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.


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


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