Journal of Inequalities and Applications
Volume 2011 (2011), Article ID 485730, 17 pages
doi:10.1155/2011/485730
Research Article

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

Metin Başarır and Mahpeyker Öztürk

Department of Mathematics, Sakarya University, 54187 Sakarya, Turkey

Received 29 November 2010; Accepted 18 January 2011

Academic Editor: Radu Precup

Copyright © 2011 Metin Başarır and Mahpeyker Öztürk. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We define the new generalized difference Riesz sequence spaces 𝑟 𝑞 ∞ ( 𝑝 , 𝐵 𝑚 ) , 𝑟 𝑞 𝑐 ( 𝑝 , 𝐵 𝑚 ) , and 𝑟 𝑞 0 ( 𝑝 , 𝐵 𝑚 ) which consist of all the sequences whose 𝐵 𝑚 -transforms are in the Riesz sequence spaces 𝑟 𝑞 ∞ ( 𝑝 ) , 𝑟 𝑞 𝑐 ( 𝑝 ) , and 𝑟 𝑞 0 ( 𝑝 ) , respectively, introduced by Altay and Başar (2006). We examine some topological properties and compute the 𝛼 -, 𝛽 -, and 𝛾 -duals of the spaces 𝑟 𝑞 ∞ ( 𝑝 , 𝐵 𝑚 ) , 𝑟 𝑞 𝑐 ( 𝑝 , 𝐵 𝑚 ) , and 𝑟 𝑞 0 ( 𝑝 , 𝐵 𝑚 ) . Finally, we determine the necessary and sufficient conditions on the matrix transformation from the spaces 𝑟 𝑞 ∞ ( 𝑝 , 𝐵 𝑚 ) , 𝑟 𝑞 𝑐 ( 𝑝 , 𝐵 𝑚 ) , and 𝑟 𝑞 0 ( 𝑝 , 𝐵 𝑚 ) to the spaces 𝑙 ∞ and 𝑐 and prove that sequence spaces 𝑟 𝑞 0 ( 𝑝 , 𝐵 𝑚 ) and 𝑟 𝑞 𝑐 ( 𝑝 , 𝐵 𝑚 ) have the uniform Opial property for 𝑝 𝑘 ≥ 1 for all 𝑘 ∈ ℕ .