On Some Lacunary Almost Convergent Double Sequence Spaces and Banach Limits

Metin Başarır,Şükran Konca

doi:10.1155/2012/426357

Abstract

The object of this paper is to introduce some new sequence spaces related with the concept of lacunary strong almost convergence for double sequences and also to characterize these spaces through sublinear functionals that both dominate and generate Banach limits and to establish some inclusion relations.

Highlights

Introduction and PreliminariesLet w2 be the set of all real or complex double sequences
The notion of almost convergence for single sequences was introduced by Lorentz 3 and for double sequences by Moricz and Rhoades 2 and some further studies are in 4–14
The notion of strong almost convergence for single sequences has been introduced by Maddox 15, and for double sequences by Basarir

Summary

Introduction and Preliminaries

Let w2 be the set of all real or complex double sequences. We mean the convergence in the Pringsheim sense, that is, a double sequence x xi,j. The notion of almost convergence for single sequences was introduced by Lorentz 3 and for double sequences by Moricz and Rhoades 2 and some further studies are in 4–14. The notion of strong almost convergence for single sequences has been introduced by Maddox 15, 16 and for double sequences by Basarir 17. The following sequence spaces were introduced and examined by Basarir 19 : wθ x: lim r sup i. 0, for some s , with respect to sublinear functionals on l∞ the set of all real or complex bounded single sequences by φθ x lim r sup i. The object of the present paper is to determine some new sublinear functionals involving double-lacunary sequence that both dominates and generates Banach limits. We extend the sequence spaces which were introduced for single sequences by Basarir 19 to the double sequences with respect to these sublinear functionals. We present some inclusion relations with these new sequence spaces between the sequence spaces which were introduced by Mursaleen and Mohiuddine 7 , earlier

Sublinear Functionals and Double-Lacunary Sequence Spaces

Inclusion Relations