In this paper, we propose and study the problem of solve non-linear inclusion problem in Banach spaces, where the involved operator can be written as sum of a Fréchet differentiable function with a continuous perturbation. We use a specific technique introduced by Robinson (Numer. Math. 19, 341–347, 1972) to obtain Newton-Kantorovich theorem, which extends the results of Rokne (Numer. Math. 18, 401–412, 1971), for instance. In our main convergence result, we assume a kind of Hölder condition. Thus, one of the major difficulties to obtain our main result is to show that the sequence of scalars associated with the Newton sequence is convergent. Numerical examples are given to justify the theoretical results.

在本文中，我们提出并研究了解决Banach空间中的非线性包含问题的问题，其中所涉及的算子可以写为具有连续扰动的Fréchet可微函数的和。我们使用由罗宾逊引入的特定技术（NUMER。数学式19，341-347，1972年），得到牛顿的Kantorovich定理，其延伸Rokne的（NUMER。数学式的结果18，401-412，1971），例如。在我们的主要收敛结果中，我们假设一种Hölder条件。因此，获得我们主要结果的主要困难之一是证明与牛顿序列相关的标量序列是收敛的。数值例子证明了理论结果的正确性。