Existence of Solutions for Generalized Vector Quasi-Equilibrium Problems by Scalarization Method with Applications

and Applied Analysis 3 Example 8. LetX = Z = R, Y = R, and E = F = [0, +∞) ⊂ X and let G,C : E → 2, e : E → Y define as G (u) = {(U, V) : U − u ≤ V ≤ u − U, 0 ≤ U ≤ u} , ∀u ∈ F, C (x) = {(U, V) : 0 ≤ V ≤ ( 3 2 + x)U,U ≥ 0} , ∀x ∈ E, e (x) = (1, 1) , ∀x ∈ E, (7) respectively. We can see that ξ G (x, u) = (1 + (2/(1 + 2x)))u. Definition 9. LetD ⊂ Y be a cone,H : F → 2, and ς : F → R. ς is calledmonotone (resp., strictly monotone) with respect to (wrt for brevity)H, if for any u 1 , u 2 ∈ F,H(u 1 ) − H(u 2 ) ⊂ intD implies that ς(u 2 ) ≤ ς(u 1 ) (resp., ς(u 2 ) < ς(u 1 )). Obviously, the strict monotonicity wrt H implies the monotonicity wrtH. Moreover, ifH(u) = {u}, for all u ∈ F, then the (strict) monotonicity wrt H of ς is equivalent to the (strict) monotonicity under the general order structures. The following examples illuminate the relationship between the monotonicity wrtH and the monotonicity in the normal sense whenH is not the identity mapping. Example 10. Let Y = Z = R, F = [0, 10], andD = R + and let ς (u) = { u, u ∈ [0, 5] ,