Journal of Global Optimization

, Volume 53, Issue 2, pp 231–241 | Cite as

Local boundedness of monotone bifunctions

  • Mohammad Hossein Alizadeh
  • Nicolas Hadjisavvas


We consider bifunctions \({F : C\times C\rightarrow \mathbb{R}}\) where C is an arbitrary subset of a Banach space. We show that under weak assumptions, monotone bifunctions are locally bounded in the interior of their domain. As an immediate corollary, we obtain the corresponding property for monotone operators. Also, we show that in contrast to maximal monotone operators, monotone bifunctions (maximal or not maximal) can also be locally bounded at the boundary of their domain; in fact, this is always the case whenever C is a locally polyhedral subset of \({\mathbb{R}^{n}}\) and F(x, ·) is quasiconvex and lower semicontinuous.


Monotone bifunction Equilibrium problem Locally bounded operator 


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Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  • Mohammad Hossein Alizadeh
    • 1
  • Nicolas Hadjisavvas
    • 1
  1. 1.Department of Product and Systems Design EngineeringUniversity of the AegeanErmoúpolis, SyrosGreece

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