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Cesaro summable sequence spaces over the non-Newtonian complex field.

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 Cesaro summable sequence spaces defined earlier to their generalized sequence spaces over the non-Newtonian complex field by using various generator functions, that is, exp and [q.sub.r] 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 R. 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 R and whose range is a subset [R.sub.a] of R where [R.sub.a] = {[alpha](x) : x [member of] R}. Each generator generates exactly one arithmetic, and conversely each arithmetic is generated by exactly one generator. If I(x) = x for all x [member of] R, then I is called identity function whose inverse is itself. In the special cases [alpha] = I and [alpha] = exp, [alpha] generates the classical and geometric arithmetic, respectively. By [alpha]-arithmetic, we mean the arithmetic whose domain is R and whose operations are defined as follows. For x,y [member of] [R.sub.[alpha]] and any generator a,

[alpha]-addition x + y = [alpha] {[[alpha].sup.-1] (x) + [[alpha].sup.-1] (y)} [alpha]-subtraction x - y = [alpha] {[[alpha].sup.-1] (x) - [[alpha].sup.-1] (y)} [alpha]-multiplication x x y = [alpha] {[[alpha].sup.-1] (x) x [[alpha].sup.-1] (y)} (1)

[alpha]-division x / y = [alpha] {[[alpha].sup.-1] (x) / [[alpha].sup.-1] (y)} [alpha]-order x < y [??][[alpha].sup.-1] (x) < [[alpha].sup.-1] (y). (1)

As an example if we choose exp function from R to the set [R.sub.exp] [subset] [R.sup.+],

[alpha]: R [right arrow] [R.sub.exp]

x [??] y = [alpha](x) = [e.sup.x], (2)

and [alpha]-arithmetic turns out to be Geometric arithmetic:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

Following Grosmann and Katz [8] we give the infinitely many [q.sub.r]-arithmetic, of which the quadratic arithmetic and harmonic arithmetic are special cases for r = 2 and r = -1, respectively. The function [q.sub.r] : R [right arrow] [R.sub.q] [subset] R and its inverse [q.sup.-1.sub.r](x) are defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

If r = 1 then the [q.sub.r]-calculus is reduced to the classical calculus.

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

Definition 1 (see [13]). Let X = (X, [d.sub.[alpha]]) be an [alpha]-metric space. Then the basic notions can be defined as follows:

(a) A sequence x = ([x.sub.k]) is a function from the set N into the set [R.sub.a]. The [alpha]-real number [x.sub.k] denotes the value of the function at k [member of] N and is called the kth term of the sequence.

(b) A sequence ([x.sub.n]) in X = (X,[d.sub.[alpha]]) is said to be [alpha] -convergent if, for every given [epsilon] > 0 ([epsilon] [member of] [R.sub.a]), there exist [n.sub.0] = [n.sub.0]([epsilon]) [member of] N and x [member of] X such that [d.sub.[alpha]]([x.sub.n], x) = [[absolute value of (xn - x)].sub.[alpha]] < [epsilon] for all n > [n.sub.0] and is denoted by [sup.[alpha]][lim.sub.n[right arrow][infinity]][x.sub.n] = x or [x.sub.n] [??] x,as n [right arrow] [infinity].

(c) A sequence ([x.sub.n]) in X = (X, [d.sub.[alpha]]) is said to be [alpha] -Cauchy if for every [epsilon] > 0 there is [n.sub.0] = [n.sub.0]([epsilon]) [member of] N such that [d.sub.[alpha]]([x.sub.n], [x.sub.m]) < [epsilon] for all m,n > [n.sub.0].

Throughout this paper, we define the pth [alpha]-exponent [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and qth [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of x [member of] [R.sub.[alpha]] by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] provided there exists y [member of] [R.sub.[alpha]] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

2.1. *-Arithmetic. Suppose that [alpha] and [beta] be two arbitrarily selected generators and "star-" also be the ordered pair of types of arithmetic ([beta]-arithmetic, [alpha]-arithmetic). The sets [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are complete ordered fields and beta(alpha)-generator generates beta(alpho)-arithmetic, respectively. Definitions given for [beta]-arithmetic are also valid for [alpha]-arithmetic. Also [alpha]-arithmetic is used for arguments and [beta]-arithmetic is used for values; in particular, changes in arguments and values are measured by a-differences and [beta]-differences, respectively.

Definition 2 (see [15]). (a) The *-limit of a function f, denoted by [sup.*][lim.sub.x[right arrow]a]f(x) = b, at an element a in [R.sub.[alpha]] is, if it exists, the unique number b in [R.sub.[beta]] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

A function f is *-continuous at a point a in [R.sub.[alpha]] if and only if a is an argument of f and [sup.*][lim.sub.x[right arrow]a]f(x) = f(a). When [alpha] and [beta] are the identity function I, the concepts of * -limit and *-continuity are identical with those of classical limit and classical continuity.

(b) The isomorphism from [alpha]-arithmetic to [beta]-arithmetic is the unique function i (iota) which has the following three properties:

(i) i is one to one.

(ii) i is from [R.sub.[alpha]] onto [R.sub.[beta]].

(iii) For any numbers u, v [member of] [R.sub.[alpha]],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

It turns out that i(x) = [beta]{[[alpha].sup.-1](x)] for every a in [R.sub.[alpha]] and that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for every [alpha] -integer [??]. Since, for example, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], it should be clear that any statement in [alpha]-arithmetic can readily be transformed into a statement in [beta]-arithmetic.

2.2. Non-Newtonian Complex Field and Some Inequalities.

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be arbitrarily chosen elements from corresponding arithmetic. Then the ordered pair (a,b) [member of] [R.sub.[alpha]] x [R.sub.[beta]] [subset or equal to] [R.sup.2] is called a *-point. The set of all *-points is called the set of *-complex numbers and is denoted by [C.sup.*]; that is,

[C.sup.*] := {[z.sup.*] = (a, b) | a [member of] [R.sub.[alpha]], b [member of] [R.sub.[beta]]}. (9)

Define the binary operations addition ([direct sum]) and multiplication ([dot encircle]) of *-complex numbers [z.sup.*.sub.1] = ([a.sub.1],[b.sub.1]) and [z.sup.*.sub.2] = ([a.sub.2], [b.sub.2]):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

for all [a.sub.1], [a.sub.1] [member of] [R.sub.a] and [b.sub.1], [b.sub.1] [member of] [R.sub.[beta]].

Theorem 3 (see [15]). (C*,[direct sum], [dot encircle]) 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 X be a nonempty set and let [d.sup.*] : X x X [right arrow] [R.sub.[beta]] be a function such that, for all x,y,z [member of] X, the following axioms hold:

(NM1) [d.sup.*] (x, y) = [??] if and only if x = y,

(NM2) [d.sup.*] (x,y) = [d.sup.*] (y,x),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then the pair (X, [d.sup.*]) and [d.sup.*] are called a non-Newtonian metric (*-metric) space and a *-metric on X, respectively.

The *-distance [d.sup.*] between two arbitrarily elements [z.sup.*.sub.1] = ([a.sub.1], [b.sub.1]) and [z.sup.*.sub.2] = ([a.sub.2], [b.sub.2]) of the set [C.sup.*] is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)

Up to now, we know that [C.sup.*] is a field and the distance between two points in [C.sup.*] is computed by the function [d.sup.*]. Let [z.sup.*] = (a, b) [member of] [C.sup.*] be an arbitrary element. The distance function [d.sup.*] ([z.sup.*], [0.sup.*]) is called *-norm of [z.sup.*] and is denoted by [??]*[??]. In other words, let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

Moreover, for all [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [d.sup.*] is the induced metric from [??]*[??] norm.

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

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

Definition 7. (a) Given a sequence [z.sup.*.sub.k] = ([a.sub.k], [b.sub.k]) of *-complex numbers, the formal notation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

is called an infinite series with *-complex terms, or simply complex *-series for all k e N. Also, for integers n [member of] N, the finite *-sums [s.sup.*.sub.n] = [sub.*][[summation].sup.n.sub.k=0] [z.sup.*.sub.k] are called the partial sums of complex *-series. If the sequence *-converges to a complex number [s.sup.*] then we say that the series *-converges and write [s.sup.*.sub.n] = [sub.*][[summation].sup.[infinity].sub.k=0] [z.sup.*.sub.k]. The number [s.sup.*] is then called the *-sum of this series. If ([s.sub.n]) *-diverges, we say that the series *-diverges or that it is *-divergent.

(b) A *-series [sub.*][[summation].sup.[infinity].sub.k=0] [z.sup.*.sub.k] is said to *-converge absolutely if [s.sup.*.sub.n] = [sub.*][[summation].sup.[infinity].sub.k=0] [??][z.sup.*.sub.k][??] = l for some number l [member of] [R.sub.[beta]].

(c) Let {[f.sub.n](x)} be a sequence of functions from A [subset or equal to] R to [C.sup.*] for each n. We say that {[f.sub.n](v)} is uniformly *-convergent to f on A if and only if, for each x [member of] A and for an arbitrary [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], there exists an integer N = N([epsilon], x) such that [d.sup.*] ([f.sub.n] (x), f(x)) [??] [epsilon] whenever n > N.

(d) The series [sub.*][[summation].sup.[infinity].sub.k=0][f.sub.k](x) is said to be uniformly *-convergent to f(x) on A if, given any [epsilon]> 0, there exists an integer [n.sub.0] (e) such that

[d.sup.*] ([sub.*][[summation].sup.n.sub.k=0][f.sub.k](x),f(x)) [??] [epsilon] whenever n [greater than or equal to] [n.sub.0] ([epsilon]). (14)

Proposition 8 (see [15]). Let p [greater than or equal to] 1 and [z.sup.*.sub.k], [t.sup.*.sub.k] [member of] [C.sup.*] for k [member of] {0,1,2,3, ..., n}. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

Remark 9. Let [z.sup.*.sub.1] = ([a.sub.1], [b.sub.1]),[z.sup.*.sub.2] = ([a.sub.2], [b.sub.2]) [member of] [C.sup.*]. Then the following statements hold:

(i) One has

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

(ii) Let a and be the same generators. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

Definition 10. Given a point [x.sub.0] [member of] X. Then, for a positive [beta]-real number r,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

are *-neighborhood (or *-open (closed) ball) of centre [x.sub.0] and radius r, respectively.

We see that an * -open ball of radius r is the set of all points in X whose beta-distance from the center of the ball is less than r and we say directly from the definition that every *neighborhood of [x.sub.0] contains [x.sub.0]; in other words, [x.sub.0] is a point of each of its *-neighborhoods.

Definition 11. Let (X,[d.sup.*]) be a *-metric space. Then the followings are valid:

(i) G [subset] X is called *-open set if and only if every point of G has a *-neighborhood contained in G. Also G [subset] X is called *-closed set if and only if its complement is *-open.

(ii) The *-interior [G.sup.0] is the largest *-open set contained in G and the *-closure [bar.G] is the smallest *-closed set contained in G.

Definition 12 (usual *-topology). Consider the set of *complex numbers with

[tau] ={S [subset] [C.sup.*] | [for all]v [member of] S, [there exists]r [??] 0 [contains as member] [B.sup.*] ([x.sub.0];r) [subset] s} (19)

for all [x.sub.0] [member of] [C.sup.*] and r [member of] [R.sub.[beta]]. Then ([C.sup.*],[tau]) is a topological space and is called *-usual topology on [C.sup.*].

Definition 13. (i) A topological *-vector (linear) space X 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 [lambda] with a *-linear topology is called a *K-space provided each of the maps [p.sub.i] : [lambda] [right arrow] [C.sup.*] defined by [p.sub.i](x) = [x.sub.i] is *-continuous for all i [member of] N. A * K-space is called a * FK-space provided [lambda] is a complete linear *-metric space (see [15]). An * FK-space whose topology is *-normable is called a * BK-space.

Definition 15 (p- * normed space). Let X be a real or complex *-linear space and let [[??]*[??].sub.p] be a function from X to the set [R.sup.+.sub.[beta]] and p > 0. Thenthepair (X, [[??]*[??].sub.p]) is called a p-*normed space and [??]*[??] p is a p-*norm for X, if the following axioms are satisfied for all elements x,y [member of] X and for all scalars [lambda]:

(N1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(N2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(N3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 16. (i) A *-linear map or *-linear operator T between real (or complex) *-linear spaces X, Y is a function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(ii) Let X and Y be two *-normed linear spaces. A *-linear map T : X [right arrow] Y is *-bounded if there is a constant [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all x [member of] X. We denote the set of all *-linear maps T : X [right arrow] Y by L(X, Y) and the set of all *-bounded linear maps by B(X, Y).

3. Non-Newtonian Infinite Matrices

Let [w.sup.*] denote the set of all *-complex sequences x = [([x.sub.k]).sup.[infinity].sub.k=0]. As usual, we write [l.sup.*.sub.[infinity]], [c.sup.*], [c.sup.*.sub.0] for the sets of all *-bounded, *-convergent, *-null sequences and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

A non-Newtonian infinite matrix A = ([a.sup.*.sub.ij]) of *-complex numbers is defined by a function A from the set N x N into [C.sup.*]. The addition ([direct sum]) and scalar multiplication ([dot encircle]) of the infinite matrices A = ([a.sup.*.sub.ij]) = ([[epsilon].sub.ij], [[delta].sub.ij]) and B = ([b.sup.*.sub.ij]) = ([[mu].sup.*.sub.ij], [[eta].sub.ij]) are defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (21)

where [[epsilon].sub.ij], [[mu].sub.ij] [member of] [R.sub.[alpha]] and [[delta].sub.ij], [[eta].sub.ij] [member of] [R.sub.[beta]] for all i, j [member of] N. Also, the product [(A [dot encircle] B).sub.ij] of A = ([a.sup.*.sub.ij]) and B = ([b.sup.*.sub.ij]) can be interpreted as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (22)

provided 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 if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (23)

are convergent classically. However the series (22) may *-diverge for some, or all, values of i, j; the product A [dot encircle] B may not exist.

Let [[mu].sup.*.sub.1], [[mu].sup.*.sub.2] [subset] [w.sup.*], and A = ([a.sup.*.sub.nk]) = ([[epsilon].sub.nk], [[delta].sub.nk]) be an infinite matrix of *-complex numbers. Then we say that A defines a matrix mapping from [[mu].sup.*.sub.1] into [[mu].sup.*.sub.2] and denote it by writing A : [[mu].sup.*.sub.1] [right arrow] [[mu].sup.*.sub.2],if for every sequence z = ([z.sup.*.sub.k]) [member of] [[mu].sup.*.sub.1] the sequence A [dot encircle] z = {[(Az).sub.n]}, the A-transform of z, exists and is in [[mu].sup.*.sub.2]. In this way, we transform the sequence z = ([z.sup.*.sub.k]) = ([[mu].sub.k], [[eta].sub.k]) with [[mu].sub.k] [member of] [R.sub.[alpha]] and [[eta].sub.k] [member of] [R.sub.[beta]], into the sequence {[(Az).sub.n]} by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (24)

for all k,n [member of] N. Thus, A [member of] ([[mu].sup.*.sub.1] : [[mu].sup.*.sub.2]) if and only if the series on the right side of (24) *-converges for each n [member of] N and every z [member of] [[mu].sup.*.sub.1], and we have A [dot encircle] z = [{[(Az).sub.n]}.sub.n[member of]N] [member of] [[mu].sup.*.sub.2] for all z [member of] [[mu].sup.*.sub.1].A sequence z is said to be A-summable to [gamma] if A [dot encircle] z *-converges to [gamma] [member of] [C.sup.*] which is called A-*lim of z.

The Cesaro transform of a sequence z = ([z.sup.*.sub.k]) [member of] [w.sup.*] is given by [C.sup.*.sub.1] [dot encircle] z = [{[([C.sup.*.sub.1] z).sub.n]}.sub.n[member of]N]. Now, following Example 17 we may state the Cesaro summability with respect to the non-Newtonian calculus which is analogous to the classical Cesaro summable.

Example 17. Define the matrix [C.sup.*.sub.1] = ([c.sup.*.sub.nk].) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (25)

If we choose the generator functions as [alpha] = exp and [beta] = exp, then the infinite matrix can be written as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (26)

The [C.sup.*.sub.1] -transform of a sequence x = ([x.sup.*.sub.k]) = ([a.sub.k], [b.sub.k]) is the sequence y = ([y.sup.*.sub.n]) defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (27)

where ([empty set] is *-complex division), [a.sub.k] [member of] [R.sub.[alpha]], and [b.sub.k] [member of] [R.sub.[beta]] for all k [member of] N. Takng [alpha] = exp and [beta] = exp, we obtain [C.sup.*.sub.1]-transform of x = ([x.sup.*.sub.k]) = ([a.sub.k], [b.sub.k]) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (28)

4. Matrix Transformations Related to the Strong Cesaro *-Summability

Let e and [e.sup.(n)] (n = 0,1, ...) be the sequences with [e.sub.k] = [1.sup.*] for all k [member of] N and [e.sup.(n).sub.n] = [1.sup.*] and [e.sup.(n).sub.k] = [0.sup.*] for k [not equal to] n. Also, if X is a *BK-space and a [member of] [w.sup.*] then we write [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Moreover, a *BK-space X [contains] 0 is said to have *AK if every sequence x = [([x.sub.k]).sup.[infinity].sub.k=1] [member of] X has a unique representation x = [sub.*][[summation].sup.[infinity].sub.k=1] [x.sub.k] [dot encircle] [e.sup.(k)] If X is a subset of [w.sup.*] then [X.sup.[beta]] = {a [member of] [w.sup.*] : A [dot encircle] x [member of] c[s.sup.*] for all x [member of] X} is called the [beta]-dual of X.

Throughout the text we use the notation [[[C.sup.*.sub.1]].sub.p] for strong Cesaro *-summability of order 1 and index _p. Maddox [19] introduced and studied the sets of sequences that are strongly summable and bounded with index p, (1 [less than or equal to] p < [infinity]) by the Cesaro method of order one. By taking into account the sets [w.sup.p.sub.0]([alpha], [beta]), [w.sup.p]([alpha], [beta]) and [w.sup.p.sub.[infinity]] ([alpha], [beta]) of sequences that are strongly * -summable to zero, *-summable, and *-bounded of index p [greater than or equal to] 1 are defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (29)

For instance,

(i) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 18. The sets [w.sup.p.sub.0]([alpha], [beta]), [w.sup.p]([alpha],[beta]), and [w.sup.p.sub.[infinity]]([alpha], [beta]) are complete * -metric spaces over the field [C.sup.*] with the metric [d.sup.([alpha],[beta]).sub.p] defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (30)

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

Corollary 19. The sets [w.sup.p.sub.0]([alpha], [beta]), [w.sup.p]([alpha], [beta]) and [w.sup.p.sub.[infinity]] ([alpha], [beta]) are Banach spaces with the induced metric [d.sup.([alpha],[beta]).sub.p] from the corresponding norm [[parallel]x[parallel].sub.[dagger]] defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (31)

Proposition 20. Let 1 [less than or equal to] p < [infinity]. The sets [w.sup.p.sub.0]([alpha], [beta]), [w.sup.p]([alpha], [beta]) and [w.sup.p.sub.[infinity]] ([alpha], [beta]) are *BK-spaces with the (equivalent) *-norms

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (32)

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

First we give the [beta]-duals of the spaces [w.sup.p.sub.0]([alpha], [beta]), [w.sup.p]([alpha], [beta]) and [w.sup.p.sub.[infinity]] ([alpha], [beta]). We prefer the notation [[summation].sub.v] and [max.sub.v] for the sum and maximum taken over all indices k with [2.sup.v] [less than or equal to] k [less than or equal to] [2.sup.v+1] - 1 and put

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (33)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (34)

In particular, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (35)

for all [z.sub.k] = ([a.sub.k], [b.sub.k]) [member of] [w.sup.*] where [a.sub.k], [b.sub.k] [member of] [R.sub.exp] = (0, [infinity]).

Now, with regard to notation, for any matrix A = ([a.sup.*.sub.nk]) we write, for 1/p + 1/q = 1, p [greater than or equal to] 1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (36)

and the case p = 1 of (36) is interpreted as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (37)

Remark 21. Let Y be a *-normed space and let T be a *-linear operator on the p- *normed space X into Y. Then it is easy to see that T is *-continuous on X if and only if there is a positive constant M [member of] [R.sub.[beta]] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on X. The *-norm on the left is the norm in Y and that on the right is p-*norm in X. It is also seen that Banach-Steinhaus theorem in the non-Newtonian sense holds in a p-*normed space defined in Definition 15.

Also one can immediately conclude that [[C.sup.*.sub.1]]p is *complete in the metric generated by the *-norm. For simplicity we denote [[C.sup.*.sub.1]]p by [OMEGA], so that, in the case p [greater than or equal to] 1, [OMEGA] is a Banach space and, in the case 0 < p < 1, [OMEGA] is a complete p-*normed space. The dual space of [OMEGA], that is, the space of continuous *-linear functionals on [OMEGA], will be denoted by [[OMEGA].sup.*].

Corollary 22 (cf. [19] (Banach-Steinhaus)). If ([T.sub.n]) is a sequence of continuous *-linear operators on a p-*normed space X into a *-normed space Y such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on a second category set in X, then [sup.sub.n][parallel][T.sub.n][parallel] [??] [infinity]. It is important to remark that [parallel][T.sub.n][parallel] denotes the norm of [T.sub.n], that is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the supremum being taken over all non-[beta]-zero elements of X.

The next theorem characterizes all infinite matrices A which map the space [[C.sup.*.sub.1]]p into the space [c.sup.*] of convergent sequences of *-complex numbers.

Theorem 23. [(Az).sub.n] is defined for each n and the sequence {[(Az).sub.n]} is *-convergent, whenever z [member of] [OMEGA], if and only if

(a) for 0 < p < 1,

(i) [d.sup.*]([a.sup.*.sub.nk], [[xi].sub.k]) = [??] (n [right arrow] [infinity],k fixed),

(ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(b) for p [greater than or equal to] 1,

(i) as in (a) above,

(ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(iii) [d.sup.*]([sub.*][[summation].sup.n.sub.k=0] [a.sup.*.sub.nk], [xi]) = [??], (n [right arrow] [infinity]).

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 0 < p < 1, leaving the necessities (ii) and (ii/ to the next. Now, consider that the condition (ii) holds. Then the series defining [(Az).sub.n] is absolutely *-convergent (see Definition 7) for each n [member of] N. By using the known inequality we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (38)

Again for 0 < p < 1 we will show that the conditions (i) and (ii) imply (iii). In fact we will prove that (ii) which implies [[summation].sup.[infinity].sub.k=0] [parallel][a.sup.*.sub.nk][parallel] is uniformly *-convergent (see Definition 7) and this together with (i) gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (39)

To prove uniformly *-convergent, by using the conjugate numbers p and q for 0 < p < 1, then, for any positive integer [n.sub.0] and for any [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (40)

Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (41)

holds where [z.sub.k] [right arrow] l, and (i)-(ii) imply [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and we deduce that [sub.*][summation] [[xi].sub.k] [dot encircle] [z.sub.k] is absolutely * -convergent and the last sum in (41) goes to [??] as n [right arrow] [infinity]. From (39),

[sub.*][summation] [a.sup.*.sub.nk] [dot encircle] [z.sub.k] [right arrow] [sub.*][summation] [[xi].sub.k] [dot encircle] [z.sub.k] (42)

for every z [member of] [[omega].sup.*]. Also, the sufficiency for the case p [greater than or equal to] 1 can be obtained in a similar way.

Conversely suppose that [(Az).sub.n] exists for each n [greater than or equal to] 1 whenever x [member of] [OMEGA]. Thus, the functionals

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (43)

for each n and each v [greater than or equal to] 0. They are *-linear and * -continuous since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (44)

From Corollary 22, it follows that [lim.sub.v] [sub.*][[summation].sub.v] [a.sup.*.sub.nk] [do encircle] [z.sub.k] = [(Az).sub.n] is in [[OMEGA].sup.*], whence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (45)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Taking any integer s > 0 and define z [member of] [OMEGA] by [z.sub.k] = [0.sup.*] for k [greater than or equal to] [2.sup.S+1] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [N.sub.v] is such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. By (45) we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for each n and implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (46)

In fact the series [(Az).sub.n] was absolutely *-convergent which gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (47)

From (46) and (47), it can easily be seen that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (48)

This leads us together with Corollary 22 to the fact that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (49)

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 [beta]-duals for [[omega].sup.p.sub.0]([alpha], [beta]), [[omega].sup.p]([alpha], [beta]) and [[omega].sup.p.sub.[infinity]] ([alpha], [beta]).

Corollary 24. The [beta]-duals of the spaces [[omega].sup.p.sub.0]([alpha], [beta]), [[omega].sup.p]([alpha], [beta]) and [[omega].sup.p.sub.[infinity]] ([alpha], [beta]).

Remark 25. The symbol "[beta]" used for [beta]-dual has different meaning from that of [beta] 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 X be a *BK-space. Then we have A [member of] (X : [l.sup.*.sub.[infinity]]) if and only [A.sub.n] [member of] [X.sup.[beta]] for all n [member of] N where [A.sub.n] = ([a.sup.*.sub.nk]) for the sequence in the nth row of A for all k [member of] N.

Lemma 27. Let A = ([a.sup.*.sub.nk]) be an innite matrix. If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] uniformly *-converges in n [member of] N.

Theorem 28. The following statements hold:

(i) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (50)

(ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (51)

(iii) A = ([a.sup.*.sub.nk]) [member of] ([[omega].sup.p.sub.0] ([alpha],[alpha]) : [c.sup.*.sub.0]) if and only if (50) holds and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (52)

for all k [member of] N.

(iv) A = ([a.sup.*.sub.nk]) [member of] ([[omega].sup.p] ([alpha],[alpha]) : [c.sup.*.sub.0]) if and only if (50), (52) hold and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (53)

Proof. (i) Condition (50) for A [member of] ([[omega].sup.p.sub.0]([alpha],[beta]) : [l.sup.*.sub.[infinity]]) follows from Proposition 26 and Corollary 24. Then the other parts follows from the fact that [[omega].sup.p.sub.0]([alpha],[beta]) [subset] [[omega].sup.p]([alpha],[beta]) [subset] [[omega].sup.p.sub.[infinity]]([alpha],[beta]) by Proposition 20.

(ii) This condition is proved in the same way as in Theorem 23 with [[xi].sub.k] = [0.sup.*] for all k [member of] N.

(iii) Since [[omega].sup.p.sub.0]([alpha],[beta]) is a *BK-space with *AK by Proposition 20 and [c.sup.*],[c.sup.*.sub.0] are closed subset of [l.sup.*.sub.[infinity]] the conditions follow from the characterization of ([[omega].sup.p.sub.0]([alpha],[beta]) : [I.sup.*.sub.[infinity]]).

(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 [[omega].sup.p.sub.0]([alpha],[beta]), [[omega].sup.p]([alpha],[beta]), and [[omega].sup.p.sub.[infinity]]([alpha],[beta]) as a generalization of the spaces [[omega].sup.p.sub.0],[ [omega].sup.p], and [[omega].sup.p.sub.[infinity]] of Maddox [19]. Our main purpose is to determine the [beta]-duals of the new spaces [[omega].sup.p.sub.0]([alpha],[beta]), [[omega].sup.p]([alpha],[beta]), and [[omega].sup.p.sub.[beta]]([alpha],[beta]) and is to characterize the classes of matrix transformations from these spaces to any one of the spaces [l.sup.*.sub.[infinity]], [c.sup.*], and [c.sup.*.sub.0]. As a future work we will try to obtain the characterizations of the classes of infinite matrices from the spaces [[omega].sup.p.sub.0]([alpha],[beta]), [[omega].sup.p]([alpha],[beta]), and [[omega].sup.p.sub.[infinity]]([alpha],[beta]) to a sequence space A over the non-Newtonian complex field different from [l.sup.*.sub.[infinity]], [c.sup.*], and [c.sup.*.sub.0].

http://dx.doi.org/10.1155/2016/5862107

Competing Interests

The author declares that there are no competing interests.

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Ugur Kadak

Department of Mathematics, Faculty of Sciences and Arts, Bozok University,

Turkey

Correspondence should be addressed to Ugur Kadak; ugurkadak@gmail.com

Received 12 September 2015; Accepted 27 October 2015

Academic Editor: Chin-Shang Li
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Date:Jan 1, 2016
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