Strong Convergence Theorems for Semigroups of Asymptotically Nonexpansive Mappings in Banach Spaces

and Applied Analysis 3 where ∂ denotes the subdifferential in the sense of convex analysis. We need the subdifferential inequality Φ( 󵄩󵄩󵄩󵄩x + y 󵄩󵄩󵄩󵄩) ≤ Φ (‖x‖) + ⟨y, j (x + y)⟩ ∀x, y ∈ X, j (x + y) ∈ Jφ (x + y) . (14) For a smoothX, we have Φ( 󵄩󵄩󵄩󵄩x + y 󵄩󵄩󵄩󵄩) ≤ Φ (‖x‖) + ⟨y, Jφ (x + y)⟩ ∀x, y ∈ X, (15) or considering the normalized duality mapping J, we have 󵄩󵄩󵄩󵄩x + y 󵄩󵄩󵄩󵄩 2 ≤ ‖x‖ 2 + 2 ⟨y, J (x + y)⟩ ∀x, y ∈ X. (16) Assume that a sequence {xn} in X converges weakly to a point x inX. Then the following identity holds; lim sup n→∞ Φ( 󵄩󵄩󵄩󵄩xn − y 󵄩󵄩󵄩󵄩) = lim sup n→∞ Φ( 󵄩󵄩󵄩󵄩xn − x 󵄩󵄩󵄩󵄩) + Φ ( 󵄩󵄩󵄩󵄩x − y 󵄩󵄩󵄩󵄩) ∀y ∈ X. (17) Remark 2. For any k with 0 ≤ k ≤ 1, we have φ (kt) ≤ φ (t) ∀t > 0,