Strong Convergence Theorems for a Common Fixed Point of a Family of Asymptotically k-Strict Pseudocontractive Mappings

and Applied Analysis 3 It is shown in [19] that if E = H, a real Hilbert space, then ρ(t) = t, for t > 0. In our general setting, throughout this paper we assume that ρ(t) ≤ 2t. Lemma 3. Let E be a real Banach space. Then the following inequality holds: 󵄩󵄩󵄩󵄩x + y 󵄩󵄩󵄩󵄩 2 ≤ ‖x‖ 2 + ⟨y, j (x + y)⟩ , ∀x, y ∈ H, j (x + y) ∈ J (x + y) . (10) Lemma 4 (see [20]). Let E be a uniformly convex Banach space and B R (0) a closed ball of E. Then, there exists a continuous strictly increasing convex function g : [0,∞) → [0,∞) with g(0) = 0 such that 󵄩󵄩󵄩󵄩α0x0 + α1x1 + α2x2 + ⋅ ⋅ ⋅ + αkxk 󵄩󵄩󵄩󵄩 2