{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:45:49Z","timestamp":1760233549902,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,1,30]],"date-time":"2021-01-30T00:00:00Z","timestamp":1611964800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Using the direct variational method together with the monotonicity approach we consider the existence of non-spurious solutions to the following Dirichlet problem \u2212x\u00a8t\u00a0=ft,xt, x0\u00a0=x1\u00a0=0, where f:\u00a00,1\u00a0\u00d7\u00a0R\u2192R is a jointly continuous and not necessarily convex function. A new approach towards deriving the discrete family of approximating problems is proposed.<\/jats:p>","DOI":"10.3390\/sym13020231","type":"journal-article","created":{"date-parts":[[2021,1,30]],"date-time":"2021-01-30T06:22:20Z","timestamp":1611987740000},"page":"231","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the Existence of Non-Spurious Solutions to Second Order Dirichlet Problem"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9653-612X","authenticated-orcid":false,"given":"Micha\u0142","family":"Be\u0142dzi\u0144ski","sequence":"first","affiliation":[{"name":"Institute of Mathematics, Lodz University of Technology, W\u00f3lcza\u0144ska 215, 90-924 Lodz, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2835-4180","authenticated-orcid":false,"given":"Tomasz","family":"Ga\u0142aj","sequence":"additional","affiliation":[{"name":"Institute of Information Technology, Lodz University of Technology, W\u00f3lcza\u0144ska 215, 90-924 Lodz, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0468-6401","authenticated-orcid":false,"given":"Rados\u0142aw","family":"Bednarski","sequence":"additional","affiliation":[{"name":"Institute of Information Technology, Lodz University of Technology, W\u00f3lcza\u0144ska 215, 90-924 Lodz, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7753-0064","authenticated-orcid":false,"given":"Filip","family":"Pietrusiak","sequence":"additional","affiliation":[{"name":"Institute of Mathematics, Lodz University of Technology, W\u00f3lcza\u0144ska 215, 90-924 Lodz, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3224-2456","authenticated-orcid":false,"given":"Marek","family":"Galewski","sequence":"additional","affiliation":[{"name":"Institute of Mathematics, Lodz University of Technology, W\u00f3lcza\u0144ska 215, 90-924 Lodz, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3786-7225","authenticated-orcid":false,"given":"Adam","family":"Wojciechowski","sequence":"additional","affiliation":[{"name":"Institute of Information Technology, Lodz University of Technology, W\u00f3lcza\u0144ska 215, 90-924 Lodz, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1137\/0711035","article-title":"Difference equations associated with boundary value problems for second order nonlinear ordinary differential equations","volume":"11","author":"Gaines","year":"1974","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_2","unstructured":"Kelley, W.G., and Peterson, A.C. (2001). Difference Equations: An Introduction with Applications, Harcourt\/Academic Press. [2nd ed.]."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3910972","DOI":"10.1155\/2016\/3910972","article-title":"Discrete approaches to continuous boundary value problems: Existence and convergence of solutions","volume":"2016","author":"Anderson","year":"2016","journal-title":"Abstr. Appl. Anal."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1016\/S0893-9659(02)00147-7","article-title":"The nonexistence of spurious solutions to discrete, two-point boundary value problems","volume":"16","author":"Thompson","year":"2003","journal-title":"Appl. Math. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1236","DOI":"10.1016\/j.na.2006.07.010","article-title":"Existence of non-spurious solutions to discrete Dirichlet problems with lower and upper solutions","volume":"67","author":"Tisdell","year":"2007","journal-title":"Nonlinear Anal."},{"key":"ref_6","first-page":"6","article-title":"Existence of non-spurious solutions to discrete boundary value problems","volume":"3","author":"Tisdell","year":"2006","journal-title":"Aust. J. Math. Anal. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1080\/10236198.2017.1405950","article-title":"Global diffeomorphism theorem applied to the solvability of discrete and continuous boundary value problems","volume":"24","author":"Galewski","year":"2018","journal-title":"J. Diff. Equat. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1234","DOI":"10.1080\/10236198.2015.1067694","article-title":"Non-spurious solutions to discrete boundary value problems through variational methods","volume":"21","author":"Galewski","year":"2015","journal-title":"J. Differ. Equ. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Chen, Y., and Zhou, Z. (2020). Existence of Three Solutions for a Nonlinear Discrete Boundary Value Problem with \u03d5c-Laplacian. Symmetry, 12.","DOI":"10.3390\/sym12111839"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Lin, L., Liu, Y., and Zhao, D. (2020). Multiple Solutions for a Class of Nonlinear Fourth-Order Boundary Value Problems. Symmetry, 12.","DOI":"10.3390\/sym12121989"},{"key":"ref_11","unstructured":"Agarwal, R.P. (1992). Difference Equations and Inequalities: Theory, Methods, and Applications, Marcel Dekker."},{"key":"ref_12","unstructured":"Elaydi, S. (2005). An Introduction to Difference Equations, Springer Science & Business Media."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Banasiak, J. (2013). Mathematical Modelling in One Dimension. An Introduction via Difference and Differential Equations, Cambridge University Press.","DOI":"10.1017\/CBO9781139565370"},{"key":"ref_14","unstructured":"Kaplan, D., and Glass, L. (1998). Understanding Nonlinear Dynamics, Springer. Corrected Reprint of the 1995 Original; Textbooks in Mathematical Sciences."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Dr\u00e1bek, P., and Milota, J. (2013). Methods of Nonlinear Analysis. Applications to Differential Equations, Springer. [2nd ed.]. Birkh\u00e4user Advanced Texts Basler Lehrb\u00fccher.","DOI":"10.1007\/978-3-0348-0387-8"},{"key":"ref_16","unstructured":"Fu\u010d\u00edk, S., and Kufner, A. (1980). Nonlinear Differential Equations. Studies in Applied Mechanics. 2, Elsevier Scientific Publishing Company."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"520","DOI":"10.1016\/0022-247X(83)90058-6","article-title":"On multipoint boundary value problems for discrete equations","volume":"96","author":"Agarwal","year":"1983","journal-title":"J. Math. Anal. Appl."},{"key":"ref_18","unstructured":"Mawhin, J. (1987). Probl\u00e8mes de Dirichlet Variationnels Non Lin\u00e9aires, Les Presses de l\u2019Universit \u00e9 de Montr\u00e9al."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Gajewski, H., Gr\u00f6ger, K., and Zacharias, K. (1974). Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie\u2013Verlag.","DOI":"10.1002\/mana.19750672207"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1515\/anona-2020-0102","article-title":"On variational nonlinear equations with monotone operators","volume":"10","author":"Galewski","year":"2021","journal-title":"Adv. Nonlinear Anal."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/2\/231\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:17:29Z","timestamp":1760159849000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/2\/231"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,30]]},"references-count":20,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2021,2]]}},"alternative-id":["sym13020231"],"URL":"https:\/\/doi.org\/10.3390\/sym13020231","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,1,30]]}}}