Modified Mann-Halpern Algorithms for Pseudocontractive Mappings
نویسندگان
چکیده
and Applied Analysis 3 We know that T is pseudocontractive if and only if T satisfies the condition Tx − Ty 2 ≤ x − y 2 + (I − T)x − (I − T)y 2 (15) for all x, y ∈ C. Since u ∈ Fix(T), we have from (15) that ‖Tx − u‖ 2 ≤ ‖x − u‖ 2 + ‖x − Tx‖ 2 , (16) for all x ∈ C. By using (13) and (16), we obtain Tyn − u 2 ≤ yn − u 2 + yn − Tyn 2 = (1 − γn)xn + γnTxn − u 2 + (1 − γn)xn + γnTxn − Tyn 2 = (1 − γn)(xn − u) + γn(Txn − u) 2 + (1 − γn)(xn − Tyn) + γn(Txn − Tyn) 2 = (1 − γ n ) xn − u 2 + γ n Txn − u 2 − γ n (1 − γ n ) xn − Txn 2 + (1 − γ n ) xn − Tyn 2 + γ n Txn − Tyn 2 − γ n (1 − γ n ) xn − Txn 2 ≤ (1 − γ n ) xn − u 2 + γ n ( xn − u 2 + xn − Txn 2 ) − γ n (1 − γ n ) xn − Txn 2 + (1 − γ n ) xn − Tyn 2 + γ n Txn − Tyn 2 − γ n (1 − γ n ) xn − Txn 2 . (17) Note that T is k-Lipschitzian and xn − yn = γn xn − Txn . (18) From (17), we have Tyn − u 2 ≤ (1 − γ n ) xn − u 2 + γ n ( xn − u 2 + xn − Txn 2 ) − γ n (1 − γ n ) xn − Txn 2 + (1 − γ n ) xn − Tyn 2 + γ n k xn − yn 2 − γ n (1 − γ n ) xn − Txn 2 = (1 − γ n ) xn − u 2 + γ n ( xn − u 2 + xn − Txn 2 ) − γ n (1 − γ n ) xn − Txn 2 + (1 − γ n ) xn − Tyn 2 + γ 3 n k xn − Txn 2 − γ n (1 − γ n ) xn − Txn 2 = xn − u 2 + (1 − γ n ) xn − Tyn 2 − γ n (1 − 2γ n − γ 2 n k 2 ) xn − Txn 2 . (19) By condition (C5), without loss of generality, we may assume that γ n ≤ a < 1/(√1 + k + 1) for all n. Then, we have 1 − 2γ n − γ 2 n L 2 > 0 for all n ≥ 0. Substituting (19) to (14) and noting condition (C3), we have (1 − α n − β n )(x n − u) 1 − α n + β n (Ty n − u) 1 − α n 2
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