Research Article
Almost
-Statistical and Strongly Almost
-Convergence of
Order of Sequences of Fuzzy Numbers
Mahmut IGJk
1
and Mikail Et
2,3
1
Faculty of Education, Harran University, Osmanbey Campus, 63190 S ¸anlıurfa, Turkey
2
Department of Mathematics, Fırat University, 23119 Elazı˘ g, Turkey
3
Department of Mathematics, Siirt University, 56100 Siirt, Turkey
Correspondence should be addressed to Mikail Et; mikailet68@gmail.com
Received 30 September 2014; Revised 21 January 2015; Accepted 22 January 2015
Academic Editor: Ismat Beg
Copyright © 2015 M. Is ¸ık and M. Et. Tis 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.
Te main purpose of this article is to introduce the concepts of almost
-statistical convergence and strongly almost
-convergence
of order of sequences of fuzzy numbers with respect to an Orlicz function. We give some relations between strongly almost
-
convergence and almost
-statistical convergence of order of sequences of fuzzy numbers.
1. Introduction
Te concept of fuzzy set was introduced by Zadeh [1].
Matloka [2] introduced sequences of fuzzy numbers and
provided that every convergent sequence of fuzzy numbers
is bounded. Recently, sequences of fuzzy numbers have been
discussed by Altınok et al. [3, 4], Aytar and Pehlivan [5],
C ¸ olak et al. [6, 7], G¨ okhan et al. [8], Nuray [9], and Talo and
Bas ¸ar [10].
Te idea of statistical convergence was introduced by Fast
[11] and the notion was linked with summability theory by
Alotaibi et al. [12], Connor [13], Et et al. [14, 15], Fridy [16], Is ¸ık
[17], Mohiuddine et al. [18, 19], and Tripathy [20]. Recently,
the notion was generalized by Gadjiev and Orhan [21], C ¸ olak
[22], and C ¸ olak and Bektas ¸[23].
In this paper, we study the concepts of and examine some
properties of almost
-statistical convergence and strongly
almost
-convergence of order of sequences of fuzzy
numbers. Te results which we obtained in this study are
much more general than those obtained by Bas ¸arır et al. [24].
2. Definitions and Preliminaries
A fuzzy set on R is called a fuzzy number if it has the
following properties:
(i) is normal;
(ii) is fuzzy convex;
(iii) is upper semicontinuous;
(iv) supp = cl{ ∈ R : () > 0} is compact, where cl
denoted the closure of the enclosed set.
A sequence = (
) of fuzzy numbers is a function
from the set N of all positive integers into (R), where (R)
is fuzzy number space. A sequence = (
) is said to be
bounded if the set {
:∈ N} is bounded. A sequence =
(
) is said to be convergent if there exists a positive integer
0
such that (
,
0
)< for >
0
, for every >0. By
F
,
ℓ
F
∞
, and
F
we denote the set of , , and V
sequences of fuzzy numbers, respectively [2].
An Orlicz function is a function : [0,∞) → [0,∞),
which is continuous, nondecreasing, and convex with (0) =
0, () > 0 for >0, and () → ∞ as →∞.
Te generalized de la Vall´ ee-Pousion mean is defned by
() = (1/
)∑
∈
, where = (
) is a nondecreasing
sequence of positive numbers such that
+1
≤
+1,
1
=1,
and
→∞ as →∞ and
= [ −
+ 1, ].
Te space was introduced by Lorentz [25] and Maddox
[26] has defned to be strongly almost convergent to a
number if lim
→∞
(1/) ∑
=1
|
+
−| = 0, uniformly
in .
Hindawi Publishing Corporation
Journal of Function Spaces
Volume 2015, Article ID 451050, 6 pages
http://dx.doi.org/10.1155/2015/451050