Some geometric properties of a new difference sequence space defined by de la Vallée-Poussin mean Mikail Et a , Murat Karakas ß b , Vatan Karakaya c,⇑ a Department of Mathematics, Firat University, 23119 Elazig, Turkey b Department of Statistics, Bitlis Eren University, 13000 Bitlis, Turkey c Yıldız Technical University, Department of Mathematical Engineering, Davutpasa, Istanbul, Turkey article info Keywords: Cesàro difference sequence space Luxemburg norm Banach Saks property Convex modular Property ðHÞ abstract In this paper, wedefine a new generalized difference sequence space C p ðÞ D m k  and consider it equipped with the Luxemburg norm under which it is a Banach space and we show that the space C p ðÞ D m k  possess Banach Saks property of type p, uniform opial property and property ðHÞ, where p ¼ðp n Þ is a bounded sequence of positive real numbers with p n > 1 for all n 2 N. Also, we give some results about the fixed point theory for the spaces C p ðÞ D m ð Þ and C p D m ð Þ 1 < p < 1 ð Þ. Ó 2014 Elsevier Inc. All rights reserved. 1. Introduction Let X be a real Banach space and let BðXÞ and SðXÞ be the closed unit ball and the unit sphere of X, respectively. A point x 2 SðXÞ is called an extreme point of BX ð Þ if for any y; z 2 BðXÞ the equality 2x ¼ y þ z implies y ¼ z. A Banach space X is said to have property ðHÞ if every weakly convergent sequence on the unit sphere is convergent in norm. For a real vector space X, a function . : X! 0; 1 ½  is called a modular if it satisfies the following conditions: (i) .ðxÞ¼ 0 if and only if x ¼ 0, (ii) .ðaxÞ¼ .ðxÞ for all scalars a with jaj¼ 1, (iii) .ðax þ byÞ 6 .ðxÞþ .ðyÞ for all x; y 2 X and all a; b P 0 with a þ b ¼ 1. The modular . is called convex if (iv) .ðax þ byÞ 6 a.ðxÞþ b.ðyÞ for all x; y 2 X and all a; b P 0 with a þ b ¼ 1. For any modular . on X, the space X . ¼ x 2 X : .ðrxÞ < 1 for some r > 0 f g is called the modular space. A sequence x n ð Þ in X . is called modular convergent to x 2 X . if there exists k > 0 such that q k x n  x ð Þ ð Þ! 0 as n !1. If . is a convex modular, the functions http://dx.doi.org/10.1016/j.amc.2014.01.122 0096-3003/Ó 2014 Elsevier Inc. All rights reserved. ⇑ Corresponding author. E-mail addresses: mikailet68@gmail.com (M. Et), m.karakas33@hotmail.com (M. Karakas ß), vkkaya@yildiz.edu.tr (V. Karakaya). Applied Mathematics and Computation 234 (2014) 237–244 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc