Rayleigh Quotient and Surjectivity of Nonlinear Operators in Hilbert space
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Abstract
We consider continuous operators acting in a real Hilbert space and indicate conditions ensuring their continuous invertibility and/or surjectivity. In the case of bounded linear operators, these facts are well-known from basic Functional Analysis. The objective of this work is to indicate how similar properties can be proved also when the operators are not necessarily linear, using as a main tool their Rayleigh quotient and especially its lower and upper bound. In particular, we focus our attention on gradient operators and show a quantitative criterion that ensures their surjectivity through the positivity of an additional constant related to the measure of noncompactness.
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Brezis H. Functional Analysis, Sobolev Spaces and Partial Differential Equations. New York: Springer; 2011. Available from: https://link.springer.com/book/10.1007/978-0-387-70914-7
Appell J, De Pascale E, Vignoli A. Nonlinear Spectral Theory. Berlin: de Gruyter; 2004. Available from: https://doi.org/10.1515/9783110199260
de Figueiredo DG. Lectures on the Ekeland Variational Principle with Applications and Detours. Bombay: Tata Institute of Fundamental Research; 1989. Available from: https://mathweb.tifr.res.in/sites/default/files/publications/ln/tifr81.pdf
Chiappinelli R. Surjectivity of coercive gradient operators in Hilbert space and nonlinear spectral theory. Ann Funct Anal. 2019;10(2):170-179. Available from: http://dx.doi.org/10.1215/20088752-2018-0003
Chiappinelli R, Edmunds DE. Measure of noncompactness, surjectivity of gradient operators and an application to the p-Laplacian. J Math Anal Appl. 2019;471:712-727. Available from: https://doi.org/10.1016/j.jmaa.2018.11.010
Chiappinelli R, Edmunds DE. Remarks on surjectivity of gradient operators. Mathematics. 2020;8:1538. Available from: https://doi.org/10.3390/math8091538