# Kannan Contraction Maps on the Space of Null Variable Exponent Second-Order Quantum Backward Difference Sequences of Fuzzy Functions and Its Pre-Quasi Ideal

Authors: Awad A. Bakery; OM Kalthum S. K. Mohamed
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

In this paper, we construct and investigate the space of null variable exponent second-order quantum backward difference sequences of fuzzy functions, which are crucial additions to the concept of modular spaces. The idealization of the mappings has been achieved through the use of extendeds− fuzzy functions and this sequence space of fuzzy functions. This new space’s topological and geometric properties and the mappings’ ideal that corresponds to them are discussed. We construct the existence of a fixed point of Kannan contraction mapping acting on this space and its associated pre-quasi ideal. To demonstrate our findings, we give a number of numerical experiments. There are also some significant applications of the existence of solutions to nonlinear difference equations of fuzzy functions.


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## Body

## 1. Introduction

We assume thatN is the set of non-negative integers. Yaying et al. [1] defined quantum second-order backward difference operator, ∇p2, where za=0, for a<0, and ∇p2za=za−1+pza−1+pza−2, for all p∈0,1 and a∈N. Note that the operator ∇p2 reduces to ∇2 when p⟶1−, which defined and studied in Reference [2]. They proved that the spaces c0∇p2 and c∇p2 are Banach spaces linearly isomorphic to c0 and c, respectively, and obtained their Schauder bases and α−, β−, and γ−duals. They determined the spectrum, the point spectrum, the continuous spectrum, and the residual spectrum of the operator ∇p2 over the Banach space c0 of null sequences. It is clear to see that(1)c0⫋c0∇p2⫋∇2.For the strict inclusion, we have1,1,…∉c0 and 1,1,…∈c0∇p2, and 0,1,2,…∉c0∇p2 and 0,1,2,…∈c0∇2.The mappings’ ideal theory is well regarded in functional analysis. Fixed-point theory, Banach space geometry, normal series theory, approximation theory, and ideal transformations all use mappings’ ideal. Usings-numbers is an essential technique. For more background details, see Pietsch [3], Constantin [4], and Tita [5]. Pre-quasi mappings’ ideals are more extensive than quasi mappings’ ideals, according to Faried and Bakery [6]. Bakery and Elmatty [7] explained a note on Nakano generalized difference sequence space under premodular. Since the booklet of the Banach fixed-point theorem [8], many mathematicians have worked on many developments. For more background and recent works on applicative approach of fixed-point theory, see Ruẑiĉka [9], Mao et al. [10], and Younis et al. [11–14]. Kannan [15] gave an example of a class of mappings with the same fixed-point actions as contractions, though that fails to be continuous. The only attempt to describe Kannan operators in modular vector spaces was once made in Reference [16]. Bakery and Mohamed [17] explored the concept of the pre-quasi-norm on Nakano sequence space such that its variable exponent belongs to 0,1. They explained the sufficient conditions on it equipped with the definite pre-quasi-norm to generate pre-quasi Banach. They examined the Fatou property of different pre-quasi-norms on it. Moreover, they showed a fixed point of Kannan pre-quasi-norm contraction maps on it and on the pre-quasi Banach operator ideal constructed by s-numbers that belong to this sequence space.Zadeh [18] established the concept of fuzzy sets and fuzzy set operations, and many researchers adopted the concept of fuzziness in cybernetics and artificial intelligence as well as in expert systems and fuzzy control. We refer the reader to the following exciting works dealing with Kannan mappings and fuzzy concepts with different applications, see Reference [19–25]. Many researchers in sequence spaces and summability theory studied fuzzy sequence spaces and their properties. In Reference [26], the Nakano sequences of fuzzy integers were defined and analyzed. Bakery and Mohamed [27] introduced the certain space of sequences of fuzzy numbers, in short (cssf), under a certain function to be pre-quasi (cssf). This space and s−numbers have been used to describe the structure of the ideal operators. They defined and studied the weighted Nakano sequence spaces of fuzzy functions. They constructed the ideal generated by extended s− fuzzy functions and the sequence spaces of fuzzy functions. They presented some topological and geometric structures of this class of ideal and multiplication mappings acting on this sequence space of fuzzy functions. Moreover, the existence of Caristi’s fixed point was examined. Many fixed-point theorems are effective when applied to a given space because they either enlarge the self-mapping acting on it or expand the space itself. Specifically, in this study, we construct and investigate the space of null variable exponent second-order quantum backward difference sequences of fuzzy functions, which are crucial additions to the concept of modular spaces. The idealization of the mappings has been achieved through the use of extended s− fuzzy functions and this sequence space of fuzzy functions. The topological and geometric properties of this new space and the mappings’ ideal that corresponds to them are discussed. We construct the existence of a fixed point of Kannan contraction mapping acting on this space and its associated pre-quasi 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 fuzzy functions are introduced.


## 2. Definitions and Preliminaries

It is worth mentioning that Matloka [28] introduced bounded and convergent fuzzy numbers, investigated some of their properties, and demonstrated that any convergent fuzzy number sequence is bounded. Nanda [29] researched fuzzy number sequences and demonstrated that the set of all convergent fuzzy number sequences forms a complete metric space. Kumar et al. [30] presented the concept limit points and cluster points of sequences of fuzzy numbers. If Ξ is the set of all closed and bounded intervals on the real line ℜ, then we assume y=y1,y2 and z=z1,z2 in Ξ, let(2)y≤z,if,only if,y1≤z1,y2≤z2.Clearly, the relation≤ is a partial order on Ξ. We define a metric ρ on Ξ by(3)ρy,z=maxy1−z1,y2−z2.Matloka [28] proved that ρ is a metric on Ξ and Ξ,ρ is a complete metric space.Definition 1.
A fuzzy numberz is a fuzzy subset of ℜ, that is, a mapping z:ℜ⟶0,1 that verifies the four conditions:(a)
z is fuzzy convex; that is, for t1,t2∈ℜ, and α∈0,1, zαt1+1−αt2≥minzt1,zt2.(b)
z is normal; that is, there is t0∈ℜ such that zt0=1.(c)
z is an upper-semicontinuous, that is, for all α>0, z−10,t+α, for all t∈0,1, is open in the usual topology of ℜ.(d)
The closure ofz0:=t∈ℜ:zt>0 is compact.
Theβ-level set of a fuzzy real number z,0<β<1, denoted by zβ, is defined as(4)zβ=t∈ℜ:zt≥β.
The set of all upper semicontinuous, normal, convex fuzzy number, andzβ is compact, is marked by ℜ0,1. The set ℜ can be embedded in ℜ0,1, if we define r∈ℜ0,1 by(5)r¯t=1,t=r,0,t≠r.
The additive identity and multiplicative identity inℜ0,1 are denoted by 0¯ and 1¯, respectively. We assume that y,z∈ℜ0,1and the β-level sets are yβ=y1β,y2β, zβ=z1β,z2β, and β∈0,1. A partial ordering for any y,z∈ℜ0,1 is as follows: y≺¯z, if and only if, yβ≤zβ, for all β∈0,1.
We assume thatρ¯:ℜ0,1×ℜ0,1⟶ℜ+∪0 is defined by ρ¯y,z=sup0≤β≤1ρyβ,zβ.
We recall that(1)
ℜ0,1,ρ¯ is a complete metric space(2)
ρ¯y+x,z+x=ρ¯y,z for all y,z,x∈ℜ0,1(3)
ρ¯y+x,z+l≤ρ¯y,z+ρ¯x,l(4)
ρ¯ξy,ξz=ξρ¯y,z, for all ξ∈ℜ
Byc0, ℓ∞, and ℓr, we denote the space of null, bounded, and r-absolutely summable sequences of real numbers, respectively. We indicate the space of all bounded, finite rank linear mappings from an infinite dimensional Banach space Ω into an infinite dimensional Banach space Λ by ℒΩ,Λ, and FΩ,Λ and when Ω=Λ, we inscribe ℒΩ and FΩ. The space of approximable and compact bounded linear mappings from Ω into Λ will be denoted by ΥΩ,Λ and ℒcΩ,Λ, and if Ω=Λ, we mark ΥΩ and ℒcΩ, respectively.Definition 2 (see [31]).
Ans-number function is a mapping s:ℒΩ,Λ⟶ℜ+N that gives all V∈ℒΩ,Λ a sdVd=0∞ holds the following conditions:(a)
V=s0V≥s1V≥s2V≥…≥0, for every V∈ℒΩ,Λ.(b)
sl+d−1V1+V2≤slV1+sdV2, for every V1,V2∈ℒΩ,Λ and l, d∈N.(c)
sdVYW≤VsdYW, for every W∈ℒΩ0,Ω, Y∈ℒΩ,Λ, and V∈ℒΛ,Λ0, where Ω0 and Λ0 are arbitrary Banach spaces.(d)
AssumeV∈ℒΩ,Λ and γ∈ℜ, then sdγV=γsdV.(e)
IfrankV≤d, then sdV=0, for all V∈ℒΩ,Λ.(f)
sl≥aIa=0 or sl<aIa=1, where Ia indicates the unit mapping on the a-dimensional Hilbert space ℓ2a.
We give here some examples ofs-numbers:(1)
Theqth Kolmogorov number, denoted by dqX, is marked by  dqX=infdimJ≤qsupf≤1infg∈JXf−g.(2)
Theq-th approximation number, indicated by αqX, is marked by  αqX=infX−Y:Y∈ℒΩ,Λand rankY≤q.Definition 3 (see [32]).
Letℒ be the class of all bounded linear operators within any two arbitrary Banach spaces. A subclass U of ℒ is said to be a mappings’ ideal, if every UΩ,Λ=U∩ℒΩ,Λ satisfies the following setups:(i)
IΓ∈U, where Γ indicates Banach space of one dimension.(ii)
The spaceUΩ,Λ is linear over ℜ.(iii)
IfW∈ℒΩ0,Ω, X∈UΩ,Λ, and Y∈ℒΛ,Λ0, then YXW∈UΩ0,Λ0.Notations 1 (see [27]).
(6)ℐ¯U:=ℐ¯UΩ,Λ,whereℐ¯UΩ,Λ:=V∈ℒΩ,Λ:sjV¯j=0∞∈U,ℐα¯U:=ℐα¯UΩ,Λ,whereℐα¯UΩ,Λ:=V∈ℒΩ,Λ:αjV¯j=0∞∈U,ℐd¯U:=ℐd¯UΩ,Λ,whereℐd¯UΩ,Λ:=V∈ℒΩ,Λ:djV¯j=0∞∈U,where(7)sjVt=¯1,t=sjV,0,t≠sjV.Definition 4 (see [6]).
A functionH∈0,∞U is said to be a pre-quasi-norm on the ideal U if the following conditions hold:(1)
AssumeV∈UΩ,Λ, HV≥0, and HV=0, if and only if, V=0(2)
One hasQ≥1 with HαV≤DαHV, for all V∈UΩ,Λ and α∈ℜ(3)
There areP≥1 such that HV1+V2≤PHV1+HV2, for all V1,V2∈UΩ,Λ(4)
There areσ≥1 so that if V∈ℒΩ0,Ω, X∈UΩ,Λ and Y∈ℒΛ,Λ0 then HYXV≤σYHXVTheorem 1 (see [6]).
H is a pre-quasi-norm on the ideal U, whenever H is a quasi-norm on the ideal U.Lemma 1 (see [33]).
Ifτa>0 and va,ta∈ℜ, for all a∈N, then va+taτa≤2K−1vaτa+taτa, where K=max1,supaτa.


## 3. Some Characteristics ofc0F∇p2,τ

This section is devoted to provide sufficient criteria for the space of null variable exponent second-order quantum backward difference sequences of fuzzy numbers,c0F∇p2,τ, endowed with definite function h, to be pre-quasi Banach. We have examined some algebraic and topological properties such as completeness, solidness, symmetry, and convergence-free. The Fatou property of various pre-quasi-norms h on c0F∇p2,τ has been presented.LetωF denote the classes of all sequence spaces of fuzzy real numbers. If τ=τa∈ℜ+N, where ℜ+N is the space of positive reals. The space of null variable exponent second-order quantum backward difference sequences of fuzzy numbers is defined as follows:c0F∇p2,τ=z=za∈ωF:lima⟶∞ρ¯∇p2μza,0¯τa/K=0, for someμ>0.Theorem 2.
Ifτa∈ℓ∞, then(8)c0F∇p2,τ=z=za∈ωF:lima⟶∞ρ¯∇p2μza,0¯τa/K=0,for anyμ>0.Proof.
(9)c0F∇p2,τ=z=za∈ωF:lima⟶∞ρ¯∇p2μza,0¯τa/K=0,for someμ>0,=z=za∈ωF:infaμτaKlima⟶∞ρ¯∇p2za,0¯τa/K≤lima⟶∞ρ¯∇p2μza,0¯τa/K=0,for someμ>0,=z=za∈ωF:lima⟶∞ρ¯∇p2za,0¯τa/K=0,=z=za∈ωF:lima⟶∞ρ¯∇p2μza,0¯τa/K=0,for anyμ>0.
It is clear to see that ifτa∈ℓ∞, then(10)c0Fτ⊊c0F∇p2,τ⊊c0F∇2,τ.
For the strict inclusion, we have1¯,1¯,1¯,…∉c0F and 1¯,1¯,1¯,…∈c0F∇p2,τ, and 0¯,1¯,2¯,…∉c0F∇p2,τ and 0¯,1¯,2¯,…∈c0F∇2,τ.
Forz=zk, a given sequence Sz denotes the set of all permutation of the elements of zk; that is, Sz=zπk.Definition 5.
(1)
A sequence space of fuzzy numbersU is said to be symmetric if Sz∈U, for all z∈U(2)
A sequence space of fuzzy numbersU is said to be convergence-free if yk∈U whenever zk∈U and zk=0¯ implies yk=0¯Theorem 3.
Ifτa∈ℓ∞, then the space c0F∇p2,τh is not symmetric.Proof.
considerxk=1¯,…,1¯︷3times,−1¯,…,−1¯︷3times,1¯,…,1¯︷6times,−1¯,…,−1¯︷6times,1¯,…,1¯︷9times,−1¯,…,−1¯︷9times,…. Then, xk∈c0F∇p2,τh. Now, if yk is the rearrangement of xk defined by yk=1¯,−1¯,1¯,−1¯,…, then yk∉c0F∇p2,τh. Therefore, the space c0F∇p2,τh is not symmetric.Theorem 4.
Ifτa∈ℓ∞, then the space c0F∇p2,τh is not convergence-free.Proof.
consider the sequencexk=1¯,1¯,…. Then, xk∈c0F∇p2,τh. Again if yk=k¯, then clearly, yk∉c0F∇p2,τh. Hence, the space c0F∇p2,τh is not convergence-free.
Let us mark the space of all functionsh:U⟶0,∞ by 0,∞U.Definition 6 (see [34]).
U is a vector space. A function h∈0,∞U is said to be modular if the following conditions hold:(a)
Assumey∈U, y=ϑ¯⇔hy=0 with hy≥0, where ϑ¯=0¯,0¯,0¯,…(b)
hηz=hz verifies for every z∈U and η=1(c)
The inequalityhαy+1−αz≤hy+hz holds for every y,z∈U and α∈0,1Definition 7 (see [27]).
The linear spaceU is called a certain space of sequences of fuzzy numbers (cssf), when(1)
b¯qq∈N⊆U, where b¯q=0¯,0¯,…,1¯,0¯,0¯,…, while 1¯ displays at the qth place(2)
U is solid; that is, if y=yq∈ωF, z=zq∈U, and yq≤zq for every q∈N, then y∈U(3)
yq/2q=0∞∈U, where q/2denotes the integral part of q/2, if yqq=0∞∈UDefinition 8 (see [27]).
A subclassUh of U is said to be a premodular (cssf), if there is h∈0,∞U, which satisfies the following conditions:(i)
Assumey∈U, y=ϑ¯⇔hy=0 with hy≥0, where ϑ¯=0¯,0¯,0¯,….(ii)
One hasQ≥1, the inequality hαy≤Qαhy holds for all y∈U and α∈ℜ.(iii)
One hasP≥1, the inequality hy+z≤Phy+hz verifies, for all y,z∈U.(iv)
Supposeyq≤zq, for all q∈N, then hyq≤hzq.(v)
The inequality,hyq≤hyq/2≤P0hyq, verifies, for some P0≥1.(vi)
IfE is the space of finite sequences of fuzzy numbers, then the closure of E=Uh.(vii)
One hasσ>0 with hα¯,0¯,0¯,0¯,…≥σαh1¯,0¯,0¯,0¯,…, where(11)α¯t=1,t=α0,t≠α.
We note that the notion of premodular vector spaces is more general than modular vector spaces. There are some examples of premodular vector spaces but not modular vector spaces.Example 1.
The functionhz=supqρ¯∇p2zq,0¯4q+1/q+4 on the vector space c0F∇p2,4q+1/q+4. As for every z,y∈c0F∇p2,4q+1/q+4, one has(12)hz+y2=supqρ¯∇p2zq+yq2,0¯4q+1/q+4≤824hz+hy.Example 2.
The functionhz=supqρ¯∇p2zq,0¯5q+2/q+1 on the vector space c0F∇p2,5q+2/q+1. As for every z,y∈c0F∇p2,5q+2/q+1, one has(13)hz+y2=supqρ¯∇p2zq+yq2,0¯5q+2/q+1≤4hz+hy.
Some examples of premodular vector spaces and modular vector spaces are as follows:Example 3.
The functionhz=supqρ¯∇p2zq,0¯q+1/3q+4 on the vector space c0F∇p2,q+1/3q+4. As for every z,y∈c0F∇p2,q+1/3q+4, one has(14)hz+y2=supqρ¯∇p2zq+yq2,0¯q+1/3q+4≤124hz+hy.Example 4.
The functionhy=infα>0:supqρ¯∇p2yq/α,0¯2q+3/q+2≤1 is a premodular (modular) on the vector space c0F∇p2,2q+3/q+2.Definition 9 (see [27]).
U is a cssf. The function h∈0,∞U is said to be a pre-quasi-norm on U, if it verifies the following settings:(i)
Supposey∈U, y=ϑ¯⇔hy=0 with hy≥0, where ϑ¯=0¯,0¯,0¯,….(ii)
We haveQ≥1, the inequality hαy≤Qαhy holds, for all y∈U and α∈ℜ.(iii)
One hasP≥1, the inequality hy+z≤Phy+hz verifies, for all y,z∈U.Theorem 5 (see [27]).
We suppose thatU is a premodular (cssf), then it is pre-quasi-normed (cssf).Theorem 6 (see [27]).
U is a pre-quasi-normed (cssf), if it is quasi-normed (cssf).Definition 10.
(a)
The functionh on c0F∇p2,τ is called h-convex, if(15)hαy+1−αz≤αhy+1−αhz,for everyα∈0,1 and y,z∈c0F∇p2,τ.(b)
yqq∈N⊆c0F∇p2,τh ish-convergent to y∈c0F∇p2,τh, if and only if, limq⟶∞hyq−y=0. When the h-limit exists, then it is unique.(c)
yqq∈N⊆c0F∇p2,τh is h-Cauchy, if limq,r⟶∞hyq−yr=0.(d)
Γ⊂c0F∇p2,τh is h-closed, when for all h-converges yqa∈N⊂Γ to y, then y∈Γ.(e)
Γ⊂c0F∇p2,τh is h-bounded, if δhΓ=suphy−z:y,z∈Γ<∞.(f)
Theh-ball of radius ε≥0 and center y, for every y∈c0F∇p2,τh, is described as follows:(16)Bhy,ε=z∈c0F∇p2,τh:hy−z≤ε.(g)
A pre-quasi-normh on c0F∇p2,τ satisfies the Fatou property, if for every sequence zq⊆c0F∇p2,τh under limq⟶∞hzq−z=0 and all y∈c0F∇p2,τh, one has hy−z≤suprinfq≥rhy−zq.
We recall that the Fatou property gives theh-closedness of the h-balls. We will denote the space of all increasing sequences of real numbers by I.Theorem 7.
c0F∇p2,τh, where hy=supqρ¯ηqq!∇p2yq1/q+1,0¯τq/K, for every y∈c0F∇p2,τ, is a premodular (cssf), if the following conditions are satisfied:(a)
τqq∈N∈ℓ∞∩I with τ0>0.(b)
∇p2 is an absolute nondecreasing; that is, if zi≤yi, for all i∈ℕ, then ∇p2zi≤∇p2yi.Proof.
Clearly,hy≥0 and hy=0⇔y=ϑ¯.(i)
Assumey,z∈c0F∇p2,τ. We have(17)hy+z=supqρ¯∇p2yq+zq,0¯τq/K≤supqρ¯∇p2yq,0¯τq/K+supqρ¯∇p2zq,0¯τq/K=hy+hz<∞.Then,y+z∈c0F∇p2,τ.(ii)
There areP≥1 with hy+z≤Phy+hz, for every y,z∈c0F∇p2,τ.(iii)
Ifα∈ℜ and y∈c0F∇p2,τ, one has(18)hαy=supqρ¯∇p2αyq,0¯τq/K≤supqατq/Ksupqρ¯∇p2yq,0¯τq/K≤Qαhy<∞.So,αy∈c0F∇p2,τ. From parts (1-i) and (1-ii), we have c0F∇p2,τ is linear. Also, b¯p∈c0F∇p2,τ, for every p∈N, as hb¯p=supqρ¯∇p2b¯pq,0¯τq/K=1.(iv)
One hasQ=max1,supqατq/K−1≥1 with hαy≤Qαhy, for every y∈c0F∇p2,τ and α∈ℜ.(v)
Ifyq≤zq, for every q∈N and z∈c0F∇p2,τ. We obtain(19)hy=supqρ¯∇p2yq,0¯τq/K≤supqρ¯∇p2zq,0¯τq/K=hz<∞.Then,y∈c0F∇p2,τ.(vi)
Evidently, from Reference (2).(vii)
Assumeyq∈c0F∇p2,τ, one can see(20)hyq/2=supqρ¯∇p2yq/2,0¯τq/K≤maxsupqρ¯∇p2yq,0¯τ2q/K,supqρ¯∇p2yq,0¯τ2q+1/K≤supqρ¯∇p2yq,0¯τq/K=hyq.Then,yq/2∈c0F∇p2,τ. (v) From (3), one has P0=2≥1.(viii)
Clearly, the closure ofE=c0F∇p2,τ.(ix)
One gets0<σ≤supqατq/K−1, for α≠0 or σ>0, for α=0 with hα¯,0¯,0¯,0¯,…≥σαh1¯,0¯,0¯,0¯,….Theorem 8.
If the conditions of Theorem7 are satisfied, then c0F∇p2,τh is a pre-quasi Banach (cssf), where hy=supqρ¯∇p2yq,0¯τq/K, for all y∈c0F∇p2,τ.Proof.
According to Theorem7 and Theorem 5, the space c0F∇p2,τh is a pre-quasi-normed (cssf). If yl=yqlq=0∞ is a Cauchy sequence in c0F∇p2,τh, hence for all ε∈0,1, then l0∈N such that for every l,m≥l0, we have(21)hyl−ym=supqρ¯∇p2yql−yqm,0¯τq/K<ε.
Therefore,ρ¯∇p2yql−yqm,0¯<ε. Since ℜ0,1,ρ¯ is a complete metric space, yqm is a Cauchy sequence in ℜ0,1, for fixed q∈N. This gives limm⟶∞yqm=yq0, for fixed q∈N. Then, hyl−y0<ε, for all l≥l0. As hy0=hy0−yl+yl≤hyl−y0+hyl<∞. Then, y0∈c0F∇p2,τ.Theorem 9.
The functionhy=supqρ¯∇p2yq,0¯τq/K satisfies the Fatou property, when the conditions of Theorem 7 are satisfied.Proof.
Letzr⊆c0F∇p2,τh such that limr⟶∞hzr−z=0. Since c0F∇p2,τh is a pre-quasi closed space, we have z∈c0F∇p2,τh. For every y∈c0F∇p2,τh, then(22)hy−z=supqρ¯∇p2yq−zq,0¯τq/K≤supqρ¯∇p2yq−zqr,0¯τq/K+supqρ¯∇p2zqr−zq,0¯τq/K≤supminfr≥mhy−zr.Theorem 10.
The functionhy=supqρ¯∇p2yq,0¯τq does not satisfy the Fatou property, for all y∈c0F∇p2,τ, if the conditions of Theorem 7 are satisfied with τ0>1.Proof.
Assumezr⊆c0F∇p2,τh such that limr⟶∞hzr−z=0. As c0F∇p2,τh is a pre-quasi closed space, we have z∈c0F∇p2,τh. For all z∈c0F∇p2,τh, then(23)hy−z=supqρ¯∇p2yq−zq,0¯τq≤2supqτq−1supqρ¯∇p2yq−zqr,0¯τq+supqρ¯∇p2zqr−zq,0¯τq≤2supqτq−1supminfr≥mhy−zr.Example 5.
Forτq∈1,∞N, the function hy=infα>0:supqρ¯∇p2yq/α,0¯τq≤1 is a norm on c0F∇p2,τ.Example 6.
The functionhy=supqρ¯∇p2yq,0¯3q+2/q+3 is a pre-quasi-norm (not a norm) on c0F∇p2,3q+2/q+3q=0∞.Example 7.
The functionhy=supqρ¯∇p2yq,0¯3q+2/q+1 is a pre-quasi-norm (not a quasi-norm) on c0F∇p2,3q+2/q+1q=0∞.


## 4. Structure of Mappings’ Ideal

The structure of the mappings’ ideal byc0F∇p2,τh, where hz=supqρ¯∇p2zq,0¯τq/K, for all z∈c0F∇p2,τ, and extended s− fuzzy functions have been explained. We study enough setups on c0F∇p2,τh such that the class ℐ¯c0F∇p2,τh is complete. We investigate conditions (not necessary) on c0F∇p2,τh such that the closure ofF=ℐα¯c0F∇p2,τh. This gives a negative answer of Rhoades [34] open problem about the linearity of s− type c0F∇p2,τh spaces. We explain enough setups on c0F∇p2,τh such that ℐ¯c0F∇p2,τh is strictly contained for different powers, weights, and backward generalized differences, the class ℐ¯c0F∇p2,τh is simple, and the space of every bounded linear mappings is which sequence of eigenvalues in c0F∇p2,τh equals ℐ¯c0F∇p2,τh.Theorem 11 (see [27]).
IfU is a (cssf), then ℐ¯U is a mappings’ ideal.
In view of Theorem7 and Theorem 11, one has the following theorem:Theorem 12.
If the conditions of Theorem7 are satisfied, then ℐ¯c0F∇p2,τh is a mappings’ ideal.Theorem 13.
If the conditions of Theorem7 are satisfied, then the function H is a pre-quasi-norm on ℐ¯c0F∇p2,τh, with HZ=supqρ¯∇p2sqZ¯,0¯τq/K, for every Z∈ℐ¯c0F∇p2,τhΩ,Λ.Proof.
(1)
SupposeX∈ℐ¯c0F∇p2,τhΩ,Λ, HX=supqρ¯∇p2sqZ¯,0¯τq/K≥0 and HX=supqρ¯∇p2sqZ¯,0¯τq/K=0, if and only if, sqX¯=0¯, for all q∈N, if and only if, X=0.(2)
One hasQ≥1 with HαX=supqρ¯∇p2sqαX¯,0¯τq/K≤QαHX, for all X∈ℐ¯c0F∇p2,τhΩ,Λ and α∈ℜ.(3)
ForX1,X2∈ℐ¯c0F∇p2,τhΩ,Λ, we have(24)HX1+X2=supqρ¯∇p2|sqX1+X2¯|,0¯τq/K≤hsq/2X1¯q=0∞+hsq/2X2¯q=0∞≤hsqX1¯q=0∞+hsqX2¯q=0∞.(4)
There areϱ≥1, if X∈ℒΩ0,Ω, Y∈ℐ¯c0F∇p2,τhΩ,Λ and Z∈ℒΛ,Λ0, then(25)HZYX=supqρ¯∇p2sqZYX¯,0¯τq/K≤hXZsqY¯q=0∞≤ϱXHYZ.
In the next theorems, we will use the notationℐ¯c0F∇p2,τh,H, where HV=hsqV¯q=0∞, for all V∈ℐ¯c0F∇p2,τh.Theorem 14.
assume that the conditions of Theorem7 are satisfied, then ℐ¯c0F∇p2,τh,H is a pre-quasi Banach mappings’ ideal.Proof.
LetVaa∈N be a Cauchy sequence in ℐ¯c0F∇p2,τhΩ,Λ. Since ℒΩ,Λ⊇Sc0F∇p2,τhΩ,Λ, then(26)HVr−Va=supqρ¯∇p2sqVr−Va¯,0¯τq/K≥hs0Vr−Va¯,0¯,0¯,0¯,…≥Vr−Vaτ0/K.
This implies thatVaa∈N is a Cauchy sequence in ℒΩ,Λ. Since ℒΩ,Λ is a Banach space, one has V∈ℒΩ,Λ such that lima⟶∞Va−V=0 and as sqVa¯q=0∞∈c0F∇p2,τh, for every a∈N, and c0F∇p2,τh is a premodular (cssf). Then, we have(27)HV=hsqV¯q=0∞≤hsq/2V−Va¯q=0∞+hsq/2Va¯q=0∞≤hVa−V1¯q=0∞+hsqVa¯q=0∞<ε.
Hence, one hassqV¯q=0∞∈c0F∇p2,τh, then V∈ℐ¯c0F∇p2,τhΩ,Λ.Definition 11.
A pre-quasi-normH on the ideal ℐ¯Uh satisfies the Fatou property if for all Tqq∈N⊆ℐ¯UhΩ,Λ such that limq⟶∞HTq−T=0 and M∈ℐ¯UhΩ,Λ, then(28)HM−T≤supqinfj≥qHM−Tj.Theorem 15.
If the conditions of Theorem7 are satisfied, then ℐ¯c0F∇p2,τh,H does not satisfy the Fatou property.Proof.
LetTqq∈N⊆ℐ¯c0F∇p2,τhΩ,Λ with limq⟶∞HTq−T=0. Since ℐ¯c0F∇p2,τh is a pre-quasi closed ideal, then T∈ℐ¯c0F∇p2,τhΩ,Λ; hence, for all M∈ℐ¯c0F∇p2,τhΩ,Λ, we have(29)HM−T=supqρ¯∇p2sqM−T¯,0¯τq/K≤supqρ¯∇p2sq/2M−Ti¯,0¯τq/K+supqρ¯∇p2sq/2Ti−T¯,0¯τq/K≤suprinfi≥rsupqρ¯∇p2sqM−Ti¯,0¯τq/K.Theorem 16.
ℐα¯c0F∇p2,τhΩ,Λ=the closure ofFΩ,Λ, if the conditions of Theorem 7 are satisfied. But the converse is not necessarily true.Proof.
Asb¯m∈c0F∇p2,τh, for all m∈N and c0F∇p2,τh is a linear space. If Z∈FΩ,Λ, one has αmZ¯m=0∞∈E. Then, the closure ofFΩ,Λ⊆ℐα¯c0F∇p2,τhΩ,Λ. Suppose Z∈ℐα¯c0F∇p2,τhΩ,Λ, one has αmZ¯m=0∞∈c0F∇p2,τh. Since hαmZ¯m=0∞<∞, if ρ∈0,1, one has m0∈N−0 so that hαmZ¯m=m0∞<ρ/4. As αmZ¯m=0∞ is decreasing, one gets(30)supm=m0+12m0ρ¯∇p2α2m0Z¯,0¯τm/K≤supm=m0+12m0ρ¯∇p2αmZ¯,0¯τm/K≤supm=m0∞ρ¯∇p2αmZ¯,0¯τm/K<ρ4.
Then, one hasY∈F2m0Ω,Λ such that rank Y≤2m0 and(31)supm=2m0+13m0ρ¯∇p2Z−Y¯,0¯τm/K≤supm=m0+12m0ρ¯∇p2Z−Y¯,0¯τm/K<ρ4.
Asτqq∈N∈ℓ∞∩I with τ0>0, we take(32)supm=0m0ρ¯∇p2Z−Y¯,0¯τm/K<ρ4.
According to inequalities (1–3), then(33)dZ,Y=supm=0∞ρ¯∇p2αmZ−Y¯,0¯τm/K≤supm=03m0−1ρ¯∇p2αmZ−Y¯,0¯τm/K+supm=3m0∞ρ¯∇p2αmZ−Y¯,0¯τm/K≤supm=03m0ρ¯∇p2Z−Y¯,0¯τm/K+supm=m0∞ρ¯∇p2αm+2m0Z−Y¯,0¯τm+2m0/K≤supm=03m0ρ¯∇p2Z−Y¯,0¯τm/K+supm=m0∞ρ¯∇p2αmZ¯,0¯τm/K≤3supm=0m0ρ¯∇p2Z−Y¯,0¯τm/K+supm=m0∞ρ¯∇p2αmZ¯,0¯τm/K<ρ.
This impliesℐα¯c0F∇p2,τhΩ,Λ⊆the closure ofFΩ,Λ. Contrarily, one has a counterexample as I3∈ℐα¯c0F∇p2,0,0,1,1,…Ω,Λ, but τ0>0 is not satisfied.Theorem 17.
assume the conditions of Theorem7 are satisfied with τm1<τm2, for every m∈N, then(34)ℐ¯c0F∇q2,τm1nΩ,Λ⊊ℐ¯c0F∇2,τm2nΩ,Λ⊊ℒΩ,Λ.Proof.
suppose thatZ∈ℐ¯c0F∇q2,τm1hΩ,Λ, then smZ¯∈c0F∇q2,τm1h. We have(35)limm⟶∞ρ¯∇2smZ¯,0¯τm2=limm⟶∞ρ¯∇q2smZ¯,0¯τm1=0.
Then,Z∈ℐ¯c0F∇2,τm2hΩ,Λ. Next, if we take smZ¯m=0∞=0¯,1¯,2¯,…, one has Z∈ℒΩ,Λ so that(36)limm⟶∞ρ¯∇q2smZ¯,0¯τm1≠0,limm⟶∞ρ¯∇2smZ¯,0¯τm2=0.
Therefore,Z∉ℐ¯c0F∇q2,τm1hΩ,Λ and Z∈ℐ¯c0F∇2,τm2hΩ,Λ.
Evidently,ℐ¯c0F∇2,τm2hΩ,Λ⊂ℒΩ,Λ. After, if we choose smZ¯m=0∞, then ∇2smZ¯=1¯,1¯,…. One has Z∈ℒΩ,Λ such that Z∉ℐ¯χ0F∇q2,τm2hΩ,Λ.Lemma 2 (see [3]).
If we supposeB∈ℒΩ,Λ and B∉ΥΩ,Λ, then D∈ℒΩ and M∈ℒΛ with MBDeb=eb, with b∈N.Theorem 18 (see [3]).
In general, one has(37)FΩ⊊YΩ⊊ℒcΩ⊊ℒΩ.Theorem 19.
If the conditions of Theorem7 are satisfied with τm1<τm2, for all m∈N, then(38)ℒℐ¯c0F∇2,τm2hΩ,Λ,ℐ¯c0F∇q2,τm1hΩ,Λ=Υℐ¯c0F∇2,τm2hΩ,Λ,ℐ¯c0F∇q2,τm1hΩ,Λ.Proof.
LetX∈ℒℐ¯c0F∇2,τm2hΩ,Λ,ℐ¯c0F∇q2,τm1hΩ,Λ and
X∉Υℐ¯c0F∇2,τm2hΩ,Λ,ℐ¯c0F∇q2,τm1hΩ,Λ. In view of Lemma 2, one has
Y∈ℒℐ¯c0F∇2,τm2hΩ,Λand Z∈ℒℐ¯c0F∇q2,τm1hΩ,Λ so that ZXYIb=Ib, and then, with b∈N, we have(39)Ibℐ¯c0F∇q2,τm1hΩ,Λ=supmρ¯∇q2smIb¯,0¯τm1≤ZXYIbℐ¯c0F∇2,τm2hΩ,Λ≤supmρ¯∇2smIb¯,0¯τm2.
This contradicts Theorem18. As X∈Υℐ¯c0F∇2,τm2hΩ,Λ,ℐ¯c0F∇q2,τm1hΩ,Λ.Corollary 1.
suppose that the conditions of Theorem7 are satisfied with τm1<τm2, for every m∈N, then(40)ℒℐ¯c0F∇2,τm2hΩ,Λ,ℐ¯c0F∇q2,τm1hΩ,Λ=ℒcℐ¯c0F∇2,τm2hΩ,Λ,ℐ¯c0F∇q2,τm1hΩ,Λ.Proof.
Obviously, sinceΥ⊂ℒc.Definition 12.
[3] A Banach space Ω is said to be simple, if there is only one nontrivial closed ideal in ℒΩ.Theorem 20.
assume that the conditions of Theorem7 are verified, then ℐ¯c0F∇p2,τh is simple.Proof.
LetX∈ℒcℐ¯c0F∇p2,τhΩ,Λ and X∉Υℐ¯c0F∇p2,τhΩ,Λ. From Lemma 2, there exist Y,Z∈ℒℐ¯c0F∇p2,τhΩ,Λ with ZXYIb=Ib. This implies Iℐ¯c0F∇p2,τhΩ,Λ∈ℒcℐ¯c0F∇p2,τhΩ,Λ. If ℒℐ¯c0F∇p2,τhΩ,Λ=ℒcℐ¯c0F∇p2,τhΩ,Λ, then ℐ¯c0F∇p2,τh is a simple Banach space.Notations 2.
(41)ℐ¯Uλ:=ℐ¯UλΩ,Λ;ΩandΛare Banach Spaces,whereℐ¯UλΩ,Λ:=X∈ℒΩ,Λ:λmXm=0∞∈UandX−ρ¯λmX,0¯Iis not invertible,withm∈N.Theorem 21.
If the conditions of Theorem7 are satisfied, then(42)ℐ¯c0F∇p2,τhλΩ,Λ=ℐ¯c0F∇p2,τhΩ,Λ.Proof.
LetX∈ℐ¯c0F∇p2,τhλΩ,Λ, then λmXm=0∞∈c0F∇p2,τh and X−ρ¯λmX,0¯I=0, for all m∈N. Therefore, limm⟶∞ρ¯∇p2λmX,0¯τm/K=0. One has X=ρ¯λmX,0¯I, for every m∈N, so(43)ρ¯smX¯,0¯=ρ¯smρ¯λmX,0¯I¯,0¯=ρ¯λmX,0¯,
for everym∈N. Hence, smX¯m=0∞∈c0F∇p2,τh, and then, X∈ℐ¯c0F∇p2,τhΩ,Λ. After, we assume X∈ℐ¯c0F∇p2,τhΩ,Λ. Hence, smX¯m=0∞∈c0F∇p2,τh. We have(44)limm⟶∞ρ¯∇p2smX¯,0¯τm/K=0.
As∇p2 is continuous, then limm⟶∞ρ¯smX¯,0¯=0. If X−ρ¯smX¯,0¯I−1 exists, with m∈N, then X−ρ¯smX¯,0¯I−1 exists and bounded, for every m∈N. As limm⟶∞X−ρ¯smX¯,0¯I−1=X−1 exists and bounded. As ℐ¯c0F∇p2,τh,H is a pre-quasi mappings’ ideal, one gets(45)I=XX−1∈ℐ¯c0F∇p2,τhΩ,Λ⇒smI¯m=0∞∈c0F∇p2,τ⇒limm⟶∞ρ¯smI¯,0¯=0.
We have a contradiction, sincelimm⟶∞ρ¯smI¯,0¯=1. Then, X−ρ¯smX¯,0¯I=0, with m∈N, which proves that X∈ℐ¯c0F∇p2,τhλΩ,Λ.Theorem 22.
Fors− type, Uh:=z=srX¯∈ωF:X∈ℒΩ,Λandhz<∞. If ℐ¯Uh is a mappings’ ideal, then the following conditions are verified:(1)
E⊂s− type Uh.(2)
AssumesrX1¯r=0∞∈s− type Uh and srX2¯r=0∞∈s− type Uh, then srX1+X2¯r=0∞∈s− type Uh.(3)
Ifλ∈ℜ and srX¯r=0∞∈s− type Uh, then λsrX¯r=0∞∈s− type Uh.(4)
The sequence spaceUh is solid; that is, if srY¯r=0∞∈s− type Uh and srX¯≤srY¯, for all r∈N and X,Y∈ℒΩ,Λ, then srX¯r=0∞∈s− type Uh.Proof.
Ifℐ¯Uh is a mappings’ ideal.(i)
We haveFΩ,Λ⊂ℐ¯UhΩ,Λ. Hence, for all X∈FΩ,Λ, we have srX¯r=0∞∈E. This gives srX¯r=0∞∈s−typeUh. Hence, E⊂s− type Uh.(ii)
The spaceℐ¯UhΩ,Λ is linear over ℜ. Hence, for each λ∈ℜ and X1,X2∈ℐ¯UhΩ,Λ, we have X1+X2∈ℐ¯UhΩ,ΛandλX1∈ℐ¯UhΩ,Λ. This implies(46)srX1¯r=0∞∈s−typeUhandsrX2¯r=0∞∈s−typeUh⇒srX1+X2¯r=0∞∈s−typeUh,λ∈ℜandsrX1¯r=0∞∈s−typeUh⇒λsrX1¯r=0∞∈s−typeUh.(iii)
IfA∈ℒΩ0,Ω, B∈ℐ¯UhΩ,Λ and D∈ℒΛ,Λ0, then DBA∈ℐ¯UhΩ0,Λ0, where Ω0 and Λ0 are arbitrary Banach spaces. Therefore, since srB¯r=0∞∈s−typeUh, then srDBA¯r=0∞∈s−typeUh. Since srDBA¯≤DsrB¯A. By using condition 3, if DAsrB¯r=0∞∈Uh, we have srDBA¯r=0∞∈s−typeUh. This means s−typeUh is solid.
In view of Theorem12 and Theorem 23, we conclude the following properties of the s−typec0F∇p2,τh space.Theorem 23.
Ifs−typec0F∇p2,τh:=z=srX¯∈ωF:X∈ℒΩ,Λandhz<∞, then the following conditions are verified:(1)
E⊂s− type c0F∇p2,τh.(2)
AssumesrX1¯r=0∞∈s− type c0F∇p2,τh and srX2¯r=0∞∈s− type c0F∇p2,τh, then srX1+X2¯r=0∞∈s− type c0F∇p2,τh.(3)
Ifλ∈ℜ and srX¯r=0∞∈s− type c0F∇p2,τh, then λsrX¯r=0∞∈s− type c0F∇p2,τh.(4)
The sequence spacec0F∇p2,τh is solid; that is, if srY¯r=0∞∈s− type c0F∇p2,τh and srX¯≤srY¯, for all r∈N and X,Y∈ℒΩ,Λ, then srX¯r=0∞∈s− type c0F∇p2,τh.Theorem 24.
The spaceI¯0F∇p2,τ is not mappings’ ideal, if the conditions (a) and (c) of Theorem 7 are satisfiedProof.
If we choosem=1, n=1, zk=1¯, yk=zk for k=3s or yk=0¯, otherwise, for all s,k∈N. We have yk≤zk, for all k∈N, z∈c0F∇p2,τh and y∉c0F∇p2,τh. Hence, the space c0F∇p2,τh is not solid.


## 5. Kannan Contraction Mapping onc0F∇p2,τ

In this section, we look at how to configurec0F∇p2,τh with different h so that there is only one fixed point of Kannan contraction mapping. We construct the existence of a fixed point of Kannan contraction mapping acting on this space and its associated pre-quasi ideal. Interestingly, several numerical experiments are presented to illustrate our results.Definition 13.
An operatorV:Uh⟶Uh is said to be a Kannan h-contraction, if one gets α∈0,1/2 with hVy−Vz≤αhVy−y+hVz−z, for all y,z∈Uh.
An elementy∈Uh is called a fixed point of V, when Vy=y.Theorem 25.
If the conditions of Theorem7 are satisfied, and V:c0F∇p2,τh⟶c0F∇p2,τh is Kannan h-contraction mapping, where hy=supqρ¯∇p2yq,0¯τq/K, for all y∈c0F∇p2,τ, then V has a unique fixed point.Proof.
Ify∈c0F∇p2,τ, one has Vpy∈c0F∇p2,τ. As V is a Kannan h-contraction mapping, one gets(47)hVl+1y−Vly≤αhVl+1y−Vly+hVly−Vl−1y⇒hVl+1y−Vly≤α1−αhVly−Vl−1y≤α1−α2hVl−1y−Vl−2y≤⋯≤α1−αlhVy−y.
So, for alll,m∈N with m>l, one gets(48)hVly−Vmy≤αhVly−Vl−1y+hVmy−Vm−1y≤αα1−αl−1+α1−αm−1hVy−y.
Then,Vly is a Cauchy sequence in c0F∇p2,τh. As the space c0F∇p2,τh is pre-quasi Banach space. One has z∈c0F∇p2,τh with liml⟶∞Vly=z to prove that Vz=z. Since h verifies the Fatou property, one obtains(49)hVz−z≤supiinfl≥ihVl+1y−Vly≤supiinfl≥iα1−αlhVy−y=0.
Then,Vz=z. So, z is a fixed point of V to show the uniqueness. Let y,z∈c0F∇p2,τh be two not equal fixed points of V. One has(50)hy−z≤hVy−Vz≤αhVy−y+hVz−z=0.
So,y=z.Corollary 2.
If the conditions of Theorem7 are satisfied, and V:c0F∇p2,τh⟶c0F∇p2,τh is Kannan h-contraction mapping, where hy=supqρ¯∇p2yq,0¯τq/K, for all y∈c0F∇p2,τ, one has unique fixed point z of V so that hVly−z≤αα/1−αl−1hVy−y.Proof.
In view of Theorem26, one has a unique fixed point z of V. So(51)hVly−z=hVly−Vz≤αhVly−Vl−1y+hVz−z=αα1−αl−1hVy−y.Example 8.
assumeV:c0F∇p2,2q+3/q+2h⟶c0F∇p2,2q+3/q+2h, where hz=supqρ¯∇p2zq,0¯2q+3/2q+4, for every z∈c0F∇p2,2q+3/q+2 and(52)Vz=z4,hz∈0,1,z5,hz∈1,∞.
As for eachx,y∈c0F∇p2,2q+3/q+2h with hx,hy∈0,1, one has(53)hVx−Vy=hx4−y4≤1274h3x4+h3y4=1274hVx−x+hVy−y.
For allx,y∈c0F∇p2,2q+3/q+2h with hx,hy∈1,∞, one has(54)hVx−Vy=hx5−y5≤1644h4x5+h4y5=1644hVx−x+hVy−y.
For allx,y∈c0F∇p2,2q+3/q+2h with hx∈0,1 and hy∈1,∞, we get(55)hVx−Vy=hx4−y5≤1274h3x4+1644h4y5≤1274h3x4+h4y5=1274hVx−x+hVy−y.
Hence,V is Kannan h-contraction as h satisfies the Fatou property. From Theorem 26, one has V holds one fixed point ϑ¯∈c0F∇p2,2q+3/q+2h.Definition 14.
pick upUh be a pre-quasi-normed (cssf), V:Uh⟶Uh and z∈Uh. The operator V is called h-sequentially continuous at z, if and only if, when limq⟶∞hyq−z=0, then limq⟶∞hVyq−Vz=0.Example 9.
suppose thatV:c0F∇p2,q+1/2q+4h⟶c0F∇p2,q+1/2q+4h, where hz=supqρ¯∇p2zq,0¯4q+4/2q+4, for every z∈c0F∇p2,q+1/2q+4h and(56)Vz=118b¯0+z,z0t∈0,117,117b¯0,z0t=117,118b¯0,z0t∈117,1.V is clearly both h-sequentially continuous and discontinuous at 1/17b¯0∈c0F∇p2,q+1/2q+4h.Example 10.
assume thatV is defined as in Example 8. Suppose zn⊆c0F∇p2,2q+3/q+2h is such that limn⟶∞hzn−z0=0, where z0∈c0F∇p2,2q+3/q+2h with hz0=1.
As the pre-quasi-normh is continuous, we have(57)limn⟶∞hVzn−Vz0=limn⟶∞hzn4−z05=hz020>0.
Therefore,V is not h-sequentially continuous at z0.Theorem 26.
If the conditions of Theorem7 are satisfied with τ0>1, and V:c0F∇p2,τh⟶c0F∇p2,τh, where hy=supqρ¯∇p2yq,0¯τq, for all y∈c0F∇p2,τ, then we suppose that(1)
V is Kannan h-contraction mapping.(2)
V is h-sequentially continuous at z∈c0F∇p2,τh.(3)
There isy∈c0F∇p2,τh with Vly has Vljy converging to z.
Then,z∈c0F∇p2,τh is the only fixed point of V.Proof.
If we assume thatz is not a fixed point of V, one has Vz≠z. From parts (2) and (3), we get(58)limlj⟶∞hVljy−z=0,limlj⟶∞hVlj+1y−Vz=0.
AsV is Kannan h-contraction, one obtains(59)0<hVz−z=hVz−Vlj+1y+Vljy−z+Vlj+1y−Vljy≤22supiτi−2hVlj+1y−Vz+22supiτi−2hVljy−z+2supiτi−1αα1−αlj−1hVy−y.
Aslj⟶∞, one has a contradiction. Then, z is a fixed point of V to show that the uniqueness. Let z,y∈c0F∇p2,τh be two not equal fixed points of V. One obtains(60)hz−y≤hVz−Vy≤αhVz−z+hVy−y=0.
Hence,z=y.Example 11.
assume thatV is defined as in Example 8. Let hz=supqρ¯∇p2zq,0¯2q+3/q+2, for all z∈c0F∇p2,2q+3/q+2h. Since for all x,y∈c0F∇p2,2q+3/q+2h with hx,hy∈0,1, one gets(61)hVx−Vy=hx4−y4≤227h3x4+h3y4=227hVx−x+hVy−y.
For allx,y∈c0F∇p2,2q+3/q+2h with hx,hy∈1,∞, one gets(62)hVx−Vy=hx5−y5≤14h4x5+h4y5=14hVx−x+hVy−y.
For allx,y∈c0F∇p2,2q+3/q+2h with hx∈0,1 and hy∈1,∞, one gets(63)hVx−Vy=hx4−y5≤227h3x4+14h4y5≤227h3x4+h4y5=227hVx−x+hVy−y.
So,V is Kannanh-contraction and Vpz=z/4p,hz∈0,1,z/5p,hz∈1,∞.
Obviously,V is h-sequentially continuous at ϑ¯∈c0F∇p2,2q+3/q+2h and Vpz holds Vljz converges to ϑ¯. By Theorem 27, the point ϑ¯∈c0F∇p2,2q+3/q+2h is the only fixed point of V.Definition 15.
An operatorV:ℐ¯UhΩ,Λ⟶ℐ¯UhΩ,Λ is said to be a Kannan H-contraction, if one has α∈0,1/2 with HVT−VM≤αHVT−T+HVM−M, for all T,M∈ℐ¯UhΩ,Λ.Definition 16.
An operatorV:ℐ¯UhΩ,Λ⟶ℐ¯UhΩ,Λ is said to be H-sequentially continuous at M, where M∈ℐ¯UhΩ,Λ, if and only if, limr⟶∞HTr−M=0⇒limr⟶∞HVTr−VM=0.Example 12.
IfV:ℐ¯c0F∇p2,2q+3/q+2hΩ,Λ⟶ℐ¯c0F∇p22q+3/q+2hΩ,Λ,where HT=supqρ¯∇p2sqT¯,0¯2q+3/2q+4, for every T∈ℐ¯c0F∇p2,2q+3/q+2hΩ,Λ and(64)VT=T6,HT∈0,1,T7,HT∈1,∞.
Evidently,V is H-sequentially continuous at the zero operator Θ∈ℐ¯c0F∇p2,2q+3/q+2hΩ,Λ. Let Tj⊆ℐ¯c0F∇p2,2q+3/q+2hΩ,Λ be such that limj⟶∞HTj−T0=0, where T0∈ℐ¯c0F∇p2,2q+3/q+2hΩ,Λ with HT0=1. Since the pre-quasi-norm H is continuous, one gets(65)limj⟶∞HVTj−VT0=limj⟶∞HT06−T07=HT042>0.
Therefore,V is not H-sequentially continuous at T0.Theorem 27.
the conditions of Theorem7 are satisfied and V:ℐ¯c0F∇p2,τhΩ,Λ⟶ℐ¯c0F∇p2,τhΩ,Λ, then we assume that(i)
V is Kannan H-contraction mapping.(ii)
V is H-sequentially continuous at an element M∈ℐ¯c0F∇p2,τhΩ,Λ.(iii)
There areG∈ℐ¯c0F∇p2,τhΩ,Λ such that the sequence of iterates VrG has a VrmG converging to M.
Then,M∈ℐ¯c0F∇p2,τhΩ,Λ is the unique fixed point of V.Proof.
LetM be not a fixed point of V; hence, VM≠M. By using parts (ii) and (iii), we get(66)limrm⟶∞HVrmG−M=0andlimrm⟶∞HVrm+1G−VM=0.
SinceV is Kannan H-contraction, one obtains(67)0<HVM−M=HVM−Vrm+1G+VrmG−M+Vrm+1G−VrmG≤2HVrm+1G−VM+4HVrmG−M+4αα1−αrm−1HVG−G.
Asrm⟶∞, there is a contradiction. Hence, M is a fixed point of V to prove that the uniqueness of the fixed point M. We suppose that one has two not equal fixed points M,J∈ℐ¯c0F∇p2,τhΩ,Λ of V. So, one gets HM−J≤HVM−VJ≤αHVM−M+HVJ−J=0. Then, M=J.Example 13.
In view of Example12. Since for all T1,T2∈ℐ¯c0F∇p2,2q+3/q+2hΩ,Λ with HT1,HT2∈0,1, we have(68)HVT1−VT2=HT16−T26≤11254H5T16+H5T26=11254HVT1−T1+HVT2−T2.
For allT1,T2∈ℐ¯c0F∇p2,2q+3/q+2hΩ,Λ with HT1,HT2∈1,∞, we have(69)HVT1−VT2=HT17−T27≤12164H6T17+H6T27=12164HVT1−T1+HVT2−T2.
For allT1,T2∈ℐ¯c0F∇p2,2q+3/q+2hΩ,Λ with HT1∈0,1 and HT2∈1,∞, we have(70)HVT1−VT2=HT16−T27≤11254H5T16+12164H6T27≤11254HVT1−T1+HVT2−T2.
Hence,V is Kannan H-contraction and VrT=T/6r,HT∈0,1,T/7r,HT∈1,∞.
Obviously,V is H-sequentially continuous at Θ∈ℐ¯c0F∇p2,2q+3/q+2hΩ,Λ and VrT has a subsequence VrmT converges to Θ. By Theorem 28, Θ is the only fixed point of G.


## 6. Applications

In this section, some successful applications to the existence of solutions of nonlinear difference equations of fuzzy functions are introduced.Theorem 28.
consider the summable equation(71)yq=Rq+∑r=0∞Dq,rmr,yr,
which presented by Salimi et al. [35], and assume V:c0F∇p2,τh⟶c0F∇p2,τh, where the conditions of Theorem 7 are satisfied and hy=supqρ¯∇p2yq,0¯τq/K, for every y∈c0F∇p2,τ, defined by(72)Vyqq∈N=Rq+∑r=0∞Dq,rmr,yrq∈N.
The summable equation (4) has a unique solution in c0F∇p2,τh, if D:N2⟶ℜ, m:N×ℜ0,1⟶ℜ0,1, R:N⟶ℜ0,1, z:N⟶ℜ0,1, there is ε so that supqετq/K∈0,0.5, and for all q∈N, we have(73)∑r∈NDq,rmr,yr−mr,zr≤εRq−yq+∑r=0∞Dq,rmr,yr+Rq−zq+∑r=0∞Dq,rmr,zr.Proof.
One has(74)hVy−Vz=supqρ¯∇p2Vyq−Vzq,0¯τq/K=supqρ¯∇p2∑r∈NDq,rmr,yr−mr,zr,0¯τq/K≤supqετq/Ksupqρ¯∇p2Rq−yq+∑r=0∞Dq,rmr,yr,0¯τq/K+supqετq/Ksupqρ¯∇p2Rq−zq+∑r=0∞Dq,rmr,zr,0¯τq/K=supqετq/KhVy−y+hVz−z.
By Theorem26, one gets a unique solution of equation (4) in c0F∇p2,τh.Example 14.
supposec0F∇p2,2q+3/q+2h, where hy=supqρ¯∇p2yq,0¯2q+3/2q+4, for all y∈c0F∇p2,2q+3/q+2. We consider the summable equation(75)yq=Rq+∑r=0∞−1q+ryqq2+r2+1t,
witht>0. Let V:c0F∇p2,2q+3/q+2⟶c0F∇p2,2q+3/q+2 defined by(76)Vyq=Rq+∑r=0∞−1q+ryqq2+r2+1t.
Obviously,(77)∑r=0∞−1qyqq2+r2+1t−1r−−1r≤εRq−yq+∑r=0∞−1q+ryqq2+r2+1t+Rq−zq+∑r=0∞−1q+rzqq2+r2+1t.
By Theorem29, the summable equation (75) has a unique solution in c0F∇p2,2q+3/q+2.Example 15.
supposec0F∇p2,q+3/2q+4h, where hy=supqρ¯∇p2yq,0¯q+3/2q+4, for all y∈c0F∇p2,q+3/2q+4. We consider the summable equation(78)yq=Rq+∑r=0∞eq+ryq5yq3+yr2+1¯t,
witht>0. Let V:c0F∇p2,q+3/2q+4⟶c0F∇p2,q+3/2q+4 defined by(79)Vyq=Rq+∑r=0∞eq+ryq5yq3+yr2+1¯t.
Obviously,(80)∑r=0∞eqyq5yq3+yr2+1¯ter−er≤εRq−yq+∑r=0∞eq+ryq5yq3+yr2+1¯t+Rq−zq+∑r=0∞eq+rzq5zq3+zr2+1¯t.
By Theorem29, the summable equation (75) has a unique solution in c0F∇p2,q+3/2q+4.Example 16.
supposec0F∇p2,2q+3/q+2h, where hy=supqρ¯∇p2yq,0¯2q+3/2q+4, for every y∈c0F∇p2,2q+3/q+2. We consider the nonlinear difference equations:(81)yq=Rq+∑l=0∞−1q+lyq−2ryq−1w+l2+1¯,
withr,w>0, y−2t,y−1t>0, for all t∈ℜ, and assume V:c0F∇p2,2q+3/q+2⟶c0F∇p2,2q+3/q+2, defined by(82)Vyqq=0∞=Rq+∑l=0∞−1q+lyq−2ryq−1w+l2+1¯q=0∞.
Evidently,(83)∑l=0∞−1qyq−2ryq−1w+l2+1¯−1l−−1l≤εRq−yq+∑l=0∞−1q+lyq−2ryq−1w+l2+1¯+Rq−zq+∑l=0∞−1q+lzq−2rzq−1w+l2+1¯.
By Theorem29, the nonlinear difference (81) have a unique solution in c0F∇p2,2q+3/q+2.Theorem 29.
consider the summable equation (4) and assume V:c0F∇p2,τh⟶c0F∇p2,τh is defined by (5), where the conditions of Theorem 7 are satisfied with τ0>1 and hy=supqρ¯∇p2yq,0¯τq, for every y∈c0F∇p2,τ. The summable equation (4) has a unique solution z∈c0F∇p2,τh, if the following conditions are satisfied:(1)
IfD:N2⟶ℜ, m:N×ℜ0,1⟶ℜ0,1, R:N⟶ℜ0,1, z:N⟶ℜ0,1, there is ε so that 2K−1supqετq∈0,0.5, and for all q∈N, we have(84)∑r∈NDq,rmr,yr−mr,zr≤εRq−yq+∑r=0∞Dq,rmr,yr+Rq−zq+∑r=0∞Dq,rmr,zr.(2)
V is h-sequentially continuous at z∈c0F∇p2,τh.(3)
There isy∈c0F∇p2,τh with Vly has Vljy converging to z.Proof.
One has(85)hVy−Vz=supqρ¯∇p2Vyq−Vzq,0¯τq=supqρ¯∇p2∑r∈NDq,rmr,yr−mr,zr,0¯τq≤2K−1supqετqsupqρ¯∇p2Rq−yq+∑r=0∞Dq,rmr,yr,0¯τq+2K−1supqετqsupqρ¯∇p2Rq−zq+∑r=0∞Dq,rmr,zr,0¯τq=2K−1supqετqhVy−y+hVz−z.
By Theorem27, one gets a unique solution z∈c0F∇p2,τh of (4).Example 17.
supposec0F∇p2,2q+3/q+2h, where hy=supqρ¯∇p2yq,0¯2q+3/q+2, for all y∈c0F∇p2,2q+3/q+2. We consider the summable equation(86)yq=Rq+∑r=0∞−1q+ryqq2+r2+1t,
witht>0. Let V:c0F∇p2,2q+3/q+2⟶c0F∇p2,2q+3/q+2 defined by(87)Vyq=Rq+∑r=0∞−1q+ryqq2+r2+1t.
We assumeV is h-sequentially continuous at z∈c0F∇p2,τh, and there is y∈c0F∇p2,τh with Vly has Vljy converging to z. Obviously,(88)∑r=0∞−1qyqq2+r2+1t−1r−−1r≤εRq−yq+∑r=0∞−1q+ryqq2+r2+1t+Rq−zq+∑r=0∞−1q+rzqq2+r2+1t.
By Theorem29, the summable equation (86) has a unique solution z∈c0F∇p2,2q+3/q+2.Example 18.
supposec0F∇p2,5q+3/q+1h, where hy=supqρ¯∇p2|yq|,0¯5q+3/q+1, for all y∈c0F∇p2,5q+3/q+1. We consider the summable equation(89)yq=Rq+∑r=0∞eq+ryq5yq3+yr2+1¯t,
witht>0. Let V:c0F∇p2,5q+3/q+1⟶c0F∇p2,5q+3/q+1 defined by(90)Vyq=Rq+∑r=0∞eq+ryq5yq3+yr2+1¯t.
We assume thatV is h-sequentially continuous at z∈c0F∇p2,τh, and there is y∈c0F∇p2,τh with Vly has Vljy converging to z. Obviously,(91)∑r=0∞eqyq5yq3+yr2+1¯ter−er≤εRq−yq+∑r=0∞eq+ryq5yq3+yr2+1¯t+Rq−zq+∑r=0∞eq+rzq5zq3+zr2+1¯t.
By Theorem29, the summable equation (89) has a unique solution z∈c0F∇p2,5q+3/q+1.Example 19.
supposec0F∇p2,2q+3/q+2h, where hy=supqρ¯∇p2yq,0¯2q+3/q+2, for every y∈c0F∇p2,2q+3/q+2. We consider the nonlinear difference equation:(92)yq=Rq+∑l=0∞−1q+lyq−2ryq−1w+l2+1¯,
withr,w>0, y−2t,y−1t>0, for all t∈ℜ, and assume V:c0F∇p2,2q+3/q+2⟶c0F∇p2,2q+3/q+2, defined by(93)Vyqq=0∞=Rq+∑l=0∞−1q+lyq−2ryq−1w+l2+1¯q=0∞.
We suppose thatV is h-sequentially continuous at z∈c0F∇p2,τh, and there is y∈c0F∇p2,τh with Vly has Vljy converging to z. Evidently,(94)∑l=0∞−1qyq−2ryq−1w+l2+1¯−1l−−1l≤εRq−yq+∑l=0∞−1q+lyq−2ryq−1q+l2+1¯+Rq−zq+∑l=0∞−1q+lzq−2rzq−1w+l2+1¯.
By Theorem29, the nonlinear difference equation (18) has a unique solution z∈c0F∇p2,2q+3/q+2.
In this part, we search for a solution to nonlinear matrix (81) at D∈ℐ¯c0F∇p2,τhΩ,Λ, where Ω and Λ are Banach spaces, the conditions of Theorem 7 are satisfied, and
ΨG=supqρ¯∇p2sqG¯,0¯τq/K, for all G∈ℐ¯c0F∇p2,τhΩ,Λ. We consider the summable equation(95)saG¯=saP¯+∑m=0∞Aa,mfm,smG.
And we suppose thatW:ℐ¯c0F∇p2,τhΩ,Λ⟶ℐ¯c0F∇p2,τhΩ,Λ is defined by(96)WG=saP¯+∑m=0∞Aa,mfm,smGI.Theorem 30.
The summable equation (18) has one solution D∈ℐ¯c0F∇p2,τhΩ,Λ, if the following conditions are satisfied:(a)
A:N2⟶ℜ, f:N×ℜ0,1⟶ℜ0,1, P∈ℒΩ,Λ, T∈ℒΩ,Λ, and for every a∈N, there is κ so that supaκτa∈0,0.5, with(97)∑m∈NAa,mfm,smG¯−fm,smT¯≤κKsaP¯−saG¯+∑m∈NAa,mfm,smG¯+saP¯−saT¯+∑m∈NAa,mfm,smT¯.(b)
W is Ψ-sequentially continuous at a point D∈ℐ¯c0F∇p2,τhΩ,Λ.(c)
There isB∈ℐ¯c0F∇p2,τhΩ,Λ so that the sequence of iterates WaB has a subsequence WaiB converging to D.Proof.
suppose the settings are verified. We consider the mappingW:ℐ¯c0F∇p2,τhΩ,Λ⟶ℐ¯c0F∇p2,τhΩ,Λ defined by (19). We have(98)ΨWG−WT=supaρ¯∇p2∑m∈NAa,mfm,smG¯−fm,smT¯,0¯τa/K≤supaκτasupaρ¯∇p2saP¯−saG¯+∑m∈NAa,mfm,smG¯,0¯τa/K+supaκτasupaρ¯∇p2saT¯−saG¯+∑m∈NAa,mfm,smT¯,0¯τa/K=supaκτaΨWG−G+ΨWT−T.
In view of Theorem27, one obtains a unique solution of equation (18) at D∈ℐ¯c0F∇p2,τhΩ,Λ.Example 20.
We assume the classℐ¯c0F∇p2,a+1/a+2a=0∞hΩ,Λ, where ΨG=supaρ¯∇p2saG¯,0¯a+1/a+2, for all G∈ℐ¯c0F∇p2,a+1/a+2a=0∞hΩ,Λ.
We consider the nonlinear difference equations:(99)saG¯=e−2a+3¯+∑m=0∞tan2m+1cosh3m−acosrsa−2G¯sinhqsa−1G¯+sinma¯+1¯,where a≥2 and r,q>0 and let W:ℐ¯c0F∇p2,a+1/a+2a=0∞hΩ,Λ⟶ℐ¯c0F∇p2,a+1/a+2a=0∞hΩ,Λ be defined as(100)WG=e−2a+3¯+∑m=0∞tan2m+1cosh3m−acosrsa−2G¯sinhqsa−1G¯+sinma¯+1¯I.
We supposeW is Ψ-sequentially continuous at a point D∈ℐ¯c0F∇p2,a+1/a+2a=0∞hΩ,Λ, and there is B∈ℐ¯c0F∇p2,a+1/a+2a=0∞hΩ,Λ so that the sequence of iterates WaB has a subsequence WaiB converging to D. It is easy to see that(101)∑m=0∞cosh3m−acosrsa−1G¯sinhqsa−1G¯+sinma¯+1¯tan2m+1−tan2m+1a+1/a+2≤15e−2a+3¯−saG¯+∑m=0∞tan2m+1cosh3m−acosrsa−2G¯sinhqsa−1G¯+sinma¯+1¯a+1/a+2+15e−2a+3¯−saT¯+∑m=0∞tan2m+1cosh3m−acosrsa−2T¯sinhqsa−1T¯+sinma¯+1¯a+1/a+2.
By Theorem30, the nonlinear difference (99) has one solution D∈ℐ¯c0F∇p2,a+1/a+2a=0∞hΩ,Λ.Example 21.
assume the classℐ¯c0F∇p2,2a+3/a+2a=0∞hΩ,Λ, where ΨG=supaρ¯∇p2saG¯,0¯2a+3/2a+4, for all G∈ℐ¯c0F∇p2,2a+3/a+2a=0∞hΩ,Λ. We consider the nonlinear difference equation (20) and let W:ℐ¯c0F∇p2,2a+3/a+2a=0∞hΩ,Λ⟶ℐ¯c0F∇p2,2a+3/a+2a=0∞hΩ,Λ be defined as (21). Suppose W is Ψ-sequentially continuous at a point D∈ℐ¯c0F∇p2,2a+3/a+2a=0∞hΩ,Λ, and there is B∈ℐ¯c0F∇p2,2a+3/a+2a=0∞hΩ,Λ so that the sequence of iterates WaB has a subsequence WaiB converging to D. It is easy to see that(102)∑m=0∞cosh3m−acosrsa−2G¯sinhqsa−1G¯+sinma¯+1¯tan2m+1−tan2m+12a+3/2a+4≤125e−2a+3¯−saG¯+∑m=0∞tan2m+1cosh3m−acosrsa−2G¯sinhqsa−1G¯+sinma¯+1¯2a+3/2a+4+15e−2a+3¯−saT¯+∑m=0∞tan2m+1cosh3m−acosrsa−2T¯sinhqsa−1T¯+sinma¯+1¯2a+3/2a+4.
By Theorem30, the nonlinear difference (99) has one solution D∈ℐ¯c0F∇p2,2a+3/a+2a=0∞hΩ,Λ.


## 7. Conclusion

In this paper, we have explained sufficient settings of the spacec0F∇p2,τ equipped with definite function h to be pre-quasi Banach. The Fatou property of various pre-quasi-norms h on c0F∇p2,τ has been investigated. The geometric and topological structures of the mappings’ ideal by this space and extended s− fuzzy functions have been explained. We construct the existence of a fixed point of Kannan contraction mapping acting on this space and its associated pre-quasi 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 fuzzy functions are introduced. As a future project, we can build the domain of second-order quantum backward difference in Nakano sequences of fuzzy functions space and look at its properties.



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