# Convergence Methods for Double Sequences and Applications

New Type Continuities via Abel Convergence

Chapter 1: Almost and statistical convergence of ordinary sequences: A preview. Du kanske gillar.

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Lifespan David Sinclair Inbunden. Spara som favorit. Skickas inom vardagar. This book exclusively deals with the study of almost convergence and statistical convergence of double sequences. The notion of "almost convergence" is perhaps the most useful notion in order to obtain a weak limit of a bounded non-convergent sequence. There is another notion of convergence known as the "statistical convergence", introduced by H.

Fast, which is an extension of the usual concept of sequential limits. By Corollary to Karamata's Hauptsatz on page in [ 18 ], f p n is Cesaro convergent to f l. Thus f is Cesaro continuous at the point l. Hence f is Cesaro continuous at any point in the domain. The proof follows from the preceding theorem, and the theorem on page 73 in [ 23 ].

It is well known that uniform limit of a sequence of continuous functions is continuous. This is also true for Abel continuity; that is, uniform limit of a sequence of Abel continuous functions is Abel continuous. If f n is a sequence of Abel continuous functions defined on a subset E of R and f n is uniformly convergent to a function f , then f is Abel continuous on E. Let p n be an Abel convergent sequence of real numbers in E.

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Since f n is uniformly convergent to f , there exists a positive integer N such that. In the following theorem we prove that the set of Abel continuous functions is a closed subset of the space of continuous functions. Let f be any element in the closure of A C E. To show that f is Abel continuous, take any Abel convergent sequence p n of points E with Abel limit l.

Since f n is convergent to f , there exists a positive integer N such that. The set of all Abel continuous functions on a subset E of R is a complete subspace of the space of all continuous functions on E. The proof follows from the preceding theorem and the fact that the set of all continuous functions on E is complete.

Abel continuous image of any Abel sequentially compact subset of R is Abel sequentially compact. Although the proof follows from Theorem 7 in [ 2 ], we give a short proof for completeness. Let f be any Abel continuous function defined on a subset E of R and let F be any Abel sequentially compact subset of E. If a function f is Abel continuous on a subset E of R , then. The proof follows from the regularity of Abel method and Theorem 8 on page of [ 3 ].

For any regular subsequential method G , if a subset of R is G -sequentially compact, then it is Abel sequentially compact.

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The proof can be obtained by noticing the regularity and subsequentiality of G see [ 2 ] for the detail of G -sequential compactness. It is clear that any bounded and closed subset of R is Abel sequentially compact. It is easily seen that the sequence p has no Abel convergent subsequence. If it is unbounded below, then similarly we construct a sequence of points in E which has no Abel convergent subsequence. Hence E is not Abel sequentially compact. Every subsequence of p also converges to l. Since Abel method is regular, every subsequence of p Abel converges to l.

Since l is not a member of E , E is not Abel sequentially compact. This contradiction completes the proof that Abel sequentially compactness implies boundedness and closedness. In this paper we introduce a concept of Abel continuity and a concept of Abel sequential compactness and present theorems related to this kind of sequential continuity, this kind of sequential compactness, continuity, statistical continuity, lacunary statistical continuity, and uniform continuity.

One may expect this investigation to be a useful tool in the field of analysis in modeling various problems occurring in many areas of science, dynamical systems, computer science, information theory, and biological science.

## New Type Continuities via Abel Convergence

On the other hand, we suggest to introduce a concept of fuzzy Abel sequential compactness and investigate fuzzy Abel continuity for fuzzy functions see [ 26 ] for the definitions and related concepts in fuzzy setting. However due to the change in settings, the definitions and methods of proofs will not always be the same. An investigation of Abel continuity and Abel compactness can be done for double sequences see [ 27 ] for basic concepts in the double sequences case. It seems both double Abel continuity and Abel continuity coincides but it needs proving.

The authors would like to thank the referees for a careful reading and several constructive comments that have improved the presentation of the results. The authors declare that there is no conflict of interests regarding the publication of this paper. National Center for Biotechnology Information , U.

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Journal List ScientificWorldJournal v. Published online Apr Author information Article notes Copyright and License information Disclaimer. Received Jan 24; Accepted Mar Cakalli and M. 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 We investigate the concept of Abel continuity. Introduction The concept of continuity and any concept involving continuity play a very important role not only in pure mathematics but also in other branches of sciences involving mathematics especially in computer sciences, information theory, and dynamical systems. Main Results First we introduce two notions. Theorem 5 — A subset A of R is bounded if and only if it is Abel ward compact. Proof — It is an easy exercise to check that bounded subsets of R are Abel ward compact. Theorem 7 — If a function f is Abel continuous on a subset E of R , then it is continuous on E in the ordinary sense. Proof — Suppose that a function f is not continuous on E. Corollary 8 — If f is Abel continuous, then it is statistically continuous. Corollary 9 — If f is Abel continuous, then it is lacunarily statistically sequentially continuous.

Proof — The proof follows from Theorem 7 see [ 20 ].

## Convergence Methods For Double Sequences And Applications

Corollary 10 — If p n is slowly oscillating, Abel convergent, and f is an Abel continuous function, then f p n is a convergent sequence. Proof — If p n is slowly oscillating and Abel convergent, then p n is convergent by the generalized Littlewood Tauberian theorem for the Abel summability method. Corollary 11 — For any regular subsequential method G , any Abel continuous function is G -continuous.

Proof — Let f be an Abel continuous function and G be a regular subsequential method. Theorem 12 — Any bounded Abel continuous function is Cesaro continuous. Volume 67 Issue 6 Nov , pp. Volume 66 Issue 6 Dec , pp. Volume 65 Issue 6 Dec , pp. Volume 64 Issue 6 Dec , pp. Volume 63 Issue 6 Dec , pp. Volume 62 Issue 6 Dec , pp. Volume 61 Issue 6 Dec , pp. Volume 60 Issue 6 Dec , pp. Volume 59 Issue 6 Dec , pp. Volume 58 Issue 6 Dec , pp. Volume 57 Issue 6 Dec , pp. Previous Article.

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