Let X, Y be Banach spaces and let \(A,B :X \rightarrow Y\) be two operators, where A is a linear homeomorphism and B is a \(C^{1} \)-compact operator. Sufficient weakly coerciveness conditions are provided to assert that the perturbed operator \(A+B\) is a \(C^{1} \)-diffeomorphism. The proof of our result is based on properties of Fredholm mappings as well as on local and global inverse mapping theorems. A corollary is provided. It shows that the operator B has one and only one fixed point if some weak coerciveness hypotheses are verified. As an application of our results, two examples are given for integral equations.

中文翻译：

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线性同胚的紧致微扰

设X , Y为巴拿赫空间，并设\(A,B :X \rightarrow Y\)为两个算子，其中A是线性同胚，B是\(C^{1} \) -紧算子。提供了足够的弱强制条件来断言扰动算子\(A+B\)是\(C^{1} \) -微分同胚。我们结果的证明基于 Fredholm 映射的性质以及局部和全局逆映射定理。提供了一个推论。这表明操作员B如果某些弱强制假设得到验证，则有一个且只有一个固定点。作为我们结果的应用，给出了积分方程的两个例子。