On Some New Uncertain Spaces using Generalized Difference Operator
DOWLATH FATHIMA
Basic Science Department,
College of Science and Theoretical Studies,
Saudi Electronic University,
Riyadh,
KINGDOM OF SAUDI ARABIA
Abstract: - It was C. You, who has given the concept of uncertainty theory in 2009. The scenario of
this paper is to define a new notion of spaces using generalized ∆- operator and the uncertainty
theory. Also, certain basic structures will be given. Moreover, inclusion relations and their counter
examples will be given.
Key-Words: - ∆-operator; uncertain variable; convergence
Received: January 15, 2023. Revised: July 16, 2023. Accepted: August 11, 2023. Published: September 13, 2023.
1 Introduction
Uncertainty theory was introduced in [1] and many
researchers have shown wings in this area of study.
The random event is given by probability measure. It
was Zadeh, [2], who introduced possibility Fuzzy
measure. In, [3], the authors have introduced a self-
dual measure and the measure of credibility and in,
[4], the axiomatic structure for credibility theory
have been determined and can be seen in, [5]. In
order to analyze the concept that fuzziness and
randomness simultaneously in system, a fuzzy
random variable have been given in, [6]. In, [7], the
results of convergence were given. Further, in, [8],
the notion of chance measure was established. The
various applications of these structures have been
given in, [9], [10].
For a non-void set ℧, and a σ-algebra L over ℧,
we call each entry v L as an event. The uncertain
measure has been well structured as a set function S
which satisfies the following axioms:
a. M{℧} = 1 (normality).
b. M{v1} ≤ M{v2} whenever v1 ≤ v2 (monotonicity).
c. for any v1 (self-duality).
d. For every countable sequence of events {vr} gives
(Countable subadditivity).
Using, [11], we have following definitions:
For uncertain variable ζ, the map ψ(ϱ) is said to
be uncertainty distribution if
An uncertain sequence {ϱr} is said to converge
almost surely (a.s.) to ϱ if we can find an event ν
having M(ν) = 1 in such a way that
for all w ν.
An uncertain sequence {ζ} is called as convergent in
measure to ζ if
A sequence {ζj} is said to be convergence in
average to ζ if
For uncertain variable ϑ,ϑ1,ϑ2,··· having
respectively the expected values as ζ,ζ1,ζ2,··· .
Then, {ζj} is said to be convergence in
distribution to ζ if corresponding to any
continuous point ϑ, we have ϑn → ϑ.
For a sequence ϑ = (ηr) of whole numbers
having η0 = 0, 0 < ηr < ηr+1 and hr = ηr −ηr−1 → ∞
for r → ∞, we call ϑ to be as lacunary sequence
as can be seen in, [12], [13], [14], [15], [16],
and many others.
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Space as normal (or solid) if (ζj) yields
(βjvj) for each scalar sequence (βj) having
|βj| ≤ 1 j as in, [17], [18], [19], and many
others.
Lemma 1.1. Every solid sequence space is
monotone.
In, [20], we have following for T {ℓ∞, c, C0}:
T () = {v = (vi) Λ : (vi) T }, where and
vi = vi − vi−1.
Also, in, [21], we have following for integer s ≥
0 :
∆s(T ) = {v = (vk) : (∆sv) T },
with ∆svr = ∆s−1vr − ∆s−1vr+1 for every r .
Further, consider sequence g = (gj) of non-
zero complex numbers, then as in, [22], we have
,
where
And
is complete having norm
The various well-structured properties
concerning this space can be found in, [17],
[23], [24], [25], [26], [27], [28], and many
others.
This space was further studied and according
to, [9], we have following in hand for fixed
integers s, t ≥ 0:
where
Clearly, by taking s = 1 = t, we have results
obtained in, [20], and choosing t = 1, we get
results of, [21].
Following these, we involve the ∆-definition
with uncertain variables by joining of lacunary
sequences and will determine some new results.
2 Main Results
This section deals with the introduction of new
sequences of uncertain variables using ∆-
operator.
Following the cited references as, [2], [9], [29],
[30], [31], [32], [33], [34], [35], we define the
following new spaces:
and
where σ(v) (℧, L, S).
Theorem 2.1.
The sets and
are linear.
Proof: We prove the result for
only and rest will follow on similar steps.
Therefore, suppose that
, then,
and
Now for any , we have
Consequently,
and the result follows.
Theorem 2.2. The sets
and of complex certain sequences are
normed linear spaces with norm
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Proof: It can be proved by using classical
techniques.
Theorem 2.3. If = 0, c, ∞, t ≥ 1 and s ≥ 1, then
The inclusions are proper.
Proof: Only the case of will be proved
and rest will follow on similar lines. So, let
, then we see
(1)
We can write
Using (1) and approaching j → ∞ in above inequality
yields
,
Consequently, to establish the result is proper, we
choose a lacunary sequence θ = (2i) and take the
sequence of uncertain variables as (ζi) = (is−1) with gi
= 1 for each i . Consequently, for all i , we
have
Hence, but not
as desired.
Theorem 2.4. If T = 0, c, ∞, then, in general, the sets
is not symmetric.
Proof: The result will be proved for
only and others will be
proved similarly by taking s = 2 = t and for
each i and by considering θ = (2i) as lacunary
sequence.
Setting the uncertainty space (℧ , L, M) as {τ1, τ2,
···} with the power set and taking any event ν L
such that
Now consider,
Clearly, {ζr} for r Im and is in
. Consider the rearrangement of {ζr} as
{ωm} given by ωm(τ) = {ζ1,ζ4,ζ9,ζ2,ζ10,···} and is
consequently not in . From which, we
conclude that is not symmetric.
On similar lines, the following theorem is obvious.
Theorem 2.5. The sets is not
monotone in general for
3 Structure of Lacunary Convergence
using Operator
This portion of the manuscript deals with the
lacunary convergence structure of ∆-operator for
uncertain variables. Also, we will establish certain
new relation corresponding to them.
Definition 3.1. The uncertain sequence {ζj} is called
as lacunary strongly convergent almost surely to ζ
with respect to difference sequence if for ε > 0, we
can find an event ν with M(ν) = 1 such that
for all η ν.
Definition 3.2. An uncertain sequence {ζj} is
called as lacunary strongly convergent in measure to
ζ if
for all ε > 0.
Definition 3.3. Let ζ, ζ1, ζ2,··· be the uncertainty
distributions of uncertain variables. Then the
sequence {ζj} is called convergence in mean to ζ if
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Definition 3.4. Let ϑ1, ϑ2, ϑ3,··· be uncertain
variables with finite expected values ζ, ζ1, ζ2,···,
respectively. Then the sequence {ζj} is said to be
lacunary strong convergent in distribution to to ζ
w.r.t. difference sequence if
,
for every complex λ in which ϑ(λ) is continuous.
Definition 3.5. The uncertain sequence {ϑi} is said
to be convergent uniformly almost surely to ζ if we
can find a sequence of events {Ej },
M{Ej } → 0 such that {ϑi} converges uniformly to ζ
in ℧− ε j for some fixed j .
Theorem 3.6. Consider an uncertain sequence {ζi}.
If it is lacunary strongly convergent in average to ζ
w.r.t. difference sequence, then it converges
lacunary strongly in measure to ζ, but not
conversely.
Proof: By Markov’s inequality, we have for ε
> 0 that
as i Ij. Thus {ζi} is converges in measure to σ w.r.t.
difference sequence.
Now for converse, define uncertainty space (℧, L,
M) as {τ1, τ2,···} and consider the event ν L such
that
Also, set the uncertain variables as
for every m Ij and σ ≡ 0. Now for ε > 0, we
see
This shows that {ζi} converges in measure to ζ.
But, for all i Ij, we see the uncertainty distribution
of uncertain variable ζi − ζ = ζi is
if ϱ < 0,
if 0 ≤ ϱ < m,
if elsewhere.
Consequently, {ζi(ϱ) does not converge in mean to
ζ(ϱ) w.r.t. difference sequence.
4 Conclusion
In this work, we have discussed the uncertainty of
variables and have defined new type of spaces
using
generalized ∆-operator. We have computed their
linear structure and some of their topological
structures. The inclusions relations corresponding to
these spaces have been given. To support the
properness, some example have been given.
Symmetry and monotonicity corresponding to these
spaces are analyzed. Further, we have constructed
some basic ideas of Lacunary convergence using
uncertain variables and ∆-operator. In the future, we
will continue to study the above proposed structures
by applying statistical convergence and the modulus
function to obtain the new results.
Acknowledgement:
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We are thankful to the reviewers for the careful
reading and suggestions which improved the
presentation of the paper.
References:
[1] C. You, On the convergence of uncertain
sequences, Mathematical and computer
modelling, 49(3-4), 2009, p.482-487.
[2] L. A. Zadeh, Fuzzy sets as a basis for a theory
of possibility, Fuzzy Sets and Systems, 1
1978, p.3-28.
[3] B. Liu, A survey of credibility theory, Fuzzy
Optimization and Decision Making, 5(4),
2006, p.387-408.
[4] M. AlBaidani, Notion of new structure of
uncertain sequences using Δ-Spaces, Journal
of Mathematics, vol. 2022, Article ID
2615772,
https://doi.org/10.1155/2022/2615772.
[5] B. Liu, Uncertainty Theory, 2nd ed., Springer-
Verlag, Berlin, 2007.
[6] H. Kizmaz, On certain sequence spaces.
Canad Math Bull., 24(1), 1981, p.169-176.
[7] P. J. Dowari and B. C. Tripathy, Lacunary
convergence of sequences of complex
unertain variables, Bol. Soc. Paran. Mat.,
3s.(41), (2023, p.1-10.
[8] H. Kwakernaak, Fuzzy random variables - I:
Definition and theorem, Information Sciences,
15, 1978, p.1-29.
[9] C. Barton, C. Liu and S. P. Pissis, On-Line
Pattern Matching on Uncertain Sequences and
Applications, Combinatorial Optimization and
Applications, 2016, p.547-562.
[10] A. H. Ganie and N. A. Sheikh, Some new
difference sequence spaces defined on Orlicz
function, Nonlinear Funct. Anal. Appl., 17(3)
2012, p.397-404.
[11] X. Li and B. Liu, Chance measure for hybrid
events with fuzziness and randomness, Soft
Computing, 13, Article number: 105 (2009).
[12] A. H. Ganie, Sequences of Ces`aro type using
lacunary notion, Int. J. Nonlinear Anal. Appl.,
13, 2022, p.1399–1405.
[13] A. H. Ganie, Lacunary sequences with almost
and statistical convergence, Annals Commu.
Maths., 3(1), 2020, p.46-53.
[14] A. H. Ganie, New notion of sequence spaces
of modulus function, Palestine J. Maths.,
11(4), 2022, p.331-335.
[15] A. H. Ganie and N. A. Sheikh, Generalized
difference sequence spaces of Fuzzy numbers,
New York J. Math., 19, 2013, p.431-438.
[16] B. Liu, Y. Liu, Expected value of fuzzy
variable and fuzzy expected value models,
IEEE Transactions on Fuzzy Systems, 10(4) ,
2002, p.445-450.
[17] B. Liu, Uncertainty Theory: An Introduction
to its Axiomatic Foundations, Springer-
Verlag, Berlin, 2004.
[18] N. A. Sheikh and A. H. Ganie, A new type of
sequence space of non-absolute type and
matrix transformation, WSEAS Trans. Math.,
8(12), 2013, p.852-859.
[19] A. G¨okhan, M. Gu¨ng¨or, and Y. Bulut, On
the strong lacunary convergence and strong
ces´aro summability of sequences of real-
valued functions, Applied Sciences, 8, 2006,
p.70-77.
[20] M. Et and R. C¸olak, On some generalized
difference sequence spaces, Soochow J.
Math., 21, 1995, p.377-386.
[21] M. Et and A. Esi, On K¨othe Toeplitz duals of
generalized difference sequence spaces, Bull.
Malaysian Math. Sc. Soc., (2)23, 2000, p.25-
32.
[22] M. Et, Generalized ces`aro difference
sequence spaces of non-absolute type
involving lacunary sequences, Appl Math
Comput., 219(17), 2013, p. 9372-9376.
[23] A. H. Ganie, New spaces over modulus
function, Boletim Sociedade Paranaense de
Matematica, (3s.)(41), 2023, p.1-6.
[24] A. H. Ganie, New approach for structural
behavior of variables, J. Nonlinear Sci, Appl.,
14(5), 2021, p.351-358
[25] A. H. Ganie and S. A. Gupkari, Generalized
difference sequence space of fuzzy numbers,
Annals of Fuzzy Math. and Informatics,
21(2), 2021, p.161-168.
[26] A. H. Ganie and S. A. Lone, Some sequence
spaces of sigma means defined by Orlicz
function, Appl. Math. Inf. Sci., 6(14), 2020,
p.1-4.
[27] A. H. Ganie, Mobin Ahmad and N. A. Sheikh,
New generalized difference sequence spaces
of fuzzy numbers, Ann. Fuzzy Math. Inform.,
12(2), 2016, 255263.
[28] A. Esi, B. C. Tripathy and B. Sarma, On some
type of generalized difference sequence
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spaces, Mathematica Slovaca, 57(5), 2007,
p.475-482.
[29] B. Liu, Why is there a need for uncertainty
theory?, Journal uncertain system, 6(1), 2012,
p.3-10.
[30] D. Fathima and A. H. Ganie, On some new
scenario of ∆-spaces, J. Nonlinear Sci. Appl.,
14, 2021, p.163-167.
[31] A. R. Freedman, J.J. Sember and M. Raphael,
Some Ces`aro-type summability spaces, Proc.
London Math. Soc., 37(3), 1978, p.508-520.
[32] X. Gao, Y. Gao, Connectedness index of
uncertain graphs, Int. J. Uncer. Fuzzy. Knowl.
Based Syst., 21(1), 2013, p.127-137.
[33] M. Sugeno, Theory of fuzzy integrals and its
applications, Ph.D. Dissertation, Tokyo
Institute of Technology, 1974.
[34] B. C. Tripathy and A. Esi, A new type Of
difference sequence spaces, International J.
Sci. Tech., 1(1), 2006, p.11-14.
[35] M. Mursaleen, A. H. Ganie, N. A. Sheikh,
New type of generalized difference sequence
space of non-absolute type and some matrix
transformations, Filomat, 28, 2014, p.1381–
1392.
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