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Compact perturbations of a linear homeomorphism

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Abstract

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.

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Authors and Affiliations

  1. Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, c/ Tarfia s/n, Sevilla, 41012, Spain

    M. O. Cabrera & J. M. Soriano Arbizu

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  1. M. O. Cabrera
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  2. J. M. Soriano Arbizu
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Correspondence to J. M. Soriano Arbizu.

Additional information

The authors have been partially supported by the Plan Andaluz de Investigación de la Junta de Andalucía FQM-127 Grant P08-FQM-03543, and by MEC Grant MTM2015-65242-C2-1-P.

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Cabrera, M.O., Soriano Arbizu, J.M. Compact perturbations of a linear homeomorphism. Acta Math. Hungar. 164, 522–532 (2021). https://doi.org/10.1007/s10474-021-01141-x

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  • DOI: https://doi.org/10.1007/s10474-021-01141-x

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