Farajzadeh, Ali; Jafari, Somaye; Pang, Chin-Tzong On \(\eta\)-upper sign property and upper sign continuity and their applications in equilibrium-like problems. (English) Zbl 1469.49011 Abstr. Appl. Anal. 2014, Article ID 207502, 6 p. (2014). 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. MSC: 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) Citations:Zbl 1302.90249; Zbl 1064.49004; Zbl 1235.47060 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Hanson, M. A., On sufficiency of the Kuhn-Tucker conditions, Journal of Mathematical Analysis and Applications, 80, 2, 545-550 (1981) · Zbl 0463.90080 · doi:10.1016/0022-247X(81)90123-2 [2] Blum, E.; Oettli, W., From optimization and variational inequalities to equilibrium problems, The Mathematics Student, 63, 1-4, 123-145 (1994) · Zbl 0888.49007 [3] Noor, M. A.; Oettli, W., On general nonlinear complementarity problems and quasi-equilibria, Le Matematiche, 49, 2, 313-331 (1994) · Zbl 0839.90124 [4] Giannessi; Maugeri A, F.; Pardalos, P. M., Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models (2001), Dordrecht, The Netherlands: Kluwer Academic, Dordrecht, The Netherlands · Zbl 0992.49001 [5] Noor, M. A., Auxiliary principle technique for equilibrium problems, Journal of Optimization Theory and Applications, 122, 2, 371-386 (2004) · Zbl 1092.49010 · doi:10.1023/B:JOTA.0000042526.24671.b2 [6] Noor, M. A., Invex equilibrium problems, Journal of Mathematical Analysis and Applications, 30, 463-475 (2005) · Zbl 1058.49007 [7] Bianchi, M.; Pini, R., Coercivity conditions for equilibrium problems, Journal of Optimization Theory and Applications, 124, 1, 79-92 (2005) · Zbl 1064.49004 · doi:10.1007/s10957-004-6466-9 [8] Hadjisavvas, N., Continuity and maximality properties of pseudomonotone operators, Journal of Convex Analysis, 10, 459-469 (2003) · Zbl 1063.47041 [9] Castellani, M.; Giuli, M., Refinements of existence results for relaxed quasimonotone equilibrium problems, Journal of Global Optimization, 57, 4, 1213-1227 (2013) · Zbl 1302.90249 · doi:10.1007/s10898-012-0021-2 [10] Farajzadeh, A. P.; Zafarani, J., Equilibrium problems and variational inequalities in topological vector spaces, Optimization, 59, 3-4, 485-499 (2010) · Zbl 1235.47060 · doi:10.1080/02331930801951090 [11] Farajzadeh, A. P.; Noor, M. A., On dual invex Ky Fan inequalities, Journal of Optimization Theory and Applications, 145, 2, 407-413 (2010) · Zbl 1198.90389 · doi:10.1007/s10957-010-9675-4 [12] Fan, K.; Shisha, O., A minimax inequality and applications, Inequalities III, 103-113 (1972), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0302.49019 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.