In the present paper, we have studied the local convergence of a three-step Newton-type method for solving nonlinear equations in Banach spaces under weak L-average. More precisely, we have derived the two existence theorems when the first-order Fréchet derivative of nonlinear operator satisfies the radius and center Lipschitz condition with a weak L-average; particularly, it is assumed that L is positive integrable function but not necessarily non-decreasing, which was assumed in the earlier discussion.

中文翻译：

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弱L平均下三步牛顿型方法的存在性定理

在本文中，我们研究了弱L平均下求解Banach空间非线性方程的三步牛顿型方法的局部收敛性。更准确地说，我们推导了非线性算子的一阶Fréchet导数满足弱L平均的半径和中心Lipschitz条件时的两个存在性定理；特别是，假设L是正可积函数，但不一定是非减函数，这是在前面的讨论中假设的。