Mehmet Dik

New Types of Continuities

A new concept of quasi slowly oscillating continuity is introduced. Furthermore, it is shown that this kind of continuity implies ordinary continuity, but the converse is not always true.

Applications of subsequential Tauberian theory to classical Tauberian theory

AbstractIn this work, we introduce some classical and neoclassical Tauberian-like conditions to retrieve subsequential convergence of areal sequence {u n } and some other sequences related to it out of the boundedness of the sequence {u n }. Consequently we obtainsignificant information about subsequential behavior of the sequence.c 2007 Elsevier Ltd. All rights reserved. Keywords: Subsequentially convergent sequences; Slowly oscillating sequences 1. IntroductionClassical Tauberian theory studies the class of all sequences {u n } for whichlim x→1− (1− x)X ∞n=0 u n x n (1)exists. The existence of the limit (1) does not necessarily imply convergence of {u n }. However, it controls thedivergence of {u n }. Therefore, the class of all sequences for which the limit (1) exists is the class of all sequenceswhose divergence is manageable. One of the main objectives of classical Tauberian theory is to retrieve convergentsequences out of the existence of the limit (1) by restricting the oscillatory behavior of sequences {u

Δ-quasi-slowly oscillating continuity

In this paper, a new concept of Δ-quasi-slowly oscillating continuity is introduced. Furthermore, it is shown that this kind of continuity implies ordinary continuity. A new type of compactness is also defined and some new results related to compactness are proved.

One-sided Tauberian conditions for (A,k) summability method

In this paper, some one-sided Tauberian conditions for (A,k) summability method have been obtained.

ON A THEOREM OF W. MEYER-KÖNIG AND H. TIETZ

Let (un) be a sequence of real numbers and let L be an additive limitable method with some property. We prove that if the classical control modulo of the oscillatory behavior of (un) belonging to some class of sequences is a Tauberian condition for L, then convergence or subsequential convergence of (un) out of L is recovered depending on the conditions on the general control modulo of the oscillatory behavior of different order.

Subsequential Convergence Conditions

Let be a sequence of real numbers and let be any regular limitable method. We prove that, under some assumptions, if a sequence or its generator sequence generated regularly by a sequence in a class of sequences is a subsequential convergence condition for, then for any integer, the repeated arithmetic means of,, generated regularly by a sequence in the class, is also a subsequential convergence condition for.

Conditions for convergence and subsequential convergence

Abstract Let ( u n ) be a sequence, regularly generated by another sequence ( α n ) where either ( α n ) or ( Δ α n ) = ( α n − α n − 1 ) is slowly oscillating. We investigate conditions under which the sequence ( u n ) converges or converges subsequentially.

Some conditions under which subsequential convergence follows from boundedness

Abstract Let ( u n ) be a sequence of real numbers. We obtain some Tauberian-like conditions in terms of the general control modulo of integer order to retrieve subsequential convergence of ( u n ) from the boundedness of ( u n ) .

One-sided Tauberian conditions for a general summability method

Abstract Let ( u n ) be a sequence of real numbers and L be an additive summability method with some property. We show that if slow decrease of ( u n ) or one-sided boundedness of the classical control modulo of the oscillatory behavior of ( u n ) is a Tauberian condition for a general summability method L , then one-sided boundedness by a sequence with certain conditions of the general control modulo of the oscillatory behavior of integer order m of ( u n ) is also a Tauberian condition for L .

On Tauberian theorems for (A, k) summability method

Let (un) be a sequence of real numbers. In this paper we introduce some Tauberian conditions in terms of regularly generated sequences for (A, k) summability method.