Utilizing Proposition 1.7, we have (1.17).
Indeed, utilizing Proposition 2.8, we have, for each, (3.52).
Thus, utilizing Proposition 1.7 ii) we know from (2.5) that (2.7).
Utilizing Proposition 4.6 we can prove the following.
In addition, utilizing Proposition 2.1 and rn+1- r n → 0 we can obtain lim n → ∞ ǁ T r n + 1 ( x n - r n A x n ) - T r n ( x n - r n A x n ) ǁ = 0.
Note that 0 < lim infn→∞r n ≤ lim supn→∞r n < 2δ and limn→∞(rn+1- r n ) = 0. Then utilizing Proposition 2.1 we have lim n → ∞ ǁ T r n + 1 ( x n - r n A x n ) - T r n ( x n - r n A x n ) ǁ = 0. (3.13).
However, following Colao, Marino and Xu's Step 7 of the proof in [[14], Theorem 3.1] and utilizing Proposition 2.3 (i.e., Lemma 2.8 in [14]), we successively derive w ∈ ∩ i = 1 N Fix ( S i ) by the condition { λ n, i } i = 1 N ⊂ a, b with 0 < a ≤ b < 1. .
However, following Colao, Marino and Xu's Step 7 of the proof in [[14], Theorem 3.1] and utilizing Proposition 2.3 (i.e., Lemma 2.8 in [14]), we successively derive w ∈ ∩ i = 1 N Fix ( S i ) by the condition { λ n, i } i = 1 N ⊂ a, b with 0 < a ≤ b < 1. Theorem 3.2.
Accordingly, utilizing Proposition 2.1 (i) we deduce from the α-inverse strong monotonicity of A1 that x ̄ ∈ VI Fix ( T ), A 1. Therefore, from {x*} = VI VI(Fix(T), A1), A2), we have lim sup n → ∞ A 2 x *, x * - x n = lim k → ∞ A 2 x *, x * - x n k = A 2 x *, x * - x ̄ ≤ 0. (3.7).
Noticing that (|alpha|>rho+|w|) and utilizing Proposition 2.2(3), we have the following fact: (P) if (widetilde{t}in[a,b]), (alpha(a geqrho+|w|), and (alpha(b geqrho+|w|), then (z(t geqrho) on ([a,b]). . if (widetilde{t}in[a,b]), (alpha(a geqrho+|w|), and (alpha(b geqrho+|w|), then (z(t geqrho) on ([a,b]).
