Iterative algorithms for solving generalized nonlinear mixed variational inequalities (Q704199)

The aim of this very important and useful paper is to investigate the iterative algorithms for solving generalized nonlinear mixed variational inequalities. Here the generalized nonlinear variational inequality problem: Find \(x\in H\), \(u\in T(x)\) and \(v\in A(x)\) such that \(\langle N(u, v),g(y)- g(x)\rangle+ \varphi(g(y))- \varphi(g(x))\geq 0\), \(\forall g(y)\in H\) is considered, where \(H\) is a real Hilbert space, \(C(H)\) are the families of all nonempty compact subsets of \(H\), \(T,A: H\to C(H)\), the set-valued mappings, \(N: H\times H\to H\) and \(g: H\to H\) are single-valued mappings, and \(\varphi: H\to(-\infty, +\infty]\) be a real function. The author introduces a concept of \(g\)-partially relaxed strong monotonicity for mappings. By applying the concept of auxiliary variational inequality technique some new predictor-corrector iterative schemes for solving a class of generalized nonlinear variational inequalities are suggested. Main result: The convergence of the iterative sequence generated by the suggested iterative algorithm is proved. The precise proof is proposed. We note that the convergence analysis of the predictor-corrector algorithm only requires (crucial point) that the underlying mappings are continuous and \(g\)-partially relaxed strongly monotone. Moreover, these algorithms and convergence results are new.