Statistical and lacunary statistical convergence of interval numbers in topological groups

In this paper we introduce the concepts of statistical and lacunary statistical convergence of interval numbers in topological groups. We prove some inclusion relations and study some of their properties.


Introduction
Interval arithmetic was first suggested by Dwyer (1951).Development of interval arithmetic as a formal system and evidence of its value as a computational device was provided by Moore (1959) and Moore and Yang (1962).Furthermore, Moore and others Dwyer (1951), Dwyer (1953), and Moore and Yang (1958) have developed applications to differential equations.Chiao (2002) introduced sequence of interval numbers and defined usual convergence of sequences of interval number.Şengönül and Eryilmaz (2010) introduced and studied bounded and convergent sequence spaces of interval numbers and showed that these spaces are complete metric space.Recently Esi (2011) introduced and studied strongly almost   convergence and statistically almost   convergence of interval numbers.
The idea of statistical convergence for single sequences was introduced by Fast (1951).Schoenberg (1959) studied statistical convergence as a summability method and listed some of elementary properties of statistical convergence.Both of these authors noted that if bounded sequence is statistically convergent, then it is Cesaro summable.Existing work on statistical convergence appears to have been restricted to real or complex sequence, but several authors extended the idea to apply to sequences of fuzzy numbers and also introduced and discussed the concept of statistically sequences of fuzzy numbers.

 
is called an interval number.A real interval can also be considered as a set.Thus we can investigate some properties of interval numbers, for instance arithmetic properties or analysis properties.
We denote the set of all real valued closed intervals by I .

R Any elements of I .
R is called closed interval and denoted by .
An interval number A is a closed subset of real numbers (CHIAO, 2002).Let l a and r a be first and last points of A interval number, respectively.For  B A, The set of all interval numbers I .
R is a complete metric space defined by of all interval numbers with real terms and the algebraic properties of i w can be found in (ŞENGÖNÜL; ERYILMAZ, 2010).Now we give the definition of convergence of interval numbers: By X , we will denote an abelian topological Hausdorff group, written additively which satisfies the first axiom of countability.In Cakalli (1995), a single sequence where the vertical bars indicate the number of elements in the enclosed set.In this case we write In Cakalli (1996), for single sequence In this paper, we introduce and study the concepts of statistical convergence, strongly convergence, lacunary statistical convergence and lacunary strongly convergence for interval numbers in topological groups as follows.

Main results
In this section we give some definition and prove the results of this paper.
Definition 3.1 -An interval numbers sequence . The set of all statistically convergent sequences of interval number sequences is denoted by  .
In this case we write The set of all lacunary statistically convergent sequences of interval number sequences is denoted by  .

X s
In the special case (iii) It follows from (i) and (ii).

Conclusion
The concept of interval arithmetic was first suggested by Dwyer (1951).After then Chiao (2002) introduced usual convergence of sequences of interval numbers.Recently, interval numbers sequences studied by several authors.The results obtained in this paper are much more general than those obtained earlier.
the concept of lacunary statistical convergence was defined byCakallı (1996)  as follows: Let It follows from (i) and (ii).