Compact perturbations of a linear homeomorphism

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.