Iterative Methods for Pseudocontractive Mappings in Banach Spaces

and Applied Analysis 3 Lemma 1 (see [1, 2]). Let E be a Banach space and let J be the normalized duality mapping on E. Then for any x, y ∈ E, the following inequality holds: 󵄩󵄩󵄩󵄩x + y 󵄩󵄩󵄩󵄩 2 ≤ ‖x‖ 2 + 2⟨y, j (x + y)⟩, ∀j (x + y) ∈ J (x + y) . (14) Lemma 2 (see [20]). Let {s n } be a sequence of nonnegative real numbers satisfying s n+1 ≤ (1 − λ n ) s n + λ n δ n , ∀n ≥ 0, (15) where {λ n } and {δ n } satisfy the following conditions: (i) {λ n } ⊂ [0, 1] and ∑∞ n=0 λ n = ∞ or, equivalently,