{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:51:52Z","timestamp":1760147512708,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,2,8]],"date-time":"2023-02-08T00:00:00Z","timestamp":1675814400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11571207","ZR2018MA011"],"award-info":[{"award-number":["11571207","ZR2018MA011"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Shandong Natural Science Foundation","award":["11571207","ZR2018MA011"],"award-info":[{"award-number":["11571207","ZR2018MA011"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we consider the existence of extremal solutions for the nonlinear fourth-order differential equation. By use of a new comparison result, some sufficient conditions for the existence of extremal solutions are established by combining the monotone iterative technique and the methods of lower and upper solutions. Finally, an example is given to illustrate the validity of our main results.<\/jats:p>","DOI":"10.3390\/axioms12020178","type":"journal-article","created":{"date-parts":[[2023,2,9]],"date-time":"2023-02-09T01:37:07Z","timestamp":1675906627000},"page":"178","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Monotonically Iterative Method for the Cantilever Beam Equations"],"prefix":"10.3390","volume":"12","author":[{"given":"Yujun","family":"Cui","sequence":"first","affiliation":[{"name":"College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Huiling","family":"Chen","sequence":"additional","affiliation":[{"name":"College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7213-3107","authenticated-orcid":false,"given":"Yumei","family":"Zou","sequence":"additional","affiliation":[{"name":"College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1016\/S0022-247X(86)80006-3","article-title":"Existence and uniqueness theorems for fourth-order boundary value problems","volume":"116","author":"Aftabizadeh","year":"1986","journal-title":"J. Math. Anal. Appl."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Agarwal, R.P. (1986). Boundary Value Problems for Higher Order Differential Equations, World Scientific. [3rd ed.].","DOI":"10.1142\/0266"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"4191","DOI":"10.1016\/j.amc.2009.12.040","article-title":"Positive solutions of some nonlocal fourth-order boundary value problem","volume":"215","author":"Bai","year":"2010","journal-title":"Appl. Math. Comput."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1161","DOI":"10.1216\/RMJ-2013-43-4-1161","article-title":"Existence and uniqueness of positive solutions for fourth-order m-point boundary value problems with two parameters","volume":"43","author":"Hao","year":"2013","journal-title":"Rocky Mt. J.Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"5261","DOI":"10.1016\/j.amc.2012.11.066","article-title":"Multiplicity of solutions of a two point boundary value problem for a fourth-order equation","volume":"219","author":"Cabada","year":"2013","journal-title":"Appl. Math. Comput."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"209","DOI":"10.5269\/bspm.v36i4.33584","article-title":"New fixed point approach for a fully nonlinear fourth order boundary value problem","volume":"36","author":"Dang","year":"2018","journal-title":"Bol. Soc. Parana. Mat."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"323","DOI":"10.4208\/eajam.231017.250118a","article-title":"The unique solvability and approximation of BVP for a nonlinear fourth order Kirchhoff type equation","volume":"8","author":"Dang","year":"2018","journal-title":"East Asian J. Appl. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1016\/j.nonrwa.2017.01.001","article-title":"Existence results and iterative method for solving the cantilever beam equation with fully nonlinear term","volume":"36","author":"Dang","year":"2017","journal-title":"Nonlinear Anal. RWA"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"6700","DOI":"10.1016\/j.amc.2011.01.071","article-title":"Nontrivial solutions of singular fourth-order Sturm-Liouville boundary value problems with a sign-changing nonlinear term","volume":"217","author":"Fan","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_10","first-page":"53","article-title":"Nontrivial solutions of fourth-order singular boundary value problems with sign-changing nonlinear terms","volume":"40","author":"Zhang","year":"2012","journal-title":"Topol. Methods Nonl. An."},{"key":"ref_11","first-page":"14","article-title":"A cantilever equation with nonlinear boundary conditions","volume":"15","author":"Infante","year":"2009","journal-title":"Electron. J. Qual. Theory Differ. Equ."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1186\/s13661-019-1155-7","article-title":"Positive solutions of fourth-order problems with dependence on all derivatives in nonlinearity under Stieltjes integral boundary conditions","volume":"2019","author":"Ma","year":"2019","journal-title":"Bound. Value Prob."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1016\/S0168-9274(03)00065-5","article-title":"Existence results and numerical solutions for a beam equation with nonliear boundary conditions","volume":"47","author":"Ma","year":"2003","journal-title":"Appl. Numer. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"911","DOI":"10.1080\/00036811.2011.556623","article-title":"Multiplicity results for a class of fourth order semipositone m-point boundary value problems","volume":"91","author":"Liu","year":"2012","journal-title":"Appl. Anal."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2085","DOI":"10.1090\/S0002-9939-00-05320-X","article-title":"Twin solutions to singular boundary value problems","volume":"128","author":"Agarwal","year":"2000","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/S0362-546X(98)00341-1","article-title":"Singular (p,n\u2013p) focal and (n,p) higher order boundary value problems","volume":"42","author":"Agarwal","year":"2000","journal-title":"Nonlinear Anal."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"142","DOI":"10.1006\/jdeq.2000.3808","article-title":"Multiplicity results for singular conjugate, focal and (n,p) problems","volume":"170","author":"Agarwal","year":"2001","journal-title":"J. Differ. Equ."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1195","DOI":"10.1016\/j.mcm.2004.06.001","article-title":"Positive solutions for a beam equation on a nonlinear elastic foundation","volume":"39","author":"Ma","year":"2004","journal-title":"Math. Comput. Model."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3834","DOI":"10.1016\/j.na.2009.02.051","article-title":"Monotone positive solutions for a fourth order equation with nonlinear boundary conditions","volume":"71","author":"Alves","year":"2009","journal-title":"Nonlinear Anal."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"432","DOI":"10.1016\/j.amc.2008.08.044","article-title":"Monotonically iterative method of nonlinear cantilever beam equations","volume":"205","author":"Yao","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/j.amc.2003.08.088","article-title":"Iterative solutions for a beam equation with nonlinear boundary conditions of third order","volume":"159","author":"Ma","year":"2004","journal-title":"Appl. Math. Comput."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1016\/j.nonrwa.2015.07.016","article-title":"Existence of positive solutios for the cantilever beam equations with fully nonlinear terms","volume":"27","author":"Li","year":"2016","journal-title":"Nonlinear Anal-Real."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1186\/s13661-019-1200-6","article-title":"Solvability for fully cantilever beam equations with superlinear nonlinearities","volume":"2019","author":"Li","year":"2019","journal-title":"Bound. Value Probl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1186\/s13660-019-2088-5","article-title":"The method of lower and upper solutions for the cantilever beam equations with fully nonlinear terms","volume":"2019","author":"Li","year":"2019","journal-title":"J. Inequal. Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1704","DOI":"10.1016\/j.na.2006.08.009","article-title":"The upper and lower solution method for some fourth-order boundary value problems","volume":"67","author":"Bai","year":"2007","journal-title":"Nonlinear Anal-Theor."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1006","DOI":"10.1515\/math-2020-0056","article-title":"Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis","volume":"18","author":"Almuthaybiri","year":"2020","journal-title":"Open Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1186\/s13662-021-03402-z","article-title":"Lower and upper solutions method to the fully elastic cantilever beam equation with support","volume":"2021","author":"Wei","year":"2021","journal-title":"Adv. Differ. Equ."},{"key":"ref_28","first-page":"641617","article-title":"Existence results for a fully fourth-order boundary value problem","volume":"2013","author":"Li","year":"2013","journal-title":"J. Funct. Spaces"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/2\/178\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:28:40Z","timestamp":1760120920000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/2\/178"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,8]]},"references-count":28,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["axioms12020178"],"URL":"https:\/\/doi.org\/10.3390\/axioms12020178","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2023,2,8]]}}}