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On generalized variational-like inequalities with generalized monotone multivalued mappings. (English) Zbl 1158.49010

Summary: Let \(E\) be a reflexive Banach space with the dual space \(E^{*}\) and \(K\) be a nonempty closed convex subset of \(E\). Let us have \(\varPsi :K\times K\times E^{*}\rightarrow R\) and \(A:E^{*}\rightarrow E^{*}\). We introduce the class of generalized \(\alpha \)-monotone multifunctions \(T:K\rightarrow 2^{E^{*}}\) with respect to \(\varPsi \) and \(A\) where \(\alpha :E\times E\rightarrow R\). By using the KKM technique and the concept of the Hausdorff metric, we establish some existence results for generalized variational-like inequalities with generalized monotone multivalued mappings in \(E\).

MSC:

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