Perturbation of nonlinear operators in the theory of nonlinear multifrequency electromagnetic wave propagation

Tikhov, Stanislav; Валовик, Д. В.

doi:10.1016/j.cnsns.2019.03.020

Cited by 5 publications

(2 citation statements)

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“…The idea is to use the suitable inequalities appearing in the description of the operator to control the error committed. For instance, in [31] the authors analyze a nonlinear multiparameter eigenvalue problem arising in the theory of nonlinear multifrequency electromagnetic wave propagation. Under certain conditions, the problem can be written as a system of n nonlinear single parameter eigenvalue problems, with linear and non-linear components.…”

Section: Topological Rulesmentioning

confidence: 99%

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Approximate Diagonal Integral Representations and Eigenmeasures for Lipschitz Operators on Banach Spaces

Erdoğan

Pérez

2022

Mathematics

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A new stochastic approach for the approximation of (nonlinear) Lipschitz operators in normed spaces by their eigenvectors is shown. Different ways of providing integral representations for these approximations are proposed, depending on the properties of the operators themselves whether they are locally constant, (almost) linear, or convex. We use the recently introduced notion of eigenmeasure and focus attention on procedures for extending a function for which the eigenvectors are known, to the whole space. We provide information on natural error bounds, thus giving some tools to measure to what extent the map can be considered diagonal with few errors. In particular, we show an approximate spectral theorem for Lipschitz operators that verify certain convexity properties.

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Section: Topological Rulesmentioning

confidence: 99%

“…with f belonging to the space of functions with continuous second derivatives (see Equation ( 6) in [31]; see also Equation (15) in [32]). We can then search for the eigenvectors of the operator T( f ) = f + α f 3 + f .…”

Section: Topological Rulesmentioning

confidence: 99%