Abstract
We develop a unified convergence analysis of three-step iterative schemes for solving nonlinear Banach space valued equations. The local convergence order has been shown before to be five on the finite dimensional Euclidean space assuming Taylor expansions and the existence of the sixth derivative not on these schemes. So, the usage of them is restricted six or higher differentiable mappings. But in our paper only the first Frèchet derivative is utilized to show convergence. Consequently, the scheme is expanded. Numerical applications are also given to test convergence.
Highlights
We study the problem of finding a simple solution x∗ of math9233106Academic Editors: José ManuelGutiérrez and Ángel Alberto MagreñánReceived: 3 November 2021F ( x ) = 0, provided F : D ⊂ E −→ E1 is an operator acting between Banach spaces E and E1 withD 6= ∅
Many iterative schemes are studied for approximating x∗
In order to extend the utilization of these schemes, we study the local convergence of scheme (2): in a Banach space, under generalized continuity conditions and using hypotheses only on the operators appearing on these schemes
Summary
Notice that convergence can be obtained without the fifth derivative, which does not appear on the scheme. In order to extend the utilization of these schemes, we study the local convergence of scheme (2): in a Banach space, under generalized continuity conditions and using hypotheses only on the operators appearing on these schemes. Notice that the convergence of most high convergence order schemes was given only on the finite dimensional Euclidean space with no error bounds on k xn − x∗ k or information about the uniqueness of the solution. We give such results in this paper. The analysis is provided in the three sections followed by the examples and conclusions in the last two sections
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