Convergence Theorems for Common Fixed Points of a Finite Family of Relatively Nonexpansive Mappings in Banach Spaces

and Applied Analysis 3 for each x, y ∈ U. It is also said to be uniformly smooth if the limit is attainted uniformly for each x, y ∈ U. It is well known that ifE is smooth, then the dualitymapping J is single valued. It is also known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset ofE. Someproperties of the dualitymapping have been given in [22]. A Banach space E is said to have Kadec-Klee property if a sequence {x n } of E satisfying that x n ⇀ x ∈ E and ‖x n ‖ → ‖x‖, then x n → x. It is known that if E is uniformly convex, then E has the Kadec-Klee property; see [22] for more details. Let E be a smooth Banach space. The function Φ : E × E → R is defined by φ (y, x) = 󵄩󵄩󵄩󵄩y 󵄩󵄩󵄩󵄩 2 − 2 ⟨y, Jx⟩ + ‖x‖ 2 , (11) for all x, y ∈ E. It is obvious from the definition of the function φ that (1) (‖x‖ − ‖y‖)2 ≤ φ(y, x) ≤ (‖y‖2 + ‖x‖2), (2) φ(x, y) = φ(x, z) + φ(z, y) + 2⟨x − z, Jz − Jx⟩, (3) φ(x, y) = ⟨x, Jx − Jy⟩ + ⟨y − x, Jy⟩ ≤ ‖x‖‖Jx − Jy‖ + ‖y − x‖‖y‖, for all x, y ∈ E; see [4, 7, 23] for more details. Lemma 1 (see [4]). IfE is a strictly convex and smooth Banach space, then for x, y ∈ E, φ(x, y) = 0 if and only if x = y. Lemma 2 (see [23]). Let E be a uniformly convex and smooth Banach space and let {y n }, {z n } be two sequences of E. If φ(y n , z n ) → 0 and either {y n } or {z n } is bounded, then y n − z n → 0. Let C be a closed convex subset of E. Suppose that E is reflexive, strictly convex, and smooth. Then, for any x ∈ E, there exists a point x 0 ∈ C such that φ(x 0 , x) = min y∈C φ(y, x). The mapping Π C : E → C defined by Π C x = x 0 is called the generalized projection (see [4, 7, 23]). Lemma3 (see [7]). LetC be a closed convex subset of a smooth Banach space E and x ∈ E. Then, x 0 = Π C x if and only if ⟨x 0 − y, Jx − Jx 0 ⟩ ≥ 0, ∀y ∈ C. (12) Lemma 4 (see [7]). Let E be a reflexive, strictly convex, and smooth Banach space and let C be a closed convex subset of E and x ∈ E. Then, φ(y, Π C x) + φ(Π C x, x) ≤ φ(y, x) for all y ∈ C. Lemma 5 (see [24]). Let E be a uniformly convex Banach space and B r (0) = {x ∈ E : ‖x‖ ≤ r} a closed ball of E. Then, there exists a continuous strictly increasing convex function g : [0,∞) → [0,∞) with g(0) = 0 such that 󵄩󵄩󵄩󵄩λx + μy + ]z 󵄩󵄩󵄩󵄩 2 ≤ λ‖x‖ 2 + μ 󵄩󵄩󵄩󵄩y 󵄩󵄩󵄩󵄩 2 + ]‖z‖ 2 − λμg ( 󵄩󵄩󵄩󵄩x − y 󵄩󵄩󵄩󵄩) , (13) for all x, y, z ∈ B r (0) and λ, μ, ] ∈ [0, 1] with λ + μ + ] = 1. Lemma 6 (see [19]). Let E be a uniformly convex and uniformly smooth Banach space and let C be a closed convex subset of E. Then, for points w, x, y, z ∈ E and a real number a ∈ R, the setK := {V ∈ C : φ(V, y) ≤ φ(V, x)+⟨V, Jz−Jw⟩+a} is closed and convex. 3. Main Results In this section, we will prove the strong convergence theorem for a common fixed point of a finite family of relatively nonexpansive mappings in a Banach space by using the hybrid method in mathematical programming. Let us prove a proposition first. Proposition 7. Let E be a uniformly convex Banach space and B r (0) = {x ∈ E : ‖x‖ ≤ r} a closed ball of E. Then, there exists a continuous strictly increasing convex function g : [0,∞) → [0,∞) with g(0) = 0 such that 󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 n ∑ i=1 λ i x i 󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 2 ≤ n