(2) A mapping T : C → C is said to be quasi-ϕ-asymptotically nonexpansive, if F T) ≠ ∅ and there exists a real sequence {k n } ⊂ [1, ∞), k n → 1 such that ϕ ( p, T n x ) ≤ k n ϕ ( p, x ), ∀ n ≥ 1, x ∈ C, p ∈ F ( T ). Remark 2.5[23].
A multi-valued mapping T : C → 2 C is said to be quasi-ϕ-asymptotically nonexpansive if F(T ) ≠ Ø and there exists a real sequence {k n } ⊂ [1, ∞) with k n → 1 such that ϕ ( p, w n ) ≤ k n ϕ ( p, x ), ∀ n ≥ 1, x ∈ C, w n ∈ T n x, p ∈ F ( T ). (1.8).
A multi-valued mapping T : D → CB ( D ) is said to be quasi-ϕ-asymptotically nonexpansive if F ( T ) ≠ Φ and there exists a real sequence k n ⊂ [ 1, + ∞ ), k n → 1, such that ϕ ( p, z n ) ≤ k n ϕ ( p, x ), ∀ x ∈ D, p ∈ F ( T ), z n ∈ T n x. (2.2).
A nonself multi-valued mapping T : D → X is said to be quasi-ϕ-asymptotically nonexpansive, if F ( T ) ≠ Φ and there exists a real sequence k n ⊂ [ 1, + ∞ ), k n → 1 (as n → ∞ ) such that ϕ ( p, z n ) ≤ k n ϕ ( p, x ), ∀ x ∈ D, p ∈ F ( T ), z n ∈ T ( P T ) n − 1 x, ∀ n ≥ 1 ; (1.8).
(2) A multi-valued mapping T : C → 2 C is said to be quasi-ϕ-asymptotically nonexpansive if F(T ) ≠ Ø and there exists a real sequence {k n } ⊂ [1, ∞) with k n → 1 such that ϕ ( p, w n ) ≤ k n ϕ ( p, x ), ∀ n ≥ 1, x ∈ C, w n ∈ T n x, p ∈ F ( T ).
A mapping T : C → C is said to be ( { k n } ) -quasi-ϕ-asymptotically nonexpansive, if F ( T ) ≠ ∅ and there exists a real sequence { k n } ⊂ [ 1, ∞ ), k n → 1 such that ϕ ( p, T n x ) ≤ k n ϕ ( p, x ), ∀ n ≥ 1, x ∈ C, p ∈ F ( T ).
(15) (b) There exists a real solution sequence { ω ( n ) } of equation (10) which satisfies the inequality β ( n ) ≤ ω ( n ) ≤ γ ( n ) for n ∈ N M. .
Since {u n }, {v n } are bounded sequences, there exists a real constant M > 0 such that u n ≤ M, v n ≤ M (n = 0, 1, 2, …).
Indeed, suppose that there exist a real number (rho> 0), a sequence ((xi _{n}, varepsilon_{n}) downarrow 0, 0)) and (mathbf{x}_{n} in M xi_{n}, varepsilon_{n})) such that dbigl(mathbf{x}_{n},mathbf{S}_{g}(bar{lambda} bigr) geqrhoquad text{for all } n in mathbb{N}.
Definition 2.1 We say that the r.v.s { η n, n ≥ 1 } are widely upper orthant dependent (WUOD) if there exists a finite real sequence { g U ( n ), n ≥ 1 } satisfying for each n ≥ 1 and for all x i ∈ , 1 ≤ i ≤ n, P r ( ⋂ i = 1 n { η i > x i } ) ≤ g U ( n ) ∏ i = 1 n P r ( η i > x i ) ; (2.5).
we say that the r.v.s { η n, n ≥ 1 } are widely lower orthant dependent (WLOD) if there exists a finite real sequence { g L ( n ), n ≥ 1 } satisfying for each n ≥ 1 and for all x i ∈ , 1 ≤ i ≤ n, P r ( ⋂ i = 1 n { η i ≤ x i } ) ≤ g L ( n ) ∏ i = 1 n P r ( η i ≤ x i ) ; (2.6).
