Strong Convergence Theorems for Asymptotically Weak G-Pseudo-Ψ-Contractive Non-Self-Mappings with the Generalized Projection in Banach Spaces

and Applied Analysis 3 or the modified Mann iterative sequence method xn 1 QG ( 1 − αn xn αnT ΠGT xn ) , n 1, 2, . . . , x1 ∈ G, 1.6 where QG : B → G is a sunny nonexpansive retraction. So, in some ways, our results extend and improve some results of other authors such as, see 1–5, 7, 9–13 , from self mappings to non-self-mappings, from Hilbert spaces to Banach spaces. 2. Preliminaries In the sequel, we will assume that B is a real uniformly convex and uniformly smooth hence reflexive Banach space, then B∗ will be the same. If we denote by δB ε the modulus of convexity of the Banach space B and by ρB τ its modulus of smoothness, then δB ε , ρB τ , gB ε ε−1δB ε , hB τ τ−1ρB τ 2.1 are all continuous and increasing on their domains, respectively, and δB 0 ρB 0 gB 0 hB 0 0 see 9 . Also, under the conditions the normalized duality operator J : Jx 1 2 grad { ‖x‖ } 2.2 is single-valued, strictly monotone, continuous, coercive, bounded, and homogeneous, but not addible. In a Hilbert space, J is the Identity operator I : Ix x. Definition 2.1 see 10, 11 . The operator PG : B → G ⊆ B is called metric projection operator if it assigns to each x ∈ B its nearest point x ∈ G, that is, the solution x for the minimization problem PGx x; x : ‖x − x‖ infξ∈G‖x − ξ‖. 2.3 The operator ΠG : B → G ⊆ B is called the generalized projection operator if it assigns to each x ∈ B a minimum point x̂ ∈ G of the Lapunov function V x, ξ : B × B → 0,∞ : V x, ξ ‖x‖ − 2〈Jx, ξ〉 ‖ξ‖, 2.4 that is, a solution of the following minimization problem: ΠGx x̂; x̂ : V x, x̂ infξ∈GV x, ξ . 2.5 Lemma 2.2 see 10, 11 . The point x PGx is the metric projection of x ∈ B on G ⊆ B if and only if the following inequality is satisfied: 〈J x − x , x − ξ〉 ≥ 0, ∀ξ ∈ G, 2.6 and the operator PG is nonexpansive in Hilbert spaces. 4 Abstract and Applied Analysis The point x̂ ΠGx is the generalized projection of x ∈ B on G ⊆ B if and only if the following inequality is satisfied: 〈Jx − Jx̂, x̂ − ξ〉 ≥ 0, ∀ξ ∈ G. 2.7 Furthermore, the inequality below also holds: V ΠGx, ξ ≤ V x, ξ − V x,ΠGx , ∀ξ ∈ G. 2.8