Abstract
The paper studies regularity properties of set-valued map**s between metric spaces. In the context of metric regularity, nonlinear models correspond to nonlinear dependencies of estimates of error bounds in terms of residuals. Among the questions addressed in the paper are equivalence of the corresponding concepts of openness and “pseudo-Hölder” behavior, general and local regularity criteria with special emphasis on “regularity of order \(k\)”, for local settings, and variational methods to extimate regularity moduli in case of length range spaces. The majority of the results presented in the paper are new.
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Notes
That is for any \(y\in B(F(x),\mu (t))\cap V\) there is a \(v\in F(B(x,t))\) such that \(d(y,v)<\lambda \mu (t)\).
A set is locally complete at a point if its intersection with a closed neighborhood of the point is a complete space in the induced metric.
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I wish to express my thanks to the reviewers for the detailed analysis of the text and many good suggestions.
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The paper is dedicated to Jon Borwein’s sixtieth anniversary.
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Ioffe, A.D. Nonlinear regularity models. Math. Program. 139, 223–242 (2013). https://doi.org/10.1007/s10107-013-0670-z
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DOI: https://doi.org/10.1007/s10107-013-0670-z