Applications of an HIDS Theorem to the Existence of Fixed Point, Abstract Equilibria and Optimization Problems

and Applied Analysis 3 then HIDS will reduce to the following fixed point problem P3 : P3 Find v v i∈I ∈ Y such that vi ∈ Fi vi and v ∈ Gi v, yi for all yi ∈ Ti v for all i ∈ I. Example 1.4. For each i ∈ I, let Fi : X × Yi Ui and Gi : Y × Yi Zi be multivalued maps with nonempty values. If Hi and Ai are defined as follows: Hi { yi ∈ Yi : θUi / ∈ Fi ( u, yi )} , Ai ( y ) { zi ∈ Yi : θZi / ∈ Gi ( y, zi )} , 1.5 then HIDS will reduce to the following system of mixed type of parametric variational inclusion and disclusion problem P4 : P4 Find v v i∈I ∈ Y such that θUi / ∈ Fi u, vi and θZi ∈ Gi v, yi for all yi ∈ Ti v and for all i ∈ I. In this paper, we study the existence theorems of system of parametric vectorial quasi-equilibrium problems, vectorial variants of system of Ekeland’s variational principle VSEVP, for short , and the existence theorems of system of the Stampacchia-type vectorial equilibrium problem by using an HIDS theorem established by the author 11 . Our results improve and generalize some theorems in 19 to the vector case. Till now, to my knowledge, there are extremely few results about Stampacchia-type vectorial equilibrium problem in the literature. The existence of VSEVP and the Stampacchia-type vectorial equilibrium problem are established by applying the HIDS theorem without the use of any scalarization method. So our results are completely different from 3, 8, 9, 12, 13, 15, 23 . As an application, a vectorial minimization theorem is also proved. Moreover, we prove some equivalence relations between our VSEVP, common fixed point theorem, and maximal element theorem.