An Overview of Generalized Cesàro Sequence Spaces in Geometric
Calculus
MELİSA TİFTİKÇİER1, NİHAN GÜNGÖR2
1Department of Mathematics Engineering
Gumushane University
Gumushane
TURKEY
2Department of Basic Science
Samsun University
Samsun
TURKEY
Abstract: In this study, we introduce the generalized Cesàro sequence space in geometric calculus and establish
a -modular on this space. The generalized geometric Cesàro sequence space is equipped with the Luxemburg
-norm induced by the -modular. The connections of the -modular and Luxemburg -norm on this
space are studied. In addition, we show that the generalized geometric Cesàro sequence space is a -Banach
space under the Luxemburg -norm and it is -nonsquare where for all .
Key-Words: non-Newtonian calculus, geometric calculus, Cesàro sequence spaces, geometric Cesàro
sequence spaces, -modular, Luxemburg -norm, -nonsquare
Received: March 7, 2024. Revised: October 9, 2024. Accepted: November 7, 2024. Published: December 2, 2024.
1 Introduction
Grossman and Katz [11] introduced non-Newtonian
calculus, which is a novel framework composed of
the branches of geometric, bigeometric, harmonic,
biharmonic, quadratic, and biquadratic calculus.
Non-Newtonian calculus encompasses a diverse
range of uses that include subjects like interest rates,
the theory of economic elasticity, blood viscosity,
biology, and computer science, including image
processing and artificial intelligence, functional
analysis, probability theory, and differential
equations. One of the most well-known classes of
non-Newtonian calculus is geometric calculus, which
offers a variety of viewpoints that are helpful for
applications in the fields of science and engineering.
It offers differentiation and integration methods
grounded in multiplication rather than addition. In
general, geometric calculus is a methodology that
allows for a different perspective on problems that
can be studied through calculus. Geometric calculus
is preferred over a traditional Newtonian one in
specific cases, particularly when dealing with issues
related to price elasticity and growth. To have a
deeper understanding of non-Newtonian calculus,
one must be familiar with several forms of arithmetic
and their generators. The all, -absolutely summable,
boundedness, convergent and null sequence spaces in
the context of non-Newtonian calculus denoted by
, respectively, are
defined and it is shown that these sets constitute a
complete metric space by Çakmak and Başar [4].
Güngör [10] investigated some geometric properties
of the non-Newtonian geometric sequence spaces
. Boruah and Hazarika [2] introduced the
generalized geometric difference sequence spaces
with some properties.
Mahto et al. [16] introduced bigeometric Cesàro
difference sequence spaces and investigated the -
duals of these sequence spaces. More information on
the non-Newtonian calculus may be found for the
reader in [1, 6-9,12-14,17-20,23,26].
Sequence spaces have applications in a wide
variety of disciplines, including economics and
engineering. Studies on the geometric and topologic
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aspects of sequence spaces have been the focus of
research in both pure and applied analysis due to the
significance of sequence spaces as an example of
function spaces and their involvement in the study of
the theory of Banach spaces. The Cesàro sequence
spaces, ( and were established
in 1968 as a part of the Dutch Mathematical Society's
challenge to find duals. Shiue [24] investigated some
properties of these spaces and gave the first norm-
based description of them. Leibowitz [15] showed
that are separable reflexive Banach
spaces for and the spaces are in
for . Sanhan and Suantai [21] defined the
generalized Cesàro sequence spaces . They
examined the space for completeness and also
discussed its rotundity. Suantai [25] showed that the
space has property (H) and property (G), and
it is rotund. Many mathematicians have extensively
researched the Cesàro sequence spaces via geometric
and topological properties.
Motivated essentially by the aforementioned
publications above, this study considers the
geometric calculus concept of generalized Cesàro
sequence space a novel and intriguing addition to the
current literature in this field. We investigate
geometric calculus versions of some concepts and
properties given for classical generalized Cesàro
sequence spaces. We hope this study will shed fresh
light on how to approach solving issues in contexts
where the theory of sequence spaces in fields ranging
from engineering to economics and the theory of
geometric calculus have a wide variety of uses.
Now, we offer a brief introduction to geometric
calculus that emphasizes the terminology required for
this discussion.
The building blocks of every arithmetic system
are the four operations on the set (addition,
subtraction, multiplication and division) and an
ordering relation that follows the rules of a
completely ordered field. The set is referred to as
the realm, and the elements of the set are termed
the numbers of the system. A generator is an injective
function whose domain is and whose range is a
subset of . The range of the generator is called
non-Newtonian real line and it is demonstrated by
. arithmetic operations and ordering relations
are described as follows [11]:
addition
subtraction
multiplication
division
order
Particularly, the identity function generates classical
arithmetic.
In calculus, the paired arithmetics
(arithmetic, arithmetic) are utilized for
arguments and values, respectively. The subsequent
particular calculi are derived when and are
chosen as either and , representing the identity
and exponential functions, respectively [11]:
Calculus
Classical
Geometric
Anageometric
Bigeometric
.
The classical arithmetic is derived from the identity
function. If the generator is chosen as exponential
function defined by for , then
, arithmetic turns into geometric
arithmetic. The definitions of geometric operations
and ordering relation are [2, 11]:
Geometric additon
Geometric subtraction
Geometric multiplication
Geometric division
Geometric order
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The set of geometric real numbers which is denoted
by , is defined as . is a field
with geometric zero and geometric identity
. The sets of geometric positive real numbers
and geometric negative real numbers are defined as
and
, respectively. The set of all geometric
integers are as follows:
The geometric absolute value of is defined by
and this is equivalent to expression . For
any , the subsequent statements are valid:
i)
ii)
iii)
iv)
For any , and
[2, 11].
Definition 1.1. [3, 5] A geometric vector space (-
vector space) over a geometric field is a non-
empty set equipped with two operations and ,
called geometric vector addition and geometric scalar
multiplication, respectively, which satisfy the
following properties:
GV1) Closure: If , then belong to .
GV2) Associative law:
for all .
GV3) Additive identity: contains an additive
identity element denoted by , such that
for all .
GV4) Additive inverse: For all , there is a vector
with and
.
GV5) Commutative law: for all
.
GV6) Closure: If and , then
belong to .
GV7) Distributive laws:
, for all and
.
, for all and
GV8) Associative law:
for all and .
GV9) Unitary law: for all .
The set of all sequences of the geometric real
numbers demonstrated by , i.e.,
. Based on the algebraic
operations addition and multiplication, is a
-vector space over [2,4].
Definition 1.2. [4] Let be a -vector space. If the
function holds the following
properties for all and,
GN1)
GN2)
GN3)
then is said to be a -normed space.
Definition 1.3. [4, 27] Let be -normed
space and be a sequence in . If for every given
, there exist and such
that for all , then is said
to be -convergent and it is denoted by
as
.
Definition 1.4. [27] Let be -normed
space and be a sequence in . If for every given
, there is such that
for all , then is said to be
-Cauchy sequence.
If every -Cauchy sequence in converges, then
it is said that is a -Banach. For example,
is a -Banach space.
Definition 1.5. [10] Let be a
function. The function is said to be -convex, if for
every it satisfies
with .
Proposition 1.6. [22] Let , then
for .
2 Main Results
This section introduces the idea of generalized
geometric Cesàro sequence space with a new
perspective on the concept of Cesàro sequence space
and it provides a basic explanation of the theory
behind this sequence space.
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First, we will present the concepts of modular,
modular spaces and Luxemburg norm according to
geometric calculation style.
Definition 2.1. Let be a -vector space on
field. The function is called -
modular provided that it satisfies the given
requirements:
i) if and only if ,
ii) for all scalar with
,
iii) for all
and with.
Moreover, the -modular is called -convex (-
convex modular) if
iv)
for all and with.
Definition 2.2. If is a -modular in , we define
and it is called -modular spaces.
Theorem 2.3. If is a -convex modular, then the
function defined as
is -norm on .
Proof.
GN1) Let . Since
for every
, then we find .
Conversely, let . Since is a -convex
modular
for all with . Therefore, we can
write
for every . Hence, it is obtained that
which implies .
GN2) Take any and . If , then
clearly . Assume that
. Based on the provided definition of ,
we can write
Let . Considering
, then we
obtain
Let . If we take it as
, we get
By using (1) and (2), we obtain
for all . If we take ,
then we can see that
Using the expressions (3) and (4), we obtain
for all .
GN3) Let any be given. Let
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and
Since is a -convex modular, we find
for arbitrary . Selecting
, then we can see that
. Because of ,
holds. As a result, we get
Now, the concept of generalized geometric
Cesàro sequence space is given, which forms the
basis of this paper:
Definition 2.4. Let represent a bounded
sequence of positive real numbers with for all
. The generalized geometric Cesàro sequence
space is defined as
whenever
.
In the present study, we make the assumption that
be a sequence of positive real numbers
where for every and
.
Proposition 2.5. Let and , then
the following statement holds:
Proof. We can write
for any . Therefore, we get
Theorem 2.6. is a -vector spaces under
geometric addition and geometric scalar
multiplication operations for geometric real
sequences.
Proof. Given any
and . Let
. If it is choosen
for all , we have
from Proposition.1.6. We get
from Proposition 2.5. This demonstrates that
, i.e., .
Taken
, it is obtained that
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This shows that . Consequently,
is -vector space.
Proposition 2.7. The function ,
is -convex where for all .
Proof. Let’s taken . We have
t
for all . Since implies
, we get
by aid of the convexity of the function in
classical calculus. From (5) and (6), we find
Hence, the proof is completed.
Theorem 2.8. is a -convex modular on
.
Proof. Let .
i) It is obvious that .
ii) For with , we write
iii) Let with , .
Since
is -convex function for all
, we have
Proposition 2.9. The -modular on has
the subsequent properties:
i) If , then
and .
ii) If , then
.
iii) If , then
.
Proof.
i) Let . Hence we find
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Since for all , we have
ii) Let, then we get
iii) Let . Since is -convex modular, we
can write . Also, we find
Hence, we obtain the required inequalities.
Theorem 2.10. is -normed space with
regard to the Luxemburg -norm
Proof.
GN1) Let . Since
for every , then we
find .
Conversely, let . Since
for all with , we can write
for all . Hence, it is obtained that
which
implies .
GN2) Let’s take any and . If
, then it is obvious that
. Suppose that . We can write
If it is taken as , then we obtain
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.
GN3) Let any be given. , and
be sets of the positive geometric numbers ,
and hold the following inequalities
and
respectively. Taken as
. For any , we get
Therefore, we have . It is obtained that
, because of
. So, we get .
Now, we discuss some relations between the -
modular and Luxemburg -norm on .
Proposition.2.11. For any , we have
i) If , then
ii) If , then
iii) If
iv) If
v) If .
Proof.
i) Let be such that .
Hence we can write . By the
definiton of , there exists such that
and
. From
Proposition.2.9.(i) and (iii), we find
Thus , for all .
Taken as ,
then we see that
. Since is a
lower bound of , we have .
ii) Let be such that
,
hence we can write
. By the definiton of and
Proposition.2.9.(i), we find
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So, we can write for all
. Taken as
,
then we see that
. Since is an
upper bound of , we have .
iii) Let . Suppose that . From the
definiton of , there exists such that
and
. Since
,
by Proposition.2.9 (ii). Hence
for all which implies . Assume that
, we can choose such that
. Hence we have
by Proposition.2.9.(i). Therefore, we get
which contradicts to our
assumption that . Hence .
Conversely, suppose that . The definition of
Luxemburg -norm , we conclude
that. If , then we have by (i)
that which contradicts to our
assumption that . Therefore, .
iv) If , then we have by (i) that
.
Conversely, assume that . It follows from
(i) and (ii) .
v) It follows from (iii) and (iv).
Proposition 2.12. Let and
.
i) If and , then
.
ii) If and , then .
Proof.
i) Assume that and . Hence
we can write
We have
by using
Proposition.2.11.(ii). Since , we obtain
from
Proposition.2.9.(i).
ii) Assume that and . Hence we
can write
. We find
from
Proposition.2.11.(i). If , then we get
. If , we obtain
from Proposition.2.9.(ii).
Proposition 2.13. Let be a sequence in
.
i) If
, then
.
ii) If
, then
.
Proof.
i) Assume that
. Let .
Then there exists such that
for all . By
Proposition.2.12 (i) and (ii), we find
for all which
implies that
as .
ii) Assume that
. Then there exists
and a subsequence of such that
for all . By Proposition.2.12.(i),
we have for all . This implies
as .
Proposition.2.14..Let
for all . If
as
and
as for all , then
as .
Proof.
Let be given. Since
, there exists such
that
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Since
as and
as , there exists
such that
for all , and
for all . It follows from (7), (8) and (9) that for
all ,
This shows that
as .
Therefore, by Proposition 2.13 (ii)
as .
Theorem 2.15. is a -Banach space under
the Luxemburg -norm.
Proof. Let be -Cauchy sequence
in . Given any . Hence there
exists such that
for all . By Proposition.2.11.(i), we have
for all . Hence, we can write
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Because of
for
all , we find
This implies
for all and for all . Since be a
-Cauchy sequence in , there exists
such that
for all . Let , we
shall show that . Taken as
. For all and
, we find
by using (10). Since
for all
, we find
as . Hence, we get
for all and . This implies
for all . By Proposition.2.12.(i), we obtain
for all . This means that
where
. Also, we see that
, because of
Therefore, is -Banach space.
Definition 2.16. Let be a -Banach space. A point
is referred to as
-nonsquare point if for every the
condition
holds true.
Definition 2.17. A -Banach space is called -
nonsquare, if all element in is -nonsquare
point.
Proposition 2.18. Let be a bounded
sequence of positive real numbers with for
all . Then is a -nonsquare
point .
Proof. From Proposition 2.11 (iii), it seen that if
, then . Now let and
suppose that is not -nonsquare point. Then
there exits such that
. Hence, we can write
by using Proposition 2.11 (iii).
Since for all
is obtained due to strict -convexity. This
constitutes a contradiction.
Theorem 2.19. Let be a bounded
sequence of positive real numbers with for
all . Then is -nonsquare.
Proof. It follows from Proposition 2.11 (iii) and
Proposition 2.18.
3 Conclusion
The concepts of modular, modular spaces, and
Luxemburg norm are given from a new perspective
using geometric arithmetic. We define the
generalized geometric Cesàro sequence space and
construct a -modular on this space. Luxemburg -
norm, produced by the -modular, is built into the
generalized geometric Cesàro sequence space. The
relationships between -modular and Luxemburg -
norm are investigated. Also, we provide evidence that
the generalized geometric Cesàro sequence space is,
in fact, a -Banach space under the Luxemburg -
norm. Moreover, one gets that the generalized
geometric Cesàro sequence space is -nonsquare
when for all . This sets the way for our
future work, which will look into the dual spaces of
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the geometric Cesàro sequence space and establish
the relevant matrix transformations. Since the theory
of sequence space and geometric calculus is quite
active and has extensive applications, we believe
many researchers will use our newly acquired results
for future works and applications in related fields.
References:
[1] D. Aniszewska and M. Rybaczuk, Analysis of
the multiplicative Lorenz system, Chaos
Solitons Fractals, 25, 2005, 79–90.
[2] K. Boruah and B. Hazarika, On some
generalized geometric difference sequence
spaces, Proyecciones Journal of Mathematics,
36(3), 2017, 373-395.
[3] K. Boruah and B. Hazarika, Topology on
geometric sequence spaces, Springer Nature
Singapore Pte Ltd., S. A. Mohiuddine et al.
(eds.), Approximation theory, sequence spaces
and applications, Industrial and Applied
Mathematics, 2022.
https://doi.org/10.1007/978-981-19-6116-8
[4] A. F. Çakmak and F. Başar, Some new results
on sequence spaces with respect to non-
Newtonian calculus, Journal of Inequalities
and Applications, 228, 2012, 1-12.
[5] N. Değirmen, Non-Newtonian approach
to algebras, Communications in Algebra,
50(6), 2022, 2653-2671.
[6] C. Duyar and M. Erdoğan, On non-Newtonian
real number series, IOSR Journal of
Mathematics, 12(6), 2016, 34-48.
[7] D. Filip and C. Piatecki, A non-Newtonian
examination of the theory of exogenous growth,
Mathematica Aeterna, 4, 2014, 101-117.
[8] L. Florak and H.V. Assen, Multiplicative
calculus in biomedical image analysis, J. Math.
Imaging Vis., 42(64), 2012, 64–75.
[9] N. Güngör, A note on linear non-Newtonian
Volterra integral equations, Mathematical
Sciences, 16, 2022, 373-387.
[10] N. Güngör, Some geometric properties of the
non-Newtonian sequence spaces , Math.
Slovaca, 70(3), 2020, 689-696.
[11] M. Grossman and R. Katz, Non-Newtonian
calculus, Pigeon Cove Massachusetts: Lee
Press, 1972
[12] U. Kadak, On multiplicative difference
sequence spaces and related dual properties,
Bol. Soc. Paran. Mat., 35(3), 2017, 181-193.
[13] U. Kadak, Cesàro summable sequence spaces
over the non-Newtonian complex field,
Journal of Probability and Statistics, Volume
2016, Article ID 5862107, 10 pages
http://dx.doi.org/10.1155/2016/5862107
[14] M. Kirişci, Topological structures of non-
Newtonian metric spaces, Electronic Journal
of Mathematical Analysis and Applications,
5(2), 2017, 156-169.
[15] G.M. Leibowitz, A note on the Cesàro
sequence spaces, Tamkang J. Math., 2, 1971,
151-157.
[16] S.K. Mahto, A. Manna and P. D. Srıvastava,
Bigeometric Cesàro difference sequence
spaces and Hermite interpolation, Asian-
European Journal of Mathematics, 13(4),
2020.
https://doi.org/10.1142/S1793557120500849
[17] O. Ogur and Z. Gunes, A Note on non-
Newtonian Isometry, WSEAS Transactions on
Mathematics, 23, 2024, 80-86.
[18] O. Ogur and Z. Gunes, Vitali Theorems in
Non-Newtonian Sense and Non-Newtonian
Measurable Functions, WSEAS Transactions
on Mathematics, 23, 2024, 627-632.
[19] K. Raj and C. Sharma, Applications of non-
Newtonian calculus for classical spaces and
Orlicz functions, Afrika Mathematica, 30,
2019, 297-309.
[20] M. Rybaczuk and P. Stoppel, The fractal
growth of fatigue defects in materials, Int. J.
Fract., 103, 2000, 71–94.
[21] W. Sanhan and S. Suantai, Some geometric
properties of Cesaro sequence spaces,
Kyungpook Math. J, 43, 2003, 191-197.
[22] B. Sağır and İ. Eyüpoğlu, Some geometric
properties of Lebesgue sequence spaces
according to geometric calculation style,
GUFBD / GUJS, 12(2), 2022, 395-403.
[23] B. Sağır and F. Erdoğan, On the function
sequences and series in the non-Newtonian
calculus, Journal of Science and Arts, 4(49),
2019, 915-936.
[24] J. S. Shiue, On the Cesàro sequence spaces,
Tamkang Journal of Mathematics, 1, 1970, 19-
25.
[25] S. Suantai, On the H-property of some Banach
sequence spaces, Archivum Mathematicum,
39(4), 2003, 309-316.
[26] D.F.M. Torres, On a non-Newtonian calculus
of variations, Axioms, 10, 171, 2021, 1-15.
https://doi.org/10.3390/axioms10030171
[27] C. Türkmen and F. Başar, Some basic results
on the sets of sequences with geometric
calculus, Commun. Fac. Sci.Univ. Ank. Series
A1, 61(2), 2012, 17-34.
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The authors equally contributed in the present
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