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Abstract
In this paper we introduce the fuzzy numbers defined by an Orlicz function and study some of their properties and inclusion results.
Public Interest Statement
In this paper, we introduced the fuzzy numbers defined by an Orlicz function and study some of their properties with inclusion results. Furthermore we provided an example of triple sequence of gai which is not symmetric, not solid, not monotone and not convergent free.
Our result unifies the results of several author’s in the case of classical Orlicz spaces. One can extend our results for more general spaces.
1. Introduction
A triple sequence (real or complex) can be defined as a function where
and
denote the set of natural numbers, real numbers and complex numbers, respectively.
Some initial work on double series is found in Apostol (Citation1978), Alzer, Karayannakis, and Srivastava (Citation2006), Bor, Srivastava, and Sulaiman (Citation2012), Choi and Srivastava (Citation1991), Liu and Srivastava (Citation2006 and double sequence spaces are found in Hardy (Citation1917), Deepmala Subramanian, and Mishra (Citationin press), Deepmala, Mishra, and Subramanian (Citation2016) and many others. Later on some initial work on triple sequence spaces is found in Sahiner, Gurdal, and Duden (Citation2007), Esi (Citation2014), Esi and Necdet Catalbas (Citation2014),Esi and Savas (Citation2015), Subramanian and Esi (Citation2015) and many others.
A sequence is said to be triple analytic if
The vector space of all triple analytic sequences are usually denoted by
.
A sequence is called triple entire sequence if
as
A sequence is called triple chi sequence if
as
The triple gai sequences will be denoted by
.
This paper deals with introducing the -fuzzy number defined by an Orlicz function and study some topological properties, inclusion relations and give some examples. Some interesting results may be seen in Alzer et al. (Citation2006), Bor et al. (Citation2012), Choi and Srivastava (Citation1991), Liu and Srivastava (Citation2006).
2. Definitions and preliminaries
Definition 2.1
An Orlicz function (see Kamthan & Gupta, Citation1981) is a function which is continuous, non-decreasing and convex with
for
and
as
If convexity of Orlicz function M is replaced by
then this function is called modulus function. Lindenstrauss and Tzafriri (Citation1971) used the idea of Orlicz function to construct Orlicz sequence space.
Throughout a triple sequence is denoted by a triple infinite array of fuzzy real numbers.
Let D denote the set of all closed and bounded intervals on the real line
For
and
define
It is known that is a complete metric space.
A fuzzy real number X is a fuzzy set on that is, a mapping
associating each real number t with its grade of membership
The -level set
of the fuzzy real number X, for
is defined by
The 0-level set is th closure of the strong 0-cut that is,
A fuzzy real number X is called convex if where
If there exists
such that
then, the fuzzy real number X is called normal.
A fuzzy real number X is said to be upper-semi continuous if, for each is open in the usual topology of
for all
The set of all upper-semi continuous, normal, convex fuzzy real numbers is denoted by
The absolute value, of
is defined by
Let be defined by
Then, defines a metric on
and it is well-known that
is a complete metric space.
A sequence is said to be null if
A triple sequence of fuzzy real numbers is said to be gai in Pringsheim’s sense to a fuzzy number 0 if
A triple sequence is said to
regularly if it converges in the Prinsheim’s sense and the following limts zero:
A fuzzy real-valued double sequence space is said to be solid if
whenever
and
for all
Let and
be a triple sequence space. A K-step space of
is a sequence space
A canonical pre-image of a sequence is a sequence
defined as follows:
A canonical pre-image of a step space is a set of canonical pre-images of all elements in
A sequence set is said to be monotone if
contains the canonical pre-images of all its step spaces.
A sequence set is said to be symmetric if
whenever
where
is a permutation of
A fuzzy real-valued sequence set is said to be convergent free if
whenever
and
implies
We define the following classes of sequences:
Also, we define the classes of sequences as follows :
A sequence if
and the following limits hold
3. Main results
Theorem 3.1
Let
and
(i) |
| ||||
(ii) |
|
Proof
(i) | Necessity: Let |
Proof
(ii) | Let |
Proposition 3.2
The class of sequences is symmetric but the classes of sequences
and
are not symmetric.
Proof
Obviously the class of sequences is symmetric. For the other classes of sequences, consider the following example
Example
Consider the class of sequences Let
and consider the sequence
be defined by
and for
Let be a rearrangement of
defined by
and for
Then, but
Hence,
is not symmetric. Similarly other sequences are also not symmetric.
Proposition 3.3
The classes of sequences and
are solid.
Proof
Consider the class of sequences . Let
and
be such that
As f is non-decreasing, we have
Hence, the class of sequence is solid. Simlarly it can be shown that the other classes of sequences are also solid.
Proposition 3.4
The classes of sequences and
are not monotone and hence not solid.
Proof
The result follows from the following example.
Example
Consider the class of sequences and
. Let
Let
be defined by
for all
Then Let
be the canonical pre-image of
for the subsequence J of
Then
Then, Hence
is not monotone. Similarly, it can be shown that the other classes of sequences are also not monotone. Hence, the classes of sequences
and
are not solid.
Proposition 3.5
(i) (ii)
Proof
It is easy, so omitted.
Proposition 3.6
Let and
be three Orlicz functions, then, (i)
, (ii)
, (iii)
Proof
We prove the result for the case the other cases are similar. Let
be given. As f is continuous and non-decreasing, so there exists
such that
Let
Then, there exist
such that
Hence, Thus,
Proposition 3.7
(i) , (ii)
the inclusions are strict.
Proof
The inclusion (i) (ii)
is obvious. For establishing that the inclusions are proper, consider the following example.
Example
We prove the result for the case the other case similar. Let
Let the sequence
be defined by for
and for
Then, but
Proposition 3.8
The classes of sequences and
are not convergent free.
Proof
The result follows from the following example.
Example
Consider the classes of sequences Let
and consider the sequence
defined by
and for other values,
Let the sequence be defined by
and for other values,
Then, but
Hence, the classes of sequences
are not convergent free. Similarly, the other spaces are also not convergent free.
4. Conclusion
The fuzzy numbers defined by an Orlicz function and discuss inclusion relation. Furthermore, the given example of triple sequence of gai is not symmetric, not solid, not monotone and not convergent free.
Acknowledgements
The authors are extremely grateful to the anonymous learned referee(s) for their keen reading, valuable suggestion and constructive comments for the improvement of the manuscript. The authors are thankful to the editor(s) and reviewers of Cogent Mathematics.The third author NS wish to thank the Department of Science and Technology, Government of India for the financial sanction towards this work under FIST program SR/FST/MSI-107/2015.The research of the second author Deepmala is supported by the Science and Engineering Research Board (SERB), Department of Science and Technology (DST), Government of India under SERB National Post-Doctoral fellowship scheme File Number: PDF/2015/000799.
Additional information
Funding
Notes on contributors
Vandana
Vandana is a research scholar at School of Studies in Mathematics, Pt. Ravishankar Shukla University, Raipur-492010, (C.G.) India. Her research interests are in the areas of applied mathematics including Optimization, Mathematical Programming, Inventory control, Supply Chain Management, Operation Research, etc. She is member of several scientific committees, advisory boards as well as member of editorial board of a number of scientific journals.
Deepmala
Deepmala is Visiting Scientist at SQC & OR Unit at Indian Statistical Institute, Kolkata, India. Her research interests are in the areas of Optimization, Mathematical Programming, Fixed Point Theory and Applications, Operator theory, Approximation Theory etc. She is member of several scientific committees and also member of editorial board of a number of scientific journals.
N. Subramanian
N Subramanian received PhD degree in Mathematics from Alagappa University at Karaikudi,Tamil Nadu,India and also getting Doctor of Science (D.Sc) degree in Mathematics from Berhampur University, Berhampur, India. His research interests are in the areas of summability through functional analysis of applied mathematics and pure mathematics.
References
- Alzer, H., Karayannakis, D., & Srivastava, H. M. (2006). Series representations for some mathematical constants. Journal of Mathematical Analysis and Applications, 320, 145–162.
- Apostol, T. (1978). Mathematical analysis. London: Addison-Wesley.
- Bor, H., Srivastava, H. M., & Sulaiman, W. T. (2012). A new application of certain generalized power increasing sequences. Filomat, 26, 871–879. doi:10.2298/FIL1204871B
- Choi, J., & Srivastava, H. M. (1991). Certain classes of series involving the Zeta function. Journal of Mathematical Analysis and Applications, 231, 91–117.
- Deepmala, Mishra, L. N., & Subramanian, N. (2016). Characterization of some Lacunary X2AUV- Convergence of order α with p-metric defined by sequence of moduli Musielak. Applied Mathematics & Information Sciences Letters, 4(3).
- Deepmala, Subramanian, N., & Mishra, V.N. (in press). Double almost (λmμn in X2-Riesz space. Southeast Asian Bulletin of Mathematics.
- Esi, A. (2014). On some triple almost lacunary sequence spaces defined by Orlicz functions. Research and Reviews: Discrete Mathematical Structures, 1, 16–25.
- Esi, A., & Necdet Catalbas, M. (2014). Almost convergence of triple sequences. Global Journal of Mathematical, Analysis, 2, 6–10.
- Esi, A., & Savas, E. (2015). On lacunary statistically convergent triple sequences in probabilistic normed space. Applied Mathematics & Information Sciences, 9, 2529–2534.
- Hardy, G. H. (1917). On the convergence of certain multiple series. Proceedings of the Cambridge Philosophical Society, 19, 86–95.
- Kamthan, P. K., & Gupta, M. (1981). Sequence spaces and series, Lecture notes, Pure and Applied Mathematics. New York, NY: 65 Marcel Dekker Inc.
- Lindenstrauss, J., & Tzafriri, L. (1971). On Orlicz sequence spaces. Israel Journal of Mathematics, 10, 379–390.
- Liu, G. D., & Srivastava, H. M. (2006). Explicit formulas for the Nörlund polynomials B(x)n and b(x)n. Computers and Mathematics with Applications, 51, 1377–1384.
- Sahiner, A., Gurdal, M., & Duden, F. K. (2007). Triple sequences and their statistical convergence. Selcuk Journal of Applied Mathematics, 8, 49–55.
- Subramanian, N., & Esi, A. (2015). Some new semi-normed triple sequence spaces defined by a sequence of moduli. Journal of Analysis & Number Theory, 3, 79–88.