On the Classical Paranormed Sequence Spaces and Related Duals over the Non-Newtonian Complex Field

Abstract

The studies on sequence spaces were extended by using the notion of associated multiplier sequences. A multiplier sequence can be used to accelerate the convergence of the sequences in some spaces. In some sense, it can be viewed as a catalyst, which is used to accelerate the process of chemical reaction. Sometimes the associated multiplier sequence delays the rate of convergence of a sequence. In the present paper, the classical paranormed sequence spaces have been introduced and proved that the spaces are -complete. By using the notion of multiplier sequence, the α-, β-, and γ-duals of certain paranormed spaces have been computed and their basis has been constructed.

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, operator theory, the theory of orthogonal series, and approximation theory. The classical summability theory deals with the generalization of the convergence of sequences or series of real or complex numbers. Besides this, the studies on paranormed sequence spaces were initiated by Nakano [1] and Simons [2] at the initial stage. Later on it was further studied by Maddox [3], Lascarides [4], and Lascarides and Maddox [5]. In recent years, Mursaleen et al. [6–8] have investigated some matrix transformations of paranormed sequence spaces. Also Kirişçi and Başar [9, 10] motivated the notion of generalized difference matrix and Demiriz and Çakan [11] determined some new paranormed sequence spaces.

In the period from 1967 till 1972, Grossman and Katz [12] introduced the non-Newtonian calculus consisting of the branches of geometric, bigeometric, quadratic, biquadratic calculus, and so forth. Also Grossman extended this notion to the other fields in [13, 14]. All these calculi can be described simultaneously within the framework of a general theory. We prefer to use the name non-Newtonian to indicate any of the 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.

Bashirov et al. [15, 16] have recently concentrated on non-Newtonian calculus and gave the results with applications corresponding to the well-known properties of derivatives and integrals in classical calculus. Further Misirli and Gurefe have introduced multiplicative Adams Bashforth-Moulton method for numerical solution of differential equations in [17]. Also some authors have also worked on classical sequence spaces and related topics by using non-Newtonian calculus [18, 19]. Further Kadak [20] and Kadak et al. [21–23] have determined Kothe-Toeplitz duals and 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.

2. Preliminaries, Background, and Notations

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. For example, the identity function generates classical arithmetic, and exponential function generates geometric (multiplicative) arithmetic. As a generator, we choose the function such that those basic algebraic operations are defined as follows:for all . As an example if we choose the function : -arithmetic can be interpreted as geometric arithmetic: By an arithmetic, we mean a complete ordered field whose realm is a subset of . There are infinitely many arithmetics, all of which are isomorphic, that is, structurally equivalent. The -positive real numbers, denoted by , are the numbers in such that ; the -negative real numbers, denoted by , are those for which . The -zero, , and the -one, , turn out to be and . Also holds for all . Thus the set of all -integers can be given by One can immediately conclude that the set of -integer can be written as Besides, the -summation is defined byfor all .

Definition 1 (see [18]). Let be a nonempty set and be a function such that, for all , the following axioms hold: (NM1) if and only if , (NM2) , (NM3) .Then, the pair and are called an -metric space and an -metric on , respectively.

Throughout this paper, we define the th -exponent and th -root of asHence provided there exists an such that . For each -nonnegative number , the symbol will be used to denote which is the unique -nonnegative number whose -square is equal to . For each number , where the absolute value of is defined by

Definition 2 (see [18]). 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 th term of the sequence.(b)A sequence in is said to be -convergent if, for every given (), there exist an and such that for all which is denoted by or , as .(c)A sequence in is said to be -Cauchy if for every there is an such that for all .

Following [12], we give a new type of calculus by using the notion of non-Newtonian complex numbers, denoted by -calculus (“star-”), which is a branch of non-Newtonian calculus. From now on we will use the notation -calculus corresponding calculus which is based on two arbitrarily selected generator functions.

2.1. -Arithmetic (“Star”-Arithmetic)

Suppose that and are two arbitrarily selected generators and (“star-”) also is the ordered pair of arithmetics, that is, -arithmetic and -arithmetic. The sets and are complete ordered fields (see [19]) and beta- (alpha-) generator generates beta- (alpha-) arithmetics, respectively. Definitions given for -arithmetic are also valid for -arithmetic. The important point to note here is that -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 3 (see [13]). The -limit of a function at an element in is, if it exists, the unique number in such thatand is denoted by . Also we can give the definition for every sequence of arguments of distinct from ; if is -convergent to , then   -converges to .

A 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.

The isomorphism from -arithmetic to -arithmetic is the unique function (iota) that possesses the following three properties.(i) is one to one.(ii) is from to .(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 -arithmetic can readily be transformed into a statement in -arithmetic.

2.2. Non-Newtonian Complex Field

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

Theorem 4 (see [19]). is a field.

Following Grossman and Katz [12] we can give the definition of -distance regarding -calculus.

Definition 5 (see [19]). The -distance between two arbitrarily elements and of the set is defined by

Definition 6 (see [20]). Given a sequence of -complex numbers, the formal notation for all , is called an infinite series with -complex terms or simply complex -series. 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.

Definition 7 (see [22]). Let be a real or complex linear space and let be a function from to the set of nonnegative -real numbers. 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 :  (NN1) (), (NN2) ,  (NN3).It is trivial that a -norm on defines a -metric on which is given by , , and is called the -metric induced by the -norm.

Let be an arbitrary element. The distance function is called -norm of . In other words, where and .

In particular, in multiplicative calculus by taking , the identity function and , the exponential function, and the axioms of -normed space turn into (N(MC)1) (), (N(MC)2) , (N(MC)3) .

Then we say that is multiplicative normed space.

Definition 8 (see [21]). Let . We define the -complex conjugate of by . Conjugation changes the sign of the imaginary part of but leaves the real part the same. Thus

Remark 9 (see [21]). The following conditions hold.(i)Let , . We can give the -division of two -complex numbers and as (ii)Let and be the same generators and let . Then, the relation holds. Really,

Theorem 10 (see [19]). is a complete metric space, where is defined by (13).

Corollary 11 (see [19]). is a Banach space with the -norm defined by ;

Following Tekin and Başar [19], we can give some examples of -normed sequence spaces. First, consider the following relations which are derived from the corresponding metrics given in (13) by putting, as usual, .

Theorem 12 (see [19]). The following statements hold. (a)The spaces , , and are Banach spaces with the norm defined by (b)The space is Banach spaces with the norm defined by

Theorem 13 (see [20]). (a) The spaces , , and are Banach spaces with the norm defined by (b) The spaces , , and are Banach spaces with the corresponding norms defined by where and , for all .

Analogous to classical analysis, a sequence space with a linear -metric topology (cf. [19]) is called a -space provided that each of the maps defined by is -continuous by (9) for all . Additionally, a -space is called an -space provided that is a complete linear non-Newtonian metric space, denoted by -linear (see [20]). An -space whose non-Newtonian topology is normable and is called a -space.

3. Some Inequalities and Inclusion Relations

Definition 14 (Schauder basis). If a -normed sequence space contains a sequence with the property that for every there is a unique sequence of scalars such that with corresponding norm, then is called a Schauder basis (in non-Newtonian sense) briefly -basis, for . The series which has the sum is then called the expansion of with respect to and is written as . The concepts of Schauder and algebraic -bases coincide for finite dimensional spaces. Nevertheless, there are -linear spaces without a Schauder -basis.

Let and , () be the sequences with for all , and , where denotes the non-Newtonian Kronecker delta defined by

Example 15. The sequence is a Schauder -basis for the space and any in has a unique representation of the form

Theorem 16. The space is norm isomorphic to the space ; that is, .

Proof. To prove this, we should show the existence of a -norm preserving linear bijection between the spaces and .
Consider the transformation defined from to by . By using the corresponding operations and , the -linearity of is obvious. Further, it is trivial that whenever and hence is injective. Let and define the sequence by for all with . Then, we obtain that Thus, we observe that and hence . Consequently is surjective and is norm preserving. Hence, is a linear bijection which therefore says that the spaces and are norm isomorphic, as desired.

Theorem 17. Then the following relations are satisfied. (i) holds for each .(ii) and , where .(iii)If the inverse function is bounded in classical mean, then holds.

Proof. Since the proof is trivial for the conditions (i) and (ii), we prove only (iii).
(iii) Using (i) and (ii) we need only to show , , and . Now, consider is given. Then for every there exist an and such that for all . Since is a bounded function there exists an element such that for all . On the other hand, by applying the well-known inequalitywhich implies that . Therefore, by taking into account the boundedness of there exists such that ; we obtain that is bounded in classical mean. Thus . Hence . The remaining part can be obtained similarly.

Corollary 18. The spaces , , , , , , and are -norm isomorphic to the spaces , , , , , , and , respectively.

Now, we give some well-known inequalities in the non-Newtonian sense which are essential in the study.

Lemma 19 (Young’s inequality). Let and be conjugate real numbers. Then,holds for all and .

Proof. For any generator function , we must show that the following inequality holds:It is trivial that (29) holds for or . Let be nonzero -real numbers. Consider the function defined bywhere and . Then, the -derivative of (see [23]) can be written as From the first derivative test in non-Newtonian sense, the condition holds and is a critial point of . Besides this, and, by using the second derivative test in non-Newtonian sense we have which implies that has a maximum at ; that is, . In other words, we say thatNow taking and in (33), we get Hence the inclusion (29) holds. This step completes the proof.

Theorem 20 (Hölder’s inequality). Let and be conjugate positive real numbers and for . Then, the following inequality holds:

Proof. The inequality clearly holds when or . We may assume in the following proof. Letand, , where and . By taking into account Lemma 19 for each , we obtain which implies that Then, as is easy to see,Therefore, we deduce by combining this with the inclusion (39) that (35) holds for every .

In particular, for the inequality (35) turns out to bedenoted by Cauchy-Schwartz inequality in non-Newtonian sense.

Theorem 21 (Minkowski’s inequality). Let and for all . Then,

Proof. The case is trivial. Let and . One can immediately conclude thatThis leads us with Theorem 20 to the consequence that This concludes the proof.

4. Non-Newtonian Paranormed Sequence Spaces

Firstly, we give the definition of non-Newtonian paranorm, briefly -paranorm.

Definition 22. Let be a real or complex -linear space and let be a subadditive function from to the subset . Then the pair is called a -paranormed space and is a -paranorm, for , if the following axioms are satisfied for all elements and for all scalars . (N(PN)1)  if , . (N(PN)2) , ( is opposite -vector of ). (N(PN)3) . (N(PN)4) If is a sequence of complex scalars, that is, with as and , for all with , then as .

In particular, in bigeometric calculus case, that is, , the conditions (N(PN)1), (N(PN)2), and (N(PN)4) also hold with zero -vector , and (N(PN)3) turns into (BG(PN)3) 

Assume hereafter that is a bounded sequence of strictly positive real numbers, so that and . We will assume throughout that provided that for all .

Quite recently Tekin and Başar [19] have introduced the sets , and of sequences over the complex field which correspond to the sets , and over the complex field , respectively. It is natural to expect that the Banach spaces , and can be extended to the complete -paranormed sequence spaces so as the Maddox's spaces are derived on the real or complex field from the classical sequence spaces. Now, we may give the spaces , and in non-Newtonian sense which correspond to the well-known examples of the paranormed sequence spaces in (CC):

Following Kadak [20], we define the several sets , and of sequences in the sense of non-Newtonian calculus as follows: It is a routine verification that each of the sets , and is a -linear space.

Theorem 23. The following statements hold. (i)Define the functions and by Then and are complete -paranormed spaces by if . Also the spaces and are complete -paranormed spaces paranormed by and , respectively if and only if .(ii)The sets , , and of sequences are the complete -paranormed spaces paranormed by by (iii)The sets and are the complete -paranormed spaces by and defined by respectively, where ; for all .

Proof. To avoid repetition of similar statements, we give the proof only for the space in case (iii). The remaining parts can be obtained similarly.
The -linearity of with respect to coordinatewise addition and scalar multiplication follows from the following inequalities which are satisfied for (see Theorem 21):and the conditionholds for any scalar (cf. [3]). It is clear that and for all . Hence, by combining the inclusions (49) and (50) with subadditivity of we get the inequality .
Let be any sequence of the points of the space such that and let be any sequence of -complex scalars such that with corresponding -metric. Then, since the -triangle inequality holds, the sequence is -bounded and we thus have which tends to as . That is to say that the scalar multiplication is -continuous. Hence, is a -paranorm on the space .
It remains to prove the -completeness of the space . Let be any Cauchy sequence in the space , where . Then, for a given there exists a positive integer such thatBy taking into account the definition of for each fixed , we have that which leads us to the fact that is a Cauchy sequence for every fixed . Since is complete (see [19]), it -converges; that is, as . Using these infinitely many limits , we define the sequence . From the inclusion (52) for each and we haveTake any . First let in (54) and then , to obtain . We have by Minkowski’s inequality for each that which implies that . Since for all it follows that as . Therefore, we have shown that is -complete.

It is trivial to show that the -paranormed spaces , and may be reduced to some new sequence spaces in the special cases of the sequences and generator functions. For instance, the sequence space corresponds in the case for all to the sequence space of -bounded variation sequences in [20].

Now, as a consequence of Corollary 18, the following corollary presents the relations between -paranormed and classic paranormed spaces.

Corollary 24. The following statements hold. (i)The space is norm isomorphic to the usual paranormed space where .(ii) and .

4.1. Duality Properties

Following [24], we give the alpha-, beta-, and gamma-duals of a -paranormed sequence space which are, respectively, denoted by , , and as follows: where is the coordinatewise product of -complex numbers and for all . Throughout the text, we also use the notation “” for a -linear subspace which was created in [18].

Theorem 25. Let . Then the following statements are valid. (a) is a sequence space if .(b)If , then .(c).(d).(e).

Proof. Since the proofs are trivial for the conditions (b) and (c), we prove only (a), (d), and (e). Let , , and . (a)It is trivial that holds from the hypothesis. We show that for . Suppose that . Then and for all . We can deduce that  Hence, . Now, we show that for any and . Since for all and combining this with for all , we get . Therefore, we have proved that is a subspace of the space .(d)Obviously by Theorem 25(b). Then we must show that and . Now, consider and are given. By taking into account the cases ((a)-(b)) of Theorem 12, we have  where which implies that . So the condition holds. Conversely, for a given we prove the existence of an with . According to we can choose an index sequence which is strictly increasing with and ; By taking into account Remark 9(i) we define by where the -complex signum function is defined by  for all . Finally, by using Remark 9(ii) and taking the generators we get  Therefore, and thus . Hence . The other part of this case can be obtained similarly.(e)From the condition (c) we have since . Now we assume the existence of a . Since is an -unbounded there exists a subsequence of and we can find a number such that for all . The sequence is defined by if , and otherwise. Then . However,  Hence , which contradicts our assumption and . This step completes the proof.

In addition to Theorem 25, we give the following corollary which is immediate consequences of the -duals .

Corollary 26. For each , the following statements hold. (a); in particular, is a sequence space over .(b) and .(c)If , then .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors record their pleasure to the anonymous referee for his/her constructive report and many helpful suggestions on the main results.