Relaxed Quasimonotone Operators and Relaxed Quasiconvex Functions

In this paper, we introduce the class of multivalued relaxed μ quasimonotone operators and establish the existence of solutions of variational inequalities for such operators. This result is compared with a recent result of Bai et al. on densely relaxed pseudomonotone operators. A similar comparison regarding an existence result of Luc on densely pseudomonotone operators is provided. Also, we introduce a broad class of functions, called relaxed quasiconvex functions, and show that they are characterized by the relaxed μ quasimonotonicity of their subdifferentials. The results strengthen a variety of other results in the literature. This work is supported by NNSF of China (10571046) and by the GSRT of Greece (06FR-062).