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

Abdul Latif,Wutiphol Sintunavarat,Chirasak Mongkolkeha

doi:10.1186/1029-242x-2014-375

Abstract

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.

Highlights

1 Introduction A number of mathematicians are studying the geometric properties of Banach spaces, because such properties were identified as an important characteristic of the Banach spaces
In, Şimşek et al [ ] introduced a new modular sequence space which is more general than the Cesàro sequence space defined by Shiue [ ] and the generalized Cesàro sequence space defined by Suantai
The main purpose of this paper is to investigate the property (β) and the uniform Opial property equipped with the Luxemburg norm of the new modular sequence space, which is defined by using the generalized de la Vallée-Poussin mean with generalized Riesz transformation

Summary

Introduction

Space defined by Shiue [ ] and the generalized Cesàro sequence space defined by Suantai. In , Mongkolkeha and Kumam [ ] defined the generalized Cesàro sequence space ces(p)(q) for a bounded sequence p = (pk) with pk ≥ for all k ∈ N and q = (qk) of positive real numbers. Şimşek et al [ , ] defined it by the modular sequence space with de la Vallée-Poussin’s mean and studied some geometric properties in these spaces. The generalized Cesàro sequence space ces(p) for p = (pk) a bounded sequence of positive real numbers with pk ≥ for all k ∈ N of Suantai [ , ] is defined by ces(p) = x ∈ l : (λx) < ∞ for some λ > , where. The modular sequence space V (λ; p) of Şimşek et al [ , ] is defined by de la ValléePoussin’s mean, namely

Then the

Then for each x

Since again