Proximal Point Algorithms for Finding a Zero of a Finite Sum of Monotone Mappings in Banach Spaces

and Applied Analysis 3 where J is the normalized duality mapping from E into 2E ∗ . If E = H, a Hilbert space, then (13) reduces to φ(x, y) = ‖x − y‖ 2, for x, y ∈ H. Let E be a reflexive, strictly convex, and smooth Banach space, and letC be a nonempty closed and convex subset ofE. The generalized projectionmapping, introduced byAlber [29], is a mapping Π C : E → C that assigns an arbitrary point x ∈ E to the minimizer, x, of φ(⋅, x) over C; that is, Π C x = x, where x is the solution to the minimization problem φ (x, x) = min {φ (y, x) , y ∈ C} . (14) We know the following lemmas. Lemma 4 (see [23]). Let E be a real smooth and uniformly convex Banach space, and let {x n } and {y n } be two sequences of E. If either {x n } or {y n } is bounded and φ(x n , y n ) → 0, as n → ∞, then x n − y n → 0, as n → ∞. Lemma 5 (see [29]). Let C be a convex subset of a real smooth Banach space E, and let x ∈ E. Then x 0 = Π C x if and only if ⟨z − x 0 , Jx − Jx 0 ⟩ ≤ 0, ∀z ∈ C. (15) We make use of the function V : E × E∗ → R defined by V (x, x ∗ ) = ‖x‖ 2 − 2 ⟨x, x ∗ ⟩ + ‖x‖ 2 , ∀x ∈ E, x ∗ ∈ E,