Let C be a bounded closed convex subset of a uniformly convex Banach space X and let = = {T (t) : t ∈ G} be a commutative semigroup of asymptotically nonexpansive in the intermediate mapping from C into itself. In this paper, we provide the strong mean ergodic convergence theorem for the almost-orbit of =.

Let E be a real Banach space with norm ∥.∥ and let J be the normalized duality mapping from E into 2E ∗ given by Jx = {x∗ ∈ E∗ : ⟨x, x∗⟩ = ∥x∥∥x∗∥, ∥x∥ = ∥x∗∥} for all x ∈ E, where E∗ denotes the dual space of E and ⟨., .⟩ denotes the generalized duality pairing between E and E∗. A Banach space E is said to be strictly convex if ∥ 2 ∥ < 1 for all x, y ∈ E with ∥x∥ = ∥y∥ = 1 and x ̸= y. It is sai...

We use techniques of proof mining to obtain a computable and uniform rate metastability (in the sense Tao) for mean ergodic theorem finite number commuting linear contractive operators on uniformly convex Banach space.

let $omega_x$ be a bounded, circular and strictly convex domain of a banach space $x$ and $mathcal{h}(omega_x)$ denote the space of all holomorphic functions defined on $omega_x$. the growth space $mathcal{a}^omega(omega_x)$ is the space of all $finmathcal{h}(omega_x)$ for which $$|f(x)|leqslant c omega(r_{omega_x}(x)),quad xin omega_x,$$ for some constant $c>0$, whenever $r_{omega_x}$ is the m...

It is known that a uniformly convex Banach space is reflexive and strictly convex. E is said to be smooth provided limt→ ‖x+ty‖–‖x‖ t exists for all x, y ∈UE . It is also said to be uniformly smooth if the limit is attained uniformly for all x, y ∈UE . It is well known that if E* is strictly convex, then J is single valued; if E* is reflexive, and smooth, then J is single valued and demicontin...

We discuss renorming properties of the dual of a James tree space JT . We present examples of weakly Lindelöf determined JT such that JT ∗ admits neither strictly convex nor Kadec renorming and of weakly compactly generated JT such that JT ∗ does not admit Kadec renorming although it is strictly convexifiable. The norm of a Banach space is said to be locally uniformly rotund (LUR) if for every ...

Let X be a Banach space with closed unit ball B. Given k ∈ N, X is said to be k-β, repectively, (k + 1)-nearly uniformly convex ((k + 1)-NUC), if for every ε > 0, there exists δ, 0 < δ < 1, so that for every x ∈ B, and every ε-separated sequence (xn) ⊆ B, there are indices (ni) k i=1, respectively, (ni) k+1 i=1 , such that 1 k+1 ‖x + ∑k i=1 xni‖ ≤ 1 − δ, respectively, 1 k+1 ‖ ∑k+1 i=1 xni‖ ≤ 1−...

Constructive properties of uniform convexity, strict convexity, near convexity, and metric convexity in real normed linear spaces are considered. Examples show that certain classical theorems, such as the existence of points of osculation, are constructively invalid. The methods used are in accord with principles introduced by Errett Bishop.