Abstract and Applied Analysis

Volume 2012 (2012), Article ID 680456, 8 pages

doi:10.1155/2012/680456

Research Article

-Ward Continuity

Huseyin Cakalli

Department of Mathematics, Maltepe University, Marmara Education Village, 34857 Istanbul, Turkey

Received 8 March 2012; Accepted 19 April 2012

Academic Editor: Ljubisa Kocinac

Copyright © 2012 Huseyin Cakalli. 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.

A function is continuous if and only if preserves convergent sequences; that is, is a convergent sequence whenever is convergent. The concept of -ward continuity is defined in the sense that a function is -ward continuous if it preserves -quasi-Cauchy sequences; that is, is an -quasi-Cauchy sequence whenever is -quasi-Cauchy. A sequence of points in , the set of real numbers, is -quasi-Cauchy if , where , and is a lacunary sequence, that is, an increasing sequence of positive integers such that and . A new type compactness, namely, -ward compactness, is also, defined and some new results related to this kind of compactness are obtained.

1. Introduction

It is well known that a real function is continuous if and only if, for each point in the domain, whenever . This is equivalent to the statement that is a convergent sequence whenever is. This is also equivalent to the statement that is a Cauchy sequence whenever is Cauchy provided that the domain of the function is either whole or a bounded and closed subset of where is the set of real numbers. These well known results for continuity for real functions in terms of sequences suggested to introduce and study new types of continuities such as slowly oscillating continuity [1], quasi-slowly oscillating continuity [2], -quasi-slowly oscillating continuity [3], forward continuity [4], statistical ward continuity [5] which enabled some authors to obtain some characterizations of uniform continuity in terms of sequences in the sense that a function preserves either quasi-Cauchy sequences or slowly oscillating sequences (see [6–8]).

The purpose of this paper is to introduce a new kind of continuity and a new type of compactness, namely, -ward continuity and -ward compactness, respectively, in the senses that a function is -ward continuous if preserves -quasi-Cauchy sequences, and a subset of is -ward compact if any sequence of points in has an -quasi-Cauchy subsequence and to investigate relations among this kind of continuity, compactness, and some other types of continuities.

2. Preliminaries

We will use boldface letters for sequences , and ,… of points in for the sake of abbreviation. and will denote the set of all sequences and the set of convergent sequences of points in .

A subset of is compact if and only if it is closed and bounded. A subset of is bounded if for all where is a positive real constant number. This is equivalent to the statement that any sequence of points in has a Cauchy subsequence. The concept of a Cauchy sequence involves far more than that the distance between successive terms is tending to zero. Nevertheless, sequences which satisfy this weaker property are interesting in their own right. A sequence of points in is quasi-Cauchy if is a null sequence where . These sequences were named as quasi-Cauchy by Burton and Coleman [8, page 328], while they were called as forward convergent to 0 sequences in [4, page 226].

It is known that a sequence of points in is slowly oscillating if

where denotes the integer part of (see [9, Definition 2 page 947]). Any Cauchy sequence is slowly oscillating, and any slowly oscillating sequence is quasi-Cauchy. There are quasi-Cauchy sequences which are not Cauchy. For example, the sequence is quasi-Cauchy, but not Cauchy. Any subsequence of a Cauchy sequence is Cauchy. The analogous property fails for quasi-Cauchy sequences, and fails for slowly oscillating sequences as well. A counter example for the case, quasi-Cauchy, is again the sequence with the subsequence . A counter example for the case slowly oscillating is the sequence with the subsequence . Furthermore we give more examples without neglecting: the sequences and combinations like that are all slowly oscillating, but not Cauchy. The bounded sequence is slowly oscillating, but not Cauchy. The sequences and are quasi-Cauchy, but not slowly oscillating.

By a method of sequential convergence, or briefly a method, we mean a linear function defined on a subspace of , denoted by , into . A sequence is said to be -convergent to if and [10]. In particular, denotes the limit function on the space of convergent sequences of points in . A method is called regular if ; that is, every convergent sequence is -convergent with . A point in is in the -sequential closure of a subset of if there is a sequence of points in such that . A subset is called -sequentially closed if it contains all of the points in its -sequential closure.

Consider an infinite matrix of real numbers. Then, for any sequence the sequence is defined as

provided that each of the series converges. A sequence is called -convergent (or -summable) to if exists and is convergent with

Then is called the -limit of . We have thus defined a method of sequential convergence, that is, , called a matrix method or a summability matrix.

The concept of statistical convergence is a generalization of the usual notion of convergence that, for real-valued sequences, parallels the usual theory of convergence. A sequence of points in is called statistically convergent to an element of if for each

and this is denoted by st- (see [11–15]). This defines a method of sequential convergence, that is, .

Now we recall the concepts of ward compactness, and slowly oscillating compactness: a subset of is called ward compact if whenever is a sequence of points in , there is a quasi-Cauchy subsequence of [4]. A subset of is called slowly oscillating compact if whenever is a sequence of points in , there is a slowly oscillating subsequence of [1].

A function is called -sequentially continuous at if, given a sequence of points in , implies that .

Recently, Cakalli (see [16, page 594], [17]) gave a sequential definition of compactness, which is a generalization of ordinary sequential compactness, as in the following: a subset of is -sequentially compact if for any sequence of points in there exists a subsequence of the sequence such that . His idea enables us obtaining new kinds of compactness via most of the nonmatrix sequential convergence methods as well as all matrix sequential convergence methods.

3. -Quasi-Cauchy Sequences

A lacunary sequence is an increasing sequence of positive integers such that and . The intervals determined by will be denoted by , and the ratio will be abbreviated by . Sums of the form frequently occur, and will often be written for convenience as . Throughout this paper, we will assume that .

The notion of convergence was introduced and studied by Freedman et al. in [18]. Basarir and Altundag studied --asymptotically equivalent sequences in [19]. Using the idea of Sember and Raphael, Fridy and Orhan introduced lacunary statistical convergence (see [20, 21]).

A sequence of points in is called -convergent to an element of if

and it is denoted by . This defines a method of sequential convergence, that is, . Any convergent sequence is -convergent, but the converse is not always true. Throughout the paper will denote the set of convergent sequences of points in .

For example, limit of the sequence of the ratios of Fibonacci numbers converge, to the golden mean. This property ensures the regularity of lacunary sequential method obtained via the sequence of Fibonacci numbers; that is, is the lacunary sequence defined by writing and where is the Fibonacci sequence, that is, , , and for .

Now we modify the definition of -sequential compactness to the special case, [16] as in the following: a subset , of is called -sequentially compact if whenever is a sequence of points in there is an -convergent subsequence of whose -limit is in .

Adopting the technique in the proof of the necessity of Theorem  6 in [22], we see that the sequential method is subsequential. It follows from [16, Corollary 5, page 597] that a subset of is sequentially compact if and only if it is -sequentially compact. A subset of is closed and bounded if and only if it is -sequentially compact. A subset of is -sequentially compact if and only if it is -sequentially compact for any regular subsequential method .

In connection with -convergent sequences and convergent sequences the problem arises to investigate the following types of continuity of functions on :

We see that is -sequential continuity of , and is the ordinary continuity of . It is easy to see that implies , and does not imply ; implies , and does not imply ; implies , and does not imply ; and is equivalent to .

If a function is -sequentially continuous at a point , then it is continuous at . If a function is -sequentially continuous on a subset of , then it is statistically continuous on . We obtain from [16, Theorem 7, page 597] that -sequentially continuous image of any -sequentially compact subset of is -sequentially compact.

In [23] a nonempty subset of a is called -sequentially connected if there are no nonempty and disjoint -sequentially closed subsets and such that , and and are nonempty. As far as -sequentially connectedness is considered, we see that -sequentially continuous image of any -sequentially connected subset of is -sequentially connected, so -sequentially continuous image of any interval is an interval. Furthermore it can be easily seen that a subset of is -sequentially connected if and only if it is connected in the ordinary sense, and so it is an interval.

Definition 3.1. A sequence of points in is called -quasi-Cauchy if is -convergent to 0. will denote the set of all -quasi-Cauchy sequences of points in .

We note that -quasi-Cauchy sequences were studied in [19] in a different point of view.

Now we give the definition of -ward compactness.

Definition 3.2. A subset of is called -ward compact if whenever is a sequence of points in , there is an -quasi-Cauchy subsequence of .

Theorem 3.3. A subset of is bounded if and only if it is -ward compact.

Proof. Let be any bounded subset of and let be any sequence of points in . is also a sequence of points in where denotes the closure of . As is sequentially compact, there is a convergent subsequence of (no matter the limit is in or not). This subsequence is -convergent since -method is regular. Hence is -quasi-Cauchy. Thus (a) implies (b). To prove that (b) implies (a), suppose that is unbounded. If it is unbounded above, then one can construct a sequence of numbers in such that for each positive integer . Then the sequence does not have any -quasi-Cauchy subsequence, so is not -ward compact. If is bounded above and unbounded below, then similarly we obtain that is not -ward compact. This completes the proof of the theorem.

It easily follows from the preceding theorem that a closed subset of is -ward compact if and only if it is -sequentially compact and a closed subset of is -ward compact if and only if it is statistically ward compact.

A sequence is -quasi-Cauchy if where [3]. A subset of is called -ward compact if whenever is a sequence of points in , there is a subsequence of with . It follows from the previous theorem that any -ward compact subset of is -ward compact.

We see that for any regular subsequential method defined on , if a subset of is -sequentially compact, then it is -ward compact. But the converse is not always true.

Now we give the definition of -ward continuity in the following.

Definition 3.4. A function defined on a subset of is called -ward continuous if it preserves -quasi-Cauchy sequences; that is, is an -quasi-Cauchy sequence whenever is.

Sum of two -ward continuous functions is -ward continuous, but product of -ward continuous functions need not be -ward continuous.

In connection with -quasi-Cauchy sequences and convergent sequences the problem arises to investigate the following types of continuity of functions on :

We see that is -ward continuity of , is -sequential continuity of , and is the ordinary continuity of . It is easy to see that implies , and does not imply ; implies , and does not imply ; implies , and does not imply ; clearly implies as we have seen in Section 3.

Now we give the implication that implies ; that is, any -ward continuous function is -sequentially continuous.

Theorem 3.5. If is -ward continuous on a subset of , then it is -sequentially continuous on .

Proof. Assume that is an -ward continuous function on a subset of . Let be any -convergent sequence with . Then the sequence

is also -convergent to . Hence it is -quasi-Cauchy. As is -ward continuous, the sequence

is -quasi-Cauchy. It follows from this that the sequence -converges to . This completes the proof of the theorem.

The converse is not always true for the function is an example since the sequence is -quasi-Cauchy while is not.

Corollary 3.6. If is -ward continuous on a subset of , then it is continuous on .

Proof. The proof immediately follows from the preceding theorem, so it is omitted.

Corollary 3.7. If is -ward continuous on a subset of , then it is statistically continuous on .

It is well known that any continuous function on a compact subset of is uniformly continuous on . It is also true for a regular subsequential method that any -ward continuous function on a -sequentially compact subset of is also uniformly continuous on (see [6]). Furthermore, for -ward continuous functions defined on an -ward compact subset of , we have the following.

Theorem 3.8. Let be an -ward compact subset of and let be an -ward continuous function on . Then is uniformly continuous on .

Proof. Suppose that is not uniformly continuous on so that there exists an such that for any there are with but . For each positive integer , there exist and such that , and . Since is -ward compact, there exists an -quasi-Cauchy subsequence of the sequence . It is clear that the corresponding subsequence of the sequence is also -quasi-Cauchy, since is a sum of three -null sequences, that is,

On the other hand, it follows from the equality that the sequence is -convergent to 0. Hence the sequence

is -quasi-Cauchy. But the transformed sequence

is not -quasi-Cauchy. Thus does not preserve -quasi-Cauchy sequences. This contradiction completes the proof of the theorem.

Corollary 3.9. If a function is -ward continuous on a bounded subset of , then it is uniformly continuous on .

Proof. The proof follows from the preceding theorem and Theorem 3.3.

Theorem 3.10. -ward continuous image of any -ward compact subset of is -ward compact.

Proof. Assume that is an -ward continuous function on a subset of and is an -ward compact subset of . Let be any sequence of points in . Write where for each positive integer . -ward compactness of implies that there is a subsequence of with . Write . As is -ward continuous, is -quasi-Cauchy. Thus we have obtained a subsequence of the sequence with . Thus is -ward compact. This completes the proof of the theorem.

Corollary 3.11. -ward continuous image of any compact subset of is -ward compact.

The proof follows from the preceding theorem.

Corollary 3.12. -ward continuous image of any bounded subset of is bounded.

The proof follows from Theorems 3.3 and 3.10.

Corollary 3.13. -ward continuous image of a -sequentially compact subset of is -ward compact for any regular subsequential method .

For a further study, we suggest to investigate -quasi-Cauchy sequences of fuzzy points and -ward continuity for the fuzzy functions (see [24] 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 analogous to those of the present work.

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