Iterative algorithms with perturbations for Lipschitz pseudocontractive mappings in Banach spaces

Iterative Methods for Pseudocontractive Mappings in Banach Spaces

and Applied Analysis 3 Lemma 1 (see [1, 2]). Let E be a Banach space and let J be the normalized duality mapping on E. Then for any x, y ∈ E, the following inequality holds: 󵄩󵄩󵄩󵄩x + y 󵄩󵄩󵄩󵄩 2 ≤ ‖x‖ 2 + 2⟨y, j (x + y)⟩, ∀j (x + y) ∈ J (x + y) . (14) Lemma 2 (see [20]). Let {s n } be a sequence of nonnegative real numbers satisfying s n+1 ≤ (1 − λ n ) s n + λ n δ n , ∀n ≥ 0, (15) where {λ n } and ...

Viscosity approximation methods for pseudocontractive mappings in Banach spaces

Strong convergence of implicit viscosity approximation methods for pseudocontractive mappings in Banach spaces Lu-Chuan Ceng a b , Adrian Petruşel c , Mu-Ming Wong d & Su-Jane Yu e a Department of Mathematics, Shanghai Normal University, Shanghai 200234, China b Scientific Computing Key Laboratory of Shanghai Universities, Shanghai, China c Department of Applied Mathematics, Babeş-Bolyai Univer...

On the Convergence of Iterative Processes for Generalized Strongly Asymptotically ϕ-Pseudocontractive Mappings in Banach Spaces

Throughout this paper, we assume that X is a uniformly convex Banach space and X∗ is the dual space of X. Let J denote the normalized duality mapping form X into 2 ∗ given by J x {f ∈ X∗ : 〈x, f〉 ‖x‖2 ‖f‖2} for all x ∈ X, where 〈·, ·〉 denotes the generalized duality pairing. It is well known that if X is uniformly smooth, then J is single valued and is norm to norm uniformly continuous on any b...

On the Convergence of Implicit Picard Iterative Sequences for Strongly Pseudocontractive Mappings in Banach Spaces

⟨(I − T) x − (I − T) y, j (x − y)⟩ ≥ k 󵄩󵄩󵄩󵄩x − y 󵄩󵄩󵄩󵄩 2 (4) for all x, y ∈ D(T), where k = (t − 1)/t ∈ (0, 1). Consequently, from inequality (4) it follows easily that T is strongly pseudocontractive if and only if 󵄩󵄩󵄩󵄩x − y 󵄩󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩󵄩x − y + s [(I − T − kI) x − (I − T − kI) y] 󵄩󵄩󵄩󵄩 (5) for all x, y ∈ D(T) and s > 0. Closely related to the class of pseudocontractive maps is the class of accret...

On Convergence Results for Lipschitz Pseudocontractive Mappings