","authenticated-orcid":false,"given":"Jo\u00e3o","family":"Fialho","sequence":"first","affiliation":[{"name":"Department of Mathematics, British University of Vietnam, Ecopark Campus, 160000 Hung Yen, Hanoi, Vietnam"},{"name":"Centro de Investiga\u00e7\u00e3o em Matem\u00e1tica e Aplica\u00e7\u00f5es (CIMA), Instituto de Investiga\u00e7\u00e3o e Forma\u00e7\u00e3o Avan\u00e7ada, Universidade de \u00c9vora, Rua Rom\u00e3o Ramalho, 59, 7000-671 \u00c9vora, Portugal"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7485-2500","authenticated-orcid":false,"given":"Feliz","family":"Minh\u00f3s","sequence":"additional","affiliation":[{"name":"Centro de Investiga\u00e7\u00e3o em Matem\u00e1tica e Aplica\u00e7\u00f5es (CIMA), Instituto de Investiga\u00e7\u00e3o e Forma\u00e7\u00e3o Avan\u00e7ada, Universidade de \u00c9vora, Rua Rom\u00e3o Ramalho, 59, 7000-671 \u00c9vora, Portugal"},{"name":"Departamento de Matem\u00e1tica, Escola de Ci\u00eancias e Tecnologia, Universidade de \u00c9vora, 7000-812 \u00c9vora, Portugal"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,2,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Al-Moqbali, M.K.A., Al-Salti, N.S., and Elmojtaba, I.M. (2018). Prey\u2013Predator Models with Variable Carrying Capacity. Mathematics, 6.","DOI":"10.3390\/math6060102"},{"key":"ref_2","first-page":"56","article-title":"On the Stability of Delay Population Dynamics Related with Allee Effects","volume":"17","author":"Gumus","year":"2012","journal-title":"Math. Comput. Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"711","DOI":"10.3390\/pr2040711","article-title":"Mathematical Modeling of Microbial Community Dynamics: A Methodological Review","volume":"2","author":"Song","year":"2014","journal-title":"Processes"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Osakwe, C.J.U. (2018). Incentive Compatible Decision Making: Real Options with Adverse Incentives. Axioms, 7.","DOI":"10.3390\/axioms7010009"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Deuflhard, P. (2000). Differential equations in technology and medicine: Computational concepts, adaptive algorithms, and virtual labs. Computational Mathematics Driven by Industrial Problems, Springer.","DOI":"10.1007\/BFb0103918"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"690","DOI":"10.1002\/cjce.20557","article-title":"Parameter estimation in models with hidden variables: An application to a biotech process","volume":"90","author":"Jang","year":"2012","journal-title":"Can. J. Chem. Eng."},{"key":"ref_7","first-page":"19","article-title":"Monte Carlo Simulation for Electron Dynamics in Semiconductor Devices","volume":"10","author":"Akarsu","year":"2005","journal-title":"Math. Comput. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Malinzi, J., and Quaye, P.A. (2018). Exact Solutions of Non-Linear Evolution Models in Physics and Biosciences Using the Hyperbolic Tangent Method. Math. Comput. Appl., 23.","DOI":"10.3390\/mca23030035"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1223","DOI":"10.1016\/S0362-546X(01)00889-6","article-title":"Periodic boundary value problems for first-order impulsive ordinary differential equations","volume":"51","author":"Nieto","year":"2002","journal-title":"Nonlinear Anal."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1016\/S0895-7177(04)90519-5","article-title":"Periodicity in a generalized ecological competition system governed by impulsive differential equations with delays","volume":"39","author":"Zhang","year":"2004","journal-title":"Math. Comput. Model."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1080\/00036819808840668","article-title":"A coupled system of boundary value problems","volume":"69","author":"Agarwal","year":"1998","journal-title":"Appl. Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1186\/s13661-015-0397-2","article-title":"Existence of solutions for first-order coupled system with nonlinear coupled boundary conditions","volume":"2015","author":"Asif","year":"2015","journal-title":"Bound. Val. Prob."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"848","DOI":"10.1016\/j.jmaa.2011.08.039","article-title":"Positive solutions to singular system with four-point coupled boundary conditions","volume":"386","author":"Asif","year":"2012","journal-title":"J. Math. Anal. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1744","DOI":"10.1002\/mana.201100239","article-title":"Extremal solutions to fourth order discontinuous functional boundary value problems","volume":"286","author":"Cabada","year":"2013","journal-title":"Math. Nachr."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2157","DOI":"10.1016\/j.nonrwa.2008.03.026","article-title":"Extremal solutions to fourth-order functional boundary value problems including multipoint condition","volume":"10","author":"Cabada","year":"2009","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1186\/1687-2770-2013-81","article-title":"Higher order functional boundary value problems without monotone assumptions","volume":"2013","author":"Fialho","year":"2013","journal-title":"Bound. Val. Prob."},{"key":"ref_17","first-page":"453","article-title":"Multiplicity and location results for second order functional boundary value problems","volume":"23","author":"Fialho","year":"2014","journal-title":"Dyn. Syst. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1016\/j.camwa.2010.10.044","article-title":"Higher order boundary value problems with \u03d5 -Laplacian and functional boundary conditions","volume":"61","author":"Graef","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"133","DOI":"10.2298\/AADM110221010G","article-title":"On the lower and upper solution method for higher order functional boundary value problems","volume":"5","author":"Graef","year":"2011","journal-title":"Appl. Anal. Discret. Math."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Zeidler, E. (1986). Nonlinear Functional Analysis and Its Applications, I: Fixed-Point Theorems, Springer.","DOI":"10.1007\/978-1-4612-4838-5"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Angstmann, C.N., Henry, B.I., and McGann, A.V. (2017). A Fractional-Order Infectivity and Recovery SIR Model. Fract. Fract., 1.","DOI":"10.3390\/fractalfract1010011"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Cui, Q., Qiu, Z., Liu, W., and Hu, Z. (2017). Complex Dynamics of an SIR Epidemic Model with Nonlinear Saturate Incidence and Recovery Rate. Entropy, 19.","DOI":"10.3390\/e19070305"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Secer, A., Ozdemir, N., and Bayram, M. (2018). A Hermite Polynomial Approach for Solving the SIR Model of Epidemics. Mathematics, 6.","DOI":"10.3390\/math6120305"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1794","DOI":"10.1137\/040604947","article-title":"Bifurcation analysis of an SIRS epidemic model with generalized incidence","volume":"65","author":"Alexander","year":"2005","journal-title":"SIAM J. Appl. Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1007\/s11766-004-0027-8","article-title":"An SIRS epidemic model","volume":"19","author":"Chen","year":"2004","journal-title":"Appl. Math. J. Chin. Univ."},{"key":"ref_26","first-page":"93","article-title":"Bifurcations of an SIRS epidemic model with nonlinear incidence rate","volume":"15","author":"Hu","year":"2011","journal-title":"Discret. Contin. Dyn. Syst. Ser. B"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/j.chaos.2007.07.047","article-title":"Global stability of an SIRS epidemic model with transport-related infection","volume":"40","author":"Liu","year":"2009","journal-title":"Chaos Solitons Fract."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2599","DOI":"10.1016\/j.na.2007.08.036","article-title":"Persistence and extinction of disease in non-autonomous SIRS epidemic models with disease-induced mortality","volume":"69","author":"Teng","year":"2008","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1482","DOI":"10.1016\/j.chaos.2006.04.022","article-title":"An SIRS model with a nonlinear incidence rate","volume":"34","author":"Jin","year":"2007","journal-title":"Chaos Solitons Fract."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1016\/j.aml.2017.03.005","article-title":"Threshold dynamics of an SIRS model with nonlinear incidence rate and transfer from infectious to susceptible","volume":"70","author":"Li","year":"2017","journal-title":"Appl. Math. Lett."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/0025-5564(78)90006-8","article-title":"A generalization of the Kermack-McKendrick deterministic epidemic model","volume":"42","author":"Capasso","year":"1978","journal-title":"Math. Biosci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1016\/j.cam.2016.01.044","article-title":"A class of Lyapunov functions and the global stability of some epidemic models with nonlinear incidence","volume":"6","author":"Li","year":"2016","journal-title":"J. Appl. Anal. 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