Relaxed η-α quasimonotone and application to the generalized variational-like inequality problem

In this paper, some new mappings called relaxed η-α quasimonotone and a relaxed η-α properly quasimonotone operator are first introduced. The relationships between them are obtained. After this, the variational-like inequality problem and the relaxed Minty variational-like inequality problem are discussed by use of the proposed generalized monotone operators. Furthermore, we give the gap function of the two variational-like inequalities and two kinds of optimization problems. Finally, we point out that the two optimization problems are equivalent under some conditions.