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

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 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...

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 ...

Convergence of Ishikawa iterative sequence for strongly pseudocontractive operators in arbitrary Banach spaces

Under the condition of removing the restriction any bounded, we give the convergence of the Ishikawa iteration process to a unique fixed point of a strongly pseudocontractive operator in arbitrary real Banach space. Furthermore, general convergence rate estimate is given in our results, which extend the recent results of Ciric [3] and Soltuz [12]. AMS subject classifications: Primary 47H10; Sec...

Convergence of an Implicit Iteration for Affine Mappings in Normed and Banach Spaces