Abstract
We give explicit formulas for the subdifferential set of the conjugate of not necessarily convex functions defined on general Banach spaces. Even if such a subdifferential mapping takes its values in the bidual space, we show that, up to a weak∗∗ closure operation, it is still described by using only elements of the initial space relying on the behavior of the given function at the nominal point. This is achieved by means of formulas using the ε-subdifferential and an appropriate enlargement of the subdifferential of this function, revealing a useful relationship between the subdifferential of the conjugate function and its part lying in the initial space.
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Acknowledgements
Research supported by Fondecyt Projects Nos. 1080173 and 1110019 and ECOS-Conicyt project No. C10E08.
We are grateful to two anonymous referees for many very helpful suggestions and constructive comments that have substantially improved the paper. We also would like to thank Professor C. Zălinescu for making valuable suggestions and carefully reading a previous version of this paper, namely for kindly pointing out to us the gap in the proof given in Correa and Hantoute (2010a, Corollary 7) of Corollary 6. Finally, our grateful thanks also go to the Guest Editors of this Special Issue, Profs. M.J. Cánovas and J. Parra, for their nice work.
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A Marco López amb mottu del seu seixanté aniversari (To Marco López, on the occasion of his sixtieth birthday).
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Correa, R., Hantoute, A. Subdifferential of the conjugate function in general Banach spaces. TOP 20, 328–346 (2012). https://doi.org/10.1007/s11750-011-0238-0
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DOI: https://doi.org/10.1007/s11750-011-0238-0