# Kannan Nonexpansive Operators on Variable Exponent Cesàro Sequence Space of Fuzzy Functions

Authors: Awad A. Bakery; Mustafa M. Mohammed
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In general, we have constructed the operator ideal generated by extendeds-fuzzy numbers and a certain space of sequences of fuzzy numbers. An investigation into the conditions sufficient for variable exponent Cesàro sequence space of fuzzy functions furnished with the definite function to create pre-quasi-Banach and closed is carried out. The R and the normal structural properties of this space are shown. Fixed points for Kannan contraction and nonexpansive mapping have been introduced. Lastly, we explore whether the Kannan contraction mapping has a fixed point in its associated pre-quasioperator ideal. The existence of solutions to nonlinear difference equations is illustrated with a few real-world examples and applications.


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## Body

## 1. Introduction

Probability theory, fuzzy set theory, soft sets, and rough sets have contributed substantially to the study of uncertainty. But there are drawbacks to these theories that must be considered. After Zadeh [1] established the concept of fuzzy sets and fuzzy set operations, many researchers adopted the concept of fuzziness in cybernetics and artificial intelligence as well as in expert systems and fuzzy control. For more information and real-world examples, some comparable fixed point results were discussed by Javed et al. [2] to ensure that a fixed point exists and is unique in R-fuzzy b-metric spaces. The viability of the proposed methodologies was demonstrated through a challenging case study. There was no doubt about the superiority of the findings delivered. For the first type of Fredholm-type integral equation, an application was described. In [3], Al-Masarwah and Ahmad defined and investigated the m-Polar α,β-Fuzzy Ideals in BCK/BCI-Algebras and explored some pertinent properties. There are many other orthogonal fuzzy metric spaces; however, Javed et al. [4] expanded the orthogonal image fuzzy metric space concept. In the context of the newly specified structure, they displayed some fixed point outcomes. Fuzzy sequence spaces were introduced, and their various features were studied by many workers on sequence spaces and summability theory. Nuray and Savas [5] defined and studied the Nakano sequences of fuzzy numbers, ℓFτ equipped with the function h. The operator ideal is very important in fixed point theory, Banach space geometry, normal series theory, approximation theory, and ideal transformations. See [6–8] for further proof. Pre-quasioperator ideals are more extensive than quasioperator ideals, according to Faried and Bakery [9]. The learning about the variable exponent Lebesgue spaces obtained impetus from the mathematical description of the hydrodynamics of non-Newtonian fluids (see [10, 11]). There are numerous uses for electrorheological fluids, which include military science, civil engineering, and orthopedic. There have been many developments in mathematics since the Banach fixed point theorem [12] was first published. While contractions have fixed point actions, Kannan [13] cited an example of a type of mapping that is not continuous. In Reference [14], the only attempt was made to explain Kannan operators in modular vector spaces. For more details on Kannan’s fixed point theorems, see [15–20]. 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 options are viable. Hence, we have constructed the Cesàro sequence spaces of fuzzy functions and have presented the solutions of a fuzzy nonlinear dynamical system in this newly created space. This work is aimed at introducing 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. We explain the sufficient conditions of variable exponent Cesàro sequence space of fuzzy functions, which is denoted by Cτ.F, equipped with the definite function h to be pre-quasi-Banach and closed (cssf). The R and the normal structure property of this space are shown. Fixed points for Kannan contraction and nonexpansive mapping have been introduced. Lastly, we explore whether the Kannan contraction mapping has a fixed point in its associated pre-quasioperator ideal. The existence of solutions to nonlinear difference equations is illustrated with a few real-world examples and applications.


## 2. Definitions and Preliminaries

As a reminder, Matloka [21] presented the notion of ordinary convergence of sequences of fuzzy numbers, where he introduced bounded and convergent fuzzy numbers, explored some of their features, and proved that any convergent fuzzy number sequence is bounded. Nanda [22] studied the sequences of fuzzy numbers and showed the set of all convergent sequences of fuzzy numbers from a complete metric space. Kumar et al. [23] investigated the notion of limit points and cluster points of sequences of fuzzy numbers. Assume Ω is the set of all closed and bounded intervals on the real line R. For f=f1,f2 and g=g1,g2 in Ω, suppose
(1)f≤g,ifandonlyiff1≤g1andf2≤g2.Define a metricρ on Ω by
(2)ρf,g=maxf1−g1,f2−g2.Matloka [21] showed that ρ is a metric on Ω and Ω,ρ is a complete metric space. Also, the relation ≤ is a partial order on Ω.Definition 1.
A fuzzy numberg is a fuzzy subset of R, i.e., a mapping g:R⟶0,1 which verifies the following four settings:
(a)
g is fuzzy convex, i.e., for x,y∈R and α∈0,1, gαx+1−αy≥mingx,gy(b)
g is normal, i.e., there is y0∈R such that gy0=1(c)
g is an upper semicontinuous, i.e., for all α>0, g−10,x+α for all x∈0,1 is open in the usual topology of R(d)
the closure ofg0≔y∈R:gy>0 is compact
Theβ-level set of a fuzzy real number g,0<β<1, indicated by gβ is defined as
(3)gβ=y∈R:gy≥β.
The set of every upper semicontinuous, normal, convex fuzzy number, andgβ is compact is denoted by R0,1. The set R can be embedded in R0,1, if we define r∈R0,1 by
(4)r¯t=1,t=r,0,t≠r.
The additive identity and multiplicative identity inR0,1 are denoted by 0¯ and 1¯, respectively.
The arithmetic operations onR0,1 are defined as follows:
(5)f⊕gy=supy∈Rminfx,gy−x,f!gy=supy∈Rminfx,gx−y,f⊗gy=supy∈Rminfx,gyx,fgy=supy∈Rminfxy,gx,xfy=fx−1y,x≠0,0,x=0.
The absolute valuef of f∈R0,1 is defined by
(6)fy=maxfy,f−y,ify≥0,0,ify<0.
Supposef,g∈R0,1 and the β-level sets are fβ=f1β,f2β, gβ=g1β,g2β, and β∈0,1. A partial ordering for any f,g∈R0,1 as follows: f°g, if and only if fβ≤gβ, for all β∈0,1. Then, the above operations can be defined in terms of β-level sets as follows:
(7)f⊕gβ=f1β+g1β,f2β+g2β,f!gβ=f1β−g2β,f2β−g1β,f⊗gβ=minj∈1,2fjβgjβ,maxj∈1,2fjβgjβ,f−1β=f2β−1,f1β−1,fjβ>0,foreveryβ∈0,1,xfβ=xf1β,xf2β,x≥0,xf2β,xf1β,x<0.
Assumeρ¯:R0,1×R0,1⟶R+∪0 is defined by ρ¯f,g=sup0≤β≤1ρfβ,gβ.
Recall that(1)
R0,1,ρ¯ is a complete metric space(2)
ρ¯f+k,g+k=ρ¯f,gforallf,g,k∈R0,1(3)
ρ¯f+k,g+l≤ρ¯f,g+ρ¯k,l.(4)
ρ¯ξf,ξg=ξρ¯f,g,forallξ∈R.Definition 2.
A sequencef=fj of fuzzy numbers is said to be
(a)
bounded if the setfj:j∈N of fuzzy numbers is bounded, i.e., if a sequence fj is bounded, then there are two fuzzy numbers g,l such that g≤fj≤l(b)
convergent to a fuzzy real numberf0 if for every ε>0, there exists n0∈N such that ρ¯fj,f0<ε, for all j≥j0Lemma 3 (see [24]).
Supposeτa≥1 and va,ta∈R, for every a∈N, then va+taτa≤2K−1vaτa+taτa, where K=max1,supaτa.


## 3. Main Results

### 3.1. Some Properties ofCτ.F

In this section, we have introduced the certain space of sequences of fuzzy numbers or in short (cssf), under the definite function to form pre-quasi (cssf). We explain the sufficient setting ofCτ.F equipped with the definite function h to construct pre-quasi-Banach and closed (cssf). The Fatou property of various pre-quasinorms h on Cτ.F has been investigated. We have presented this space’s k-nearly uniformly convex, the property R, and the h-normal structure-property, which are connected with the fixed point theorem.Byℓ∞ and ℓr, we denote the spaces of bounded and r-absolutely summable sequences of real numbers, respectively. Let ωF denote the classes of all sequence spaces of fuzzy real numbers. Suppose τ=τa∈R+N, where R+N is the space of positive real sequences. The variable exponent Cesàro sequence space of fuzzy functions is denoted by the following: Cτ.F=v=va∈ωF:hμv<∞,forsomeμ>0, when hv=∑a=0∞∑k=0aρ¯vk,0¯/a+1τa. If τa∈ℓ∞, then
(8)Cτ.F=v=va∈ωF:hμv<∞,forsomeμ>0=v=va∈ωF:infaμτa∑a=0∞∑k=0aρ¯vk,0¯a+1τa≤∑a=0∞∑k=0aρ¯μvk,0¯a+1τa<∞,forsomeμ>0=v=va∈ωF:∑a=0∞∑k=0aρ¯vk,0¯a+1τa<∞=v=va∈ωF:hμv<∞,foranyμ>0.Definition 4 (see [25]).
The linear spaceU is said to be a certain space of sequences of fuzzy numbers (cssf), if
(1)
b¯qq∈N⊆U, where b¯q=0¯,0¯,⋯,1¯,0¯,0¯,⋯, while 1¯ displays at the qth place(2)
supposeY=Yq∈ωF, Z=Zq∈U and Yq≤Zq, for all q∈N, then Y∈U(3)
Yq/2q=0∞∈U, where q/2 marks the integral part of q/2, if Yqq=0∞∈UDefinition 5 (see [25]).
A subclassUh of U is called a premodular (cssf), if there is h∈0,∞U satisfies the next settings:
(i)
IfY∈U, Y=ϑ¯⇔hY=0 with hY≥0, where ϑ¯=0¯,0¯,0¯,(ii)
There isQ≥1, and the inequality hαY≤QαhY holds, for every Y∈U and α∈R(iii)
There isP≥1, and the inequality hY+Z≤PhY+hZ holds, for every Y,Z∈U(iv)
IfYq≤Zq, for every q∈N, one has hYq≤hZq(v)
The inequalityhYq≤hYq/2≤P0hYq holds, for some P0≥1(vi)
LetE be the space of finite sequences of fuzzy numbers; then, the closure of E=Uh(vii)
There isσ>0 with hα¯,0¯,0¯,0¯,⋯≥σαh1¯,0¯,0¯,0¯,⋯, where(9)α¯y=1,y=α,0,y≠α.Definition 6 (see [25]).
SupposeU is a (cssf). The function h∈0,∞U is called a pre-quasinorm on U, if it satisfies the following conditions:
(i)
IfY∈U, Y=ϑ¯⇔hY=0 with hY≥0, where ϑ¯=0¯,0¯,0¯,(ii)
There isQ≥1, and the inequality hαY≤QαhY satisfies, for every Y∈U and α∈R(iii)
There isP≥1, and the inequality hY+Z≤PhY+hZ holds, for each Y,Z∈UClearly, from the last two definitions, we conclude the following two theorems:Theorem 7 (see [25]).
IfU is a premodular (cssf), then it is pre-quasinormed (cssf).Theorem 8 (see [25]).
U is a pre-quasinormed (cssf) if it is quasinormed (cssf).Definition 9.
(a)
The functionh on Cτ.F is named h-convex, if
(10)hαY+1−αZ≤αhY+1−αhZ,
for everyα∈0,1 and Y,Z∈Cτ.F.
(b)
Yqq∈N⊆Cτ.Fh is h-convergent to Y∈Cτ.Fh, if and only if limq⟶∞hYq−Y=0. When the h-limit exists, then it is unique(c)
Yqq∈N⊆Cτ.Fh is h-Cauchy, if limq,r⟶∞hYq−Yr=0(d)
Γ⊂Cτ.Fh is h-closed, when for all h-converges Yqa∈N⊂Γ to Y, then Y∈Γ(e)
Γ⊂Cτ.Fh is h-bounded, if δhΓ=suphY−Z:Y,Z∈Γ<∞(f)
Theh-ball of radius ε≥0 and center Y, for every Y∈Cτ.Fh, is described as follows:
(11)BhY,ε=Z∈Cτ.Fh:hY−Z≤ε.(g)
A pre-quasinormh on Cτ.F satisfies the Fatou property, if for every sequence Zq⊆Cτ.Fh under limq⟶∞hZq−Z=0 and all Y∈Cτ.Fh, one has hY−Z≤suprinfq≥rhY−Zq
Note that the Fatou property implies theh-closed of the h-balls. We will denote the space of all increasing sequences of real numbers by I.Theorem 10.
Cτ.Fh, where hY=∑q=0∞∑p=0qρ¯Yp,0¯/q+1τq1/K, for all Y∈Cτ.F, is a premodular (cssf), when τqq∈N∈ℓ∞∩I with τ0>1.Proof.
(i) Evidently,hY≥0 and hY=0⇔Y=ϑ¯
(1-i) LetY,Z∈Cτ.F. One has
(12)hY+Z=∑q=0∞∑p=0qρ¯Yp+Zp,0¯q+1τq1/K≤∑q=0∞∑p=0qρ¯Yp,0¯q+1τq1/K+∑q=0∞∑p=0qρ¯Zp,0¯q+1τq1/K=hY+hZ<∞,
and then,Y+Z∈Cτ.F.
(iii) One getsP≥1 with hY+Z≤PhY+hZ, for all Y,Z∈Cτ.F
(1-ii) Assumeα∈R and Y∈Cτ.F, and we obtain
(13)hαY=∑q=0∞∑p=0qρ¯αYp,0¯q+1τq1/K≤supqατq/K∑q=0∞∑p=0qρ¯Yp,0¯q+1τq1/K≤QαhY<∞.
AsαY∈Cτ.F. Hence, from conditions (1-i) and (1-ii), one has Cτ.F is linear. Also, b¯r∈Cτ.F, for all r∈N, since hb¯r=∑q=0∞∑p=0qρ¯b¯r,0¯/q+1τq1/K≤∑q=0∞1/q+1τ01/K<∞.
(ii) There isQ=max1,supqατq/K−1≥1 with hαY≤QαhY, for all Y∈Cτ.F and α∈R
(2) AssumeYq≤Zq, for all q∈N and Z∈Cτ.F. One finds
(14)hY=∑q=0∞∑p=0qρ¯Yp,0¯q+1τq1/K≤∑q=0∞∑p=0qρ¯Zp,0¯q+1τq1/K=hZ<∞,
and then,Y∈Cτ.F.
(iv) Obviously, from (2)
(3) LetYq∈Cτ.F, and we get
(15)hYq/2=∑q=0∞∑p=0qρ¯Yp/2,0¯q+1τq1/K=∑q=0∞∑p=02qρ¯Yp/2,0¯2q+1τ2q+∑q=0∞∑p=02q+1ρ¯Yp/2,0¯2q+2τ2q+11/K≤21/K∑q=0∞∑p=0qρ¯Yp,0¯q+1τq1/K≤∑q=0∞ρ¯Yq,0¯+2∑p=0qρ¯Yp,0¯q+1τq+∑q=0∞2∑p=0qρ¯Yp,0¯q+1τq1/K≤21/K∑q=0∞∑p=0qρ¯Yp,0¯q+1τq1/K≤∑q=0∞3∑p=0qρ¯Yp,0¯q+1τq+∑q=0∞2∑p=0qρ¯Yp,0¯q+1τq1/K≤3K+2K1/K∑q=0∞∑p=0qρ¯Yp,0¯q+1τq1/K=3K+2K1/KhYq,
and then,Yp/2∈Cτ.F.
(v) From (4), we obtain P0=3K+2K1/K≥1
(vi) Evidently the closure ofE=Cτ.F
(vii) There is0<σ≤supqατq/K−1, for α≠0 or σ>0, for α=0 with hα¯,0¯,0¯,0¯,⋯≥σαh1¯,0¯,0¯,0¯,⋯Theorem 11.
Ifτqq∈N∈ℓ∞∩I with τ0>1, then Cτ.Fh is a pre-quasi-Banach (cssf), where hY=∑q=0∞∑p=0qρ¯Yp,0¯/q+1τq1/K, for every Y∈Cτ.F.Proof.
In view of Theorem10 and Theorem 7, the space Cτ.Fh is a pre-quasinormed (cssf). Assume Yl=Yqlq=0∞ is a Cauchy sequence in Cτ.Fh. Hence, for every ε∈0,1, one has l0∈N such that for all l,m≥l0, one gets
(16)hYl−Ym=∑q=0∞∑p=0qρ¯Ypl−Ypm,0¯q+1τq1/K<ε.
That impliesρ¯Yql−Yqm,0¯<ε. As R0,1,ρ¯ is a complete metric space. Then, Yqm is a Cauchy sequence in R0,1, for fixed q∈N, which implies limm⟶∞Yqm=Yq0, for constant q∈N. Hence, hYl−Y0<ε, for every l≥l0, since hY0=hY0−Yl+Yl≤hYl−Y0+hYl<∞. So Y0∈Cτ.F.Theorem 12.
Supposeτqq∈N∈ℓ∞∩I with τ0>1, then Cτ.Fh is a pre-quasiclosed (cssf), where hY=∑q=0∞∑p=0qρ¯Yp,0¯/q+1τq1/K, for every Y∈Cτ.F.Proof.
In view of Theorem10 and Theorem 7, the space Cτ.Fh is a pre-quasinormed (cssf). Assume Yl=Yqlq=0∞∈Cτ.Fh and liml⟶∞hYl−Y0=0; then, for all ε∈0,1, there is l0∈N such that for all l≥l0, we obtain
(17)ε>hYl−Y0=∑q=0∞∑p=0qρ¯Ypl−Yp0,0¯q+1τq1/K,
which impliesρ¯Yql−Yq0,0¯<ε. As R0,1,ρ¯ is a complete metric space, therefore, Yql is a convergent sequence in R0,1, for fixed q∈N. So, liml⟶∞Yql=Yq0, for fixed q∈N. Since hY0=hY0−Yl+Yl≤hYl−Y0+hYl<∞, one has Y0∈Cτ.F.Theorem 13.
The functionhY=∑q=0∞∑p=0qρ¯Yp,0¯/q+1τq1/K verifies the Fatou property, when τqq∈N∈ℓ∞∩I with τ0>1, for all Y∈Cτ.F.Proof.
LetZr⊆Cτ.Fh such that limr⟶∞hZr−Z=0. Since Cτ.Fh is a pre-quasiclosed space, one has Z∈Cτ.Fh. For all Y∈Cτ.Fh, one gets
(18)hY−Z=∑q=0∞∑p=0qρ¯Yp−Zp,0¯q+1τq1/K≤∑q=0∞∑p=0qρ¯Yp−Zpr,0¯q+1τq1/K+∑q=0∞∑p=0qρ¯Zpr−Zp,0¯q+1τq1/K≤supminfr≥mhY−Zr.Theorem 14.
The functionhY=∑q=0∞∑p=0qρ¯Yp,0¯/q+1τq does not satisfy the Fatou property, for all Y∈Cτ.F, when τq∈ℓ∞ and τq>1, for all q∈N.Proof.
LetZr⊆Cτ.Fh so that limr⟶∞hZr−Z=0. Since Cτ.Fh is a pre-quasiclosed space, one gets Z∈Cτ.Fh. For every Z∈Cτ.Fh, we obtain
(19)hY−Z=∑q=0∞∑p=0qρ¯Yp−Zp,0¯q+1τq≤2K−1∑q=0∞∑p=0qρ¯Yp−Zpr,0¯q+1τq+∑q=0∞∑p=0qρ¯Zpr−Zp,0¯q+1τq≤2K−1supminfr≥mhY−Zr.Example 1.
Forτq∈1,∞N, the function hY=infα>0:∑q∈N∑p=0qρ¯Yp/α,0¯/q+1τq≤1 is a norm on Cτ.F.Example 2.
The functionhY=∑q∈N∑p=0qρ¯Yp,0¯/q+13q+2/q+13 is a pre-quasinorm (not a norm) on CF3q+2/q+1q=0∞.Example 3.
The functionhY=∑q∈N∑p=0qρ¯Yp,0¯/q+13q+2/q+1 is a pre-quasinorm (not a quasinorm) on CF3q+2/q+1q=0∞.Example 4.
The functionhY=∑q∈N∑p=0qρ¯Yp,0¯/q+1dd is a pre-quasinorm, quasinorm, and not a norm on CdF, for 0<d<1.
In the next part of this section, we will use the functionh as hY=∑q=0∞∑p=0qρ¯Yp,0¯/q+1τq1/K, for every Y∈Cτ.F.Definition 15 [26].
The functionh is said to be strictly convex, (SC), if for all Y,Z∈Uh such that hY=hZ and hY+Z/2=hY+hZ/2, we get Y=Z.Definition 16 [27].
A sequenceYp⊆U is said to be ε-separated sequence for some ε>0, if
(20)sepYp=infhYp−Yq:p≠q>ε.Definition 17 (see [27]).
Letk≥2 be an integer, and a Banach space U is called k-nearly uniformly convex (k-NUC), if for any ε>0, there exists δ∈0,1 such that for any sequence Yp⊆Bh0,1, with sepYp≥ε, there are p1,p2,p3,⋯,pk∈N, such that
(21)hYp1+Yp2+Yp3+⋯+Ypkk<1−δ.Definition 18 (see [28]).
A functionh is said to satisfy the δ2-condition (h∈δ2), if for any ε>0, there exists a constant k≥2 and a>0 such that h2u≤khu+ε, for each u∈Xh, with hu≤a.
Ifh satisfies the δ2-condition for any a>0 with k≥2 depending on a, we say that h satisfies the strong δ2-condition (ρ∈δ2s).The following known results are very important for our consideration.Theorem 19 (see [28], Lemma 2.1).
Ifh∈δ2s, then for any L>0 and ε>0, there exists δ>0 such that hx+y−hx<ε, where x,y∈Xh, with hx≤L and hy≤δ.Theorem 20.
Pick anτqq∈N∈ℓ∞∩I with τ0>1; then, for any L>0 and ε>0, there exists δ>0 such that hx+y−hx<ε, for all x,y∈Cτ.Fh, with hx≤L and hy≤δ.Proof.
Sinceτq is bounded, it is easy to see that h∈δ2s. Hence, the proposition is obtained directly from Theorem 19.Theorem 21.
Supposeτqq∈N∈ℓ∞∩I with τ0>1; then, Cτ.Fh is k-NUC, for any integer k≥2.Proof.
Letε∈0,1 and xn⊆Bh0,1 with sepxn≥ε, for each m∈N, and let xnm=0,0,0,⋯,xnm,xnm+1,⋯. Since for each i∈N, xnin=0∞ is bounded, and by using the diagonal method, we can find a subsequence xnj of xn such that xnji converges for each i∈N, 0≤i≤m. Therefore, there exists an increasing sequence of positive integers tm such that sepxnjmj>tm≥ε. Hence, there is a sequence of positive integers rmm=0∞ with r0<r1<r2<⋯, such that
(22)hKxrmm≥ε2,
for eachm∈N. For fixed integer k≥2, let ε1=kp0−1−1/k−1kp0ε/4; then, by Theorem 20, there exists δ>0 such that
(23)hKx+y−hKx<ε1,
wheneverhKx≤1 and hKy≤δ. Since hKxn≤1, for any n∈N, then there exist positive integers mii=0,1,2,⋯,k−2 with m0<m1<m2<⋯<mk−2 such that hKximi≤δ. Define mk−1=mk−2+1. By inequality (1), we have hxrmkmk≥ε/2. Let si=i for 0≤i≤k−2 and sk−1=rmk−1. Then, in virtue of inequality (1), inequality (2), and convexity of the function fnu=uτn for any n∈N, we have
(24)hKxs0+xs1+xs2+.⋯+xsk−1k=∑n=0∞∑i=0nρ¯xs0i+xs1i++xsk−1i/k,0¯n+1τn=∑n=0m1−1∑i=0nρ¯xs0i+xs1i++xsk−1i/n+1,0¯n+1τn+∑n=m1∞∑i=0nρ¯xs0i+xs1i++xsk−1i/k,0¯n+1τn≤∑n=0m1−1∑i=0nρ¯xs0i+xs1i++xsk−1i/k,0¯n+1τn+∑n=m1∞∑i=0nρ¯xs1i+xs2i++xsk−1i/k,0¯n+1τn+ε1≤∑n=0m1−11k∑j=0k−1∑i=0nρ¯xsji,0¯n+1τn+∑n=m1m2−1∑i=0nρ¯xs1i+xs2i++xsk−1i/k,0¯n+1τn+∑n=m2∞∑i=0nρ¯xs1i+xs2i++xsk−1i/k,0¯n+1τn+ε1≤∑n=0m1−11k∑j=0k−1∑i=0nρ¯xsji,0¯n+1τn+∑n=m1m2−1∑i=0nρ¯xs1i+xs2i++xsk−1i/k,0¯n+1τn+∑n=m2∞∑i=0nρ¯xs2i+xs3i++xsk−1i/k,0¯n+1τn+2ε1≤∑n=0m1−11k∑j=0k−1∑i=0nρ¯xsji,0¯n+1τn+∑n=m1m2−11k∑j=1k−1∑i=0nρ¯xsji,0¯n+1τn+∑n=m2m3−11k∑j=2k−1∑i=0nρ¯xsji,0¯n+1τn++∑n=mk−1mk−11k∑j=k−2k−1∑i=0nρ¯xsji,0¯n+1τn+∑n=mk∞∑i=0nρ¯xski/k,0¯n+1τn+k−1ε1≤hKxs0+xs1+xs2+.⋯+xsk−2k+1k∑n=0mk−1∑i=0nρ¯xski,0¯n+1τn+∑n=mk∞∑i=0nρ¯xski/k,0¯n+1τn+k−1ε1≤k−1k+1k∑n=0mk−1∑i=0nρ¯xski,0¯n+1τn+1kp0∑n=mk∞∑i=0nρ¯xski,0¯n+1τn+k−1ε1≤1−1k+1k1−∑n=mk∞∑i=0nρ¯xski,0¯n+1τn+1kp0∑n=mk∞∑i=0nρ¯xski,0¯n+1τn+k−1ε1=1+k−1ε1−kp0−1−1kp0∑n=mk∞∑i=0nρ¯xski,0¯n+1τn≤1+k−1ε1−kp0−1−1kp0ε2=1−kp0−1−1kp0ε4.
Therefore,Cτ.Fh is k-NUC.Recall thatk-NUC implies reflexivity.Definition 22.
The spaceUh satisfies the property R, if and only if, for all decreasing sequence Γjj∈N of h-closed and h-convex nonempty subsets of Uh with supj∈NKhY,Γj<∞, for some Y∈Uh, one has ⋂j∈NΓj≠∅.
By fixingΓ a nonempty h-closed and h-convex subset of Cτ.Fh.Theorem 23.
Ifτqq∈N∈ℓ∞∩I with τ0>1, one has the following:
(i)
SupposeY∈Cτ.Fh with KhY,Γ=infhY−Z:Z∈Γ<∞. There is a unique λ∈Γ so that KhY,Γ=hY−λ(ii)
Cτ.Fh verifies the property R.Proof.
To prove (i), assumeY∉Γ as Γ is h-closed. One has C≔KhY,Γ>0. Hence, for all r∈N, one has Zr∈Γ with hY−Zr<C1+1/r. If Zr/2 is not h-Cauchy, one gets a subsequence Zgr/2 and l0>0 with hZgr−Zgj/2≥l0, for every r>j≥0, since
(25)maxhY−Zgr,hY−Zgj≤C1+1gj,hZgr−Zgj2≥l0≥C1+1gjl02C,
for everyr>j≥0. Since τqq∈N∈ℓ∞∩I with τ0>1, then the function fnu=uτn is strictly convex, for any n∈N. Therefore, the space Cτ.Fh is strictly convex; hence,
(26)hY−Zgr+Zgj2<C1+1gj.
Then,(27)C=KhY,Γ<C1+1gj,
for allj∈N. By putting j⟶∞, one has a contradiction. So Zr/2 is h-Cauchy. As Cτ.Fh is h-complete, then Zr/2h-converges to some Z. For all j∈N, one gets Zr+Zj/2h-converges to Z+Zj/2. Since Γ is h-closed and h-convex, then Z+Zj/2∈Γ. Since Z+Zj/2h-converges to 2Z, then 2Z∈Γ. Let λ=2z, and from Theorem 13, since h satisfies the Fatou property, one has
(28)KhY,Γ≤hY−λ≤supiinfj≥ihY−Z+Zj2≤supiinfj≥isupiinfr≥ihY−Zr+Zj2≤12supiinfr≥isupiinfr≥ihY−Zr+hY−Zj=KhY,Γ.
ThenhY−λ=KhY,Γ. Since h is (SC), this implies the uniqueness of λ. To prove (ii), assume Y∉Γr0, for some r0∈N. Since KhY,Γrr∈N∈ℓ∞ is increasing, put limr⟶∞KhY,Γr=C, when C>0. Otherwise, Y∈Γr, for all r∈N. According to (i), there is one point Zr∈Γr with KhY,Γr=hY−Zr, for every r∈N. A similar proof will prove that Zr/2h-converges to some Z∈Cτ.Fh. As Γr is h-convex, decreasing, and h-closed, one has 2Z∈∩r∈NΓr.Definition 24.
The spaceUh verifies the h-normal structure-property, if and only if, for all nonempty h-bounded, h-convex and h-closed subset Γ of Uh not decreased to one point, and one has Y∈Γ with
(29)supZ∈ΓhY−Z<δhΓ≔suphY−Z:Y,Z∈Γ<∞.Definition 25 (see [29]).
Uh is a real Banach space, and SUh is the unit sphere of Uh. The weakly convergent sequence coefficient of Uh, denoted by WCSUh, is defined as follows:
(30)WCSUh=infAxn:xnn=1∞⊂SUh,Axn=A1xn,xnw⟶0,where
(31)Axn=limsupn⟶∞xi−xj:i,j≥n,i≠j,A1xn=liminfn⟶∞xi−xj:i,j≥n,i≠j.Theorem 26 (see [30]).
A reflexive Banach spaceUh with WCSUh>1 has normal structure-property.Theorem 27.
Ifτqq∈N∈ℓ∞∩I with τ0>1, then Cτ.Fh holds the h-normal structure-property.Proof.
Take anyε>0 and an asymptotic equidistant sequence xn⊂SCτ.Fh with xnw⟶0 and put v1=x1. There exists i1∈N such that h∑i=i1+1∞v1ib¯i<ε. Since xn⟶0 coordinate-wise, there exists n2∈N such that h∑i=1i1xnib¯i<ε, whenever n≥n2. Take v2=xn2; then, there is i2>i1 such that h∑i=i2+1∞v1ib¯i<ε. Since xni⟶0 coordinate-wise, there exists n3∈N such that h∑i=1i2xnib¯i<ε, whenever n≥n3. Continuing this process in such a way by induction, we get a subsequence vn of xn such that
(32)h∑i=in+1∞vnib¯i<ε,h∑i=1invn+1ib¯i<ε.
Putzn=∑i=in−1+1invnib¯i, for n=2,3,⋯ Then,
(33)1≥hzn=h∑i=1∞vnib¯i−∑i=1in−1vnib¯i−∑i=in+1∞vnib¯i≥h∑i=1∞vnib¯i−h∑i=1in−1vnib¯i−h∑i=in+1∞vnib¯i>1−2ε.
Moreover, for anyn,m∈N with n≠m, we have
(34)hvn−vm=h∑i=1∞vnib¯i−∑i=1∞vmib¯i≥h∑i=in−1+1invnib¯i−∑i=im−1+1imvmib¯i−h∑i=1in−1vnib¯i−h∑i=in+1∞vnib¯i−h∑i=1im−1vmib¯i−h∑i=im+1∞vmib¯i≥hzn−zm−4ε.
This means thatAxn=Avn≥Azn−4ε. Put un=zn/zn, for n=2,3,⋯ Then,
(35)un∈SCτ.Fh,(36)Axn≥1−εAun−4ε.
On the other hand,(37)hvn−vm≤hzn−zm+4ε≤hun−um+4ε,
for anyn,m∈N with n≠m. Therefore,
(38)Aun≥Axn−4ε.
By the arbitrariness ofε>0, we have from the relations (35), (36), and (38) that
(39)WCSCτ.Fh=infAun,
such that(40)un=∑i=in−1+1inunib¯i∈SCτ.Fh,0=i0<i1<⋯,unw⟶0andunisasymptoticequidistant.
Takem∈N large enough such that ∑k=im−1+1∞b/kτk<ε, where b≔∑i=in−1+1inuni. We have for n<m that
(41)hKun−um=∑k=in−1+1im−11k∑i=1kρ¯uni,0¯τk+∑k=im−1+1∞1kb+∑i=1kρ¯umi,0¯τk≥∑k=in−1+1im−11k∑i=1kρ¯uni,0¯τk+∑k=im−1+1∞1k∑i=1kρ¯umi,0¯τk=∑k=in−1+1∞1k∑i=1kρ¯uni,0¯τk−∑k=im−1+1∞bkτk+∑k=im−1+1∞1k∑i=1kρ¯umi,0¯τk>1−ε+1=2−ε,
that is,Anun≥2−ε1/K. Note that
(42)∑k=im−1+1∞1kb+∑i=1kρ¯umi,0¯τk1/K≤∑k=im−1+1∞bkτk1/K+∑k=im−1+1∞1k∑i=1kρ¯umi,0¯τk1/K<ε1/K+1.
Therefore,(43)hKun−um=∑k=in−1+1im−11k∑i=1kρ¯umi,0¯τk+∑k=im−1+1∞1kb+∑i=1kρ¯umi,0¯τk≤∑k=in−1+1∞1k∑i=1kρ¯umi,0¯τk+∑k=im−1+1∞1kb+∑i=1kρ¯umi,0¯τk≤1+1+ε1/KK,
for anyn,m∈N with n≠m. Therefore, Anun≤1+1+ε1/KK1/K, and by the arbitrariness of ε>0, we obtain WCSCτ.Fh=21/K. From Theorem 21 and Theorem 26, the sequence space Cτ.Fh has the h-normal structure-property.


## 3.1. Some Properties ofCτ.F

In this section, we have introduced the certain space of sequences of fuzzy numbers or in short (cssf), under the definite function to form pre-quasi (cssf). We explain the sufficient setting ofCτ.F equipped with the definite function h to construct pre-quasi-Banach and closed (cssf). The Fatou property of various pre-quasinorms h on Cτ.F has been investigated. We have presented this space’s k-nearly uniformly convex, the property R, and the h-normal structure-property, which are connected with the fixed point theorem.Byℓ∞ and ℓr, we denote the spaces of bounded and r-absolutely summable sequences of real numbers, respectively. Let ωF denote the classes of all sequence spaces of fuzzy real numbers. Suppose τ=τa∈R+N, where R+N is the space of positive real sequences. The variable exponent Cesàro sequence space of fuzzy functions is denoted by the following: Cτ.F=v=va∈ωF:hμv<∞,forsomeμ>0, when hv=∑a=0∞∑k=0aρ¯vk,0¯/a+1τa. If τa∈ℓ∞, then
(8)Cτ.F=v=va∈ωF:hμv<∞,forsomeμ>0=v=va∈ωF:infaμτa∑a=0∞∑k=0aρ¯vk,0¯a+1τa≤∑a=0∞∑k=0aρ¯μvk,0¯a+1τa<∞,forsomeμ>0=v=va∈ωF:∑a=0∞∑k=0aρ¯vk,0¯a+1τa<∞=v=va∈ωF:hμv<∞,foranyμ>0.Definition 4 (see [25]).
The linear spaceU is said to be a certain space of sequences of fuzzy numbers (cssf), if
(1)
b¯qq∈N⊆U, where b¯q=0¯,0¯,⋯,1¯,0¯,0¯,⋯, while 1¯ displays at the qth place(2)
supposeY=Yq∈ωF, Z=Zq∈U and Yq≤Zq, for all q∈N, then Y∈U(3)
Yq/2q=0∞∈U, where q/2 marks the integral part of q/2, if Yqq=0∞∈UDefinition 5 (see [25]).
A subclassUh of U is called a premodular (cssf), if there is h∈0,∞U satisfies the next settings:
(i)
IfY∈U, Y=ϑ¯⇔hY=0 with hY≥0, where ϑ¯=0¯,0¯,0¯,(ii)
There isQ≥1, and the inequality hαY≤QαhY holds, for every Y∈U and α∈R(iii)
There isP≥1, and the inequality hY+Z≤PhY+hZ holds, for every Y,Z∈U(iv)
IfYq≤Zq, for every q∈N, one has hYq≤hZq(v)
The inequalityhYq≤hYq/2≤P0hYq holds, for some P0≥1(vi)
LetE be the space of finite sequences of fuzzy numbers; then, the closure of E=Uh(vii)
There isσ>0 with hα¯,0¯,0¯,0¯,⋯≥σαh1¯,0¯,0¯,0¯,⋯, where(9)α¯y=1,y=α,0,y≠α.Definition 6 (see [25]).
SupposeU is a (cssf). The function h∈0,∞U is called a pre-quasinorm on U, if it satisfies the following conditions:
(i)
IfY∈U, Y=ϑ¯⇔hY=0 with hY≥0, where ϑ¯=0¯,0¯,0¯,(ii)
There isQ≥1, and the inequality hαY≤QαhY satisfies, for every Y∈U and α∈R(iii)
There isP≥1, and the inequality hY+Z≤PhY+hZ holds, for each Y,Z∈UClearly, from the last two definitions, we conclude the following two theorems:Theorem 7 (see [25]).
IfU is a premodular (cssf), then it is pre-quasinormed (cssf).Theorem 8 (see [25]).
U is a pre-quasinormed (cssf) if it is quasinormed (cssf).Definition 9.
(a)
The functionh on Cτ.F is named h-convex, if
(10)hαY+1−αZ≤αhY+1−αhZ,
for everyα∈0,1 and Y,Z∈Cτ.F.
(b)
Yqq∈N⊆Cτ.Fh is h-convergent to Y∈Cτ.Fh, if and only if limq⟶∞hYq−Y=0. When the h-limit exists, then it is unique(c)
Yqq∈N⊆Cτ.Fh is h-Cauchy, if limq,r⟶∞hYq−Yr=0(d)
Γ⊂Cτ.Fh is h-closed, when for all h-converges Yqa∈N⊂Γ to Y, then Y∈Γ(e)
Γ⊂Cτ.Fh is h-bounded, if δhΓ=suphY−Z:Y,Z∈Γ<∞(f)
Theh-ball of radius ε≥0 and center Y, for every Y∈Cτ.Fh, is described as follows:
(11)BhY,ε=Z∈Cτ.Fh:hY−Z≤ε.(g)
A pre-quasinormh on Cτ.F satisfies the Fatou property, if for every sequence Zq⊆Cτ.Fh under limq⟶∞hZq−Z=0 and all Y∈Cτ.Fh, one has hY−Z≤suprinfq≥rhY−Zq
Note that the Fatou property implies theh-closed of the h-balls. We will denote the space of all increasing sequences of real numbers by I.Theorem 10.
Cτ.Fh, where hY=∑q=0∞∑p=0qρ¯Yp,0¯/q+1τq1/K, for all Y∈Cτ.F, is a premodular (cssf), when τqq∈N∈ℓ∞∩I with τ0>1.Proof.
(i) Evidently,hY≥0 and hY=0⇔Y=ϑ¯
(1-i) LetY,Z∈Cτ.F. One has
(12)hY+Z=∑q=0∞∑p=0qρ¯Yp+Zp,0¯q+1τq1/K≤∑q=0∞∑p=0qρ¯Yp,0¯q+1τq1/K+∑q=0∞∑p=0qρ¯Zp,0¯q+1τq1/K=hY+hZ<∞,
and then,Y+Z∈Cτ.F.
(iii) One getsP≥1 with hY+Z≤PhY+hZ, for all Y,Z∈Cτ.F
(1-ii) Assumeα∈R and Y∈Cτ.F, and we obtain
(13)hαY=∑q=0∞∑p=0qρ¯αYp,0¯q+1τq1/K≤supqατq/K∑q=0∞∑p=0qρ¯Yp,0¯q+1τq1/K≤QαhY<∞.
AsαY∈Cτ.F. Hence, from conditions (1-i) and (1-ii), one has Cτ.F is linear. Also, b¯r∈Cτ.F, for all r∈N, since hb¯r=∑q=0∞∑p=0qρ¯b¯r,0¯/q+1τq1/K≤∑q=0∞1/q+1τ01/K<∞.
(ii) There isQ=max1,supqατq/K−1≥1 with hαY≤QαhY, for all Y∈Cτ.F and α∈R
(2) AssumeYq≤Zq, for all q∈N and Z∈Cτ.F. One finds
(14)hY=∑q=0∞∑p=0qρ¯Yp,0¯q+1τq1/K≤∑q=0∞∑p=0qρ¯Zp,0¯q+1τq1/K=hZ<∞,
and then,Y∈Cτ.F.
(iv) Obviously, from (2)
(3) LetYq∈Cτ.F, and we get
(15)hYq/2=∑q=0∞∑p=0qρ¯Yp/2,0¯q+1τq1/K=∑q=0∞∑p=02qρ¯Yp/2,0¯2q+1τ2q+∑q=0∞∑p=02q+1ρ¯Yp/2,0¯2q+2τ2q+11/K≤21/K∑q=0∞∑p=0qρ¯Yp,0¯q+1τq1/K≤∑q=0∞ρ¯Yq,0¯+2∑p=0qρ¯Yp,0¯q+1τq+∑q=0∞2∑p=0qρ¯Yp,0¯q+1τq1/K≤21/K∑q=0∞∑p=0qρ¯Yp,0¯q+1τq1/K≤∑q=0∞3∑p=0qρ¯Yp,0¯q+1τq+∑q=0∞2∑p=0qρ¯Yp,0¯q+1τq1/K≤3K+2K1/K∑q=0∞∑p=0qρ¯Yp,0¯q+1τq1/K=3K+2K1/KhYq,
and then,Yp/2∈Cτ.F.
(v) From (4), we obtain P0=3K+2K1/K≥1
(vi) Evidently the closure ofE=Cτ.F
(vii) There is0<σ≤supqατq/K−1, for α≠0 or σ>0, for α=0 with hα¯,0¯,0¯,0¯,⋯≥σαh1¯,0¯,0¯,0¯,⋯Theorem 11.
Ifτqq∈N∈ℓ∞∩I with τ0>1, then Cτ.Fh is a pre-quasi-Banach (cssf), where hY=∑q=0∞∑p=0qρ¯Yp,0¯/q+1τq1/K, for every Y∈Cτ.F.Proof.
In view of Theorem10 and Theorem 7, the space Cτ.Fh is a pre-quasinormed (cssf). Assume Yl=Yqlq=0∞ is a Cauchy sequence in Cτ.Fh. Hence, for every ε∈0,1, one has l0∈N such that for all l,m≥l0, one gets
(16)hYl−Ym=∑q=0∞∑p=0qρ¯Ypl−Ypm,0¯q+1τq1/K<ε.
That impliesρ¯Yql−Yqm,0¯<ε. As R0,1,ρ¯ is a complete metric space. Then, Yqm is a Cauchy sequence in R0,1, for fixed q∈N, which implies limm⟶∞Yqm=Yq0, for constant q∈N. Hence, hYl−Y0<ε, for every l≥l0, since hY0=hY0−Yl+Yl≤hYl−Y0+hYl<∞. So Y0∈Cτ.F.Theorem 12.
Supposeτqq∈N∈ℓ∞∩I with τ0>1, then Cτ.Fh is a pre-quasiclosed (cssf), where hY=∑q=0∞∑p=0qρ¯Yp,0¯/q+1τq1/K, for every Y∈Cτ.F.Proof.
In view of Theorem10 and Theorem 7, the space Cτ.Fh is a pre-quasinormed (cssf). Assume Yl=Yqlq=0∞∈Cτ.Fh and liml⟶∞hYl−Y0=0; then, for all ε∈0,1, there is l0∈N such that for all l≥l0, we obtain
(17)ε>hYl−Y0=∑q=0∞∑p=0qρ¯Ypl−Yp0,0¯q+1τq1/K,
which impliesρ¯Yql−Yq0,0¯<ε. As R0,1,ρ¯ is a complete metric space, therefore, Yql is a convergent sequence in R0,1, for fixed q∈N. So, liml⟶∞Yql=Yq0, for fixed q∈N. Since hY0=hY0−Yl+Yl≤hYl−Y0+hYl<∞, one has Y0∈Cτ.F.Theorem 13.
The functionhY=∑q=0∞∑p=0qρ¯Yp,0¯/q+1τq1/K verifies the Fatou property, when τqq∈N∈ℓ∞∩I with τ0>1, for all Y∈Cτ.F.Proof.
LetZr⊆Cτ.Fh such that limr⟶∞hZr−Z=0. Since Cτ.Fh is a pre-quasiclosed space, one has Z∈Cτ.Fh. For all Y∈Cτ.Fh, one gets
(18)hY−Z=∑q=0∞∑p=0qρ¯Yp−Zp,0¯q+1τq1/K≤∑q=0∞∑p=0qρ¯Yp−Zpr,0¯q+1τq1/K+∑q=0∞∑p=0qρ¯Zpr−Zp,0¯q+1τq1/K≤supminfr≥mhY−Zr.Theorem 14.
The functionhY=∑q=0∞∑p=0qρ¯Yp,0¯/q+1τq does not satisfy the Fatou property, for all Y∈Cτ.F, when τq∈ℓ∞ and τq>1, for all q∈N.Proof.
LetZr⊆Cτ.Fh so that limr⟶∞hZr−Z=0. Since Cτ.Fh is a pre-quasiclosed space, one gets Z∈Cτ.Fh. For every Z∈Cτ.Fh, we obtain
(19)hY−Z=∑q=0∞∑p=0qρ¯Yp−Zp,0¯q+1τq≤2K−1∑q=0∞∑p=0qρ¯Yp−Zpr,0¯q+1τq+∑q=0∞∑p=0qρ¯Zpr−Zp,0¯q+1τq≤2K−1supminfr≥mhY−Zr.Example 1.
Forτq∈1,∞N, the function hY=infα>0:∑q∈N∑p=0qρ¯Yp/α,0¯/q+1τq≤1 is a norm on Cτ.F.Example 2.
The functionhY=∑q∈N∑p=0qρ¯Yp,0¯/q+13q+2/q+13 is a pre-quasinorm (not a norm) on CF3q+2/q+1q=0∞.Example 3.
The functionhY=∑q∈N∑p=0qρ¯Yp,0¯/q+13q+2/q+1 is a pre-quasinorm (not a quasinorm) on CF3q+2/q+1q=0∞.Example 4.
The functionhY=∑q∈N∑p=0qρ¯Yp,0¯/q+1dd is a pre-quasinorm, quasinorm, and not a norm on CdF, for 0<d<1.
In the next part of this section, we will use the functionh as hY=∑q=0∞∑p=0qρ¯Yp,0¯/q+1τq1/K, for every Y∈Cτ.F.Definition 15 [26].
The functionh is said to be strictly convex, (SC), if for all Y,Z∈Uh such that hY=hZ and hY+Z/2=hY+hZ/2, we get Y=Z.Definition 16 [27].
A sequenceYp⊆U is said to be ε-separated sequence for some ε>0, if
(20)sepYp=infhYp−Yq:p≠q>ε.Definition 17 (see [27]).
Letk≥2 be an integer, and a Banach space U is called k-nearly uniformly convex (k-NUC), if for any ε>0, there exists δ∈0,1 such that for any sequence Yp⊆Bh0,1, with sepYp≥ε, there are p1,p2,p3,⋯,pk∈N, such that
(21)hYp1+Yp2+Yp3+⋯+Ypkk<1−δ.Definition 18 (see [28]).
A functionh is said to satisfy the δ2-condition (h∈δ2), if for any ε>0, there exists a constant k≥2 and a>0 such that h2u≤khu+ε, for each u∈Xh, with hu≤a.
Ifh satisfies the δ2-condition for any a>0 with k≥2 depending on a, we say that h satisfies the strong δ2-condition (ρ∈δ2s).The following known results are very important for our consideration.Theorem 19 (see [28], Lemma 2.1).
Ifh∈δ2s, then for any L>0 and ε>0, there exists δ>0 such that hx+y−hx<ε, where x,y∈Xh, with hx≤L and hy≤δ.Theorem 20.
Pick anτqq∈N∈ℓ∞∩I with τ0>1; then, for any L>0 and ε>0, there exists δ>0 such that hx+y−hx<ε, for all x,y∈Cτ.Fh, with hx≤L and hy≤δ.Proof.
Sinceτq is bounded, it is easy to see that h∈δ2s. Hence, the proposition is obtained directly from Theorem 19.Theorem 21.
Supposeτqq∈N∈ℓ∞∩I with τ0>1; then, Cτ.Fh is k-NUC, for any integer k≥2.Proof.
Letε∈0,1 and xn⊆Bh0,1 with sepxn≥ε, for each m∈N, and let xnm=0,0,0,⋯,xnm,xnm+1,⋯. Since for each i∈N, xnin=0∞ is bounded, and by using the diagonal method, we can find a subsequence xnj of xn such that xnji converges for each i∈N, 0≤i≤m. Therefore, there exists an increasing sequence of positive integers tm such that sepxnjmj>tm≥ε. Hence, there is a sequence of positive integers rmm=0∞ with r0<r1<r2<⋯, such that
(22)hKxrmm≥ε2,
for eachm∈N. For fixed integer k≥2, let ε1=kp0−1−1/k−1kp0ε/4; then, by Theorem 20, there exists δ>0 such that
(23)hKx+y−hKx<ε1,
wheneverhKx≤1 and hKy≤δ. Since hKxn≤1, for any n∈N, then there exist positive integers mii=0,1,2,⋯,k−2 with m0<m1<m2<⋯<mk−2 such that hKximi≤δ. Define mk−1=mk−2+1. By inequality (1), we have hxrmkmk≥ε/2. Let si=i for 0≤i≤k−2 and sk−1=rmk−1. Then, in virtue of inequality (1), inequality (2), and convexity of the function fnu=uτn for any n∈N, we have
(24)hKxs0+xs1+xs2+.⋯+xsk−1k=∑n=0∞∑i=0nρ¯xs0i+xs1i++xsk−1i/k,0¯n+1τn=∑n=0m1−1∑i=0nρ¯xs0i+xs1i++xsk−1i/n+1,0¯n+1τn+∑n=m1∞∑i=0nρ¯xs0i+xs1i++xsk−1i/k,0¯n+1τn≤∑n=0m1−1∑i=0nρ¯xs0i+xs1i++xsk−1i/k,0¯n+1τn+∑n=m1∞∑i=0nρ¯xs1i+xs2i++xsk−1i/k,0¯n+1τn+ε1≤∑n=0m1−11k∑j=0k−1∑i=0nρ¯xsji,0¯n+1τn+∑n=m1m2−1∑i=0nρ¯xs1i+xs2i++xsk−1i/k,0¯n+1τn+∑n=m2∞∑i=0nρ¯xs1i+xs2i++xsk−1i/k,0¯n+1τn+ε1≤∑n=0m1−11k∑j=0k−1∑i=0nρ¯xsji,0¯n+1τn+∑n=m1m2−1∑i=0nρ¯xs1i+xs2i++xsk−1i/k,0¯n+1τn+∑n=m2∞∑i=0nρ¯xs2i+xs3i++xsk−1i/k,0¯n+1τn+2ε1≤∑n=0m1−11k∑j=0k−1∑i=0nρ¯xsji,0¯n+1τn+∑n=m1m2−11k∑j=1k−1∑i=0nρ¯xsji,0¯n+1τn+∑n=m2m3−11k∑j=2k−1∑i=0nρ¯xsji,0¯n+1τn++∑n=mk−1mk−11k∑j=k−2k−1∑i=0nρ¯xsji,0¯n+1τn+∑n=mk∞∑i=0nρ¯xski/k,0¯n+1τn+k−1ε1≤hKxs0+xs1+xs2+.⋯+xsk−2k+1k∑n=0mk−1∑i=0nρ¯xski,0¯n+1τn+∑n=mk∞∑i=0nρ¯xski/k,0¯n+1τn+k−1ε1≤k−1k+1k∑n=0mk−1∑i=0nρ¯xski,0¯n+1τn+1kp0∑n=mk∞∑i=0nρ¯xski,0¯n+1τn+k−1ε1≤1−1k+1k1−∑n=mk∞∑i=0nρ¯xski,0¯n+1τn+1kp0∑n=mk∞∑i=0nρ¯xski,0¯n+1τn+k−1ε1=1+k−1ε1−kp0−1−1kp0∑n=mk∞∑i=0nρ¯xski,0¯n+1τn≤1+k−1ε1−kp0−1−1kp0ε2=1−kp0−1−1kp0ε4.
Therefore,Cτ.Fh is k-NUC.Recall thatk-NUC implies reflexivity.Definition 22.
The spaceUh satisfies the property R, if and only if, for all decreasing sequence Γjj∈N of h-closed and h-convex nonempty subsets of Uh with supj∈NKhY,Γj<∞, for some Y∈Uh, one has ⋂j∈NΓj≠∅.
By fixingΓ a nonempty h-closed and h-convex subset of Cτ.Fh.Theorem 23.
Ifτqq∈N∈ℓ∞∩I with τ0>1, one has the following:
(i)
SupposeY∈Cτ.Fh with KhY,Γ=infhY−Z:Z∈Γ<∞. There is a unique λ∈Γ so that KhY,Γ=hY−λ(ii)
Cτ.Fh verifies the property R.Proof.
To prove (i), assumeY∉Γ as Γ is h-closed. One has C≔KhY,Γ>0. Hence, for all r∈N, one has Zr∈Γ with hY−Zr<C1+1/r. If Zr/2 is not h-Cauchy, one gets a subsequence Zgr/2 and l0>0 with hZgr−Zgj/2≥l0, for every r>j≥0, since
(25)maxhY−Zgr,hY−Zgj≤C1+1gj,hZgr−Zgj2≥l0≥C1+1gjl02C,
for everyr>j≥0. Since τqq∈N∈ℓ∞∩I with τ0>1, then the function fnu=uτn is strictly convex, for any n∈N. Therefore, the space Cτ.Fh is strictly convex; hence,
(26)hY−Zgr+Zgj2<C1+1gj.
Then,(27)C=KhY,Γ<C1+1gj,
for allj∈N. By putting j⟶∞, one has a contradiction. So Zr/2 is h-Cauchy. As Cτ.Fh is h-complete, then Zr/2h-converges to some Z. For all j∈N, one gets Zr+Zj/2h-converges to Z+Zj/2. Since Γ is h-closed and h-convex, then Z+Zj/2∈Γ. Since Z+Zj/2h-converges to 2Z, then 2Z∈Γ. Let λ=2z, and from Theorem 13, since h satisfies the Fatou property, one has
(28)KhY,Γ≤hY−λ≤supiinfj≥ihY−Z+Zj2≤supiinfj≥isupiinfr≥ihY−Zr+Zj2≤12supiinfr≥isupiinfr≥ihY−Zr+hY−Zj=KhY,Γ.
ThenhY−λ=KhY,Γ. Since h is (SC), this implies the uniqueness of λ. To prove (ii), assume Y∉Γr0, for some r0∈N. Since KhY,Γrr∈N∈ℓ∞ is increasing, put limr⟶∞KhY,Γr=C, when C>0. Otherwise, Y∈Γr, for all r∈N. According to (i), there is one point Zr∈Γr with KhY,Γr=hY−Zr, for every r∈N. A similar proof will prove that Zr/2h-converges to some Z∈Cτ.Fh. As Γr is h-convex, decreasing, and h-closed, one has 2Z∈∩r∈NΓr.Definition 24.
The spaceUh verifies the h-normal structure-property, if and only if, for all nonempty h-bounded, h-convex and h-closed subset Γ of Uh not decreased to one point, and one has Y∈Γ with
(29)supZ∈ΓhY−Z<δhΓ≔suphY−Z:Y,Z∈Γ<∞.Definition 25 (see [29]).
Uh is a real Banach space, and SUh is the unit sphere of Uh. The weakly convergent sequence coefficient of Uh, denoted by WCSUh, is defined as follows:
(30)WCSUh=infAxn:xnn=1∞⊂SUh,Axn=A1xn,xnw⟶0,where
(31)Axn=limsupn⟶∞xi−xj:i,j≥n,i≠j,A1xn=liminfn⟶∞xi−xj:i,j≥n,i≠j.Theorem 26 (see [30]).
A reflexive Banach spaceUh with WCSUh>1 has normal structure-property.Theorem 27.
Ifτqq∈N∈ℓ∞∩I with τ0>1, then Cτ.Fh holds the h-normal structure-property.Proof.
Take anyε>0 and an asymptotic equidistant sequence xn⊂SCτ.Fh with xnw⟶0 and put v1=x1. There exists i1∈N such that h∑i=i1+1∞v1ib¯i<ε. Since xn⟶0 coordinate-wise, there exists n2∈N such that h∑i=1i1xnib¯i<ε, whenever n≥n2. Take v2=xn2; then, there is i2>i1 such that h∑i=i2+1∞v1ib¯i<ε. Since xni⟶0 coordinate-wise, there exists n3∈N such that h∑i=1i2xnib¯i<ε, whenever n≥n3. Continuing this process in such a way by induction, we get a subsequence vn of xn such that
(32)h∑i=in+1∞vnib¯i<ε,h∑i=1invn+1ib¯i<ε.
Putzn=∑i=in−1+1invnib¯i, for n=2,3,⋯ Then,
(33)1≥hzn=h∑i=1∞vnib¯i−∑i=1in−1vnib¯i−∑i=in+1∞vnib¯i≥h∑i=1∞vnib¯i−h∑i=1in−1vnib¯i−h∑i=in+1∞vnib¯i>1−2ε.
Moreover, for anyn,m∈N with n≠m, we have
(34)hvn−vm=h∑i=1∞vnib¯i−∑i=1∞vmib¯i≥h∑i=in−1+1invnib¯i−∑i=im−1+1imvmib¯i−h∑i=1in−1vnib¯i−h∑i=in+1∞vnib¯i−h∑i=1im−1vmib¯i−h∑i=im+1∞vmib¯i≥hzn−zm−4ε.
This means thatAxn=Avn≥Azn−4ε. Put un=zn/zn, for n=2,3,⋯ Then,
(35)un∈SCτ.Fh,(36)Axn≥1−εAun−4ε.
On the other hand,(37)hvn−vm≤hzn−zm+4ε≤hun−um+4ε,
for anyn,m∈N with n≠m. Therefore,
(38)Aun≥Axn−4ε.
By the arbitrariness ofε>0, we have from the relations (35), (36), and (38) that
(39)WCSCτ.Fh=infAun,
such that(40)un=∑i=in−1+1inunib¯i∈SCτ.Fh,0=i0<i1<⋯,unw⟶0andunisasymptoticequidistant.
Takem∈N large enough such that ∑k=im−1+1∞b/kτk<ε, where b≔∑i=in−1+1inuni. We have for n<m that
(41)hKun−um=∑k=in−1+1im−11k∑i=1kρ¯uni,0¯τk+∑k=im−1+1∞1kb+∑i=1kρ¯umi,0¯τk≥∑k=in−1+1im−11k∑i=1kρ¯uni,0¯τk+∑k=im−1+1∞1k∑i=1kρ¯umi,0¯τk=∑k=in−1+1∞1k∑i=1kρ¯uni,0¯τk−∑k=im−1+1∞bkτk+∑k=im−1+1∞1k∑i=1kρ¯umi,0¯τk>1−ε+1=2−ε,
that is,Anun≥2−ε1/K. Note that
(42)∑k=im−1+1∞1kb+∑i=1kρ¯umi,0¯τk1/K≤∑k=im−1+1∞bkτk1/K+∑k=im−1+1∞1k∑i=1kρ¯umi,0¯τk1/K<ε1/K+1.
Therefore,(43)hKun−um=∑k=in−1+1im−11k∑i=1kρ¯umi,0¯τk+∑k=im−1+1∞1kb+∑i=1kρ¯umi,0¯τk≤∑k=in−1+1∞1k∑i=1kρ¯umi,0¯τk+∑k=im−1+1∞1kb+∑i=1kρ¯umi,0¯τk≤1+1+ε1/KK,
for anyn,m∈N with n≠m. Therefore, Anun≤1+1+ε1/KK1/K, and by the arbitrariness of ε>0, we obtain WCSCτ.Fh=21/K. From Theorem 21 and Theorem 26, the sequence space Cτ.Fh has the h-normal structure-property.


## 4. Kannan Contraction Mapping onCτ.F

In this section, we look at how to configureCτ.Fh with different h so that there is only one fixed point of Kannan contraction mapping.Definition 28.
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. The operator V is called Kannan h-nonexpansive, when α=1/2.
An elementY∈Uh is called a fixed point of V when VY=Y.Theorem 29.
Ifτqq∈N∈ℓ∞∩I with τ0>1, and V:Cτ.Fh⟶Cτ.Fh is Kannan h-contraction mapping, where hY=∑q=0∞∑p=0qρ¯Yp,0¯/q+1τq1/K, for all Y∈Cτ.F, then V has a unique fixed point.Proof.
IfY∈Cτ.F, one has VpY∈Cτ.F. As V is a Kannan h-contraction mapping, one gets
(44)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
(45)hVlY−VmY≤αhVlY−Vl−1Y+hVmY−Vm−1Y≤αα1−αl−1+α1−αm−1hVY−Y.
Then,VlY is a Cauchy sequence in Cτ.Fh. As the space Cτ.Fh is pre-quasi-Banach space, one has Z∈Cτ.Fh with liml⟶∞VlY=Z. To prove that VZ=Z, since h has the Fatou property, one obtains
(46)hVZ−Z≤supiinfl≥ihVl+1Y−VlY≤supiinfl≥iα1−αlhVY−Y=0,
and then,VZ=Z. So Z is a fixed point of V. To show the uniqueness. Let Y,Z∈Cτ.Fh be two not equal fixed points of V. One has
(47)hY−Z≤hVY−VZ≤αhVY−Y+hVZ−Z=0.
So,Y=Z.Corollary 30.
Ifτqq∈N∈ℓ∞∩I with τ0>1, and V:Cτ.Fh⟶Cτ.Fh is Kannan h-contraction mapping, where hY=∑q=0∞∑p=0qρ¯Yp,0¯/q+1τq1/K, for all Y∈Cτ.F, one has V has unique fixed point Z so that hVlY−Z≤αa/1−αl−1hVY−Y.Proof.
In view of Theorem29, one has a unique fixed point Z of V. So
(48)hVlY−Z=hVlY−VZ≤αhVlY−Vl−1Y+hVZ−Z=αα1−αl−1hVY−Y.Example 5.
AssumeV:CF2q+3/q+2q=0∞h⟶CF2q+3/q+2q=0∞h, where hg=∑q=0∞∑p=0qρ¯gp,0¯/q+12q+3/q+2, for every g∈CF2q+3/q+2q=0∞ and
(49)Vg=g4,hg∈0,1,g5,hg∈1,∞.
As for eachg1,g2∈CF2q+3/q+2q=0∞h with hg1,hg2∈0,1, one has
(50)hVg1−Vg2=hg14−g24≤1274h3g14+h3g24=1274hVg1−g1+hVg2−g2.
For allg1,g2∈CF2q+3/q+2q=0∞h with hg1,hg2∈1,∞, one has
(51)hVg1−Vg2=hg15−g25≤1644h4g15+h4g25=1644hVg1−g1+hVg2−g2.
For allg1,g2∈CF2q+3/q+2q=0∞h with hg1∈0,1 and hg2∈1,∞, we get
(52)hVg1−Vg2=hg14−g25≤1274h3g14+1644h4g25≤1274h3g14+h4g25=1274hVg1−g1+hVg2−g2.
Hence,V is Kannan h-contraction. As h satisfies the Fatou property, from Theorem 29, one has V holds one fixed point ϑ¯∈CF2q+3/q+2q=0∞h.Definition 31.
Pick upUh be a pre-quasinormed (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 6.
SupposeV:CFq+1/2q+4q=0∞h⟶CFq+1/2q+4q=0∞h, where hZ=∑q=0∞∑p=0qρ¯Zp,0¯/q+1q+1/2q+44, for every Z∈CFq+1/2q+4q=0∞ and
(53)VZ=118b¯0+Z,Z0y∈0,117,117b¯0,Z0y=117,118b¯0,Z0y∈117,1.
V is clearly both h-sequentially continuous and discontinuous at 1/17b¯0∈CFq+1/2q+4q=0∞h.Example 7.
AssumeV is defined as in Example 5. Suppose Zn⊆CF2q+3/q+2q=0∞h such that limn⟶∞hZn−Z0=0, where Z0∈CF2q+3/q+2q=0∞h with hZ0=1.
As the pre-quasinormh is continuous, we have
(54)limn⟶∞hVZn−VZ0=limn⟶∞hZn4−Z05=hZ020>0.
Therefore,V is not h-sequentially continuous at Z0.Theorem 32.
Ifτqq∈N∈ℓ∞∩I with τ0>1, V:Cτ.Fh⟶Cτ.Fh, where hY=∑q=0∞∑p=0qρ¯Yp,0¯/q+1τq, for all Y∈Cτ.F. Suppose
(1)
V is Kannan h-contraction mapping(2)
V is h-sequentially continuous at Z∈Cτ.Fh(3)
there isY∈Cτ.Fh with VlY has VljY converging to Z
Then,Z∈Cτ.Fh is the only fixed point of V.Proof.
AssumeZ is not a fixed point of V, and one has VZ≠Z. From parts (2) and (4), we get
(55)limlj⟶∞hVljY−Z=0,limlj⟶∞hVlj+1Y−VZ=0.
AsV is Kannan h-contraction, one obtains
(56)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 the uniqueness, let Z,Y∈Cτ.Fh be two not equal fixed points of V. One obtains
(57)hZ−Y≤hVZ−VY≤αhVZ−Z+hVY−Y=0.
Hence,Z=Y.Example 8.
AssumeV is defined as in Example 5. Let hY=∑q∈N∑p=0qρ¯Yp,0¯/q+12q+3/q+2, for all v∈CF2q+3/q+2q=0∞. Since for all Y1,Y2∈CF2q+3/q+2q=0∞h with hY1,hY2∈0,1, one gets hVY1−VY2=hY1/4−Y2/4≤2/27h3Y1/4+h3Y2/4=2/27hVY1−Y1+hVY2−Y2. For all Y1,Y2∈CF2q+3/q+2q=0∞h with hY1,hY2∈1,∞, one gets
(58)hVY1−VY2=hY15−Y25≤14h4Y15+h4Y25=14hVY1−Y1+hVY2−Y2.
For allY1,Y2∈CF2q+3/q+2q=0∞h with hY1∈0,1 and hY2∈1,∞, one gets
(59)hVY1−VY2=hY14−Y25≤227h3Y14+14h4Y25≤227h3Y14+h4Y25=227hVY1−Y1+hVY2−Y2.
SoV is Kannan h-contraction and VpY=Y/4p,hY∈0,1,Y/5p,hY∈1,∞.
Obviously,V is h-sequentially continuous at ϑ¯∈CF2q+3/q+2q=0∞h, and VpY holds VljY converges to ϑ¯. By Theorem 32, the point ϑ¯∈CF2q+3/q+2q=0∞h is the only fixed point of V.


## 5. Kannan Nonexpansive Mapping onCτ.Fh

We introduce the sufficient conditions ofCτ.Fh, where hg=∑m=0∞ρ¯gm,0¯τm1/K, for every g∈Cτ.F, such that the Kannan nonexpansive mapping on it has a fixed point, by fixing Γ a nonempty h-bounded, h-convex, and h-closed subset of Cτ.Fh.Lemma 33.
IfCτ.Fh verifies the R property and the h-quasinormal property. Assume V:Γ⟶Γ is a Kannan h-nonexpansive mapping. For t>0, let Gt=Y∈Γ:hY−VY≤t≠∅. Put
(60)Γt=⋂Bhr,j:VGt⊂Bhr,j∩Γ.
Then,Γt≠∅, h-convex, h-closed subset of Γ, and VΓt⊂Γt⊂GtandδhΓt≤t.Proof.
SinceVGt⊂Γt, then Γt≠∅. As the h-balls are h-convex and h-closed, then Γt is a h-closed and h-convex subset of Γ. To show that Γt⊂Gt, assume Y∈Γt. When hY−VY=0, one has Y∈Gt. Else, assume hY−VY>0. Put
(61)r=suphVZ−VY:Z∈Gt.
From the definition ofr, one gets VGt⊂BhVY,r. Therefore, Γt⊂BhVY,r, then hY−VY≤r. Let l>0. One has Z∈Gt with r−l≤hVZ−VY. So
(62)hY−VY−l≤r−l≤hVZ−VY≤12hY−VY+hZ−VZ≤12hY−VY+t.
Asl is an arbitrary positive, one obtains hY−VY≤t; then, Y∈Gt. Since VGt⊂Γt, one gets VΓt⊂VGt⊂Γt, so Γt is V-invariant, to show that δhΓt≤t, since
(63)hVY−VZ≤12hY−VY+hZ−VZ,
for allY,Z∈Gt. Let Y∈Gt. Then, VGt⊂BhVY,t. The definition of Γt gives Γt⊂BhVY,t. Therefore, VY∈⋂t∈ΓtBhZ,t. One has hZ−Y≤t, for all Z,Y∈Γt, so δhΓt≤t.Theorem 34.
IfCτ.Fh satisfies the h-quasinormal property and the R property, let V:Γ⟶Γ be a Kannan h-nonexpansive mapping. Then, V has a fixed point.Proof.
Lett0=infhY−VY:Y∈Γ and tr=t0+1/r, for every r≥1. By the definition of t0, one gets Gtr=Y∈Γ:hY−VY≤tr≠∅, for every r≥1. Assume Γtr is defined as in Lemma 33. Clearly, Γtr is a decreasing sequence of nonempty h-bounded, h-closed, and h-convex subsets of Γ. The property R investigates that Γ∞=⋂r≥1Γtr≠∅. Let Y∈Γ∞, and one has hY−VY≤tr, for all r≥1. Suppose r⟶∞; then, hY−VY≤t0, so hY−VY=t0. Therefore, Gt0≠∅. Then, t0=0. Else, t0>0; then, V fails to have a fixed point. Let Γt0 be defined in Lemma 33. As V fails to have a fixed point and Γt0 is V-invariant, then Γt0 has more than one point, so δhΓt0>0. By the h-quasinormal property, one has Y∈Γt0 with
(64)hY−Z<δhΓt0≤t0,
for allZ∈Γt0. From Lemma 33, we get Γt0⊂Gt0. From definition of Γt0, VY∈Gt0⊂Γt0. Then,
(65)hY−VY<δhΓt0≤t0,
which contradicts the definition oft0. Then, t0=0 which gives that any point in Gt0 is a fixed point of V.According to Theorems23, 27, and 34, we conclude the following:Corollary 35.
Assumeτqq∈N∈ℓ∞∩I with τ0>1, and V:Γ⟶Γ is a Kannan h-nonexpansive mapping. Then, V has a fixed point.Example 9.
AssumeV:Γ⟶Γ with VY=Y/4,hY∈0,1,Y/5,hY∈1,∞, where Γ=Y∈CF2q+3/q+2q=0∞h:Y0=Y1=0¯ and hY=∑q∈Nρ¯Yq,0¯2q+3/q+2, for every Y∈CF2q+3/q+2q=0∞h. By using Example 8, V is Kannan h-contraction. So it is Kannan h-nonexpansive. By Corollary 35, V has a fixed point ϑ¯ in Γ.


## 6. Kannan Contraction and Structure of Operator Ideal

The structure of the operator ideal byCτ.Fh equipped with the definite function h, where hg=∑q=0∞∑p=0qρ¯gp,0¯/q+1τq1/K, for every g∈Cτ.F, and s-numbers has been explained. Finally, we examine the idea of Kannan contraction mapping in its associated pre-quasioperator ideal. As well, the existence of a fixed point of Kannan contraction mapping has been introduced. We indicate the space of all bounded, finite rank linear operators from a Banach space Δ into a Banach space Λ by LΔ,Λ, and FΔ,Λ, and if Δ=Λ, we inscribe LΔ and FΔ.Definition 36 (see [31]).
Ans-number function is s:LΔ,Λ⟶R+N which sorts every V∈LΔ,ΛasdVd=0∞ verifies the following settings:
(a)
V=s0V≥s1V≥s2V≥⋯≥0,forallV∈LΔ,Λ(b)
sl+d−1V1+V2≤slV1+sdV2, for all V1,V2∈LΔ,Λ and l, d∈N(c)
sdVYW≤VsdYW, for all W∈LΔ0,Δ, Y∈LΔ,Λ, and V∈LΛ,Λ0, where Δ0 and Λ0 are arbitrary Banach spaces(d)
IfV∈LΔ,Λ and γ∈R, then sdγV=γsdV(e)
SupposerankV≤d, and then, sdV=0, for each V∈LΔ,Λ(f)
sl≥aIa=0 or sl<aIa=1, where Ia denotes the unit map on the a-dimensional Hilbert space ℓ2aDefinition 37 (see [8]).
(i)
L is the class of all bounded linear operators within any two arbitrary Banach spaces. A subclass U of L is said to be an operator ideal, if all UΔ,Λ=U∩LΔ,Λ verifies the following conditions: IΓ∈U, where Γ denotes Banach space of one dimension(ii)
The spaceUΔ,Λ is linear over R(iii)
AssumeW∈LΔ0,Δ, X∈UΔ,Λ, and Y∈LΛ,Λ0, then YXW∈UΔ0,Λ0Notation 38.
(66)✠¯U≔✠¯UΔ,Λ,where
(67)✠¯UΔ,Λ≔V∈LΔ,Λ:sdV¯d=0∞∈U,where
(68)sdV¯x=1,x=sdV,0,x≠sdV.Theorem 39.
SupposeU is a (cssf); then, ✠¯U is an operator ideal.Proof.
(i)
AssumeV∈FΔ,Λ and rankV=n for all n∈N; as b¯i∈U for all i∈N and U is a linear space, one has
siV¯i=0∞=s0V¯,s1V¯,⋯,sn−1V¯,0¯,0¯,0¯,⋯=∑i=0n−1siV¯b¯i∈U; for that V∈✠¯UΔ,Λ then FΔ,Λ⊆✠¯EΔ,Λ.
(ii)
SupposeV1,V2∈✠¯UΔ,Λ and β1,β2∈R, then by Definition 4 condition (33), one has si/2V1¯i=0∞∈U and si/2V1¯i=0∞∈U, as i≥2i/2; by the definition of s-numbers and siP is a decreasing sequence, one gets siβ1V1+β2V2¯≤s2i/2β1V1+β2V2¯≤si/2β1V1+si/2β2V2¯=β1si/2V1¯+β2si/2V2¯, for each i∈N. In view of Definition 4 condition (23) and U is a linear space, one obtains siβ1V1+β2V2¯i=0∞∈U; hence, β1V1+β2V2∈✠¯UΔ,Λ.(iii)
SupposeP∈LΔ0,Δ, T∈✠¯UΔ,Λ, and R∈LΛ,Λ0, one has siT¯i=0∞∈U, and as siRTP¯≤RsiT¯P, by Definition 4 conditions (22) and (23), one gets
siRTP¯i=0∞∈U, and then, RTP∈✠¯UΔ0,Λ0.According to Theorems10 and 39, one concludes the following theorem.Theorem 40.
Letτqq∈N∈ℓ∞∩I with τ0>1, and one has ✠¯Cτ.Fh is an operator ideal.Definition 41 (see [9]).
A functionH∈0,∞U is called a pre-quasinorm on the ideal U if the next conditions hold:
(1)
LetV∈UΔ,Λ, HV≥0, and HV=0, if and only if V=0(2)
We haveQ≥1 so as to HαV≤DαHV, for every V∈UΔ,Λ and α∈R(3)
We haveP≥1 so that HV1+V2≤PHV1+HV2, for each V1,V2∈UΔ,Λ(4)
We haveσ≥1 for V∈LΔ0,Δ, X∈UΔ,Λ, and Y∈LΛ,Λ0; then, HYXV≤σYHXV.Theorem 42 (see [9]).
H is a pre-quasinorm on the ideal U if H is a quasinorm on the ideal U.Theorem 43.
Ifτqq∈N∈ℓ∞∩I with τ0>1, then the function H is a pre-quasinorm on ✠¯Cτ.Fh, with HZ=hsqZ¯q=0∞, for all Z∈✠¯Cτ.FhΔ,Λ.Proof.
(1)
WhenX∈✠¯Cτ.FhΔ,Λ, HX=hsqX¯q=0∞≥0 and HX=hsqX¯q=0∞=0, if and only if sqX¯=0¯, for all q∈N, if and only if X=0(2)
There isQ≥1 with HαX=hsqαX¯q=0∞≤QαHX, for all X∈✠¯Cτ.FhΔ,Λ and α∈R(3)
One hasPP0≥1 so that for X1,X2∈✠¯Cτ.FhΔ,Λ, one can see
(69)HX1+X2=hsqX1+X2¯q=0∞≤Phsq/2X1¯q=0∞+hsq/2X2¯q=0∞≤PP0hsqX1¯q=0∞+hsqX2¯q=0∞(4)
We haveρ≥1, if X∈LΔ0,Δ, Y∈✠¯Cτ.FhΔ,Λ, and Z∈LΛ,Λ0, and then, HZYX=hsqZYX¯q=0∞≤hXZsqY¯q=0∞≤ρXHYZ.In the next theorems, we will use the notation✠¯Cτ.Fh,H, where HV=hsqV¯q=0∞, for all V∈✠¯Cτ.Fh.Theorem 44.
Supposeτqq∈N∈ℓ∞∩I with τ0>1, and one has ✠¯Cτ.Fh,H is a pre-quasi-Banach operator ideal.Proof.
SupposeVaa∈N is a Cauchy sequence in ✠¯Cτ.FhΔ,Λ. As LΔ,Λ⊇SCτ.FhΔ,Λ, one has
(70)HVr−Va=hsqVr−Va¯q=0∞≥hs0Vr−Va¯,0¯,0¯,0¯,⋯≥infqVr−Vaτq/K∑q=0∞1q+1τq1/K.
Hence,Vaa∈N is a Cauchy sequence in LΔ,Λ. LΔ,Λ is a Banach space, so there exists V∈LΔ,Λ so that lima⟶∞Va−V=0 and since sqVa¯q=0∞∈Cτ.Fh, for all a∈N, and Cτ.Fh is a premodular (cssf). Hence, one can see
(71)HV=hsqV¯q=0∞≤hsq/2V−Va¯q=0∞+hsq/2¯Vaq=0∞≤hVa−V1¯q=0∞+3K+2K1/KhsqVa¯q=0∞<ε.
We obtainsqV¯q=0∞∈Cτ.Fh, and hence, V∈✠¯Cτ.FhΔ,Λ.Theorem 45.
Ifτqq∈N∈ℓ∞∩I with τ0>1, one has ✠¯Cτ.Fh,H is a pre-quasiclosed operator ideal.Proof.
SupposeVa∈✠¯Cτ.FhΔ,Λ, for all a∈N and lima⟶∞HVa−V=0. As LΔ,Λ⊇SCτ.FhΔ,Λ, one has
(72)HVa−V=hsqVa−V¯q=0∞≥hs0Va−V¯,0¯,0¯,0¯,⋯≥infqVa−Vτq/K∑q=0∞1q+1τq1/K.
SoVaa∈N is convergent in LΔ,Λ. i.e., lima⟶∞Va−V=0, and since sqVa¯q=0∞∈Cτ.Fh, for all q∈N and Cτ.Fh is a premodular (cssf). Hence, one can see
(73)HV=hsqV¯q=0∞≤hsq/2V−Va¯q=0∞+hsq/2¯Vaq=0∞≤hVa−V1¯q=0∞+3K+2K1/KhsqVa¯q=0∞<ε.We obtain sqV¯q=0∞∈Cτ.Fh, and hence, V∈✠¯Cτ.FhΔ,Λ.Definition 46.
A pre-quasinormH on the ideal ✠¯Uh verifies the Fatou property if for every Tqq∈N⊆✠¯UhΔ,Λ so that limq⟶∞HTq−T=0 and M∈✠¯UhΔ,Λ, one gets
(74)HM−T≤supqinfj≥qHM−Tj.Theorem 47.
Supposeτqq∈N∈ℓ∞∩I with τ0>1, then ✠¯Cτ.Fh,H does not satisfy the Fatou property.Proof.
AssumeTqq∈N⊆✠¯Cτ.FhΔ,Λ with limq⟶∞HTq−T=0. Since ✠¯Cτ.Fh is a pre-quasiclosed ideal, then T∈✠¯Cτ.FhΔ,Λ. So for every M∈✠¯Cτ.FhΔ,Λ, one has
(75)HM−T=∑q=0∞∑p=0qρ¯spM−T¯,0¯q+1τq1/K≤∑q=0∞∑p=0qρ¯sp/2M−Tj¯,0¯q+1τq1/K+∑q=0∞∑p=0qρ¯sp/2Tj−T¯,0¯q+1τq1/K≤3K+2K1/Ksuprinfj≥r∑q=0∞∑p=0qρ¯spM−Tj¯,0¯q+1τq1/K.Definition 48.
An operatorV:✠¯UhΔ,Λ⟶✠¯UhΔ,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∈✠¯UhΔ,Λ.Definition 49.
An operatorV:✠¯UhΔ,Λ⟶✠¯UhΔ,Λ is said to be H-sequentially continuous at M, where M∈✠¯UhΔ,Λ, if and only if limr⟶∞HTr−M=0⇒limr⟶∞HVTr−VM=0.Example 10.
V:✠¯CF2q+3/q+2q=0∞hΔ,Λ⟶✠¯CF2q+3/q+2q=0∞hΔ,Λ,
whereHT=∑q=0∞∑p=0qρ¯spT¯,0¯/q+12q+3/q+2, for every T∈✠¯CF2q+3/q+2q=0∞hΔ,Λ and
(76)VT=T6,HT∈0,1,T7,HT∈1,∞.
Evidently,V is H-sequentially continuous at the zero operator Θ∈✠¯CF2q+3/q+2q=0∞h. Let Tj⊆✠¯CF2q+3/q+2q=0∞h be such that limj⟶∞HTj−T0=0, where T0∈✠¯CF2q+3/q+2q=0∞h with HT0=1. Since the pre-quasinorm H is continuous, one gets
(77)limj⟶∞HVTj−VT0=limj⟶∞HT06−T07=HT042>0.
Therefore,V is not H-sequentially continuous at T0.Theorem 50.
Pick upτqq∈N∈ℓ∞∩I with τ0>1 and V:✠¯Cτ.FhΔ,Λ⟶✠¯Cτ.FhΔ,Λ. Assume
(i)
V is Kannan H-contraction mapping(ii)
V is H-sequentially continuous at an element A∈✠¯Cτ.FhΔ,Λ(iii)
there areG∈✠¯Cτ.FhΔ,Λ such that the sequence of iterates VrG has a VrmG converging to M
Then,M∈✠¯Cτ.FhΔ,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
(78)limrm⟶∞HVrmG−M=0,limrm⟶∞HVrm+1G−VM=0.
SinceV is Kannan H-contraction, one obtains
(79)0<HVM−M=HVM−Vrm+1G+VrmGminus;M+Vrm+1G−VrmG≤3K+2K1/KHVrm+1G−VM+3K+2K2/KHVrmG−M+3K+2K2/Kαα1−αrm−1HVG−G.
Asrm⟶∞, there is a contradiction. Hence, M is a fixed point of V. To prove the uniqueness of the fixed point M, suppose one has two not equal fixed points M,J∈✠¯Cτ.FhΔ,Λ of V. So, one gets HM−J≤HVM−VJ≤αHVM−M+HVJ−J=0. Then, M=J.Example 11.
Given Example10, since for all T1,T2∈✠¯CF2q+3/q+2q=0∞h with HT1,HT2∈0,1, we have
(80)HVT1−VT2=HT16−T26≤21254H5T16+H5T26=21254HVT1−T1+HVT2−T2.
For allT1,T2∈✠¯CF2q+3/q+2q=0∞h with HT1,HT2∈1,∞, we have
(81)HVT1−VT2=HT17−T27≤22164H6T17+H6T27=22164HVT1−T1+HVT2−T2.
For allT1,T2∈✠¯CF2q+3/q+2q=0∞h with HT1∈0,1 and HT2∈1,∞, we have
(82)HVT1−VT2=HT16−T27≤21254H5T16+22164H6T27≤21254HVT1−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∈✠¯CF2q+3/q+2q=0∞h, and VrT has a subsequence VrmT converges to Θ. By Theorem 50, Θ is the only fixed point of G.


## 7. Applications

Theorem 51.
Consider the summable equation(83)Yp=Rp+∑r=0∞Dp,rmr,Yr,
which presented by many authors [32, 33, 34], and assume V:Cτ.Fh⟶Cτ.Fh, where τqq∈N∈ℓ∞∩I with τ0>1 and hY=∑q=0∞∑p=0qρ¯Yp,0¯/q+1τq1/K, for all Y∈Cτ.F, is defined by
(84)VYpp∈N=Rp+∑r=0∞Dp,rmr,Yrp∈N.
The summable equation (83) has a unique solution in Cτ.Fh, if D:N2⟶R, m:N×R0,1⟶R0,1, R:N⟶R0,1, and Z:N⟶R0,1; assume there is δ∈R such that supqδτq/K∈0,0.5, and for all q∈N, let
(85)∑p=0q∑r=0∞Dp,rmr,Yr−mr,Zr≤δ∑p=0qRp−Yp+∑r=0∞Dp,rmr,Yr+∑p=0qRp−Zp+∑r=0∞Dp,rmr,Zr.Proof.
One has(86)hVY−VZ=∑q=0∞∑p=0qρ¯VYp−VZp,0¯q+1τq1/K=∑q=0∞∑p=0qρ¯∑r=0∞Dp,rmr,Yr−mr,Zr,0¯q+1τq1/K≤supqδτq/K∑q=0∞∑p=0qρ¯Rp−Yp+∑r=0∞Dp,rmr,Yr,0¯q+1τq1/K+supqδτq/K∑q=0∞∑p=0qρ¯Rp−Zp+∑r=0∞Dp,rmr,Zr,0¯q+1τq1/K=supqδτq/KhVY−Y+hVZ−Z.
By Theorem29, one gets a unique solution of equation (83) in Cτ.Fh.Example 12.
SupposeCF2q+3/q+2q=0∞h, where hY=∑q=0∞∑p=0qρ¯Yp,0¯/q+12q+3/q+2, for all Y∈CF2q+3/q+2q=0∞. Consider the summable equation
(87)Yp=Rp+∑r=0∞−1p+rYpp2+r2+1t,
withp≥2 and t>0. Suppose Γ=Y∈CF2q+3/q+2q=0∞h:Y0=Y1=0¯. Indeed, Γ is a nonempty, h-convex, h-closed, and h-bounded subset of CF2q+3/q+2q=0∞h. Let V:Γ⟶Γ be defined by
(88)VYpp≥2=Rp+∑r=0∞−1p+rYpp2+r2+1tp≥2.
Obviously,(89)∑p=0q∑r=0∞−1pYpp2+r2+1t−1r−−1r≤12∑p=0qRp−Yp+∑r=0∞−1p+rYpp2+r2+1t+∑p=0qRp−Zp+∑r=0∞−1p+rZpp2+r2+1t.
By Corollary35 and Theorem 51, the summable equation (87) has a solution in Γ.Example 13.
SupposeCF2q+3/q+2q=0∞h, where hY=∑q=0∞∑p=0qρ¯Yp,0¯/q+12q+3/q+2, for every Y∈CF2q+3/q+2q=0∞. Consider the following nonlinear difference equation:
(90)Yp=Rp+∑l=0∞−1p+lYp−2rYp−1p+l2+1,
withr,p>0, Y−2x,Y−1x>0, for all x∈R, and assume V:CF2q+3/q+2q=0∞⟶CF2q+3/q+2q=0∞ is defined by
(91)VYpp=0∞=Rp+∑l=0∞−1p+lYp−2rYp−1p+l2+1p=0∞.
Evidently,(92)∑p=0q∑l=0∞−1pYp−2rYp−1p+l2+1−1l−−1l≤12∑p=0qRp−Yp+∑l=0∞−1p+lYp−2rYp−1p+l2+1+∑p=0qRp−Zp+∑l=0∞−1p+lZp−2rZp−1p+l2+1.
By Theorem51, the nonlinear difference equation (90) has a unique solution in CF2q+3/q+2q=0∞.


## 8. Conclusion

Rather than simply referring to a “quasi-normed” place, we used the term “prequasi-normed.” It is the concept of a fixed point of the Kannan pre-quasinorm contraction mapping in the pre-quasi-Banach variable exponent Cesàro sequence spaces of fuzzy functions (cssf). Pre-quasinormal structure andR are supported. The Kannan nonexpansive mapping’s presence of a fixed point was investigated. The presence of a fixed point of Kannan contraction mapping in the pre-quasi-Banach operator ideal produced by variable exponent Cesàro sequence spaces of fuzzy functions (cssf) and s-fuzzy numbers has also been examined. To put our findings to the test, we introduce several numerical experiments. In addition, various effective implementations of the stochastic nonlinear dynamical system are discussed. The fixed points of any Kannan contraction and nonexpansive mappings on this new fuzzy functions space, its associated pre-quasi-ideal, and a new general space of solutions for many stochastic nonlinear dynamical systems are investigated.



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