# Decision-Making on the Solution of a Stochastic Nonlinear Dynamical System of Kannan-Type in New Sequence Space of Soft Functions

Authors: Meshayil M. Alsolmi; Awad A. Bakery
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we construct and investigate the space of weighted Gamma matrix of orderr in Nakano sequence space of soft functions. The idealization of the mappings has been achieved through the use of extended s-soft functions and this sequence space of soft functions. This new space’s topological and geometric properties, the multiplication mappings that stand in on it, and the mappings’ ideal that correspond to them are discussed. We construct the existence of a fixed point of Kannan contraction mapping acting on this space and its associated prequasi ideal. Interestingly, several numerical experiments are presented to illustrate our results. Additionally, some successful applications to the existence of solutions of nonlinear difference equations of soft functions are introduced.


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## Body

## 1. Introduction

Probability theory, fuzzy set theory, soft sets, and rough sets have all contributed substantially to the study of uncertainty. But there are drawbacks to these theories that must be considered. For more information and real-world examples, please refer to [1–10]. Numerous mathematicians have investigated potential expansions to the theorem and its applications in various contexts since the publication of the book [11] on the Banach fixed point theorem. The Banach contraction principle is an important part of nonlinear analysis, which uses it as a powerful tool [12–15]. Kannan [16] presented a collection of mappings with the same actions at fixed places as contractions. However, this collection is discontinuous. In Reference [17], an explanation of Kannan operators in modular vector spaces was once tried. Only this one try was ever made as [18–23] show that much attention has been paid to the s-number mapping ideal and the multiplication operator hypothesis in functional analysis. Bakery and Mohamed [24] offered the idea of a prequasi norm on the Nakano sequence space with a variable exponent that fell somewhere in the range 0,1. They talked about the conditions that must be met to generate prequasi Banach and closed space when it is endowed with a specified prequasi norm and the Fatou property of various prequasi norms on it. They also determined a fixed point for Kannan prequasi norm contraction mappings on it, in addition to the ideal of prequasi Banach mappings derived from s-numbers in this sequence space. Both of these ideals were established. In addition, several fixed point findings of Kannan nonexpansive mappings on generalized Cesàro backward difference sequence space of a nonabsolute type were discovered in [25]. Assume that R is the set of real numbers and N is the set of nonnegative integers. We denote the collection of all nonempty bounded subsets of R by BR, and E is the set of parameters. By RA∗ and RA, we indicate the set of nonnegative and all soft real numbers (corresponding to A), where A⊂E. The additive identity and multiplicative identity in RA are denoted by 0~ and 1~, respectively. For more details on the arithmetic operations on RA, see [26]. Let μ:RA×RA⟶RA∗, where μf~,g~=f~−g~, for all f~,g~∈RA. Assume ρ~:RA×RA⟶R+ is defined by
(1)ρ~f~,g~=maxλ∈Aμf~,g~λ.Given that the proof of many fixed point theorems in a given space requires either growing the space itself or expanding the self-mapping that acts on it, both of these options are viable; we have constructed the space,ΓrSq,vτ, which is the domain of weighted Gamma matrix of order r in Nakano soft sequence space since it is constructed by the domain of weighted Gamma matrix of order r defined in ℓvlS, where the weighted Gamma matrix of order r, WΓr=λlzrq, is defined as
(2)λlzrq=r+z−1zqzr+ll,0≤z≤l,0,z>l,where r is a positive integer, qz∈0,∞, for all z∈N and
(3)r+z−1z=r+z−1!z!r−1!.In [27], Roopaei and Basar studied the Gamma spaces, including the spaces of absolutely p-summable, null, convergent, and bounded sequences.In this article, we have introduced a new general space calledΓrSq,vτ and the mappings’ ideal space of solutions for many stochastic nonlinear and matrix systems of Kannan contraction type. We have offered some geometric and topological structures of the soft function space, ΓrSq,vτ, multiplication operator acting on it, and its operators’ ideal. A fixed point of the Kannan contraction operator exists in this space, and its prequasi operator ideal is confirmed. Finally, we discuss many applications of solutions to nonlinear stochastic dynamical systems and illustrative examples of our findings.


## 2. Properties ofΓrSq,vτ and Its Operators’ Ideal

Some geometric and topological structures of the soft function space,ΓrSq,vτ, and its operators’ ideals are presented in this section.Byc0, ℓ∞, and ℓr, we denote the space of null, bounded, and r-absolutely summable sequences of reals. We indicate the space of all bounded, finite rank linear mappings from an infinite-dimensional Banach space G into an infinite-dimensional Banach space V by DG,V and FG,V, and if G=V, we write DG and FG. The space of approximable and compact bounded linear operators from G into V will be marked by AG,V and KG,V, respectively. The ideal of bounded, approximable, and compact mappings between every two infinite-dimensional Banach spaces will be denoted by D, A, and K, respectively. Suppose ωS is the class of all sequence spaces of soft reals.Definition 1.
Ifvl∈R+N, R+N is the space of all sequences of positive reals. The sequence space ΓrSq,vτ with the function τ is defined by
(4)ΓrSq,vτ=h~=hm~∈ωS:τδh~<∞,forsomeε>0,whereτh~=∑m=0∞ρ~∑z=0mz+r−1zqzhz~,0~r+mmvm.Lemma 2 (see [28]).
Ifvb>0 and xb,zb∈R, for all b∈N, and ℏ=max1,supbvb, then
(5)xb+zbvb≤2ℏ−1xbvb+zbvb.Theorem 3.
Supposevl∈ℓ∞∩R+N, then
(6)ΓrSq,vτ=h~=hb~∈ωS:τδh~<∞,forallδ>0.Proof.
Obviously,vl is a bounded sequence.Theorem 4.
The spaceΓrSq,vτ is a nonabsolute type, whenever vl∈1,∞N∩ℓ∞.Proof.
Clearly, since(7)τ1~,−1~,0~,0~,0~,⋯=q0v0+q0−rq11+rv1+q0−rq1r+22v2+⋯≠q0v0+q0+rq11+rv1+q0+rq1r+22v2+⋯=τ1~,1~,0~,0~,0~,⋯.Definition 5.
Assumevb∈R+N and vb≥1, for all b∈N:
(8)ΓrSq,vφ≔h~=hb~∈ωS:φδf<∞,forsomeδ>0,where
(9)φh~=∑b=0∞ρ~∑z=0bz+r−1zqzhz~,0~r+bbvb.Theorem 6.
Supposevl∈1,∞N∩ℓ∞ with
(10)l+1r+ll∉ℓvl,hence ΓrSq,vφ⊊ΓrSq,vτ.Proof.
Assumef~∈ΓrSq,vφ, as
(11)∑b=0∞ρ~∑z=0bz+r−1zqzfz~,0~r+bbvb≤∑b=0∞ρ~∑z=0bz+r−1zqzfz~,0~r+bbvb<∞.
Thenf~∈ΓrSq,vτ. If we choose
(12)g~=−1~zz+r−1zqzz∈N,one gets g~∈ΓrSq,vτ and g~∉ΓrSq,vφ.
SupposeES is a linear space of sequences of soft functions, and p describes an integral part of the real number p.Definition 7.
The spaceES is said to be a private sequence space of soft functions psssf if it satisfies the next setups:
(a1)
For allb∈N, then eb~∈ES, where eb~=0~,0~,⋯,1~,0~,0~,⋯, while 1~ displays at the bth place(a2)
Iff~=fb~∈ωS, g~=gb~∈ES and fb~≤gb~, with b∈N, then f~∈ES(a3)
hb/2~b=0∞∈ES, whenever hb~b=0∞∈ESDefinition 8 (see [29]).
Ans-number is a function s:DG,V⟶R+N that gives all V∈DG,V a sdVd=0∞ holds the following conditions:
(1)
V=s0V≥s1V≥s2V≥⋯≥0,forallV∈DG,V(2)
sdVYW≤VsdYW, for every W∈DG0,G, Y∈DG,V and V∈DV,V0, where G0 and V0 are arbitrary Banach spaces(3)
sl+d−1V1+V2≤slV1+sdV2, for every V1,V2∈DG,V and l, d∈N(4)
AssumeV∈DG,V and γ∈R, then sdγV=γsdV(5)
IfrankV≤d, then sdV=0, for all V∈DG,V(6)
sl≥aIa=0 or sl<aIa=1, where Ia indicates the unit mapping on the a-dimensional Hilbert space ℓ2a
Some examples ofs-numbers:
(a)
Thebth approximation number is defined as αbX=infX−Y:Y∈DG,VandrankY≤b(b)
Thebth Kolmogorov number is defined as dbX=infdimJ≤bsupf≤1infg∈JXf−gNotation 9 (see [30]).
(13)Ds~ES≔Ds~ESG,V,whereDs~ESG,V≔V∈DG,V:sjV~j=0∞∈ES,Dα~ES≔Dα~ESG,V,whereDα~ESG,V≔V∈DG,V:αjV~j=0∞∈ES,Dd~ES≔Dd~ESG,V,whereDd~ESG,V≔V∈DG,V:djV~j=0∞∈ES,Ds~ESγ≔Ds~ESγG,V,whereDs~ESγG,V≔V∈DG,V:γbV~b=0∞∈ESandV−ρ~γbV~,0~I=0,forallb∈N.Theorem 10.
Assume the linear sequence spaceES is a psssf, then Ds~ES is an operator ideal.Proof.
(i)
AssumeV∈FG,V and rankV=n with n∈N, as ei~∈ES for all i∈N and ES is a linear space, one has siV~i=0∞=s0V~,s1V~,⋯,sn−1V~,0~,0~,0~,⋯=∑i=0n−1siV~ei~∈ES, for that V∈Ds~ESG,V then FG,V⊆Ds~ESG,V(ii)
SupposeV1,V2∈Ds~ESG,V and β1,β2∈R then by Definition 7 condition (iii), one has si/2V1~i=0∞∈ES and si/2V1~i=0∞∈ES, as i≥2i/2, by the definition of s~-numbers and siP~ is a decreasing sequence, we have(14)siβ1V1+β2V2~≤s2i/2β1V1+β2V2~≤si/2β1V1+si/2β2V2~=β1si/2V1~+β2si/2V2~,for each i∈N. In view of Definition 7 condition (ii) and ES is a linear space, one obtains siβ1V1+β2V2~i=0∞∈ES, then β1V1+β2V2∈Ds~ESG,V(iii)
IfP∈DG0,G, T∈Ds~ESG,V, and R∈DV,V0, one has siT~i=0∞∈ES and as siRTP~≤RsiT~P, by Definition 7 conditions (i) and (ii), one gets siRTP~i=0∞∈ES, hence RTP∈Ds~ESG0,V0
Assumeθ~=0~,0~,0~,⋯ and F is the space of finite sequences of soft numbers.Definition 11.
A subspace of thepsssf is called a premodular psssf, if there is a function τ:ES⟶0,∞ satisfies the next setups:
(i)
Ifh~∈ES, h~=θ~⇔τh~=0, and τh~≥0(ii)
Assumeh~∈ES and ε∈R, one has E0≥1 so that τεh~≤εE0τh~(iii)
There areG0≥1 so that τf~+g~≤G0τf~+τg~, for all f~,g~∈ES(iv)
Assumefb~≤gb~, for all b∈N, then τfb~≤τgb~(v)
One getsD0≥1 such that τf~≤τf·~≤D0τf~(vi)
The closure ofF=EτS(vii)
There areε>0 with τν~,0~,0~,0~,⋯≥εντ1~,0~,0~,0~,⋯Definition 12.
ThepsssfEτS is said to be a prequasi normed psssf, if τ confirms the setups (i)-(iii) of Definition 11. The space EτS is called a prequasi Banach psssf, whenever ES is complete under τ.Theorem 13.
The spaceEτS is a prequasi normed psssf, whenever it is premodular psssf. By ↑ and ↓, we denote the space of all monotonic increasing and decreasing sequences of positive reals, respectively.Theorem 14.
ΓrSq,vτ is a prequasi Banach psssf, if the next setups are confirmed:
(f1)
vl∈↑∩ℓ∞ with v0>1/r(f2)
b+r−1bqbb=0∞∈↓ or b+r−1bqbb=0∞∈↑∩ℓ∞ and there exists C≥1 such that(15)2b+r2b+1q2b+1≤Cb+r−1bqbProof.
First, we have to show thatΓrSq,vτ is a premodular psss.
(i)
Obviously,τh~=0⇔h~=θ~ and τh~≥0
(a1) and (iii) Iff~,g~∈ΓrSq,vτ, then
(16)τf~+g~=∑l=0∞ρ~∑z=0lz+r−1zqzfz~+gz~,0~r+llvl≤2ℏ−1∑l=0∞ρ~∑z=0lz+r−1zqzfz~,0~r+llvl+∑l=0∞ρ~∑z=0lz+r−1zqzgz~,0~r+llvl=2ℏ−1τf~+τg~<∞,hence f~+g~∈ΓrSq,vτ(ii)
Next, supposeλ∈R, f~∈ΓrSq,vτ and as vl∈↑∩ℓ∞, we get(17)τλf~=∑m=0∞ρ~∑z=0mz+r−1zqzλfz~,0~r+mmvm≤supmλvm∑m=0∞ρ~∑z=0mz+r−1zqzfz~,0~r+mmvm≤E0λτf~<∞,where E0=max1,suplλvl−1≥1. So, λf~∈ΓrSq,vτ.
Asvl∈↑∩ℓ∞ and v0>1/r, one obtains
(18)∑m=0∞ρ~∑z=0mz+r−1zqzebz~,0~r+mmvm=∑m=b∞b+r−1bqbr+mmvm≤supm=b∞b+r−1bqbvm∑m=b∞1r+mmvm<∞.
Therefore,eb~∈ΓrSq,vτ, for every b∈N.
(a2) and (iv) Iffm~≤gm~, for all m∈N and g~∈ΓrSq,vτ, then
(19)τf~=∑m=0∞ρ~∑z=0mz+r−1zqzfz~,0~r+mmvm≤∑m=0∞ρ~∑z=0mz+r−1zqzgz~,0~r+mmvm=τg~<∞,hence f~∈ΓrSq,vτ
(a3) and (v) Assumefz~∈ΓrSq,vτ, with vl∈↑∩ℓ∞ and
(20)z+r−1zqzz=0∞∈↓,we get
(21)τfz/2~=∑l=0∞ρ~∑z=0lz+r−1zqzfz/2~,0~r+llvl=∑l=0∞ρ~∑z=02lz+r−1zqzfz/2~,0~r+2l2lv2l+∑l=0∞ρ~∑z=02l+1z+r−1zqzfz/2~,0~r+2l+12l+1v2l+1≤∑l=0∞ρ~∑z=02lz+r−1zqzfz/2~,0~r+llvl+∑l=0∞ρ~∑z=02l+1z+r−1zqzfz/2~,0~r+llvl≤∑l=0∞ρ~2l+r−12lq2lfl~+∑z=0l2z+r−1zq2z+2z+r2z+1q2z+1fz~,0~r+llvl+∑l=0∞ρ~∑z=0l2z+r−1zq2z+2z+r2z+1q2z+1fz~,0~r+llvl≤2ℏ−1∑l=0∞ρ~∑z=0lz+r−1zqzfz~,0~r+llvl+∑l=0∞2ρ~∑z=0lz+r−1zqzfz~,0~r+llvl+∑l=0∞2ρ~∑z=0lz+r−1zqzfz~,0~r+llvl≤D0τf~<∞,where D0≥22ℏ−1+2ℏ−1+2ℏ≥1. Hence, fz/2~∈ΓrSq,vτ
(vi) It is clear that the closure ofF=ΓrSq,v
(vii)There are0<δ≤suplλvl−1 so that τλ~,0~,0~,0~,⋯≥δλτ1~,0~,0~,0~,⋯, for all λ≠0 and δ>0, if λ=0
By Theorem13, the space ΓrSq,vτ is a prequasi normed psssf. Second, to prove that ΓrSq,vτ is a Banach space, suppose hi~=hki~k=0∞ is a Cauchy sequence in ΓrSq,vτ, hence for every γ∈0,1, one has i0∈N with i,j≥i0, we have
(22)τhi~−hj~=∑l=0∞ρ~∑z=0lz+r−1zqzhzi~−fzj~,0~r+llvl<γℏ.
That implies(23)ρ~∑z=0lz+r−1zqzhzi~−hzj~,0~<γ.
AsRA,ρ~ is a complete metric space. Therefore, hkj~ is a Cauchy sequence in RA, for constant k∈N. So, it is convergent to hk0~∈RA. This implies τhi~−h0~<γℏ, for every i≥i0. Clearly, from condition (iii) that h0~∈ΓrSq,vτ.
In view of Theorems10 and 14, we have the next theorem.Theorem 15.
The spaceDs~ΓrSq,v is an operator ideal, if the conditions of Theorem 14 are verified.Theorem 16.
Ifs-type EτS≔h~=sjH~∈RN:H∈DG,Vandτh~<∞. Assume Ds~Eτ is an operator ideal, one has the next setups:
(a)
s-type EτS⊃F(b)
SupposesjH1~j=0∞∈s-type EτS and sjH2~j=0∞∈s-type EτS, then sjH1+H2~j=0∞∈s-type EτS(c)
Ifε∈R and sjH~j=0∞∈s-type EτS, one has εsjH~j=0∞∈s-type EτS(d)
SupposesjU~j=0∞∈s-type EτS and sjT~≤sjU~, for all j∈N and T,U∈DG,V, one gets sjT~j=0∞∈s-type EτS, i.e., EτS is a solid spaceProof.
IfDs~Eτ is a mappings’ ideal.
(a)We haveFG,V⊂Ds~EτSG,V. Hence, for all X∈FG,V, we have srX~r=0∞∈F. This gives srX~r=0∞∈s‐typeEτS. Hence, F⊂s-type EτS
(b) and (c)The spaceDs~EτSG,V is linear over R. Hence, for each λ∈R and X1,X2∈Ds~EτSG,V, we have X1+X2∈Ds~EτSG,VandλX1∈Ds~EτSG,V. That implies
(24)srX1~r=0∞∈s‐typeEτSandsrX2~r=0∞∈s‐typeEτS⇒srX1+X2~r=0∞∈s‐typeEτS,λ∈RandsrX1~r=0∞∈s‐typeEτS⇒λsrX1~r=0∞∈s‐typeEτS
(d)IfA∈DG0,G, B∈Ds~EτSG,V, and D∈DV,V0, then DBA∈Ds~EτSG0,V0. Therefore, since srB~r=0∞∈s‐typeEτS, then srDBA~r=0∞∈s‐typeEτS. Since srDBA~≤DsrB~A. By using condition (c), if DAsrB~r=0∞∈EτS, we have srDBA~r=0∞∈s‐typeEτS. This means s‐typeEτS is solid
Some properties ofs‐type ΓrSq,vτ are presented in the next theorem according to Theorems 16 and 15.Theorem 17.
(a)
s-type ΓrSq,vτ⊃F(b)
IfsnX1~n=0∞∈s-type ΓrSq,vτ and snX2~n=0∞∈s-type ΓrSq,vτ, then snX1+X2~n=0∞∈s-type ΓrSq,vτ(c)
Assumeλ∈R and snX~n=0∞∈s-type ΓrSq,vτ, hence λsnX~n=0∞∈s-type ΓrSq,vτ(d)
s-type ΓrSq,vτ is a solid spaceDefinition 18 (see [31]).
A subclassU of D is said to be a mappings’ ideal, if every UG,V=U∩DG,V satisfies the following setups:
(i)
IΓ∈U, where Γ indicates Banach space of one dimension(ii)
The spaceUG,V is linear over R(iii)
IfW∈DG0,G, X∈UG,V, and Y∈DV,V0, then YXW∈UG,V0Definition 19 (see [32]).
A functionH∈0,∞U is said to be a prequasi norm on the ideal U if the following conditions hold:
(1)
AssumeV∈UG,V, HV≥0 and HV=0, if and only if, V=0(2)
One hasQ≥1 with HαV≤DαHV, for all V∈UG,V and α∈R(3)
There areP≥1 such that HV1+V2≤PHV1+HV2, for all V1,V2∈UG,V(4)
There areσ≥1 so that if V∈DG0,G, X∈UG,V, and Y∈DV,V0, then HYXV≤σYHXVTheorem 20 (see [32]).
Every quasi norm on the idealU is a prequasi norm.
We have discussed some properties of the ideal constructed by our soft space and extendeds-numbers, supposing that the conditions of Theorem 14 are verified.Theorem 21.
The conditions of Theorem14 are sufficient only for Dα~ΓrSq,vτG,V=theclosureofFG,V.Proof.
Clearly,theclosureofFG,V⊆Dα~ΓrSq,vτG,V from the linearity of the space ΓrSq,vτ and em~∈ΓrSq,vτ, for all m∈N. Next, to show that Dα~ΓrSq,vτG,V⊆theclosureofFG,V. If H∈Dα~ΓrSq,vτG,V, one has αlH~m=0∞∈ΓrSq,vτ. As ταmH~m=0∞<∞, assume γ∈0,1, we have l0∈N−0 so that ταmH~m=l0∞<γ/2ℏ+3δj, for some j≥1 and
(25)δ=max1,∑l=l0∞1r+llvl.
SinceαlH~∈I↘S, we get
(26)∑l=l0+12l0ρ~∑z=0lz+r−1zqzα2l0H~,0~r+llvl≤∑l=l0+12l0ρ~∑z=0lz+r−1zqzαzH~,0~r+llvl≤∑l=l0∞ρ~∑z=0lz+r−1zqzαzH~,0~r+llvl<γ2ℏ+3δj.
We getU∈F2l0G,V with rank U≤2l0 and
(27)∑l=2l0+13l0ρ~∑z=0lz+r−1zqzH−U~,0~r+llvl≤∑l=l0+12l0ρ~∑z=0lz+r−1zqzH−U~,0~r+llvl<γ2ℏ+3δj,since vl∈↑∩ℓ∞, we have
(28)supl=l0∞ρ~vl∑z=0l0z+r−1zqzH−U~,0~<γ22ℏ+2δ.
Therefore, one has(29)∑l=0l0ρ~∑z=0lz+r−1zqzH−U~,0~r+llvl<γ2ℏ+3δj.
Because of inequalities (5), (26), (27), (28), and (29), one gets
(30)dH,U=ταlH−U~l=0∞=∑l=03l0−1ρ~∑z=0lz+r−1zqzαzH−U~,0~r+llvl+∑l=3l0∞ρ~∑z=0lz+r−1zqzαzH−U~,0~r+ll≤∑l=03l0ρ~∑z=0lz+r−1zqzH−U~,0~r+llvl+∑l=l0∞ρ~∑z=0l+2l0z+r−1zqzαzH−U~,0~r+l+2l0l+2l0vl+2l0vl≤∑l=03l0ρ~∑z=0lz+r−1zqzH−U~,0~r+llvl+∑l=l0∞ρ~∑z=0l+2l0z+r−1zqzαzH−U~,0~r+llvl≤3∑l=0l0ρ~∑z=0lz+r−1zqzH−U~,0~r+llvl+∑l=l0∞ρ~∑z=02l0−1z+r−1zqzαzH−U~+∑z=2l0l+2l0z+r−1zqzαzH−U~,0~r+llvl≤3∑l=0l0ρ~∑z=0lz+r−1zqzH−U~,0~r+llvl+2ℏ−1∑l=l0∞ρ~∑z=02l0−1z+r−1zqzαzH−U~,0~r+llvl+2ℏ−1∑l=l0∞ρ~∑z=2l0l+2l0z+r−1zqzαzH−U~,0~r+llvl≤3∑l=0l0ρ~∑z=0lz+r−1zqzH−U~,0~r+llvl+2ℏ−1∑l=l0∞ρ~∑z=02l0−1z+r−1zqzαzZ−U~,0~r+llvl+2ℏ−1∑l=l0∞ρ~∑z=0lz+2l0+r−1z+2l0qz+2l0αz+2l0H−U~,0~r+llvl≤3∑l=0l0ρ~∑z=0lz+r−1zqzH−U~,0~r+llvl+22ℏ−1supl=l0∞ρ~vl∑z=0l0z+r−1zqzH−U~,0~∑l=l0∞1r+llvl+2ℏ−1∑l=l0∞ρ~∑z=0lz+r−1zqzαzH~,0~r+llvl<γ.
On the other hand, one has a negative example asI2∈Dα~ΓrSq,vτG,V, where z+r−1zqz=1, for all z∈N and v=0,−1,2,2,2, but vl∉↑. This gives a negative answer to the Rhoades [33] open problem about the linearity of s-type ΓrSq,vτ spaces.Theorem 22.
The classDs~ΓrSq,vτ,Ξ is a prequasi Banach ideal, where ΞH=τsbH~b=0∞.Proof.
Evidently,Ξ is a prequasi norm on Ds~ΓrSq,vτ since τ is a prequasi norm on ΓrSq,vτ. Assume Hmm∈N is a Cauchy sequence in Ds~ΓrSq,vτG,V. Since DG,V⊇Ds~ΓrSq,vτG,V, we have
(31)ΞHj−Hm=∑l=0∞ρ~∑z=0lz+r−1zqzszHj−Hm~,0~r+llvl≥Hj−Hmv0,then Hmm∈N is a Cauchy sequence in DG,V. As DG,V is a Banach space, one has H∈DG,V so that limm⟶∞Hm−H=0. As slHm~l=0∞∈ΓrSq,vτ, for all m∈N. By Definition 11 conditions (ii), (iii), and (v), we have
(32)ΞH=∑l=0∞ρ~∑z=0lz+r−1zqzszH~,0~r+llvl≤2ℏ−1∑l=0∞ρ~∑z=0lz+r−1zqzsz/2H−Hm~,0~r+llvl+2ℏ−1∑l=0∞ρ~∑z=0lz+r−1zqzsz/2Hm~,0~r+llvl≤2ℏ−1∑l=0∞ρ~∑z=0lz+r−1zqzH−Hm~,0~r+llvl+2ℏ−1D0∑l=0∞ρ~∑z=0lz+r−1zqzszHm~,0~r+llvl<∞.
Hence,sbH~b=0∞∈ΓrSq,vτ, so H∈Ds~ΓrSq,vτG,V.Theorem 23.
If1<vb1<vb2, and 0<qb2≤qb1, for every b∈N, then
(33)Ds~ΓrSqb1,vb1τG,V⫋Ds~ΓrSqb2,vb2τG,V⫋DG,V.Proof.
LetH∈Ds~ΓrSqb1,vb1τG,V, then sbH~∈ΓrSqb1,vb1τ. One obtains
(34)∑b=0∞ρ~∑z=0bz+r−1zqz2szH~,0~r+bbvb2<∑b=0∞ρ~∑z=0bz+r−1zqz1szH~,0~r+bbvb1<∞,then H∈Ds~ΓrSqb2,vb2τG,V. Take sbH~b=0∞ with
(35)ρ~∑z=0bz+r−1zqz1szH~,0~=r+bbb+1vb1,we have H∈DG,V with
(36)∑b=0∞ρ~∑z=0bz+r−1zqz1szH~,0~r+bbvb1=∑b=0∞1b+1=∞,∑b=0∞ρ~∑z=0bz+r−1zqz2szH~,0~r+bbvb2≤∑b=0∞ρ~∑z=0bz+r−1zqz1szH~,0~r+bbvb2=∑b=0∞1b+1vb2/vb1<∞.
Hence,H∉Ds~ΓrSqb1,vb1τG,V and H∈Ds~ΓrSqb2,vb2τG,V.
Clearly,Ds~ΓrSqb2,vb2τG,V⊂DG,V. Take sbH~b=0∞ with
(37)ρ~∑z=0bz+r−1zqz2szH~,0~=r+bbb+1vb2.
ThenH∈DG,V and H∉Ds~ΓrSqb2,vb2τG,V.
Recall that ifG and V are infinite-dimensional, by Dvoretzky’s theorem [34], there are G/Yj and Mj⊆V operated onto ℓ2j through isomorphisms Vj and Xj such that VjVj−1≤2 and XjXj−1≤2, for all j∈N. Assume Tj is the quotient mapping from G onto G/Yj, Ij is the identity operator on ℓ2j and Jj is the natural embedding operator from Mj into V. Assume mj is the Bernstein numbers [18].Theorem 24.
The classDα~ΓrSq,vτ is minimum, whenever
(38)∑z=0lz+r−1zqzr+lll=0∞∉ℓvl.Proof.
AssumeDα~ΓrSq,vG,V=DG,V, one has γ>0 so that ΞH≤γH, for all H∈DG,V and
(39)ΞH=∑b=0∞ρ~∑z=0bz+r−1zqzαzH~,0~r+bbvb.
We have(40)1=mzIj=mzXjXj−1IjVjVj−1≤XjmzXj−1IjVjVj−1=XjmzJjXj−1IjVjVj−1≤XjdzJjXj−1IjVjVj−1=XjdzJjXj−1IjVjTjVj−1≤XjαzJjXj−1IjVjTjVj−1.
Take0≤m≤j, one has
(41)∑z=0mz+r−1zqz≤ρ~∑z=0mXjz+r−1zqzαzJjXj−1IjVjTj~Vj−1,0~⇒∑z=0mz+r−1zqzr+mmvm≤XjVj−1vmρ~∑z=0mz+r−1zqzαzJjXj−1IjVjTj~,0~r+mmvm.
Therefore, for someλ≥1, we obtain
(42)∑m=0j∑z=0mz+r−1zqzr+mmvm≤λXjVj−1∑m=0jρ~∑z=0mz+r−1zqzαzJjXj−1IjVjTj~,0~r+mmvm⇒∑m=0j∑z=0mz+r−1zqzr+mmvm≤λXjVj−1ΞJjXj−1IjVjTj≤λγXjVj−1JjXj−1IjVjTj≤4λγ.
Whenj⟶∞, one has a contradiction. So, G and V both cannot be infinite-dimensional when Dα~ΓrSq,vG,V=DG,V.Theorem 25.
The classDd~ΓrSq,v is minimum, whenever
(43)∑z=0lz+r−1zqzr+lll=0∞∉ℓvl.Lemma 26 (see [19]).
SupposeW∈DG,V and W∉AG,V, one has P∈DG and A∈DV with AWPej=ej, for every j∈N.Theorem 27 (see [19]).
IfES is an infinite-dimensional Banach space, then
(44)FES⫋AES⫋KES⫋DES.Theorem 28.
If1<vl1<vl2 and 0<ql2≤ql1, for every l∈N, then
(45)DDs~ΓrSql2,vl2τG,V,Ds~ΓrSql2,vl2τG,V=ADs~ΓrSql2,vl2τG,V,Ds~ΓrSql2,vl2τG,V.Proof.
LetX∈DDs~ΓrSql2,vl2τG,V,Ds~ΓrSql1,vl1τG,V and X∉ADs~ΓrSql2,vl2τG,V,Ds~ΓrSql1,vl1τG,V. In view of Lemma 26, there are Y∈DDs~ΓrSql2,vl2τG,V and Z∈DDs~ΓrSql1,vl1τG,V with ZXYIb=Ib. Therefore, for every b∈N, one has
(46)IbDs~ΓrSql1,vl1τG,V=∑l=0∞ρ~∑z=0lz+r−1zqz1szIb~,0~r+llvl1≤ZXYIbDs~ΓrSql2,vl2τG,V≤∑l=0∞ρ~∑z=0lz+r−1zqz2szIb~,0~r+llvl2.
This contradicts Theorem23; hence, X∈ADs~ΓrSql2,vl2τG,V,Ds~ΓrSql1,vl1τG,V.Corollary 29.
Suppose1<vl1<vl2, and 0<ql2≤ql1, for every l∈N, then
(47)DDs~ΓrSql2,vl2τG,V,Ds~ΓrSql1,vl1τG,V=KDs~ΓrSql2,vl2τG,V,Ds~ΓrSql1,vl1τG,V.Proof.
Evidently, asA⊂K.Definition 30 (see [19]).
A Banach spaceESis said to be simple whenDEShas a unique nontrivial closed ideal.Theorem 31.
The classDs~ΓrSq,vτ is simple.Proof.
Let the closed idealKDs~ΓrSq,vτG,V contain a mapping H∉ADs~ΓrSq,vτG,V. By Lemma 26, there are P,A∈DDs~ΓrSq,vτG,V so that AHPIj=Ij. Therefore, IDs~ΓrSq,vτG,V∈KDs~ΓrSq,vτG,V. Hence, DDs~ΓrSq,vτG,V=KDs~ΓrSq,vτG,V. Therefore, Ds~ΓrSq,vτ is a simple Banach space.Theorem 32.
Assume(48)infl∑z=0lz+r−1zqzr+llvl>0,then Ds~ΓrSq,vτγG,V=Ds~ΓrSq,vτG,V.Proof.
LetH∈Ds~ΓrSq,vτγG,V, then γmH~m=0∞∈ΓrSq,vτ and H−ρ~γmH~,0~I=0, for every m∈N. One has H=ρ~γmH~,0~I, for all m∈N, then
(49)ρ~smH~,0~=ρ~smρ~γmH~,~0~I,0~=ρ~γmH~,0~,for all m∈N. Hence smH~m=0∞∈ΓrSq,vτ, so H∈Ds~ΓrSq,vτG,V.
Next, assumeH∈Ds~ΓrSq,vτG,V. Hence, smH~m=0∞∈ΓrSq,vτ. Therefore, one has
(50)∑m=0∞ρ~∑z=0mz+r−1zqzszH~,0~r+mmvm≥infm∑z=0mz+r−1zqzr+mmvm∑m=0∞ρ~smH~,0~vm.
Hence,limm⟶∞smH~=0~. If H−ρ~smH~,0~I−1 exists, for all m∈N. Then H−ρ~smH~,0~I−1 exists and bounded, for all m∈N. So, limm⟶∞H−ρ~smH~,0~I−1=H−1 exists and bounded. Since Ds~ΓrSq,vτ,Ξ is a prequasi ideal, one obtains
(51)I=HH−1∈Ds~ΓrSq,vτG,V⇒smI~m=0∞∈ΓrSq,v⇒limm⟶∞smI~=0~.
One has a contradiction, aslimm⟶∞smI~=1~. Then, H−ρ~smH~,0~I=0, for all m∈N. So, H−ρ~γmH~,0~I=0, for all m∈N. Therefore, H∈Ds~ΓrSq,vτγG,V.


## 3. Multiplication Mappings onΓrSq,vτ

Under the conditions of Theorem14, we have presented in this section some properties of the multiplication mapping acting on ΓrSq,vτ.LetRangeVc indicate the complement of RangeV. Let I be the space of all sets with a finite number of elements. Assume ℓ∞S is the space of bounded sequences of soft functions.Definition 33.
SupposeEτS is a prequasi normed psssf and λ=λk∈RN. The mappingHλ:EτS⟶EτSis said to be a multiplication mapping onEτS, ifHλf~=λbfb~∈EτS, for all f∈EτS. The multiplication mapping is called constructed by λ, if Hλ∈DEτS.Definition 34 (see [35]).
A mappingV∈DE is said to be Fredholm if dimRangeVc<∞, RangeV is closed and dimkerV<∞.Theorem 35.
(1)
λ∈ℓ∞⇔Hλ∈DΓrSq,vτ(2)
λa=1, for every a∈N, if and only if, Hλ is an isometry(3)
Hλ∈AΓrSq,vτ⇔λaa=0∞∈c0(4)
Hλ∈KΓrSq,vτ⇔λbb=0∞∈c0(5)
KΓrSq,vτ⫋DΓrSq,vτ(6)
0<α<λa<η, for every a∈kerλc, if and only if, RangeHλ is closed(7)
0<α<λa<η, for all a∈N, if and only if, Hλ∈DΓrSq,vτ is invertible(8)
Hλ is Fredholm operator, if and only if (g1) kerλ⫋N∩I and (g2) λa≥ρ, for all a∈kerλcProof.
(1)
Supposeλ∈ℓ∞, one has ν>0 with λa≤ν, for all a∈N. If f~∈ΓrSq,vτ, we have(52)τHλf~=τλf~=∑l=0∞ρ~∑z=0lλzz+r−1zqzfz~,0~r+llvl≤suplνvl∑l=0∞ρ~∑z=0lz+r−1zqzfz~,0~r+llvl=suplνvlτf~.
Therefore,Hλ∈DΓrSq,vτ.
Next, ifHλ∈DΓrSq,vτ and λ∉ℓ∞. One has xb∈N, for every b∈N with λxb>b. Then,
(53)τHλexb~=τλexb~=∑l=0∞ρ~∑z=0lλzz+r−1zqzexbz~,0~r+llvl=∑l=xb∞λxbxb+r−1xbqxbr+llvl>∑l=xb∞bxb+r−1xbqxbr+llvl>bv0τexb~.
Hence,Hλ∉DΓrSq,vτ. So, λ∈ℓ∞.
(2)
Letf~∈ΓrSq,vτ and λb=1, for every b∈N. One obtains(54)τHλf~=τλf~=∑l=0∞ρ~∑z=0lz+r−1zqzλzfz~,0~r+llvl=∑l=0∞ρ~∑z=0lz+r−1zqzfz~,0~r+llvl=τf~,then Hλ is an isometry.
Next, if for someb=b0 that λb<1, one has
(55)τHλeb0~=τλeb0~=∑l=0∞ρ~∑z=0lz+r−1zqzλzeb0z~,0~r+llvl=∑l=b0∞λb0b0+r−1b0qb0r+llvl<∑l=b0∞b0+r−1b0qb0r+llvl=τeb0~.
Whenλb0>1, so τHλeb0~>τeb0~. Hence, λa=1, for every a∈N.
(3)
AssumeHλ∈AΓrSq,vτ, so Hλ∈KΓrSq,vτ. If limb⟶∞λb≠0. One has ρ>0 with Kϱ=a∈N:λa≥ρ⫋I. Let αaa∈N⊂Kρ. We have eαa~:αa∈Kϱ∈ℓ∞S be an infinite set in ΓrSq,vτ. For all αa,αb∈Kρ, one gets(56)τHλeαa~−Hλeαb~=τλeαa~−λeαb~=∑l=0∞ρ~∑z=0lz+r−1zqzλzeαaz~−eαbz~,0~r+llvl≥∑l=0∞ρ~∑z=0lz+r−1zqzρeαaz~−eαbz~,0~r+llvl≥inflρvlτeαa~−eαb~.
Hence,eαb~:αb∈Kρ∈ℓ∞S has not a convergent subsequence under Hλ. So, Hλ∉KΓrSq,vτ. Therefore, Hλ∉AΓrSq,vτ; this is a contradiction. So, limb⟶∞λb=0. Next, let lima⟶∞λa=0. Hence, for every ρ>0, we have Kρ=b∈N:λb≥ρ⊂I. Therefore, for all ρ>0, one gets dimΓrSq,vτKρ=dimRKρ<∞. So, Hλ∈FΓrSq,vτKρ. If λa∈RN, for all a∈N, where
(57)λab=λb,b∈K1/a+1,0,otherwise.
Obviously,Hλa∈FΓrSq,vτK1/a+1, since dimΓrSq,vτK1/a+1<∞, for all a∈N. According to vl∈↑∩ℓ∞ with v0>1/r, we have
(58)τHλ−Hλaf~=τλb−λabfb~b=0∞=∑l=0∞ρ~∑z=0lz+r−1zqzλz−λazfz~,0~r+llvl=∑l=0,l∈K1/a+1∞ρ~∑z=0lz+r−1zqzλz−λazfz~,0~r+llvl+∑l=0,l∉K1/a+1∞ρ~∑z=0lz+r−1zqzλz−λazfz~,0~r+llvl=∑l=0,l∉K1/a+1∞ρ~∑z=0lz+r−1zqzλzfz~,0~r+llvl≤1a+1v0∑l=0,l∉K1/a+1∞ρ~∑z=0lz+r−1zqzfz~,0~r+llvl<1a+1v0∑l=0∞ρ~∑z=0lz+r−1zqzfz~,0~r+llvl=1a+1v0τf~.
Therefore,Hλ−Hλa≤1/a+1v0. This implies Hλ is a limit of finite rank mappings.
(4)
AsAΓrSq,vτ⫋KΓrSq,vτ, the proof follows(5)
SinceI=Iλ, where λ=1,1,, one has I∉KΓrSq,vτ and I∈DΓrSq,vτ(6)
Let the sufficient setups be verified. One hasρ>0 with λa≥ρ, for every a∈kerλc. We have to show that RangeHλ is closed; let g~ be a limit point of RangeHλ. One has Hλfb~∈ΓrSq,vτ, for all b∈N with limb⟶∞Hλfb~=g~. Clearly, Hλfb~ is a Cauchy sequence. Since vl∈↑∩ℓ∞, we have(59)τHλfa~−Hλfb~=∑l=0∞ρ~∑z=0lz+r−1zqzλzfaz~−λzfbz~,0~r+llvl=∑l=0,l∈kerλc∞ρ~∑z=0lz+r−1zqzλzfaz~−λzfbz~,0~r+llvl+∑l=0,l∉kerλc∞ρ~∑z=0lz+r−1zqzλzfaz~−λzfbz~,0~r+llvl≥∑l=0,l∈kerλc∞ρ~∑z=0lz+r−1zqzλzfaz~−λzfbz~,0~r+llvl=∑l=0∞ρ~∑z=0lz+r−1zqzλzuaz~−λzubz~,0~r+llvl>∑l=0∞ρ~ρ∑z=0lz+r−1zqzuaz~−ubz~,0~r+llvl≥inflρvlτua~−ub~,where
(60)uak~=fak~,k∈kerλc,0,k∉kerλc.
Therefore,ua~ is a Cauchy sequence in ΓrSq,vτ. Since ΓrSq,vτ is complete. One has f~∈ΓrSq,vτ with limb⟶∞ub~=f~. As Hλ∈DΓrSq,vτ, we have limb⟶∞Hλub~=Hλf~. As limb⟶∞Hλub~=limb⟶∞Hλfb~=g~. So, Hλf~=g~. Then, g~∈RangeHλ, i.e., RangeHλ is closed. Next, suppose the necessary condition is satisfied. One has ρ>0 with τHλf~≥ρτf~ and f~∈ΓrSq,vτkerλc. Let K=b∈kerλc:λb<ρ≠∅, then for a0∈K, we have
(61)τHλea0~=τλbea0b~b=0∞=∑l=0∞ρ~∑z=0lz+r−1zqzλzea0~z,0~r+llvl<∑l=0∞ρ~ρ∑z=0lz+r−1zqzea0z~,0~r+llvl≤suplρvlτea0~,which introduces a contradiction. So K=ϕ, we have λa≥ρ, for all a∈kerλc.
(7)
First, assumeκ∈RN so that κa=1/λa. By Theorem 35 part (1), we have Hλ,Hκ∈DΓrSq,vτ. One has Hλ·Hκ=Hκ·Hλ=I. So, Hκ=Hλ−1. Second, if Hλ is invertible. Then RangeHλ=ΓrSq,vτN. Therefore, RangeHλ is closed. From Theorem 35 part (5), one has α>0 with λa≥α, for all a∈kerλc. Then, kerλ=∅, when λa0=0, where a0∈N; this implies ea0∈kerHλ, which is a contradiction, since kerHλ is trivial. Then, λa≥α, for all a∈N. As Hλ∈ℓ∞. From Theorem 35 part (1), one has η>0 with λa≤η, for all a∈N. So α≤λa≤η, for all a∈N(8)
First, ifkerλ⫋N and kerλ∉I, one has ea~∈kerHλ, for all a∈kerλ. As ea~’s are linearly independent, we have dimkerHλ=∞; this is a contradiction. Therefore, kerλ⫋N∈I. The condition (g2) comes from Theorem 35 part (6). Next, assume the setups (g1) and (g2) are satisfied. According to Theorem 35 part (6), the setup (g2) gives that RangeHλ is closed. The condition (g1) implies that dimRangeHλc<∞ and dimkerHλ<∞. Therefore, Hλ is Fredholm


## 4. Fixed Points of Kannan Contraction Type

In this section, we offer the existence of a fixed point of Kannan contraction mapping acting on this new space under the conditions of Theorem14 and its associated prequasi ideal. Interestingly, several numerical experiments are presented to illustrate our results.Definition 36.
A prequasi normedpsssfτ on ES confirms the Fatou property, if for every sequence hb~⊆EτS so that limb⟶∞τhb~−h~=0~ and every g~∈EτS, one has τg~−h~≤suppinfb≥pτg~−hb~.
Throughout the next part of this article, we will use the two functionsτ1 and τ2 as
(62)τ1f~=∑l=0∞ρ~∑z=0lz+r−1zqzfz~,0~r+llvl1/ℏ,τ2f~=∑l=0∞ρ~∑z=0lz+r−1zqzfz~,0~r+llvl,for all f~∈ΓrSq,v.Theorem 37.
The functionτ1 satisfies the Fatou property.Proof.
Assumegb~⊆ΓrSq,vτ1 so that limb⟶∞τ1gb~−g~=0. Clearly, g~∈ΓrSq,vτ1. For every f~∈ΓrSq,vτ1, one has
(63)τ1f~−g~=∑l=0∞ρ~∑z=0lz+r−1zqzfz~−gz~,0~r+llvl1/ℏ≤∑l=0∞ρ~∑z=0lz+r−1zqzfz~−gzb~,0~r+llvl1/ℏ+∑l=0∞ρ~∑z=0lz+r−1zqzgzb~−gz~,0~r+llvl1/ℏ≤supjinfb≥jτ1f~−gb~.Theorem 38.
Supposev0>1, then τ2 does not verify the Fatou property.Proof.
Ifgb~⊆ΓrSq,vτ2 so that limb⟶∞τ2gb~−g~=0. Clearly, g~∈ΓrSq,vτ2. For every f~∈ΓrSq,vτ2, one has
(64)τ2f~−g~=∑l=0∞ρ~∑z=0lz+r−1zqzfz~−gz~,0~r+llvl≤2ℏ−1∑l=0∞ρ~∑z=0lz+r−1zqzfz~−gzb~,0~r+llvl+∑l=0∞ρ~∑z=0lz+r−1zqzgzb~−gz~,0~r+llvl≤2ℏ−1supjinfb≥jτ2f~−gb~.Hence,τ2 does not satisfy the Fatou property.Definition 39 (see [30]).
A mappingG:EτS⟶EτS is called a Kannan τ-contraction, if one has ζ∈0,1/2, with τGg~−Gh~≤ζτGg~−g~+τGh~−h~, for all g~,h~∈EτS. When Gg~=g~, then g~∈EτS is called a fixed point of G.Theorem 40.
SupposeG:ΓrSq,vτ1⟶ΓrSq,vτ1 is Kannan τ1-contraction operator, then G has a unique fixed point.Proof.
Ifh~∈ΓrSq,v, one has Gmh~∈ΓrSq,v. As G is a Kannan τ1-contraction, one has
(65)τ1Gm+1h~−Gmh~≤ζτ1Gm+1h~−Gmh~+τ1Gmh~−Gm−1h~⇒τ1Gm+1h~−Gmh~≤ζ1−ζτ1Gmh~−Gm−1h~≤ζ1−ζ2τ1Gm−1h~−Gm−2h~≤≤ζ1−ζmτ1Gh~−h~.
We get for allm,n∈N so that n>m that
(66)τ1Gmh~−Gnh~≤ζτ1Gmh~−Gm−1h~+τ1Gnh~−Gn−1h~≤ζζ1−ζm−1+ζ1−ζn−1τ1Gh~−h~.Therefore,Gmh~ is a Cauchy sequence in ΓrSq,vτ1. As ΓrSq,vτ1 is prequasi Banach space. One has J~∈ΓrSq,vτ1 with limm⟶∞Gmh~=J~. To show that GJ~=J~. Since τ1 satisfies the Fatou property, one can see
(67)τ1GJ~−J~≤supiinfm≥iτ1Gm+1h~−Gmh~≤supiinfm≥iζ1−ζmτ1Gh~−h~=0,then GJ~=J~. Therefore, J~ is a fixed point of G. To indicate the uniqueness of the fixed point. Let us have two different fixed points f~,J~∈ΓrSq,vτ1 of G. We have
(68)τ1f~−J~≤τ1Gf~−GJ~≤ζτ1Gf~−f~+τ1GJ~−J~=0.Therefore,f~=J~.Corollary 41.
IfG:ΓrSq,vτ1⟶ΓrSq,vτ1 is Kannan τ1-contraction, then G has a unique fixed point J~ so that τ1Gmh~−J~≤ζζ/1−ζm−1τ1Gh~−h~.Proof.
By Theorem40, one has a unique fixed point J~ of G. Hence,
(69)τ1Gmh~−J~=τ1Gmh~−GJ~≤ζτ1Gmh~−Gm−1h~+τ1GJ~−J~=ζζ1−ζm−1τ1Gh~−h~.Definition 42.
IfEτS is a prequasi normed psssf, G:EτS⟶EτS and j~∈EτS. The mapping G is called τ-sequentially continuous at j~, if and only if, when limi⟶∞τgi~−j~=0, then limi⟶∞τGgi~−Gj~=0.Theorem 43.
Ifv0>1, and G:ΓrSq,vτ2⟶ΓrSq,vτ2. The element h~∈ΓrSq,vτ2 is the unique fixed point of G, when the following conditions are confirmed:
(i)
G is Kannan τ2-contraction(ii)
G is τ2-sequentially continuous at h~∈ΓrSq,vτ2(iii)
One hasj~∈ΓrSq,vτ2 with Gmj~ has Gmij~ converges to h~Proof.
Assumeh~ is not a fixed point of G, one has Gh~≠h~. According to conditions (ii) and (iii), we have
(70)limmi⟶∞τ2Gmij~−h~=0,limmi⟶∞τ2Gmi+1j~−Gh~=0.
AsG is Kannan τ2-contraction, one has
(71)0<τ2Gh~−h~=τ2Gh~−Gmi+1j~+Gmij~−h~+Gmi+1j~−Gmij~≤22ℏ−2τ2Gmi+1j~−Gh~+22ℏ−2τ2Gmij~−h~+2ℏ−1ζζ1−ζmi−1τ2Gj~−j~.
Takemi⟶∞, one obtains a contradiction. Therefore, h~ is a fixed point of G. To explain the uniqueness of h~. Suppose we have two different fixed points h~,g~∈ΓrSq,vτ2 of G. Then
(72)τ2h~−g~≤τ2Gh~−Gg~≤ζτ2Gh~−h~+τ2Gg~−g~=0.
Soh~=g~.Example 44.
IfT:ΓrS1/l+5l+r−1ll=0∞,2l+3/l+2l=0∞τ1⟶ΓrS1/l+5l+r−1ll=0∞,2l+3/l+2l=0∞τ1 and
(73)Tf~=f~4,τ1f~∈0,1,f~5,τ1f~∈1,∞.
For allf~,g~∈ΓrS1/l+5l+r−1ll=0∞,2l+3/l+2l=0∞τ1. If τ1f~,τ1g~∈0,1, we have
(74)τ1Tf~−Tg~=τ1f~4−g~4≤1274τ13f~4+τ13g~4=1274τ1Tf~−f~+τ1Tg~−g~.
For everyτ1f~,τ1g~∈1,∞, we have
(75)τ1Tf~−Tg~=τ1f~5−g~5≤1644τ14f~5+τ14g~5=1644τ1Tf~−f~+τ1Tg~−g~.
For everyτ1f~∈0,1 and τ1g~∈1,∞, one has
(76)τ1Tf~−Tg~=τ1f~4−g~5≤1274τ13f~4+1644τ14g~5≤1274τ13f~4+τ14g~5=1274τ1Tf~−f~+τ1Tg~−g~.
Hence,T is Kannan τ1-contraction, as τ1 satisfies the Fatou property. By Theorem 40, T has a unique fixed point θ~. Assume
(77)hk~⊆ΓrS1l+5l+r−1ll=0∞,2l+3l+2l=0∞τ1,so that limk⟶∞τ1hk~−h0~=0, where
(78)h0~∈ΓrS1l+5l+r−1ll=0∞,2l+3l+2l=0∞τ1,such that τ1h0~=1. As τ1 is continuous, one has
(79)limk⟶∞τ1Thk~−Th0~=limk⟶∞τ1hk~4−h0~5=τ1h0~20>0.
SoT is not τ1-sequentially continuous at h0~. This implies T is not continuous at h0~.
For everyf~,g~∈ΓrS1/l+5l+r−1ll=0∞,2l+3/l+2l=0∞τ2. If τ2f~,τ2g~∈0,1, one has
(80)τ2Tf~−Tg~=τ2f~4−g~4≤227τ23f~4+τ23g~4=227τ2Tf~−f~+τ2Tg~−g~.
Letτ2f~,τ2g~∈1,∞, one has
(81)τ2Tf~−Tg~=τ2f~5−g~5≤14τ24f~5+τ24g~5=14τ2Tf~−f~+τ2Tg~−g~.
For everyτ2f~∈0,1 and τ2g~∈1,∞, one has
(82)τ2Tf~−Tg~=τ2f~4−g~5≤227τ23f~4+14τ24g~5≤227τ23f~4+τ24g~5=227τ2Tf~−f~+τ2Tg~−g~.
Hence,T is Kannan τ2-contraction and
(83)Tmf~=f~4m,τ2f~∈0,1,f~5m,τ2f~∈1,∞.Evidently,T is τ2-sequentially continuous at θ~ and Tmf~ has a subsequence Tmjf~ converges to θ~. According to Theorem 43, the element θ~ is the only fixed point of T.Example 45.
LetT:ΓrS1/l+5l+r−1ll=0∞,2l+3/l+2l=0∞τ2⟶ΓrS1/l+5l+r−1ll=0∞,2l+3/l+2l=0∞τ2 and
(84)Tf~=14e1~+f~,f0~t∈0,13,13e1~,f0~t=13,14e1~,f0~t∈13,1.
Asf0~t,g0~t∈0,1/3, we get
(85)τ2Tf~−Tg~=τ214f0~−g0~,f1~−g1~,f2~−g2~,⋯≤227τ23f~4+τ23g~4≤227τ2Tf~−f~+τ2Tg~−g~.
For allf0~t,g0~t∈1/3,1, hence for all ε>0, we have
(86)τ2Tf~−Tg~=0≤ετ2Tf~−f~+τ2Tg~−g~.
For allf0~t∈0,1/3 and g0~t∈1/3,1, one has
(87)τ2Tf~−Tg~=τ2f~4≤127τ23f~4=127τ2Tf~−f~≤127τ2Tf~−f~+τ2Tg~−g~.
Hence,T is Kannan τ2-contraction. Obviously, T is τ2-sequentially continuous at 1/3e1~, and there is f~∈ΓrS1/l+5l+r−1ll=0∞,2l+3/l+2l=0∞τ2 with f0~t∈0,1/3 such that the sequence of iterates Tmf~=∑a=1m1/4ae1~+1/4mf~ includes a subsequence Tmjf~=∑a=1mj1/4ae1~+1/4mjf~ converges to 1/3e1~. In view of Theorem 43, the operator T has one fixed point 1/3e1~. Note that T is not continuous at 1/3e1~.
For allf~,g~∈ΓrS1/l+5l+r−1ll=0∞,2l+3/l+2l=0∞τ1. If f0~t,g0~t∈0,1/3, we have
(88)τ1Tf~−Tg~=τ114f0~−g0~,f1~−g1~,f2~−g2~,⋯≤1274τ13f~4+τ13g~4≤1274τ1Tf~−f~+τ1Tg~−g~.
For allf0~t,g0~t∈1/3,1, hence for all ε>0, one has
(89)τ1Tf~−Tg~=0≤ετ1Tf~−f~+τ1Tg~−g~.
For allf0~t∈0,1/3 and g0~t∈1/3,1, we have
(90)τ1Tf~−Tg~=τ1f~4≤1274τ13f~4=1274τ1Tf~−f~≤1274τ1Tf~−f~+τ1Tg~−g~.
Therefore, the operatorT is Kannan τ1-contraction. Since τ1 confirms the Fatou property. By Theorem 40, the operator T has a unique fixed point 1/3e1~.
In this part, we will use(91)ΞV=τsbV~b=0∞=∑l=0∞ρ~∑z=0lz+r−1zqzszV~,0~r+llvl1/ℏ,for every V∈Ds~ΓrSq,vτG,V.Definition 46.
A functionΞ on Ds~ES satisfies the Fatou property if for all Vbb∈N⊆Ds~ESZ,M so that limb⟶∞ΞVb−V=0 and all T∈Ds~ESZ,M, one has ΞT−V≤supbinfj≥bΞT−Vj.Theorem 47.
The functionΞ does not verify the Fatou property.Proof.
AssumeWmm∈N⊆Ds~ΓrSq,vτG,V so that limm⟶∞ΞWm−W=0. Clearly, W∈Ds~ΓrSq,vτG,V. Hence, for every V∈Ds~ΓrSq,vτG,V, we have
(92)ΞV−W=∑l=0∞ρ~∑z=0lz+r−1zqzszV−W~,0~r+llvl1/ℏ≤∑l=0∞ρ~∑z=0lz+r−1zqzsz/2V−Wi~,0~r+llvl1/ℏ+∑l=0∞ρ~∑z=0lz+r−1zqzsz/2W−Wi~,0~r+llvl1/ℏ≤22ℏ−1+2ℏ−1+2ℏ1/ℏsupminfi≥m∑l=0∞ρ~∑z=0lz+r−1zqzszV−Wi~,0~r+llvl1/ℏ.Therefore,Ξ does not satisfy the Fatou property.Definition 48 (see [30]).
A mappingW:Ds~ESZ,M⟶Ds~ESZ,M is said to be a Kannan Ξ-contraction, assume there is ζ∈0,1/2 with ΞWV−WT≤ζΞWV−V+ΞWT−T, for all V,T∈Ds~ESZ,M.Definition 49.
AssumeG:Ds~ESZ,M⟶Ds~ESZ,M and B∈Ds~ESZ,M. The mapping G is called Ξ-sequentially continuous at B, if and only if, when limm⟶∞ΞWm−B=0, one has limm⟶∞ΞGWm−GB=0.Theorem 50.
IfG:Ds~ΓrSq,vτG,V⟶Ds~ΓrSq,vτG,V. The operator A∈Ds~ΓrSq,vτG,V is the only fixed point of G, when the following conditions are confirmed:
(i)
G is Kannan Ξ-contraction(ii)
G is Ξ-sequentially continuous at A∈Ds~ΓrSq,vτG,V(iii)
One hasB∈Ds~ΓrSq,vτG,V with GmB has GmiB converges to AProof.
SupposeA is not a fixed point of G, then GA≠A. By conditions (ii) and (iii), one has
(93)limmi⟶∞ΞGmiB−A=0,limmi⟶∞ΞGmi+1B−GA=0.
AsG is Kannan Ξ-contraction operator, we get
(94)0<ΞGA−A=ΞGA−Gmi+1B+GmiB−A+Gmi+1B−GmiB≤22ℏ−1+2ℏ−1+2ℏ1/ℏΞGmi+1B−GA+22ℏ−1+2ℏ−1+2ℏ2/ℏΞGmiB−A+22ℏ−1+2ℏ−1+2ℏ2/ℏζζ1−ζmi−1ΞGB−B.
Bymi⟶∞, we have a contradiction. Then, A is a fixed point of G. To show the uniqueness of the fixed point A. If one has two different fixed points A,D∈Ds~ΓrSq,vτG,V of G. So
(95)ΞA−D≤ΞGA−GD≤ζΞGA−A+ΞGD−D=0.
Therefore,A=D.Example 51.
Assume(96)M:SΓrS1/l+4l+r−1ll=0∞,2l+3/l+2l=0∞τG,V⟶SΓrS1/l+4l+r−1ll=0∞,2l+3/l+2l=0∞τG,V,(97)MH=H6,ΞH∈0,1,H7,ΞH∈1,∞.
For all(98)H1,H2∈SΓrS1/l+4l+r−1ll=0∞,2l+3/l+2l=0∞τ.
IfΞH1,ΞH2∈0,1, we have
(99)ΞMH1−MH2=ΞH16−H26≤21254Ξ5H16+Ξ5H26=21254ΞMH1−H1+ΞMH2−H2.
SupposeΞH1,ΞH2∈1,∞, one has
(100)ΞMH1−MH2=ΞH17−H27≤22164Ξ6H17+Ξ6H27=22164ΞMH1−H1+ΞMH2−H2.
AssumeΞH1∈0,1 and ΞH2∈1,∞, one gets
(101)ΞMH1−MH2=ΞH16−H27≤21254Ξ5H16+22164Ξ6H27≤21254ΞMH1−H1+ΞMH2−H2.
Hence,M is Kannan Ξ-contraction and
(102)MmH=H6m,ΞH∈0,1,H7m,ΞH∈1,∞.
Evidently,M is Ξ-sequentially continuous at the zero operator Θ and MmH has a subsequence MmjH converges to Θ. According to Theorem 50, the zero operator is the only fixed point of M.
If(103)Ha⊆SΓrS1/l+4l+r−1ll=0∞,2l+3/l+2l=0∞τ,with lima⟶∞ΞHa−H0=0, where
(104)H0∈SΓrS1/l+4l+r−1ll=0∞,2l+3/l+2l=0∞τ,so that ΞH0=1. As Ξ is continuous, one has
(105)lima⟶∞ΞMHa−MH0=lima⟶∞ΞH06−H07=ΞH042>0.
Therefore,M is not Ξ-sequentially continuous at H0. This implies M is not continuous at H0.


## 5. Applications on Stochastic Nonlinear Dynamical System

We investigate in this section a solution inΓrSq,vτ1 to stochastic nonlinear dynamical system (106) under the conditions of Theorem 14. For every f~∈ΓrSq,v.Consider the stochastic nonlinear dynamical system [36]:
(106)fz~=yz~+∑m=0∞Πz,mgm,fm~,and assumeW:ΓrSq,vτ1⟶ΓrSq,vτ1 is constructed by
(107)Wfz~z∈N=yz~+∑m=0∞Πz,mgm,fm~z∈N.Theorem 52.
The stochastic nonlinear dynamical system (106) has one and only one solution in ΓrSq,vτ1,ifΠ:N2⟶R,g:N×RA⟶RA,f~:N⟶RA,y~:N⟶RA,η~:N⟶RA, one has λ∈R with suplλvl/ℏ∈0,1/2 and for every l∈N, one obtains
(108)∑z=0l∑m∈NΠz,mgm,fm~−gm,ηm~z+r−1zqz≤~λ∑z=0lyz~−fz~+∑m=0∞Πz,mgm,fm~z+r−1zqz+λ∑z=0lyz~−ηz~+∑m=0∞Πz,mgm,ηm~z+r−1zqz.Proof.
Let the conditions be established. Assume the mappingW:ΓrSq,vτ1⟶ΓrSq,vτ1 is defined by equation (11). Hence,
(109)τ1Wf~−Wη~=∑l=0∞ρ~∑z=0lz+r−1zqzWfz~−Wηz~,0~r+llvl1/ℏ=∑l=0∞ρ~∑z=0l∑m∈NΠz,mgm,fm~−gm,ηm~z+r−1zqz,0~r+llvl1/ℏ≤suplλvl/ℏ∑l=0∞ρ~∑z=0lyz~−fz~+∑m=0∞Πz,mgm,fm~z+r−1zqz,0~r+llvl1/ℏ+suplλvl/ℏ∑l=0∞ρ~∑z=0lyz~−ηz~+∑m=0∞Πz,mgm,ηm~z+r−1zqz,0~r+llvl1/ℏ=suplλvl/ℏτ1Wf~−f~+τ1Wη~−η~.From Theorem40, one has one and only one solution of (106) in ΓrSq,vτ1.Example 53.
Consider(110)ΓrS1l+1l+r−1ll=0∞,2l+3l+2l=0∞τ1.
Suppose the stochastic nonlinear dynamical system:(111)fz~=e−3z+6~+∑m=0∞−1z+mfz−2b~fz−1d~+m2+1~,
withb,d,f−2~t,f−1~t>0, for all t∈A and suppose
(112)W:ΓrS1l+1l+r−1ll=0∞,2l+3l+2l=0∞τ1⟶ΓrS1l+1l+r−1ll=0∞,2l+3l+2l=0∞τ1is defined by
(113)Wfz~z=0∞=e−3z+6~+∑m=0∞−1z+mfz−2b~fz−1d~+m2+1~z=0∞.Evidently, one hasλ∈R with suplλ2l+3/2l+4∈0,1/2 and for every l∈N, we have
(114)∑z=0l∑m=0∞−1zfz−2b~fz−1d~+m2+1~−1m−−1mz+r−1zqz≤~λ∑z=0le−3z+6~−fz+∑m=0∞−1z+mfz−2b~fz−1d~+m2+1~z+r−1zqz+λ∑z=0le−3z+6~−ηz~+∑m=0∞−1z+mηz−2b~ηz−1d~+m2+1~z+r−1zqz.From Theorem52, system (111) has one and only one solution in
(115)ΓrS1l+1l+r−1ll=0∞,2l+3l+2l=0∞τ1.Example 54.
Suppose the sequence space(116)ΓrS1l+1l+r−1ll=0∞,2l+3l+2l=0∞τ1.
Assume the stochastic nonlinear dynamical system:(117)fz~=yz~+∑m=0∞ez+mfz−2b~fz−1d~+fz−1b~+2~,with b,d,f−2~t,f−1~t>0, for all t∈A and suppose
(118)W:ΓrS1l+1l+r−1ll=0∞,2l+3l+2l=0∞τ1⟶ΓrS1l+1l+r−1ll=0∞,2l+3l+2l=0∞τ1is defined by
(119)Wfz~z=0∞=yz~+∑m=0∞ez+mfz−2b~fz−1d~+fz−1b~+2~z=0∞.Evidently, there isλ∈R such that suplλ2l+3/2l+4∈0,1/2 and for every l∈N, we have
(120)∑z=0l∑m=0∞ezfz−2b~fz−1d~+fz−1b~+2~em−emz+r−1zqz≤~λ∑z=0lyz~−fz+∑m=0∞ez+mfz−2b~fz−1d~+fz−1b~+2~z+r−1zqz+λ∑z=0lyz~−ηz~+∑m=0∞ez+mηz−2b~ηz−1d~+ηz−1b~+2~z+r−1zqz.According to Theorem52, the stochastic nonlinear dynamical system (14) contains a unique solution in
(121)ΓrS1l+1l+r−1ll=0∞,2l+3l+2l=0∞τ1.Theorem 55.
IfW:ΓrSq,vτ2⟶ΓrSq,vτ2 is defined by (11) and v0>1. The stochastic nonlinear dynamical system (106) has a unique solution Z~∈ΓrSq,vτ2, when the following conditions are satisfied:
(1)
IfΠ:N2⟶R,g:N×RA⟶RA,f~:N⟶RA,y~:N⟶RA,η~:N⟶RA, assume there is λ∈R so that 2ℏ−1suplλvl∈0,1/2 and for every l∈N, one has(122)∑z=0l∑m∈NΠz,mgm,fm~−gm,ηm~z+r−1zqz≤~λ∑z=0lyz~−fz~+∑m=0∞Πz,mgm,fm~z+r−1zqz+λ∑z=0lyz~−ηz~+∑m=0∞Πz,mgm,ηm~z+r−1zqz(2)
W is τ2-sequentially continuous at Z~∈ΓrSq,vτ2(3)
There isY~∈ΓrSq,vτ2 with WmY~ has WmjY~ converging to Z~Proof.
One has(123)τ2Wf~−Wη~=∑l=0∞ρ~∑z=0lr+z−1zqzWfz~−Wηz~,0~r+llvl=∑l=0∞ρ~∑z=0l∑m∈NΠz,mgm,fm~−gm,ηm~z+r−1zqz,0~r+llvl≤2ℏ−1suplλvl∑l=0∞ρ~∑z=0lyz~−fz~+∑m=0∞Πz,mgm,fm~z+r−1zqz,0~r+llvl+2ℏ−1suplλvl∑l=0∞ρ~∑z=0lyz~−ηz~+∑m=0∞Πz,mgm,ηm~z+r−1zqz,0~r+llvl=2ℏ−1suplλvlτ2Wf~−f~+τ2Wη~−η~.By Theorem43, one gets a unique solution Z~∈ΓrSq,vτ2 of equation (106).Example 56.
Suppose the sequence space(124)ΓrS1l+1l+r−1ll=0∞,2l+3l+2l=0∞τ2.
Consider the summable equation (111):
Let(125)W:ΓrS1l+1l+r−1ll=0∞,2l+3l+2l=0∞τ2⟶ΓrS1l+1l+r−1ll=0∞,2l+3l+2l=0∞τ2defined by (13). Assume W is τ2-sequentially continuous at
(126)Z~∈ΓrS1l+1l+r−1ll=0∞,2l+3l+2l=0∞τ2,and there is
(127)Y~∈ΓrS1l+1l+r−1ll=0∞,2l+3l+2l=0∞τ2with WmY~ has WmjY~ converging to Z~. Evidently, there is λ∈R such that 2ℏ−1suplλ2l+3/l+2∈0,1/2 and for all l∈N, one has
(128)∑z=0l∑m=0∞−1zfz−2b~fz−1d~+m2+1~−1m−−1mz+r−1zqz≤~λ∑z=0le−3z+6~−fz+∑m=0∞−1z+mfz−2b~fz−1d~+m2+1~z+r−1zqz+λ∑z=0le−3z+6~−ηz~+∑m=0∞−1z+mηz−2b~ηz−1d~+m2+1~z+r−1zqz.
By Theorem57, the stochastic nonlinear dynamical system (111) has one and only one solution
(129)Z~∈ΓrS1l+1l+r−1ll=0∞,2l+3l+2l=0∞τ2.
In this part, we search for a solution to nonlinear matrix equation (131) at D∈Ds~ΓrSq,vτG,V, the conditions of Theorem 14 are satisfied, and
(130)ΞG=∑l=0∞ρ~∑z=0lr+z−1zqzszG~,0~r+llvl1/ℏ,for every G∈Ds~ΓrSq,vτG,V. Consider the stochastic nonlinear dynamical system:
(131)szG~=szP~+∑m=0∞Πz,mfm,smG~,and suppose W:Ds~ΓrSq,vτG,V⟶Ds~ΓrSq,vτG,V is defined by
(132)WG=szP~+∑m=0∞Πz,mfm,smG~I.Theorem 57.
The stochastic nonlinear dynamical system (131) has one and only one solution D∈Ds~ΓrSq,vτG,V, if the following conditions are satisfied:
(1)
Π:N2⟶R,f:N×RA⟶RA,P∈DG,V,T∈DG,V, and for every z∈N, there is a positive real number κ so that supzκtz/ℏ∈0,0.5, with(133)∑m∈NΠz,mfm,smG~−fm,smT~≤~κszP~−szG~+∑m∈NAa,mfm,smG~+szP~−szT~+∑m∈NΠz,mfm,smT~(2)
W is Ξ-sequentially continuous at a point D∈Ds~ΓrSq,vτG,V(3)
There isB∈Ds~ΓrSq,vτG,V so that the sequence of iterates WaB has a subsequence WaiB converging to DProof.
Suppose the settings are verified. Consider the mappingW:Ds~ΓrSq,vτG,V⟶Ds~ΓrSq,vτG,V defined by (132). We have
(134)ΞWG−WT=∑l=0∞ρ~∑z=0lz+r−1zqzszG~−szT~,0~r+llvl1/ℏ=∑l=0∞ρ~∑z=0lz+r−1zqz∑m∈NAa,mfm,smG~−fm,smT~,0~r+llvl1/ℏ≤supzκtz/ℏ∑l=0∞ρ~∑z=0lz+r−1zqzszP~−szG~+∑m∈NΠz,mfm,smG~,0~r+llvl1/ℏ+supzκtz/ℏ∑l=0∞ρ~∑z=0lz+r−1zqzszT~−szG~+∑m∈NΠz,mfm,smT~,0~r+llvl1/ℏ=supzκtz/ℏΞWG−G+ΞWT−T.
In view of Theorem50, one obtains a unique solution of equation (131) at D∈Ds~ΓrSq,vτG,V.Example 58.
Assume the classDs~ΓrS1/l!,2l+3/l+2τG,V, where
(135)ΞG=∑l=0∞ρ~∑z=0lz+r−1zqzszG~,0~l!∑z=0lz+r−1zqz2l+3/l+2,forallG∈Ds~ΓrS1/l!,2l+3/l+2τG,V..
Consider the stochastic nonlinear dynamical system:(136)szG~=e−2z+3~+∑m=0∞tan2m+1cosh3m−zcosbsz−2G~sinhdsz−1G~+sinmz~+1~,where z≥2 and b,d>0 and let W:Ds~ΓrS1/l!,2l+3/l+2τG,V⟶Ds~ΓrS1/l!,2l+3/l+2τG,V be defined as
(137)WG=e−2z+3~+∑m=0∞tan2m+1cosh3m−zcosbsz−2G~sinhdsz−1G~+sinmz~+1~I.SupposeW is Ξ-sequentially continuous at a point D∈Ds~ΓrS1/l!,2l+3/l+2τG,V, and there is B∈Ds~ΓrS1/l!,2l+3/l+2τG,V so that the sequence of iterates WaB has a subsequence WaiB converging to D. It is easy to see that
(138)∑m=0∞cosh3m−zcosbsz−2G~sinhdsz−1G~+sinmz~+1~tan2m+1−tan2m+1≤~125e−2z+3~−szG~+∑m=0∞tan2m+1cosh3m−zcosbsz−2G~sinhdsz−1G~+sinmz~+1~+125e−2z+3~−szT~+∑m=0∞tan2m+1cosh3m−zcosbsz−2T~sinhdsz−1T~+sinmz~+1~.By Theorem57, the stochastic nonlinear dynamical system (18) has one solution D.


## 6. Conclusion

In this article, we introduced a new general space calledΓrSq,vτ and the mappings’ ideal space of solutions for many stochastic nonlinear and matrix systems of Kannan contraction type. We have presented some topological and geometric properties of it, of the multiplication operators acting on it, of the mappings’ ideal, and of the spectrum of its mappings’ ideal. The existence of a fixed point in the Kannan contraction mapping on these spaces is explored. To put our findings to the test, we introduced several numerical experiments. In addition, various effective implementations of the stochastic nonlinear dynamical and matrix system are discussed. The ideal spectrum of mappings, multiplication operators, and the fixed points of any contraction mappings in this new soft functions space are investigated.



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*Source: 9011506-2022-08-28.xml*