Abstract
In this paper we show that the \(G\) – subdifferential of a lower semicontinuous function is contained in the limit superior of the \(G\) – subdifferential of lower semicontinuous uniformly convergent family to this function. It happens that this result is equivalent to the corresponding normal cones formulas for family of sets which converges in the sense of the bounded Hausdorff distance. These results extend to the infinite dimensional case those of Ioffe for \(C^2\) – functions and of Benoist for Clarke’s normal cone. As an application we characterize the subdifferential of any function which is bounded from below by a negative quadratic form in terms of its Moreau–Yosida proximal approximation.
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Jourani, A. Limit Superior of Subdifferentials of Uniformly Convergent Functions. Positivity 3, 33–47 (1999). https://doi.org/10.1023/A:1009740914637
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DOI: https://doi.org/10.1023/A:1009740914637