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This paper will present some results on quasivariational inequality {C, E , P , 'p} in topological linear locally convex Hausdorff spaces. We shall be concerning with quasivariatioiial inequalities defined on subsets which are convexe closed, or only closed. The compactness of the subset C is replaced by the condensing property of the mapping E . Further, we also obtain some results for quasivariational inequality {C, E , P , v}, where the multivalued mapping E maps CI into Zx and satkfiev n general inwnrd boundary condition.

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In this article we study a general variational inclusion problem with constraints. An existence theorem is given for the scalar problem which allows us to derive several results on existence of solutions of variational inequality and variational inclusion models in which the set-valued data are present.

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