Uniform Opial Property Research Articles

Opial properties in interpolation spaces

AbstractWe study interpolation spaces obtained via a general discrete interpolation method based on a Banach space with an unconditional basis. We find conditions which guarantee that such interpolation spaces have the Opial property and the uniform Opial property.

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.

Some vector valued sequence spaces of Musielak-Orlicz functions and their operator ideals

AbstractIn the present paper we introduce generalized vector-valued Musielak-Orlicz sequence spacel(A,𝓜,u,p,Δr,∥·,… ,·∥)(X) and study some geometric properties like uniformly monotone, uniform Opial property for this space. Further, we discuss the operators ofs-type and operator ideals by using the sequence ofs-number (in the sense of Pietsch) under certain conditions on matrixA.

On some geometric properties of generalized Musielak-Orlicz sequence space and corresponding operator ideals

Let $\bold{\Phi}=(\phi_n)$ be a Musielak-Orlicz function, $X$ be a real Banach space and $A$ be any infinite matrix. In this paper, a generalized vector-valued Musielak-Orlicz sequence space $l_{\bold {\Phi}}^{A}(X)$ is introduced. It is shown that the space is complete normed linear space under certain conditions on the matrix $A$. It is also shown that $l_{\bold{\Phi}}^{A}(X)$ is a $\sigma$- Dedikind complete whenever $X$ is so. We have discussed some geometric properties, namely, uniformly monotone, uniform Opial property for this space. Using the sequence of $s$-number (in the sense of Pietsch), the operators of $s$-type $l_{\bold{\Phi}}^{A}$ and operator ideals under certain conditions on the matrix $A$ are discussed.

Some Geometric Properties of Generalized Cesàro-Musielak-Orlicz Spaces Equipped with the Amemiya Norm

A generalized Cesaro-Musielak-Orlicz sequence space CesΦ(q) endowed with the Amemiya norm is introduced. Criteria for the coordinatewise uniformly Kadec-Klee property and the uniform Opial property of the space CesΦ(q) with respect to the Amemiya norm are obtained.

Some geometric properties of generalized modular sequence space derived by the generalized de la Vallée-Poussin mean

In this paper, we define a generalized modular sequence space by using the generalized de la Vallée-Poussin mean with a generalized Riesz transformation. Moreover, we investigate the property and the uniform Opial property which is equipped with the Luxemburg norm. Finally, we show that this space has the fixed point property.

Some geometric properties of a new difference sequence space defined by de la Vallée-Poussin mean

In this paper, wedefine a new generalized difference sequence space CpΔλm and consider it equipped with the Luxemburg norm under which it is a Banach space and we show that the space CpΔλm possess Banach Saks property of type p, uniform opial property and property (H), where p=(pn) is a bounded sequence of positive real numbers with pn>1 for all n∈N. Also, we give some results about the fixed point theory for the spaces CpΔm and CpΔm1<p<∞.

Some geometric properties of N ( λ , p ) -spaces

In this paper, we introduce the sequence spaces N(λ,p) and we show some geometric properties of that spaces. The main purpose of this paper is to show that N(λ,p) is a Banach space and has the rotund property, the Kadec-Klee property, the uniform Opial property, the (β)-property, the k-NUC property and the Banach-Saks property of type p.MSC:46A45, 40C05, 46B45, 46A35.

Some geometric properties of a new modular space defined by Zweier operator

Abstract In this paper, we define the modular space Z σ ( s , p ) by using the Zweier operator and a modular. Then, we consider it equipped with the Luxemburg norm and also examine the uniform Opial property and property β. Finally, we show that this space has the fixed point property. MSC:40A05, 46A45, 46B20.

On Geometrical Properties of Some Banach Spaces

In this work, we prove that modular spaces V ρ(λ; p) defined in (26) have the k-nearly uniform convexity(k − N U C property) when they are endowed with the Luxemburg norm. We also prove that these spaces have the uniform Opial property with the Luxemburg norm. The above investigated geometric properties will enable us to obtain some fixed point results in modular spaces.

Some topological and geometrical properties of the sequence space er (u,p)

In this paper, we introduce the sequence space er(u,p) and investigate its some topological and geometrical properties such as basis, α-, β-, γ- duals and the uniform Opial property.

On H-property and uniform Opial property of generalized cesàro sequence spaces

In this article, we define the generalized cesaro sequence spaces ces(p)(q) and consider it equipped with the Luxemburg norm. We show that the spaces ces(p)(q) has the H-property and Uniform Opial property. The results of this article, we improve and extend some results of Petrot and Suantai.

Cesaro序列空间cesp (1 < p

Opial模是序列空间最重要数字指标，丰富了Opial性质的内涵，是序列空间的讨论与应用的有力工具，给出了Cesaro序列空间cesp(1 < p < ∞) 的Opial模的计算公式并予以证明；验证了cesp(1 < p < ∞) 具有一致Opial性质。 The Opial modular is the most important digital index in sequence spaces, it riches the intension of the Opial property, and it is a powerful tool for the discussion and the applications in sequence spaces. The calculation formula of Opial modular in Cesaro sequence spacescesp(1 < p< ∞) was given and proved, the fact that cesp(1 < p < ∞) has the uniform Opial property was also proved.

Some Geometric Properties of Lacunary Sequence Spaces Related to Fixed Point Property

The main purpose of this paper is considering the lacunary sequence spaces defined by Karakaya (2007), by proving the property (β) and Uniform Opial property.

On Some Generalized -Difference Riesz Sequence Spaces and Uniform Opial Property

We define the new generalized difference Riesz sequence spaces , , and which consist of all the sequences whose -transforms are in the Riesz sequence spaces , , and , respectively, introduced by Altay and Basar (2006). We examine some topological properties and compute the -, -, and -duals of the spaces , , and . Finally, we determine the necessary and sufficient conditions on the matrix transformation from the spaces , , and to the spaces and and prove that sequence spaces and have the uniform Opial property for for all .

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 .

Weak Convergence of Greedy Algorithms in Banach Spaces

We study the convergence of certain greedy algorithms in Banach spaces. We introduce the WN property for Banach spaces and prove that the algorithms converge in the weak topology for general dictionaries in uniformly smooth Banach spaces with the WN property. We show that reflexive spaces with the uniform Opial property have the WN property. We show that our results do not extend to algorithms which employ a ‘dictionary dual’ greedy step.

Uniform Opial properties in generalized Cesàro sequence spaces

The main purpose of this paper is considering the generalized Cesàro sequence spaces defined by S. Suantai [On some convexity properties of generalized Cesàro sequence spaces, Georgian Math. J. 10(1) (2003) 193–200] by giving some topological properties and finding condition for ces ( p ) equipped with both the Amemiya norm and the Luxemburg norm to possesses uniform Opial property.

On uniform opial condition, Uniform Kadec-Klee property in modular spaces and application to fixed point theory

In this paper we define an uniform opial condition and uniform Kadec-Klee property in sense of modular spaces. The main purpose of this paper is to extend the results of uniform opial condition and uniform Kadec-Klee property in Banach spaces to modular spaces Xρ.

On Opial properties and Opial modulus for Orlicz sequence spaces

Criteria in order that a Orlicz sequence space l Φ (or its subspace h Φ ) has the Opial property, the locally uniform Opial property and the uniform Opial property are given for the Luxemburg norm. For the Orlicz norm only criteria for the positive Opial property and for the uniform Opial property are presented. Formulae for the Opial modulus in l Φ are obtained for both norms. In consequence, Opial modulus is calculated for l Φ generated by Φ(u)=u p (1⩽p<∞) and Φ( u)= u 2+ u 4.