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Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.36 no.2 Antofagasta June 2017

http://dx.doi.org/10.4067/S0716-09172017000200325 

Articles

On some spaces of Lacunary I-convergent sequences of interval numbers defined by sequence of moduli

Mohd Shafiq1 

Ayhan Esi2 

1 Govt. Degree College, India, e-mail : shafiqmaths7@gmail.com

2Adiyaman University, Turkey, e-mail: aesi237@hotmail.com

Abstract:

In this article we introduce and study some spaces of I-convergent sequences of interval numbers with the help of a sequence of moduli, a bounded sequence of positive real numbers and a lacunary sequence of increasing integers. We study some topological and algebraic properties and some inclusion relations on these spaces.

Keywords and Phrases :  Interval numbers; Ideal; filter; I-convergent sequence; solid and monotone space; Banach space; modulus function

1. Introduction and Preliminaries11 34

Let R , R and C , be the sets of all natural, real and complex numbers, respectively.

Let ℓ ∞ , c and c0 be denote the Banach spaces of bounded, convergent and null sequences, respectively with norm

We denote

the space of all real or complex sequences.

It is an admitted fact that the real and complex numbers are playing a vital role in the world of mathematics. Many mathematical structures have been constructed with the help of these numbers. In recent years, since 1965 fuzzy numbers and interval numbers also managed their place in the world of mathematics and credited into account some alike structures . Interval arithmetic was first suggested by P.S.Dwyer (12) in 1951. Further development of interval arithmetic as a formal system and evidence of its value as a computational device was provided by R.E.Moore (22) in 1959 and Moore and Yang (23) and others and have developed applications to differential equations.

Recently, Chiao (11) introduced sequences of interval numbers and defined usual convergence of sequences of interval numbers. Şengönül and Eryilmaz (34) introduced and studied bounded and convergent sequence spaces of interval numbers and showed that these spaces are complete.

A set (closed interval) of real numbers X such that is called an interval number (11). 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. Let us denote the set of all real valued closed intervals by IR. Any element of IR is called a closed interval and it is denoted by . IR is a quasilinear space under the algebraic operations and a partial order relation for IR found in (34) and any subspace of IR is called quasilinear subspace (see (34)). Further research on interval numbers was carried out by Esi and Braha (6), Esi and Hazarika (8), Esi (1) -(9) and the references therein.

The set of all interval numbers IR is a complete metric space defined by

of R with

Let us denote the set of sequences of interval numbers with real terms

The following definitions were given by and Eryilmaz in (34).

Let us denote the space of all convergent, null and bounded sequences of 34

In this paper, we assume that a norm of the sequence of interval numbers is the distance from to and satisfies the following properties:

of sequences of interval numbers are normed interval spaces normed by

Throughout, represent zero and identity interval numbers according to addition and multiplication, respectively.

As a generalisation of usual convergence for the sequences of real or complex numbers, the concept of statistical convergent was first introduced by Fast (13) and also independently by Buck (10) and Schoenberg (33). Later on, it was further investigated from a sequence space point of view and linked with the Summability Theory by Fridy (15), Šalát (30), Tripathy (35) and many others. The statistical convergence has been extended to interval numbers by Esi and others as follows in (1) - (9).

Let us suppose that

Then, the sequence is said to be statistically convergent to an interval number where vertical lines denote the cardinality of the enclosed set. That is,

The notion of ideal convergence (I-convergence) was introduced and studied by Kostyrko, Mačaj, Salǎt and Wilczyński 20, 21. Later on, it was studied by Šalát , Tripathy and Ziman 31, 32, Esi and Hazarika 8, Tripathy and Hazarika 36, Khan etal 16, 17, Mursaleen and Sunil 25 and many others.

Definition 1.1. Let N be the set of natural numbers. Then, a family of sets is said to be an ideal, if

Definition 1.2. A sequence space of interval numbers is is a permutation on N

A canonical preimage of a step space

Definition 1.5. A sequence space is said to be monotone, if it contains the canonical preimages of its step space.

Definition 1.6. A function is called a modulus function if

The idea of modulus function was introduced by Nakano in 1953, (See 26, Nakano, 1953).

Ruckle 27- 29 used the idea of a modulus function f to construct the sequence space

After then, E.Kolk 18- 19 gave an extension of by considering a sequence of moduli and defined the sequence space

Mursaleen and Noman 24 introduced the notion of λ — convergent and λ —bounded sequences. We extended this concept to the sequence of interval numbers as follows.

be a strictly increasing sequence of positive real numbers tending to infinity. That is,

Here and in the sequel, we shall use the convention that any term with a

Let us denote the classes of I — convergent, I — null,bounded I — convergent and bounded I — null sequences of interval numbers with and respectively.

The space of lacunary strongly convergent sequence NƟ was defined by Freedman etal (14) as:

We need the following popular inequalities throughout the paper.

For any modulus function ƒ we have the inequalities

Now, we give some important Lemmas.

Lemma.1.7. Every solid space is monotone.

Lemma.1.8.Let

Lemma.1.9.

Let us give a most important definitions for this paper.

Definition 1.10 (see 37). Let × be a quasilinear space of interval numbers.

Definition 1.11 (see 37). Suppose that X is a quasilinear space and . Y is called a subspace of X whenever Y is a quasilinear space with the same partial ordering and same operations on X

Theorem 1.12 (see 37). Y is a subspace of a quasilinear space X if and

In this article, we introduce and study the following classes of sequences;

2. Main Results

Theorem 2.1. Let be a sequence of modulus functions and p = (pk) be the bounded sequence of positive real numbers. Then, the sets are quasilinear spaces over the field of real numbers.

Proof. We shall prove the result for the space . Rests will follow similarly.

The result follows from the above inclusion and the definition of quasilinear space.

Theorem 2.2. Let be a sequence of modulus functions and p = (pk) be the bounded sequence of positive real numbers. Then, the classes of are paranormed spaces, paranormed by

as That is to say that scalar multiplication is continuous. Since each is an increasing function, it is clear that

Hence is a paranormed space.

For the result is similar.

Theorem 2.3. The set is closed subspace of

Proof. Let be a Cauchy sequence in such that

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Received: 2016; Accepted: 2017

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