{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:47:35Z","timestamp":1760150855015,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,1,29]],"date-time":"2022-01-29T00:00:00Z","timestamp":1643414400000},"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>We study a variable end-points calculus of variations problem of Bolza containing inequality and equality constraints. The proof of the principal theorem of the paper has a direct nature since it is independent of some classical sufficiency approaches invoking the Hamiltonian-Jacobi theory, Riccati equations, fields of extremals or the theory of conjugate points. In contrast, the algorithm employed to prove the principal theorem of the article is based on elementary tools of the real analysis.<\/jats:p>","DOI":"10.3390\/axioms11020055","type":"journal-article","created":{"date-parts":[[2022,1,29]],"date-time":"2022-01-29T23:02:06Z","timestamp":1643497326000},"page":"55","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Straightforward Sufficiency Proof for a Nonparametric Problem of Bolza in the Calculus of Variations"],"prefix":"10.3390","volume":"11","author":[{"given":"Gerardo S\u00e1nchez","family":"Licea","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Facultad de Ciencias, Universidad Nacional Aut\u00f3noma de M\u00e9xico, Ciudad de Mexico 04510, Mexico"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,29]]},"reference":[{"key":"ref_1","unstructured":"Bliss, G.A. (1946). 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Variational Analysis: Critical Extremals and Sturmian Extensions, John Wiley & Sons."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Rindler, F. (2018). Calculus of Variations, Springer.","DOI":"10.1007\/978-3-319-77637-8"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Troutman, J.L. (1983). Variational Calculus with Elementary Convexity, Springer.","DOI":"10.1007\/978-1-4684-0158-5"},{"key":"ref_16","unstructured":"Wan, F.Y.M. (1995). Introduction to the Calculus of Variations and Its Applications, Chapman & Hall."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2153","DOI":"10.1002\/mana.201800053","article-title":"Oscillatory and periodic solutions of diffential equations with piecewise constant generalized mixed arguments","volume":"292","author":"Chiu","year":"2019","journal-title":"Math. Nachr."},{"key":"ref_18","first-page":"316","article-title":"Boundedness for a nonlocal reaction chemotaxis model even in the attraction-dominated regime","volume":"34","author":"Li","year":"2021","journal-title":"Differ. Integral Equ."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1093\/imamci\/dnp009","article-title":"Sufficiency by a direct method in the variable state problem of calculus of variations: Singular extremals","volume":"26","author":"Licea","year":"2009","journal-title":"IMA J. Math. Control. Inf."},{"key":"ref_20","first-page":"7281","article-title":"Sufficiency for singular arcs in two isoperimetric calculus of variations problems","volume":"9","author":"Callejas","year":"2015","journal-title":"Appl. Math. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"111","DOI":"10.3934\/math.2020008","article-title":"Sufficiency for singular trajectories in the calculus of variations","volume":"5","author":"Licea","year":"2019","journal-title":"AIMS Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/j.sysconle.2018.12.003","article-title":"The broken link between normality and regularity in the calculus of variations","volume":"124","author":"Cortez","year":"2019","journal-title":"Syst. Control. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"759","DOI":"10.1007\/s10957-017-1070-y","article-title":"The importance of being normal, regular and proper in the calculus of variations","volume":"172","author":"Becerril","year":"2017","journal-title":"J. Optim. 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