Abstract and Applied Analysis

Volume 2013 (2013), Article ID 739319, 11 pages

http://dx.doi.org/10.1155/2013/739319

Research Article

Certain Sequence Spaces over the Non-Newtonian Complex Field

Sebiha Tekin and Feyzi Başar

Department of Mathematics, Faculty of Arts and Sciences, Fatih University, The Hadimköy Campus, Büyükçekmece, 34500 Istanbul, Turkey

Received 4 February 2013; Accepted 5 April 2013

Academic Editor: Victor Kovtunenko

Copyright © 2013 Sebiha Tekin and Feyzi Başar. This 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.

Abstract.

It is known from functional analysis that in classical calculus, the sets , , , and of all bounded, convergent, null and -absolutely summable sequences are Banach spaces with their natural norms and they are complete according to the metric reduced from their norm, where . In this study, our main goal is to construct the spaces , , , and over the non-Newtonian complex field and to obtain the corresponding results for these spaces, where .

1. Preliminaries, Background, and Notations

A complete ordered field is a system consisting of a set , four binary operations , , , for , and an ordering relation for , all of which behave with respect to the set exactly as +, −, , , behave with respect to the set of real numbers. We call the realm of the complete ordered field, [1, page 32]. A complete ordered field is called arithmetic if its realm is a subset of . A bijective function with domain and range a subset of is called a generator. For example, the identity function , exponential function, and the function are generators.

Bashirov et al. [2] have recently emphasized on the multiplicative calculus and gave the results with applications corresponding to the well-known properties of derivative and integral in the classical calculus. Quite recently, Uzer [3] has extended the multiplicative calculus to the complex valued functions and gave the statements of some fundamental theorems and concepts of multiplicative complex calculus, and demonstrated some analogies between the multiplicative complex calculus and classical calculus by theoretical and numerical examples. Bashirov and Rıza [4] have studied the multiplicative differentiation for complex-valued functions and established the multiplicative Cauchy-Riemann conditions. Bashirov et al. [5] have investigated various problems from different fields which can be modelled more efficiently using multiplicative calculus, in place of Newtonian calculus.

Let be a generator with range . An arithmetic with range , and operations and ordering relation defined as follows, is called -arithmetic. Let . Then, we define the operations addition , subtraction , multiplication , division , and ordering () as follows:

With the above new operations, is an -arithmetic. In other words, one can easily show that is a complete ordered field. As was seen, -generator generates -arithmetic. For example, the identity function generates classical arithmetic, and exponential function generates geometric arithmetic. Each generator generates exactly one arithmetic and each arithmetic is generated by exactly one generator. We denote -zero by   and -one by which are obtained from and , respectively. and numbers obtained by successive -addition of to with numbers obtained by successive of from are called . Thus, are given as follows:

Thus, we have for all that . Let . If then is called -positive and if then is called . The -absolute value of is defined by

For any elements and in with , the set of all elements in such that is called an , is denoted by , has -extent of , and has the -interior consisting of all elements in such that . Let be an infinite sequence of the elements in . Then there is at most one element in such that every with in its contains all but finitely many terms of . If there is such a number , then is said to be to , which is called the -limit of . In other words,

and be two arbitrarily selected generators and let (“star”)  also be the ordered pair of arithmetics . is a . Definitions given for are also valid for . For example, is defined by means of and their .

In the   is used for arguments and is used for values; in particular, changes in arguments and values are measured by and , respectively. The operators of the are applied only to functions with arguments in and values in . The of a function at an element in is, if it exists, the unique number in such that for every sequence of arguments of distinct from , if is to , then to and is denoted by . That is,

A function is at a point in if and only if is an argument of and When and are the identity function , the concepts of and are identical with those of classical limit and classical continuity.

The isomorphism from to is the unique function that possesses the following three properties:

(i)is one to one. (ii) is from onto . (iii)For any numbers and in ,

It turns out that for every in , and that for every integer . Since, for example, , it should be clear that any statement in can readily be transformed into a statement in .

2. Non-Newtonian Complex Field and Some Inequalities

Let and be arbitrarily chosen elements from corresponding arithmetics. Then the ordered pair is called as a -point. The set of all -points is called the set of non-Newtonian complex numbers and is denoted by ; that is,

The binary operations addition and multiplication of non-Newtonian complex numbers and are defined, as follows:

where and with

Then we have the following.

Theorem 1. is a field.

Proof. A straightforward calculation leads to the following statements:

 (i) is an Abelian group; (ii) is an Abelian group;  (iii) the operation is distributive over the operation

which conclude that is a field.

Let . Then the number is called the -square of and is denoted by . Let be a non-negative number in . Then, is called the -square root of and is denoted by . The -distance between two arbitrarily elements and of the set is defined by

Up to now, we know that is a field and the distance between two points in is computed by the function , defined by (10). Now, we define the -norm and next derive some required inequalities in the sense of non-Newtonian complex calculus.

Let be an arbitrary element. is called -norm of and is denoted by . In other words,

where and . Moreover, since for all we have , is the induced metric from the norm .

Lemma 2 (-Triangle inequality). Let . Then,

Proof. Let . Then, a straightforward calculation gives that

Hence, the inequality (12) holds.

Lemma 3. for all .

Proof. Let . In this case, one can observe by a routine verification that

as required.

Lemma 4. Let . Then, the following inequality holds:

Proof. Let . Then, one can see that

This means that the inequality (15) holds.

Lemma 5 (-Minkowski inequality). Let and for all . Then,

Proof. Let and for all . Then,

Let us take . Then, since the equality

holds for every fixed , we obtain

by (18) and Minkowski inequality in the complex field leads us to

On the other hand, let us prove

Indeed,

Substituting the relation (24) in (22) we obtain,

By using this equality in (21), we get

as desired.

Theorem 6. is a complete metric space, where is defined by (10).

Proof. First, we show that , defined by (10), is a metric on .

It is immediate for that

(i)Now we show that if and only if for . Indeed,

(ii)One can easily establish for all that

(iii)We show that the inequality holds for all . In fact,

Therefore, is a metric over .

Now, we can show that the metric space is complete. Let be an arbitrary Cauchy sequence in . In this case, for all there exists an such that for all . Let and . Then,

Thus we obtain that

On the other hand, since the following inequalities:

hold we therefore have by (32) that and . This means that and are Cauchy sequences with real terms. Since is complete, it is clear that for every there exists an such that for all and for every there exists an such that for all .

Define, by . Then, we have

Hence, is a complete metric space.

Since is a complete metric space with the metric defined by (10) induced by the norm , as a direct consequence of Theorem 6, we have the following.

Corollary 7. is a Banach space with the norm defined by

3. Sequence Spaces over Non-Newtonian Complex Field

In this section, we define the sets , , , , and of all, bounded, convergent, null, and absolutely sequences over the non-Newtonian complex field which correspond to the sets , , , , and over the complex field , respectively. That is to say that

For simplicity in notation, here and in what follows, the summation without limits runs from to . One can easily see that the set forms a vector space over with respect to the algebraic operations addition and scalar multiplication defined on , as follows:

Theorem 8. Define the function by

where such that is convergent with for all . Then, is a metric space.

Proof. We show that satisfies the metric axioms on the space of all non-Newtonian complex valued sequences.

(i) First we show that for all .

Because is a metric space, we have ; and . Hence, we obtain that . Moreover, since for all , we have

This means that

(ii) We show that iff . In this situation, one can see that

(iii) We show that for . First, we know that is a metric over . Thus,

(iv) We show that holds for . Again, using the fact that is a metric space, it is easy to see by Lemma 4 that

as required.

Theorem 9. The set is a sequence space.

Proof. It is trivial that the inclusion holds.

(i) We show that for . Indeed, combining the hypothesis

with the fact obtained from Lemma 2, we can easily derive that

Hence, .

(ii) We show that for any and for .

Since by Lemma 3 and , it is immediate that

which means that .

Therefore, we have proved that is a subspace of the space .

Theorem 10. Define the relation by

Then, is a complete metric space.

Proof. One can easily show by a routine verification that satisfies the metric axioms on the space . So, we omit the details.

Now, we prove the second part of the theorem. Let be a Cauchy sequence in , where . Then, there exists a positive integer such that for all with . For any fixed , if then

In this case for any fixed , is a Cauchy sequence of non-Newtonian complex numbers and since is complete, it converges to a . Define with infinitely many limits . Let us show and , as . Indeed, by (48), by letting , for we obtain that

On the other hand, since , there exists such that for all . Hence, by triangle inequality (12), the inequality

holds for all which is independent of . Hence, . By (49), since , we obtain . Therefore, the sequence converges to which means that is complete.

Since it is known by Theorem 10 that is a complete metric space with the metric induced by the norm , defined by

we have the following.

Corollary 11. is a Banach space with the norm defined by (51).

Now, we give the following lemma required in proving the fact that is a sequence space in the case .

Lemma 12. Let . Then, the inequality holds for all .

Proof. Let and . Then, one can easily see that

as desired.

Theorem 13. The sets and are sequence spaces, where .

Proof. It is not hard to establish by the similar way that and are the sequence spaces. So, to avoid the repetition of the similar statements, we consider only the set .

It is obvious that the inclusion strictly holds.

(i) Let . Then, there exist such that and . Thus, there exist such that

Thus if we set , by (53) we obtain for all that

which means that

Therefore, .

(ii) Let and . Since there exists an such that , we have

Thus, for , we have

which implies that Hence, .

That is to say that is a subspace of .

Theorem 14. , , and are complete metric spaces, where is defined as follows:

Proof. We consider only the space with .

(i) For , we establish that the two sided implication holds. In fact,

(ii) For , we show that . In this situation, since holds for every fixed it is immediate that

(iii)  By Minkowski inequality in Lemma 5, we have for that

Hence, triangle inequality is satisfied by on the space . Therefore, the function is a metric over the space .

Now we show that the metric space is complete. Let be an arbitrary Cauchy sequence in the space , where . Then, for any there exists an such that

for all . Hence, for and every fixed , we obtain

If we set fixed then, it follows by (63) that is a Cauchy sequence. Since is complete, this sequence converges to a point say . Let us define the sequence with these limits and show and , as . Indeed, by (62), we obtain the inequality for all with that

and thus we have by letting and that

which gives as and for all that

Setting and applying Lemma 5 we obtain by (66) and the fact that

which means that . Therefore, we see from (66) that . Since the arbitrary Cauchy sequence is convergent, the space is complete.

Corollary 15. and are the Banach spaces equipped with the norm defined in (51).

Since it is known by Theorem 14 that is a complete metric space with the metric induced in the case by the norm and in the case by the -norm , defined for by

we have the following.

Corollary 16. The space is a Banach space with the norm and -norm defined by (68).

4. Conclusion

We present some important inequalities such as triangle, Minkowski, and some other inequalities in the sense of non-Newtonian complex calculus which are frequently used. We state the classical sequence spaces over the non-Newtonian complex field and try to understand their structure of being non-Newtonian complex vector space. There are lots of techniques that have been developed in the sense of non-Newtonian complex calculus. If the non-Newtonian complex calculus is employed instead of the classical calculus in the formulations, then many of the complicated phenomena in physics or engineering may be analyzed more easily.

As an alternative to the classical (additive) calculus, Grossman and Katz [1] introduced some new kind of calculus named as non-Newtonian calculus, geometric calculus, anageometric calculus, and bigeometric calculus. Türkmen and Başar [6, 7] have recently studied the classical sequence spaces and related topics, in the sense of geometric calculus. Quite recently, Çakmak and Başar [8] have also worked on the same subject by using non-Newtonian calculus. In the present paper, we use the non-Newtonian complex calculus instead of non-Newtonian real calculus and geometric calculus. It is trivial that in the special cases of the generators and , the non-Newtonian complex calculus gives the special kind of the following calculus:

(i)if , the identity function, then the non-Newtonian complex calculus is reduced to the classical calculus;(ii)if and , then the non-Newtonian complex calculus is reduced to the geometric calculus;(iii)if and , then the non-Newtonian complex calculus is reduced to the anageometric calculus;(iv)if , then the non-Newtonian complex calculus is reduced to the bigeometric calculus.

Since our results are obtained by using the non-Newtonian complex calculus, they are much more general and comprehensive than those of Türkmen and Başar [6, 7], Çakmak and Başar [8].

Quite recently, Talo and Başar have studied the certain sets of sequences of fuzzy numbers and introduced the classical sets , , , and consisting of the bounded, convergent, null, and absolutely -summable sequences of fuzzy numbers in [9]. Nextly, they have defined the alpha-, beta-, and gamma-duals of a set of sequences of fuzzy numbers and gave the duals of the classical sets of sequences of fuzzy numbers together with the characterization of the classes of infinite matrices of fuzzy numbers transforming one of the classical set into another one. Following Bashirov et al. [2] and Uzer [3], we give the corresponding results for multiplicative calculus to the results derived for the sets of fuzzy valued sequences in Talo and Başar [9], as a beginning. As a natural continuation of this paper, we should record from now on that it is meaningful to define the alpha-, beta-, and gamma-duals of a set of sequences over the non-Newtonian complex field and to determine the duals of classical spaces , , , and . Further, one can obtain the similar results by using another type of calculus instead of non-Newtonian complex calculus.

Acknowledgment

The authors are very grateful to the referees for many helpful suggestions and comments about the paper which improved the presentation and its readability.

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