## Variational-like inequalities with generalized monotone mappings in Banach spaces.(English)Zbl 1041.49006

Let a Banach space $$E$$ with dual $$E^{\ast}$$ and a convex set $$K\subseteq E$$ be given. Let further $$T:K\rightarrow E^{\ast}$$, $$\eta:K\times K\rightarrow E$$ be maps and $$f:K\rightarrow\mathbb{R}\cup\{+\infty\}$$ be a proper convex function. This paper investigates the existence of solutions for the following “variational-like” inequality problem: find $$x\in K$$ such that $$\left\langle Tx,\eta(y,x)\right\rangle +f(y)-f(x)\geq0$$, for all $$y\in K$$. It is assumed that $$T$$ is relaxed $$\eta$$-$$\alpha$$ monotone in the sense that for some function $$\alpha:E\rightarrow\mathbb{R}$$, the inequality $$\left\langle Tx-Ty,\eta(x,y)\right\rangle \geq\alpha(x-y)$$ holds for all $$x,y\in K$$. The proof makes use of the Ky Fan Lemma. A similar result is established for another class of maps introduced in the paper, the relaxed $$\eta$$-$$\alpha$$ semimonotone maps.

### MSC:

 49J40 Variational inequalities 47J20 Variational and other types of inequalities involving nonlinear operators (general)

### References:

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