# On the Nonlinearity of Extendeds-Type Weighted Nakano Sequence Spaces of Fuzzy Functions with Some Applications

Authors: Awad A. Bakery; Elsayed A. E. Mohamed
Journal: Journal of Function Spaces
 (2022)

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## Abstract

We have defined and studied the weighted Nakano sequence spaces of fuzzy functions. We have constructed the ideal generated by extendeds-fuzzy functions and the sequence spaces of fuzzy functions. We present 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 is examined. To show how the work is done, some examples and applications to the existence of solutions for a class of nonlinear summable and matrix equations are also talked about.


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## Body

## 1. Introduction

The mathematical description of the hydrodynamics of non-Newtonian fluids provided additional impetus to the learning about variable exponent Lebesgue spaces (see [1, 2]). Electric rheological fluids have many applications, including military technology, civil engineering, and orthopedics. In cybernetics, artificial intelligence, and fuzzy control, the concept of fuzziness was widely embraced after Zadeh [3] introduced fuzzy sets and fuzzy set operations. Javed et al. [4] investigated the Banach contraction in R-fuzzy b-metric spaces and discussed some related fixed point results to ensure a fixed point’s existence and uniqueness. A nontrivial example is given to illustrate the feasibility of the proposed methods. They offered an application to solve the first kind of Fredholm-type integral equation. In [5], Rehman and Aydi proved some common fixed point theorems for mappings involving generalized rational-type fuzzy cone-contraction conditions in fuzzy cone metric spaces. They gave a common solution of two definite Fredholm integral equations. The concept of orthogonal partial b-metric spaces was pioneered by Javed et al. [6]. They presented a unique fixed point for some orthogonal contractive-type mappings with some examples and an application. Humaira et al. [7] discussed the existence theorem for a unique solution to a coupled system of impulsive fractional differential equations in complex-valued fuzzy metric spaces and the fuzzy version of some fixed point results by using the definition and presented some properties of a complex-valued fuzzy metric space with some applications. In this study, Sarwar and Rodríguez-López [8] looked into the concept of extended fuzzy rectangular b-metric space. They explained that some fixed point results in the literature could be generalized by α-admittance in this space. They used this to show solutions for a group of integral equations. Many researchers in sequence spaces and summability theory were active in studying fuzzy sequence spaces and their properties. Different classes of sequences of fuzzy real numbers have been discussed by Nanda [9], Nuray and Savas [10], Matloka [11], Altinok et al. [12], Colak et al. [13], Hazarika and Savas [14], and many others. In [10], the Nakano sequences of fuzzy integers were defined and analyzed. The mappings’ ideal theory is well regarded in functional analysis. Using s-numbers is an essential technique. Pietsch [15–18] developed and studied the theory of s-numbers of linear bounded mappings between Banach spaces. He offered and explained some topological and geometric structures of the quasi-ideals of ℓp-type mappings. Then, Constantin [19] generalized the class of ℓp-type mappings to the class of cesp-type mappings. Makarov and Faried [20] showed some inclusion relations of ℓp-type mappings. As a generalization of ℓp-type mappings, Stolz mappings and mappings’ ideal were examined by Tita [21, 22]. In [23], Maji and Srivastava studied the class Aps of s-type cesp mappings using s-number sequence and Cesàro sequence spaces and they introduced a new class Ap,qs of s-type cesp,q mappings by weighted cesp with 1<p<∞. In [24], the class of s-type Zu,v;ℓp mappings was defined and some of their properties were explained. Yaying et al. [25] defined and studiedχrη, withr-Cesàro matrix inℓη, withr∈0,1and1≤η≤∞. They explained the quasi-Banach ideal of type χrη, with r∈0,1 and 1<η<∞. Komal et al. [26] explained the multiplication mappings defined on cesp equipped with the Luxemburg norm. The multiplication mappings acting on Cesàro second-order function spaces discussed by Ilkhan et al. [27]. Many fixed point theorems in a particular space work by either expanding the self-mapping acting on it or expanding the space itself. In this paper, we have introduced the concept of premodular spaces of fuzzy numbers, which are important extensions of the concept of modular spaces. We also extended s-fuzzy numbers to build large spaces of solutions to many nonlinear summable and matrix equations of fuzzy numbers. It is the first attempt to examine Caristi’s fixed point in certain premodular vector spaces. This work is aimed at introducing the particular space of sequences of fuzzy numbers, in short (pssf), under a particular function to be pre-quasi (pssf). We have defined and analyzed weighted Nakano sequence spaces of fuzzy functions. Extended s-fuzzy functions and weighted Nakano sequence spaces of fuzzy functions have been used to create the mappings’ ideal. The topological and geometric characteristics of mappings’ ideal and multiplication mappings acting on this sequence space of fuzzy functions are offered. Caristi’s fixed point is also discussed in this paper. Some supporting examples and applications to the existence of solutions for a class of nonlinear summable and matrix equations are also explored to provide a better understanding of the work that has been done.


## 2. Definitions and Preliminaries

As a reminder, Matloka [11] presented the concept 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 [9] explained the sequences of fuzzy numbers and proved the set of all convergent sequences of fuzzy numbers from a complete metric space. Kumar et al. [28] investigated the 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. Let f=f1,f2 and g=g1,g2 in Ω; suppose
(1)f≤gifandonlyiff1≤g1andf2≤g2.Define a metricρ on Ω by
(2)ρf,g=maxf1−g1,f2−g2.Matloka [11] showed that ρ is a metric on Ω, Ω,ρ is a complete metric space, and the relation ≤ is a partial order on Ω.Definition 1.
A fuzzy numberg is 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., one has y0∈R such that gy0=1(c)
gis upper semicontinuous; i.e., for allα>0, g−10,x+αand for allx∈0,1, which is open in the usual topology ofR(d)
The closure ofg0≔y∈R:gy>0 is compactTheβ-level set of 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 and is denoted by R0,1. The set R can be embedded in R0,1, when we define k∈R0,1 by
(4)k¯x=1,x=k,0,x≠k.0¯ and 1¯ denote the additive identity and multiplicative identity in R0,1 in R0,1, respectively.The arithmetic operations onR0,1 are defined as
(5)f⊕gq=supq∈Rminfp,gq−p,f⊖gq=supq∈Rminfp,gp−q,f⊗gq=supq∈Rminfp,gqp,fgq=supminq∈Rfpq,gp,pfq=fp−1q,p≠00,p=0.The absolute valuef of f∈R0,1 is defined as
(6)fq=maxfq,f−q,ifq≥0,0,ifq<0.Supposef,g∈R0,1 and the β-level sets are fβ=f1β,f2β and gβ=g1β,g2β, 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. Hence, the above operations can be defined by β-level sets as
(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,g for all f,g,k∈R0,1(3)
ρ¯f+k,g+l≤ρ¯f,g+ρ¯k,l(4)
ρ¯ξf,ξg=ξρ¯f,g, for all ξ∈RDefinition 2.
A sequence of fuzzy numbersf=fj is called
(a)
bounded if there are two fuzzy numbersg,l such that g≤fj≤l(b)
convergent to a fuzzy real numberf0 if for all ε>0, one has n0∈N such that ρ¯fj,f0<ε, for all j≥j0Byℓ∞ and ℓr, we denote the spaces of bounded and r-absolutely summable sequences of R, respectively.Lemma 3 (see [29]).
Supposeτq>0, K=max1,supqτq, and Yq,Zq∈R with q∈N; hence,
(8)Yq+Zqτq≤2K−1Yqτq+Zqτq.
We will explain our main results.


## 3. Some Properties ofℓτ.F

This section introduces the particular space of sequences of fuzzy functions (pssf), under definite function to be pre-quasi (pssf). We investigate sufficient setup ofℓτ.F equipped with definite function h to be pre-quasi closed and Banach (pssf). We also present the Fatou property of various h on ℓτ.F.LetωF and ℓ∞F mark the classes of all and bounded sequence spaces of fuzzy functions, respectively. Suppose τ=τq∈R+N, where R+N is the space of positive real sequences. The variable exponent sequence space of fuzzy functions is denoted by
(9)ℓτ.F=Z=Zq∈ωF:hrZ<∞,forsomer>0,whenhZ=∑q=0∞1τqρ¯Zq,0¯τq.Theorem 4.
Ifτq∈ℓ∞, then
(10)ℓτ.F=Z=Zq∈ωF:hrZ<∞,foranyr>0.Proof.
(11)ℓτ.F=Z=Zq∈ωF:hrZ<∞,forsomer>0=Z=Zq∈ωF:infqrτq∑q=0∞1τqρ¯Zq,0¯τq≤∑q=0∞1τqρ¯rZq,0¯τq<∞,forsomer>0=Z=Zq∈ωF:∑q=0∞1τqρ¯Zq,0¯τq<∞=Z=Zq∈ωF:hrZ<∞,foranyr>0.By0,∞U, we denote the space of all functions h:U⟶0,∞. Nakano [30] introduced the concept of modular vector spaces.Definition 5.
SupposeU is a vector space. A function h∈0,∞U is called modular if the next conditions hold:
(a)
IfY∈U, Y=ϑ¯⇔hY=0 with hY≥0, where ϑ¯=0¯,0¯,0¯,⋯(b)
hηZ=hZ holds, for all Z∈U and η=1(c)
The inequalityhαY+1−αZ≤hY+hZ satisfies, for all Y,Z∈U and α∈0,1Definition 6.
The linear spaceU is said to be a particular space of sequences of fuzzy functions (pssf), if
(a)
b¯qq∈N⊆U, where b¯q=0¯,0¯,⋯,1¯,0¯,0¯,⋯, while 1¯ displays at the qth place(b)
U is solid; i.e., suppose Y=Yq∈ωF, Z=Zq∈U and Yq≤Zq, for all q∈N, then Y∈U(c)
Yq/2q=0∞∈U, where q/2 marks the integral part of q/2, if Yqq=0∞∈UDefinition 7.
A subclassUhofUis said to be a premodular (pssf), if one hash∈0,∞U which satisfies the next settings:
(i)
IfY∈U, Y=ϑ¯⇔hY=0 with hY≥0(ii)
There isQ≥1; the inequality hαY≤QαhY holds, for every Y∈U and α∈R(iii)
There isP≥1; 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)
The closure ofE=Uh, where E is the space of finite sequences of fuzzy functions(vii)
There isσ>0 with hα¯,0¯,0¯,0¯,⋯≥σαh1¯,0¯,0¯,0¯,⋯, where(12)α¯y=1,y=α,0,y≠αClearly, the concept of premodular vector spaces is more general than modular vector spaces. Some examples of premodular vector spaces but not modular vector spaces are shown.Example 1.
The functionhZ=∑q=0∞3q+4/q+1ρ¯Zq,0¯q+1/3q+44 is a premodular (not a modular) on the vector space ℓFq+1/3q+4q=0∞. As for every Z,Y∈ℓFq+1/3q+4q=0∞, one has
(13)hZ+Y2=∑q=0∞3q+4q+1ρ¯Zq+Yq2,0¯q+1/3q+44≤4hZ+hY.Example 2.
The functionhZ=∑q=0∞q+4/2q+3ρ¯Zq,0¯2q+3/q+4 is a premodular (not a modular) on the vector space ℓF2q+3/q+4q=0∞. As for every Z,Y∈ℓF2q+3/q+4q=0∞, one has
(14)hZ+Y2=∑q=0∞q+42q+3ρ¯Zq+Yq2,0¯2q+3/q+4≤284hZ+hY,an example of premodular vector space and modular vector space.Example 3.
The functionhY=infα>0:∑q=0∞q+2/2q+3ρ¯Yq/α,0¯2q+3/q+2≤1 is a premodular (modular) on the vector space ℓF2q+3/q+2q=0∞.Definition 8.
SupposeU is a (pssf). The function h∈0,∞U is said to be a pre-quasi-norm on U, if the following setups are verified:
(i)
IfY∈U, Y=ϑ¯⇔hY=0 with hY≥0, where ϑ¯=0¯,0¯,0¯,(ii)
There isQ≥1; the inequality hαY≤QαhY satisfies, for every Y∈U and α∈R(iii)
There isP≥1; the inequality hY+Z≤PhY+hZ holds, for each Y,Z∈U.Clearly, from the last two definitions, we conclude the following two theorems.Theorem 9.
Every premodular (pssf) is a pre-quasi-normed (pssf).Theorem 10.
Every quasi-normed (pssf) is a pre-quasi-normed (pssf).Definition 11.
(a)
The functionh on ℓτ.F is named h-convex, if(15)hαY+1−αZ≤αhY+1−αhZ,for every α∈0,1 and Y,Z∈ℓτ.F.
(b)
Whenlimq⟶∞hYq−Y=0,we calledYqq∈N⊆ℓτ.Fhh-convergent toY∈ℓτ.Fh(c)
Yqq∈N⊆ℓτ.Fh is h-Cauchy, if limq,r⟶∞hYq−Yr=0(d)
Γ⊂ℓτ.Fh is h-closed, when for all h-converges Yqq∈N⊂Γ to Y, then Y∈Γ(e)
Γ⊂ℓτ.Fh is h-bounded, if δhΓ=suphY−Z:Y,Z∈Γ<∞(f)
A pre-quasi-normh on ℓτ.F verifies the Fatou property, when for all Zq⊆ℓτ.Fh under limq⟶∞hZq−Z=0 and Y∈ℓτ.Fh, one has hY−Z≤suprinfq≥rhY−ZqTheorem 12.
ℓτ.Fh, where hY=∑q=0∞1/τqρ¯Yq,0¯τq1/K, for all Y∈ℓτ.F, is a premodular (pssf), when τqq∈N∈ℓ∞∩I with τ0>0, where I is the space of every increasing sequences of reals.Proof.
(i)
Evidently,hY≥0 and hY=0⇔Y=ϑ¯.
(1-i) LetY,Z∈ℓτ.F. One has
(16)hY+Z=∑q=0∞1τqρ¯Yq+Zq,0¯τq1/K≤∑q=0∞1τqρ¯Yq,0¯τq1/K+∑q=0∞1τqρ¯Zq,0¯τq1/K=hY+hZ<∞,then Y+Z∈ℓτ.F(ii)
One getsP≥1 with hY+Z≤PhY+hZ, for all Y,Z∈ℓτ.F.
(1-ii) Assumingα∈R and Y∈ℓτ.F, we obtain
(17)hαY=∑q=0∞1τqρ¯αYq,0¯τq1/K≤supqατq/K∑q=0∞1τqρ¯Yq,0¯τq1/K≤Qαhv<∞
AsαY∈ℓτ.F, hence, from setups (1-i) and (1-ii), we get ℓτ.F is linear. Also, b¯p∈ℓτ.F, for all p∈N, since hb¯p=∑q=0∞1/τqρ¯b¯p,0¯τq1/K=1/τb.(iii)
There isQ=max1,supqατq/K−1≥1 with hαY≤QαhY, for all Y∈ℓτ.F and α∈R(1)
AssumeYq≤Zq, for all q∈N and Z∈ℓτ.F. One finds(18)hY=∑q=0∞1τqρ¯Yq,0¯τq1/K≤∑q=0∞1τqρ¯Zq,0¯τq1/K=hZ<∞,then Y∈ℓτ.F(iv)
Obviously, from (2)(1)
LetYq∈ℓτ.F; we get(19)hYq/2=∑q=0∞1τqρ¯Yq/2,0¯τq1/K=∑q=0∞1τ2qρ¯Yq,0¯τ2q+∑q=0∞1τ2q+1ρ¯Yq,0¯τ2q+11/K≤21K∑q=0∞1τqρ¯Yq,0¯τq1/K=21/KhYq,then Yq/2∈ℓτ.F(v)
From (3), we obtain P0=21/K≥1(vi)
Evidently, the closure ofE=ℓτ.F(vii)
There is0<σ≤ατ0/K−1, for α≠0 or σ>0, for α=0 with(20)hα¯,0¯,0¯,0¯,⋯≥σαh1¯,0¯,0¯,0¯,⋯Example 4.
Forτq∈1,∞N, the function hY=infα>0:∑q∈N1/τqρ¯Yq/α,0¯τq≤1 is a norm on ℓτ.F.Example 5.
The functionhY=∑q∈Nq+1/3q+2ρ¯Yq,0¯3q+2/q+13 is a pre-quasi-norm (not a norm) on ℓF3q+2/q+1q=0∞.Example 6.
The functionhY=∑q∈Nq+1/3q+2ρ¯Yq,0¯3q+2/q+1 is a pre-quasi=norm (not a quasinorm) on ℓF3q+2/q+1q=0∞.Example 7.
The functionhY=∑q∈N1/dρ¯Yq,0¯dd is a pre-quasi-norm, quasinorm, and not a norm on ℓdF, with 0<d<1.Theorem 13.
Ifτqq∈N∈ℓ∞∩I with τ0>0, then ℓτ.Fh is a pre-quasi-Banach (pssf), where hY=∑q=0∞1/τqρ¯Yq,0¯τq1/K, for every Y∈ℓτ.F.Proof.
By Theorem9 and Theorem 12, one obtainsℓτ.Fhwhich is a pre-quasi-normed (pssf). If Yl=Yqlq=0∞ is a Cauchy sequence in ℓτ.Fh. Then, for every ε∈0,1, one has l0∈N such that for all l,m≥l0, one gets
(21)hYl−Ym=∑q=0∞1τqρ¯Yql−Yqm,0¯τq1/K<ε.
This impliesρ¯Yql−Yqm,0¯<ε.AsR0,1,ρ¯is a complete metric space, thenYqmis a Cauchy sequence in R0,1, for fixed q∈N. Then, 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∈ℓτ.F.Theorem 14.
Supposeτqq∈N∈ℓ∞∩I with τ0>0; then, ℓτ.Fh is a pre-quasi-closed (pssf), where hY=∑q=0∞1/τqρ¯Yq,0¯τq1/K, for every Y∈ℓτ.F.Proof.
In view of Theorem12 and Theorem 9, the space ℓτ.Fh is a pre-quasi-normed (pssf). Assume Yl=Yqlq=0∞∈ℓτ.Fh and liml⟶∞hYl−Y0=0; then, for all ε∈0,1, there is l0∈N such that for all l≥l0, we obtain
(22)ε>hYl−Y0=∑q=0∞1τqρ¯Yql−Yq0,0¯τq1/K.
This impliesρ¯Yql−Yq0,0¯<ε. As R0,1,ρ¯is a complete metric space, therefore,Yqlis a convergent sequence in R0,1, for fixedq∈N. So, liml⟶∞Yql=Yq0, for fixed q∈N. Since hY0=hY0−Yl+Yl≤hYl−Y0+hYl<∞, one has Y0∈ℓτ.F.Theorem 15.
The functionhY=∑q=0∞1/τqρ¯Yq,0¯τq1/K holds the Fatou property, if τqq∈N∈ℓ∞∩I with τ0>0, for all Y∈ℓτ.F.Proof.
LetZr⊆ℓτ.Fh such that limr⟶∞hZr−Z=0. Since ℓτ.Fh is a pre-quasi-closed space, one has Z∈ℓτ.Fh. For all Y∈ℓτ.Fh, one gets
(23)hY−Z=∑q=0∞1τqρ¯Yq−Zq,0¯τq1/K≤∑q=0∞1τqρ¯Yq−Zqr,0¯τq1/K+∑q=0∞1τqρ¯Zqr−Zq,0¯τq1/K≤supminfr≥mhY−Zr.Theorem 16.
The functionhY=∑q=0∞1/τqρ¯Yq,0¯τq does not hold the Fatou property, for all Y∈ℓτ.F, when τq∈ℓ∞ and τq>1 with q∈N.Proof.
LetZr⊆ℓτ.Fh so that limr⟶∞hZr−Z=0. Since ℓτ.Fh is a pre-quasi-closed space, one gets Z∈ℓτ.Fh. For every Z∈ℓτ.Fh, we obtain
(24)hY−Z=∑q=0∞1τqρ¯Yq−Zq,0¯τq≤2supqτq−1∑q=0∞1τqρ¯Yq−Zqr,0¯τq+∑q=0∞1τqρ¯Zqr−Zq,0¯τq≤2supqτq−1supminfr≥mhY−Zr.


## 4. Caristi’s Fixed Point Theorem inℓτ.Fh

In this section, the existence of Caristi’s fixed point inℓτ.Fh is presented according to Farkas [31], where hY=∑q=0∞1/τqρ¯Yq,0¯τq1/K, for all Y∈ℓτ.F.Definition 17.
The functionΨ1:ℓτ.Fh⟶−∞,∞ is said to be lower semicontinuous at G0∈ℓτ.Fh if liminfG⟶G0Ψ1G=Ψ1G0, where liminfG⟶G0Ψ1G=supV∈VG0infG∈VΨ1G, where VG0 is a neighborhood system of G0.Definition 18.
The functionΨ1:ℓτ.Fh⟶−∞,∞ is called proper, when
(25)DΨ1=G∈ℓτ.Fh:Ψ1G<∞≠∅.Theorem 19.
AssumeΞ≠∅ and Ξ is a h-closed subset of ℓτ.Fh and Ψ1:Ξ⟶−∞,∞ is a proper, h-lower semicontinuous function with infG∈ΞΨ1G>−∞. Suppose γ>0, ϖq⊂0,∞, and G0∈Ξ with Ψ1G0≤infG∈ΞΨ1G+γ. Therefore, one has Gq∈Ξ which h-converges to some Gγ, and
(i)
hGγ−Gq≤γ/2qϖ0, with q∈N(ii)
Ψ1Gγ+∑q=0∞ϖqhGγ−Gq≤Ψ1G0(iii)
whenG≠Gγ, we have(26)Ψ1Gγ+∑q=0∞ϖqhGγ−Gq<Ψ1G+∑q=0∞ϖqhG−GqProof.
LetSG0=G∈Ξ:Ψ1G+ϖ0hG−G0≤Ψ1G0. As G0∈SG0, hence, SG0≠∅. Since Ψ1 is h-lower semicontinuous, h verifies the Fatou property, and Ξ is h-closed, one has SG0 which is h-closed. Choose G1∈SG0 and
(27)Ψ1G1+ϖ0hG1−G0≤infG∈SG0Ψ1G+ϖ0hG−G0+γϖ12ϖ0.
Take(28)SG1=G∈SG0:Ψ1G+∑j=01ϖjhG−Gj≤Ψ1G1+ϖ0hG1−G0.
AsSG0, we get SG1≠∅ and h-closed. Assume that one has built G0,G1,G2,,Gq and SG0,SG1,SG2,⋯,SGq. Next, choose Gq+1∈SGq and
(29)Ψ1Gq+1+∑j=0qϖjhGq+1−Gj≤infG∈SGqΨ1G+∑j=0qϖjhG−Gj+γϖq2qϖ0.
Let(30)SGq+1≔G∈SGq:Ψ1G+∑j=0q+1ϖjhG−Gj≤Ψ1Gq+1+∑j=0qϖjhGq+1−Gj.
Hence, we form by induction the sequencesGq and SGq. Fix q∈N. Suppose W∈SGq. One obtains
(31)Ψ1W+∑j=0qϖjhW−Gj≤Ψ1Gq+∑j=0q−1ϖjhGq−Gj.
Hence,(32)ϖqhW−Gq≤Ψ1Gq+∑j=0q−1ϖjhGq−Gj−Ψ1W+∑j=0q−1ϖjhW−Gj≤Ψ1Gq+∑j=0q−1ϖjhGq−Gj−infG∈SGq−1Ψ1G+∑j=0q−1ϖjhG−Gj≤γϖq2qϖ0.
AsSGq is decreasing with Gq∈SGq, for every q∈N, we obtain
(33)hGq+p−Gq≤γ2qϖ0,with q,p∈N. This implies Gq is h-Cauchy. Since ℓτ.Fh is a h-Banach space, hence, Gq has h-limits Gγ and ⋂q∈NSGq=Gγ. Since Gq+1∈SGq, we can see
(34)Ψ1Gq+1+∑j=0qϖjhGq+1−Gj≤Ψ1Gq+∑j=0q−1ϖjhGq−Gj.
Hence,Ψ1Gq+∑j=0q−1ϖjhGq−Gj is decreasing. After, let G≠Gγ. One gets m∈N with G∉SGq, with q≥m, i.e.,
(35)Ψ1Gq+∑j=0q−1ϖjhGq−Gj<Ψ1G+∑j=0qϖjhG−Gj.
SinceGγ∈SGq, with q≥m, we get
(36)Ψ1Gγ+∑j=0qϖjhGγ−Gj≤Ψ1Gq+∑j=0q−1ϖjhGq−Gj≤Ψ1Gm+∑j=0m−1ϖjhGm−Gj.
Puttingq⟶∞ in the previous inequality, one can see
(37)Ψ1Gγ+∑j=0∞ϖjhGγ−Gj≤Ψ1xm+∑j=0m−1ϖjhGm−Gj<Ψ1G+∑j=0mϖjhG−Gj≤Ψ1G+∑j=0∞ϖjhG−Gj.
This gives(38)Ψ1Gγ+∑q=0∞ϖqhGγ−Gq<Ψ1G+∑q=0∞ϖqhG−Gq.Example 8.
SupposeΞ=Y∈ℓF2q+3/q+2q=0∞h:Y0=0¯ and hY=∑q∈Nq+2/2q+3ρ¯Yq,0¯2q+3/q+2, for every Y∈ℓF2q+3/q+2q=0∞h. Suppose γ>0, ϖq⊂0,∞, and G0∈Ξ with supG∈ΞlnhG≤lnhG0+γ.SinceΨ1:Ξ⟶−∞,∞, whereΨ1G=−lnhG, clearly,Ξ≠∅andΞis ah-closed subset ofℓτ.Fh, andΨ1is a proper,h-lower semicontinuous function withinfG∈ΞΨ1G>−∞. From Theorem 19, one has Gq∈Ξ which h-converges to some Gγ, and
(i)
hGγ−Gq≤γ/2qϖ0, with q∈N(ii)
lnhG0+∑q=0∞ϖqhGγ−Gq≤lnhGγ(iii)
whenG≠Gγ, we have(39)lnhG+∑q=0∞ϖqhGγ−Gq<lnhGγ+∑q=0∞ϖqhG−GqExample 9.
SupposeΞ=Y∈ℓFq+1/3q+4q=0∞h:Y0=0¯ and hY=∑q∈N3q+4/q+1ρ¯Yq,0¯q+1/3q+4, for every Y∈ℓFq+1/3q+4q=0∞h. Suppose γ>0, ϖq⊂0,∞, and G0∈Ξ with hG0≤γ. Since Ψ1:Ξ⟶−∞,∞, where Ψ1G=hG, clearly, Ξ≠∅ and Ξ is a h-closed subset of ℓτ.Fh, and Ψ1 is a proper, h-lower semicontinuous function with infG∈ΞΨ1G>−∞. From Theorem 19, one has Gq∈Ξ which h-converges to some Gγ, and
(i)
hGγ−Gq≤γ/2qϖ0, with q∈N(ii)
hGγ+∑q=0∞ϖqhGγ−Gq≤hG0(iii)
whenG≠Gγ, we have(40)hGγ+∑q=0∞ϖqhGγ−Gq<hG+∑q=0∞ϖqhG−GqTheorem 20.
IfΞ≠∅ and Ξ is a h-closed subset of ℓτ.Fh, by choosing γ>0 and ϖn and 0<ω=∑n=0∞ϖn<∞, suppose H:Ξ⟶Ξ is a mapping and there is a function Ψ1:Ξ⟶−∞,∞ that holds a proper andh-lower semicontinuous with infG∈ΞΨ1G>−∞ and
(1)
hHG−Y−hG−Y≤hHG−G, for any G,Y∈Ξ(2)
hHG−G≤Ψ1G−Ψ1HG, with G∈Ξ
Hence,H has a fixed point in Ξ.Proof.
Since0<ω=∑n=0∞ϖn<∞, we get Ψ2≔ωΨ1 which is also proper, h-lower semicontinuous, and bounded from below. If G∈Ξ, one gets
(41)ωhHG−G≤Ψ2G−Ψ2HG.
AsinfG∈ΞΨ2G>−∞, one obtains G0∈Ξ with Ψ2G0<infG∈ΞΨ2G+γ. From Theorem 19, there is Gq which h-converges to some Gγ∈Ξ, and
(42)Ψ2Gγ+∑q=0∞ϖqhGγ−Gq<Ψ2G+∑q=0∞ϖqhG−Gq,for every G≠Gγ. Assume that HGγ≠Gγ; we have
(43)Ψ2Gγ+∑q=0∞ϖqhGγ−Gq<Ψ2HGγ+∑q=0∞ϖqhHGγ−Gq.
Then,(44)Ψ2Gγ−Ψ2HGγ<∑q=0∞ϖqhHGγ−Gq−∑q=0∞ϖqhGγ−Gq=∑q=0∞ϖqhHGγ−Gq−hGγ−Gq.
From condition (41), one can see
(45)Ψ2Gγ−Ψ2HGγ<∑q=0∞ϖqhHGγ−Gγ=ωhHGγ−Gγ.
The inequality (1) implies that
(46)ωhHGγ−Gγ≤Ψ2Gγ−Ψ2HGγ<ωhHGγ−Gγ.
This is a disagreement. Therefore,HGγ=Gγ.Example 10.
SupposeΞ=Y∈ℓF2q+3/q+2q=0∞h:Y0=0¯ and hY=∑q∈Nq+2/2q+3ρ¯Yq,0¯2q+3/q+2, for every Y∈ℓF2q+3/q+2q=0∞h. Suppose H:Ξ⟶Ξ is a mapping and hHG−G≤lnhHG−lnhG, with G∈Ξ. From Theorem 20, H has a fixed point in Ξ.Example 11.
SupposeΞ=Y∈ℓFq+1/3q+4q=0∞h:Y0=0¯ and hY=∑q∈N3q+4/q+1ρ¯Yq,0¯q+1/3q+4, for every Y∈ℓFq+1/3q+4q=0∞h. Suppose H:Ξ⟶Ξ is a mapping and hHG−G≤hG−hHG, with G∈Ξ. From Theorem 20, H has a fixed point in Ξ.Definition 21.
Pick upUhas a pre-quasi-normed (pssf),V:Uh⟶UhandZ∈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 12.
SupposeV:ℓFq+1/2q+4q=0∞h⟶ℓFq+1/2q+4q=0∞h, where hZ=∑q=0∞2q+4/q+1ρ¯Zq,0¯q+1/2q+44, for every Z∈ℓFq+1/2q+4q=0∞ and
(47)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∈ℓFq+1/2q+4q=0∞h.Example 13.
AssumeV:ℓF2q+3/q+2q=0∞h⟶ℓF2q+3/q+2q=0∞h, where hg=∑q=0∞q+2/2q+3ρ¯gq,0¯2q+3/q+2, for every g∈ℓF2q+3/q+2q=0∞ and
(48)Vg=g4,hg∈0,1,g5,hg∈1,∞.
SupposeZn⊆ℓF2q+3/q+2q=0∞h is such that limn⟶∞hZn−Z0=0, where Z0∈ℓF2q+3/q+2q=0∞h with hZ0=1.
As the pre-quasi-normh is continuous, we have
(49)limn⟶∞hVZn−VZ0=limn⟶∞hZn4−Z05=hZ020>0.
Therefore,V is not h-sequentially continuous at Z0.Theorem 22.
Ifτqq∈N∈ℓ∞∩I with τ0>1 and V:ℓτ.Fh⟶ℓτ.Fh, where hY=∑q=0∞1/τqρ¯Yq,0¯τq,for all Y∈ℓFτ, suppose
(1)
V:ℓτ.Fh⟶ℓτ.Fh is a mapping, and there is a function Ψ1:ℓτ.Fh⟶−∞,∞ that holds a proper and h-lower semicontinuous with infG∈ℓτ.FhΨ1G>−∞, and there is α∈0,1 so that hVl+1G−VlG≤αlΨ1G−Ψ1VG, with G∈ℓτ.Fh(2)
V is h-sequentially continuous at Z∈ℓτ.Fh(3)
there isY∈ℓτ.Fh with VlY which has VljY converging to Z
Then,Z∈ℓτ.Fh is a fixed point of V.Proof.
AssumeZ is not a fixed point of V; one has VZ≠Z. From parts (2) and (3), we get
(50)limlj⟶∞hVljY−Z=0,limlj⟶∞hVlj+1Y−VZ=0.
From part (1), one obtains
(51)0<hVZ−Z=hVZ−Vlj+1Y+VljY−Z+Vlj+1Y−VljY≤22supiτi−2hVlj+1Y−VZ+22supiτi−2hVljY−Z+2supiτi−1αljΨ1Y−Ψ1VY.
Aslj⟶∞, one has a contradiction. Then, Z is a fixed point of V.


## 5. Multiplication Mappings onℓτ.F

In this section, we examine the sufficient conditions onℓτ.Fh such that the multiplication mapping defined on it is bounded, isometry, approximable, compact, closed range, invertible, and Fredholm, where hY=∑q=0∞1/τqρ¯Yq,0¯τq1/K, for every Y∈ℓτ.F.The space of approximable and compact bounded linear mappings from a Banach spaceΔ into a Banach space Λ will be marked by YΔ,Λ and LcΔ,Λ, and if Δ=Λ, we mark YΔ and LcΔ, respectively.Definition 23.
Letκ∈ℂN∩ℓ∞ and Uh be a pre-quasi-normed (pssf). A mapping Vκ:Uh⟶Uh is called multiplication mapping if VκY=κY=κrYrr=0∞∈U, with Y∈U. When Vκ∈LU, we call it a multiplication mapping generated by κ.Theorem 24.
Supposeκ∈ℂN and τqq∈N∈ℓ∞∩I with τ0>0; then, κ∈ℓ∞, if and only if Vκ∈Lℓτ.Fh.Proof.
Let the conditions be satisfied. Assumeκ∈ℓ∞; hence, one gets ε>0 with κr≤ε, with r∈N. If x∈ℓτ.Fh, since τqq∈N∈ℓ∞∩I with τ0>0, then
(52)hVκY=hκY=hκrYrr=0∞=∑r=0∞1τrρ¯κrYr,0¯τr1/K≤suprετr/K∑r=0∞1τrρ¯Yr,0¯τr1/K=suprετr/KhY.
One hasVκ∈Lℓτ.Fh. Conversely, assume that Vκ∈Lℓτ.Fh. Suppose κ∉ℓ∞, then, for each j∈N, there is ij∈N such that κij>j. Since τqq∈N∈ℓ∞∩I with τ0>0, we have
(53)hVκb¯ij=hκb¯ij=hκrb¯ijrr=0∞=∑r=0∞1τrρ¯κrb¯ijr,0¯τr1/K=1τijκijτij1/K>1τijjτij1/Khb¯ij.
This shows thatVκ∉Lℓτ.Fh. Therefore, κ∈ℓ∞.Theorem 25.
Pick upκ∈ℂNandℓτ.Fhas a pre-quasi-normed (pssf). Then, κq=1, for every q∈N, if and only if Vκ is an isometry.Proof.
Ifκr=1, for all r∈N, one gets
(54)hVκY=hκY=hκrYrr=0∞=∑r=0∞1τrρ¯κrYr,0¯τr1/K=∑r=0∞1τrρ¯Yr,0¯τr1/K=hY,for all Y∈ℓτ.Fh. Therefore, Vκ is an isometry. Conversely, Assume that ∣κi∣<1 for some i=i0. We obtain
(55)hVκb¯i0=hκb¯i0=hκrb¯i0rr=0∞=∑r=0∞1τrρ¯κrei0r,0¯τr1/K<∑r=0∞1τrρ¯ei0r,0¯τr1/K=hb¯i0.
Whenκi0>1, we can prove that hVκb¯i0>hb¯i0. One gets a contradiction. So, κr=1, with r∈N.
By cardA, we indicate the cardinality of the set A.Theorem 26.
Supposeκ∈ℂN and τqq∈N∈ℓ∞∩I with τ0>0. Then, Vκ∈Yℓτ.Fh if and only if κrr=0∞∈c0.Proof.
Assume thatVκ∈Yℓτ.Fh. One gets Vκ∈Lcℓτ.Fh. To showκrr=0∞belongs toc0, supposeκrr=0∞∉c0. One gets ν>0 such that Λν=r∈N:κr≥ν verifies card Λν=∞. If zi∈Λν, with i∈N, then b¯zi:zi∈Λν is an infinite bounded in ℓτ.Fh. Let
(56)hVκb¯zi−Vκb¯zj=hκb¯zi−κb¯zj=hκrb¯zir−b¯zjrr=0∞=∑r=0∞1τrρ¯κrb¯zir−b¯zjr,0¯τr1/K≥infrντr/Khb¯zi−b¯zj,for all zi,zj∈Λν. One obtains b¯zi:zi∈Λν∈ℓ∞F which cannot have a convergent subsequence under Vκ, which gives Vκ∉Lcℓτ.Fh. Then, Vκ∉Yℓτ.Fh; this gives a contradiction. Then, limi⟶∞κi=0. On the other hand, if limi⟶∞κi=0, hence, for each ν>0, the set Λν=i∈N:κi≥ν holds card Λν<∞. So, for all ν>0, the space ℓτ.FhΛν=Y=Yi∈ℂΛν is finite dimensional. Then, Vκ∣ℓτ.FhΛν is a finite rank mapping. For every i∈N, define κi∈ℂN by
(57)κij=κj,j∈Λ1/i,0,otherwise.
Obviously,Vκi has rankVκi<∞ as dimℓτ.FhΛ1/i<∞, with i∈N; hence, as τqq∈N∈ℓ∞∩I with τ0>0, we get
(58)hVκ−VκiY=hκj−κijYjj=0∞=∑j=0∞1τjρ¯κj−κijYj,0¯τj1/K=∑j=0,j∈Λ1/i∞1τjρ¯κj−κijYj,0¯τj+∑j=0,j∉Λ1/i∞1τjρ¯κj−κijYj,0¯τj1/K=∑j=0,j∉Λ1/i∞1τjρ¯κjYj,0¯τj1/K≤supj1iτj/K∑j=0,j∉Λ1/i∞1τjρ¯Yj,0¯τj1/K<1iτ0/KhY.
This implies thatVκ−Vκi≤1/iτ0/K; hence, Vκ is an approximable mapping.Theorem 27.
Ifκ∈ℂN and τqq∈N∈ℓ∞∩I with τ0>0, then, Vκ∈Lcℓτ.Fh, if and only if κii=0∞∈c0.Proof.
Easy.Corollary 28.
Assumeκ∈ℂN and τqq∈N∈ℓ∞∩I with τ0>0; then, Lcℓτ.FhÞLℓτ.Fh.Proof.
AsI is a multiplication mapping on ℓτ.Fh generated by κ=1,1,⋯, therefore, I∉Lcℓτ.Fh and I∈Lℓτ.Fh.Definition 29 (see [32]).
A mappingD∈LV is called Fredholm if it satisfies that dimkerD<∞, dimRDc<∞ and D has closed range, where RDc denotes the complement of the range D.Theorem 30.
Ifκ∈ℂN and τqq∈N∈ℓ∞∩I with τ0>0, then κ is bounded away from zero on kerκc, if and only if RVκ is closed., where Vκ∈Lℓτ.Fh.Proof.
Let the sufficient condition be satisfied. Therefore, there isε>0 with κi≥ε, for all i∈kerκc. To explainRVκis closed, assumeZis a limit point ofRVκ. One has VκXi in ℓτ.Fh, with i∈N such that limi⟶∞VκXi=Z. Clearly, VκXi is a Cauchy sequence. Since τqq∈N∈ℓ∞∩I with τ0>0, one has
(59)hVκXi−VκXj=∑q=0∞1τqρ¯κqXiq−κqXjq,0¯τq1/K=∑q=0,q∈kerκc∞1τqρ¯κqXiq−κqXjq,0¯τq+∑q=0,q∉kerκc∞1τqρ¯κqXiq−κqXjq,0¯τq1/K≥∑q=0,q∈kerκc∞1τqρ¯κqXiq−κqXjq,0¯τq1/K=∑q=0∞1τqρ¯κqYiq−κqYjq,0¯τq1/K>infqετqK∑q=0∞1τqρ¯Yiq−Yjq,0¯τq1/K,where
(60)Yiq=Xiq,q∈kerκc,0,q∉kerκc.
SoYi is a Cauchy sequence in ℓτ.Fh. Since ℓτ.Fh is complete, one has X∈ℓτ.Fh such that limi⟶∞Yi=X. Since Vκ is continuous, limi⟶∞VκYi=VκX. But limi⟶∞VκXi=limi⟶∞VκYi=Z; hence, VκX=Z. So, Z∈RVκ. One obtainsRVκwhich is closed. On the other hand, assume RVκ is closed; hence, Vκ is bounded away from zero on ℓτ.Fhkerκc. One gets ε>0 such that hVκX≥εhX, with X∈ℓτ.Fhkerκc.
AssumeΛ=q∈kerκc:∣κq∣<ε. If Λ≠ϕ, then for i0∈Λ, we obtain
(61)hVκb¯i0=hκqb¯i0qq=0∞=∑q=0∞1τqρ¯κqb¯n0q,0¯τq1/K<∑q=0∞1τqρ¯εb¯n0q,0¯τq1/K≤supqετq/Khb¯n0.
One gets a contradiction. Then,Λ=ϕ with ∣κq∣≥ε, and q∈kerκc.Theorem 31.
Ifκ∈ℂN and and τqq∈N∈ℓ∞∩I with τ0>0, one has d>0 and D>0 with d<κq<D, and q∈N, if and only if Vκ∈Lℓτ.Fh is invertible.Proof.
Let the conditions be verified. Defineγ∈ℂN by γq=1/κq. From Theorem 7.5, we have Vκ,Vγ∈Lℓτ.Fh and Vκ.Vγ=Vγ.Vκ=I; hence, Vγ is the inverse of Vκ. On the other side, assume Vκ is invertible. Hence, RVκ=ℓτ.FhN. One gets, RVκ which is closed. By Theorem 30, we have d>0 with κq≥d, and q∈kerκc. We have kerκ=ϕ, if κq0=0; for several q0∈N, one has eq0∈kerVκ, which is a contradiction, as kerVκ is trivial. Therefore, κq≥d, for all q∈N. As Vκ is bounded, by Theorem 7.5, one gets D>0 such that κq≤D, for all q∈N.Theorem 32.
Ifκ∈ℂN and τqq∈N∈ℓ∞∩I with τ0>0, then Vκ∈Lℓτ.Fh is Fredholm mapping, if and only if (a) cardkerκ<∞ and (b) κq≥ε, with q∈kerκc.Proof.
AssumeVκ is Fredholm. When cardkerκ=∞, one has b¯n∈kerVκ, with n∈kerκ. As b¯n’s are linearly independent, one obtains cardkerVκ=∞, which implies a contradiction. Hence, cardkerκ<∞. In view of Theorem 30, the condition (b) is confirmed. After, when the necessary conditions hold, prove thatVκis Fredholm. According to Theorem 30, the condition (b) proves that RVκ is closed. The condition (a) gives that dimkerVκ<∞ and dimRVκc<∞. So, Vκ is Fredholm.


## 6. Mappings’ Ideal

The structure of the mappings’ ideal byℓτ.Fh, where hg=∑m=0∞1/τmρ¯gm,0¯τm1/K, for every g∈ℓτ.F, and extended s-fuzzy functions has been explained. We study the enough setups on ℓτ.Fh such that the class ✠¯ℓτ.Fh is complete and closed. We investigate the enough setups (not necessary) on ℓτ.Fh such that the closure of F=✠α¯ℓτ.Fh. This gives a negative answer of Rhoades [33] open problem about the linearity of s-type ℓτ.Fh spaces. We explain the enough setups on ℓτ.Fh such that ✠¯ℓτ.Fh is strictly contained for different powers, ✠α¯ℓτ.Fh is minimum, the class ✠¯ℓτ.Fh is simple, and the space of every bounded linear mappings in which the sequence of eigenvalues inℓτ.Fhequals✠¯ℓτ.Fh.We indicate the space of all bounded, finite rank linear mappings from an infinite dimensional Banach spaceΔ into an infinite dimensional Banach space Λ by LΔ,Λ and FΔ,Λ, and when Δ=Λ, we inscribe LΔ and FΔ.Definition 33. (see [34]).
Ans-number function iss:LΔ,Λ⟶R+Nwhich sorts everyV∈LΔ,ΛandsdVd=0∞which verifies the following settings:
(a)
V=s0V≥s1V≥s2V≥⋯≥0, for all V∈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)
WhenV∈LΔ,Λ and γ∈R, then sdγV=∣γ∣sdV(e)
SupposerankV≤d; then, sdV=0, for each V∈LΔ,Λ(f)
sl≥qIq=0 or sl<qIq=1, where Iq denotes the unit map on the q-dimensional Hilbert space ℓ2q
We mention here some examples ofs-numbers:
(1)
Theqth Kolmogorov number, described by dqX, is marked by(62)dqX=infdimJ≤qsupf≤1infg∈JXf−g(2)
Theqth approximation number, described by αqX, is marked by(63)αqX=infX−Y:Y∈LΔ,Λ,rankY≤qDefinition 34 (see [17]).
AssumeL is the class of all bounded linear mappings within any two arbitrary Banach spaces. A subclass U of L is said to be a mappings’ ideal, if all UΔ,Λ=U∩LΔ,Λ verifies the following conditions:
(i)
IΓ∈U, where Γ marks the Banach space of one dimension(ii)
The spaceUΔ,Λ is linear over R(iii)
IfW∈LΔ0,Δ, X∈UΔ,Λ and Y∈LΛ,Λ0 then, YXW∈UΔ0,Λ0Notations 1.
(64)✠¯U≔✠¯UΔ,Λ,where✠¯UΔ,Λ≔V∈LΔ,Λ:sjV¯j=0∞∈U,✠α¯U≔✠α¯UΔ,Λ,where✠α¯UΔ,Λ≔V∈LΔ,Λ:αjV¯j=0∞∈U,✠d¯U≔✠d¯UΔ,Λ,where✠d¯UΔ,Λ≔V∈LΔ,Λ:djV¯j=0∞∈U,where
(65)sjV¯x=1,x=sjV,0,x≠sjV.Theorem 35.
SupposeU is a (pssf); then, ✠¯U is a mappings’ ideal.Proof.
(i)
AssumeV∈FΔ,Λ and rankV=n with n∈N, as b¯i∈U for all i∈N and U is a linear space, one hassiV¯i=0∞=s0V¯,s1V¯,⋯,sn−1V¯,0¯,0¯,0¯,⋯=∑i=0n−1siV¯b¯i∈U, forV∈✠¯UΔ,Λ, thenFΔ,Λ⊆✠¯EΔ,Λ(ii)
SupposeV1,V2∈✠¯UΔ,Λ and β1,β2∈R; then, by Definition 6 condition (78), one has si/2V1¯i=0∞∈U and si/2V1¯i=0∞∈U, as i≥2i/2;by the definition ofs-numbers andsiPwhich is a decreasing sequence, we havesiβ1V1+β2V2¯≤s2i/2β1V1+β2V2¯≤si/2β1V1+si/2β2V2¯=β1si/2V1¯+β2si/2V2¯ for each i∈N. In view of Definition 6 condition (77) and U whcih 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¯≤∥R∥siT¯∥P∥; by Definition 6 conditions (41) and (77), one gets siRTP¯i=0∞∈U, then RTP∈✠¯UΔ0,Λ0
According to Theorem12 and Theorem 35, one concludes the next theorem.Theorem 36.
Letτqq∈N∈ℓ∞∩Iwithτ0>0; one has✠¯ℓτ.Fhwhich is a mappings’ ideal.Definition 37 (see [35]).
A functionH∈0,∞U is called a pre-quasi-norm 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, when V∈LΔ0,Δ, X∈UΔ,Λ and Y∈LΛ,Λ0 then HYXV≤σYHXVTheorem 38 (see [36]).
Every quasinorm on the idealU is a pre-quasi-norm on the same ideal.Theorem 39.
Ifτqq∈N∈ℓ∞∩I with τ0>0, then H is a pre-quasi-norm on ✠¯ℓτ.Fh, with HZ=hsqZ¯q=0∞, for all Z∈✠¯ℓτ.FhΔ,Λ.Proof.
(1)
WhenX∈✠¯ℓτ.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∈✠¯ℓτ.FhΔ,Λ and α∈R(3)
One hasPP0≥1 so that for X1,X2∈✠¯ℓτ.FhΔ,Λ, one can see(66)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∈✠¯ℓτ.FhΔ,Λ, and Z∈LΛ,Λ0, then HZYX=hsqZYX¯q=0∞≤hXZsqY¯q=0∞≤ρXHYZ.
In the next theorems, we will use the notation✠¯ℓτ.Fh,H, where HV=hsqV¯q=0∞, for all V∈✠¯ℓτ.Fh.Theorem 40.
Supposeτqq∈N∈ℓ∞∩I with τ0>0; one has ✠¯ℓτ.Fh,H which is a pre-quasi-Banach ideal.Proof.
SupposeVaa∈N is a Cauchy sequence in ✠¯ℓτ.FhΔ,Λ. Since LΔ,Λ⊇Sℓτ.FhΔ,Λ, one has
(67)HVr−Va=hsqVr−Va¯q=0∞≥hs0Vr−Va¯,0¯,0¯,0¯,⋯=1τ0Vr−Vaτ0.
Hence,Vaa∈N is a Cauchy sequence in LΔ,Λ. As LΔ,Λ is a Banach space, so there exists V∈LΔ,Λ so that lima⟶∞Va−V=0and sincesqVa¯q=0∞∈ℓτ.Fh, for all a∈N, and ℓτ.Fhis a premodular (pssf); hence, one can see
(68)HV=hsqV∞¯q=0∞≤hsq/2V−Va¯q=0∞+hsq/2Vaq=0∞¯≤hVa−V1¯q=0∞+21/KhsqVa¯q=0∞<ε.
We obtainsqV¯q=0∞∈ℓτ.Fh; hence, V∈✠¯ℓτ.FhΔ,Λ.Theorem 41.
Ifτqq∈N∈ℓ∞∩I with τ0>0, one has ✠¯ℓτ.Fh,H which is a pre-quasi-closed ideal.Proof.
SupposeVa∈✠¯ℓτ.FhΔ,Λ, for all a∈N and lima⟶∞HVa−V=0. Therefore, there is ς>0 and as LΔ,Λ⊇Sℓτ.FhΔ,Λ, one has
(69)HVa−V=hsqVa−V¯q=0∞≥hs0Va−V¯,0¯,0¯,0¯,⋯=1τ0Vr−Vaτ0.
SoVaa∈N is convergent in LΔ,Λ, i.e., lima⟶∞Va−V=0, and since sqVa¯q=0∞∈ℓτ.Fh, for all q∈N, and ℓτ.Fh is a premodular (pssf), hence, one can see
(70)HV=hsqV¯q=0∞≤hsq/2V−Va¯q=0∞+hsq/2Va¯q=0∞≤hVa−V1¯q=0∞+21/KhsqVa¯q=0∞<ε.
We obtainsqV¯q=0∞∈ℓτ.Fh; hence, V∈✠¯ℓτ.FhΔ,Λ.Definition 42.
A pre-quasi-normH 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
(71)HM−T≤supqinfj≥qHM−Tj.Theorem 43.
Supposeτqq∈N∈ℓ∞∩I with τ0>0; then, ✠¯ℓτ.Fh,H does not satisfy the Fatou property.Proof.
IfTqq∈N⊆✠¯ℓτ.FhΔ,Λ with limq⟶∞HTq−T=0, since ✠¯ℓτ.Fh is a pre-quasi-closed ideal, hence, T∈✠¯ℓτ.FhΔ,Λ. So with M∈✠¯ℓτ.FhΔ,Λ, one has
(72)HM−T=∑q=0∞1τqρ¯sqM−T¯,0¯τq1/K≤∑q=0∞1τqρ¯sq/2M−Ti¯,0¯τq1/K+∑q=0∞1τqρ¯sq/2Ti−T¯,0¯τq1/K≤21Ksuprinfj≥r∑q=0∞1τqρ¯sqM−Tj¯,0¯τq1/K.Definition 44.
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 14.
IfV:✠¯ℓF2q+3/q+2q=0∞hΔ,Λ⟶✠¯ℓF2q+3/q+2q=0∞hΔ,Λ, where HT=∑q=0∞q+2/2q+3ρ¯sqT¯,0¯2q+3/q+2, for every T∈✠¯ℓF2q+3/q+2q=0∞hΔ,Λ and
(73)VT=T6,HT∈0,1,T7,HT∈1,∞,evidently, V is H-sequentially continuous at the zero operator Θ∈✠¯ℓF2q+3/q+2q=0∞h. Let Tj⊆✠¯ℓF2q+3/q+2q=0∞h be such that limj⟶∞HTj−T0=0, where T0∈✠¯ℓF2q+3/q+2q=0∞h with HT0=1. Since the pre-quasi-norm H is continuous, one gets
(74)limj⟶∞HVTj−VT0=limj⟶∞HT06−T07=HT042>0.
Therefore,V is not H-sequentially continuous at T0.Theorem 45.
Pick upτqq∈N∈ℓ∞∩I with τ0>0 and V:✠¯ℓτ.FhΔ,Λ⟶✠¯ℓτ.FhΔ,Λ. Assume
(i)
there is a functionΨ1:✠¯ℓτ.FhΔ,Λ⟶−∞,∞ that holds a proper and h-lower semicontinuous with infG∈✠¯ℓτ.FhΔ,ΛΨ1G>−∞ and there is α∈0,1 so that HVl+1G−VlG≤αlΨ1G−Ψ1VG, with G∈✠¯ℓτ.FhΔ,Λ(ii)
V is H-sequentially continuous at an element M∈✠¯ℓτ.FhΔ,Λ(iii)
there areG∈✠¯ℓτ.FhΔ,Λ such that the sequence of iterates VrG has a VrmG converging to M
Then,M∈✠¯ℓτ.FhΔ,Λ is a fixed point of V.Proof.
LetM be not a fixed point of V; hence, VM≠M. By using parts (ii) and (iii), we get
(75)limrm⟶∞HVrmG−M=0andlimrm⟶∞HVrm+1G−VM=0.
By using part (i), one obtains(76)0<HVM−M=HVM−Vrm+1G+VrmG−M+Vrm+1G−VrmG≤21/KHVrm+1G−VM+22/KHVrmG−M+22/KαrmΨ1G−Ψ1VG.
Asrm⟶∞, there is a contradiction. Hence, M is a fixed point of V.Theorem 46.
✠α¯ℓτ.FhΔ,Λ=theclosureofFΔ,Λ, if τqq∈N∈ℓ∞∩I with τ0>0. But the converse is not necessarily true.Proof.
Asb¯x∈ℓτ.Fh, for every x∈N, and ℓτ.Fh is a linear space, suppose Z∈FΔ,Λ; one has αxZ¯x=0∞∈E. Therefore, the closure of FΔ,Λ⊆✠α¯ℓτ.FhΔ,Λ. Assume Z∈✠α¯ℓτ.FhΔ,Λ; we have αxZ¯x=0∞∈ℓτ.Fh. As hαxZ¯x=0∞<∞, assume ρ∈0,1; then, there is x0∈N−0 with hαxZ¯x=x0∞<ρ/4. Since αxZ¯x=0∞ is decreasing, we have
(77)∑x=x0+12x01τxρ¯α2x0Z¯,0¯τx≤∑x=x0+12x01τxρ¯αxZ¯,0¯τx≤∑x=x0∞1τxρ¯αxZ¯,0¯τx<ρ4.
Hence, there isY∈F2x0Δ,Λ so that rankY≤2x0 and
(78)∑x=2x0+13x01τxρ¯∥Z−Y∥¯,0¯τx≤∑x=x0+12x01τxρ¯Z−Y¯,0¯τx<ρ4.
Sinceτqq∈N∈ℓ∞∩I with τ0>0, we can choose
(79)∑x=0x01τxρ¯Z−Y¯,0¯τx<ρ4.
In view of inequalities (2)-(4), one has
(80)dZ,Y=hαxZ−Y¯x=0∞=∑x=03x0−11τxρ¯αxZ−Y¯,0¯τx+∑x=3x0∞1τxρ¯αxZ−Y¯,0¯τx≤∑x=03x01τxρ¯Z−Y¯,0¯τx+∑x=x0∞1τx+2x0ρ¯αx+2x0Z−Y¯,0¯τx+2x0≤∑x=03x01τxρ¯Z−Y¯,0¯τx+∑x=x0∞1τxρ¯αxZ¯,0¯τx≤3∑x=0x01τxρ¯Z−Y¯,0¯τx+∑x=x0∞1τxρ¯αxZ¯,0¯τx<ρ.
Therefore,✠α¯ℓτ.FhΔ,Λ⊆theclosureofFΔ,Λ. Contrarily, one has a counter example as I6∈✠α¯ℓF0,0,1,1,1,hΔ,Λ, but η0>0 is not verified.Theorem 47.
Supposeτqq∈N∈ℓ∞∩I with 0<τx1<τx2, for all x∈N; hence,
(81)✠¯ℓFτx1hΔ,ΛÞ✠¯ℓFτx2hΔ,ΛÜLΔ,Λ.Proof.
LetZ∈✠¯ℓFτx1hΔ,Λ; hence, sxZ¯∈ℓFτx1h. One gets
(82)∑x=0∞1τx2ρ¯sxZ¯,0¯τx2<∑x=0∞1τx1ρ¯sxZ¯,0¯τx1<∞,then Z∈✠¯ℓFτx2hΔ,Λ. After, if we choose sxZ¯x=0∞ with ρ¯sxZ¯,0¯=τx1/x+1τx1, we have Z∈LΔ,Λ such that
(83)∑x=0∞1τx1ρ¯sxZ¯,0¯τx1=∑x=0∞1x+1=∞,(84)∑x=0∞1τx2ρ¯sxZ¯,0¯τx2≤∑x=0∞1τx1τx1x+1τx2/τx1≤supxτx1τx2/τx1−1∑x=0∞1x+1τx2/τx1<∞.
Then,Z∉✠¯ℓFτx1hΔ,Λ and Z∈✠¯ℓFτx2hΔ,Λ.
Clearly,✠¯ℓFτx2hΔ,Λ⊂LΔ,Λ. Next, if we put sxZ¯x=0∞ such that ρ¯sxZ¯,0¯=τx2/x+1τx2, we have Z∈LΔ,Λ such that Z∉✠¯ℓFτx2hΔ,Λ.Theorem 48.
Assumeτqq∈N∈ℓ∞∩I with τ0>0; hence, ✠α¯ℓτ.Fh is minimum.Proof.
Let✠α¯ℓτ.FΔ,Λ=LΔ,Λ; then, there is η>0 with HZ≤ηZ, where HZ=∑q=0∞1/τqρ¯αqZ¯,0¯τq, for every Z∈LΔ,Λ. By using Dvoretzky’s theorem [37], with r∈N, we get quotient spaces Δ/Yr and subspaces Mr of Λ which can be mapped onto ℓ2r by isomorphisms Vr and Xr with VrVr−1≤2 and XrXr−1≤2. If Ir is the identity map on ℓ2r, Tr is the quotient map from Δ onto Δ/Yr and Jr is the natural embedding map from Mr into Λ. Assume mq is the Bernstein numbers [16]; then,
(85)1=mqIr=mqXrXr−1IrVrVr−1≤XrmqXr−1IrVrVr−1=XrmqJrXr−1IrVrVr−1≤XrdqJrXr−1IrVrVr−1=XrdqJrXr−1IrVrTrVr−1≤XrαqJrXr−1IrVrTrVr−1,for 0≤x≤r. Then, we have
(86)1≤XrVr−1τqρ¯αqJrXr−1IrVrTr¯,0¯τq.
So, there areρ≥1; we obtain
(87)∑q=0r1τq≤ρXrVr−1∑q=0r1τqρ¯αqJrXr−1IrVrTr¯,0¯τq⇒∑q=0r1τq≤ρXrVr−1HJrXr−1IrVrTr⇒∑q=0r1τq≤ρηXrVr−1JrXr−1IrVrTr⇒∑q=0r1τq≤ρηXrVr−1JrXr−1IrVrTr=ρηXrVr−1Xr−1IrVr≤4ρη.
So there is a contradiction, ifr⟶∞. Therefore, Δ and Λ both cannot be infinite dimensional if ✠α¯ℓτ.FΔ,Λ=LΔ,Λ.
As with the previous theorem, we can easily prove the next theorem.Theorem 49.
Supposeτqq∈N∈ℓ∞∩I with τ0>0; hence, ✠d¯ℓτ.F is minimum.Lemma 50 (see [17]).
IfB∈LΔ,Λ and B∉YΔ,Λ, then D∈LΔ and M∈LΛ with MBDeb=eb, with b∈N.Theorem 51 (see [17]).
In general, we have(88)FΔÞYΔÞLcΔÞLΔ.Theorem 52.
Letτqq∈N∈ℓ∞∩I with 0<τx1<τx2, for all x∈N; hence,
(89)L✠¯ℓFτx2hΔ,Λ,✠¯ℓFτx1hΔ,Λ=Y✠¯ℓFτx2hΔ,Λ,✠¯ℓFτx1hΔ,Λ.Proof.
AssumeX∈L✠¯ℓFτx2hΔ,Λ,✠¯ℓFτx1hΔ,Λ and X∉Y✠¯ℓFτx2hΔ,Λ,✠¯ℓFτx1hΔ,Λ. By using Lemma 50, we have Y∈L✠¯ℓFτx2hΔ,Λ and Z∈L✠¯ℓFτx1hΔ,Λ so that ZXYIb=Ib; hence, with b∈N, one has
(90)Ib✠¯ℓFτx1hΔ,Λ=∑x=0∞1τx1ρ¯sxIb¯,0¯τx1≤ZXYIb✠¯ℓFτx2hΔ,Λ≤∑x=0∞1τx2ρ¯sxIb¯,0¯τx2.
This fails Theorem47. So X∈Y✠¯ℓFτx2hΔ,Λ,✠¯ℓFτx1hΔ,Λ.Corollary 53.
Assumeτqq∈N∈ℓ∞∩I with 0<τx1<τx2, for all x∈N; hence,
(91)L✠¯ℓFτx2hΔ,Λ,✠¯ℓFτx1hΔ,Λ=Lc✠¯ℓFτx2hΔ,Λ,✠¯ℓFτx1hΔ,Λ.Proof.
Evidently, asY⊂Lc.Definition 54 (see [17]).
A Banach spaceΔ is called simple, if there is only one nontrivial closed ideal in LΔ.Theorem 55.
Letτqq∈N∈ℓ∞∩I with τ0>0; hence, ✠¯ℓτ.Fh is simple.Proof.
LetX∈Lc✠¯ℓτ.FhΔ,Λ and X∉Y✠¯ℓτ.FhΔ,Λ. From Lemma 50, there exist Y,Z∈L✠¯ℓτ.FhΔ,Λ with ZXYIb=Ib, which gives that I✠¯ℓτ.FhΔ,Λ∈Lc✠¯ℓτ.FhΔ,Λ. Then, L✠¯ℓτ.FhΔ,Λ=Lc✠¯ℓτ.FhΔ,Λ; hence, ✠¯ℓτ.Fh is a simple Banach space.Notations 2.
(92)✠¯Uλ≔✠¯UλΔ,Λ;ΔandΛareBanachspaces,where✠¯UλΔ,Λ≔X∈LΔ,Λ:λxXx=0∞∈UandX−ρ¯λxX,0¯Iisnotinvertible,withx∈N.Theorem 56.
Assumeτqq∈N∈ℓ∞∩I with τ0>0; hence,
(93)✠¯ℓτ.FhλΔ,Λ=✠¯ℓτ.FhΔ,Λ.Proof.
SupposeX∈✠¯ℓτ.FhλΔ,Λ; hence, λxXx=0∞∈ℓτ.Fh and X−ρ¯λxX,0¯I=0, for all x∈N. We have X=ρ¯λxX,0¯I, for all x∈N, so
(94)ρ¯sxX¯,0¯=ρ¯sxρ¯λxX,0¯I¯,0¯=ρ¯λxX,0¯,for every x∈N. Therefore, sxX¯x=0∞∈ℓτ.Fh; hence, X∈✠¯ℓτ.FhΔ,Λ. Next, suppose X∈✠¯ℓτ.FhΔ,Λ. Hence, sxX¯x=0∞∈ℓτ.Fh. One gets ∑x=0∞1/τxρ¯sxX¯,0¯τx<∞. Then, limx⟶∞ρ¯sxX¯,0¯=0. Assume X−ρ¯sxX¯,0¯I−1 exists, with x∈N. Then, X−ρ¯sxX¯,0¯I−1 exists and is bounded, for all x∈N. So, limx⟶∞X−ρ¯sxX¯,0¯I−1=X−1 exists and is bounded. As ✠¯ℓτ.Fh,H is a pre-quasi-mappings’ ideal, we have
(95)I=XX−1∈✠¯ℓτ.FhΔ,Λ⇒sxI¯x=0∞∈ℓτ.F⇒limx⟶∞ρ¯sxI¯,0¯=0.
This gives a contradiction, aslimx⟶∞ρ¯sxI¯,0¯=1. Therefore, X−ρ¯sxX¯,0¯I=0, with x∈N. This explains X∈✠¯ℓτ.FhλΔ,Λ.


## 7. Applications

Consider the summable equations which are presented by many authors [38–40]:
(96)Yq=Rq+∑r=0∞Dq,rmr,Yr,where D:N2⟶R,m:N×R0,1⟶R0,1,R:N⟶R0,1, and assume V:ℓτ.Fh⟶ℓτ.Fh, where τqq∈N∈ℓ∞∩I with τ0>1 and hY=∑j=0∞1/τjρ¯Yj,0¯τj1/K, for every Y∈ℓFτ, defined by
(97)VYqq∈N=Rq+∑r=0∞Dq,rmr,Yrq∈N.Example 15.
The summable equation (96) has a solution in ℓτ.Fh, if
(98)K∑q=0∞1τqρ¯Rq−Yq+∑r=0∞Dq,rmr,Yr,0¯τq1/K≤ln∑q=0∞1/τqρ¯Rq+∑r=0∞Dq,rmr,Yr,0¯τq∑q=0∞1/τqρ¯Yq,0¯τq.
Evidently, we have(99)hVY−Y=∑q∈N1τqρ¯VYq−Yq,0¯τq1/K=∑q=0∞1τqρ¯Rq−Yq+∑r=0∞Dq,rmr,Yr,0¯τq1/K≤1Kln∑q=0∞1/τqρ¯Rq+∑r=0∞Dq,rmr,Yr,0¯τq∑q=0∞1/τqρ¯Yq,0¯τq=lnhVY−lnhY.
By Theorem20, one gets a solution of equation (96) in ℓτ.Fh.Example 16.
The summable equation (96) has a solution in ℓτ.Fh, if
(100)∑q=0∞1τqρ¯Rq−Yq+∑r=0∞Dq,rmr,Yr,0¯τq1/K≤∑q=0∞1τqρ¯Yq,0¯τq1/K−∑q=0∞1τqρ¯Rq+∑r=0∞Dq,rmr,Yr,0¯τq1/K.
Clearly, we have(101)hVY−Y=∑q∈N1τqρ¯VYq−Yq,0¯τq1/K=∑q=0∞1τqρ¯Rq−Yq+∑r=0∞Dq,rmr,Yr,0¯τq1/K≤∑q=0∞1τqρ¯Yq,0¯τq1/K−∑q=0∞1τqρ¯Rq+∑r=0∞Dq,rmr,Yr,0¯τq1/K=hY−hVY.
By Theorem20, one gets a solution of equation (96) in ℓτ.Fh.
We conclude the following two applications in view of Theorem22.Example 17.
The summable equation (96) has a solution in ℓτ.Fh, where hY=∑q=0∞1/τqρ¯Yq,0¯τq, for all Y∈ℓFτ, if
(1)
hVl+1G−VlG≤αllnhVG/hG, with G∈ℓτ.Fh(2)
V is h-sequentially continuous at Z∈ℓτ.Fh(3)
there isY∈ℓτ.Fh with VlY which has VljY converging to Z
Then,Z∈ℓτ.Fh is a fixed point of V.Example 18.
The summable equation (96) has a solution in ℓτ.Fh, where hY=∑q=0∞1/τqρ¯Yq,0¯τq, for all Y∈ℓFτ, if
(1)
hVl+1G−VlG≤αlhG−hVG, with G∈ℓτ.Fh(2)
V is h-sequentially continuous at Z∈ℓτ.Fh(3)
there isY∈ℓτ.Fh with VlY which has VljY converging to Z
Then,Z∈ℓτ.Fh is a fixed point of V.In this part, we search for a solution to nonlinear matrix equations (102) at M∈✠¯ℓτ.FhΔ,Λ, where Δ and Λ are Banach spaces, τqq∈N∈ℓ∞∩I with τ0>0, and HG=∑q=0∞1/τqρ¯sqG¯,0¯τq1/K, for all G∈✠¯ℓτ.FhΔ,Λ. Consider the summable equations
(102)sqG¯=sqP¯+∑r=0∞Dq,rmr,srG¯,where D:N2⟶R,m:N×R0,1⟶R0,1, and suppose V:✠¯ℓτ.FhΔ,Λ⟶✠¯ℓτ.FhΔ,Λ defined by
(103)HVG=∑q=0∞1τqρ¯sqP¯+∑r=0∞Dq,rmr,srG¯,0¯τq1/K.We conclude the following two applications in view of Theorem45.Example 19.
If(i)
there isα∈0,1 so that HVl+1G−VlG≤αllnHVG/HG, with G∈✠¯ℓτ.FhΔ,Λ(ii)
V is H-sequentially continuous at an element M∈✠¯ℓτ.FhΔ,Λ(iii)
there areG∈✠¯ℓτ.FhΔ,Λ such that the sequence of iterates VrG has a VrmG converging to M
Then,M∈✠¯ℓτ.FhΔ,Λ is a fixed point of V.Example 20.
Suppose(i)
there isα∈0,1 so that HVl+1G−VlG≤αlHG−HVG, with G∈✠¯ℓτ.FhΔ,Λ(ii)
V is H-sequentially continuous at an element M∈✠¯ℓτ.FhΔ,Λ(iii)
there areG∈✠¯ℓτ.FhΔ,Λ such that the sequence of iterates VrG has a VrmG converging to M
Then,M∈✠¯ℓτ.FhΔ,Λ is a fixed point of V.


## 8. Conclusion

We proposed in this paper the notions of premodular spaces of fuzzy numbers and extendeds-fuzzy numbers to construct large spaces of solutions to many nonlinear summable and matrix equations of fuzzy numbers. We discuss some topological and geometric structures of ℓτ.Fh, of the multiplication mappings defined on ℓτ.Fh, of the class ✠¯ℓτ.Fh, and of the class ✠¯ℓτ.Fhλ. Moreover, the existence of Caristi’s fixed point in ℓτ.Fh is investigated. We also presented some examples and illustrated the implication of the new results in the study of the existence of solutions for a class of nonlinear summable and matrix equations.



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