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Ideal quasi-Cauchy sequences. (English) Zbl 1283.40004

Summary: An ideal \(I\) is a family of subsets of the positive integers \(\mathbb N\) which is closed under taking finite unions and subsets of its elements. A sequence \((x_n)\) of real numbers is said to be \(I\)-convergent to a real number \(L\) if for each \(\varepsilon > 0\) the set \(\{n\:|x_n-L| \geq \varepsilon\}\) belongs to \(I\). We introduce \(I\)-ward compactness of a subset of \(\mathbb R\), the set of real numbers, and \(I\)-ward continuity of a real function in the sense that a subset \(E\) of \(\mathbb R\) is \(I\)-ward compact if any sequence \((x_n)\) 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 \((x_n)\) is called to be \(I\)-quasi-Cauchy when \((\Delta x_n)\) is \(I\)-convergent to 0. We obtain results related to \(I\)-ward continuity, \(I\)-ward compactness, ward continuity, ward compactness, ordinary compactness, ordinary continuity, \(\delta\)-ward continuity, and slowly oscillating continuity.

MSC:

40A35 Ideal and statistical convergence
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
40G15 Summability methods using statistical convergence
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References:

[1] doi:10.1016/j.aml.2011.04.029 · Zbl 1223.26004
[2] doi:10.4169/000298910X480793 · Zbl 1204.26003
[3] doi:10.1016/j.mcm.2011.04.037 · Zbl 1228.40003
[4] doi:10.1016/j.mcm.2010.09.010 · Zbl 1211.40001
[5] doi:10.1090/S0002-9939-1961-0121591-X
[6] doi:10.2307/2308747 · Zbl 0089.04002
[7] doi:10.2307/2372456 · Zbl 0050.05901
[8] doi:10.1112/plms/s3-59.3.417 · Zbl 0694.10040
[9] doi:10.1090/S0002-9947-1995-1260176-6
[10] doi:10.2140/pjm.1981.95.293 · Zbl 0504.40002
[11] doi:10.1017/S0305004100065312 · Zbl 0674.40008
[12] doi:10.2140/pjm.1993.157.201 · Zbl 0794.40004
[13] doi:10.2307/1911532 · Zbl 0356.90006
[14] doi:10.1006/jmaa.2000.6725 · Zbl 0982.46007
[15] doi:10.1016/j.topol.2008.01.015 · Zbl 1155.54004
[16] doi:10.1016/j.aml.2010.10.021 · Zbl 1216.40009
[17] doi:10.1016/j.aml.2011.12.022 · Zbl 1255.54010
[18] doi:10.2140/pjm.1993.160.43 · Zbl 0794.60012
[19] doi:10.1006/jmaa.1993.1082 · Zbl 0786.40004
[20] doi:10.2478/s12175-011-0059-5 · Zbl 1289.40028
[21] doi:10.2478/s12175-011-0071-9 · Zbl 1274.40034
[22] doi:10.1016/j.jmaa.2007.09.016 · Zbl 1139.40002
[23] doi:10.1016/j.camwa.2009.11.002 · Zbl 1189.40003
[24] doi:10.2478/s12175-008-0096-x · Zbl 1199.40026
[25] doi:10.1216/rmjm/1181069988 · Zbl 1040.26001
[26] doi:10.1016/j.aml.2007.07.011 · Zbl 1145.54001
[27] doi:10.1016/j.camwa.2010.12.044 · Zbl 1217.54013
[28] doi:10.1016/j.camwa.2010.11.006 · Zbl 1211.40002
[29] doi:10.1016/j.camwa.2011.09.004 · Zbl 1236.40005
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