The generalized set-valued strongly nonlinear implicit variational inequalities (Q1963042)

A class of generalized set-valued strongly nonlinear implicit variational inequalities of the following type in a real Hilbert space \(H\) is studied: \[ \begin{gathered} u \in H,\;y \in Tu,\;z \in Au, \\ (gv-gu,N(y,z)) \geq f(u)-f(v) \;\text{ for all }v \in H. \end{gathered} \] Here \(g:H\to H\), \(f:H\to R\cup \{+\infty\}\), \(N:H\times H \to H\) are single-valued mappings, \(T,\;A:H\to 2^H\) are set-valued mappings. By a suitable choice of the mappings, a number of usual classes of variational inequalities and complementarity problems can be obtained. Some particular cases are briefly mentioned. An algorithm for solving the general problem mentioned by using the auxiliary principle technique of Glowinski, Lions and Tremolieres is given. Particularly, the convergence of the corresponding iterative solutions and the existence of solutions to the original problem is proved.