In this study, the new semilocal convergence for the family of Chebyshev Kurchatov type methods is proposed under weaker conditions. The convergence analysis demands conditions on the initial approximation, auxiliary point, and the underlying operator (Argyros et al. (2017) ). By utilizing the notion of auxiliary point in convergence conditions, the convergence domains are obtained where the existing results are not applicable. The theoretical results are validated using numerical examples, such as nonlinear PDE and mixed Hammerstein type integral equations originating in biology, vehicular traffic theory, and queuing theory, to determine the applicability of the proposed framework. The numerical testing shows that the new approach performed better (accurately and converge faster) than Steffensen's method, Chebyshev type methods and two step secant method.

中文翻译：

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不可微算子的Chebyshev Kurchatov型方法的半局部收敛


在本研究中，在较弱的条件下提出了切比雪夫·库尔恰托夫型方法族的新半局部收敛性。收敛分析需要初始近似、辅助点和底层算子的条件（Argyros et al. (2017)）。利用收敛条件中辅助点的概念，得到了现有结果不适用的收敛域。使用非线性偏微分方程和源自生物学的混合 Hammerstein 型积分方程、车辆交通理论和排队论等数值实例验证了理论结果，以确定所提出框架的适用性。数值测试表明，新方法比Steffensen方法、Chebyshev型方法和两步割线法表现更好（准确且收敛更快）。