J Glob Optim (2013) 56:647–667 DOI 10.1007/s10898-012-9891-6 Mixed generalized quasi-equilibrium problems Truong Thi Thuy Duong Received: 19 May 2011 / Accepted: 13 March 2012 / Published online: 28 March 2012 © Springer Science+Business Media, LLC. 2012 Abstract In this paper, we introduce mixed generalized quasi-equilibrium problems and show some sufficient conditions on the existence of their solutions. As special cases, we obtain several results for different mixed quasi-equilibrium problems, mixed quasi-varia- tional inclusions problems and mixed quasi-relation problems etc. Keywords Generalized quasi-equilibrium problems · Upper and lower quasivariational inclusions · Upper and lower C -convex · Upper and lower C -quasiconvex-like multivalued mappings · Upper and lower C -continuous multivalued mappings · KKM multivalued mappings Mathematics Subject Classification 49J27 · 49J53 · 91B50 · 90C48 1 Introduction Throughout this paper, unless otherwise specify, X , Y 1 , Y 2 , Z are supposed to be locally convex Hausdorff topological vector spaces. Assume that D ⊂ X , K ⊂ Z are nonempty subsets. Given multivalued mappings S : D × K → 2 D , T : D × K → 2 K ; P : D → 2 D , Q : D × D → 2 K and F : K × K × D × D → 2 Y 1 , G : K × D × D → 2 Y 2 , we consider the following problem: Find ( ¯ x , ¯ y ) ∈ D × K such that ¯ x ∈ S( ¯ x , ¯ y );¯ y ∈ T ( ¯ x , ¯ y ); 0 ∈ F ( ¯ y , ¯ y , ¯ x , t ), for all t ∈ S( ¯ x , ¯ y ); 0 ∈ G( y , ¯ x , t ), for all t ∈ P ( ¯ x ), y ∈ Q( ¯ x , t ). This problem is called a mixed generalized quasi-equilibrium problem and denoted by ( MGQEP ). In [2], Duong and Tan obtained several results on the existence of solutions for the problem of finding ( ¯ x , ¯ y ) ∈ D × K such that T. T. T. Duong(B ) Vinh Technical Teachers Training University, Vinh City, Vietnam e-mail: thuyduongktv@yahoo.com.vn 123