The semilocal convergence using recurrence relations of a family of iterations for solving nonlinear equations in Banach spaces is established. It is done under the assumption that the second order Fréchet derivative satisfies the Hölder continuity condition. This condition is more general than the usual Lipschitz continuity condition used for this purpose. Examples can be given for which the Lipschitz continuity condition fails but the Hölder continuity condition works on the second order Fréchet derivative. Recurrence relations based on three parameters are derived. A theorem for existence and uniqueness along with the error bounds for the solution is provided. The R-order of convergence is shown to be equal to 3 + q when θ = ±1; otherwise it is 2 + q , where q ∈ (0, 1]. Numerical examples involving nonlinear integral equations and boundary value problems are solved and improved convergence balls are found for them. Finally, the dynamical study of the family of iterations is also carried out.

中文翻译：

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假设 Hölder 连续二阶 Fréchet 导数的迭代族的递归关系

利用迭代族的递推关系建立求解Banach空间非线性方程组的半局部收敛性。它是在二阶 Fréchet 导数满足 Hölder 连续性条件的假设下完成的。此条件比用于此目的的常用 Lipschitz 连续性条件更通用。可以给出 Lipschitz 连续性条件失败但 Hölder 连续性条件适用于二阶 Fréchet 导数的示例。导出基于三个参数的递归关系。提供了存在性和唯一性的定理以及解的误差界限。当 θ = ±1 时，收敛的 R 阶等于 3 + q；否则为 2 + q ，其中 q ∈ (0, 1]。求解了涉及非线性积分方程和边值问题的数值例子，并找到了改进的收敛球。最后，还进行了迭代族的动力学研究。