## 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 FrĂ©chet-differentiable on $$K$$ and $$\sigma^k> 0$$ is a number.

### MSC:

 49J40 Variational inequalities