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Mean-Value Theorem with Small Subdifferentials

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  • J. P. Penot

    (University of Pau)

Abstract

We prove a mean-value theorem for lower semicontinuous functions on a large class of Banach spaces which contains the class of Asplund spaces, in particular reflexive Banach spaces and Banach spaces with a separable dual. It involves the lower subdifferential (or contingent subdifferential) and the Fréchet subdifferentials, which are among the smallest subdifferentials known to date. It follows that the estimates which it provides require weak assumptions and are accurate. When the function is locally Lipschitzian, we get a simple statement which refines the Lebourg mean-value theorem.

Suggested Citation

  • J. P. Penot, 1997. "Mean-Value Theorem with Small Subdifferentials," Journal of Optimization Theory and Applications, Springer, vol. 94(1), pages 209-221, July.
    • Handle: RePEc:spr:joptap:v:94:y:1997:i:1:d:10.1023_a:1022672005994
    DOI: 10.1023/A:1022672005994
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    References listed on IDEAS

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    1. WEGGE, Leon L., 1974. "Mean value theorem for convex functions," LIDAM Reprints CORE 185, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Wegge, Leon L., 1974. "Mean value theorem for convex functions," Journal of Mathematical Economics, Elsevier, vol. 1(2), pages 207-208, August.
    3. J. P. Penot & P. H. Sach, 1997. "Generalized Monotonicity of Subdifferentials and Generalized Convexity," Journal of Optimization Theory and Applications, Springer, vol. 94(1), pages 251-262, July.
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    Cited by:

    1. J. P. Penot & P. H. Sach, 1997. "Generalized Monotonicity of Subdifferentials and Generalized Convexity," Journal of Optimization Theory and Applications, Springer, vol. 94(1), pages 251-262, July.

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