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Generalized partial relaxed monotonicity and nonlinear variational inequalities. (English) Zbl 1025.49008

Summary: The approximation-solvability of the following class of nonlinear variational inequality problems involving the generalized partially relaxed monotone mappings based on a general framework for the auxiliary problem principle is presented.
Find an element \(x^*\in K\) such that \[ \langle T(x^*), \eta(x,x^*)\rangle f(x)- f(x^*)\geq 0\quad\text{for all }x\in K, \] where \(T: K\to \mathbb{R}^n\) is a mapping from a nonempty closed convex subset \(K\) of \(\mathbb{R}^n\) into \(\mathbb{R}^n\), \(\eta: K\times K\to\mathbb{R}^n\) is a mapping, and \(f: K\to \mathbb{R}\) is a continuous convex functional on \(K\). The general auxiliary problem principle is described as follows: for a given iterate \(x^k\in K\) and for a parameter \(p> 0\), determine \(x^{k+1}\) such that \[ \langle\rho T(x^k)+ h'(x^{k+1})- h'(x^k), \eta(x, x^{k+1})\rangle+ \rho[f(x)- f(x^{k+1})]\geq (-\sigma^k) \text{ for all }x\in K, \] where \(h: K\to \mathbb{R}\) is continuously Fréchet-differentiable on \(K\) and \(\sigma^k> 0\) is a number.

MSC:

49J40 Variational inequalities
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