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Minimax Theory and its Applications 02 (2017), No. 2, 175--229
Copyright Heldermann Verlag 2017

Equilibrium and Quasi-Equilibrium Problems under φ-Quasimonotonicity and φ-Quasiconvexity

Mohamed Ait Mansour
Dép. de Mathématiques et Informatiques, Faculté Poly-Disciplinaire, Université Cadi Ayyad, Route Sidi Bouzid, 4600 Safi, Morocco

R.-A. Elakri
Dép. de Mathématique, Faculté des Sciences, Université Chouaib Doukkali, B.P. 20, El Jadida -- Morocco

Mohamed Laghdir
Dép. de Mathématique, Faculté des Sciences, Université Chouaib Doukkali, B.P. 20, El Jadida -- Morocco

We introduce a further generalization of quasimonotonicity of real-valued bifunctions and quasiconvexity of real-valued functions what we call φ-quasimonotonicity and φ-quasiconvexity respectively, φ being a bivariate function under appropriate conditions. Our generalization is on the one hand motivated by several counter-examples and examples such as the Clarke-Rockafellar directional derivative, and on the other hand includes the weak or relaxed and strong counterparts of these concepts and moreover leads to new definitions of generalized quasiconvexity and quasimonotonicity. Thereafter, this new material is exploited in the main focus of the paper, i. e., equilibrium problems. To deal with them we first consider a converse/opposed dual problem defined by a Min or Max formulation. Then, by providing conditions ensuring the emptiness of the set of solutions to the converse problem, we obtain global or local Minty solutions. Thereafter, the recourse is made to a very weak concept of sign-regularity, the so-called upper-sign property in the sense of M. Castellani and M. Giuli [Refinements of existence results for relaxed quasimonotone equilibrium problems, J. Global Optim. 57 (2013) 1213--1227], to ensure the passage from Minty's type solutions to standard ones. Our method discusses several kinds of classic solutions local and/or global, weak or relaxed, standard and strong ones, and moreover leads to new strong Minty and eigenvalue-equilibrium points for which we show the intimate link to φ-quasiconvexity, representing by the meantime solutions to an adequate penalized problem of the original one. This new class of solutions meets the so-called strict and star solutions for the particular case of set-valued Stampacchia variational inequalities in their quasi-convex programming context, and moreover disposes at nice stability properties under mild assumptions whenever the objective bifunction is subject to a deterministic parametric perturbation. Furthermore, the treatment we propose here contains both the compact and coercive setting under a variety of assumptions including also a noncoercive case. After that, we turn our attention into quasi-equilibrium problems, wherein the constraints set is not a fixed set but a set-valued map. In this quasi case, we make appeal to the Himmelberg Fixed Point Theorem which is well adapted to the coercive case instead of the Kakutani's one widely used in literature under compactness assumption as in the very recent paper by D. Aussel and J. Cotrina [Quasimonotone quasivariational inequalities: existence results and applications, JOTA 158 (2013) 637--652]. The obtained pattern is then applied to coercive set-valued quasi-variational inequalities and coercive quasi-minimization of second type semistrictly quasiconvex functions including the Nikaido-Isoda functions that arise in generalized Nash equilibria. Our optimization approach here is a direct one and doesn't need the passage via the normal operator to adjusted sublevels of the underlying function which not only complements but also extends and improves in many directions the study proposed in D. Aussel and J. Cotrina [see above].

Keywords: Clarke subdifferential, Clarke generalized derivative, Clarke-Rockafellar subdiffernetial, eigenvalues equilibrium points, eigenvalues minimizers, equilibrium problems, Minty equilibrium problems, fixed points, Lipschitz functions, parametric perturbati

MSC: 90C47, 49J35, 49J52, 49K40

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