On \(\eta\)-upper sign property and upper sign continuity and their applications in equilibrium-like problems. (English) Zbl 1469.49011

Summary: We first introduce the notion of \(\eta\)-upper sign property which is an extension of the upper sign property introduced in [M. Castellani and M. Giuli, J. Glob. Optim. 57, No. 4, 1213–1227 (2013; Zbl 1302.90249)], by relaxing convexity on the set. Afterwards, we establish a link between the solution sets of local dual equilibrium problem (Minty local equilibrium problem) and equilibrium problem for mappings whose domains are not necessarily convex by relaxing the upper sign continuity on the map, as it is assumed in the literature [M. Bianchi and R. Pini, J. Optim. Theory Appl. 124, No. 1, 79–92 (2005; Zbl 1064.49004); Castellani and Giuli, loc. cit.; the first author and J. Zafarani, Optimization 59, No. 3–4, 485–499 (2010; Zbl 1235.47060)]. Accordingly, it allows us to extend and obtain some existence results for equilibrium-like problems.


49J40 Variational inequalities
47J20 Variational and other types of inequalities involving nonlinear operators (general)
90C30 Nonlinear programming
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
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