- Research
- Open access
- Published:
Ideal quasi-Cauchy sequences
Journal of Inequalities and Applications volume 2012, Article number: 234 (2012)
Abstract
An ideal I is a family of subsets of positive integers N which is closed under taking finite unions and subsets of its elements. A sequence of real numbers is said to be I-convergent to a real number L if for each , the set belongs to I. We introduce I-ward compactness of a subset of R, the set of real numbers, and I-ward continuity of a real function in the senses that a subset E of R is I-ward compact if any sequence of points in E has an I-quasi-Cauchy subsequence, and a real function is I-ward continuous if it preserves I-quasi-Cauchy sequences where a sequence is called to be I-quasi-Cauchy when is I-convergent to 0. We obtain results related to I-ward continuity, I-ward compactness, ward continuity, ward compactness, ordinary compactness, ordinary continuity, δ-ward continuity, and slowly oscillating continuity.
MSC: 40A35, 40A05, 40G15, 26A15.
1 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 science, information theory, and biological science.
A subset E of R, the set of real numbers, is compact if any open covering of E has a finite subcovering where R is the set of real numbers. This is equivalent to the statement that any sequence of points in E has a convergent subsequence whose limit is in E. A real function f is continuous if and only if is a convergent sequence whenever is. Regardless of limit, this is equivalent to the statement that is Cauchy whenever is. Using the idea of continuity of a real function and the idea of compactness in terms of sequences, Çakallı [1] introduced the concept of ward continuity in the sense that a function f is ward continuous if it preserves quasi-Cauchyness, i.e., is a quasi-Cauchy sequence whenever is, and a subset E of R is ward compact if any sequence of points in E has a quasi-Cauchy subsequence of the sequence x where a sequence is called quasi-Cauchy if (see also [2], [3] and [4]). In [5] a real-valued function defined on a subset E of R is called δ-ward continuous if it preserves δ-quasi Cauchy sequences where a sequence is defined to be δ-quasi Cauchy if the sequence is quasi-Cauchy. A subset E of R is said to be δ-ward compact if any sequence of points in E has a δ-quasi-Cauchy subsequence of the sequence x.
A sequence of points in R is slowly oscillating if for any given , there exists a and an such that if and . A function defined on a subset E of R is called slowly oscillating continuous if it preserves slowly oscillating sequences (see [6]). A function defined on a subset E of R is called quasi-slowly oscillating continuous on E if it preserves quasi-slowly oscillating sequences of points in E where a sequence is called quasi-slowly oscillating if is a slowly oscillating sequence [7].
The purpose of this paper is to investigate the concept of ideal ward compactness of a subset of R and the concept of ideal ward continuity of a real function which cannot be given by means of any summability matrix and to prove related theorems.
2 Preliminaries
First of all, some definitions and notation will be given in the following. Throughout this paper, N and R will denote the set of all positive integers and the set of all real numbers, respectively. We will use boldface letters , , , … for sequences , , , … of terms in R. c and S will denote the set of all convergent sequences and the set of all statistically convergent sequences of points in R, respectively.
Following the idea given in a 1946 American Mathematical Monthly problem [8], a number of authors such as Posner [9], Iwinski [10], Srinivasan [11], Antoni [12], Antoni and Salat [13], Spigel and Krupnik [14] have studied A-continuity defined by a regular summability matrix A. Some authors, Öztürk [15], Savaş and Das [16], Borsik and Salat [17], have studied A-continuity for methods of almost convergence or for related methods.
The idea of statistical convergence was formerly given under the name ‘almost convergence’ by Zygmund in the first edition of his celebrated monograph published in Warsaw in 1935 [18]. The concept was formally introduced by Fast [19] and later was reintroduced by Schoenberg [20], and also independently by Buck [21]. Although statistical convergence was introduced over nearly the last seventy years, it has been an active area of research for twenty years. This concept has been applied in various areas such as number theory [22], measure theory [23], trigonometric series [18], summability theory [24], locally convex spaces [25], in the study of strong integral summability [26], turnpike theory [27–29], Banach spaces [30], metrizable topological groups [31], and topological spaces [32, 33]. It should be also mentioned that the notion of statistical convergence has been considered, in other contexts, by several people like R.A. Bernstein and Z. Frolik et al. 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. For a subset M of N, the asymptotic density of M denoted by is given by
if this limit exists where denotes the cardinality of the set . A sequence is statistically convergent to ℓ if
for every . In this case, ℓ is called the statistical limit of x. Schoenberg [20] studied some basic properties of statistical convergence and also studied the statistical convergence as a summability method. Fridy [34] gave characterizations of statistical convergence. Caserta, Maio, and Kočinac [35] studied statistical convergence in function spaces, while Caserta and Kočinac [36] investigated statistical exhaustiveness.
By a lacunary sequence , we mean an increasing sequence of positive integers such that and . The intervals determined by θ will be denoted by , and the ratio will be abbreviated as . In this paper, we assume that . The notion of lacunary statistical convergence was introduced and studied by Fridy and Orhan in [37] and [38] (see also [39] and [40]). A sequence of points in R is called lacunary statistically convergent to an element ℓ of R if
for every positive real number ε. The condition assumed a few lines above ensures the regularity of the lacunary statistical sequential method.
The concept of I-convergence, which is a generalization of statistical convergence, was introduced by Kostyrko, S̆alàt, and Wilczyński [41] by using the ideal I of subsets of N and further studied in [42–46], and [47]. The concept was also studied for double sequences in [48–51], and [52]. Although an ideal is defined as a hereditary and additive family of subsets of a non-empty arbitrary set X, here in our study, it suffices to take I as a family of subsets of N such that for each , and each subset of an element of I is an element of I. A non-trivial ideal I is called admissible if and only if . A non-trivial ideal I is maximal if there cannot exist any non-trivial ideal containing I as a subset. Further details on ideals can be found in Kostyrko et al. (see [41]). Throughout this paper, we assume I is a non-trivial admissible ideal in N. Recall that a sequence of points in R is said to be I-convergent to a real number ℓ if for every . In this case, we write . We say that a sequence of points in R is I-quasi-Cauchy if . We see that I-convergence of a sequence implies I-quasi-Cauchyness of . Throughout the paper, and ΔI will denote the set of all I-convergent sequences and the set of all I-quasi-Cauchy sequences of points in R, respectively.
If we take , then will be a non-trivial admissible ideal in and the corresponding convergence will coincide with the usual convergence. If we take , then will be a non-trivial admissible ideal of N, and the corresponding convergence will coincide with the statistical convergence.
Connor and Grosse-Erdman [53] gave sequential definitions of continuity for real functions calling G-continuity instead of A-continuity. Their results cover the earlier works related to A-continuity where a method of sequential convergence, or briefly a method, is a linear function G defined on a linear subspace of s, denoted by , into R. A sequence is said to be G-convergent to ℓ if and . In particular, lim denotes the limit function on the linear space c, and denotes the statistical limit function on the linear space S. Also denotes the I-limit function on the linear space . A method G is called regular if every convergent sequence is G-convergent with . A method is called subsequential if whenever x is G-convergent with , then there is a subsequence of x with . Since the ordinary convergence implies ideal convergence, I is a regular sequential method [54]. Recently, Cakalli studied new sequential definitions of compactness in [55, 56] and slowly oscillating compactness in [6].
3 Ideal sequential compactness
First, we recall the definition of G-sequentially compactness of a subset E of R. A subset E of R is called G-sequentially compact if whenever is a sequence of points in E, there is subsequence of such that in E (see [57]). For regular methods, any sequentially compact subset E of R is also G-sequentially compact and the converse is not always true. For any regular subsequential method G, a subset E of R is G-sequentially compact if and only if it is sequentially compact in the ordinary sense.
Although I-sequential compactness is a special case of G-sequential compactness when , we state the definition of I-sequential compactness of a subset E of R as follows.
Definition 1 A subset E of R is called I-sequentially compact if whenever is a sequence of points in E, there is an I-convergent subsequence of such that is in E.
Lemma 1 [41]
Sequential method I is regular, i.e., if , then .
Lemma 2 ([58], Proposition 3.2)
Any I-convergent sequence of points in R with an I-limit ℓ has a convergent subsequence with the same limit ℓ in the ordinary sense.
We have the following result which states that any non-trivial admissible ideal I is a regular subsequential sequential method.
Theorem 1 The sequential method I is regular and subsequential.
Proof Regularity of I follows from Lemma 1, and subsequentiality of I follows from Lemma 2. □
Theorem 2 ([55], Corollary 3)
A subset of R is sequentially compact if and only if it is I-sequentially compact.
Although I-sequential continuity is a special case of G-sequential continuity when , we state the definition of I-sequential continuity of a function defined on a subset E of R as follows.
Definition 2 ([54], Definition 2)
A function is I-sequentially continuous at a point if, given a sequence of points in E, implies that .
Theorem 3 Any I-sequentially continuous function at a point is continuous at in the ordinary sense.
Proof Although there is a proof in [54], we give a different proof for completeness. Let f be any I-sequentially continuous function at a point . Since any proper admissible ideal is a regular subsequential method by Theorem 1, it follows from [57], Theorem 13, that f is continuous in the ordinary sense. □
Theorem 4 ([54], Theorem 2.2)
Any continuous function at a point is I-sequentially continuous at .
Combining Theorem 3 and Theorem 4 we have the following corollary.
Corollary 1 A function is I-sequentially continuous at a point if and only if it is continuous at .
As statistical limit is an I-sequential method, we get ([56], Theorem 2).
Corollary 2 A function is statistically continuous at a point if and only if it is continuous at in the ordinary sense.
As lacunary statistical limit is an I-sequential method, we get ([56], Theorem 5).
Corollary 3 A function is lacunarily statistically continuous at a point if and only if it is continuous at in the ordinary sense.
Corollary 4 For any regular subsequential method G, a function is G-sequentially continuous at a point , then it is I-sequentially continuous at .
Proof The proof follows from [57], Theorem 13. □
Corollary 5 Any ward continuous function on a subset E of R is I-sequentially continuous on E.
Theorem 5 If a function is slowly oscillating continuous on a subset E of R, then it is I-sequentially continuous on E.
Proof Let f be any slowly oscillating continuous function on E. It follows from [6], Theorem 2.1, that f is continuous. By Theorem 4, we see that f is I-sequentially continuous on E. This completes the proof. □
Theorem 6 If a function is δ-ward continuous on a subset E of R, then it is I-sequentially continuous on E.
Proof Let f be any δ-ward continuous function on E. It follows from [5], Corollary 2, that f is continuous. By Theorem 4, we obtain that f is I-sequentially continuous on E. This completes the proof. □
Corollary 6 If a function is quasi-slowly oscillating continuous on a subset E of R, then it is I-sequentially continuous on E.
Proof Let f be any quasi-slowly oscillating continuous function on E. It follows from [7], Theorem 3.2, that f is continuous. By Theorem 4, we deduce that f is I-sequentially continuous on E. This completes the proof. □
4 Ideal ward continuity
We say that a sequence is I-ward convergent to a number ℓ if . For the special case , we say that x is ideal quasi-Cauchy or I-quasi-Cauchy.
Now, we give the definition of I-ward compactness of a subset of R.
Definition 3 A subset E of R is called I-ward compact if whenever is a sequence of points in E, there is an I-quasi-Cauchy subsequence of x.
We note that this definition of I-ward compactness cannot be obtained by any G-sequential compactness, i.e., by any summability matrix A, even by the summability matrix defined by if and if and
Despite the fact that G-sequential compact subsets of R should include the singleton set , I-ward compact subsets of R do not have to include the singleton .
Firstly, we note that any finite subset of R is I-ward compact, the union of two I-ward compact subsets of R is I-ward compact, and the intersection of any I-ward compact subsets of R is I-ward compact. Furthermore, any subset of an I-ward compact set is I-ward compact, and any bounded subset of R is I-ward compact. Any compact subset of R is also I-ward compact, and the set N is not I-ward compact. We note that any slowly oscillating compact subset of R is I-ward compact (see [6] for the definition of slowly oscillating compactness). These observations suggest that we have the following result.
Theorem 7 A subset E of R is ward compact if and only if it is I-ward compact.
Proof Let us suppose first that E is ward compact. It follows from [2], Lemma 2, that E is bounded. Then for any sequence , there exists a convergent subsequence of whose limit may be in E or not. Then the sequence is a null sequence. Since I is a regular method, is I-convergent to 0, so it is I-quasi-Cauchy. Thus, E is I-ward compact.
Now, to prove the converse, suppose that E is I-ward compact. Take any sequence of points in E. Then there exists an I-quasi-Cauchy subsequence of . Since I is subsequential, there exists a convergent subsequence of . The sequence is a quasi-Cauchy subsequence of the sequence . Thus, E is ward compact. This completes the proof of the theorem. □
Theorem 8 A subset E of R is bounded if and only if it is I-ward compact.
Proof Using an idea in the proof of [2], Lemma 2, and the preceding theorem, the proof can be obtained easily, so it is omitted. □
Now, we give the definition of I-ward continuity of a real function.
Definition 4 A function f is called I-ward continuous on a subset E of R if whenever for a sequence of terms in E.
We note that this definition of continuity cannot be obtained by any A-continuity, i.e., by any summability matrix A, even by the summability matrix defined by (1). However, for this special summability matrix A, if A-continuity of f at the point 0 implies I-ward continuity of f, then ; and if I-ward continuity of f implies A-continuity of f at the point 0, then .
We note that the sum of two I-ward continuous functions is I-ward continuous, but the product of two I-ward continuous functions need not be I-ward continuous as it can be seen by considering a product of the I-ward continuous function with itself.
In connection with I-quasi-Cauchy sequences and I-convergent sequences, the problem arises to investigate the following types of ‘continuity’ of functions on R:
(δi): ,
(): ,
(c): ,
(): ,
(i): .
We see that (δi) is I-ward continuity of f, (i) is I-continuity of f and (c) states the ordinary continuity of f. It is easy to see that () implies (δi) and (δi) does not imply (); (δi) implies () and () does not imply (δi); () implies (c) and (c) does not imply (); and (c) is equivalent to ().
Now, we give the implication (δi) implies (i), i.e., any I-ward continuous function is I-sequentially continuous.
Theorem 9 If f is I-ward continuous on a subset E of R, then it is I-sequentially continuous on E.
Proof Although the following proof is similar to that of [59], Theorem 1, and that of [7], Theorem 3.2, we give it for completeness. Suppose that f is an I-ward continuous function on a subset E of R. Let be an I-quasi-Cauchy sequence of points in E. Then the sequence
is an I-quasi-Cauchy sequence. Since f is I-ward continuous, the sequence
is an I-quasi-Cauchy sequence. Therefore, . Hence, . It follows that the sequence I-converges to . This completes the proof of the theorem.
The converse is not always true for the function is an example since for the sequence . But , because . □
Theorem 10 If f is I-ward continuous on a subset E of R, then it is continuous on E in the ordinary sense.
Proof Let f be an I-ward continuous function on E. By Theorem 9, f is I-sequentially continuous on E. It follows from Theorem 3 that f is continuous on E in the ordinary sense. Thus, the proof is completed. □
Theorem 11 An I-ward continuous image of any I-ward compact subset of R is I-ward compact.
Proof Suppose that f is an I-ward continuous function on a subset E of R, and E is an I-ward compact subset of R. Let be a sequence of points in . Write , where for each . I-ward compactness of E implies that there is an I-quasi-Cauchy subsequence of . As f is I-ward continuous, is an I-quasi-Cauchy subsequence of y. Thus, is I-ward compact. This completes the proof of the theorem. □
Corollary 7 An I-ward continuous image of any compact subset of R is compact.
Proof The proof of this theorem follows from Theorem 3. □
Corollary 8 An I-ward continuous image of any bounded subset of R is bounded.
Proof The proof follows from Theorem 8 and Theorem 10. □
Corollary 9 An I-ward continuous image of a G-sequentially compact subset of R is G-sequentially compact for any regular subsequential method G.
It is a well-known result that a uniform limit of a sequence of continuous functions is continuous. It is true for slowly oscillating continuous functions ([56], Theorem 12), and quasi-slowly oscillating continuous functions ([7], Theorem 3.5), (see also [59], Theorem 5). This is also true in the case of I-ward continuity, i.e., a uniform limit of a sequence of I-ward continuous functions is I-ward continuous.
Theorem 12 If is a sequence of I-ward continuous functions defined on a subset E of R, and is uniformly convergent to a function f, then f is I-ward continuous on E.
Proof Let and be a sequence of points in E such that . By the uniform convergence of , there exists a positive integer N such that for all whenever . By the definition of ideal convergence, for all , we have
As is I-ward continuous on E, we have
On the other hand, we have
Since I is an admissible ideal, the right-hand side of the relation (3) belongs to I, we have
This completes the proof of the theorem. □
Theorem 13 The set of all I-ward continuous functions on a subset E of R is a closed subset of the set of all continuous functions on E, i.e., , where is the set of all I-ward continuous functions on E, denotes the set of all cluster points of .
Proof Let f be an element in . Then there exists a sequence of points in such that . To show that f is I-ward continuous, consider an I-quasi-sequence of points in E. Since converges to f, there exists a positive integer N such that for all and for all , . By the definition of an ideal for all , we have
As is I-ward continuous on E, we have
On the other hand, we have
Since I is an admissible ideal, the right-hand side of the relation (3) belongs to I, we have
This completes the proof of the theorem. □
We note that [59], Theorem 6, [56], Theorem 10, and [7], Theorem 2.2, Theorem 3.9, supply some other closed subsets of the set of all continuous functions.
Corollary 10 The set of all I-ward continuous functions on a subset E of R is a complete subspace of the space of all continuous functions on E.
Proof The proof follows from the preceding theorem. □
In this paper, two new concepts, namely the concept of I-ward continuity of a real function and the concept of I-ward compactness of a subset of R, were introduced and investigated. In this investigation, we have obtained theorems related to I-ward continuity, I-ward compactness, compactness, sequential continuity, and uniform continuity. We also introduced and studied some other continuities involving I-quasi-Cauchy sequences, statistical sequences, and convergent sequences of points in R. The present work also contains a generalization of results of the paper [1], and some results in [2] and [4].
References
Çakallı H: Forward continuity. J. Comput. Anal. Appl. 2011, 13(2):225–230. MR 2012c:26004
Çakallı H: Statistical ward continuity. Appl. Math. Lett. 2011, 24(10):1724–1728. MR 2012f:40020 10.1016/j.aml.2011.04.029
Burton D, Coleman J: Quasi-Cauchy sequences. Am. Math. Mon. 2010, 117(4):328–333. MR 2011c:40004 10.4169/000298910X480793
Çakallı H: Statistical-quasi-Cauchy sequences. Math. Comput. Model. 2011, 54(5–6):1620–1624. MR 2012f:40006 10.1016/j.mcm.2011.04.037
Çakallı H: Delta quasi-Cauchy sequences. Math. Comput. Model. 2011, 53: 397–401. MR 2011m:26004 10.1016/j.mcm.2010.09.010
Çakallı H: Slowly oscillating continuity. Abstr. Appl. Anal. 2008., 2008: Article ID 485706. doi:10.1155/2008/485706 MR 2009b:26004
Dik M, Canak I: New types of continuities. Abstr. Appl. Anal. 2010., 2010: Article ID 258980. doi:10.1155/2010/258980 MR 2011c:26005, MR 2011c:26005
Buck RC: Solution of problem 4216. Am. Math. Mon. 1948., 55: Article ID 36 MR 15: 26874
Posner EC: Summability preserving functions. Proc. Am. Math. Soc. 1961, 12: 73–76. MR 22 12327 10.1090/S0002-9939-1961-0121591-X
Iwinski TB: Some remarks on Toeplitz methods and continuity. Ann. Soc. Math. Pol., 1 Comment. Math. 1972, 17: 37–43.
Srinivasan VK: An equivalent condition for the continuity of a function. Tex. J. Sci. 1980, 32: 176–177. MR 81f:26001
Antoni J: On the A -continuity of real functions II. Math. Slovaca 1986, 36(3):283–287. MR 88a:26001
Antoni J, Salat T: On the A -continuity of real functions. Acta Math. Univ. Comen. 1980, 39: 159–164. MR 82h:26004
Spigel E, Krupnik N: On the A -continuity of real functions. J. Anal. 1994, 2: 145–155. MR 95h:26004
Öztürk E: On almost-continuity and almost A -continuity of real functions. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 1983, 32: 25–30. MR 86h:26003
Savaş E, Das G: On the A -continuity of real functions. Istanb. Üniv. Fen Fak. Mat. Derg. 1994, 53: 61–66. MR 97m:26004
Borsik J, Salát T: On F -continuity of real functions. Tatra Mt. Math. Publ. 1993, 2: 37–42. MR 94m:26006
Zygmund A: Trigonometric Series. I, II. 3rd edition. Cambridge University Press, Cambridge; 2002. With a foreword by Robert A. Fefferman. Cambridge Mathematical Library. MR 2004h:01041
Fast H: Sur la convergence statistique. Colloq. Math. 1951, 2: 241–244. MR 14:29c
Schoenberg IJ: The integrability of certain functions and related summability methods. Am. Math. Mon. 1959, 66: 361–375. MR 21:3696 10.2307/2308747
Buck RC: Generalized asymptotic density. Am. J. Math. 1953, 75: 335–346. MR 14854f 10.2307/2372456
Erdos P, Tenenbaum G: Sur les densités de certaines suites d’entiers. Proc. Lond. Math. Soc. 1989, 59(3):417–438. MR 90h:11087 10.1112/plms/s3-59.3.417
Miller HI: A measure theoretical subsequence characterization of statistical convergence. Trans. Am. Math. Soc. 1995, 347(5):1811–1819. MR 95h:40010 10.1090/S0002-9947-1995-1260176-6
Freedman AR, Sember JJ: Densities and summability. Pac. J. Math. 1981, 95(2):293–305. MR 82m:10081 10.2140/pjm.1981.95.293
Maddox IJ: Statistical convergence in a locally convex space. Math. Proc. Camb. Philos. Soc. 1988, 104(1):141–145. MR 89k:40012 10.1017/S0305004100065312
Connor J, Swardson MA: Strong integral summability and the Stone-Cech compactification of the half-line. Pac. J. Math. 1993, 157(2):201–224. MR 94f:40007 10.2140/pjm.1993.157.201
Makarov VL, Levin MJ, Rubinov AM Advanced Textbooks in Economics 33. In Mathematical Economic Theory: Pure and Mixed Types of Economic Mechanisms. North-Holland, Amsterdam; 1995.
Mckenzie LW: Turnpike theory. Econometrica 1976, 44(5):841–865. MR 56:17735 10.2307/1911532
Pehlivan S, Mamedov MA: Statistical cluster points and turnpike. Optimization 2000, 48(1):93–106. MR 2001b:49008
Connor J, Ganichev M, Kadets V: A characterization of Banach spaces with separable duals via weak statistical convergence. J. Math. Anal. Appl. 2000, 244(1):251–261. MR 2000m:46042 10.1006/jmaa.2000.6725
Çakallı H: Lacunary statistical convergence in topological groups. Indian J. Pure Appl. Math. 1995, 26(2):113–119. MR 95m:40016
Di Maio G, Kočinac LDR: Statistical convergence in topology. Topol. Appl. 2008, 156: 28–45. MR 2009k:54009 10.1016/j.topol.2008.01.015
Çakallı H, Khan MK: Summability in topological spaces. Appl. Math. Lett. 2011, 24(3):348–352. MR 2011m:40026 10.1016/j.aml.2010.10.021
Fridy JA: On statistical convergence. Analysis 1985, 5: 301–313. MR 87b:40001
Caserta A, Di Maio G, Kočinac LDR: Statistical convergence in function spaces. Abstr. Appl. Anal. 2011., 2011: Article ID 420419. doi:10.1155/2011/420419 MR 2012k:40003
Caserta A, Kočinac LDR: On statistical exhaustiveness. Appl. Math. Lett. 2012, 25(10):1447–1451. doi:10.1016/j.aml.2011.12.022 10.1016/j.aml.2011.12.022
Fridy JA, Orhan C: Lacunary statistical convergence. Pac. J. Math. 1993, 160(1):43–51. MR 94j:40014 10.2140/pjm.1993.160.43
Fridy JA, Orhan C: Lacunary statistical summability. J. Math. Anal. Appl. 1993, 173(2):497–504. MR 95f:40004 10.1006/jmaa.1993.1082
Freedman AR, Sember JJ, Raphael M: Some Cesaro-type summability spaces. Proc. Lond. Math. Soc. 1978, 3(37):508–520. MR 80c:40007
Karakaya V: Some geometric properties of sequence spaces involving lacunary sequence. J. Inequal. Appl. 2007., 2007: Article ID 81028 MR 2009k:46037
Kostyrko P, S̆alàt T, Wilczyński W: I -convergence. Real Anal. Exch. 2000–2001, 26(2):669–686. MR 2002e:54002
Mursaleen M, Alotaibi A: On I -convergence in random 2-normed spaces. Math. Slovaca 2011, 61(6):933–940. MR 2012k:40006 10.2478/s12175-011-0059-5
Sahiner A, Gurdal M, Saltan S, Gunawan H: Ideal convergence in 2-normed spaces. Taiwan. J. Math. 2007, 11: 1477–1484. MR 2008j:46006
Savas E: On some new sequence spaces in 2-normed spaces using ideal convergence and an Orlicz function. J. Inequal. Appl. 2010., 2010: Article ID 482392 MR 2011k:46003
Mursaleen M, Mohiuddine SA: On ideal convergence in probabilistic normed spaces. Math. Slovaca 2012, 62: 49–62. 10.2478/s12175-011-0071-9
Mursaleen M, Abdullah A, Mohammed AA: I -Summability and I -approximation through invariant mean. J. Comput. Anal. Appl. 2012, 14(6):1049–1058.
Komisarski A: Pointwise I -convergence and -convergence in measure of sequences of functions. J. Math. Anal. Appl. 2008, 340: 770–779. MR 2009j:28010 10.1016/j.jmaa.2007.09.016
Kumar V: On I and -convergence of double sequences. Math. Commun. 2007, 12: 171–181. MR 2009c:40007
Mursaleen M, Mohiuddine SA: On ideal convergence of double sequences in probabilistic normed spaces. Math. Rep. 2010, 12(62)(4):359–371. MR 2012a:40007
Mursaleen M, Mohiuddine SA, Edely OHH: On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces. Comput. Math. Appl. 2010, 59(2):603–611. MR 2010k:40004 10.1016/j.camwa.2009.11.002
Das P, Kostyrko P, Wilczynski W, Malik P: I and -convergence of double sequences. Math. Slovaca 2008, 58: 605–620. MR 2010e:40007 10.2478/s12175-008-0096-x
Savas E: On generalized A -difference strongly summable sequence spaces defined by ideal convergence on a real n -normed space. J. Inequal. Appl. 2012., 2012: Article ID 87. doi:10.1186/1029–242X-2012–87
Connor J, Grosse-Erdmann KG: Sequential definitions of continuity for real functions. Rocky Mt. J. Math. 2003, 33(1):93–121. MR 2004e:26004 10.1216/rmjm/1181069988
Sleziak M: I -continuity in topological spaces. Acta Mathematica, Faculty of Natural Sciences, Constantine the Philosopher University Nitra 2003, 6: 115–122.
Çakallı H: Sequential definitions of compactness. Appl. Math. Lett. 2008, 21(6):594–598. MR 2009b:40005 10.1016/j.aml.2007.07.011
Çakallı H: New kinds of continuities. Comput. Math. Appl. 2011, 61(4):960–965. MR 2011j:54008 10.1016/j.camwa.2010.12.044
Çakallı H: On G -continuity. Comput. Math. Appl. 2011, 61(2):313–318. MR 2011m:40002 10.1016/j.camwa.2010.11.006
Salát T, Tripathy BC, Zimon M: On some properties of I -convergence. Tatra Mt. Math. Publ. 2004, 28: 279–286. MR 2005h:40004
Çakallı H: On Δ-quasi-slowly oscillating sequences. Comput. Math. Appl. 2011, 62(9):3567–3574. 10.1016/j.camwa.2011.09.004
Acknowledgements
The authors would like to thank the referees for a careful reading and several constructive comments that have improved the presentation of the results.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
BH drafted the manuscript. HC checked and organized the manuscript to be its final form and make the revision as the corresponding author.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Cakalli, H., Hazarika, B. Ideal quasi-Cauchy sequences. J Inequal Appl 2012, 234 (2012). https://doi.org/10.1186/1029-242X-2012-234
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2012-234