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Statistical and lacunary statistical convergence of interval numbers in topological groups/Convergencia estatistica e estatistico-lacunar de numeros de intervalos em grupos topologicos.

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. sengonul 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 [lambda]--convergence and statistically almost [lambda]--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.

Preliminaries

Let p = ([p.sub.k]) be a positive sequence of real numbers. If 0 < h = [imf.sub.k] [p.sub.k] [less than or equal to] [p.sub.k] [less than or equal to] H = [sup.sub.k] [P.sub.k] < [infinity] and D = max(1,[2.sup.H-1]), then for [a.sub.k], [b.sub.k] [member of] C for all k [member of] N we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By a lacunary sequence [theta] = ([k.sub.r]), r = 0,1,2,..., where:

[k.sub.o] = 0, we shall mean increasing sequence of nonnegative integers [h.sub.r] = [k.sub.r] - [t.sub.r-1] [right arrow] [infinity] as r [right arrow] [infinity].

The intervals determined by 0 are denoted by [I.sub.r] = ([k.sub.r-1], [k.sub.r]] and the ratio [[k.sub.r]/[k.sub.r-1]] will be denoted by [q.sub.r]. The space of lacunary strongly convergent sequence No was defined by Freedman et al. (1978) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A set consisting of a closed interval of real numbers such that a [less than or equal to] x [less than or equal to] b 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 IR. Any elements of IR. is called closed interval and denoted by [bar.A]. That is [bar.A] = {x [member of] R : a [less than or equal to] X [less than or equal to] b}.

An interval number [bar.A] is a closed subset of real numbers (CHIAO, 2002). Let [a.sub.l] and [a.sub.r] be first and last points of A interval number, respectively. For A, B g IR, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The set of all interval numbers IR. is a complete metric space defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (CHIAO, 2002).

In the special case [bar.A] = [a, a] and [bar.B] = [b, b], we obtain usual metric of R.

Let us define transformation f : N [right arrow] R by K [right arrow] f (k) = [bar.A], [bar.A] = ([[bar.A].sub.k]) Then [bar.A] = ([[bar.A].sub.k]) is called sequence of interval numbers. The [[bar.A].sub.k] is called kth term of sequence [bar.A] = ([[bar.A].sub.k]). [w.sup.i] denotes the set of all interval numbers with real terms and the algebraic properties of [w.sup.i] can be found in (SENGONUL; ERYILMAZ, 2010).

Now we give the definition of convergence of interval numbers:

A sequence [bar.A] = ([[bar.A].sub.k]) of interval numbers is said to be convergent to the interval number [[bar.A].sub.o] if for each [epsilon] > 0 there exists a positive integer [k.sub.o] such that d([[bar.A].sub.k], [[bar.A].sub.o]) < [epsilon] for all k [greater than or equal to] [k.sub.o] and we denote it by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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 x = ([X.sub.k]) in X is said to be statistically convergent to an element L [member of] X if for each neighborhood U of 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the vertical bars indicate the number of elements in the enclosed set. In this case we write st(X)- [lim.sub.k] [X.sub.k] = L or [x.sub.k] [right arrow] L(st(X)).

In Cakalli (1996), for single sequence x = ([x.sub.k]) in X the concept of lacunary statistical convergence was defined by Cakalli (1996) as follows: Let [theta] = ([k.sub.r]) be a lacunary sequence; the single sequence x = ([x.sub.k]) in X is said to be [st.sup.[theta]]--convergent to L (or lacunary statistically convergent to L in X) if for each neighborhood

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In this case we write [st.sup.[theta]] (X) - [lim.sub.k][x.sub.k] = L or [x.sub.k] [right arrow] L(st (x))

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 [bar.A] = ([[bar.A].sub.k]) is said to be statistically convergent to an element of [[bar.A].sub.o] of X if for each neighborhood system U of 0 = [0,0]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

in this case we write [[bar.A].sub.k] [right arrow] [[bar.A].sub.o] ([bar.s](x)} or [bar.s](X) - lim [[bar.A].sub.k] = [[bar.A].sub.o]. The set of all statistically convergent sequences of interval number sequences is denoted by [bar.s](X)

Definition 3.2--An interval numbers sequence A = A) is called strongly convergent in X if for each neighborhood system U of [bar.0] = [0,0] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In this case we write [[bar.A].sub.k] [right arrow] [[bar.A].sub.o] ([[bar.s].sub.[theta]](X)) or [bar.N] (X)- lim[[bar.A].sub.k] = [[bar.A].sub.o].

Definition 3.3--Let [theta] = ([h.sub.r]) be a lacunary sequence. A sequence [bar.A] = ([[bar.A].sub.k]) of interval numbers is said to be lacunary statistically convergent to interval number [[bar.A].sub.o] in X, if for each neighborhood system of [bar.0] = [0,0]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In this case we write [[bar.A].sub.k] [right arrow] [[bar.A].sub.o] ([[bar.s].sub.[theta]](X)) or [[bar.S].sub.[theta]](X)- lim [[bar.A].sub.k] = [[bar.X].sub.o]. The set of all lacunary statistically convergent sequences of interval number sequences is denoted by [[bar.S].sub.o](X). In the special case [theta]--([2.sup.r]), we shall write [bar.s](X) instead of [[bar.S].sub.[theta]](X).

Definition 3.4.--An interval numbers sequence [bar.X] = ([[bar.x].sub.h]) is called strongly lacunary convergent in X if for each neighborhood system U of [bar.0] - [0,0] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in this case we write [[bar.A].sub.k] [right arrow] [[bar.A].sub.o] ([[bar.N].sup.p.sub.[theta]] (X)) or [[bar.N].sup.p.sub.[theta]] (X)-lim[[bar.x].sub.k] = [[bar.X].sub.o]. In the special case [theta] = ([2.sup.r]), we shall write [bar.N]p (X) instead of [[bar.N].sup.p.sub.[theta]] (X).

Theorem 3.1--Let [theta] = ([k.sub.r]) be a lacunary sequence and [bar.A] = ([[bar.A].sub.k]) be a sequence of interval numbers. Then

(i) [[bar.A].sub.k] [right arrow] [[bar.A].sub.o] ([[bar.N].sup.p.sub.[theta]](X)) implies [[bar.A].sub.k] [right arrow] [[bar.A].sub.o] ([[[bar.s].sub.[theta]](X)),

(ii) [bar.A] = ([[bar.A].sub.k])[member of] [bar.m](X) and [[bar.A].sub.k] [right arrow] [[bar.A].sub.o]([[bar.s].sub.[theta]](X)) imply [[bar.A].sub.k] [right arrow] [[bar.A].sub.o]([[bar.N].sup.p.sub.[theta]](X)),

(iii) If then [bar.A] = ([[bar.A].sub.k])[member of][bar.m](X), then [[bar.A].sub.k] [right arrow] [[bar.A].sub.o]([[bar.N].sup.p.sub.[theta]](X))

and [[bar.A].sub.k] [right arrow] [[bar.A].sub.o] ([[bar.s].sub.[theta]](X)),

where:

[bar.m](X) = {[bar.A] = ([[bar.A].sub.k]): [sup.sub.k] d([[bar.A].sub.k], [[bar.A].sub.o]) < [infinity]}.

Proof. (i) Let [epsilon] > 0 and [[bar.A].sub.k] [right arrow] [[bar.A].sub.o] ([[bar.N].sup.p.sub.[theta]](X))

Then we write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence [[bar.A].sub.h] [right arrow] [[bar.A].sub.o] ([[bar.s].sub.[theta]](X))

(ii) Suppose that [bar.A] = ([[bar.A].sub.k])[member of] [bar.m](X) and [[bar.A].sub.k] [right arrow] [[bar.A].sub.o] ([[bar.s].sub.[theta]] (X)) Since [bar.A] = ([[bar.A].sub.h]) [member of] [bar.m] (X), there is a constant C > 0 such that d([[bar.A].sub.k], [[bar.A].sub.o]) [less than or equal to] C. Given [epsilon] > 0, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus we obtain [[bar.A].sub.k] [right arrow] [[bar.A].sub.o] ([[bar.N].sup.p.sub.[theta]](X)) (iii) It follows from (i) and (ii).

Theorem 3.2.--Let [theta] = ([k.sub.r]) be a lacunary sequence and [bar.A] = ([[bar.A].sub.k]) be a sequence of interval numbers. Then

(i) For lim [inf.sub.r][q.sub.r] > 1, then [[bar.A].sub.k] [right arrow] [[bar.A].sub.o] ([bar.s](x)) implies [[bar.A].sub.k] [right arrow] [[[bar.A].sub.o] ([[bar.s].sub.[theta]](X)),

(ii) For lim[sup.sub.r] [q.sub.r] < [infinity], then [[bar.A].sub.k] [right arrow] [[bar.A].sub.o] ([[bar.s].sub.[theta]](x)) implies [[bar.A].sub.k] [right arrow] [[bar.A].sub.o] ([bar.s](x))

(iii) If 1 <lim [inf.sub.r] [q.sub.r] [less than or equal to] lim [sup.sub.r] [q.sub.r] < [infinity], then [[bar.A].sub.k] [right arrow] [[bar.A].sub.o] ([bar.s](x)) if and only if [[bar.A].sub.k] = [[[bar.A].sub.o].sub.[theta]]([bar.s](x))

Proof. (i) Suppose that lim [inf.sub.r] [q.sub.r] > 1, then there exists a [delta] > 0 such that [q.sub.r] [greater than or equal to] 1 + [delta] for sufficiently large r, which implies

[[h.sub.r]/[k.sub.r]] [greater than or equal to] [[delta]/1+[delta]]

if [[bar.A].sub.k] [right arrow] [[bar.A].sub.o] ([[bar.s].sub.[theta]](X)), then for sufficiently large r, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence [[bar.A].sub.k] [right arrow] [[bar.A].sub.o] ([[bar.S].sub.[theta]](X))

(ii) If lim [sup.sub.r] [q.sub.r] < [infinity], then there exists C > 0 such that [q.sub.r] < C for all r [greater than or equal to] 1. Let [[bar.A].sub.k] [right arrow] [[bar.A].sub.o] ([[bar.s].sub.[theta]](X) and set [A.sub.r] - [absolute value of ({k[member of][I.sub.r]: ([[bar.A].sub.k], [[bar.A].sub.o]) [greater than or equal to] [epsilon]})]. Then there exists an [r.sub.o] [member of] N such that

[[A.sub.r]/[h.sub.r]] < [epsilon] for all r > [r.sub.o]. (3.1)

Now let N = max{[A.sub.r] :1 [less than or equal to] r [less than or equal to] [r.sub.o]} and choose n such that [k.sub.r-1] < n [less than or equal to] [k.sub.r]. Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus we obtain [[bar.A].sub.k] [right arrow] [[bar.A].sub.o] ([bar.s]X) (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.

Doi: 10.4025/actascitechnol.v36i3.16545

References

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CHIAO, K. P. Fundamental properties of interval vector max-norm. Tamsui Oxford Journal of Mathematics, v. 18, n. 2, p. 219-233, 2002.

DWYER, P. S. Linear computation. New York: Wiley, 1951.

DWYER, P. S. Errors of matrix computation, simultaneous equations and eigenvalues. National Bureu of Standarts, Applied Mathematics Series, v. 29, p. 49-58, 1953.

ESI, A. Strongly almost [lambda]--convergence and statistically almost [lambda]--convergence of interval numbers. Scientia Magna, v. 7, n. 2, p. 117-122, 2011.

FAST, H. Sur la convergence statistique. Colloquium Mathematics, v. 2, p. 241-244, 1951.

FREEDMAN, A. R.; SEMBER, J. J.; RAPHAEL, M. Some Cesaro type summability. Proceedings of the London Mathematical Society, v. 37, n. 3, p. 508-520, 1978.

MOORE, R. E. Automatic error analysis in digital computation. Washington, D.C.: Lockheed Missiles and Space Company, 1959. (LSMD-48421).

MOORE, R. E.; YANG, C. T. Theory of an interval algebra and its application to numeric analysis. RAAG Memories II. Tokyo: Gaukutsu Bunken Fukeyu-kai, 1958.

MOORE, R. E.; YANG, C. T. Interval analysis I. Sunnyvale: Lockheed Missiles and Space Company, 1962. (LMSD-285875).

SCHOENBERG, I. J. The integrability of certain functions and related summability methods. American Mathematical Monthly, v. 66, p. 361-375, 1959.

SENGONUL, M.; ERYILMAZ, A. On the sequence spaces of interval numbers. Thai Journal of Mathematics, v. 8, n. 3, p. 503-510, 2010.

Received on March 28, 2012.

Accepted on October 4, 2012.

License information: This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Ayhan Esi

Department of Mathematics, Science and Art Faculty, Adiyaman University, 02040, Adiyaman, Turkey. E-mail: aesi23@hotmail.com
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