Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Subsequential Convergence Conditions ˙ ¨ Ibrahim Canak, Umit Totur, and Mehmet Dik | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 87414 8 pages doi 2007 87414 Research Article Subsequential Convergence Conditions Ibrahim Canak Umit Totur and Mehmet Dik Received 27 April 2007 Accepted 19 August 2007 Recommended by Martin J. Bohner 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 Au 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 V 0 Au V m Au generated regularly by a sequence in the class A m is also a subsequential convergence condition for L. Copyright 2007 Ibrahim Canak 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. 1. Introduction Let u n be a sequence of real numbers. Let co tx tf and M denote the space of sequences converging to 0 bounded slowly oscillating and moderately oscillating respectively. The classical control modulo of the oscillatory behavior of un is denoted by w j0 u nAun where Au n un - un-1 and u-1 0 and the general control modulo of the oscillatory behavior of integer order m of un is defined 1 inductively by Am u . u - ffn te m-1 u where ffn u 1 n 1 zn 0 uk. The Kronecker identity un-ffn u V Au where Vn Au 1 n 1 n 0 kAuk is well known and used in various steps of proofs of theorems. For each integer m 1 and for all nonnegative integers n we inductively define sequences related to un such as Vnm Au ffn V m-1 Au and ơnm u ơni ơ m-1 u where ơn u un. Throughout this work a different definition of slow oscillation better tailored for our purposes will be used. A sequence u un is slowly oscillating 2 if limA .plim maxn 1 k An uk - un 0 where An denotes the integer part of .
