On the existence of solutions to generalized quasi-equilibrium problems

Abstract

In this paper, we apply new results on variational relation problems obtained by D. T. Luc (J Optim Theory Appl 138:65-76, 2008) to generalized quasi-equilibrium problems. Some sufficient conditions on the existence of its solutions of generalized quasiequilibrium problems are shown. As special cases, we obtain several results on the existence of solutions of generalized Pareto and weak quasi-equilibrium problems concerning C-pseudomonotone multivalued mappings. We deduce also some results on the existence of solutions to generalized vector Pareto and weakly quasivariational inequality and vector Pareto quasi-optimization problems with multivalued mappings.Keywords Generalized quasi-equilibrium problems · Upper and lower quasivariational inclusions · Quasi-optimization problems · Upper and lower C-quasiconvex · Upper and lower-quasiconvex-like multivalued mappings · Upper and lower C-continuous multivalued mappings · C-pseudomonotone · C-strong pseudomonotone multivalued mappings Mathematics Subject Classification (2000) 49J27 · 49J53 · 91B50 · 90C48

Quasi-equilibrium Problems and Fixed Point Theorems of the Product Mapping of Lower and Upper Semicontinuous Mappings

“…In this section we shall apply Theorem 8 in Section 2 above on partition of unity and our result in [13] to obtain sufficient conditions for solutions of (QEP). Before proving the main results in this section, we recall the following notions.…”

“…and φ satisfy all conditions of Corollary 3.4 in [13]. It implies that there is (x,w), (v,ȳ) ∈D ×K such that (x,w) ∈P (x,w), (v,ȳ)), (v,ȳ) ∈ Q(x,w), (v,ȳ)) and φ ((v,ȳ), (x,w), (t, z)) ≥ 0,for all (t, z) ∈P ((x,w)(v,ȳ)).…”

“…The existence of solutions to this problem is studied in [ [13], [14]] for the case the multivalued mapping P is continuous, and the multivalued mapping F is upper semicontinuous. All these mapping P and F need to have nonempty convex and closed values.…”

“…For the last decade there has been a number of generalizations of these problems to different directions such as quasi-equilibrium problems with constraint sets depending on parameters, quasi-variational and quasi-equilibrium inclusion problems with multi-valued data (see, for examples, in [8], [13], [14], [15]). Problem (QEP) described above is quite general.…”

“…It encompasses a large class of problems of applied mathematics including quasi-optimization problems, quasi-variational inclusion, quasi-equilibrium problems, quasi-variational relation problems etc. Typical instances of (QEP) are shown in [13], [14] and [15] involving upper semi-continuous utility multivalued mappings with nonempty convex closed values. In this paper, we consider the above (QEP) in the product spaces as follows.…”