Let T : D → CB ( D ) be a closed and totally quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences { v n }, { μ n } and a strictly increasing continuous function ζ : R + → R + with ζ ( 0 ) = 0.
Let S : C → C be a closed and totally quasi-ϕ-asymptotically nonexpansive mapping with nonnegative real sequences { ν n } and { μ n } with ν n → 0 and μ n → 0 as n → ∞, respectively, and a strictly increasing continuous function ζ : R + → R + with ζ ( 0 ) = 0.
Let S : C → C be a closed and total quasi-ϕ-asymptotically nonexpansive mapping with nonnegative real sequences ν n and μ n with ν n → 0, μ n → 0 as n → ∞ and a strictly increasing continuous function ζ : R + → R + with ζ ( 0 ) = 0.
Let T : D → C B ( D ) be a closed and quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences { k n } with { k n } ⊂ [ 1, ∞ ) and k n → 1 (as n → ∞ ), then F ( T ) is a closed and convex subset of D. Proof Let { x n } be a sequence in F ( T ) such that x n → x ∗.
Let S : C → 2 C be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequence ν n and μ n with ν n → 0, μ n → 0 as n → ∞ and a strictly increasing continuous function φ : R + → R + with φ ( 0 ) = 0.
Let T: C → C be a closed and total quasi-ϕ-asymptotically nonexpansive mapping with nonnegative real sequences {ν n }, {μ n } and a strictly increasing continuous functions ς : R + → R + such that ν n → 0, μ n → 0 (as n → ∞) and ζ(0) = 0, and T is uniformly L- Lipschitzian.
Let T : C → C be a closed and totally quasi-ϕ-asymptotically nonexpansive mapping with nonnegative real sequences { ν n }, { μ n } and a strictly increasing continuous function ζ : R + → R + such that ν n, μ n → 0 and ζ ( 0 ) = 0.
Let T : C → N(C) be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences {ν n }, {μ n } and a strictly increasing continuous function ζ : ℛ+ → ℛ+ such that ν n → 0, μ n → 0 as n → ∞) and ζ(0) = 0.
Let S : C → 2 C be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences ν n, μ n with ν n → 0, μ n → 0 as n → ∞ and a strictly increasing continuous function ψ : R + → R + with ψ ( 0 ) = 0. Assume that S is uniformly asymptotically regular on C with μ 1 = 0 and F : = F ( S ) ∩ EP ( f ) ∩ A − 1 0 ≠ ∅.
Let S : C → 2 C be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences ν n, μ n with ν n → 0, μ n → 0 as n → ∞ and a strictly increasing continuous function ψ : R + → R + with ψ ( 0 ) = 0. Assume that S is uniformly asymptotically regular on C with μ 1 = 0 and F : = F ( S ) ∩ EP ( f ) ∩ VI ( A, C ) ≠ ∅.
Let S : C → 2 C be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences ν n, μ n with ν n → 0, μ n → 0 as n → ∞ and a strictly increasing continuous function ψ : R + → R + with ψ ( 0 ) = 0. Assume that S is uniformly asymptotically regular on C with μ 1 = 0 and F : = F ( S ) ∩ F ( T ) ∩ EP ( f ) ∩ A − 1 0 ≠ ∅.
