A system of generalized auxiliary problems principle and a system of variational inequalities (Q2745658)

The author obtains in an elegant manner the approximation-solvability of a system of nonlinear variational and quasivariational inequalities NEWLINE\[NEWLINE\langle F_1(x^*, y^*), x-x^*\rangle\geq 0\quad\text{for all }x\in X,NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\langle F_2(x^*, y^*), g(y)- g(y^*)\rangle\geq 0\quad\text{for all }g(y)\in Y,NEWLINE\]NEWLINE where \(X\) and \(Y\), respectively, are nonempty closed convex subsets of \(\mathbb{R}^m\) and \(\mathbb{R}^n\) and related \(F_1: X\times Y\to \mathbb{R}^m\) and \(F_2: X\times Y\to \mathbb{R}^n\) are any mappings such that \(F= (F_1,F_2)\) is \(g-\gamma\)-partially relaxed monotone. In the paper \(g:\mathbb{R}^n\to \mathbb{R}^n\) can be any mapping. The obtained results complement similar investigations by \textit{G. Cohen} [J. Optimization Theory Appl. 59, No. 2, 325-334 (1988; Zbl 0628.90066)], \textit{D. L. Zhu} and \textit{P. Marcotte} [SIAM J. Optim. 6, No. 3, 714-726 (1996; Zbl 0855.47043)] and \textit{R. U. Verma} (1999, 2000, and on-going research).