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.
Only unique mappings were considered.
Only unambiguous mappings were considered, yielding a total of 14 916 human Entrez gene IDs with an associated expression pattern from the mouse brain.
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].
In this paper, based on the class of generalized asymptotically nonexpansive mappings, an Ishikawa-type implicit iterative algorithm with errors for two families of mappings is considered.
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.
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].
