{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:56:00Z","timestamp":1760144160439,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2024,3,27]],"date-time":"2024-03-27T00:00:00Z","timestamp":1711497600000},"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>Due to the restrictive growth and\/or monotonicity requirements inherent in their employment, classical iterative fixed-point theorems are rarely used to approximate solutions to an integral operator with Green\u2019s function kernel whose fixed points are solutions of a boundary value problem. In this paper, we show how one can decompose a fixed-point problem into multiple fixed-point problems that one can easily iterate to approximate a solution of a differential equation satisfying one boundary condition, then apply a bisection method in an intermediate value theorem argument to meet a second boundary condition. Error estimates on the iterates are also established. The technique will be illustrated on a second-order right focal boundary value problem, with an example provided showing how to apply the results.<\/jats:p>","DOI":"10.3390\/axioms13040222","type":"journal-article","created":{"date-parts":[[2024,3,27]],"date-time":"2024-03-27T13:39:56Z","timestamp":1711546796000},"page":"222","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Iteration with Bisection to Approximate the Solution of a Boundary Value Problem"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2270-2527","authenticated-orcid":false,"given":"Richard","family":"Avery","sequence":"first","affiliation":[{"name":"College of Arts and Sciences, Dakota State University, Madison, SD 57042, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3069-2816","authenticated-orcid":false,"given":"Douglas R.","family":"Anderson","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Concordia College, Moorhead, MN 56562, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0808-802X","authenticated-orcid":false,"given":"Jeffrey","family":"Lyons","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, The Citadel, Charleston, SC 29409, USA"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,27]]},"reference":[{"key":"ref_1","first-page":"131","article-title":"A novel fixed point iteration method for the solution of third order boundary value problems","volume":"271","author":"Abushammala","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2515","DOI":"10.1016\/j.aej.2017.09.010","article-title":"A fixed point iterative method for the solution of two-point boundary value problems for a second order differential equations","volume":"57","author":"Bello","year":"2017","journal-title":"Alexandria Eng. J."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1016\/j.camwa.2010.04.037","article-title":"Iterative method for solving a nonlinear fourth order boundary value problem","volume":"60","author":"Dang","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_4","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_5","doi-asserted-by":"crossref","unstructured":"Granas, A., and Dugundji, J. (2003). Fixed Point Theory, Springer.","DOI":"10.1007\/978-0-387-21593-8"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/j.cpc.2015.09.006","article-title":"Bratu\u2019s problem: A novel approach using fixed-point iterations and Green\u2019s functions","volume":"198","author":"Kafri","year":"2015","journal-title":"Comput. Phys. Commun."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1080\/15502287.2016.1157646","article-title":"A new approach based on embedding Green\u2019s functions into fixed-point iterations for highly accurate solution to Troesch\u2019s problem","volume":"17","author":"Kafri","year":"2016","journal-title":"Int. J. Comput. Methods Eng. Sci. Mech."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1016\/j.aml.2018.02.016","article-title":"A novel Ishikawa-Green\u2019s fixed point scheme for the solution of BVPs","volume":"82","author":"Khuri","year":"2018","journal-title":"Appl. Math. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1303","DOI":"10.1080\/00207160.2019.1615618","article-title":"A fixed point iteration method using Green\u2019s functions for the solution of nonlinear boundary value problems over semi-infinite intervals","volume":"97","author":"Khuri","year":"2020","journal-title":"Int. J. Comput. Math."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Duffy, D. (2015). Green\u2019s Functions with Applications, CRC Press.","DOI":"10.1201\/9781315371412"},{"key":"ref_11","first-page":"3131","article-title":"Numerical solution of fourth order boundary-value problems using Haar wavelets","volume":"5","author":"Haq","year":"2011","journal-title":"Appl. Math. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"103","DOI":"10.3329\/dujs.v62i2.21973","article-title":"Numerical solutions of general fourth order two point boundary value problems by Galerkin method with Legendre polynomials","volume":"62","author":"Hossain","year":"2014","journal-title":"Dhaka Univ. J. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Kumar, V., Latif, A., Rafiq, A., and Hussain, N. (2013). S-iteration process for quasi-contractive mappings. J. Inequalities Appl., 206.","DOI":"10.1186\/1029-242X-2013-206"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1047","DOI":"10.1007\/s12190-021-01557-2","article-title":"Solution of nonlinear boundary value problem by S-iteration","volume":"68","author":"Thenmozhi","year":"2022","journal-title":"J. Appl. Math. Comput."},{"key":"ref_15","first-page":"19","article-title":"Fixed points, differential equations, and proper mappings","volume":"3","author":"Burton","year":"2002","journal-title":"Semin. Fixed Point Theory-Cluj-Napoca"},{"key":"ref_16","first-page":"1","article-title":"Multiple positive solutions of a discrete second order conjugate problem","volume":"8","author":"Avery","year":"1998","journal-title":"Panam. Math. J."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"506","DOI":"10.1090\/S0002-9939-1953-0054846-3","article-title":"Mean Value Methods in Iteration","volume":"4","author":"Mann","year":"1953","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_18","first-page":"255","article-title":"Decomposing a Conjugate Fixed-Point Problem into Multiple Fixed-Point Problems","volume":"32","author":"Avery","year":"2023","journal-title":"Dyn. Syst. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Avery, R., Anderson, D., and Henderson, J. (Rocky Mt. J. Math., 2023). Decomposing a Fixed-Point Problem into Multiple Fixed-Point Problems, Rocky Mt. J. Math., accepted.","DOI":"10.46719\/dsa2023.32.14"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"133","DOI":"10.4064\/fm-3-1-133-181","article-title":"Sur les operations dans les ensembles abstraits et leur applications aux equations integrales","volume":"3","author":"Banach","year":"1922","journal-title":"Fund. Math."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Agarwal, R.P. (1998). Focal Boundary Value Problems for Differential and Difference Equations, Kluwer Academic Publishers. Mathematics and Its Applications 436.","DOI":"10.1007\/978-94-017-1568-3"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"171","DOI":"10.4064\/sm-2-1-171-180","article-title":"Der Fixpunktsatz in Funktionalraumen","volume":"2","author":"Schauder","year":"1930","journal-title":"Stud. 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