{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:27:19Z","timestamp":1760149639030,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"17","license":[{"start":{"date-parts":[[2023,8,23]],"date-time":"2023-08-23T00:00:00Z","timestamp":1692748800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>This paper deals with two types of systems of second-order differential equations with parameters: coupled systems with the boundary conditions of the Sturm\u2013Liouville type and classical systems with Dirichlet boundary conditions. We discuss an Ambosetti\u2013Prodi alternative for each system. For the first type of system, we present sufficient conditions for the existence and non-existence of its solutions, and for the second type of system, we present sufficient conditions for the existence and non-existence of a multiplicity of its solutions. Our arguments apply the lower and upper solutions method together with the properties of the Leary\u2013Schauder topological degree theory. To the best of our knowledge, the present study is the first time that the Ambrosetti\u2013Prodi alternative has been obtained for such systems with different parameters.<\/jats:p>","DOI":"10.3390\/math11173645","type":"journal-article","created":{"date-parts":[[2023,8,24]],"date-time":"2023-08-24T10:23:40Z","timestamp":1692872620000},"page":"3645","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Ambrosetti\u2013Prodi Alternative for Coupled and Independent Systems of Second-Order Differential Equations"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7485-2500","authenticated-orcid":false,"given":"Feliz","family":"Minh\u00f3s","sequence":"first","affiliation":[{"name":"Department of Mathematics, School of Science and Technology, University of \u00c9vora, Rua Rom\u00e3o Ramalho, 59, 7000-671 \u00c9vora, Portugal"},{"name":"Center for Research in Mathematics and Applications, Institute for Advanced Studies and Research, University of \u00c9vora, Rua Rom\u00e3o Ramalho, 59, 7000-671 \u00c9vora, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6956-3635","authenticated-orcid":false,"given":"Gracino","family":"Rodrigues","sequence":"additional","affiliation":[{"name":"Center for Research in Mathematics and Applications, Institute for Advanced Studies and Research, University of \u00c9vora, Rua Rom\u00e3o Ramalho, 59, 7000-671 \u00c9vora, Portugal"},{"name":"Coordination of Degree in Mathematics, Campus Macei\u00f3, Federal Institute of Education, Science and Technology of Alagoas, R. Mizael Domingues, 530, Centro, Macei\u00f3 57020-600, AL, Brazil"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1007\/BF02412022","article-title":"On the inversion of some differentiable mappings with singularities between Banach spaces","volume":"93","author":"Ambrosetti","year":"1972","journal-title":"Ann. Mat. Pura Appl."},{"key":"ref_2","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_3","doi-asserted-by":"crossref","first-page":"113","DOI":"10.21136\/MB.1994.126080","article-title":"Existence of multiple solutions for a third-order three-point regular boundary value problem","volume":"119","author":"Senkyrik","year":"1994","journal-title":"Math. Bohem."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1463","DOI":"10.4236\/jamp.2019.77098","article-title":"Existence of Multiple Positive Solutions for Third-Order Three-Point Boundary Value Problem","volume":"7","author":"Chen","year":"2019","journal-title":"J. Appl. Math. Phys."},{"key":"ref_5","first-page":"345","article-title":"Ambrosetti-Prodi type result to a Neumann problem via a topological approach","volume":"11","author":"Sovrano","year":"2018","journal-title":"Discret. Contin. Dyn. Syst. S"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1112\/blms\/18.2.173","article-title":"A Multiplicity Result for Periodic Solutions of Forced Nonlinear Second Order Ordinary Differential Equations","volume":"18","author":"Fabry","year":"1986","journal-title":"Bull. Lond. Math. Soc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1007\/s00030-019-0585-3","article-title":"Periodic solutions to parameter-dependent equations with a \u03d5-Laplacian type operator","volume":"26","author":"Feltrin","year":"2019","journal-title":"Nonlinear Differ. Equ. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"375","DOI":"10.4171\/JEMS\/58","article-title":"The periodic Ambrosetti-Prodi problem for nonlinear perturbations of the p-Laplacian","volume":"8","author":"Mawhin","year":"2006","journal-title":"J. Eur. Math. Soc."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"12986","DOI":"10.3934\/math.2023654","article-title":"On periodic Ambrosetti-Prodi-type problems","volume":"8","author":"Oliveira","year":"2023","journal-title":"AIMS Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2150012","DOI":"10.1142\/S0219199721500127","article-title":"A singular periodic Ambrosetti-Prodi problem of Rayleigh equations without coercivity conditions","volume":"24","author":"Yu","year":"2021","journal-title":"Commun. Contemp. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"103640","DOI":"10.1016\/j.nonrwa.2022.103640","article-title":"Ambrosetti\u2013Prodi problems for the Robin (p,q)-Laplacian","volume":"67","author":"Papageorgiou","year":"2022","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"106648","DOI":"10.1016\/j.aml.2020.106648","article-title":"A remark on the Ambrosetti-Prodi type problem","volume":"111","author":"Ding","year":"2021","journal-title":"Appl. Math. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1515\/ans-2017-6040","article-title":"Ambrosetti-Prodi Periodic Problem Under Local Coercivity Conditions","volume":"18","author":"Sovrano","year":"2018","journal-title":"Adv. Nonlinear Stud."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1007\/s10986-019-09419-y","article-title":"Solvability of second-order coupled systems on the half-line","volume":"59","author":"Fialho","year":"2019","journal-title":"Lith. Math. J."},{"key":"ref_15","unstructured":"Minh\u00f3s, F., and Sousa, R. (2022). Trends in Abstract and Applied Analysis, World Scientific."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1007\/s40314-021-01556-w","article-title":"Heteroclinic and homoclinic solutions for nonlinear second-order coupled systems with \u03d5-Laplacians","volume":"40","author":"Sousa","year":"2021","journal-title":"Comput. Appl. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1554","DOI":"10.1016\/j.jfa.2008.05.003","article-title":"The classification of self-adjoint boundary conditions: Separated, coupled","volume":"255","author":"Wang","year":"2008","journal-title":"J. Funct. Anal."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1016\/0022-0396(88)90156-8","article-title":"Parabolic evolution and nonlinear boundary conditions","volume":"72","author":"Amann","year":"1988","journal-title":"J. Differ. Equ."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1137\/1020038","article-title":"A comparison method for stability analysis of nonlinear parabolic problems","volume":"20","author":"Aronson","year":"1978","journal-title":"SIAM Rev."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1512\/iumj.1982.31.31020","article-title":"A semilinear reaction-diffusion prey-predator system with nonlinear coupled boundary conditions: Equilibrium and Stability","volume":"31","author":"Leung","year":"1982","journal-title":"Indiana Univ. Math. J."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/0362-546X(93)90183-S","article-title":"Nonlinear interaction problems","volume":"20","author":"Mehmeti","year":"1993","journal-title":"Nonlinear Anal."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"19049","DOI":"10.3934\/math.2023972","article-title":"Coupled systems with Ambrosetti-Prodi-type differential equations","volume":"8","author":"Carapau","year":"2023","journal-title":"AIMS Math."},{"key":"ref_23","first-page":"861","article-title":"\u00dcber die diferencialgleichung y\u201d = f (x, y, y\u2019)","volume":"364","author":"Nagumo","year":"1937","journal-title":"Proc. Phys. Math. Soc. Jpn."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"110872","DOI":"10.1016\/j.jtbi.2021.110872","article-title":"Enhanced species coexistence in Lotka-Volterra competition models due to nonlocal 366 interactions","volume":"530","year":"2021","journal-title":"J. Theor. Biol."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1016\/j.mbs.2015.09.010","article-title":"The diffusive Lotka\u2013Volterra predator-prey system with delay","volume":"270","author":"Marchant","year":"2015","journal-title":"Math. Biosci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1016\/j.jmaa.2018.11.032","article-title":"Qualitative analysis of a Lotka\u2013Volterra predator\u2013prey system with migration","volume":"472","author":"Wang","year":"2019","journal-title":"J. Math. Anal. Appl."}],"container-title":["Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2227-7390\/11\/17\/3645\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:41:09Z","timestamp":1760128869000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2227-7390\/11\/17\/3645"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,23]]},"references-count":26,"journal-issue":{"issue":"17","published-online":{"date-parts":[[2023,9]]}},"alternative-id":["math11173645"],"URL":"https:\/\/doi.org\/10.3390\/math11173645","relation":{},"ISSN":["2227-7390"],"issn-type":[{"type":"electronic","value":"2227-7390"}],"subject":[],"published":{"date-parts":[[2023,8,23]]}}}