Ann. Acad. Rom. Sci.

Ser. Math. Appl.

ISSN 2066-6594

Vol. 18, No. 2/2026

ON ρ-STRONG CONVERGENCE FOR

DIFFERENCE SEQUENCES OF FRACTIONAL

ORDER IN q-RUNG ORTHOPAIR FUZZY

NORMED SPACES^∗

Nesar Hossain^†

Rahul Mondal^‡

Communicated by G. Moro¸sanu

DOI

10.56082/annalsarscimath.2026.2.43

Abstract

In this paper, we propose the concept of ρ-strong convergence for

diﬀerence sequences of fractional order, denoted by ∆^α-ρ-strong con-

vergence, in a q-rung orthopair fuzzy normed space. We establish the

uniqueness of this convergence and provide its algebraic characteriza-

tion. A convergence criterion for subsequences is derived, and the rela-

tionship between ∆^α-strong convergence and ∆^α-ρ-strong convergence

s

ρ_s

is examined under conditions on lim inf

. Moreover, an inclusion re-

sult is obtained by employing a positive non-decreasing sequence µ_s

satisfying ρ_s< µ_sunder suitable assumptions. Finally, we introduce

the notion of ρ-strongly Cauchy diﬀerence sequences of fractional order

and investigate their connection with ∆^α-ρ-strong convergence.

Keywords: q-rung orthopair fuzzy normed space, t-norm, t-conorm,

∆^α-ρ-strong convergence, ∆^α-ρ-strong Cauchy sequence.

MSC: 40A05, 03E72, 46A45, 40C05.

^∗Accepted for publication on October 29, 2025

^†nesarhossain24@gmail.com, Department of Basic Science and Humanities, Dumkal

Institute of Engineering and Technology, West Bengal 742406, India

^‡imondalrahul@gmail.com, rahulmath@vsm.org.in, Department of Mathematics,

Vivekananda Satavarshiki Mahavidyalaya, Manikpara, Jhargram 721513, West Bengal,

India

43

ρ-strong convergence for diﬀerence sequences of fractional order

44

1 Introduction and preliminaries

Since its introduction in 1965, fuzzy set theory [40] has developed into a

powerful tool with applications in artiﬁcial intelligence, computer science,

medicine, control engineering, decision theory, management science, opera-

tions research, pattern recognition, and robotics. Unlike classical sets with

binary membership, fuzzy sets allow gradual degrees of membership, making

them well-suited for modeling imprecision in human reasoning. Formally,

for a nonempty set U, a fuzzy set A is deﬁned as

A = {hϑ, Φ_A(ϑ)i : ϑ ∈ U},

where Φ_A: U → [0, 1] is the membership function.

To overcome the limitation that non-membership is not always the com-

plement of membership, Atanassov [1] introduced intuitionistic fuzzy sets

(IFS). An IFS A in U is given by

A = {hϑ, Φ_A(ϑ), Ψ_A(ϑ)i : ϑ ∈ U},

where Φ_Aand Ψ_Adenote, respectively, membership and non-membership

functions satisfying

0 ≤ Φ_A(ϑ) + Ψ_A(ϑ) ≤ 1.

The hesitation margin is deﬁned as

π_A(ϑ) = 1 − Φ_A(ϑ) − Ψ_A(ϑ),

which quantiﬁes the degree of uncertainty.

Yager [33] extended this idea to Pythagorean fuzzy sets (PFS), where

membership and non-membership degrees satisfy

(Φ_A(ϑ))²+ (Ψ_A(ϑ))²≤ 1.

This provides greater ﬂexibility in handling uncertainty compared to IFS.

To generalize further, Yager [34 ] introduced q-rung orthopair fuzzy sets (q-

ROFS), characterized by the condition

(Φ_A(ϑ))^q+ (Ψ_A(ϑ))^q≤ 1,

where q ≥ 1. Clearly, IFS and PFS appear as special cases for q = 1 and

q = 2, respectively. Increasing q enlarges the admissible domain, allowing a

richer representation of fuzzy information.

N. Hossain, R. Mondal

45

This framework has been further advanced in functional analysis and

topology. Yager and Alajlan [35] developed approximate reasoning tech-

niques within q-ROFS, while Turkarslan et al. [30 ] introduced q-rung or-

thopair fuzzy topological spaces. Parimala et al. [23] studied their supra-

topological applications, and Saeed and Ibrahim [28] investigated n, m^th

power root fuzzy sets. Uluc¸ay introduced q-rung orthopair fuzzy normed

spaces and established new notions of statistical convergence, statistical

Cauchy sequences, and completeness criteria, thereby deepening the un-

derstanding of convergence behavior in this setting.

1.1

Identiﬁed research gaps and underlying motivation

The concept of diﬀerence sequence spaces was ﬁrst introduced by Kızmaz [15]

and later generalized by Et et al. [13,14] as follows:

∆^m(X) = {~ = (~_k) : (∆^m~_k) ∈ X},

where X is any sequence space, m ∈ N,

ꢀ

ꢁ

∆⁰~ = (~_k),

∆~ = (~_k−~_k+1),

∆^m~ = (∆^m~_k) = ∆^m−1~_k−∆^m−1

~

.

k+1

Thus,

ꢂ ꢃ

m

X

m

v

∆^m~_k=

(−1)^v

~

.

k+v

v=0

If ~ ∈ ∆^m(X), then there exists a unique sequence ϑ = (ϑ_k) ∈ X such

that ϑ_k= ∆^m~_kand

ꢂ

ꢃ

ꢂ

ꢃ

k−m

k

X

k − v − 1

k + m − v − 1

~_k=

(−1)^m

ϑ_v=

(−1)^m

ϑ_v−m

,

(1)

m − 1

v=1

with ϑ_1−m= ϑ_2−m= · · · = ϑ₀= 0 for suﬃciently large k (e.g., k > 2m).

Since then, several properties of diﬀerence sequence spaces have been studied

in [2,3,14,17,26,29].

For a proper fraction α, the fractional diﬀerence operator ∆^α: W → W

is deﬁned by

∞

X

Γ(α + 1)

∆^α(~_k) =

(−1)ⁱ

~

.

(2)

k+i

i! Γ(α − i + 1)

i=0

ρ-strong convergence for diﬀerence sequences of fractional order

46

In particular,

1/2

1

2

1

8

1

16

5

128

∆

~_k= ~_k− ~_k+1− ~_k+2

−

~

−

k+3

~

_k+4− · · · ,

k+3

−1/2

1/3

1

3

5

16

35

128

~_k= ~_k+ ~_k+1+ ~_k+2

+

~

+

~

_k+4+ · · · ,

2

8

1

3

1

9

5

81

10

243

~_k= ~_k− ~_k+1− ~_k+2

−

~

−

~

_k+4− · · · ,

_k+4− · · · .

k+3

2/3

2

3

1

9

4

81

7

243

~_k= ~_k− ~_k+1− ~_k+2

~

k+3

Here, Γ(r) denotes the Gamma function for real numbers

r ∈/ {0, −1, −2, −3, . . . },

deﬁned by the improper integral

Z

∞

Γ(r) =

e^−tt^r−1dt.

0

Its well-known properties include:

• For any natural number n, Γ(n + 1) = n!,

• For real n ∈/ {0, −1, −2, . . . }, Γ(n + 1) = nΓ(n),

• Special cases: Γ(1) = Γ(2) = 1, Γ(3) = 2!, Γ(4) = 3!, etc.

By deﬁnition, the series in (2) is convergent. Moreover, if α is a positive

integer, the inﬁnite sum in (2) reduces to a ﬁnite sum:

α

X

Γ(α + 1)

(−1)ⁱ

~

.

k+i

i! Γ(α − i + 1)

i=0

Thus, the operator ∆^αgeneralizes the classical diﬀerence operator intro-

duced by Et and C¸olak [13].

Recently, using this fractional diﬀerence operator ∆^α, Baliarsingh et

al. [7,8,22] introduced the sequence space

∆^α(X) = {~ = (~_k) : (∆^α~_k) ∈ X},

where X is any sequence space.

The concept of ρ-statistical convergence was ﬁrst introduced by C¸akallı

[10] where ρ = (ρ_s) is a non-decreasing sequence of positive reals tending to

ρ_s

∞ such that lim sup_s

< ∞, ∆ρ_s= O(1) and ∆ρ_s= ρ_s+1− ρ_sfor each

s

positive integer s. Several authors have extensively investigated convergence

theory through the use of the sequence {ρ_s}, including Barlak [9], Debnath

N. Hossain, R. Mondal

47

and Debnath [12], Kandemir [19], among others. In recent years, the study

of fractional order diﬀerence sequence spaces has emerged as an active and

signiﬁcant area of research. Notable contributions in this ﬁeld include the

works of Yaying and Hazarika [38], Kadak [18], Srivastava and Mahato [27],

Yaying [39 ], and many others, where investigations have been carried out

in various settings such as neutrosophic normed spaces [4] and n-normed

spaces [11]. Furthermore, for the development of diﬀerence sequence spaces

of fractional order, we refer the readers to [36, 37], while a comprehensive

account of sequence convergence theory can be found in [5, 20, 21]. These

works form part of the fundamental concepts that underpin and enrich our

present study. Notably, our study draws its primary motivation from the

seminal work of Aral et al. [6].

Although the concepts of convergence based on the sequence {ρ_s} and

diﬀerence sequences of fractional order have been extensively studied by nu-

merous researchers, there remains a strong need to explore these notions

within broader and more abstract frameworks, particularly in the setting of

q-rung orthopair fuzzy normed spaces. Despite several notable contributions

such as investigations into statistical convergence, the study of summability

theory and sequence convergence in q-rung orthopair fuzzy normed spaces

is still in its early stages. In particular, the extension of statistical conver-

gence to generalized forms, such as ρ-statistical convergence for diﬀerence

sequences of fractional order, remains largely unexplored. This indicates a

signiﬁcant gap in understanding the interplay between ρ-strong convergence

for diﬀerence sequences of fractional order under appropriate conditions.

Moreover, there exists ample scope to extend this line of research by incor-

porating modulus functions in a q-rung orthopair fuzzy normed space.

1.2

Key contributions

The main contributions of this paper are as follows. We introduce the no-

tion of ρ-strong convergence for diﬀerence sequences of fractional order (de-

noted by ∆^α-ρ-strong convergence) in the framework of q-rung orthopair

fuzzy normed spaces, thereby extending classical convergence concepts into

a broader and more abstract setting. We establish the uniqueness of ∆^α-

ρ-strong convergence and provide its algebraic characterization, along with

a criterion for the ∆^α-ρ-strong convergence of subsequences, ensuring sta-

bility of convergence under subsequential analysis. By imposing suitable

s

ρ_s

conditions on lim inf

, we further examine the relationship between ∆^α-

strong convergence and ∆^α-ρ-strong convergence. In addition, we present a

signiﬁcant inclusion result involving ∆^α-ρ-strong convergence by employing

ρ-strong convergence for diﬀerence sequences of fractional order

48

a positive non-decreasing sequence (µ_s) with ρ_s< µ_sunder appropriate as-

sumptions. Finally, we introduce and investigate the concept of ρ-strongly

Cauchy diﬀerence sequences of fractional order in q-rung orthopair fuzzy

normed spaces, and explore its connection with ∆^α-ρ-strong convergence.

To proceed, we ﬁrst revisit essential deﬁnitions concerning q-rung or-

thopair fuzzy normed spaces. Unless otherwise stated, N and R will denote

the sets of natural and real numbers, respectively.

Deﬁnition 1. [24] A mapping ꢀ, named as binary operation, from O × O

to O, where O = [0, 1], is referred to as a continuous t-norm if for each

ϑ₁, ϑ₂, ϑ₃, ϑ₄∈ O, the conditions listed below are met:

1. ꢀ exhibits both associativity and commutativity;

2. ꢀ exhibits continuous behavior;

3. ϑ₁ꢀ 1 = ϑ₁, ∀ ϑ₁∈ O;

4. ϑ₁ꢀ ϑ₂≤ ϑ₃ꢀ ϑ₄whenever ϑ₁≤ ϑ₃and ϑ₂≤ ϑ₄.

Deﬁnition 2. [24] A mapping ⊗, named as binary operation, from O × O

to O, where O = [0, 1], is referred to as a continuous t-conorm if for each

ϑ₁, ϑ₂, ϑ₃, ϑ₄∈ O, the conditions listed below are met:

1. ⊗ exhibits both associativity and commutativity;

2. ⊗ exhibits continuous behavior;

3. ϑ₁⊗ 0 = ϑ₁, ∀ ϑ₁∈ O;

4. ϑ₁⊗ ϑ₂≤ ϑ₃⊗ ϑ₄whenever ϑ₁≤ ϑ₃and ϑ₂≤ ϑ₄.

Example 1. [16] Some illustrations of ꢀ and ⊗ are: ϑ₁ꢀϑ₂= min{ϑ₁, ϑ₂}

and ϑ₁ꢀ ϑ₂= ϑ₁.ϑ₂. ϑ₁⊗ ϑ₂= max{ϑ₁, ϑ₂} and ϑ₁⊗ ϑ₂= ϑ₁+ ϑ₂− ϑ₁.ϑ₂.

Lemma 1. [25] If ꢀ is a continuous t-norm, ⊗ is a continuous t-conorm,

ϑ_i∈ (0, 1) and 1 ≤ i ≤ 7, then the following statements hold:

1. If ϑ₁> ϑ₂, there are ϑ₃, ϑ₄∈ (0, 1) such that ϑ₁ꢀ ϑ₃≥ ϑ₂and

ϑ₁≥ ϑ₂⊗ ϑ₄

2. If ϑ₅∈ (0, 1), there are ϑ₆, ϑ₇∈ (0, 1) such that ϑ₆ꢀ ϑ₆≥ ϑ₅and

ϑ₅≥ ϑ₇⊗ ϑ₇.

N. Hossain, R. Mondal

49

Deﬁnition 3. [40] A set C of the form

C = {(ϑ, R_C(ϑ)) : ϑ ∈ S}

is considered to be a fuzzy set on a non-empty set S. For each element

ϑ ∈ S, the function R_C(ϑ) represents the membership degree of ϑ in C,

where R_C(ϑ) ∈ [0, 1].

Deﬁnition 4. Let C be a set deﬁned as:

C = {(ϑ, R_C(ϑ), V_C(ϑ)) : ϑ ∈ S}

where S is a non-empty set. Depending on the constraints imposed on the

membership R_C(ϑ) and non-membership V_C(ϑ) functions, C can be classiﬁed

as follows:

1. Intuitionistic fuzzy set [1]: when 0 ≤ R_C(ϑ) + V_C(ϑ) ≤ 1.

2

2. Pythagorian fuzzy set [32]: when 0 ≤ R_C(ϑ) + V_C(ϑ) ≤ 1.

q

3. q-rung orthopair fuzzy set [34]: when 0 ≤ R_C(ϑ) + V_C(ϑ) ≤ 1.

for each ϑ ∈ S, R_C(ϑ) represents the membership function, while V_C(ϑ)

denotes the non-membership function of the set C.

Remark 1. q-rung orthopair fuzzy set turns into intuitionistic fuzzy set

whenever q = 1 while it coincides with the notion of Pythagorian fuzzy set

whenever q = 2.

Deﬁnition 5. [31] Let Q be a vector space, ꢀ be a continuous t-norm

and ⊗ be a continuous t-conorm. Also, let R and V be two fuzzy sets on

Q × (0, ∞) and q ≥ 1 be a real number. Then, the 5-tuple (Q, R, V, ꢀ, ⊗)

is termed a q-rung orthopair fuzzy normed space (abbreviated q-ROFNS) if

for every ϑ, κ ∈ Q and ζ, γ > 0 the conditions listed below are met:

q

1. R (ϑ, ζ) + V (ϑ, ζ) ≤ 1,

2. R(ϑ, ζ) > 0,

q

3. R (ϑ, ζ) = 1 if and only if ϑ = θ,

ꢄ

ꢅ

ζ

q

4. R (αϑ, ζ) = R

ϑ, _|α|, for each α = 0,

q

5. R (ϑ, ζ) ꢀ R (κ, γ) ≤ R (ϑ + κ, ζ + γ),

ρ-strong convergence for diﬀerence sequences of fractional order

50

6. R(ϑ, ·) : (0, ∞) → [0, 1] is continuous,

7. lim_ζ→∞R(ϑ, ζ) = 1 and lim_ζ→0R(ϑ, ζ) = 0,

8. V(ϑ, ζ) < 1,

q

9. V (ϑ, ζ) = 0 if and only if ϑ = θ,

ꢄ

ꢅ

ζ

q

10. V (αϑ, ζ) = V

ϑ, _|α|, for each α = 0,

q

11. V (ϑ, ζ) ⊗ V (κ, γ) ≥ V (ϑ + κ, ζ + γ),

12. V(ϑ, ·) : (0, ∞) → [0, 1] is continuous,

13. lim_ζ→∞V(ϑ, ζ) = 0 and lim_ζ→0V(ϑ, ζ) = 1.

In this case, (R, V) is termed a q-rung orthopair fuzzy norm (abbreviated

q-ROFN). When q = 2, this structure corresponds to a Pythagorean fuzzy

normed space, whereas for q = 1, it simpliﬁes to an intuitionistic fuzzy

normed space. In the sequel Q will stands for the 5-tuple (Q, R, V, ꢀ, ⊗) for

the sake of abbreviation.

Notably, every intuitionistic fuzzy normed space inherently qualiﬁes as a

q-rung orthopair fuzzy normed space. However, the reverse does not always

hold true. The following example illustrates this distinction.

Example 2. [31] Let Q = R on which usual norm is imposed. Consider

t-norm and t-conorm as ϑ₁ꢀ ϑ₂= ϑ₁ϑ₂and ϑ₁⊗ ϑ₂= min{ϑ₁+ ϑ₂, 1},

q

|ϑ|

ζ

q

∀ϑ₁, ϑ₂∈ [0, 1]. deﬁne R₀(ϑ, ζ) =

and V₀(ϑ, ζ) =

_ζ+|ϑ|. Then

ζ+|ϑ|

(R, R₀, V₀, ꢀ, ⊗) becomes q-rung orthopair fuzzy normed space but not intu-

itionistic fuzzy normed space.

Deﬁnition 6. [31] Consider {ϑ_i} to be a sequence in a q-ROFNS Q. Then,

q

{ϑ_i} is termed convergent to % ∈ Q if R (ϑ_i−%, ζ) → 1 and V (ϑ_i−%, ζ) → 0

whenever i → ∞ for every ζ > 0.

We now introduce the concept of convergence for diﬀerence sequences of

fractional order with respect to (R, V).

Deﬁnition 7. Consider a sequence ~_kin a q-rung orthopair fuzzy normed

space Q. Then, ~_kis referred to as ∆^α-convergent to τ ∈ Q in relation to

(R, V) (brieﬂy, (R, V)_q(∆^α)-convergence) if, for every λ > 0 and $ ∈ (0, 1),

there can be found k₀∈ N such that

q

R (∆^α~_k− τ, λ) > 1 − $ and V (∆^α~_k− τ, λ) < $, ∀k ≥ k₀.

(R,V)_q(∆^α)

In this scenario, we express ~_k−−−−−−−→ τ.

N. Hossain, R. Mondal

51

Lemma 2. [31] A sequence {ϑ_i} in q-ROFNS (R, R₀, V₀, ꢀ, ⊗) is conver-

gent if and only if it is convergent in (R, | · |).

2 Main results

This section is devoted to presenting our main results. Throughout this

section, Q denotes a q-rung orthopair fuzzy normed space, and α represents

a proper fraction, unless speciﬁed otherwise. Henceforth, ∆^αis regarded as

closed linear operator on the sequence space Q, ensuring that both ∆^α~_k

and the diﬀerence ∆^α~_k− τ are well deﬁned.

Deﬁnition 8. Consider a sequence {~_k} in Q. Then, {~_k} is referred to as

∆^α− ρ-strongly convergent to τ ∈ Q in relation to (R, V) if for every λ > 0

and $ ∈ (0, 1) there can be found s₀∈ N such that

s

X

1

q

R (∆^α~_k−τ, λ) > 1−$ and

V (∆^α~_k−τ, λ) < $, ∀s ≥ s₀.

ρ_s

k=1

(R,V)

In this scenario, we express ~_k−−−→ τ(∆^α, ρ). When ρ_s= s, it follows that

(R,V)

~_k−−−→ τ(∆^α).

Next, we present an example illustrating the sequence of ∆^α-ρ-strong

convergence for diﬀerence sequences of fractional order in a q-rung orthopair

fuzzy normed space Q.

Example 3. Let Q = R equipped with the usual norm. Deﬁne the t-norm

and t-conorm, respectively, by

ϑ₁ꢀ ϑ₂= ϑ₁ϑ₂,

ϑ₁⊗ ϑ₂= min{ϑ₁+ ϑ₂, 1} ∀ϑ₁, ϑ₂∈ [0, 1].

We consider the q-ROFN given as in Example 2. With this structure, Q

becomes a q-ROFNS. Now, deﬁne a sequence {~_k} by

(

1,

if k = b², b ∈ N

∆^α~_k=

0,

otherwise.

For any λ > 0 and 0 < $ < 1, set

q

B = {k ≤ s : R (∆^α~_k, λ) > 1 − $ and V (∆^α~_k, λ) < $}.

ρ-strong convergence for diﬀerence sequences of fractional order

52

We now compute:

q

B = {k ≤ s : R (∆^α~_k, λ) > 1 − $ and V (∆^α~_k, λ) < $}

ꢆ

ꢇ

λ

|∆^α~_k|

=

k ≤ s :

> 1 − $ and

< $

λ + |∆^α~_k|

k ≤ s : |∆^α~_k| <

λ + |∆^α~_k|

ꢇ

$λ

=

1 − $

⊆ {k ≤ s : |∆^α~_k| = 1}

= {k ≤ s : k = b²}.

Hence,

(

)

s

X

1

s

X

1

q

s ∈ N :

R (∆^α~_k, λ) > 1 − $ and

V (∆^α~_k, λ) < $

ρ_s

k=1

(R,V)

is a ﬁnite set. Therefore, we conclude that ~_k−−−→ 0(∆^α, ρ).

In the following theorem, we determine the limit of ∆^α-ρ-strong conver-

gence in a q-rung orthopair fuzzy normed space.

Theorem 1. Consider a sequence {~_k} in a q-rung orthopair fuzzy normed

(R,V)

spaces Q. If ~_k−−−→ τ(∆^α, ρ), then the limit τ is uniquely determined.

Proof. Let $ ∈ (0, 1) be ﬁxed. Select γ ∈ (0, 1) satisfying

(1 − γ) ꢀ (1 − γ) > 1 − $ γ ⊗ γ < $.

Suppose that

(R,V)

~_k−−−→ τ₁(∆^α, ρ) and ~_k−−−→ τ₂(∆^α, ρ)

with τ₁= τ₂. By the deﬁnition of convergence, for any λ > 0 and γ ∈ (0, 1),

there exist s₁, s₂∈ N such that

ꢂ

ꢃ

ꢂ

ꢃ

s

X

1

λ

1

λ

q

R

∆^α~_k− τ₁,

> 1 − γ and

V

∆^α~_k− τ₁,

< γ,

ρ_s

2

ρ_s

2

k=1

for all s ≥ s₁, and

ꢂ

ꢃ

ꢂ

ꢃ

s

X

1

λ

1

λ

q

R

∆^α~_k− τ₂,

> 1 − γ and

∆^α~_k− τ₂,

< γ,

ρ_s

2

ρ_s

2

k=1

N. Hossain, R. Mondal

53

for all s ≥ s₂. Let s₀= max{s₁, s₂}. Then, for all s ≥ s₀, there exists a

positive integer t such that

ꢂ

ꢃ

ꢂ

ꢃ

λ

q

R (τ₁− τ₂, λ) ≥ R

∆^α~_t− τ₁,

ꢀ R

∆^α~_t− τ₂,

2

> (1 − γ) ꢀ (1 − γ)

> 1 − $

and similarly,

ꢂ

ꢃ

ꢂ

ꢃ

λ

∆^α~_t− τ₂,

q

V (τ₁− τ₂, λ) ≤ V

∆^α~_t− τ₁,

⊗ V

2

< γ ⊗ γ

< $.

q

Since $ > 0 is arbitrary, it follows that R (τ₁−τ₂, λ) = 1 and V (τ₁−τ₂, λ) =

0. This implies that τ₁= τ₂, contradicting our initial assumption. Thus,

the proof stands established.

Next, we turn to the algebraic characterization of ∆^α-ρ-strong conver-

gence for sequences in Q.

Theorem 2. Let {~_k} and {η_k} be sequences in a q-rung orthopair fuzzy

normed space Q over the ﬁeld R. Then, the following properties hold:

(R,V)

1. If ~_k−−−→ τ₁(∆^α, ρ) and η_k−−−→ τ₂(∆^α, ρ), then ~_k+ η_k−−−→

τ₁+ τ₂(∆^α, ρ).

(R,V)

2. If ~_k−−−→ τ(∆^α, ρ), then for any scalar κ ∈ R \ {0}, κ~_k−−−→

κτ(∆^α, ρ).

Proof. Since the proof is routine, we omit the details.

We now proceed to establish the ∆^α-ρ-strong convergence criterion for

subsequences of a given sequence.

(R,V)

Theorem 3. If ~_k−−−→ τ(∆^α, ρ), then there exists a subsequence ∆^α~_j

k

(R,V)_q(∆^α)

of ∆^α~_ksuch that ~_j−−−−−−−→ τ.

k

ρ-strong convergence for diﬀerence sequences of fractional order

54

(R,V)

Proof. Suppose that ~_k−−−→ τ(∆^α, ρ). Then, for every λ > 0 and $ ∈

(0, 1) there exists s₀∈ N such that

s

X

1

q

R (∆^α~_k−τ, λ) > 1−$ and

V (∆^α~_k−τ, λ) < $, ∀s ≥ s₀.

ρ_s

k=1

It follows that for each s ≥ s₀, we may choose j_k≤ s such that

s

X

1

q

R (∆^α~_j− τ, λ) >

R (∆^α~_k− τ, λ) > 1 − $

k

ρ_s

k=1

s

X

1

V (∆^α~_j− τ, λ) <

V (∆^α~_k− τ, λ) < $.

q

k

ρ_s

k=1

By virtue of Deﬁnition 7, the subsequence {~_j} adheres to the (R, V)_q(∆^α)-

k

convergence condition sharing the same limit τ. Consequently,

(R,V)_q(∆^α)

~

−−−−−−−→ τ

j_k

. This completes the proof.

s

ρ_s

In the sequel, depending on the condition of

, we establish the connec-

tion between ∆^α-strong convergence and ∆^α-ρ-strong convergence in Q.

(R,V)

Theorem 4. Let {~_k} be a sequence in Q with ~_k−−−→ τ(∆^α). Then,

(R,V)

s

ρ_s

~_k−−−→ τ(∆^α, ρ) whenever lim inf_s

> 1.

Proof. Suppose that

(R,V)

~_k−−−→ τ(∆^α),

(3)

(4)

and

s

lim inf

> 1.

s→∞

ρ_s

We aim to show that this implies

(R,V)

~_k−−−→ τ(∆^α, ρ).

For any λ > 0 and 0 < $ < 1, consider

s

X

1

q

R (∆^α~_k− τ, λ).

ρ_s

k=1

N. Hossain, R. Mondal

55

This expression can be rewritten as

s

X

1

s

X

s

1

q

R (∆^α~_k− τ, λ) =

·

R (∆^α~_k− τ, λ).

(5)

ρ_s

s

k=1

From (3 ), we know that ~_kconverges to τ in the (R, V)-sense with respect

to ∆^α. This means that for suﬃciently large s,

s

X

1

q

R (∆^α~_k− τ, λ) > 1 − $.

s

k=1

s

ρ_s

By assumption (4), for suﬃciently large s we have

≥ 1. Substituting into

(5), we obtain

s

X

1

q

R (∆^α~_k− τ, λ) ≥

R (∆^α~_k− τ, λ) > 1 − $.

ρ_s

s

k=1

P

s

k=1

1

s

q

Similarly, using (3), for large s we have

V (∆^α~_k− τ, λ) < $. Since

P

s

k=1

s

ρ_s

1

ρ_s

q

≥ 1, it follows that

V (∆^α~_k− τ, λ) < $. Thus, under the

(R,V)

conditions (3) and (4), we conclude that ~_k−−−→ τ(∆^α, ρ).

s

ρ_s

Remark 2. Condition (4) ensures that the ratio

does not fall below 1 in

the limit. In other words, the sequence {ρ_s} does not reduce the eﬀect of the

terms in the summation. Since convergence already holds in the (∆^α)-sense,

this condition guarantees that the same convergence persists in the more

generalized form (∆^α, ρ)-convergence. The above result shows that (∆^α)-

s

ρ_s

convergence implies (∆^α, ρ)-convergence whenever lim inf

> 1. Thus, the

(∆^α, ρ)-convergence can be viewed as a natural extension of the standard

(∆^α)-convergence under a suitable condition imposed on ρ_s.

Now, we investigate the relationship between ∆^α-ρ-strong convergence

and ∆^α-µ-strong convergence with respect to two non-decreasing positive

sequences (ρ_s) and (µ_s) where ρ_s< µ_sfor all s ∈ N. The inclusion and

ρ_s

µ_s

equality of these convergence classes depend on how the ratio

behaves

when s becomes very large.

Theorem 5. Let {ρ_s} and {µ_s} be two sequences such that ρ_s< µ_sfor all

s ∈ N, and let B ⊆ Q be a q-rung orthopair fuzzy bounded set. Then, every

sequence that is ∆^α-ρ-strongly convergent is also ∆^α-µ-strongly convergent,

provided that

µ_s

lim

> 0.

(6)

s→∞

ρ_s

ρ-strong convergence for diﬀerence sequences of fractional order

56

(R,V)

Proof. Since ~_k−−−→ τ(∆^α, ρ) and B is q-rung orthopair fuzzy bounded,

there exists some λ > 0 such that for each (∆^α~_k− τ) ∈ B, we have

s

X

1

q

R (∆^α~_k− τ, λ) > 1 − $ and

V (∆^α~_k− τ, λ) < $,

ρ_s

k=1

for some 0 < $ < 1.

Now, since ρ_s≤ µ_sfor all s ∈ N, we may rewrite the ﬁrst inequality as

s

X

1

µ_s

1

q

R (∆^α~_k− τ, λ) =

·

R (∆^α~_k− τ, λ).

ρ_s

ρ_sµ_s

k=1

By assumption (6), we know that

µ_s

ρ_s

lim

> 0,

s→∞

µ_s

ρ_s

which guarantees that

Hence, the inequality

remains strictly positive for suﬃciently large s.

s

X

1

q

R (∆^α~_k− τ, λ) > 1 − $

ρ_s

k=1

directly implies

s

X

1

q

R (∆^α~_k− τ, λ) > 1 − $.

µ_s

k=1

A similar argument applies to the second inequality involving V:

s

X

1

q

V (∆^α~_k− τ, λ) < $

=⇒

V (∆^α~_k− τ, λ) < $.

ρ_s

µ_s

k=1

Thus, both conditions required for ∆^α-µ-strong convergence are satisﬁed.

Therefore, we conclude that {~_k} is also ∆^α-µ-strongly convergent.

Remark 3. The condition ρ_s< µ_smeans that (µ_s) increases faster than

(ρ_s). In simple terms, dividing by the larger sequence µ_smakes the expres-

sions

s

X

1

q

R (∆^α~_k− τ, λ),

V (∆^α~_k− τ, λ)

µ_s

k=1

represent weaker (less strict) conditions compared to using ρ_s. Thus, if a

sequence satisﬁes the stronger requirement of being in the class of ∆^α-ρ-

strongly convergent sequences, it will automatically satisfy the weaker re-

quirement of being in the class of ∆^α-µ-strongly convergent sequences pro-

µ_s

ρ_s

µ_s

ρ_s

vided the ratio

does not vanish (i.e., lim_s→∞

> 0).

N. Hossain, R. Mondal

57

The following class is deﬁned as:

ꢆ

ꢇ

(R,V)

(R, V)_ρ[∆^α] = {~_k} : ∃ τ ∈ Q : ~_k−−−→ τ(∆^α, ρ) .

Corollary 1. Let (ρ_s) and (µ_s) be two non-decreasing sequences of positive

µ_s

real numbers such that ρ_s< µ_sfor all s ∈ N. If lim_s→∞

= c > 0, c is a

ρ_s

ﬁnite positive constant and if B ⊆ Q is q-rung orthopair fuzzy bounded, then

(R, V)_ρ[∆^α] = (R, V)_µ[∆^α].

Proof. From Theorem 5, we already have

(R, V)_ρ[∆^α] ⊆ (R, V)_µ[∆^α].

Now, interchanging the roles of ρ and µ, and noting that

1

ρ_s

µ_s

lim

=

> 0,

s→∞

c

we obtain the reverse inclusion

(R, V)_µ[∆^α] ⊆ (R, V)_ρ[∆^α].

Therefore,

(R, V)_ρ[∆^α] = (R, V)_µ[∆^α].

Remark 4. An interesting question naturally arises: what can be said about

the inclusion relations between the above convergence classes in the case

µ_s

ρ_s

when lim_s→∞

explore.

= ∞? We leave this as an open problem for the reader to

We now introduce the notion of a ∆^α-ρ-strongly Cauchy sequence in a

q-rung orthopair fuzzy normed space, and investigate its relationship with

∆^α-strong convergence.

Deﬁnition 9. Let {~_k} be a sequence in a q-rung orthopair fuzzy normed

space Q. The sequence {~_k} is said to be a ∆^α-ρ-strongly Cauchy sequence

with respect to (R, V) if, for every λ > 0 and $ ∈ (0, 1), there exist s₀, t ∈ N

such that

s

X

1

q

R (∆^α~_k−∆^α~_t, λ) > 1−$ and

V (∆^α~_k−∆^α~_t, λ) < $,

ρ_s

k=1

for all s ≥ s₀.

ρ-strong convergence for diﬀerence sequences of fractional order

58

Theorem 6. Let {~_k} be a sequence in a q-rung orthopair fuzzy normed

space Q. If {~_k} is ∆^α-ρ-strongly convergent, then it is ∆^α-ρ-strongly

Cauchy sequence.

Proof. Let $ ∈ (0, 1) be given. Choose γ ∈ (0, 1) such that

(1 − γ) ꢀ (1 − γ) > 1 − $ and γ ⊗ γ < $.

Suppose that {~_k} is ∆^α-ρ-strongly convergent to τ ∈ Q. Then, for every

λ > 0, there exists s₀∈ N such that

ꢂ

ꢃ

ꢂ

ꢃ

s

X

1

λ

1

λ

∆^α~_k− τ,

> 1 − γ and

V

∆^α~_k− τ,

< γ,

q

R

ρ_s

2

ρ_s

2

k=1

for all s ≥ s₀. For s ≥ s₀, one can select t ∈ N such that

ꢂ

ꢃ

ꢂ

ꢃ

s

X

λ

1

λ

q

R

∆^α~_t− τ,

>

R

∆^α~_k− τ,

> 1 − γ

2

ρ_s

2

k=1

and

ꢂ

ꢃ

ꢂ

ꢃ

s

X

λ

1

λ

q

V

∆^α~_t− τ,

<

V

∆^α~_k− τ,

< γ.

2

ρ_s

2

k=1

We now prove that

s

X

1

s

X

1

q

R (∆^α~_k−∆^α~_t, λ) > 1−$ and

V (∆^α~_k−∆^α~_t, λ) < $,

ρ_s

k=1

∀ s ≥ s₀. Indeed, we have

ꢂ

ꢃ

ꢂ

ꢃ

λ

q

R (∆^α~_k− ∆^α~_t, λ) ≥ R

∆^α~_k− τ,

ꢁ R

∆^α~_t− τ,

2

> (1 − γ) ꢀ (1 − γ)

> 1 − $.

This leads to

s

X

1

q

R (∆^α~_k− ∆^α~_t, λ) > 1 − $.

(7)

ρ_s

k=1

N. Hossain, R. Mondal

59

And,

ꢂ

ꢃ

ꢂ

ꢃ

λ

q

V (∆^α~_k− ∆^α~_t, λ) ≤ V

∆^α~_k− τ,

⊗ V

∆^α~_t− τ,

2

< γ ⊗ γ

< $.

We deduce that

1

s

X

q

V (∆^α~_k− ∆^α~_t, λ) < $.

(8)

ρ_s

k=1

Hence, from (7) and (8), it follows that the sequence {~_k} is ∆^α-ρ-strongly

Cauchy sequence. Thus, the proof stands established.

Conclusion and future scope

In this paper, we introduced the novel concept of ∆^α-ρ-strong convergence

in a q-rung orthopair fuzzy normed space. We established the uniqueness

of the ∆^α-ρ-strong convergence of a sequence and provided an algebraic

characterization of this convergence. Further, we derived the ∆^α-ρ-strong

convergence criterion for subsequences of a given sequence. Depending on

s

ρ_s

the conditions imposed on lim inf

, we also examined the relationship be-

tween ∆^α-strong convergence and ∆^α-ρ-strong convergence. Moreover, we

presented a signiﬁcant inclusion result involving ∆^α-ρ-strong convergence us-

ing an alternative a positive non-decreasing sequence µ_sand the inequality

ρ_s< µ_s, under suitable assumptions. In addition, we introduced the notion

of a ∆^α-ρ-strong Cauchy sequence in a q-rung orthopair fuzzy normed space

and explored its relationship with ∆^α-ρ-strong convergence.

Since research on sequence convergence in q-rung orthopair fuzzy normed

spaces is still at an early stage, we believe this work provides a solid founda-

tion for further developments. Future research directions include extending

these ideas in connection with modulus functions and double sequences of

order 0 < α ≤ 1 within the setting of q-rung orthopair fuzzy normed spaces.

Another promising direction would be the construction of new sequence

spaces based on this convergence concept in association with Orlicz func-

tions, followed by an investigation of their topological properties. These

advancements could lead to powerful tools for addressing a wide range of

convergence related problems across mathematics, applied sciences, and en-

gineering disciplines.

ρ-strong convergence for diﬀerence sequences of fractional order

60

Acknowledgements. We extend our sincere gratitude to the reviewers for

their thoughtful suggestions and constructive feedback, which have signiﬁ-

cantly contributed to enhancing the quality and clarity of our paper.

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