Journal of Inequalities and Applications 
Volume 2007 (2007), Article ID 87414, 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 61108, IL, USA
Received 27 April 2007; Accepted 19 August 2007

Recommended by Martin J. Bohner

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.