Utilizing the arguments similar to those of (3.17) in the proof of Theorem 3.1, we can obtain lim sup n → ∞ 〈 f ( q ) − q, J ( x n − q ) 〉 ≤ 0, (4.17).
Indeed, utilizing the arguments similar to those of Step 5 in the proof of Theorem 3.1, we can obtain the desired conclusion.
Utilizing the arguments similar to those of (3.29) in the proof of Theorem 3.1, we can deduce that lim sup n → ∞ 〈 f ( q ) − q, J ( y n − q ) 〉 ≤ 0. (4.27).
Step 4. lim n → ∞ ∥ S y n − y n ∥ = 0. Indeed, utilizing the arguments similar to those of Step 4 in the proof of Theorem 3.1, we can obtain the desired conclusion.
Step 6. lim n → ∞ ∥ x n − x ¯ ∥ = 0. Indeed, utilizing the arguments similar to those of Step 6 in the proof of Theorem 3.1, we can obtain the desired conclusion.
Step 3. lim n → ∞ ∥ B 1 z ˜ n − B 1 q ∥ = 0 and lim n → ∞ ∥ B 2 z n − B 2 p ∥ = 0, where q = P C ( p − μ 2 B 2 p ). Indeed, utilizing the arguments similar to those of Step 3 in the proof of Theorem 3.1, we can obtain the desired conclusion.
Utilizing the arguments similar to those of Step 5 in the proof of Theorem 3.1 and the relations z n = P C ( x n − λ n ∇ f α n ( x n ) ) and v ∈ C, we can derive 〈 v − p ˆ, w 〉 ≥ 0 as i → ∞.
Utilizing the arguments similar to those of Step 5 in the proof of Theorem 4.1 and the relations u ˜ n = P C ( u n − λ n ∇ f α n ( u n ) ) and v ∈ C, we can derive 〈 v − p ˆ, w 〉 ≥ 0. Since T is maximal monotone, we have p ˆ ∈ T − 1 0 and hence p ˆ ∈ VI ( C, ∇ f ).
For simplicity, we write q = P C ( p − μ 2 B 2 p ), x ˜ n = P C ( x n − μ 2 B 2 x n ) and u ¯ n = P C ( u ˜ n − λ n ∇ f α n ( u ˜ n ) ). for each n ≥ 0. Then y n = σ n x n + ( 1 − σ n ) u ¯ n for each n ≥ 0. Utilizing the arguments similar to those of (4.1) and (4.2) in the proof of Theorem 4.1, from Algorithm 4.2 we can obtain ∥ u ˜ n − p ∥ ≤ ∥ u n − p ∥ + λ n α n ∥ p ∥ (4.24).
For simplicity, we write q = P C ( p − μ 2 B 2 p ), z ˜ n = P C ( z n − μ 2 B 2 z n ) and u n = P C ( z ˜ n − μ 1 B 1 z ˜ n ). for each n ≥ 0. Then y n = σ n Q x n + ( 1 − σ n ) u n for each n ≥ 0. Utilizing the arguments similar to those of (3.1) and (3.2) in the proof of Theorem 3.1, from Algorithm 3.2 we can obtain ∥ z n − p ∥ ≤ ∥ x n − p ∥ + λ n α n ∥ p ∥ (3.22).
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