Argyros, I.K.George, S.2026-02-082019Understanding Banach Spaces, 2019, Vol., , p. 125-13597815361674509781536167467https://doi.org/10.1007/s41060-024-00575-0https://idr.nitk.ac.in/handle/123456789/33839We provide a semilocal as well as a local convergence analysis of some efficient King-Werner-likemethods of order 1+2 free of derivatives for Banach space valued operators. We use our new idea of the restricted convergence region to find a smaller subset than before containing the iterates. Consequently the resulting Lipschitz parameters are smaller than in earlier works. Hence, to a finer convergence analysis is obtained. The extensions involve no new constants, since the new ones specialize to the ones in previous works. Examples are used to test the convergence criteria. © 2020 by Nova Science Publishers, Inc. All rights reserved.Banach spaceFréchet- derivativeKing’s methodSemilocal and local convergence analysisWerner’s method