Cesàro Summable Sequence Spaces over the Non-Newtonian Complex Field

Abstract

The spaces , , and can be considered the sets of all sequences that are strongly summable to zero, strongly summable, and bounded, by the Cesàro method of order with index . Here we define the sets of sequences which are related to strong Cesàro summability over the non-Newtonian complex field by using two generator functions. Also we determine the -duals of the new spaces and characterize matrix transformations on them into the sets of -bounded, -convergent, and -null sequences of non-Newtonian complex numbers.

1. Introduction

The theory of sequence spaces is the fundamental of summability. Summability is a wide field of mathematics, mainly in analysis and functional analysis, and has many applications, for instance, in numerical analysis to speed up the rate of convergence, in operator theory, the theory of orthogonal series, and approximation theory. Also, the concepts of statistical convergence have been studied by various mathematicians. In recent years, generalizations of statistical convergence have appeared in the study of strong integral summability and the structure of ideals of bounded continuous functions on locally compact spaces. Many important sequence spaces arise in a natural way from different notions of summability, that is, ordinary, absolute, and strong summability. The first two cases may be considered as the domains of the matrices that define the respective methods; the situation, however, is different and more complicated in the case of strong summability. Many authors have extensively developed the theory of the matrix transformations between some sequence spaces; we refer the reader to [1–6].

As an alternative to the classical calculus, Grossman and Katz [7–9] introduced the non-Newtonian calculus consisting of the branches of geometric, quadratic, and harmonic calculus, and so forth. All these calculi can be described simultaneously within the framework of a general theory. They decided to use the adjective non-Newtonian to indicate any of calculi other than the classical calculus. Every property in classical calculus has an analogue in non-Newtonian calculus which is a methodology that allows one to have a different look at problems which can be investigated via calculus. In some cases, for example, for wage-rate (in dollars, euro, etc.) related problems, the use of bigeometric calculus which is a kind of non-Newtonian calculus is advocated instead of a traditional Newtonian one.

Many authors have extensively developed the notion of multiplicative calculus; see [10–12] for details. Also some authors have also worked on the classical sequence spaces and related topics by using non-Newtonian calculus [13–15]. Further Kadak [16] and Kadak et al. [17, 18] have matrix transformations between certain sequence spaces over the non-Newtonian complex field and have generalized Runge-Kutta method with respect to the non-Newtonian calculus.

The main focus of this work is to extend the strong Cesàro summable sequence spaces defined earlier to their generalized sequence spaces over the non-Newtonian complex field by using various generator functions, that is, and generators.

2. Preliminaries, Background, and Notations

Arithmetic is any system that satisfies the whole of the ordered field axioms whose domain is a subset of . There are infinitely many types of arithmetic, all of which are isomorphic, that is, structurally equivalent.

A generator is a one-to-one function whose domain is and whose range is a subset of where . Each generator generates exactly one arithmetic, and conversely each arithmetic is generated by exactly one generator. If for all , then is called identity function whose inverse is itself. In the special cases and , generates the classical and geometric arithmetic, respectively. By -arithmetic, we mean the arithmetic whose domain is and whose operations are defined as follows. For and any generator , As an example if we choose function from to the set , and -arithmetic turns out to be Geometric arithmetic:Following Grosmann and Katz [8] we give the infinitely many -arithmetic, of which the quadratic arithmetic and harmonic arithmetic are special cases for and , respectively. The function and its inverse are defined as follows:If then the -calculus is reduced to the classical calculus.

One can easily conclude that the -summation can be written as follows:

Definition 1 (see [13]). Let be an -metric space. Then the basic notions can be defined as follows:(a)A sequence is a function from the set into the set . The -real number denotes the value of the function at and is called the term of the sequence.(b)A sequence in is said to be -convergent if, for every given (), there exist and such that for all and is denoted by or , as .(c)A sequence in is said to be -Cauchy if for every there is such that for all .

Throughout this paper, we define the th -exponent and th -root of byand provided there exists such that .

2.1. -Arithmetic

Suppose that and be two arbitrarily selected generators and “star-” also be the ordered pair of types of arithmetic (-arithmetic, -arithmetic). The sets and are complete ordered fields and -generator generates -arithmetic, respectively. Definitions given for -arithmetic are also valid for -arithmetic. Also -arithmetic is used for arguments and -arithmetic is used for values; in particular, changes in arguments and values are measured by -differences and -differences, respectively.

Definition 2 (see [15]). (a) The -limit of a function , denoted by , at an element in is, if it exists, the unique number in such thatA function is -continuous at a point in if and only if is an argument of and . When and are the identity function , the concepts of -limit and -continuity are identical with those of classical limit and classical continuity.
(b) The isomorphism from -arithmetic to -arithmetic is the unique function (iota) which has the following three properties: (i) is one to one.(ii) is from onto .(iii)For any numbers , It turns out that for every in and that for every -integer . Since, for example, , it should be clear that any statement in -arithmetic can readily be transformed into a statement in -arithmetic.

2.2. Non-Newtonian Complex Field and Some Inequalities

Let and be arbitrarily chosen elements from corresponding arithmetic. Then the ordered pair is called a -point. The set of all -points is called the set of -complex numbers and is denoted by ; that is, Define the binary operations addition and multiplication of -complex numbers and : for all and .

Theorem 3 (see [15]). is a field.

Following Grossman and Katz [8] we can give the definition of -distance and some applications with respect to the -calculus.

Definition 4. Let be a nonempty set and let be a function such that, for all , the following axioms hold: (NM1) if and only if ,(NM2),(NM3).Then the pair and are called a non-Newtonian metric (-metric) space and a -metric on , respectively.

The -distance between two arbitrarily elements and of the set is defined byUp to now, we know that is a field and the distance between two points in is computed by the function . Let be an arbitrary element. The distance function is called -norm of and is denoted by . In other words, let ; thenMoreover, for all we have where is the induced metric from norm.

Theorem 5 (see [15]). is a complete metric space, where is defined by (11).

Corollary 6 (see [15]). is a Banach space with the -norm which is defined by (12).

Definition 7. (a) Given a sequence of -complex numbers, the formal notationis called an infinite series with -complex terms, or simply complex -series for all . Also, for integers , the finite -sums are called the partial sums of complex -series. If the sequence -converges to a complex number then we say that the series -converges and write . The number is then called the -sum of this series. If -diverges, we say that the series -diverges or that it is -divergent.
(b) A -series is said to -converge absolutely if for some number .
(c) Let be a sequence of functions from to for each . We say that is uniformly -convergent to on if and only if, for each and for an arbitrary , there exists an integer such that whenever .
(d) The series is said to be uniformly -convergent to on if, given any , there exists an integer such that

Proposition 8 (see [15]). Let and for . Then

Remark 9. Let . Then the following statements hold:(i)One has(ii)Let and be the same generators. Then

Definition 10. Given a point . Then, for a positive -real number ,are -neighborhood (or -open (closed) ball) of centre and radius , respectively.

We see that an -open ball of radius is the set of all points in whose beta-distance from the center of the ball is less than and we say directly from the definition that every -neighborhood of contains ; in other words, is a point of each of its -neighborhoods.

Definition 11. Let be a -metric space. Then the followings are valid: (i) is called -open set if and only if every point of has a -neighborhood contained in . Also is called -closed set if and only if its complement is -open.(ii)The -interior is the largest -open set contained in and the -closure is the smallest -closed set contained in .

Definition 12 (usual -topology). Consider the set of -complex numbers with for all and . Then is a topological space and is called -usual topology on .

Definition 13. (i) A topological -vector (linear) space is a -vector space (see [17]) over the topological field that endowed with a topology such that -vector addition and scalar multiplication are -continuous functions.
(ii) A topological -vector space is called -normable if the topology of the space can be induced by a -norm.

Definition 14. A sequence space with a -linear topology is called a K-space provided each of the maps defined by is -continuous for all . A K-space is called a FK-space provided is a complete linear -metric space (see [15]). An FK-space whose topology is -normable is called a BK-space.

Definition 15 (-normed space). Let be a real or complex -linear space and let be a function from to the set and . Then the pair is called a -normed space and is a -norm for , if the following axioms are satisfied for all elements and for all scalars : (N1),(N2) ( is complex modulus),(N3).

Definition 16. (i) A -linear map or -linear operator between real (or complex) -linear spaces , is a function such that for all and similarly for all and .
(ii) Let and be two -normed linear spaces. A -linear map is -bounded if there is a constant such that for all . We denote the set of all -linear maps by and the set of all -bounded linear maps by .

3. Non-Newtonian Infinite Matrices

Let denote the set of all -complex sequences . As usual, we write for the sets of all -bounded, -convergent, -null sequences and

A non-Newtonian infinite matrix of -complex numbers is defined by a function from the set into . The addition and scalar multiplication of the infinite matrices and are defined by where and for all . Also, the product of and can be interpreted asprovided the series on the right hand side converge. On the other hand the series on the right hand side of (22) converges if and only ifare convergent classically. However the series (22) may -diverge for some, or all, values of ; the product may not exist.

Let , , and be an infinite matrix of -complex numbers. Then we say that defines a matrix mapping from into and denote it by writing , if for every sequence the sequence , the -transform of , exists and is in . In this way, we transform the sequence with and , into the sequence by for all . Thus, if and only if the series on the right side of (24) -converges for each and every , and we have for all . A sequence is said to be -summable to if -converges to which is called - of .

The Cesàro transform of a sequence is given by . Now, following Example 17 we may state the Cesàro summability with respect to the non-Newtonian calculus which is analogous to the classical Cesàro summable.

Example 17. Define the matrix byIf we choose the generator functions as and , then the infinite matrix can be written as follows:The -transform of a sequence is the sequence defined bywhere ( is -complex division),  , and for all . Taking and , we obtain -transform of as follows:

4. Matrix Transformations Related to the Strong Cesàro -Summability

Let and be the sequences with for all and and for . Also, if is a BK-space and then we write . Moreover, a BK-space is said to have AK if every sequence has a unique representation . If is a subset of then is called the -dual of .

Throughout the text we use the notation for strong Cesàro -summability of order and index . Maddox [19] introduced and studied the sets of sequences that are strongly summable and bounded with index , by the Cesàro method of order one. By taking into account the sets , , and of sequences that are strongly -summable to zero, -summable, and -bounded of index are defined byFor instance, (i), ;(ii), .

Theorem 18. The sets , , and are complete -metric spaces over the field with the metric defined by

Proof. The proof is a routine verification and hence omitted.

Corollary 19. The sets , , and are Banach spaces with the induced metric from the corresponding norm defined by

Proposition 20. Let . The sets , , and are BK-spaces with the (equivalent) -norms

Proof. The proof is straightforward (see [20]).

First we give the -duals of the spaces , , and . We prefer the notation and for the sum and maximum taken over all indices with and putwhereIn particular, we havefor all where .

Now, with regard to notation, for any matrix we write, for , ,and the case of (36) is interpreted as

Remark 21. Let be a -normed space and let be a -linear operator on the -normed space into . Then it is easy to see that is -continuous on if and only if there is a positive constant such that on . The -norm on the left is the norm in and that on the right is -norm in . It is also seen that Banach-Steinhaus theorem in the non-Newtonian sense holds in a -normed space defined in Definition 15.

Also one can immediately conclude that is -complete in the metric generated by the -norm. For simplicity we denote by , so that, in the case , is a Banach space and, in the case , is a complete -normed space. The dual space of , that is, the space of continuous -linear functionals on , will be denoted by .

Corollary 22 (cf. [19] (Banach-Steinhaus)). If is a sequence of continuous -linear operators on a -normed space into a -normed space such that on a second category set in , then . It is important to remark that denotes the norm of , that is, , the supremum being taken over all non--zero elements of .

The next theorem characterizes all infinite matrices which map the space into the space of convergent sequences of -complex numbers.

Theorem 23. is defined for each and the sequence is -convergent, whenever , if and only if,(a)for ,(i)(),(ii);(b)for ,(i)as in (a) above,(ii)′,(iii),

Proof. The proofs of necessity condition of (i) and (iii) and the sufficiency of the conditions are routine verification and hence omitted.
Firstly we prove the sufficiency when , leaving the necessities (ii) and (ii)′ to the next. Now, consider that the condition (ii) holds. Then the series defining is absolutely -convergent (see Definition 7) for each . By using the known inequality we haveAgain for we will show that the conditions (i) and (ii) imply (iii). In fact we will prove that (ii) which implies is uniformly -convergent (see Definition 7) and this together with (i) givesTo prove uniformly -convergent, by using the conjugate numbers and for , then, for any positive integer and for any , we obtainThus,holds where , and (i)-(ii) imply , and we deduce that is absolutely -convergent and the last sum in (41) goes to as . From (39),for every . Also, the sufficiency for the case can be obtained in a similar way.
Conversely suppose that exists for each whenever . Thus, the functionals for each and each . They are -linear and -continuous since From Corollary 22, it follows that is in , whencewhere . Taking any integer and define by for and , , for , where is such that . By (45) we have for each and impliesIn fact the series was absolutely -convergent which givesFrom (46) and (47), it can easily be seen that This leads us together with Corollary 22 to the fact that Similarly, one can prove that (ii)′ is necessary condition. This completes the proof.

Now, as an immediate consequence of Theorem 23, the following corollary shows the -duals for , , and .

Corollary 24. The -duals of the spaces , , and are

Remark 25. The symbol “” used for -dual has different meaning from that of generator used in the text.

We may give with quoting the following proposition and lemma without proofs (see [21, 22]) which are needed in the proof of next theorem.

Proposition 26. Let be a BK-space. Then we have if and only for all where for the sequence in the th row of for all .

Lemma 27. Let be an innite matrix. If and , then uniformly -converges in .

Theorem 28. The following statements hold: (i) if and only if(ii) if and only if(iii) if and only if (50) holds and for all .(iv) if and only if (50), (52) hold and

Proof. (i) Condition (50) for follows from Proposition 26 and Corollary 24. Then the other parts follows from the fact that by Proposition 20.
(ii) This condition is proved in the same way as in Theorem 23 with for all .
(iii) Since is a BK-space with AK by Proposition 20 and are closed subset of the conditions follow from the characterization of .
(iv) The conditions follow from those in (iii).

5. Concluding Remarks

Non-Newtonian calculus is a methodology that allows one to have a different look at problems which can be investigated via calculus. It should be clear that the non-Newtonian calculus is a self-contained system independent of any other system of calculus. Therefore the reader may be surprised to learn that there is a uniform relationship between the corresponding operators of this calculus and the classical calculus.

In this paper we have introduced the sequence spaces , , and as a generalization of the spaces , , and of Maddox [19]. Our main purpose is to determine the -duals of the new spaces , , and and is to characterize the classes of matrix transformations from these spaces to any one of the spaces , , and . As a future work we will try to obtain the characterizations of the classes of infinite matrices from the spaces , , and to a sequence space over the non-Newtonian complex field different from , , and .

Competing Interests

The author declares that there are no competing interests.