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On Diagonal Subdifferential Operators in Nonreflexive Banach Spaces

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Abstract

Consider a real-valued bifunction f defined on C ×C, where C is a closed and convex subset of a Banach space X, which is concave in its first argument and convex in its second one. We study its subdifferential with respect to the second argument, evaluated at pairs of the form (x,x), and the subdifferential of − f with respect to its first argument, evaluated at the same pairs. We prove that if f vanishes whenever both arguments coincide, these operators are maximal monotone, under rather undemanding continuity assumptions on f. We also establish similar results under related assumptions on f, e.g. monotonicity and convexity in the second argument. These results were known for the case in which the Banach space is reflexive and C = X. Here we use a different approach, based upon a recently established sufficient condition for maximal monotonicity of operators, in order to cover the nonreflexive and constrained case (C ≠ X). Our results have consequences in terms of the reformulation of equilibrium problems as variational inequality ones.

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Authors and Affiliations

  1. Instituto de Matématica Pura e Aplicada (IMPA), Estrada Dona Castorina 110, Rio de Janeiro, RJ, 22460-320, Brazil

    A. N. Iusem & Benar Fux Svaiter

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  1. A. N. Iusem
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  2. Benar Fux Svaiter
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Correspondence to A. N. Iusem.

Additional information

The work of A. N. Iusem was partially supported by CNPq grant no. 301280/86.

The work of Benar Fux Svaiter was partially supported by CNPq grants no. 474944/2010-7, 303583/2008-8 and FAPERJ grant E-26/110.821/2008.

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Iusem, A.N., Svaiter, B.F. On Diagonal Subdifferential Operators in Nonreflexive Banach Spaces. Set-Valued Anal 20, 1–14 (2012). https://doi.org/10.1007/s11228-011-0189-5

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  • DOI: https://doi.org/10.1007/s11228-011-0189-5

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