EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5641
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On a New Stochastic Space with Applications to
Nonlinear Economic Model

Meshayil M. Alsolmi1,∗, Salah H. Alshabhi1, Mustafa M. Mohammed1,
Thwiba A. Khalid2, Mona Magzoub1, Nidal E. Taha3, Khdija O. Taha3,
Awad A. Bakery1

1 University of Jeddah, Applied College, Department of Mathematics, Jeddah, Saudi Arabia
2 Department of Mathematics, Faculty of Science, Al-Baha University, Albaha 65525,
Saudi Arabia
3 Department of Mathematics, College of Science, Qassim University, Buraidah 51452,
Saudi Arabia

Abstract. This article will utilize a weighted regular matrix composed of Fibonacci numbers
and variable exponent sequence spaces to create a novel stochastic space with certain geometric
and topological properties. This area demonstrates the new form of the Kannan contraction
operator with a fixed point. In mathematical economics, we represent economic entities, processes,
and phenomena by mathematically structured functional equations, either as summable equations
or integral equations. We investigate a category of Volterra-type non-linear dynamical systems,
similar to an economic model. We employ our acquired results to formulate new solvability criteria
for a unique solution of these non-linear discrete economic dynamical systems. Ultimately, we
illustrate our findings with specific instances and applications related to the presence of solutions
in non-linear dynamical systems of Volterra-type.

2020 Mathematics Subject Classifications: 46B15, 46C05, 46E05
Key Words and Phrases: Fibonacci numbers, variable exponent, extended s−soft numbers,
new type of Kannan contraction.

Abbreviations

(i) p-q.N : pre-quasi norm.

(ii) psssf: private sequence space of soft functions.

(iii) p-m : pre-modular.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5641

Email addresses: mmralsulami@uj.edu.sa (M. M. Alsolmi), salsabieh@uj.edu.sa (S. H. Alshabhi),
mustasta@yahoo.com, mmibrahim@uj.edu.sa (M. M. Mohammed),
tabdulrhman@bu.edu.sa (T. A. Khalid), mmahmed@uj.edu.sa (M. M. Ahmed),
n.taha@qu.edu.sa (N. E. Taha), K.Taha@qu.edu.sa (K. O. Taha), aabhassan@uj.edu.sa (A. A. Bakery)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



M. M. A et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5641 2 of 20

(iv) p-q.B : pre-quasi Banach.

(v) Bs: Banach space.

(vi) Cs: Cauchy sequence.

(vii) CMs: complete metric space.

(viii) NAT : non-absolute type.

(ix) FP : Fatou property.

(x) NTK-∥.∥p−qN-C : New type of Kannan ∥.∥p−qN -contraction.

(xi) ∥.∥p−qN-Seq.C : ∥.∥p−qN -sequentially continuous.

(xii) ufp: unique fixed point.

(xiii) NLDEs: non-linear difference equations.

Notations

(i) N := {0, 1, 2, ...} and R is the set of real numbers.

(ii) R+N
: The space of all sequences of positive reals.

(iii) ℓm, ℓ∞, and c0: The spaces of m-absolutely summable, bounded, and convergent to
zero sequences of reals, respectively.

(iv) B(R) and E: The collection of all nonempty bounded subsets of R and the set of
parameters, respectively.

(v) R(A)∗ and R(A): The set of nonnegative and all soft real numbers (corresponding
to A), where A ⊂ E, respectively.

(vi) 0̃ and 1̃: The additive identity and multiplicative identity in R(A), respectively, see
[6].

(vii) µS: The space of all sequences of soft reals.

(viii) G, V: Infinite dimensional Banach spaces.

(ix) ES: The linear space of sequences of soft functions.

(x) êd := (0̂, 0̂, ..., 1̂, 0̂, 0̂, · · · ), while 1̂ displays at the dth place.

(xi) [d]: The integral part of real number d.

(xii) ϑ̂ := (0̂, 0̂, 0̂, . . .).



M. M. A et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5641 3 of 20

(xiii) F : The space of finite sequences of soft numbers.

(xiv) N+ and D−: The space of all monotonic increasing and decreasing sequences of
positive reals, respectively.

(xv) Ir: The identity mapping on ℓd2.

(xvi) Jr: The natural embedding mapping from Mr into V.

(xvii) Tr: The quotient mapping from G onto G/Yr.

(xviii) Dvoretzky’s Theorem [19]: For any d ∈ N , we have quotient spaces G/Yd and
subspaces Md of V which can be transformed onto ℓd2 by isomorphisms Vd and Xd

such that ∥Wd∥∥W−1
d ∥ ≤ 2 and ∥Xd∥∥X−1

d ∥ ≤ 2.

1. Introduction

The ability to mathematically simulate non-Newtonian fluids in hydrodynamics is
drawing more and more attention to the study of variable exponent Lebesgue spaces, as
discussed by Ruẑiĉka [20]. A variety of disciplines, including orthopedics, civil engineering,
and military science, make use of electrorheological fluids, a class of non-Newtonian fluids.
The work of Diening et al. [7] covered the topic of Lebesgue and Sobolev spaces involving
variable exponents. A particular sequence space contains the solutions of discrete dynam-
ical systems. According to [16], the construction of new sequence spaces is a subject of
significant mathematical interest. The domain of the Cesàro mean of order one in certain
spaces of double sequences was developed and studied by Mursaleen and Başar [15], while
Noman and Mursaleen [17] investigated novel non-absolute sequence spaces that are con-
nected to ℓp and ℓ∞. The notion of psssf was first proposed by Alsolmi and Bakery [4]. In
[14], the researchers examined the distinctiveness and presence of solutions inside a novel
complex function space for Kannan nonlinear dynamical systems. Supposing that (fk)

∞
k=0

is the sequence of Fibonacci numbers defined by the recurrence relation fv = fv−1 + fv−2,
v ≥ 2, so that f0 = 1 and f1 = 1. Note that in [12] that

∑l
z=0 f

2
z = flfl+1, and

∑∞
z=0

1
fz

< ∞.
Kara and Başarır further strengthen the studies on Fibonacci sequence spaces [11]. They
defined and studied the matrix domains ℓp(γf) := (ℓp)γf , c0(γf) := (c0)γf , c(γf) := (c)γf and
ℓ∞(γf) := (ℓ∞)γf .

Assume that (tl), (ql) ∈ R+N
. We have presented a novel stochastic space

(
γSf (q, t)

)
∥.∥p−qN

of soft functions as:(
γSf (q, t)

)
∥.∥p−qN

:=
{
d̂ = (d̂b) ∈ µS : ∥δd̂∥p−qN < ∞, for some δ > 0

}
,

where ∥d̂∥p−qN =
∑
l∈N

 ℏ̂
(∑l

z=0 f
2
zqzd̂z, 0̂

)
flfl+1

tl

, ℏ : R(A)×R(A) → R(A)∗, with ℏ(f̂ , ĝ) =

|f̂ − ĝ|, for all f̂ , ĝ ∈ R(A), and ℏ̂ : R(A)×R(A) → R+ is defined by

ℏ̂(f̂ , ĝ) = max
λ∈A

ℏ(f̂ , ĝ)(λ).



M. M. A et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5641 4 of 20

If (tl) ∈ R+N ∩ ℓ∞, then(
γSf (q, t)

)
∥.∥p−qN

=
{
d̂ = (d̂b) ∈ µS : ∥δd̂∥p−qN < ∞, for any δ > 0

}
.

Volterra-type summable equations are fundamental in investigating dynamical systems [1]
and stochastic processes [9, 13]. Assuming that ν̂ ∈ γSf (q, t), Γ : N 2 → R, Ψ : N×R(A) →
R(A), ν̂ : N → R(A), and β̂ : N → R(A).
Consider the Volterra-type summable equations of soft functions [4]:

ν̂d = β̂d +
∑
v∈N

Γd,vΨv,ν̂v , (1)

and when Φ :
(
γSf (q, t)

)
∥.∥p−qN

→
(
γSf (q, t)

)
∥.∥p−qN

is defined as

Φ(ν̂d)d∈N =

(
β̂d +

∑
v∈N

Γd,vΨv,ν̂v

)
d∈N

. (2)

Considering the multitude of fixed point theorems within a specific space, it is necessary
to either enlarge the space itself or to augment the self-mapping that operates within it;
both alternatives are feasible. Examples include granular systems, sweeping processes,
oscillation issues, control challenges, and decision-making problems, among others. A
particular sequence space encompasses the solutions of summable equations. So, there is
significant interest in mathematics to develop new sequence spaces. This work aims to
create a new stochastic space by utilizing a weighted regular matrix based on Fibonacci
numbers and variable exponent sequence spaces. We have applied specific geometric and

topological structures to soft functions represented as
(
γSf (q, t)

)
∥.∥p−qN

. The fixed point

of the new type of pre-quasi-Kannan contraction operator is verified in this context. We
conclude by illustrating our findings with several examples and applications related to the
existence of solutions to non-linear difference equations.

2. Definitions and Preliminaries

Definition 1. [4] ES is referred to as a psssf if it meets the following criteria:

(1c) ES is linear space and êr ∈ ES, for r ∈ N ,

(2c) ES is solid i.e., if m̂ = (m̂r) ∈ µS, |k̂| = (|k̂r|) ∈ ES and |m̂r| ≤ |k̂r|, where r ∈ N ,
then |m̂| ∈ ES,

(3c)
(∣∣∣k̂[ r

2
]

∣∣∣)
r∈N

∈ ES, if
(∣∣∣k̂x∣∣∣)

r∈N
∈ ES.

Definition 2. [4] A subspace psssf ES
∥.∥p−qN

is called a p-m psssf, if ∥.∥p−qN : ES → [0,∞)

meets the following criteria for all m̂, k̂ ∈ ES, and δ ∈ R:



M. M. A et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5641 5 of 20

(a1) k̂ = ϑ̂ ⇐⇒ ∥(|k̂|)∥p−qN = 0, and ∥k̂∥p−qN ≥ 0,

(a2) there are C1 ≥ 1 so that ∥δm̂∥p−qN ≤ |δ|C1∥m̂∥p−qN ,

(a3) ∥m̂+ k̂∥p−qN ≤ C2(∥m̂∥p−qN + ∥k̂∥p−qN ) verifies so that C2 ≥ 1,

(a4) if |m̂r| ≤ |k̂r|, then ∥(|m̂r|)∥p−qN ≤ ∥(|k̂r|)∥p−qN ,

(a5) the inequality, ∥(|k̂r|)∥p−qN ≤ ∥(|k̂[ r
2
]|)∥p−qN ≤ C3∥(|k̂r|)∥p−qN holds, for C3 ≥ 1,

(a6) the closure F of F = ES
∥.∥p−qN

,

(a7) the inequality, ∥(m̂, 0̂, 0̂, 0̂, ...)∥p−qN ≥ α|m|∥ê1∥p−qN verifies for α > 0.

Definition 3. [4] The space psssf ES
∥.∥p−qN

is called a p-q.N psssf when ∥.∥p−qN verifies

the parts (a1)-(a3) of Definition 2. The space psssf ES
∥.∥p−qN

is said to be p-q.B psssf if

the space psssf ES
∥.∥p−qN

is complete equipped with ∥.∥p−qN .

Theorem 2.1. [3] Every p-m psssf ES
∥.∥p−qN

is a p-q.N psssf.

Lemma 2.2. [5] Suppose rm > 1 and αm, δm ∈ R, for all m ∈ N , and ℶ = supm rm, one
has

|αm + δm|rm ≤ 2ℶ−1 (|αm|rm + |δm|rm) . (3)

Definition 4. [3] A function ∥.∥p−qN on ES
∥.∥p−qN

satisfies the FP when for each {k̂r} ⊆
ES
∥.∥p−qN

such that limr→∞ ∥k̂r − k̂∥p−qN = 0 and all û ∈ ES
∥.∥p−qN

, then ∥û − k̂∥p−qN ≤
supm infr≥m ∥û− k̂r∥p−qN .

Definition 5. [4] Supposing that ES
∥.∥p−qN

is a p-q.N psssf, M : ES
∥.∥p−qN

→ ES
∥.∥p−qN

and

d̂ ∈ ES
∥.∥p−qN

. The mapping M is said to be ∥.∥p−qN-Seq.C at d̂, if and only if, for any

{k̂r} ⊆ ES
∥.∥p−qN

such that limr→∞ ∥k̂r − d̂∥p−qN = 0 then limr→∞ ∥Mk̂r −Md̂∥p−qN = 0.

3. Configuration and properties of
(
γS
f (q, t)

)
∥.∥p−qN

This section introduces the definition and inclusion relations of the sequence space(
γSf (q, t)

)
∥.∥p−qN

with the function ∥.∥p−qN .

Theorem 3.1. The space
(
γSf (q, t)

)
∥.∥p−qN

is a NAT, if (tl) ∈ (0,∞)N ∩ ℓ∞.

Proof. Clearly, as

∥ê0 − ê1∥p−qN = (q0)
t0 +

(
|q0 − q1|

2

)t1

+

(
|q0 − q1|

6

)t2

+ · · ·

̸= (q0)
t0 +

(
|q0 + q1|

2

)t1

+

(
|q0 + q1|

6

)t2

+ · · · = ∥ê0 + ê1∥p−qN .



M. M. A et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5641 6 of 20

Definition 6. Supposing that (tl) ∈ [0.5,∞)N . The absolute type space
(
|γSf |(q, t)

)
φ
is

defined as (
|γSf |(q, t)

)
φ
:=
{
Ĵ = (Ĵk) ∈ µS : φ(δf) < ∞, for some δ > 0

}
,

where φ(Ĵ) =
∞∑
l=0

 ℏ̂
(∑l

z=0 f
2
zqz|Ĵz|, 0̂

)
flfl+1

tl

.

Theorem 3.2. If (tl) ∈ [0.5,∞)N ∩ ℓ∞ with
(

l+1
flfl+1

)
/∈ ℓ(tl), one gets

(
|γSf |(q, t)

)
φ

⫋(
γSf (q, t)

)
∥.∥p−qN

.

Proof. If d̂ ∈
(
|γSf |(q, t)

)
φ
, then

∑
l∈N

 ℏ̂
(∑l

z=0 f
2
zqzd̂z, 0̂

)
flfl+1

tl

≤
∑
l∈N

 ℏ̂
(∑l

z=0 f
2
zqz|d̂z|, 0̂

)
flfl+1

tl

< ∞.

Therefore, d̂ ∈
(
γSf (q, t)

)
∥.∥p−qN

. If Ĵ =
(
(−1̂)z

f2zqz

)
z∈N

, one has Ĵ ∈
(
γSf (q, t)

)
∥.∥p−qN

and

Ĵ /∈
(
|γSf |(q, t)

)
φ
.

We provide the sufficient conditions on γSf (q, t) to form a p-q.B psssf.

Theorem 3.3. γSf (q, t) is a p-m psssf, if

(o1) (tl) ∈ N+ ∩ ℓ∞ and t0 ≥ 0.5.

(o2)
(
f2zqz

)
z∈N ∈ D− or,

(
f2zqz

)
z∈N ∈ N+∩ ℓ∞ and one has A ≥ 1 such that r2z+1q2z+1 ≤

Af2zqz.

Proof. Assuming that d̂, k̂ ∈ γSf (q, t), and δ ∈ R. Assume the setups (o1) and (o2) are
verified.
The condition (a1): Clearly, ∥d̂∥p−qN ≥ 0 and ∥(|d̂|)∥p−qN = 0 ⇔ d̂ = ϑ̂.
The conditions (1c) and (a3):

∥d̂+ k̂∥p−qN =
∑
l∈N

 ℏ̂
(∑l

z=0 f
2
zqz

(
d̂z + k̂z

)
, 0̂
)

flfl+1

tl

≤ 2ℶ−1

∑
l∈N

 ℏ̂
(∑l

z=0 f
2
zqzd̂z, 0̂

)
flfl+1

tl

+
∑
l∈N

 ℏ̂
(∑l

z=0 f
2
zqzk̂z, 0̂

)
flfl+1

tl
 = C2(∥d̂∥p−qN + ∥k̂∥p−qN ) < ∞,



M. M. A et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5641 7 of 20

therefore, d̂+ k̂ ∈ γSf (q, t).
The conditions (1c) and (a2):

∥δd̂∥p−qN =
∑
l∈N

 ℏ̂
(∑l

z=0 f
2
zqzδd̂z, 0̂

)
flfl+1

tl

≤ sup
l

|δ|tl
∑
l∈N

 ℏ̂
(∑l

z=0 f
2
zqzd̂z, 0̂

)
flfl+1

tl

= C1∥d̂∥p−qN < ∞.

Hence, δd̂ ∈ γSf (q, t). Hence γSf (q, t) is a linear space. Also

∑
l∈N

 ℏ̂
(∑l

z=0 f
2
zqz (̂eb)z, 0̂

)
flfl+1

tl

=
∞∑
l=b

(
f2bqb
flfl+1

)tl

≤ ∞
sup
l=b

(
f2bqb

)tl ∞∑
l=b

(
1

flfl+1

)tl

< ∞.

So, êb ∈ γSf (q, t), for every b ∈ N .

The conditions (2c) and (a4): Assume |d̂b| ≤ |k̂b|, for b ∈ N and |k̂| ∈ γSf (q, t). Hence

∥(|d̂|)∥p−qN =
∑
l∈N

 ℏ̂
(∑l

z=0 f
2
zqz|d̂z|, 0̂

)
flfl+1

tl

≤
∑
l∈N

 ℏ̂
(∑l

z=0 f
2
zqz|k̂z|, 0̂

)
flfl+1

tl

= ∥(|k̂|)∥p−qN < ∞,

then |d̂| ∈ γSf (q, t).

The conditions (3c) and (a5): Let (|d̂z|) ∈ γSf (q, t) and
(
f2zqz

)
z∈N ∈ D−, one can see that

∥(|d̂[ z
2
]|)∥p−qN =

∑
l∈N

 ℏ̂
(∑l

z=0 f
2
zqz|d̂[ z2 ]|, 0̂

)
flfl+1

tl

=
∑
l∈N

 ℏ̂
(∑2l

z=0 f
2
zqz|d̂[ z2 ]|, 0̂

)
f2lf2l+1

t2l

+
∑
l∈N

 ℏ̂
(∑2l+1

z=0 f2zqz|d̂[ z2 ]|, 0̂
)

f2l+1f2l+2

t2l+1

≤
∑
l∈N

 ℏ̂
(∑2l

z=0 f
2
zqz|d̂[ z2 ]|, 0̂

)
flfl+1

tl

+
∑
l∈N

 ℏ̂
(∑2l+1

z=0 f2zqz|d̂[ z2 ]|, 0̂
)

flfl+1

tl

≤
∑
l∈N

 ℏ̂
(
f22lq2l|d̂l|+

∑l
z=0

(
f22zq2z + f22z+1q2z+1

)
|d̂z|, 0̂

)
flfl+1

tl

+
∑
l∈N

 ℏ̂
(∑l

z=0

(
f22zq2z + f22z+1q2z+1

)
|d̂z|, 0̂

)
flfl+1

tl

≤ 2ℶ−1

∑
l∈N

 ℏ̂
(∑l

z=0 f
2
zqz|d̂z|, 0̂

)
flfl+1

tl

+
∑
l∈N

2ℏ̂
(∑l

z=0 f
2
zqz|d̂z|, 0̂

)
flfl+1

tl
+

∑
l∈N

2ℏ̂
(∑l

z=0 f
2
zqz|d̂z|, 0̂

)
flfl+1

tl

≤ (22ℶ−1 + 2ℶ−1 + 2ℶ)
∑
l∈N

 ℏ̂
(∑l

z=0 f
2
zqz|d̂z|, 0̂

)
flfl+1

tl

= C3∥(|d̂z|)∥p−qN < ∞,



M. M. A et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5641 8 of 20

then (|d̂[ z
2
]|) ∈ γSf (q, t).

Evidently, the conditions (a6) and (a7) can be easily shown.

Assume here and after the parts of theorem 3.3 are verified.

Theorem 3.4. (γSf (q, t))∥.∥p−qN
is a p-q.B psssf.

Proof. In view of Theorem 2.1, then (γSf (q, t))∥.∥p−qN
is a p-q.N psssf. To prove that

(γSf (q, t))∥.∥p−qN
is a p-q.B psssf, assume f̂a = (f̂a

z )z∈N is a Cs in (γSf (q, t))∥.∥p−qN
, then

for λ ∈ (0, 1), we get m0 ∈ N for every m, j ≥ m0, so

∥d̂m − d̂j∥p−qN =
∑
l∈N

 ℏ̂
(∑l

z=0 f
2
zqz

(
d̂zz − d̂jz

)
, 0̂

)
flfl+1


tl

< λℶ.

So ℏ̂
(∑l

z=0 f
2
zqz

(
d̂mz − d̂jz

)
, 0̂

)
< λ. Since (R(A), ℏ̂) is a CMs. Therefore, (d̂jz) is a Cs

in R(A), for fixed z ∈ N . Therefore, ∥d̂m − d̂0∥p−qN < λℶ, for any m ≥ m0. Obviously

from the linearity, d̂0 ∈ (γSf (q, t))∥.∥p−qN
.

4. NTK-∥.∥p−qN-C

We will explore the existence and uniqueness of the fixed point of NTK-∥.∥p−qN-C
defined on γSf (q, t) in this section.
Supposing that the conditions of theorem 3.3 are confirmed. We will use in this part the
following two equivalent p-q.Ns:

∥f̂∥p−qN =

∑
l∈N

 ℏ̂
(∑l

z=0 f
2
zqz f̂z, 0̂

)
flfl+1

tl


1
ℶ

and ∥f̂∥ℶp−qN =
∑
l∈N

 ℏ̂
(∑l

z=0 f
2
zqz f̂z, 0̂

)
flfl+1

tl

,

for every f̂ ∈ γSf (q, t).

Theorem 4.1. The p-q.N ∥f̂∥p−qN satisfies the FP.



M. M. A et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5641 9 of 20

Proof. Assume {ûd} ⊆
(
γSf (q, t)

)
∥.∥p−qN

with limd→∞ ∥ûd−û∥p−qN = 0.As
(
γSf (q, t)

)
∥.∥p−qN

is a p-q.B , then û ∈
(
γSf (q, t)

)
∥.∥p−qN

. Hence

∥f̂ − û∥p−qN =

∑
l∈N

 ℏ̂
(∑l

z=0 f
2
zqz(f̂z − ûz), 0̂

)
flfl+1

tl


1
ℶ

≤

∑
l∈N

 ℏ̂
(∑l

z=0 f
2
zqz(f̂z − ûdz), 0̂

)
flfl+1

tl


1
ℶ

+

∑
l∈N

 ℏ̂
(∑l

z=0 f
2
zqz(û

d
z − ûz), 0̂

)
flfl+1

tl


1
ℶ

≤ sup
v

inf
d≥v

∥f̂ − ûd∥p−qN .

Theorem 4.2. The p-q.N ∥f̂∥ℶp−qN does not hold the FP under t0 > 1.

Proof. When {ûd} ⊆
(
γSf (q, t)

)
∥.∥ℶp−qN

with limd→∞ ∥ûd−û∥ℶp−qN = 0.As
(
γSf (q, t)

)
∥.∥ℶp−qN

is a p-q.B , then û ∈
(
γSf (q, t)

)
∥.∥ℶp−qN

. So

∥f̂ − û∥ℶp−qN =
∑
l∈N

 ℏ̂
(∑l

z=0 f
2
zqz(f̂z − ûz), 0̂

)
flfl+1

tl

≤ 2ℶ−1

∑
l∈N

 ℏ̂
(∑l

z=0 f
2
zqz(f̂z − ûdz), 0̂

)
flfl+1

tl

+
∑
l∈N

 ℏ̂
(∑l

z=0 f
2
zqz(û

d
z − ûz), 0̂

)
flfl+1

tl


≤ 2ℶ−1 sup
j

inf
d≥j

∥f̂ − ûd∥ℶp−qN .

Definition 7. A mapping M : ES
∥.∥p−qN

→ ES
∥.∥p−qN

is said to be a NTK-∥.∥p−qN-C if

there are {αi}3i=1 ⊂ [0, 1) with α1 + α2 + α3 ∈ [0, 1) so that

∥Mû−Mk̂∥p−qN ≤ α1∥Mû− û∥p−qN + α2∥Mk̂ − k̂∥p−qN + α3∥û− k̂∥p−qN

for any û, k̂ ∈ ES
∥.∥p−qN

.

If M(û) = û, we say û ∈ ES
∥.∥p−qN

is a fixed point of M .

Theorem 4.3. Assume W :
(
γSf (q, t)

)
∥.∥p−qN

→
(
γSf (q, t)

)
∥.∥p−qN

is NTK-∥.∥p−qN-C,

then W has a ufp.



M. M. A et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5641 10 of 20

Proof. When d̂ ∈ γSf (q, t), then Wmd̂ ∈ γSf (q, t). Since W is a NTK-∥.∥p−qN-C , then

∥Wm+1d̂−Wmd̂∥p−qN ≤ α1∥Wm+1d̂−Wmd̂∥p−qN + α2∥Wmd̂−Wm−1d̂∥p−qN + α3∥Wmd̂−Wm−1d̂∥p−qN ⇒

∥Wm+1d̂−Wmd̂∥p−qN ≤ α2 + α3

1− α1
∥Wmd̂−Wm−1d̂∥p−qN ≤

(
α2 + α3

1− α1

)2

∥Wm−1d̂−Wm−2d̂∥p−qN ≤ . . .

≤
(
α2 + α3

1− α1

)m

∥Wd̂− d̂∥p−qN .

For every m,n ∈ N so that n > m, we obtain

∥Wmd̂−Wnd̂∥p−qN ≤ α1∥Wmd̂−Wm−1d̂∥p−qN + α2∥Wnd̂−Wn−1d̂∥p−qN + α3∥Wm−1d̂−Wn−1d̂∥p−qN

≤

(
α1

(
α2 + α3

1− α1

)m−1

+ α2

(
α2 + α3

1− α1

)n−1
)
∥Wd̂− d̂∥p−qN + α3∥Wm−1d̂−Wn−1d̂∥p−qN .

Hence, {Wmd̂} is a Cs in
(
γSf (q, t)

)
∥.∥p−qN

. Since
(
γSf (q, t)

)
∥.∥p−qN

is p-q.B . One has

v̂ ∈
(
γSf (q, t)

)
∥.∥p−qN

with limm→∞Wmd̂ = v̂. As ∥.∥p−qN has the FP , so

∥Wv̂ − v̂∥p−qN ≤ sup
i

inf
m≥i

∥Wm+1d̂−Wmd̂∥p−qN ≤ sup
i

inf
m≥i

(
α2 + α3

1− α1

)m

∥Wd̂− d̂∥p−qN = 0,

then Wv̂ = v̂. Hence, v̂ is a fp of W . Next, when we have two fp â, v̂ ∈
(
γSf (q, t)

)
∥.∥p−qN

of W with â ̸= v̂. Then

(1− α3)∥â− v̂∥p−qN ≤ α1∥Wâ− â∥p−qN + α2∥Wv̂ − v̂∥p−qN = 0.

So â = v̂.

Corollary 4.4. Suppose W :
(
γSf (q, t)

)
∥.∥p−qN

→
(
γSf (q, t)

)
∥.∥p−qN

is NTK-∥.∥p−qN-C,

then W has a ufp â under ∥Wmd̂−â∥p−qN ≤ α1

(
α2+α3
1−α1

)m−1
∥Wd̂−d̂∥p−qN+α3∥Wm−1d̂−

â∥p−qN .

Proof. By Theorem 4.3, we have a ufp â of W . Hence

∥Wmd̂− â∥p−qN = ∥Wmd̂−Wâ∥p−qN ≤ α1∥Wmd̂−Wm−1d̂∥p−qN + α2∥Wâ− â∥p−qN + α3∥Wm−1d̂− â∥p−qN

= α1

(
α2 + α3

1− α1

)m−1

∥Wd̂− d̂∥p−qN + α3∥Wm−1d̂− â∥p−qN .



M. M. A et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5641 11 of 20

Theorem 4.5. Assuming that W :
(
γSf (q, t)

)
∥.∥ℶp−qN

→
(
γSf (q, t)

)
∥.∥ℶp−qN

, where

∥d̂∥ℶp−qN =
∑
l∈N

 ℏ̂
(∑l

z=0 f
2
zqzd̂z, 0̂

)
flfl+1

tl

, for all d̂ ∈ γSf (q, t) under t0 > 1. If the following

conditions

(k1) W is NTK-∥.∥ℶp−qN-C,

(k2) W is ∥.∥ℶp−qN-Seq.C at v̂ ∈
(
γSf (q, t)

)
∥.∥ℶp−qN

, and

(k3) there is an element d̂ ∈
(
γSf (q, t)

)
∥.∥ℶp−qN

so that the sequence of iterates {Wmd̂} has

a subsequence {Wmi d̂} converges to v̂,

are satisfied, then the vector v̂ ∈
(
γSf (q, t)

)
∥.∥ℶp−qN

is the ufp of W .

Proof. Let Wv̂ ̸= v̂. From parts (k2) and (k3), one obtains

lim
mi→∞

∥Wmi d̂− v̂∥ℶp−qN = 0 and lim
mi→∞

∥Wmi+1d̂−Wv̂∥ℶp−qN = 0.

Since W is NTK-∥.∥ℶp−qN-C , then

0 < ∥Wv̂ − v̂∥ℶp−qN = ∥(Wv̂ −Wmi+1d̂) + (Wmi d̂− v̂) + (Wmi+1d̂−Wmi d̂)∥ℶp−qN

≤ 22ℶ−2∥Wmi+1v̂ −Wv̂∥ℶp−qN + 22ℶ−2∥Wmi v̂ − v̂∥ℶp−qN + 2ℶ−1

(
α2 + α3

1− α1

)mi

∥Wd̂− d̂∥ℶp−qN .

When mi → ∞, we get a contradiction. So, Wv̂ = v̂. For the uniqueness, assume Wv̂ = v̂

and Wâ = â, where v̂, â ∈
(
γSf (q, t)

)
∥.∥ℶp−qN

and v̂ ̸= â. Hence

∥v̂ − â∥ℶp−qN ≤ ∥Wv̂ −Wâ∥ℶp−qN ≤ α1∥Wv̂ − v̂∥ℶp−qN + α2∥Wâ− â∥ℶp−qN + α3∥v̂ − â∥ℶp−qN .

So, v̂ = â.

Example 4.6. Supposing that

Φ :

(
γSf

((
1

(l+5)f2l

)
l∈N

,
(
2l+3
l+2

)
l∈N

))
∥.∥p−qN

→
(
γSf

((
1

(l+5)f2l

)
l∈N

,
(
2l+3
l+2

)
l∈N

))
∥.∥p−qN

,

where

∥d̂∥p−qN =

√√√√√√∑
l∈N

 ℏ̂
(∑l

z=0
d̂z
z+5 , 0̂

)
flfl+1


2l+3
l+2

, with , v̂, d̂ ∈
(
γSf

((
1

(l+5)f2l

)
l∈N

,
(
2l+3
l+2

)
l∈N

))
∥.∥p−qN

and

Φ(d̂) =

{
d̂
4 , ∥d̂∥p−qN ∈ [0, 1),
d̂
5 , ∥d̂∥p−qN ∈ [1,∞).



M. M. A et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5641 12 of 20

If ∥d̂∥p−qN , ∥v̂∥p−qN ∈ [0, 1), one has

∥Φd̂− Φv̂∥p−qN = ∥ d̂
4
− v̂

4
∥p−qN ≤ 1

4
√
27

(
∥3d̂
4
∥p−qN + ∥3v̂

4
∥p−qN

)
+ 0.1∥d̂− v̂∥p−qN

=
1

4
√
27

(
∥Φd̂− d̂∥p−qN + ∥Φv̂ − v̂∥p−qN

)
+ 0.1∥d̂− v̂∥p−qN .

If ∥d̂∥p−qN , ∥v̂∥p−qN ∈ [1,∞), then

∥Φd̂− Φv̂∥p−qN = ∥ d̂
5
− v̂

5
∥p−qN ≤ 1

4
√
64

(
∥4d̂
5
∥p−qN + ∥4v̂

5
∥p−qN

)
+ 0.2∥d̂− v̂∥p−qN

=
1

4
√
64

(
∥Φd̂− d̂∥p−qN + ∥Φv̂ − v̂∥p−qN

)
+ 0.2∥d̂− v̂∥p−qN .

Suppose ∥d̂∥p−qN ∈ [0, 1) and ∥v̂∥p−qN ∈ [1,∞), then

∥Φd̂− Φv̂∥p−qN = ∥ d̂
4
− v̂

5
∥p−qN ≤ 1

4
√
27

∥3d̂
4
∥p−qN +

1
4
√
64

∥4v̂
5
∥p−qN + 0.1∥d̂− v̂∥p−qN

=
1

4
√
27

∥Φd̂− d̂∥p−qN +
1

4
√
64

∥Φv̂ − v̂∥p−qN + 0.1∥d̂− v̂∥p−qN .

Therefore, Φ is NTK-∥.∥p−qN-C. Since ∥.∥p−qN verifies the FP. By Theorem 4.3, one

obtains Φ has a ufp ϑ̂.

Assume {d̂(a)} ⊆
(
γSf

((
1

(l+5)f2l

)
l∈N

,
(
2l+3
l+2

)
l∈N

))
∥.∥p−qN

under lima→∞ ∥d̂(a)−d̂(0)∥p−qN =

0, where

d̂(0) ∈
(
γSf

((
1

(l+5)f2l

)
l∈N

,
(
2l+3
l+2

)
l∈N

))
∥.∥p−qN

and ∥d̂(0)∥p−qN = 1. As the p-q.N ∥.∥p−qN

is continuous, one gets

lim
a→∞

∥Φd̂(a) − Φd̂(0)∥p−qN = lim
a→∞

∥ d̂
(a)

4
− d̂(0)

5
∥p−qN = ∥ d̂

(0)

20
∥p−qN > 0.

Hence, Φ is not ∥.∥p−qN-Seq.C at d̂(0). Therefore, Φ is not continuous at d̂(0).

Assume ∥d̂∥2p−qN =
∑
l∈N

 ℏ̂
(∑l

z=0
d̂z
z+5 , 0̂

)
flfl+1


2l+3
l+2

, so that d̂, v̂ ∈
(
γSf

((
1

(l+5)f2l

)
l∈N

,
(
2l+3
l+2

)
l∈N

))
∥.∥2p−qN

.

For ∥d̂∥2p−qN , ∥v̂∥2p−qN ∈ [0, 1), one gets

∥Φd̂− Φv̂∥2p−qN = ∥ d̂
4
− v̂

4
∥2p−qN ≤ 2√

27

(
∥3d̂
4
∥2p−qN + ∥3v̂

4
∥2p−qN

)
+ 0.05∥d̂− v̂∥2p−qN

=
2√
27

(
∥Φd̂− d̂∥2p−qN + ∥Φv̂ − v̂∥2p−qN

)
+ 0.05∥d̂− v̂∥2p−qN .



M. M. A et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5641 13 of 20

If ∥d̂∥2p−qN , ∥v̂∥2p−qN ∈ [1,∞), then

∥Φd̂− Φv̂∥2p−qN = ∥ d̂
5
− v̂

5
∥2p−qN ≤ 1

4

(
∥4d̂
5
∥2p−qN + ∥4v̂

5
∥2p−qN

)
+ 0.01∥d̂− v̂∥2p−qN

=
1

4

(
∥Φd̂− d̂∥2p−qN + ∥Φv̂ − v̂∥2p−qN

)
+ 0.01∥d̂− v̂∥2p−qN .

When ∥d̂∥2p−qN ∈ [0, 1) and ∥v̂∥2p−qN ∈ [1,∞), one obtains

∥Φd̂− Φv̂∥2p−qN = ∥ d̂
4
− v̂

5
∥2p−qN ≤ 2√

27
∥3d̂
4
∥2p−qN +

1

4
∥4v̂
5
∥2p−qN + 0.01∥d̂− v̂∥2p−qN

=
2√
27

∥Φd̂− d̂∥2p−qN +
1

4
∥Φv̂ − v̂∥2p−qN + 0.01∥d̂− v̂∥2p−qN .

Therefore, Φ is NTK-∥.∥2p−qN-C and Φm(d̂) =

{
d̂
4m , ∥d̂∥2p−qN ∈ [0, 1),
d̂
5m , ∥d̂∥2p−qN ∈ [1,∞).

Clearly, Φ is ∥.∥2p−qN-Seq.C at ϑ̂ ∈
(
γSf

((
1

(l+5)f2l

)
l∈N

,
(
2l+3
l+2

)
l∈N

))
∥.∥2p−qN

and {Φmd̂}

includes a {Φmj d̂} converges to ϑ̂. According Theorem 4.5, the element ϑ̂ is the ufp of Φ.

Example 4.7. Assuming that

Φ :

(
γSf

((
1

(l+5)f2l

)
l∈N

,
(
2l+3
l+2

)
l∈N

))
∥.∥2p−qN

→
(
γSf

((
1

(l+5)f2l

)
l∈N

,
(
2l+3
l+2

)
l∈N

))
∥.∥2p−qN

,

where

∥d̂∥2p−qN =
∑
l∈N

 ℏ̂
(∑l

z=0
d̂z
z+5 , 0̂

)
flfl+1


2l+3
l+2

, so that d̂ ∈
(
γSf

((
1

(l+5)f2l

)
l∈N

,
(
2l+3
l+2

)
l∈N

))
∥.∥2p−qN

and for all t ∈ A,

Φ(d̂) =


1
4(ê1 + d̂), d̂0(t) ∈ [0, 13),
1
3 ê1, d̂0(t) =

1
3 ,

1
4 ê1, d̂0(t) ∈ (13 , 1].

If f̂ , ĝ ∈
(
γSf

((
1

(l+5)f2l

)∞
l=0

,
(
2l+3
l+2

)∞
l=0

))
∥.∥2p−qN

with f̂0(t), ĝ0(t) ∈ [0, 13), then

∥Φf̂ − Φĝ∥2p−qN = ∥1
4
(f̂0 − ĝ0, f̂1 − ĝ1, f̂2 − ĝ2, . . .)∥2p−qN

≤ 2√
27

(
∥3f̂
4
∥2p−qN + ∥3ĝ

4
∥2p−qN

)
+ 0.03∥f̂ − ĝ∥2p−qN

≤ 2√
27

(
∥Φf̂ − f̂∥2p−qN + ∥Φĝ − ĝ∥2p−qN

)
+ 0.03∥f̂ − ĝ∥2p−qN .



M. M. A et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5641 14 of 20

For every f̂ , ĝ ∈
(
γSf

((
1

(l+5)f2l

)∞
l=0

,
(
2l+3
l+2

)∞
l=0

))
∥.∥2p−qN

under f̂0(t), ĝ0(t) ∈ (13 , 1], hence

for all εi > 0 and i = 1, 2, and 3, we have

∥Φf̂ − Φĝ∥2p−qN = 0 ≤ ε1∥Φf̂ − f̂∥2p−qN + ε2∥Φĝ − ĝ∥2p−qN + ε3∥f̂ − ĝ∥2p−qN .

If f̂ , ĝ ∈
(
γSf

((
1

(l+5)f2l

)∞
l=0

,
(
2l+3
l+2

)∞
l=0

))
∥.∥2p−qN

with f̂0(t) ∈ [0, 13) and ĝ0(t) ∈ (13 , 1], then

∥Φf̂ − Φĝ∥2p−qN = ∥ f̂
4
∥2p−qN ≤ 1√

27
∥3f̂
4
∥2p−qN =

1√
27

∥Φf̂ − f̂∥2p−qN + 0.2∥f̂ − ĝ∥2p−qN

≤ 1√
27

(
∥Φf̂ − f̂∥2p−qN + ∥Φĝ − ĝ∥2p−qN

)
+ 0.2∥f̂ − ĝ∥2p−qN .

Hence, Φ is NTK-∥.∥2p−qN-C, ∥.∥2p−qN-Seq.C at 1
3 ê1 ∈

(
γSf

((
1

(l+5)f2l

)
l∈N

,
(
2l+3
l+2

)
l∈N

))
∥.∥2p−qN

,

discontinuous at 1
3 ê1, and we have d̂ so that d̂0 ∈ [0, 13) under {Φ

md̂} =
{∑m

a=1
1
4a ê1+

1
4m d̂

}
has a {Φmj d̂} =

{∑mj

a=1
1
4a ê1+

1
4mj d̂

}
converges to 1

3 ê1. By Theorem 4.5, Φ has ufp at 1
3 ê1.

For Φ :

(
γSf

((
1

(l+5)f2l

)
l∈N

,
(
2l+3
l+2

)
l∈N

))
∥.∥p−qN

→
(
γSf

((
1

(l+5)f2l

)
l∈N

,
(
2l+3
l+2

)
l∈N

))
∥.∥p−qN

,

where ∥d̂∥p−qN =

√√√√√√∑
l∈N

 ℏ̂
(∑l

z=0
d̂z
z+5 , 0̂

)
flfl+1


2l+3
l+2

, for all d̂ ∈
(
γSf

((
1

(l+5)f2l

)
l∈N

,
(
2l+3
l+2

)
l∈N

))
∥.∥p−qN

.

If f̂ , ĝ ∈
(
γSf

((
1

(l+5)f2l

)∞
l=0

,
(
2l+3
l+2

)∞
l=0

))
∥.∥p−qN

with f̂0(t), ĝ0(t) ∈ [0, 13), then

∥Φf̂ − Φĝ∥p−qN = ∥1
4
(f̂0 − ĝ0, f̂1 − ĝ1, f̂2 − ĝ2, . . .)∥p−qN ≤ 1

4
√
27

(
∥3f̂
4
∥p−qN + ∥3ĝ

4
∥p−qN

)
+ 0.01∥f̂ − ĝ∥p−qN

≤ 1
4
√
27

(
∥Φf̂ − f̂∥p−qN + ∥Φĝ − ĝ∥p−qN

)
+ 0.01∥f̂ − ĝ∥p−qN .

Suppose f̂ , ĝ ∈
(
γSf

((
1

(l+5)f2l

)∞
l=0

,
(
2l+3
l+2

)∞
l=0

))
∥.∥p−qN

with f̂0(t), ĝ0(t) ∈ (13 , 1], hence for

all εi > 0 and i = 1, 2, and 3, then

∥Φf̂ − Φĝ∥p−qN = 0 ≤ ε1∥Φf̂ − f̂∥p−qN + ε2∥Φĝ − ĝ∥p−qN + ε3∥f̂ − ĝ∥p−qN .

If f̂ , ĝ ∈
(
γSf

((
1

(l+5)f2l

)∞
l=0

,
(
2l+3
l+2

)∞
l=0

))
∥.∥p−qN

with f̂0(t) ∈ [0, 13) and ĝ0(t) ∈ (13 , 1], one

obtains

∥Φf̂ − Φĝ∥p−qN = ∥ f̂
4
∥p−qN ≤ 1

4
√
27

∥3f̂
4
∥p−qN =

1
4
√
27

∥Φf̂ − f̂∥p−qN

≤ 1
4
√
27

(
∥Φf̂ − f̂∥p−qN + ∥Φĝ − ĝ∥p−qN

)
+ 0.01∥f̂ − ĝ∥p−qN .



M. M. A et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5641 15 of 20

Therefore, Φ is NTK-∥.∥p−qN-C. As ∥.∥p−qN verifies the FP. By Theorem 4.3, Φ has a
ufp 1

3 ê1.

5. Applications

Understanding economic models requires a firm grasp of summable equations, which of-
fer a mathematical basis for investigating issues like producer and consumer surplus, total
cost computation, and revenue functions, among others. The use of summable equations
in the formulation and solution of economic problems has recently expanded substantially.
Take a look at [2, 8, 10, 18, 21] and the citations that follow.

We present the existence and uniqueness of the soft dynamical systems (1) in
(
γSf (q, t)

)
∥.∥p−qN

,

where the conditions of theorem 3.3 are confirmed under the two equivalent p-q.Ns
∥ν̂∥p−qN and ∥ν̂∥ℶp−qN , for any ν̂ ∈ γSf (q, t).

Theorem 5.1. Assume that ξ̂ : N → R(A). The dynamical systems (1) have a unique

solution in
(
γSf (q, t)

)
∥.∥p−qN

whenever if there are εi ∈ R so that
∑3

i=1 supu |εi|
tu
ℶ ∈ [0, 1)

and for every u ∈ N , then∣∣∣∣∣
u∑

d=0

(∑
v∈N

Γd,v[Ψv,ν̂v −Ψ
v,ξ̂v

]

)
f2dqd

∣∣∣∣∣ ≤̂|ε1|

∣∣∣∣∣
u∑

d=0

(
β̂d − ν̂d +

∑
v∈N

Γd,vΨv,ν̂v

)
f2dqd

∣∣∣∣∣
+ |ε2|

∣∣∣∣∣
u∑

d=0

(
β̂d − ξ̂d +

∑
v∈N

Γd,vΨv,ξ̂v

)
f2dqd

∣∣∣∣∣+ |ε3||
u∑

d=0

(
ν̂d − ξ̂d

)
f2dqd|.

Proof. Let Φ :
(
γSf (q, t)

)
∥.∥p−qN

→
(
γSf (q, t)

)
∥.∥p−qN

is defined by equation (2). By

Theorem 4.3 and

∥Φν̂ − Φξ̂∥p−qN =

∑
u∈N

 ℏ̂
(∑u

d=0 f
2
dqd(Φν̂d − Φξ̂d), 0̂

)
fufu+1

tu


1
ℶ

=

∑
u∈N

 ℏ̂
(∑u

d=0

(∑
v∈N Γd,v[Ψv,ν̂v −Ψ

v,ξ̂v
]
)
f2dqd, 0̂

)
fufu+1

tu


1
ℶ



M. M. A et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5641 16 of 20

≤ sup
u

|ε1|
tu
ℶ

∑
u∈N

 ℏ̂
(∑u

d=0

(
β̂d − ν̂d +

∑
v∈N Γd,vΨv,ν̂v

)
f2dqd, 0̂

)
fufu+1

tu


1
ℶ

+ sup
u

|ε2|
tu
ℶ

∑
u∈N

 ℏ̂
(∑u

d=0

(
β̂d − ξ̂d +

∑
v∈N Γd,vΨv,ξ̂v

)
f2dqd, 0̂

)
fufu+1

tu


1
ℶ

+ sup
u

|ε3|
tu
ℶ

∑
u∈N

 ℏ̂
(∑u

d=0

(
ν̂d − ξ̂d

)
f2dqd, 0̂

)
fufu+1

tu


1
ℶ

= sup
u

|ε1|
tu
ℶ ∥Φν̂ − ν̂∥p−qN + sup

u
|ε2|

tu
ℶ ∥Φξ̂ − ξ̂∥p−qN + sup

u
|ε3|

tu
ℶ ∥ν̂ − ξ̂∥p−qN .

we have a unique solution of the dynamical systems (1) in
(
γSf (q, t)

)
∥.∥p−qN

.

Example 5.2. Supposing that
(
γSf

((
1

(u+1)f2u

)
u∈N

,
(
2u+3
u+2

)
u∈N

))
∥.∥p−qN

, where

∥ν̂∥p−qN =

√√√√√√∑
u∈N

 ℏ̂
(∑u

d=0
ν̂d
d+1 , 0̂

)
fufu+1


2u+3
u+2

, for all ν̂ ∈
(
γSf

((
1

(u+1)f2u

)
u∈N

,
(
2u+3
u+2

)
u∈N

))
∥.∥p−qN

.

Assume that the NLDEs:

ν̂d = ̂log2(d
4 + 1) +

∑
v∈N

cosh d cos2 v
ν̂xd−2

ν̂yd−1 +
̂tanh(2v + 3)

, (4)

for every x, y, ν̂−2(t), ν̂−1(t) > 0, under t ∈ A.
Let the mapping Φ be defined as

Φ :

(
γSf

((
1

(u+ 1)f2u

)
u∈N

,

(
2u+ 3

u+ 2

)
u∈N

))
∥.∥p−qN

→
(
γSf

((
1

(u+ 1)f2u

)
u∈N

,

(
2u+ 3

u+ 2

)
u∈N

))
∥.∥p−qN

,

where

Φ(ν̂d)d∈N =
(

̂log2(d
4 + 1) +

∑
v∈N

cosh d cos2 v
ν̂xd−2

ν̂yd−1 +
̂tanh(2v + 3)

)
d∈N

. (5)



M. M. A et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5641 17 of 20

Obviously, we have εi ∈ R with
∑3

i=1 supu |εi|
2u+3
2u+4 ∈ [0, 1) and for any u ∈ N , hence∣∣∣∣∣

u∑
d=0

(∑
v∈N

cosh d ν̂xd−2

ν̂yd−1 +
̂tanh(2v + 3)

(
cos2 v − cos2 v

))
f2dqd

∣∣∣∣∣
≤̂|ε1|

∣∣∣∣∣
u∑

d=0

(
̂log2(d

4 + 1)− ν̂d +
∑
v∈N

cosh d cos2 v
ν̂xd−2

ν̂yd−1 +
̂tanh(2v + 3)

)
f2dqd

∣∣∣∣∣+
|ε2|

∣∣∣∣∣
u∑

d=0

(
̂log2(d

4 + 1)− η̂d +
∑
v∈N

cosh d cos2 v
η̂xd−2

η̂yd−1 +
̂tanh(2v + 3)

)
f2dqd

∣∣∣∣∣+ |ε3||
u∑

d=0

(ν̂d − η̂d) f
2
dqd|.

By Theorem 5.1, the NLDEs (4) include a unique solution in
(
γSf

((
1

(u+1)f2u

)
u∈N

,
(
2u+3
u+2

)
u∈N

))
∥.∥p−qN

.

Theorem 5.3. Assume that Φ :
(
γSf (q, t)

)
∥.∥ℶp−qN

→
(
γSf (q, t)

)
∥.∥ℶp−qN

is defined by (2).

The dynamical systems (1) have a unique solution Ẑ ∈
(
γSf (q, t)

)
∥.∥ℶp−qN

, when the follow-

ing conditions are satisfied:

(k1) Suppose Γ : N 2 → R, Ψ : N × R(A) → R(A), ν̂ : N → R(A), β̂ : N → R(A),
ξ̂ : N → R(A), if one has εi ∈ R under 22ℶ−2

∑3
i=1 supu |εi|tu ∈ [0, 1) and for every

u ∈ N , one gets∣∣∣∣∣
u∑

d=0

(∑
v∈N

Γd,v[Ψv,ν̂v −Ψ
v,ξ̂v

]

)
f2dqd

∣∣∣∣∣ ≤̂|ε1|

∣∣∣∣∣
u∑

d=0

(
β̂d − ν̂d +

∑
v∈N

Γd,vΨv,ν̂v

)
f2dqd

∣∣∣∣∣
+ |ε2|

∣∣∣∣∣
u∑

d=0

(
β̂d − ξ̂d +

∑
v∈N

Γd,vΨv,ξ̂v

)
f2dqd

∣∣∣∣∣+ |ε3||
u∑

d=0

(
ν̂d − ξ̂d

)
f2dqd|.

(k2) Φ is ∥.∥ℶp−qN-Seq.C at Ẑ ∈
(
γSf (q, t)

)
∥.∥ℶp−qN

,

(k3) there is Ŷ ∈
(
γSf (q, t)

)
∥.∥ℶp−qN

with {ΦmŶ } has {Φmj Ŷ } converging to Ẑ.

Proof. By Theorem 4.5 and

∥Φν̂ − Φξ̂∥ℶp−qN =
∑
u∈N

 ℏ̂
(∑u

d=0 f
2
dqd(Φν̂d − Φξ̂d), 0̂

)
fufu+1

tu

=
∑
u∈N

 ℏ̂
(∑u

d=0

(∑
v∈N Γd,v[Ψv,ν̂v −Ψ

v,ξ̂v
]
)
f2dqd, 0̂

)
fufu+1

tu



M. M. A et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5641 18 of 20

≤ 22ℶ−2 sup
u

|ε1|tu
∑
u∈N

 ℏ̂
(∑u

d=0

(
β̂d − ν̂d +

∑
v∈N Γd,vΨv,ν̂v

)
f2dqd, 0̂

)
fufu+1

tu

+

22ℶ−2 sup
u

|ε2|tu
∑
u∈N

 ℏ̂
(∑u

d=0

(
β̂d − ξ̂d +

∑
v∈N Γd,vΨv,ξ̂v

)
f2dqd, 0̂

)
fufu+1

tu

+

2ℶ−1 sup
u

|ε3|tu
∑
u∈N

 ℏ̂
(∑u

d=0

(
ν̂d − ξ̂d

)
f2dqd, 0̂

)
fufu+1

tu

≤ 22ℶ−2

(
sup
u

|ε1|tu∥Φν̂ − ν̂∥ℶp−qN + sup
u

|ε2|tu∥Φξ̂ − ξ̂∥ℶp−qN + sup
u

|ε3|tu∥ν̂ − ξ̂∥ℶp−qN

)
.

That gives the required.

Example 5.4. Supposing that
(
γSf

((
1

(u+1)f2u

)
u∈N

,
(
2u+3
u+2

)
u∈N

))
∥.∥2p−qN

, where

∥ν̂∥2p−qN =
∑
u∈N

 ℏ̂
(∑u

d=0
ν̂d
d+1 , 0̂

)
fufu+1


2u+3
u+2

, for all ν̂ ∈
(
γSf

((
1

(u+1)f2u

)
u∈N

,
(
2u+3
u+2

)
u∈N

))
∥.∥2p−qN

.

Consider the dynamical system (4) and the mapping

Φ :
(
γSf

((
1

(u+1)f2u

)
u∈N

,
(
2u+3
u+2

)
u∈N

))
∥.∥2p−qN

→
(
γSf

((
1

(u+1)f2u

)
u∈N

,
(
2u+3
u+2

)
u∈N

))
∥.∥2p−qN

as defined by (5). Assume Φ is ∥.∥2p−qN-Seq.C at Ẑ ∈
(
γSf

((
1

(u+1)f2u

)
u∈N

,
(
2u+3
u+2

)
u∈N

))
∥.∥2p−qN

,

and one has Ŷ ∈
(
γSf

((
1

(u+1)f2u

)
u∈N

,
(
2u+3
u+2

)
u∈N

))
∥.∥2p−qN

with {ΦkŶ } has {Φkr Ŷ } con-

verging to Ẑ. Clearly, there are εi ∈ R such that 4
∑3

i=1 supu |εi|
2u+3
u+2 ∈ [0, 1) and for any

u ∈ N , hence∣∣∣∣∣
u∑

d=0

(∑
v∈N

cosh d ν̂xd−2

ν̂yd−1 +
̂tanh(2v + 3)

(
cos2 v − cos2 v

))
f2dqd

∣∣∣∣∣
≤̂|ε1|

∣∣∣∣∣
u∑

d=0

(
̂log2(d

4 + 1)− ν̂d +
∑
v∈N

cosh d cos2 v
ν̂xd−2

ν̂yd−1 +
̂tanh(2v + 3)

)
f2dqd

∣∣∣∣∣+
|ε2|

∣∣∣∣∣
u∑

d=0

(
̂log2(d

4 + 1)− η̂d +
∑
v∈N

cosh d cos2 v
η̂xd−2

η̂yd−1 +
̂tanh(2v + 3)

)
f2dqd

∣∣∣∣∣+ |ε3||
u∑

d=0

(ν̂d − η̂d) f
2
dqd|.

In view of Theorem 5.3, the dynamical systems (4) have a unique solution

Ẑ ∈
(
γSf

((
1

(u+1)f2u

)
u∈N

,
(
2u+3
u+2

)
u∈N

))
∥.∥2p−qN

.



M. M. A et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5641 19 of 20

6. Conclusion

In this article, we discussed several topological and geometric characteristics of(
γSf (q, t)

)
∥.∥p−qN

. Analyzed is the novel Kannan contraction operator in this space, along

with the potential for a fixed point. We conducted numerous numerical experiments
to validate our theories. Investigations are conducted on soft functions with non-linear
uncertainty equation implementations. Future work uses the innovative soft function space
to analyze the fixed points of the new type of Kannan contraction operator, providing a
new universal solution space for a variety of stochastic non-linear dynamical systems.

Acknowledgements

This work was funded by the University of Jeddah, Jeddah, Saudi Arabia, under
grant No. (UJ-23-DR-33). Therefore, the authors thank the University of Jeddah for its
technical and financial support.

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