EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 5, 2020, 1212-1230
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Special Issue Dedicated to
Professor Hari M. Srivastava

On the Occasion of his 80th Birthday

A Certain Class of Relatively Equi-Statistical Fuzzy
Approximation Theorems

Susanta Kumar Paikray1, Priyadarsini Parida2,∗, S. A. Mohiuddine3,4

1 Department of Mathematics, Veer Surendra Sai University of Technology, Burla 768018,
Odisha, India
2 Department of Mathematics, Kuntala Kumari Sabat Women’s College, Balasore 756003,
Odisha, India
3 Department of General Required Courses, Mathematics, Faculty of Applied Studies, King
Abdulaziz University, Jeddah 21589, Saudi Arabia
4 Operator Theory and Applications Research Group, Department of Mathematics, King
Abdulaziz University, Jeddah 21589, Saudi Arabia

Abstract. The aim of this paper is to introduce the notions of relatively deferred Nörlund uniform
statistical convergence as well as relatively deferred Nörlund point-wise statistical convergence
through the difference operator of fractional order of fuzzy-number-valued sequence of functions,
and a type of convergence which lies between aforesaid notions, namely, relatively deferred Nörlund
equi-statistical convergence. Also, we investigate the inclusion relations among these aforesaid
notions. As an application point of view, we establish a fuzzy approximation (Korovkin-type)
theorem by using our new notion of relatively deferred Nörlund equi-statistical convergence and
intimate that this result is a non-trivial generalization of several well-established fuzzy Korovkin-
type theorems which were presented in earlier works. Moreover, we estimate the fuzzy rate of
the relatively deferred Nörlund equi-statistical convergence involving a non-zero scale function by
using the fuzzy modulus of continuity.

2020 Mathematics Subject Classifications: 40A05, 40G15, 41A36, 46S40, 47S40

Key Words and Phrases: Deferred Nörlund mean, Relatively statistical uniform convergence,
Relatively equi-statistical convergence, Fuzzy positive linear operator, Fuzzy Korovkin-type theo-
rem, Fuzzy rate of relatively equi-statistical convergence

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i5.3711

Email addresses: skpaikray math@vssut.ac.in (S. K. Paikray),
priyadarsiniparida1@gmail.com (P. Parida), mohiuddine@gmail.com (S. A. Mohiuddine)

https://www.ejpam.com 1212 c© 2020 EJPAM All rights reserved.



S. K. Paikray, P. Parida, S. A. Mohiuddine / Eur. J. Pure Appl. Math, 13 (5) (2020), 1212-1230 1213

1. Introduction and Preliminaries

Moore [31] was the first who presented the notion of of uniform convergence of sequence
of functions associated with a scale function, and later Chittenden [12] studied this concept.
Recalling this concept, a sequence of functions (fn) defined over [a, b] converges relatively
uniformly to a limit function f(x), if there exists a scale function σ(x) ( 6= 0) defined over
[a, b] and for every ε > 0, there exists a positive integer nε such that∣∣∣∣fn(x)− f(x)

σ(x)

∣∣∣∣ 5 ε (∀ n > nε)

holds true (uniformly) for all x ∈ [a, b] ⊆ R. The importance of relatively uniform conver-
gence over the usual uniform convergence is discussed in the following example.

Example 1. For all n ∈ N, consider fn : [0, 1]→ R defined by

fn(x) =

{ nx
1+n2x2

(0 < x 5 1)

0 (x = 0).

The sequence (fn) of functions is not classically uniformly convergent on [0, 1]; but con-
vergent uniformly to f = 0 relative to a scale function

σ(x) =

{
1
x (0 < x 5 1)
1 (x = 0)

on [0, 1]. Here, we write fn ⇒ f = 0 ([0, 1];σ) .

Thus, uniform convergence can be viewed as a special case of relative uniform conver-
gence (or, convergent uniformly relative to a non-zero scale function). Recently, Demirci
and Orhan [16] defined the notion of relatively uniform statistical convergence of a se-
quence of functions which is based on the natural density of a set as follows:

Let E ⊂ R be compact and (fn) be a sequence of functions defined on E. The sequence
(fn) is relatively uniform statistical convergent to the limit function f defined on E, if
there exists a non-zero scale function σ(x) (σ(x) > 0) over E provided, for each ε > 0,{

k : k 5 n and sup
x∈E

∣∣∣∣fk(x)− f(x)

σ(x)

∣∣∣∣ = ε}
has natural (asymptotic) density zero; equivalently, one writes

lim
n→∞

∣∣∣∣{k : k 5 n and sup
x∈E

∣∣∣∣fk(x)− f(x)

σ(x)

∣∣∣∣ = ε}∣∣∣∣ = 0.

Here, we write stat fn(x)⇒ f(E;σ).



S. K. Paikray, P. Parida, S. A. Mohiuddine / Eur. J. Pure Appl. Math, 13 (5) (2020), 1212-1230 1214

The rapid growth of sequence spaces have been accompanied by the recent works of
many researchers in the field of statistical convergence and it has got tremendous impor-
tance over conventional convergence. The basic concept of statistical convergence was
initially studied in the year 1951 by Fast [17] and Steinhaus [41] independently. Recently,
the approximation of functions by linear operators (positive) based on statistical conver-
gence has become an energetic area of research. In the current years, the use of statistical
convergence in approximation theory has enabled the researchers to achieve more powerful
outcomes than that of the classical aspects of convergence. In this context, we refer to
the recent works [9], [10], [15], [14], [13], [20], [21], [24], [22], [23], [25], [33], [34], [35], [36],
[37], [38] and [40].

We use the symbol ω to denote the space of all real valued sequences. Also, the usual
notations `∞, c and c0 will be used to denote the classes of bounded linear spaces, con-
vergent sequences and null sequences respectively. Moreover, it is also known that any
subspace of ω is a sequence space. Note that all these spaces are Banach spaces under the
sup-norm (xk)k∈N, which a is sequence of real or complex terms, given by ‖x‖∞ = supk |xk|.

In the year 1981, Kızmaz [27] gave the preliminary notion of space of difference sequence
and subsequently, the difference sequence of order r (r ∈ N0 := N ∪ {0}) is defined as
follows:

λ(∆r) = {x = (xk) : ∆r(x) ∈ λ, λ ∈ (`∞, c0, c)} ;

∆0x = (xk); ∆rx = (∆r−1xk −∆r−1xk+1)

and

∆rxk =
r∑
i=0

(−1)i
(
r

i

)
xk+i).

The norm for difference sequence of order r is given by

‖x‖∆r =
r∑
i=1

|xi|+ supk|∆rxk|.

Note that the difference spaces obtained from λ(∆r) are Banach spaces under the above
norm.

The difference operator of fractional-order was initially used by Chapman [11] and sub-
sequently, many researchers used it with different settings (see [4], [8]). Recently, Baliars-
ingh [6] has studied a fractional-order difference sequence involving the Euler-Gamma
function as follows:

∆α(xk) =

∞∑
i=0

(−1)i
Γ(α+ 1)

i!Γ(α− i+ 1)
xk−i



S. K. Paikray, P. Parida, S. A. Mohiuddine / Eur. J. Pure Appl. Math, 13 (5) (2020), 1212-1230 1215

for k ∈ N. Moreover, in [7], Baliarsingh has introduced certain new fractional-order
difference sequence spaces. Assume that α, β and γ are real numbers and also assume
that h is a positive constant. For any (xk) ∈ ω, the generalized difference sequence with

a view of fractional-order difference operator ∆α,β,γ
h : ω → ω is defined by

(∆α,β,γ
h xk) =

∞∑
i=0

(−α)i − (β)i
i!(−γ)ihα+β−γ xk−i (k ∈ N). (1)

We now recall the sequence of fractional-order backward difference operator (see [7]) which
is required for the present investigation. Suppose that f(x) is a fractional order differ-
entiable function and also suppose that h → 0. Consider the sequence {fh(x)} which is

associated to f(x) given by fh(x) = (f(x− ih))i∈N0 . Next, in view of ∆α,β,γ
h,x , the sequence

spaces ∆α,β,γ
h,x (fh(x)) is given by

∆α,β,γ
h,x f(x) =

∞∑
i=0

(−α)i(−β)i
i!(−γ)ihα+β−γ f(x− ih). (2)

In the last equality, (α)k denotes the Pochhammer symbol of a real number α and is given
by the formula

(α)k =


1 (α = 0 or k = 0)

Γ(α+k)
Γ(α) = α(α+ 1)(α+ 2)...(α+ k − 1) (k ∈ N).

Also, without loss of generality, the summation considered in the equality (2) converges
for all γ > α+ β (see [18]).

Moreover, since in all the cases it is not possible to calculate either the simple limits
or the statistical limits with exact precision. So different approaches such as fuzzy logic,
fuzzy set theory, set valued analysis, interval analysis, etc. were introduced by various
researchers to model several mathematical structures. In this context, we recall some fuzzy
approximation (Korovkin-type) theorems recently studied by Anastassiou [2], Anastassiou
and Duman [3], Karaisa and Kadak [26], and Mohiuddine et al. ([29] and [30]). The main
objective of the proposed work is to establish a fuzzy approximation (Korovkin-type)

theorem by using relatively deferred Nörlund equi-statistical convergence based on ∆α,β,γ
h

and further to estimate its statistical fuzzy rates with the help of the fuzzy modulus of
continuity.

2. Some Basic Definitions

Consider a fuzzy number valued functionX : R→ [0, 1] which is upper semi-continuous,
normal and convex. Also, sup(X), the closure of this set is compact, where sup(X) = {x ∈
R : X(x) > 0}. Let RF denotes the set of fuzzy numbers and let [X]0 = {x ∈ R : X(x) > 0}



S. K. Paikray, P. Parida, S. A. Mohiuddine / Eur. J. Pure Appl. Math, 13 (5) (2020), 1212-1230 1216

be the closure of the set. Also, recall that the closed and bounded interval of R (see [19])
denoted by [X]r is given as [X]r = {x ∈ R : X(x) = r} (r ∈ (0, 1]).

Now, we define the fuzzy sum and the fuzzy product as follows: Let u, v ∈ RF and
suppose that λ ∈ R. For r ∈ (0, 1] the fuzzy sum and the fuzzy product respectively
denoted by u ⊕ v and u � v are given by [u ⊕ v]r = [u]r + [v]r and [λ � v]r = λ[u]r.
Next, consider the interval [u]r of the form [ur−, u

r
+], where ur− 5 ur+ and ur−, u

r
+ ∈ R for

0 5 r 5 1. Also, for u, v ∈ RF , define

u � v ⇐⇒ ur− 5 v
r
− and ur+ 5 v

r
+ ∀ r ∈ [0, 1].

Further, the metric D is such that D : RF × RF → R+ be defined as

D(u, v) = sup
05r51

max{|ur− − vr−|, |ur+ − vr+|}.

Note that, (RF ,D) is a metric space which is complete (see [42]).

Consider two fuzzy valued functions f and g such that f, g : [a, b] → R, the distance
between two functions f and g denoted as D∗(f, g) is given by

D∗(f, g) = sup
a5x5b

sup
05r51

max{|f r− − gr−|, |f r+ − gr+|}.

The fuzzy counterpart of the idea statistical convergence for sequence (xn)n∈N of fuzzy
numbers based on D was introduced by Nuray and Savaş [32]. Recall that, (xn)n∈N is
statistically convergent to a fuzzy number `, in symbols, one writes stat D(xn, `) = 0, if
for each ε > 0,

d(Kε) = lim
n→∞

|Kε|
n

= lim
n→∞

|{k : k 5 n and D(xn, `) = ε}|
n

= 0,

where | · | denotes the cardinality of the set.

Next, before presenting the notion of the deferred Nörlund mean under the generalized
differentiable function f(x) with fractional order, we recall the regularity condition (see
Agnew [1]) as follows: Let us assume two non-negative sequences (an) and (bn) of integers
such that it satisfies (i) an < bn and (ii) limn→∞ bn = +∞ (n ∈ N). We remark that these
conditions are also used to define the proposed deferred weighted mean.

We now assume that (sn) and (tn) be the sequences of non-negative real numbers such
that

Sn =

bn∑
m=an+1

sm and Tn =

bn∑
m=an+1

tm.



S. K. Paikray, P. Parida, S. A. Mohiuddine / Eur. J. Pure Appl. Math, 13 (5) (2020), 1212-1230 1217

The convolution of the above sequences can be presented as (see [39]),

Rbnan+1 = (S ∗ T )bn =

bn∑
v=an+1

svtbn−v.

Now, for defining the sequence (fn) of functions via deferred Nörlund mean depends on

∆α,β,γ
h,x , we first set

Λn =
1

Rbnan+1

bn∑
m=an+1

sbn−mtm

(
∆α,β,γ
h,x fm(x)

)
. (3)

We say that (fn) is deferred Nörlond summable to a number ` under the operator

∆α,β,γ
h,x , provided limn→∞ Λn = `.

3. Relatively Deferred Weighted Equi-statistical Convergence with
Associated Inclusion Relations

In 2018, Srivastava et al. [39] used the concept of deferred Nörlund mean to introduce
the ideas of point-wise and uniform statistical convergence as well as equi-statistical con-
vergence while these notions in classical sense were defined and studied by Balcerzak et al.
[5] and in view of weighted lacunary sequence by Mohiuddine and Alamri [28]. Analogous
to these definitions, here we present the definitions of relatively deferred Nörlund point-
wise and uniform statistical convergence and relatively deferred Nörlund equi-statistical
convergence of a FNVS (fuzzy number valued sequence) of functions as follows:

Definition 1. Assume that (an) and (bn) are sequences of integers and E ⊂ R is com-
pact. Also assume that (fm) and f are fuzzy number valued sequence (in short, FNVS)
of functions and fuzzy number valued function (in short, FNVF) respectively, both defined
on E. Then, the sequence (fm) is

(D1) relatively weighted point-wise statistically convergent to limit of FNVF f under the

fractional-order difference operator ∆α,β,γ
h,x , denoted by

fm −→ f (RW -Stpw(∆α,β,γ
h,x )) or Strwpw(∆α,β,γ

h,x ) lim fm = f (E;σ),

if there is a non-zero scale function σ(x) on E such that, for each ε > 0 and for
every x ∈ E,

lim
n→∞

1

Rbnan+1

∣∣∣∣∣∣
m : m 5 Rbnan+1 and sbn−mtm

D
(

∆α,β,γ
h,x fm(x), f(x)

)
|σ(x)|

= ε


∣∣∣∣∣∣ = 0.

(D2) relatively weighted equi-statistically convergent to the limit of FNVF f under the

fractional-order difference operator ∆α,β,γ
h,x ; here, we write

fm −→ f (RW -Stequi(∆
α,β,γ
h,x )) or Strwequi(∆

α,β,γ
h,x ) lim fm = f (E;σ),



S. K. Paikray, P. Parida, S. A. Mohiuddine / Eur. J. Pure Appl. Math, 13 (5) (2020), 1212-1230 1218

if there exists σ(x) (σ(x) > 0) on E such that, for every ε > 0 and for every x ∈ E,

lim
n→∞

Ωm(x; εσ)

Rbnan+1

= 0,

relatively uniformly with regards to x ∈ E, that is,

lim
n→∞

‖Ωm(x; εσ)‖CF (E)

Rbnan+1

= 0,

where

Ωm(x; εσ) =

∣∣∣∣∣∣
m : m 5 Rbnan+1 and sup

x∈E
sbn−mtm

D
(

∆α,β,γ
h,x fm(x), f(x)

)
|σ(x)|

= ε


∣∣∣∣∣∣ .

(D3) relatively weighted uniformly statistically convergent to the limit of FNVF f under

the fractional-order difference operator ∆α,β,γ
h,x ; here, we write

fm −→ f (RW -Stuf (∆α,β,γ
h,x )) or Strwuf (∆α,β,γ

h,x ) lim fm = f (E;σ),

if there is a scale function σ(x) (σ(x) > 0) on E such that, for each ε > 0,

lim
n→∞

1

Rbnan+1

∣∣∣∣∣∣
m : m 5 Rbnan+1 and sup

x∈E
sbn−mtm

D
(

∆α,β,γ
h,x fm(x), f(x)

)
|σ(x)|

= ε


∣∣∣∣∣∣ = 0.

As a consequence of the above definitions, we present the following inclusion relations
in the similar lines of our earlier work (see [39]).

Lemma 1. The implications,

fm −→ f(RW -Stuf (∆α,β,γ
h,x )) =⇒ fm −→ f(RW -Stequi(∆

α,β,γ
h,x ))

=⇒ fm −→ f(RW -Stpw(∆α,β,γ
h,x )) (4)

are fairly true. Moreover, reverse of the implications of (4) are not necessarily true, that
is, the above inclusions are strict.

4. A Fuzzy Korovkin-type Approximation Theorem

Based upon the proposed methods, in the present section we wish to investigate a fuzzy
approximation theorem (Korovkin-type) via relatively deferred Nörlund equi-statistical
convergence.



S. K. Paikray, P. Parida, S. A. Mohiuddine / Eur. J. Pure Appl. Math, 13 (5) (2020), 1212-1230 1219

Let f be a fuzzy number valued function, and Suppose that f : [a, b] → RF . Re-
call that, f is fuzzy continuous at a point x0 ∈ [a, b], if D(xn, x0) < ε (n→∞) whenever
xn → x0. Furthermore, if f is fuzzy continuous at each point x ∈ [a, b], then it is also fuzzy
continuous in the interval [a, b]. Let CF [a, b] be denoted as the set of all fuzzy functions
(continuous) defined in the interval [a, b] (note that, CF [a, b] is a scalar, but not usually a
vector space).

Note that, T : CF [a, b] → CF [a, b] is fuzzy linear operator, if for each µ1, µ2 ∈ R and
f1, f2 ∈ CF [a, b],

T(µ1 � f1 ⊕ µ2 � f1;x) = µ1 � T(f1)⊕ µ2 � T(f2).

Moreover, a fuzzy linear operator T is a positive fuzzy linear operator, if T(f1;x) � T(f2;x)
such that f1, f2 ∈ CF [a, b] and for all x ∈ [a, b] with f1(x) � f2(x).

In the year 2005, Anastassiou [2] has proved a classical version of fuzzy Korovkin-type
theorem. Subsequently, Anastassiou and Duman [3] established the statistical versions of
a fuzzy approximation theorem based on the fuzzy positive linear operator. Here in this
paper, we extend the result of Anastassiou and Duman [3] by using relatively Nörlund equi-
statistical convergence and accordingly established the following theorem for the same set
of functions. Throughout the paper we choose the test functions f1 = xi, where i = 0, 1, 2.

Theorem 1. Let (an) and (bn) be sequences of integers (non-negative) and let Tm :
CF [a, b] → CF [a, b] (m ∈ N) be the fuzzy sequence of positive linear operators. Suppose
that {T∗m}n∈N be the corresponding sequence of positive linear operators from C[a, b] into
itself such that

{Tm(f ;x)}r± = T∗m(f r±;x) (5)

for all x ∈ [a, b], r ∈ [0, 1], n ∈ N and f ∈ CF [a, b]. Further, assume that

T∗m(fi) −→ fi (RW -Sequi(∆
α,β,γ
h,x )) (E;σi), (6)

where fi(x) = xi (i = 0, 1, 2) and E = [a, b]. Then, for all f ∈ CF [a, b],

Strwequi(∆
α,β,γ
h,x ) limD∗ (Tm(f), f) = 0 (E;σ). (7)

Here σ(x) = max{|σi(x)| : |σi(x)| > 0, i = 0, 1, 2}.

Proof. Suppose f ∈ CF [a, b], x ∈ [a, b] and r ∈ [0, 1]. Moreover, since f r±(x) ∈ C[a, b]
(by the hypothesis), so we have for every ε > 0, there exists δ > 0, such that

|f r±(y)− f r±(x)| < ε whenever |y − x| < δ (8)

for all x, y ∈ [a, b]. Since f is fuzzy bounded, we have |f r±(x)| ≤ Kr± (a < x < b). Therefore
|f r±(y)− f r±(x)| ≤ 2Kr± (a < x, y < b). Let us choose θ(y, x) = (y − x)2. Then, we clearly
get

|f r±(y)− f r±(x)| < ε+
2Kr±
δ2

θ(y, x)



S. K. Paikray, P. Parida, S. A. Mohiuddine / Eur. J. Pure Appl. Math, 13 (5) (2020), 1212-1230 1220

which yields

−ε−
2Kr±
δ2

θ(y, x) <
(
f r±(y)− f r±(x)

)
< ε+

2Kr±
δ2

θ(y, x). (9)

Next, as the operator ∆α,β,γ
h,x T∗m is linear and monotone, so by applying ∆α,β,γ

h,x T∗m(1, x) in
(9), we obtain

∆α,β,γ
h,x T∗m(1, x)

(
−ε−

2Kr±
δ2

θ(y, x)

)
< ∆α,β,γ

h,x T∗m(1, x)
(
f r±(y)− f r±(x)

)
< ∆α,β,γ

h,x T∗m(1, x)

(
ε+

2Kr±
δ2

θ(y, x)

)
. (10)

Furthermore, x is supposed to be fixed and f r±(x) being a constant number, we thus get

−ε∆α,β,γ
h,x T∗m(1, x)−

2Kr±
δ2

∆α,β,γ
h,x T∗m(θ, x) < ∆α,β,γ

h,x T∗m(f r±, x)− f r±(x)∆α,β,γ
h,x T∗m(1, x)

< ε∆α,β,γ
h,x T∗m(1, x) +

2Kr±
δ2

∆α,β,γ
h,x T∗m(θ, x), (11)

and moreover in association with the identity (below)

∆α,β,γ
h,x T∗m(f r±, x)− f r±(x) = [∆α,β,γ

h,x T∗m(f r±, x)− f r±(x)∆α,β,γ
h,x T∗m(1, x)]

+ f r±(x)[∆α,β,γ
h,x T∗m(1, x)− 1] (12)

yields

∆α,β,γ
h,x T∗m(f r±, x)− f r±(x) < ε∆α,β,γ

h,x T∗m(1, x)

+
2Kr±
δ2

∆α,β,γ
h,x T∗m(θ, x) + f r±(x)[∆α,β,γ

h,x Trm(1, x)− 1]. (13)

Furthermore, computing ∆α,β,γ
h,x T∗m(θ, x) as,

∆α,β,γ
h,x T∗m(θ, x) = ∆α,β,γ

h,x T∗m(y2 − 2xy + x2, x)

= ∆α,β,γ
h,x T∗m(y2, x)− 2x∆α,β,γ

h,x T∗m(y, x) + x2∆α,β,γ
h,x T∗m(1, x)

= [∆α,β,γ
h,x T∗m(y2, x)− x2]− 2x[∆α,β,γ

h,x T∗m(y, x)− x]

+ x2[∆α,β,γ
h,x T∗m(1, x)− 1]

and using (13), we get

∆α,β,γ
h,x T∗m(f r±, x)− f r±(x) < ε∆α,β,γ

h,x T∗m(1, x) +
2Kr±
δ2
{[∆α,β,γ

h,x T∗m(y2, x)− x2]

− 2x[∆α,β,γ
h,x T∗m(y, x)− x] + x2[∆α,β,γ

h,x T∗m(1, x)− 1]}

+ f r±(x)[∆α,β,γ
h,x T∗m(1, x)− 1]



S. K. Paikray, P. Parida, S. A. Mohiuddine / Eur. J. Pure Appl. Math, 13 (5) (2020), 1212-1230 1221

= ε[∆α,β,γ
h,x T∗m(1, x)− 1] + ε+

2Kr±
δ2
{[∆α,β,γ

h,x T∗m(y2, x)− x2]

− 2x[∆α,β,γ
h,x Trm(y, x)− x] + x2[∆α,β,γ

h,x T∗m(1, x)− 1]}

+ f r±(x)[∆α,β,γ
h,x T∗m(1, x)− 1].

We certainly write

|∆α,β,γ
h,x T∗m(f r±, x)− f r±(x)| 5 ε+

(
ε+

2Kr±c2

δ2
+Kr±

)
|∆α,β,γ

h,x T∗m(1, x)− 1|+
4Kr±c
δ2
|∆α,β,γ

h,x Trm(y, x)− x|

+
2Kr±
δ2
|∆α,β,γ

h,x Trm(y2, x)− x2|,

where c = max{|a|, |b|}. Consequently, we obtain

|∆α,β,γ
h,x T∗m(f r±, x)− f r±(x)| 5 ε+Mr

±(ε)
(
|∆α,β,γ

h,x T∗m(1, x)− 1|

+ |∆α,β,γ
h,x T∗m(y, x)− x|+ |∆α,β,γ

h,x T∗m(y2, x)− x2|
)
, (14)

where

Mr
±(ε) = max

(
ε+

2Kr±c2

δ2
+Kr±,

4Kr±c
δ2

,
2Kr±
δ2

)
.

Now it clearly follows from (5) that,

D∗(∆α,β,γ
h,x Tm(f), f) = sup

x∈E
D
(

∆α,β,γ
h,x Tm(f ;x), f(x)

)
= sup

x∈E
sup
r∈[0,1]

max
{∣∣∣∆α,β,γ

h,x T∗m(f r−;x)− f r−
∣∣∣ , ∣∣∣∆α,β,γ

h,x T∗m(f r+;x)− f r+(x)
∣∣∣} .

Considering (14) with the last equality, one can easily write

D∗(∆α,β,γ
h,x Tm(f), f)

|σ(x)|
5 sup

x∈E

ε

σ(x)
+M(ε)

(
sup
x∈E

∣∣∣∣∣∆
α,β,γ
h,x T∗m(1, x)− 1

σ0(x)

∣∣∣∣∣
+ sup
x∈E

∣∣∣∣∣∆
α,β,γ
h,x T∗m(y, x)− x

σ1(x)

∣∣∣∣∣+ sup
x∈E

∣∣∣∣∣∆
α,β,γ
h,x T∗m(y2, x)− x2

σ2(x)

∣∣∣∣∣
)
,

where
M(ε) = sup

r∈[0,1]
max

{
Mr
−(ε),Mr

+(ε)
}

and

σ(x) = max{|σi(x)| : |σi(x)| > 0, i = 0, 1, 2}.

Therefore,

sbn−mtm
D∗(∆α,β,γ

h,x Tm(f), f)

|σ(x)|
5 sbn−mtm sup

x∈E

ε

σ(x)



S. K. Paikray, P. Parida, S. A. Mohiuddine / Eur. J. Pure Appl. Math, 13 (5) (2020), 1212-1230 1222

+M(ε)

(
sbn−mtm sup

x∈E

∣∣∣∣∣∆
α,β,γ
h,x T∗m(1, x)− 1

σ0(x)

∣∣∣∣∣
+ sbn−mtm sup

x∈E

∣∣∣∣∣∆
α,β,γ
h,x T∗m(y, x)− x

σ1(x)

∣∣∣∣∣
+ sbn−mtm sup

x∈E

∣∣∣∣∣∆
α,β,γ
h,x T∗m(y2, x)− x2

σ2(x)

∣∣∣∣∣
)
. (15)

Next, for given κ > 0, choose ε > 0 such that sbn−mtm supx∈E
ε

σ(x) < κ. Then, we can
write

Θm(x; εσ) =

∣∣∣∣∣∣
m : m 5 Rbnan+1 and sbn−mtm

D∗
(

∆α,β,γ
h,x Tm(f), f

)
|σ(x)|

 = ε′

∣∣∣∣∣∣

and

Θi,m(x, εσ) =

∣∣∣∣∣
{
m : m 5 Rbnan+1 and

sbn−mtm

D∗
(

∆α,β,γ
h,x T∗mfi(x), fi(x)

)
|σi(x)|

 = ε′ − ε

σ(x)

3Mr
±

}∣∣∣∣∣,
we easily obtain from (15) that

Θm(x, εσ(x)) 5
2∑
i=0

Θi,m(x, εσ(x)).

Thus, we fairly have

‖Θm(x, εσ(x))‖
Rbnan+1

5
2∑
i=0

‖Θi,m(x, εσ(x))‖
Rbnan+1

. (16)

Consequently, by Definition 1(D2) and under the above assumption for the implication
in (6), the right-hand side of (16) seems tend to zero as n→∞. We thus get

lim
n→∞

‖Θm(x, εσ)‖
Rbnan+1

= 0 (ε > 0).

Hence, the implication in (7) is fairly true. This completes the proof of the Theorem.



S. K. Paikray, P. Parida, S. A. Mohiuddine / Eur. J. Pure Appl. Math, 13 (5) (2020), 1212-1230 1223

5. Fuzzy Rate of Relatively Eqiu-statistical Convergence

We intend to investigate here the fuzzy rate of the relatively equi-statistical convergence
of a sequence of fuzzy positive linear operators defined from CF (E) into itself based on the
fuzzy modulus of continuity.

Definition 2. Let (an) and (bn) be sequences of non-negative integers, and also let (un)
be a positive non-increasing sequence. A fuzzy number valued sequence (fn) of functions
is relatively Λn-equi-statistical convergent to a fuzzy number valued function f on E with
rate o(un), if for every ε > 0,

lim
n→∞

Υn(x; εσ)

unR
bn
an+1

= 0

uniformly relatively with respect to x ∈ E or, otherwise if

lim
n→∞

‖Υn(x; εσ)‖CF [0,1]

unR
bn
an+1

= 0,

where

Υn(x, εσ) =

∣∣∣∣∣∣
m : m 5 Rbnan+1 and sbn−mtm

D∗
(

∆α,β,γ
h,x fm(x), f(x)

)
|σ(x)|

= ε


∣∣∣∣∣∣ .

Here, we write
Strwequi(∆

α,β,γ
h,x )D∗ (fn(x), f(x)) = o(un) on (E;σ).

We now need to prove the following Lemma.

Lemma 2. Let (un) and (vn) be two positive non-increasing sequences. Suppose the fuzzy
valued sequence of functions (fn) and (gn) ∈ CF (E) satisfy the conditions:

Strwequi(∆
α,β,γ
h,x )D∗ (fn(x), f(x)) = o(un) on (E;σ).

and
Strwequi(∆

α,β,γ
h,x )D∗ (gn(x), g(x)) = o(vn) on (E;σ1),

where σ0 > 0 and σ1 > 0; then all the following assertions are true:

(i) Strwequi(∆
α,β,γ
h,x )D∗ (fn(x) + gn(x), f(x) + g(x)) = o(wn) on (E; max{σ0, σ1});

(ii) Strwequi(∆
α,β,γ
h,x )D∗ (fn(x), f(x))D∗ (gn(x), g(x)) = o(unvn) on (E; {σ0, σ1});

(iii) Strwequi(∆
α,β,γ
h,x )µD∗ (fn(x), f(x)) = o(un) on (E;σ0), for any scalar µ;



S. K. Paikray, P. Parida, S. A. Mohiuddine / Eur. J. Pure Appl. Math, 13 (5) (2020), 1212-1230 1224

(iv) Strwequi(∆
α,β,γ
h,x ) {D∗ (fn(x), f(x))}

1
2 = o(un) on (E;

√
|σ0(x)|),

where wn = max{un, vn}.

Proof. For proving the assertion (i) of Lemma 2, we consider the following sets for
which ε > 0 and x ∈ E:

An(x, εσ) =

∣∣∣∣∣
{
m :m 5 Rbnan+1 and

sbn−mtm
D∗
([

∆α,β,γ
h,x fm + ∆

[r]
p,qgm

]
(x), (f + g)(x)

)
|σ(x)|

= ε

}∣∣∣∣∣,
A0,n(x, εσ) =

∣∣∣∣∣∣
m : m 5 Rbnan+1 and sbn−mtm

D∗
(

∆α,β,γ
h,x fm(x), f(x)

)
|σ0(x)|

=
ε

2


∣∣∣∣∣∣

and

A1,n(x, εσ) =

∣∣∣∣∣∣
m : m 5 Rbnan+1 and sbn−mtm

D∗
(

∆α,β,γ
h,x gm(x), g(x)

)
|σ1(x)|

=
ε

2


∣∣∣∣∣∣ ,

where
σ(x) = max{|σi(x)| : i = 0, 1}.

Clearly, we have

An(x, εσ) ⊆ A0,n(x, εσ) ∪ A1,n(x, εσ).

Moreover, since

wn = max{un, vn}, (17)

by using the assertion (7) of Theorem 1, we obtain

‖An(x, εσ)‖CF (E)

wnR
bn
an+1

5
‖A0,n(x, εσ)‖CF (E)

unR
bn
an+1

+
‖A1,n(x, εσ)‖CF (E)

vnR
bn
an+1

. (18)

Also, by using the assertion (6) of Theorem 1, we obtain

‖An(x, εσ)‖CF (E)

wnR
bn
an+1

= 0. (19)

Thus, assertion (i) of this Lemma is proved.



S. K. Paikray, P. Parida, S. A. Mohiuddine / Eur. J. Pure Appl. Math, 13 (5) (2020), 1212-1230 1225

Next, since all other assertions (ii) - (iv) of Lemma 2 are similar as in the assertion
(i), so these can be proved along similar lines to complete the proof of the Lemma 2.

The fuzzy modulus of continuity of f is such that f : [a, b] → RF was studied by [2],
It is defined by

ωF (f, δ) = sup
x,y∈[a,b]

{D∗(f(y), f(x)) : |y − x| 5 δ (0 < δ ≤ a− b)} . (20)

We now introduce a theorem to obtain the fuzzy rates of relatively deferred Nörlund
equi-statistical convergence based on difference sequence of functions under the support
of the fuzzy modulus of continuity.

Theorem 2. Let (an) and (bn) be sequences of integers and let Tm : CF [a, b]→ CF [a, b] (m ∈
N) be a sequence of fuzzy positive linear operators. Suppose that {T∗n}n∈N be the corre-
sponding sequence of positive linear operators from C[a, b] into itself such that (5) holds.
Assume further that (un) and (vn) be two positive non-increasing sequences and suppose
that the operators {T∗m}n∈N satisfy the conditions:

(i) Strwequi(∆
α,β,γ
h,x )T∗m(1, x)− 1 = o(un) on (E;σ0),

(ii) Strwequi(∆
α,β,γ
h,x )ωF (f, δn) = o(vn) on (E;σ1),

where

δn(x) =
{
L∗m(θ2;x)

} 1
2 and θ(y) = (y − x),

then for each f ∈ CF (E), the assertion as below holds true:

Strwequi(∆
α,β,γ
h,x )D∗ (Tm(f), f) = o(wn) on (E;σ), (21)

where (wn) defined by (17) and

σ(x) = max{|σ0(x)|, |σ1(x)|, |σ0(x)σ1(x)| : σi(x) > 0 (i = 0, 1)}.

Proof. Suppose E ⊂ R be compact and let f ∈ CF (E), x ∈ E. Then, it is obvious
that,

D∗
(

∆α,β,γ
h,x Tm(f, x); f

)
5 N|∆α,β,γ

h,x T∗m(1;x)− 1|+

(
∆α,β,γ
h,x T∗m(1;x)

+
√

∆α,β,γ
h,x T∗m(1;x)

)
ωF (f, δn),

where
N = ‖f‖CF (E).



S. K. Paikray, P. Parida, S. A. Mohiuddine / Eur. J. Pure Appl. Math, 13 (5) (2020), 1212-1230 1226

Which yields

D∗
(

∆α,β,γ
h,x Tm(f), f

)
σ(x)

5 N

(
∆α,β,γ
h,x T∗m(1;x)− 1

|σ0(x)|

)
+ 2

ωF (f, δn)

σ1(x)

+
ωF (f, δn)

σ1(x)

(
∆α,β,γ
h,x T∗m(1;x)− 1

|σ0(x)|

)

+
ωF (f, δn)

σ1(x)

√√√√(∆α,β,γ
h,x T∗m(1;x)− 1

|σ0(x)|

)
. (22)

Finally, for the conditions (i) and (ii) (of Theorem 2) along with Lemma 2, the last
inequality (22) helps us to achieve the assertion (21). Therefore, proof of Theorem 2 is
completed.

6. Concluding Remarks and Observations

In last section of our investigation, we present various further remarks and observations
relating the different outcomes which we have proved here.

Remark 1. If we put an = 0, bn = n, sn = tn = 1 and α = β = γ = 0 in our Theorem
1, then we get the statistical versions of the fuzzy Korovkin-type approximation theorem,
which was demonstrated earlier by Anastassiou and Duman (see [3]).

Remark 2. Suppose in Theorem 2, we substitute the conditions (i) and (ii) by the follow-
ing condition:

Strwequi(∆
α,β,γ
h,x )D∗(T∗m(fi, x)− fi) = o(uni) on (E;σi). (23)

Then, since

∆α,β,γ
h,x T∗m(θ2;x) = ∆α,β,γ

h,x T∗m(y2;x)− 2x∆α,β,γ
h,x T∗m(y;x) + x2∆α,β,γ

h,x T∗m(1;x),

we can write

∆α,β,γ
h,x T∗m(θ2;x) 5 κ

2∑
i=0

|∆α,β,γ
h,x T∗m(fi;x)− fi(x)|, (24)

where
κ = 1 + 2‖f1‖CF (E) + ‖f2‖CF (E).

Clearly, from (23), (24) and Lemma 2, it follows that

Strwequi(∆
α,β,γ
h,x )δn = Strwequi(∆

α,β,γ
h,x )

√
T∗m(θ2) = o(un) on (E;σ), (25)



REFERENCES 1227

where
o(un) = max{un0 , un1 , un2}.

Thus, definitely, we obtain

Strwequi(∆
α,β,γ
h,x )ωF (f, δ) = o(un) on (E;σ).

By applying (25) in Theorem 2, we instantly see that, ∀ f ∈ CF (E),

Strwequi(∆
α,β,γ
h,x )D∗ (Tm(f), f) = o(un) on (E;σ). (26)

Therefore, instead of conditions (i) and (ii) of Theorem 2, if we use the condition (23), then

we certainly find the fuzzy rates of the relatively Strwequi(∆
α,β,γ
h,x )-equi-statistical convergence

for the sequence (T∗m) of fuzzy positive linear operators in our Theorem 1.

Acknowledgements

The authors would like to keep the record of the 80th birthday of Prof. H. M. Srivas-
tava for his tremendous contribution to many significant developments in Mathematical
research. Also, the authors express their heartfelt thanks to the editors and anonymous
referees for their most valuable comments and constructive suggestions which leads to the
improvement of the earlier version of the manuscript.

Conflicts of Interest: The authors declare that they have no conflicts of interest.

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