The class of G 1 -contractive mappings was considered in [19] and that of G 2 -contractive mappings in [20].
Remark 2.2 The class of relatively asymptotically nonexpansive mappings was considered in [28] and [29]; see the references therein.
Remark 1.2 The class of asymptotically quasi-ϕ-nonexpansive mappings, which is an extension of the class of quasi-ϕ-nonexpansive mappings, was considered in [29 31].
Sensitivity analysis deals with the derivative of W ∗. In this article, limits of sequences of fuzzy sets, and limits and derivatives of fuzzy set-valued mappings are considered.
Our systems of quasi-variational inequalities are different from the existent ones in literature in the following aspects: random fuzzy mappings are considered and their values are fuzzy sets over complete countable metric spaces.
Since then several classes of variational inequalities with fuzzy mappings were considered by Chang and Haung [15], Ding [30], Ding and Park [31], Haung [36], Kumam and Petrot [48], Noor [55] and Park and Jeong [56, 57] in Hilbert spaces.
Recall that a mapping T is said to be asymptotically quasi-ϕ-nonexpansive iff there exists a sequence { μ n } ⊂ [ 0, ∞ ) with μ n → 0 as n → ∞ such that F ( T ) ≠ ∅, ϕ ( p, T n x ) ≤ ( 1 + μ n ) ϕ ( p, x ), ∀ x ∈ C, ∀ p ∈ F ( T ), ∀ n ≥ 1. Remark 1.3 The class of quasi-ϕ-nonexpansive mappings was considered in [6].
Recall that a mapping T is said to be asymptotically quasi-ϕ-nonexpansive iff there exists a sequence { μ n } ⊂ [ 0, ∞ ) with μ n → 0 as n → ∞ such that F ( T ) ≠ ∅, ϕ ( p, T n x ) ≤ ( 1 + μ n ) ϕ ( p, x ), ∀ x ∈ C, ∀ p ∈ F ( T ), ∀ n ≥ 1. Remark 2.3 The class of asymptotically quasi-ϕ-nonexpansive mappings was considered in Zhou et al. [30] and Qin et al. [31]; see also [32].
Recall that a mapping T is said to be asymptotically quasi-ϕ-nonexpansive iff there exists a sequence { μ n } ⊂ [ 0, ∞ ) with μ n → 0 as n → ∞ such that F ( T ) ≠ ∅, ϕ ( p, T n x ) ≤ ( 1 + μ n ) ϕ ( p, x ), ∀ x ∈ C, ∀ p ∈ F ( T ), ∀ n ≥ 1. Remark 2.2 The class of asymptotically quasi-ϕ-nonexpansive mappings was considered in Zhou et al. [20] and Qin et al. [21]; see also [22] and [23].
Only unique mappings were considered.
