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Existence theorems for elliptic and evolutionary variational and quasi-variational inequalities. (English) Zbl 1337.49012

Summary: This paper gives new existence results for elliptic and evolutionary variational and quasi-variational inequalities. Specifically, we give an existence theorem for evolutionary variational inequalities involving different types of pseudo-monotone operators. Another existence result embarks on elliptic variational inequalities driven by maximal monotone operators. We propose a new recessivity assumption that extends all the classical coercivity conditions. We also obtain criteria for solvability of general quasi-variational inequalities treating in a unifying way elliptic and evolutionary problems. Two of the given existence results for evolutionary quasi-variational inequalities rely on Mosco-type continuity properties and Kluge’s fixed-point theorem for set-valued maps. We also focus on the case of compact constraints in the evolutionary quasi-variational inequalities. Here a relevant feature is that the underlying space is the domain of a linear, maximal monotone operator endowed with the graph norm. Applications are also given.

MSC:

49J40 Variational inequalities
49J53 Set-valued and variational analysis
47J20 Variational and other types of inequalities involving nonlinear operators (general)
47H05 Monotone operators and generalizations
47H04 Set-valued operators
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