Abstract
A coercivity condition is usually assumed in variational inequalities over noncompact domains to guarantee the existence of a solution. We derive minimal, i.e., necessary coercivity conditions for pseudomonotone and quasimonotone variational inequalities to have a nonempty, possibly unbounded solution set. Similarly, a minimal coercivity condition is derived for quasimonotone variational inequalities to have a nonempty, bounded solution set, thereby complementing recent studies for the pseudomonotone case. Finally, for quasimonotone complementarity problems, previous existence results involving so-called exceptional families of elements are strengthened by considerably weakening assumptions in the literature.
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Bianchi, M., Hadjisavvas, N. & Schaible, S. Minimal Coercivity Conditions and Exceptional Families of Elements in Quasimonotone Variational Inequalities. Journal of Optimization Theory and Applications 122, 1–17 (2004). https://doi.org/10.1023/B:JOTA.0000041728.12683.89
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DOI: https://doi.org/10.1023/B:JOTA.0000041728.12683.89