Weak Uniform Normal Structure Research Articles

Certain geometric structures of $\Lambda$-sequence spaces

The $\Lambda$-sequence spaces $\Lambda_p$ for $1< p\leq\infty$ and its generalization $\Lambda_{\hat{p}}$ for $1<\hat{p}<\infty$, $\hat{p}=(p_n)$ is introduced. The James constants and strong $n$-th James constants of $\Lambda_p$ for $1<p\leq\infty$ is determined. It is proved that generalized $\Lambda$-sequence space $\Lambda_{\hat{p}}$ is embedded isometrically in the Nakano sequence space $l_{\hat{p}}(\mathbb{R}^{n+1})$ of finite dimensional Euclidean space $\mathbb{R}^{n+1}$. Hence it follows that sequence spaces $\Lambda_p$ and $\Lambda_{\hat{p}}$ possesses the uniform Opial property, property $(\beta)$ of Rolewicz and weak uniform normal structure. Moreover, it is established that $\Lambda_{\hat{p}}$ possesses the coordinate wise uniform Kadec-Klee property. Further necessary and sufficient conditions for element $x\in S(\Lambda_{\hat{p}})$ to be an extreme point of $B(\Lambda_{\hat{p}})$ are derived. Finally, estimation of von Neumann-Jordan and James constants of two dimensional $\Lambda$-sequence space $\Lambda_2^{(2)}$ is being carried out.

Existence of fixed points of some classes of nonlinear mappings in spaces with weak uniform normal structure

In this paper, we prove some fixed point results for some classes of nonlinear mappings recently introduced by Okeke and Olaleru [5]. Our results improves several other known results in literature, including the results of Sahu et al. [8] and Sahu [7].

Existence and approximation of fixed points of nonlinear mappings in spaces with weak uniform normal structure

It is shown that in a Banach space X with weak uniform normal structure, every demicontinuous asymptotically regular nearly Lipschitzian self-mapping T with lim supn→∞η(Tn)<WCS(X) defined on a weakly compact convex subset C of X satisfies the (ω)-fixed point property. We show that if X has a uniformly Gâteaux differentiable norm, then the set of fixed points of every asymptotically pseudocontractive nearly nonexpansive mapping T is nonempty and a sunny nonexpansive retract of C. Further, we study the approximation of fixed points of T by Halpern type iteration process: xn+1=αnu+(1−αn)Tnxnfor alln∈N, where u∈C and {αn} is a sequence in (0,1) satisfying appropriate conditions. Our results improve several known existence and convergence fixed point theorems in general Banach spaces for a wider class of nonlinear mappings which are not necessarily Lipschitzian.

Another look at Cesàro sequence spaces

We consider the Cesàro sequence space ces p as a closed subspace of the infinite ℓ p -sum of finite dimensional spaces. We easily obtain many known results, for example, ces p has property ( β) of Rolewicz, uniform Opial property, and weak uniform normal structure. We also consider some generalized Cesàro sequence spaces. Finally, we compute the von Neumann–Jordan and James constants of the two-dimensional Cesàro sequence space ces p ( 2 ) when 1 < p ⩽ 2 .

On weak uniform normal structure in weighted Orlicz sequence spaces

It is proved that a weighted Orlicz sequence space ℓ M ( w ) , equipped with Luxemburg or Amemiya norm has weak uniform normal structure iff ℓ M ( w ) ≅ h M ( w ) for wide class of weight sequences w = { w n } n = 1 ∞ . An example is constructed, where M has not Δ 2 -condition but by choosing a suitable weight sequence lim n → ∞ w n = ∞ we get that ℓ M ( w ) has weak uniform normal structure.

Weak Uniform Normal Structure and Fixed Points of Asymptotically Regular Semigroups

Let X be a Banach space with a weak uniform normal structure and C a non–empty convex weakly compact subset of X. Under some suitable restriction, we prove that every asymptotically regular semigroup T = {T(t) : t ∈¸ S} of selfmappings on C satisfying $$ {\mathop {\lim \inf }\limits_{S \mathrel\backepsilon t \to \infty } }{\left| {{\left\| {T(t)} \right\|}} \right|} < {\text{WCS}}(X) $$

On Property (M) and Its Generalizations

Properties strict (M) and uniform (M) are introduced. It is shown that if X has property (M) and is uniformly convex in every direction, then X has both strict (M) and uniform (M). It is also shown that if X* is separable, then strict (M) implies uniform (M) and property (M) implies weak uniform normal structure. Relations with other geometrical properties of Banach spaces are also discussed.

Fixed points of asymptotically regular semigroups in Banach spaces

In this paper we study in Banach spaces the existence of fixed points of (nonlinear) asymptotically regular semigroups. We establish for these semigroups some fixed point theorems in spaces with weak uniform normal structure, in a Hilbert space, inLp spaces, in Hardy spacesHp and in Sobolev spacesWr.p for 1<p<∞ andr≥0, in spaces with Lifshitz’s constant greater than one. These results are the generalizations of [8, 10, 16].

Weak uniform normal structure and iterative fixed points of nonexpansive mappings

This paper is concerned with weak uniform normal structure and iterative fixed points of nonexpansive mappings. Precisely, in Section 1, we show that the geometrical coefficient β(X) for a Banach space X recently introduced by Jimenez-Melado [8] is exactl

Iterative fixed points of non-Lipschitzian self-mappings

In this paper, we shall establish iterative fixed points of non-Lipschitzian continuous self-mappings on Banach spaces with weak uniform normal structure.

Weak uniform normal structure in direct sum spaces

Weak uniform normal structure in direct sum spaces