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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 $f:K\rightarrow\Bbb{R}\cup\{+\infty\}$ 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 $\alpha:E\rightarrow\Bbb{R}$, 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.

49J40Variational methods including variational inequalities
47J20Inequalities involving nonlinear operators
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