Muhammed Saeed, K.Krishnendu, R.George, S.Padikkal, J.2026-02-052022Journal of Analysis, 2022, , , pp. -9713611https://doi.org/10.1007/s41478-022-00449-3https://idr.nitk.ac.in/handle/123456789/22924A third-order Homeier method for solving equations in Banach space is studied. Using assumptions on the first and second derivatives, we obtained third-order convergence. Our technique does not involve Taylor series expansion and can be extended to similar higher-order methods. We have given two extensions of the method with orders five and six. Examples with radii of convergence and basins of attraction are provided. © 2022, The Author(s), under exclusive licence to The Forum D’Analystes.Banach spaceConvergence orderHomeier methodIterative methodsTaylor expansion