Journal of Inequalities and Applications
Volume 2007 (2007), Article ID 087414, 8 pages
doi:10.1155/2007/87414
Research Article

Subsequential Convergence Conditions

İbrahim Çanak,1 Ümit Totur,1 and Mehmet Dik2

1Department of Mathematics, Adnan Menderes University, Aydin 09010, Turkey
2Department of Mathematics, Rockford College, 5050 E. State Street, Rockford, IL 61108, USA

Received 27 April 2007; Accepted 19 August 2007

Academic Editor: Martin J. Bohner

Copyright © 2007 İbrahim Çanak et al. 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

Let (un) be a sequence of real numbers and let L be any (C,1) regular limitable method. We prove that, under some assumptions, if a sequence (un) or its generator sequence (Vn(0)(Δu)) generated regularly by a sequence in a class 𝒜 of sequences is a subsequential convergence condition for L, then for any integer m≥1, the mth repeated arithmetic means of (Vn(0)(Δu)), (Vn(m)(Δu)), generated regularly by a sequence in the class 𝒜(m), is also a subsequential convergence condition for L.