# Fixed Point Property of Variable Exponent Cesàro Complex Function Space of Formal Power Series under Premodular

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

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## Abstract

We have defined the variable exponent of the Cesàro complex function space of formal power series. We have constructed the prequasi-ideal generated bys-numbers and this new space of complex functions. We present some topological and geometric structures of this class of ideal. The existence of Caristi’s fixed point is examined. Some geometric properties related to the fixed point theory are presented. Finally, real-world examples and applications show solutions to some nonlinear difference equations.


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## Body

## 1. Introduction

Since the publishing of the book [1] on the Banach fixed point theorem, several mathematicians have studied possible extensions to the Banach fixed point theorem. The nonlinear analysis relies heavily on the Banach contraction principle, a powerful nonlinear analysis tool. The variable exponent Lebesgue spaces Lr contain Nakano sequence spaces. Variable exponent spaces were thought to offer adequate frameworks for the mathematical components of several issues. Standard Lebesgue spaces were inadequate throughout the second half of the twentieth century. Since these spaces and their effects have become a well-known and efficient instrument for solving a range of problems, they have become a flourishing topic of research, with ramifications that extend into a wide variety [2] of mathematical disciplines. The study of variable exponent Lebesgue spaces Lr received additional impetus from the mathematical description of non-Newtonian fluid hydrodynamics [3, 4]. Non-Newtonian fluids, also known as electrorheological fluids, have various applications ranging from military science to civil engineering and orthopedics. Guo and Zhu [5] investigated a class of stochastic Volterra-Levin equations with Poisson jumps. Mao et al. [6] were concerned with neutral stochastic functional differential equations driven by pure jumps (NSFDEwPJs). They proved the existence and uniqueness of the solution to NSFDEwPJs whose coefficients satisfy the local Lipschitz condition and established the pth exponential estimations and almost surely asymptotic estimations of the solution for NSFDEwJs. Yang and Zhu [7] concerned with a class of stochastic neutral functional differential equations of Sobolev type with Poisson jumps. The mapping ideal theory is well regarded in functional analysis. Using s-numbers is an essential technique. Pietsch [8–11] developed and studied the theory of s-numbers of linear bound mappings between Banach spaces. He offered and explained some topological and geometric structures of the quasi ideals of ℓp-type mappings. Then, Constantin [12] generalized the class of ℓp-type mappings to the class of cesp-type mappings. Makarov and Faried [13] showed some inclusion relations of ℓp-type mappings. As a generalization of ℓp-type mappings, Stolz mappings and mappings’ ideal were examined by Tita [14, 15]. In [16], 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 [17], the class of s-type Zu,v;ℓp mappings was defined and some of their properties were explained. Yaying et al. [18] defined and studied χrη, whose its r-Cesàro matrix in ℓn, with r∈0,1 and 1<η<∞. They explained the quasi-Banach ideal of type χrη, with r∈0,1 and 1<η<∞. Kannan [19] gave an example of a class of mappings with the same fixed point actions as contractions, though that fails to be continuous. The only attempt to describe Kannan operators in modular vector spaces was once made in Reference [20]. Bakery and Mohamed [21] investigated the concept of a prequasinorm on Nakano sequence space with a variable exponent in the range 0;1. They discussed the adequate circumstances for it to generate prequasi-Banach and closed space when endowed with a definite prequasinorm and the Fatou property of various prequasinorms on it. Additionally, they established a fixed point for Kannan prequasinorm contraction mappings on it and the prequasi-Banach mappings’ ideal generated from s-numbers belonging to this sequence space. Also, in [22], they found some fixed points results of Kannan nonexpansive mappings on generalized Cesàro backward difference sequence space of the nonabsolute type. The set of nonnegative integers, real, and complex numbers will be denoted by N, R, and ℂ, respectively. By RN and R+N, we denote the space of real and positive real sequences. By ℓ∞ and ℓr, we denote the spaces of bounded and r-absolutely summable sequences of R.Lemma 1 (see [23]).
Supposeτq>0andyq∈R for all q∈N, then
(1)yq+zqτq≤2K−1yqτq+zqτq,where K=max1,supqτq.Ifτ=τa∈R+N and τa≥1, for all a∈N, the variable exponent Cesàro complex function space is denoted by
(2)Cτ⋅=f∈ℂℂ:fy=∑v=0∞f^vyvandhμf<∞,forsomeμ>0,whenhf=∑a=0∞∑k=0∞f∧ka+1τa.For more information on formal power series spaces and their behaviors, see [24–27]. 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 special spaces of formal power series, which are important extensions of the concept of modular spaces. We have built large spaces of solutions to many nonlinear summable and difference equations. It is the first attempt to examine the fixed point theory and Caristi’s fixed point in certain premodular special spaces of formal power series. The purpose of this study is arranged, as follows: In Section 2, we present and study the space Cτ⋅h equipped with a definite function h. In Section 3, we suggest a generalization of Caristi’s fixed point theorem. In Section 4, the mapping ideals formed by s-numbers and this function space are constructed, and their geometric and topological properties are presented. Speciffically, we explore, in Section 5, some geometric properties connected with fixed point theory in Cτ⋅h. Finally, in Section 6, we discuss several applications of solutions to summable equations and illustrate our findings with some instances.


## 2. Some Properties ofCτ⋅

In this section, we investigate sufficient setups ofCτ⋅ equipped with definite function h to be prequasiclosed and Banach (ssfps). We also present the Fatou property of various h on Cτ⋅.Theorem 2.
Ifτq∈ℓ∞ and τa>1, for all a∈N, then
(3)Cτ⋅=f∈ℂℂ:fy=∑v=0∞f^vyvandhμf<∞,foranyμ>0.Proof.
(4)Cτ⋅=f∈ℂℂ:fy=∑v=0∞f^vyvandhμf<∞,forsomeμ>0=f∈ℂℂ:fy=∑v=0∞f^vyv,infμτa∑a=0∞∑k=0af∧ka+1τa≤∑a=0∞∑k=0aμf∧ka+1τa<∞,forsomeμ>0a=f∈ℂℂ:fy=∑v=0∞f^vyv,∑a=0∞∑k=0af∧ka+1τa<∞a=f∈ℂℂ:fy=∑v=0∞f^vyvandhμf<∞,foranyμ>0.
Let us indicateϑ, the zero function of H and the space of finite formal power series by F, i.e, when f∈F, then there is k∈N so that fy=∑a=0kf^aya. Nakano [28] introduced the concept of modular vector spaces.Definition 3.
SupposeH is a vector space. A function h:H0,∞ is said to be modular, if the next conditions hold
(a)
Ifg∈H, then hg≥0 and g=ϑ⟺hg=0(b)
hηg=hg holds, for all g∈H and η=1(c)
The inequalityhαg+1−αf≤hg+hf satisfies, for all g,f∈H and α∈0,1Definition 4 (see [29]).
The spaceH=f∈ℂℂ:fy=∑a=0∞f^aya is said to be a special space of formal power series (or in short ssfps), if it verifies the following settings:
(1)
ep∈H, for every p∈N, where epy=∑a=0∞eapya=yp(2)
For allg∈H and f^a≤g^a, for every a∈N, then f∈H(3)
Ifg∈H then g⋅∈H, where g⋅y=∑p=0∞gp/2^yp and p/2 indicates the integral part of p/2Definition 5 (see [29]).
A subspaceHh of the ssfps is said to be a premodular ssfps, if there is a function h:H⟶0,∞ verifies the following conditions:
(i)
Ifg∈H, then hg≥0 and g=ϑ⇔hg=0(ii)
Whenf∈H and λ∈ℂ, then there are Q≥1 such that hλf≤λQhf(iii)
Supposef,g∈H, then there are P≥1 such that hf+g≤Phf+hg(iv)
Supposefb^≤gb^, for all b∈N, then hf+hg(v)
There areP0≥1 such that hf≤hf⋅≤P0hf(vi)
The closure ofF=Hh(vii)
There areξ>0 so that hλe0≥ξλhe0, where λ∈ℂ
Clearly, the concept of premodular vector spaces is more general than modular vector spaces, an example of premodular vector space but not modular vector space.Example 1.
The functionhf=∑q=0∞∑p=0qf∧p/q+12q+3/q+4 is a premodular (not a modular) on the vector space C2q+3/q+4q=0∞. As for every f,g∈C2q+3/q+4q=0∞, one has
(5)hf+g2=∑q=0∞∑p=0qf∧p+g∧p/2q+12q+3/q+4≤284hf+hg,an example of premodular vector space and modular vector space.Example 2.
The functionhf=infα>−:∑q=0∞∑p=0qf∧p/α/q+12q+3/q+4≤1 is a premodular (modular) on the vector space C2q+3/q+2q=0∞.Definition 6 (see [29]).
A subspaceHh of the ssfps is said to be a prequasinormed ssfps, if there is a function h:H⟶0,∞ verifies the following conditions:
(i)
Ifg∈H, then the hg≥0 and g=ϑ⟺hg=0(ii)
Whenf∈H and λ∈ℂ, then there are Q≥1 such that hλf≤λQhf(iii)
Supposef,g∈H then there are P≥1 such that hf+g≤Phf+hgRecall thatHh is said to be a prequasi-Banach ssfps, when Hh is complete.Theorem 7 (see [30]).
All premodular ssfpsHh is a prequasinormed ssfps.Theorem 8 (see [30]).
All quasinormed (ssfps) is a prequasinormed (ssfps).Definition 9.
(a)
The functionh on Cτ⋅ is said to be h-convex, if(6)hαf+1−αg≤αhf+1−αhg,for every α∈0,1 and f,gCτ⋅(b)
gqq∈N⊆Cτ⋅h is h-convergent to g∈Cτ⋅h, if and only if, limq⟶∞hgq−g=0. When the h-limit exists, then it is unique(c)
gqq∈N⊆Cτ⋅h is h-Cauchy, if limq,r⟶∞hgq−gr=0(d)
Γ⊂Cτ⋅h is h-closed, when for all h-converges, g1a∈N⊂Γ to g, then g∈Γ(e)
Γ⊂Cτ⋅h is h-bounded, if δhΓ=suphf−g:f,g∈Γ<∞(f)
Theh-ball of radius ε≥0 and center f, for every f∈Cτ⋅h, is described as(7)Bhf,ε=g∈Cτ⋅h:hf−g≤ε(g)
A prequasinormh on Cτ⋅ holds the Fatou property, if for every sequence gq⊆Cτ⋅h under limq⟶∞hgq−g=0 and all f∈Cτ⋅h, one has hf−g≤suprinfq≥rhf−gqRecall that the Fatou property explains theh-closedness of the h-balls. We will mark the space of all increasing sequences of real numbers by I.Theorem 10.
Cτ⋅h, where hf∑q=0∞∑p=0qfp∧/q+1τq1/K. for all f∈Cτ⋅, is a premodular (ssfps), when τqq∈N∈ℓ∞∩I with τ0>1.Proof.
Evidently,hf≥0 and hf=0⟺=ϑ.
Letf,g∈Cτ⋅. One has f+gy=∑v=0∞fv^+gv^yv∈ℂ with
(8)hf+g∑q=0∞∑p=0qfp+gp∧q+1τq1/K≤∑q=0∞∑p=0qfp∧q+1τq1/K+∑q=0∞∑p=0qgp∧q+1τq1/K=hf+hg<∞.
Asαf∈Cτ⋅, hence from conditions (1-i) and (1-ii), one has Cτ⋅ is linear. Also er∈Cτ⋅, for all r∈N, since
(9)her=∑q=0∞∑p=0qepe∧q+1τq1/K=∑r=0∞1r+1τ01/K<∞.
There isQ=max1,supqατq/K−1≥1 with hαf≤Qαhf for all f∈Cτ⋅ and α∈ℂ
Assumefq≤gq, for all q∈N and g∈Cτ⋅. One finds
(10)hf=∑q=0∞∑p=0qfp∧q+1τq1/K=∑q=0∞∑p=0qgp∧q+1τq1/K=hg<∞,
thenf∈Cτ⋅.
Obviously, from (58).
Letfq∈Cτ⋅, we get
(11)hfp/2=∑q=0∞∑p=0qfp/2∧q+1τq1/K=∑q=0∞∑p=02qfp/2∧2q+1τ2q+∑q=0∞∑p=02q+1fp/2∧2q+1τ2q+11/K≤∑q=0∞fq∧+2∑p=0qfp∧q+1τq+∑q=0∞2∑p=0qfp∧q+1τq1/K≤∑q=0∞3∑p=0qfp∧q+1τq+∑q=0∞2∑p=0qfp∧q+1τq1/K≤3K+2K1/K∑q=0∞∑p=0qfp∧q+1τq1/K=3K+2K1/KhYq,
thenfp/2∈Cτ⋅.
From (59), we obtain P0=3K+2K1/K≥1.
Evidently the closure ofF=Cτ⋅.
There is0<σ≤supqατq/K−1, for α≠0 or σ>0, for α=0 with hαe0≥σαhe0.Theorem 11.
Ifτqq∈N∈ℓ∞∩I with τ0>1, then Cτ⋅h is a prequasi-Banach (ssfps), where
(12)hf=∑q=0∞∑p=0qfp∧q+1τq1/K,
for everyf∈Cτ⋅.Proof.
According to Theorems10 and 7, the space Cτ⋅h is a prequasinormed (ssfps). Assume fl=fqlq=0∞ is a Cauchy sequence in Cτ⋅h, hence for every ε∈0,1, one has l0∈N such that for all l,m≥l0, one gets
(13)hfl−fm=∑q=0∞∑p=0qfpl−fpm∧q+1τq1/K<ε.This impliesfql^−fqm^<ε. Hence, fqm^ is a Cauchy sequence in ℂ, for constant q∈N, which implies limm⟶∞fqm^−fq0^=0, for constant q∈N. Hence, hfl−f0<ε, for every l≥l0. Since hf0=hf0−fl+fl≤hfl−f0+hfl<∞. So, f0∈Cτ⋅.Theorem 12.
Supposeτqq∈N∈ℓ∞∩I with τ0>1, then Cτ⋅h is a pre-quasi closed (ssfps), where
(14)hf=∑q=0∞∑p=0qfp∧q+1τq1/K,
for everyf∈Cτ⋅.Proof.
According to Theorems10 and 7, the space Cτ⋅h is a prequasinormed (ssfps). Assume fl=fqlq=0∞∈Cτ⋅h and liml⟶∞hfl−f0=0, then for all, ε∈0,1, there is l0∈N such that for all l≥l0, we obtain
(15)ε>hfl−f0=∑q=0∞∑p=0qfpl∧−fp0∧q+1τq1/K,
which impliesfql^−fq0^<ε, as ℂ is a complete space. Therefore, fql^ is a convergent sequence in ℂ, for fixed q∈N. So liml⟶∞fql^=fq0^, for fixed q∈N. Since, hf0≤hfl−f0+hfl<∞. So, f0∈Cτ⋅.Theorem 13.
The function(16)hf=∑q=0∞∑p=0qfp∧q+1τq1/K,holds the Fatou property, when τqq∈N∈ℓ∞∩I with τ0>1, for all f∈Cτ⋅.Proof.
Letgr⊆Cτ.h such that limr⟶∞hgr−g=0. Since Cτ.h is a pre-quasi closed space, one has g∈Cτ.h. For all f∈Cτ.h, one gets
(17)hf−g=∑q=0∞∑p=0qfp∧−gp∧q+1τq1/K≤∑q=0∞∑p=0qfp∧−gp∧q+1τq1/K+∑q=0∞∑p=0qgpr∧−gp∧q+1τq1/K≤supminfr≥mhf−gr.Theorem 14.
The function(18)hf=∑q=0∞∑p=0qfp∧q+1τq,does not hold the Fatou property, for all f∈Cτ⋅, when τq∈ℓ∞ and τq>1, for all q∈N.Proof.
Letgr⊆Cτ⋅h so that limr⟶∞hgr−g=0. Since Cτ⋅h is a pre-quasi closed space, one gets g∈Cτ⋅h. For every f∈Cτ⋅, we obtain
(19)hf−g=∑q=0∞∑p=0qfp∧−gp∧q+1τq≤2K−1∑q=0∞∑p=0qfp∧−gp∧q+1τq+∑q=0∞∑p=0qgpr∧−gp∧q+1τq≤2K−1supminfr≥mhf−gr.Example 3.
Forτq∈1,∞N, the function
(20)hf=infα>0:∑q∈N∑p=0qfp∧/αq+1τq≤1,is a norm on Cτ⋅.Example 4.
The function(21)hf=∑q∈N∑p=0qfp∧q+13q+2/q+13,is a prequasinorm (not a quasinorm) on C3q+2/q+1q=0∞.Example 5.
The function(22)hf=∑q∈N∑p=0qfp∧q+13q+2/q+1,is a prequasinorm (not a quasinorm) on C3q+2/q+1q=0∞.Example 6.
The function(23)hf=∑q∈N∑p=0qfp∧q+1dd,is a prequasinorm, quasinorm, and not a norm on Cd, for 0<d<1.


## 3. Caristi’s Fixed Point Theorem inCτ⋅h

In this section, the existence of Caristi’s fixed point inCτ⋅h is presented according to Farkas [31], where
(24)hf=∑q∈N∑p=0qfp∧q+1τq1/K,for all f∈Cτ⋅.Definition 15.
The functionΨ1:Cτ⋅h⟶−∞,∞ is said to be lower semicontinuous at G0∈Cτ⋅h if limG⟶infG0Ψ1G=Ψ1G0, where limG⟶infG0Ψ1G=supv∈vG0infG∈VΨ1G, where VG0 is a neighborhood system of G0.Definition 16.
The functionΨ1:Cτ⋅h⟶−∞,∞ is said to be proper, when
(25)DΨ1=G∈Cτ⋅h:Ψ1G<∞≠0.Theorem 17.
SupposeΞ≠0 and Ξ is a h-closed subset of Cτ⋅h, and Ψ1:Ξ⟶−∞,∞ is a proper, h-lower semicontinuous function with infG∈ΞΨ1G>−∞. If γ>0, ϖq⊂0,∞, and G0∈Ξ so that Ψ1G0≤infG∈ΞΨ1G+γ. One gets Gq∈Ξ which h-converges to some Gγ, and
(i)
hGγ−Gq≤γ/2qϖ0, with q∈N(26)Ψ1Gγ+∑q=0∞ϖqhGγ−Gq≤Ψ1G0(ii)
whenG≠Gγ, then(27)Ψ1Gγ+∑q=0∞ϖqhGγ−Gq<Ψ1G+∑q=0∞ϖqhG−GqProof.
IfSG0=G∈Ξ:Ψ1G+ϖ0hG−G0≤Ψ1G0. Since G0∈SG0, then SG0≠0. As Ψ1 is h-lower semicontinuous, h holds the Fatou property and Ξ is h-closed, then SG0 is h-closed. Take G1∈SG0 and
(28)Ψ1G1+ϖ0hG1−G0≤infG∈SG0Ψ1G+ϖ0hG−G0+γϖ12ϖ0.
Choose(29)SG1=G∈SG0:Ψ1G+∑j=01ϖjhG−Gj≤Ψ1G1+ϖ0hG1−Gj.
AsSG0, we get SG1≠0 and h-closed. Suppose that one has built G0,G1,G2,⋯,Gq and SG0,SG1,SG2,⋯,SGq. Next, take Gq+1∈SGq and
(30)Ψ1Gq+1+∑j=0qϖjhGq+1−Gj≤infG∈SGqΨ1G+∑j=0qϖjhG−Gj+γϖq2qϖ0.
Let(31)SGq+1≔G∈SGq:Ψ1G+∑j=0q+1ϖjhG−Gj≤Ψ1Gq+1+∑j=0qϖjhGq+1−Gj,hence we form by induction, the sequences Gq and SGq. Fix q∈N. Suppose W∈SGq. One obtains
(32)Ψ1W+∑j=0qϖjhW−Gj≤Ψ1Gq+∑j=0q−1ϖjhGq−Gj,then
(33)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 all q∈N, one gets
(34)hGq+p−Gq≤γ2qϖ0,with q,p∈N. This implies Gq is h-Cauchy. Since Cτ⋅h is h-Banach space; hence, Gq has h–limits Gγ and ∩q∈NSGq=Gγ. Since Gq+1∈SGq, we can see
(35)Ψ1Gq+1+∑j=0qϖjhGq+1−Gj≤Ψ1Gq+∑j=0q−1ϖjhGq−Gj,hence, Ψ1Gq+∑j=0q−1ϖjhGq+1−Gj is decreasing. After, let G≠Gγ. One gets m∈N with G∈SGq, with q≥m, i.e.,
(36)Ψ1Gq+∑j=0q−1ϖjhGq−Gj<Ψ1G+∑j=0qϖjhG−Gj.
SinceGγ+∈SGq, with q≥m, we get
(37)Ψ1Gγ+∑j=0qϖjhGγ−Gj≤Ψ1Gq+∑j=0q−1ϖjhGq−Gj≤Ψ1Gm+∑j=0m−1ϖjhGm−Gj.
Putq⟶∞ in the previous inequality, then
(38)Ψ1Gγ+∑j=0∞ϖjhGγ−Gj≤Ψ1xm+∑j=0m−1ϖjhGm−Gj<Ψ1G+∑j=0mϖjhG−Gj≤Ψ1G+∑j=0∞ϖjhG−Gj.
This gives(39)Ψ1Gγ+∑q=0qϖqhGγ−Gq<Ψ1G+∑q=0∞ϖqhG−Gq.Theorem 18.
SupposeΞ≠0 and Ξ is a h-closed subset of Cτ⋅h. By taking γ>0 and ϖn and 0<ω=∑n=0∞ϖn<∞. If H:Ξ⟶Ξ is a mapping and there is a function Ψ1:Ξ⟶−∞,∞ holds a proper and h-lower semicontinuous with infG∈ΞΨ1G>−∞ and
(1)
hHG−Y−hG−Y≤hHG−Y, for any G,Y∈Ξ(2)
hHG−G≤Ψ1G−Ψ1HG, with G∈ΞThen,H has a fixed point in Ξ.Proof.
As0<ω=∑n=0∞ϖn<∞, one has Ψ2≔ωΨ1 is also proper, h-lower semicontinuous and bounded from below. If G∈Ξ, one gets
(40)ωhHG−G≤Ψ2G−Ψ2HG.
AsinfG∈ΞΨ2G>−∞, one obtains G0∈Ξ with Ψ2G0<infG∈ΞΨ2G+γ. From Theorem 17, there is Gq which h-converges to some Gγ∈Ξ, and
(41)Ψ2Gγ+∑q=0∞ϖqhGγ−Gq<Ψ2G+∑q=0∞ϖqhG−Gq,for every G≠Gγ. Assume that HGγ≠Gγ, we have
(42)Ψ2Gγ+∑q=0∞ϖqhGγ−Gq<Ψ2HGγ+∑q=0∞ϖqhHGγ−Gq,then
(43)Ψ2Gγ−Ψ2HGγ<∑q=0∞ϖqhHGγ−Gq−∑q=0∞ϖqhGγ−Gq=∑q=0∞ϖqhHGγ−Gq−hGγ−Gq.
From condition (40), then
(44)Ψ2Gγ−Ψ2HGγ<∑q=0∞ϖqhHGγ−Gγ=ωhHGγ−Gγ.
The inequality (40) implies that
(45)ωhHGγ−Gγ≤Ψ2Gγ−Ψ2HGγ<ωhHGγ−Gγ.
This is a contradiction, henceHGγ=Gγ.


## 4. Structure of Mappings’ Ideal

The structure of the mappings’ ideal byCτ⋅h, where
(46)hf=∑q=0∞∑p=0qfp∧q+1τq1/K,for allf∈Cτ⋅, and s-numbers have been explained. We study enough setups on Cτ⋅h such that the class Cτ⋅✠h is complete and closed. We investigate enough setups (not necessary) on Cτ⋅h such that the closure of F=Cτ⋅✠αh. This gives a negative answer of Rhoades’ [32] open problem about the linearity of s-type Cτ⋅h spaces. We explain enough setups on Cτ⋅h such that Cτ⋅✠h is strictly contained for different powers, Cτ⋅✠αh is the minimum, the class Cτ⋅✠h is simple, and Cτ⋅✠hλ=Cτ⋅✠h.We denote 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. The space of approximable and compact-bounded linear mappings from a Banach space Δ into a Banach space Λ will be indicated by ϒΔ,Λ and LcΔ,Λ, and if Δ=Λ, we mark ϒΔ and LcΔ, respectively.Definition 19 (see [33]).
Ans-number function is a mapping s:LΔ,Λ⟶R+N that sorts every V∈LΔ,Λ unique sequence sdVd=0∞ validates 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=γsdV(e)
suppose rankV≤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
Some examples ofs-numbers are as follows:
(1)
Theqth Kolmogorov number, described by dqX, is marked by(47)dqX=infdimsJ≤qupf≤1infg∈JXf−g.(2)
Theqth approximation number, described by αqX, is marked by(48)αqX=infX−Y:Y∈LΔ,ΛandrankY≤q.Definition 20 (see [10]).
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, when all UΔ,Λ=U∩LΔ,Λ verifies the following conditions:
(i)
IΓ∈U, where Γ marks 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 21 (see [30]).
(49)✠H≔✠HΔ,Λ,where✠HΔ,Λ≔V∈LΔ,Λ:fs∈H,wherefsy=∑n=0∞snVyn,✠Hα≔✠HαΔ,Λ,where✠HαΔ,Λ≔V∈LΔ,Λ::fα∈H,wherefαy=∑n=0∞αnVyn,✠Hd≔✠HdΔ,Λ,where✠HdΔ,Λ≔V∈LΔ,Λ::fd∈H,wherefdy=∑n=0∞dnVyn.Theorem 22. (see [29]).
SupposeH is a (ssfps), then ✠H is mappings’ ideal.
According to Theorems10 and 22, one concludes the next theorem.Theorem 23.
Supposeτqq∈N∈ℓ∞∩I with τ0>1, then Cτ⋅✠h is a mappings’ ideal.Definition 24 (see [34]).
A functionH∈0,∞U is said to be a pre-quasi norm on the ideal U, if it verifies the following setups:
(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 25 (see [35]).
Every quasinorm on the idealU is a prequasinorm on the same ideal.Theorem 26.
Ifτqq∈N∈ℓ∞∩I with τ0>1, then H is a prequasinorm on Cτ⋅✠h, so that HZ=hfs, where fsCτ⋅h and fsy=∑n=0∞snZyn.Proof.
(1)
WhenX∈Cτ⋅✠hΔ,Λ, HX=hfs≥0, and HX=hfs=0, if and only if, snX=0, for all n∈N; if and only if, X=0(2)
There isQ≥1 with HεX≤hεfs≤QεHX for every X∈Cτ⋅✠hΔ,Λ and ε∈ℂ(3)
One hasPP0≥1 so that for X1,X2∈Cτ⋅✠hΔ,Λ; hence, there are f1s,f2sCτ⋅h with f1sy=∑n=0∞snX1yn and f2sy=∑n=0∞snX2yn. Therefore, for gsy≕∑n=0∞snX1+X2yn, we have KK0≥1 so that(50)HX1+X2=hgs≤hf1s⋅+f2s⋅≤Phf1s⋅+hf2s⋅≤PP0HX1+HX2(4)
We haveϱ≥1 if X∈LΔ0,Δ, Y∈Cτ⋅✠hΔ,Λ, and Z∈LΛ,Λ0; hence, there is fs∈Cτ⋅h with fsy=∑n=0∞snYyn. Then, for gsy≕∑n=0∞snZYXyn, one has(51)HZYX=hgs≤hXZfs≤ϱXHYZ.Theorem 27.
Supposeτqq∈N∈ℓ∞∩I with τ0>1 one has Cτ⋅✠h,H is a prequasi-Banach mappings’ ideal.Proof.
SupposeVaa∈N is a Cauchy sequence in Cτ⋅✠hΔ,Λ. As LΔ,Λ⊇SCτ⋅hΔ,Λ, hence, there is fsa∈Cτ⋅h with fsa∈y=∑n=0∞snVayn for very a∈N, then
(52)HVr−Va=∑q=0∞∑p=0qspVr−Vaq+1τq1/K≥infqVr−Vaτq/K∑q=0∞1q+1τq1/K,hence, Vaa∈N is a Cauchy sequence in LΔ,Λ is a Banach space, so there exists V∈LΔ,Λ so that lima⟶∞Va−V=0 and since fsaCτ⋅h, for all a∈N and Cτ⋅h is a premodular (ssfps), hence, one can see
(53)HV=∑q=0∞∑p=0qspVq+1τq1/K≤∑q=0∞∑p=0qsp/2V−Vaq+1τq1/K+∑q=0∞∑p=0qsp/2Vaq+1τq1/K≤∑q=0∞Va−Vτq1/K3K+2K1/K∑q=0∞∑p=0qspVaq+1τq1/K<ε.We obtain fsaCτ⋅h, hence V∈Cτ⋅✠hΔ,Λ.Theorem 28.
Ifτqq∈N∈ℓ∞∩I with τ0>1, one has Cτ⋅✠h,H is a prequasiclosed mappings’ ideal.Proof.
SupposeVa∈Cτ⋅✠hΔ,Λ, for all a∈N and lima⟶∞HVa−V=0, hence, there is fsa∈Cτ⋅h with fsay=∑n=0∞snVayn, for all a∈N, there is ς>0 and as LΔ,Λ⊇SCτ⋅hΔ,Λ, one has
(54)HVa−V=∑q=0∞∑p=0qspVa−Vq+1τq1/K≤infqVa−Vτq/K∑q=0∞1q+1τq1/K.SoVaa∈N is convergent in LΔ,Λ, i.e., lima⟶∞Va−V=0 and since fsaCτ⋅h, for all a∈N and Cτ⋅h is a premodular (ssfps), hence, one can see
(55)HV=∑q=0∞∑p=0qspVq+1τq1/K≤∑q=0∞∑p=0qsp/2V−Vaq+1τq1/K+∑q=0∞∑p=0qsp/2Vaq+1τq1/K≤∑q=0∞Va−Vτq1/K+3K+2K1/K∑q=0∞∑p=0qspVaq+1τq1/K<ε.We obtain fs∈Cτ.h, hence V∈✠Cτ⋅hΔ,Λ.Definition 29.
A prequasinormH on the ideal ✠Hh verifies the Fatou property if for every Tqq∈N⊆✠HhΔ,Λ so that limq⟶∞HTq−T=0 and M∈✠HhΔ,Λ, one gets
(56)HM−T≤supqinfj≥qHM−Tj.Theorem 30.
Supposeτqq∈N∈ℓ∞∩I with τ0>1, then ✠Cτ.h,H does not verify the Fatou property.Proof.
AssumeTqq∈N⊆✠Cτ.hΔ,Λ with limq⟶∞HTq−T=0. Since ✠Cτ.h is a prequasiclosed ideal, then T∈✠Cτ.hΔ,Λ. So for every M∈✠Cτ.hΔ,Λ, one has
(57)HM−T=∑q=0∞∑p=0qspM−Tq+1τq1/K≤∑q=0∞∑p=0qsp/2M−Tjq+1τq1/K+∑q=0∞∑p=0qsp/2Tj−Tq+1τq1/K≤3K+2K1/Ksuprinfj≥r∑q=0∞∑p=0qspM−Tjq+1τq1/K.Theorem 31.
✠Cτ.hαΔ,Λ=theclosureofFΔ,Λ, if τqq∈N∈ℓ∞∩I with τ0>1. But the converse is not necessarily true.Proof.
Aseq∈Cτ.h, for every q∈N and Cτ.h is a linear space. Suppose Z∈FΔ,Λ with rank Z=m, where m∈N, hence fα∈F with fαy=∑n=0m−1αnZyn, one has fα∈Cτ.h. Therefore, theclosureofFΔ,Λ⊆✠Cτ.hαΔ,Λ. Assume Z∈✠Cτ.hαΔ,Λ, we have gα∈Cτ.h. As hgα<∞, assume ρ∈0,1, then there is q0∈N−0 with hgα−∑n=0q0−1en<ρ/2K+3ηd, for some d≥1, where η=max1,∑q=q0∞1/q+1τq. Since αqZ is decreasing, we have
(58)∑q=q0+12q0∑p=0qα2q0Zq+1τq≤∑q=q0+12q0∑p=0qαpZq+1τq≤∑q=q0∞∑p=0qαpZq+1τq<ρ2K+3ηd.
Hence, there isY∈F2q0Δ,Λ so that rankY≤2q0 and
(59)∑q=2q0+13q0∑p=0qZ−Yq+1τq≤∑q=q0+12q0∑p=0qZ−Yq+1τq<ρ2K+3ηd,since τqq∈N∈ℓ∞∩I with τ0>1, we have
(60)supq=q0∞2q0Z−Yτq<ρ22K+2η.
Therefore, one has(61)∑q=0q0Z−Yτq<ρ2K+3ηd.
AsZ−Y∈✠Cτ.hαΔ,Λ, hence gα∈Cτ.h, where gαy≔∑n=0∞αnZ−Yyn. In view of inequalities (58)–(61), one has
(62)dZ,Y=hgα=∑q=03q0−1∑p=0qαpZ−Yq+1τq+∑q=3q0∞∑p=0qαpZ−Yq+1τq≤∑q=03q0∑p=0qZ−Yq+1τq+∑q=q0∞∑p=0q+2q0αpZ−Yq+2q0+1τq+2q0≤∑q=03q0Z−Yτq+∑q=q0∞∑p=0q+2q0αpZ−Yq+1τq≤3∑q=0q0Z−Yτq+∑q=q0∞∑p=02q0−1αpZ−Y+∑p=2q0q+2q0ρ¯αpZ−Y¯,0¯q+1τq≤3∑q=0q0Z−Yτq+2K−1∑q=q0∞∑p=02q0−1αpZ−Yq+1τq+∑q=q0∞∑p=2q0q+2q0αpZ−Yq+1τq≤3∑q=0q0Z−Yτq+2K−1∑q=q0∞∑p=02q0αpZ−Yq+1τq+∑q=q0∞∑p=0qαp+2q0Z−Yq+1τq≤3∑q=0q0Z−Yτq+2K−1ηsupq=q0∞2q0Z−Yτq+∑q=q0∞∑p=0qαpZq+1τqTherefore,✠Cτ.hαΔ,Λ⊆theclosureofFΔ,Λ. Contrarily, one has a counter example as I2∈✠C0,0,2,2,2,⋯hαΔ,Λ, but τ0>1 is not verified.Theorem 32.
Supposeτqq∈N∈ℓ∞∩I with 1<τx1<τx2, for all x∈N, hence
(63)✠Cτx1hΔ,Λ✠Cτx2hΔ,Λ⫋LΔ,Λ.Proof.
LetZ∈✠Cτx1hΔ,Λ, hence gs∈Cτx1h, where gsy≔∑n=0∞snZyn. One gets
(64)∑x=0∞∑p=0xspZx+1τx2<∑x=0∞∑p=0xspZx+1τx1<∞,then gs∈Cτx2h this implies Z∈✠Cτx2hΔ,Λ. After, if we choose sxZx=0∞ with ∑p=0xspZ=x+11−1/τx1, we have Z∈LΔ,Λ such that
(65)∑x=0∞∑p=0xspZx+1τx1=∑x=0∞1x+1=∞,∑x=0∞∑p=0xspZx+1τx2≤∑x=0∞1x+1τx2/τx1<∞.Then,Z∉✠Cτx1hΔ,Λ and Z∈✠Cτx2hΔ,Λ.Clearly,✠Cτx2hΔ,Λ⊂LΔ,Λ. Next, if we put sxZx=0∞ with ∑p=0xspZ=x+11−1/τx2. We have Z∈LΔ,Λ such that Z∉✠Cτx2hΔ,Λ.Theorem 33.
Assumeτqq∈N∈ℓ∞∩I with τ0>1, hence ✠Cτ.hα is minimum.Proof.
Let✠Cτ.αΔ,Λ=LΔ,Λ, one has η>0 so that HZ≤ηZ, where
(66)HZ=∑q=0∞∑p=0qαpZq+1τq,for all Z∈LΔ,Λ. According to Dvoretzky’s theorem [36], with r∈N, we get quotient spaces Δ/Yr and subspaces Mr of Λ which can be transformed 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.Assumemq is the Bernstein numbers [9], then
(67)1=mqIr=mqXrXr−1IrVrVr−1≤XrmqXr−1IrVrVr−1=XrmqJrXr−1IrVrVr−1≤XrdqJrXr−1IrVrVr−1=XrdqJrXr−1IrVrTrVr−1≤XrαqJrXr−1IrVrTrVr−1,for 0≤q≤r. Then, we have
(68)1≤XrVr−1τq∑p=0qαpJrXr−1IrVrTrq+1τq.So, there areϱ≥1, we obtain
(69)∑q=0r1≤ϱXrVr−1∑q=0r∑p=0qαpJrXr−1IrVrTrq+1τq⟹∑q=0r1≤ϱXrVr−1HJrXr−1IrVrTr⟹∑q=0r1≤ϱηXrVr−1JrXr−1IrVrTr⟹∑q=0r1≤ϱηXrVr−1JrXr−1IrVrTr=ϱηXrVr−1Xr−1IrVr≤4ϱη.So there is a contradiction, ifr⟶∞. Therefore, Δ and Λ both cannot be infinite dimensional if ✠Cτ.αΔ,Λ=LΔ,Λ.As with the previous theorem, we can easily prove the following theorem.Theorem 34.
Ifτqq∈N∈ℓ∞∩I with τ0>1, hence ✠Cτ.d is minimum.Lemma 35 (see [10]).
IfB∈LΔ,Λ and B∉ϒΔ,Λ, then D∈LΔ and M∈LΛ with MBDIb=Ib, with b∈N.Theorem 36 (see [10]).
In general, we have(70)FΔ⫋ϒΔ⫋LcΔ⫋LΔ.Theorem 37.
Letτqq∈N∈ℓ∞∩I with 1<τx1<τx2, for all x∈N, hence
(71)L✠Cτx2hΔ,Λ,✠Cτx1hΔ,Λ=ϒ✠Cτx2hΔ,Λ,✠Cτx1hΔ,Λ.Proof.
AssumeX∈L✠Cτx2hΔ,Λ,✠Cτx1hΔ,Λ and X∉ϒ✠Cτx2hΔ,Λ,✠Cτx1hΔ,Λ. By using Lemma 35, we have Y∈L✠Cτx2hΔ,Λ and Z∈L✠Cτx1hΔ,Λ so that ZXYIb=Ib, hence with b∈N, one has
(72)Ib✠Cτx1hΔ,Λ=∑x=0∞∑p=0xspIbx+1τx1≤ZXYIb✠Cτx2hΔ,Λ≤∑x=0∞∑p=0xspIbx+1τx2.This fails Theorem32. So, X∈ϒ✠Cτx2hΔ,Λ,✠Cτx1hΔ,Λ.Corollary 38.
Assumeτqq∈N∈ℓ∞∩I with 1<τx1<τx2, for all x∈N, hence,
(73)L✠Cτx2hΔ,Λ,✠Cτx1hΔ,Λ=Lc✠Cτx2hΔ,Λ,✠Cτx1hΔ,Λ.Proof.
Evidently, asϒ⊂Lc.Definition 39 (see [10]).
A Banach spaceΔ is said to be simple, if there is an unique nontrivial closed ideal in LΔ.Theorem 40.
Letτqq∈N∈ℓ∞∩I with τ0>1, hence ✠Cτ.h is simple.Proof.
LetX∈Lc✠Cτ.hΔ,Λ and X∉ϒ✠Cτ.hΔ,Λ. From Lemma 35, there exist Y,Z∈L✠Cτ.hΔ,Λ with ZXYIb=Ib, which gives that I✠Cτ.hΔ,Λ∈Lc✠Cτ.hΔ,Λ. Then, L✠Cτ.hΔ,Λ=Lc✠Cτ.hΔ,Λ; hence, ✠Cτ.h is simple Banach space.Notations 41.
(74)✠Hλ≔✠HλΔ,Λ;ΔandΛareBanachSpaces,where
✠HλΔ,Λ≔X∈LΔ,Λ:fλ∈Hh,wherefλy=∑n=0∞λnTynandX−λxXI=0,foreveryx∈N.Theorem 42.
Assumeτqq∈N∈ℓ∞∩I with τ0>1, hence,
(75)✠Cτ.hλΔ,Λ=✠Cτ.hΔ,Λ.Proof.
LetX∈✠Cτ.hλΔ,Λ, hence fλ∈Cτ.h, where fλy=∑n=0∞λnTyn and X−λxXI=0, with x∈N. We have X=λxXI, for all x∈N, so
(76)sxX=sxλxXI=λxX,with x∈N. One gets fs∈Cτ.h; hence, X∈✠Cτ.hΔ,Λ. Next, suppose X∈✠Cτ.hΔ,Λ. Hence, fs∈Cτ.h. One gets
(77)∑x=0∞sxXτx≤∑x=0∞∑p=0xspXx+1τx<∞.Then,limx⟶∞sxX=0. If X−sxXI−1 exists, with x∈N. Then, X−sxXI−1 exists and bounded, for all x∈N. So, limx⟶∞X−sxXI−1=X−1 exists and bounded. Since ✠Cτ.h,H is a pre-quasi mappings’ ideal, one has
(78)I=XX−1∈✠Cτ.hΔ,Λ⇒gs∈Cτ.⟹limx⟶∞sxI=0,where gsy=∑n=0∞snIyn. This gives a contradiction, as limx⟶∞sxI=1. Therefore, X−sxXI=0, with x∈N, which explains X∈✠Cτ.hλΔ,Λ.


## 5. Nonexpansive Mappings onCτ.h

In this section, we have presented some geometric properties connected with the fixed point theory inCτ.h.In the next part of this section, we will use the functionh as
(79)hf=∑q=0∞∑p=0qfp∧q+1τq1/K,for all f∈Cτ..Definition 43 (see [37]).
A sequencegp⊆Hh, is said to be ε-separated sequence for some ε>0, if
(80)sepgp=infhgp−gq:p≠q>ε.Definition 44. [37].
Ifk≥2 is an integer, a Banach space Hh is said to be k-nearly uniformly convex (k-NUC) when for all ε>0 one has δ∈0,1 so that for every sequence gp⊆BHh, with sepgp≥ε, we have p1,p2,p3,⋯,pk∈N.
Such that(81)hgp1+gp2+gp3+⋯+gpkk<1−δ.Definition 45 [38].
A functionh is said to be hold the δ2-condition (h∈δ2), if for any ε>0, there exists a constant k≥2 and a>0 such that,
(82)h2g≤khg+εforeachg∈Hh,withhg≤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).Theorem 46 ((see [38]), Lemma 2.1).
Supposeh∈δ2s, then for any L>0 and ε>0 one has δ>0 with hf+g−hf<ε,f,g∈Hh, with hf≤L and hg≤δ.Theorem 47.
Pick anτqq∈N∈ℓ∞∩I with τ0>1, then for any L>0 and ε>0 one has δ>0 with hf+g−hf<ε, for every f,g∈Cτ.h, so that hf≤L and hg≤δ.Proof.
Sinceτqq∈N∈ℓ∞∩I with τ0>1, then h∈δ2s. According to Theorem 46, the proof follows.
We denoteSHh and BHh for the unit sphere and the unit ball of Hh, respectively.Theorem 48.
Supposeτqq∈N∈ℓ∞∩I with τ0>1, then is k-NUC, for any integer k≥2.Proof.
Assumeε∈0,1 and fn⊆BCτ.h, where fny=∑i=0∞fni^yi so that sepfn≥ε. For all m∈N, suppose fnmy=∑i=0∞fnmi^yi, where fnmi^i=0∞=0,0,0,⋯,fnm^,fnm+1^,⋯. As for all i∈N, fni^n=0∞∈ℓ∞, from the diagonal method, one has a subsequence fnj of fn with fnji^ converges for all i∈N, 0≤i≤m. One obtains an increasing sequence of positive integers tm so that sepfnjmj>tm≥ε. Therefore, one has a sequence of positive integers rmm=0∞ with r0<r1<r2<⋯, so that
(83)hKfrmm≥ε2,for all m∈N. For constant integer k≥2, assume ε1=kp0−1−1/k−1kp0ε/4 from Theorem 47, one gets δ>0 with
(84)hKf+g−hKf<ε1.IfhKg≤δ. As hKfn≤1, for every n∈N, one has positive integers mii=0,1,2,⋯,k−2 with m0<m1<m2<⋯<mk−2 with hKfimi≤δ. Define mk−1=mk−2+1. From inequality (83), one can see hfrmkmk≥ε/2. Suppose pi=i for 0≤i≤k−2 and pk−1=rmk−1. According to inequalities (83), (84), and convexity of Jnu=uτn for every n∈N, one has
(85)hKfp0+fp1+fp2+.⋯+fpk−1k=∑n=0∞∑i=0nfp2i∧+fp3i+⋯+fpk−1i∧/kn+1τn=∑n=0m1−1∑i=0nfp2i∧+fp3i+⋯+fpk−1i∧/kn+1τn+∑n=m1∞∑i=0nfp2i∧+fp3i+⋯+fpk−1i∧/kn+1τn≤∑n=0m1−1∑i=0nfp2i∧+fp3i+⋯+fpk−1i∧/kn+1τn+∑n=m1∞∑i=0nfp2i∧+fp3i+⋯+fpk−1i∧/kn+1τn+ε1≤∑n=0m1−11k∑j=0k−1∑i=0nfpji∧n+1τn+∑n=m1m2−1∑i=0nfp2i∧+fp3i+⋯+fpk−1i∧/kn+1τn+∑n=m2∞∑i=0nfp2i∧+fp3i+⋯+fpk−1i∧/kn+1τn+ε1≤∑n=0m1−11k∑j=0k−1∑i=0nfpji∧n+1τn+∑n=m1m2−1∑i=0nfp2i∧+fp3i+⋯+fpk−1i∧/kn+1τn+∑n=m2∞∑i=0nfp2i∧+fp3i+⋯+fpk−1i∧/kn+1τn+2ε1≤∑n=0m1−11k∑j=0k−1∑i=0nfpji∧n+1τn+∑n=m1m2−11k∑j=1k−1∑i=0nfpji∧n+1τn+∑n=m2m3−11k∑j=2k−1∑i=0nfpki∧n+1τn+⋯+∑n=mk−1mk−11k∑j=k−2k−1∑i=0nfpki∧n+1τn+∑n=mk∞∑i=0nfpki∧/kn+1τn+k−1ε1≤hKfp0+fp1+fp2+.⋯+fpk−2k+1k∑n=0mk−1∑i=0nfpki∧n+1τn+∑n=mk∞∑i=0nfpki∧/kn+1τn+k−1ε1≤k−1k+1k∑n=0mk−1∑i=0nfpki∧n+1τn+1kτ0∑n=mk∞∑i=0nfpki∧n+1τn+k−1ε1≤1−1k+1k1−∑n=mk∞∑i=0nfpki∧n+1τn+1kτ0∑n=mk∞∑i=0nfpki∧n+1τn+k−1ε1=1+k−1ε1−kτ0−1−1kτ0∑n=mk∞∑i=0nfpki∧n+1τn≤1+k−1ε1−kτ0−1−1kτ0ε2=1−kτ0−1−1kτ0ε4.So,Cτ.h is k-NUC.Recall thatk-NUC implies reflexivity.Definition 49 (see [39]).
A Banach spaceHh holds the uniform Opial property, if for all ε>0 one has γ>0 so that for every weakly null sequence fn⊂SHh and f∈Hh so that hf≥ε, then
(86)1+γ≤liminfn⟶∞hfn+f.Definition 50 (see [40]).
For a bounded subsetE⊂Hh, the set-measure of noncompactness defined by
(87)αE=infξ>0:Ecanbecoveredbyfinitelymanysetsofdiameter≤ξ.Definition 51 (see [41, 42]).
The ball-measure of noncompactness is defined by(88)βE=infξ>0:Ecanbecoveredbyfinitelymanyballsofdiameter≤ξ.Definition 52 (see [43]).
For a subsetE⊂Hh is said to be α-minimal if αC=αE, for any infinite subset C of E.Definition 53 (see [43]).
The packing rate of a Banach spaceHh is denoted by γHh, and the formula defines it
(89)γHh=δHhσHh,where δHh and σHh are defined as the supremum and the infimum, respectively, of the set
(90)βEαE:E⊂Hh,Eisα−minimal,αE>0.Definition 54 (see [41]).
The functionΔ is said to be the modulus of noncompact convexity, if for every ξ>0 define
(91)Δξ=inf1−inff∈Ehf:EisaclosedconvexsubsetofBHhwithβE≥ξ.Definition 55 (see [39]).
A Banach spaceHh is said to be hold property L, when limε⟶1−Δε=1..Definition 56.
An operatorV:Hh⟶Hh is said to be a h-contraction, if one gets α∈0,1 with hVg−Vf≤αhg−f, for all g,f∈Hh. The operator V is said to be h-non-expansive, when α=1. An element g∈Hh is said to be a fixed point of V, when Vg=g..Theorem 57 (see [39]).
(1)
Suppose a Banach spaceHh holds property L, then it has the fixed point property, i.e., for every nonexpansive self-mapping of a nonempty, closed, bounded, convex subset has a fixed point(2)
A Banach spaceHh holds property L, if and only if, it is reflexive and has the uniform Opial propertyTheorem 58.
Supposeτqq∈N∈ℓ∞∩I with τ0>1, then Cτ.h has the uniform Opial property.Proof.
Letε>0 one finds a positive number ε0∈0,ε with
(92)1+εK2>1+ε0K.
Iff∈Cτ.h and hf≥ε., one has n1∈N with
(93)∑n=n1+1∞1n+1∑i=0nfi∧τn<ε04K.
Therefore, one gets(94)h∑i=n1+1∞fi^ei<ε04<ε4.
Also, one has(95)εK≤∑n=0n11n+1∑i=0nfi∧τn+∑n=n1+1∞1n+1∑i=0nfi∧τn<∑n=0n11n+1∑i=0nfi∧τn+ε04K<∑n=0n11n+1∑i=0nfi∧τn+εK4,if
(96)3εK4≤∑n=0n11n+1∑i=0nfi∧τn.
For any weakly null sequencefm⊂SCτ.h, in virtue of fmi^⟶0 for i=0,1,2,⋯, one has m0∈N with
(97)h∑i=0n1fmi^ei<ε04,for m>m0. One can see
(98)hfm+f=h∑i=0n1fmi^+fi^ei+∑i=n1+1∞fmi^+fi^ei≥h∑i=0n1fi^ei+∑i=n1+1∞fmi^ei−h∑i=0n1fmi^ei−h∑i=n1+1∞fi^ei≥h∑i=0n1fi^ei+∑i=n1+1∞fmi^ei−ε02,if m>m0. For a≔∑i=0n1fi^, one obtains
(99)hK∑i=0n1fi^ei+∑i=n1+1∞fmi^ei=∑n=0n11n+1∑i=0nfi∧τn+∑n=n1+1∞1n+1∑i=0na+fmi∧τn≥∑n=0n11n+1∑i=0nfi∧τn+∑n=n1+1∞1n+1∑i=0nfmi∧τn≥3εK4+1−εK4=1+εK2>1+ε0K.
Combining this with the previous inequality, one has(100)hfm+f≥h∑i=0n1fi^ei+∑i=n1+1∞fmi^ei−ε02≥1+ε0−ε02=1+ε02.Therefore, the spaceCτ.h has the uniform Opial property.From Theorem58 and the reflexivity of the space Cτ.h, by applying Theorem 47, we get the following.Corollary 59.
Ifτqq∈N∈ℓ∞∩I with τ0>1, then Cτ.h has the property L and the fixed point property.Definition 60.
Hh holds the h-normal structure property, if and only if, for every nonempty h-bounded, h-convex, and h-closed subset Γ of Hh not decreased to one point, one has f∈Γ with
(101)supg∈Γhf−g<δhΓ≔suphf−g:f,g∈Γ<∞.Definition 61 (see [44]).
The weakly convergent sequence coefficient of a Banach spaceHh, denoted by WCSHh, is defined as follows:
(102)WCSHh=infAfn:fnn=1∞⊂SHh,Afn=A1fn,fn⟶w0,where
(103)Afn=limsupn⟶∞hfi−fj:i,j≥n,i≠j,A1fn=liminfn⟶∞hfi−fj:i,j≥n,i≠j.Theorem 62 (see [45]).
A reflexive Banach spaceHh such that WCSHh>1 has the normal structure property.Theorem 63.
Ifτqq∈N∈ℓ∞∩I with τ0>1, then Cτ.h holds the h-normal structure property.Proof.
Take anyε>0 and an asymptotic equidistant sequence fn⊂SCτ.h with fn⟶w0 and let v1=f1. One has i1∈N with
(104)h∑i=i1+1∞v1i^ei<ε.
Asfn⟶0 coordinate-wise, one gets n2∈N with
(105)h∑i=1i1fni^ei<ε.
Forn≥n2, put v2=fn2, one gets i2>i1 with
(106)h∑i=i2+1∞v1i^ei<ε.
Asfni⟶0 coordinate-wise, one obtains n3∈N with
(107)h∑i=1i2fni^ei<ε.
Forn≥n3. By induction, one has a subsequence vn of fn with
(108)h∑i=in+1∞vni^ei<ε,h∑i=1invn+1i^ei<ε.
Take(109)zn=∑i=in−1+1invni^ei,for n=2,3,⋯. So,
(110)1≥hzn=h∑i=1∞vni^ei−∑i=1in−1vni^ei−∑i=in+1∞vni^ei≥h∑i=1∞vni^ei−h∑i=1in−1vni^ei−h∑i=in+1∞vni^ei>1−2ε.
For everyn,m∈N so that n≠m, one can see
(111)hvn−vm=h∑i=1∞vni^ei−∑i=1∞vmi^ei≥h∑i=in−1+1invni^ei−∑i=im−1+1imvmi^ei−h∑i=1in−1vni^ei−h∑i=in+1∞vni^ei−h∑i=1im−1vmi^ei−h∑i=im+1∞vmi^ei≥hzn−zm−4ε,
which givesAfn=Avn≥Azn−4ε. Take un=zn/zn, for n=2,3,⋯. Then,
(112)un∈SCτ.h;(113)Afn≥1−εAun−4ε.
On the other hand,(114)hvn−vm≤hzn−zm+4ε≤hun−um+4ε,for any n,m∈N with n≠m. Therefore,
(115)Aun≥Afn−4ε.
By the arbitrariness ofε>0, we have from the relations (112), (113), and (115) that
(116)WCSCτ.h=infAun,such that
(117)un=∑i=in−1+1inuni^ei∈SCτ.h,0=i0<i1<⋯,un⟶w0andunisasymptoticequidistant.
Takem∈N large enough such that
(118)∑k=im−1+1∞bkτk<ε,where b≔∑i=in−1+1inuni. One gets for
(119)hKun−um=∑k=in−1+1im−11k∑i=1kuni∧τk+∑k=im−1+1∞1kb+∑i=1kumi∧τk≥∑k=in−1+1im−11k∑i=1kuni∧τk+∑k=im−1+1∞1k∑i=1kumi∧τk=∑k=in−1+1∞1k∑i=1kuni∧τk−∑k=im−1+1∞bkτk+∑k=im−1+1∞1k∑i=1kumi∧τk>1−ε+1=2−ε,that is Anun≥2−ε1/K. Note that
(120)∑k=im−1+1∞1kb+∑i=1kumi∧τk1/K≤∑k=im−1+1∞bkτk1/K+∑k=im−1+1∞1k∑i=1kumi∧τk1/K<ε1/K+1.
Therefore,(121)hKun−um=∑k=in−1+1im−11k∑i=1kumi∧τk+∑k=im−1+1∞1kb+∑i=1kumi∧τk≤∑k=in−1+1∞1k∑i=1kumi∧τk+∑k=im−1+1∞1kb+∑i=1kumi∧τk≤1+1+ε1/KK,with n,m∈N and n≠m. Therefore, Anun≤1+1+ε1/KK1/K and, by the arbitrariness of ε>0, one has WCSCτ.h=21/K. From Theorems 48 and 62, then, the function space Cτ.h has the h-normal structure property.Theorem 64 (see [46]).
IfHh is reflexive Banach space with the uniform Opial property, one has γHh=2/WCSHh..Theorem 65.
Ifτqq∈N∈ℓ∞∩I with τ0>1, then γCτ.h=21−1/K..Proof.
SinceCτ.h is reflexive Banach space with the uniform Opial property, one obtains
(122)γCτ.h=2WCSCτ.h=21−1/K.Theorem 66.
Ifτqq∈N∈ℓ∞∩I with τ0>1 and W:Cτ.h⟶Cτ.h is h-contraction mapping, where hf=∑q=0∞∑p=0qfp∧/q+1τq1/K, for every f∈Cτ., then W has a unique fixed point.Proof.
Let the setups be satisfied. For everyf∈Cτ.h, then Wpf∈Cτ.h. As W is a h-contraction mapping, one gets
(123)hWp+1f−Wpf≤αhWpf−Wp−1f≤α2hWp−1f−Wp−2f≤⋯≤αphWf−f.
So, for allp,q∈N so that q>p, one has
(124)hWqf−Wpf≤αphWq−pf−f.
Therefore,Wpf is a Cauchy sequence in Cτ.h. Since the space Cτ.h is prequasi-Banach (ssfps). One gets g∈Cτ.h with limp⟶∞Wpf=g, to prove that Wg=g. According to Theorem 13, h verifies the Fatou property; one can see
(125)hWg−g≤supiinfp≥ihWp+1f−Wpf≤supiinfp≥iαphWf−f=0,so Wg=g. Then, g is a fixed point of W. To prove that the fixed point is unique, let us have two different fixed points f,g∈Cτ.h of W. One obtains
(126)hf−g≤hWf−Wg≤αhf−g.
So,f=g.Example 7.
Assume(127)V:C2q+3q+2q=0∞h⟶C2q+3q+2q=0∞h,where
(128)hg=∑q=0∞∑p=0qgp∧q+12q+3/q+2,for every g∈C2q+3/q+2q=0∞ and Vg=g/4.
Since for allf1,f2∈C2q+3/q+2q=0∞h, one gets
(129)hVf1−Vf2=hf14−f24≤1644hf1−f2.
SoV is h-contraction. Assume V:Γ⟶Γ with Vg=g/4, where
(130)Γ=f∈C2q+3q+2q=0∞h:f0^=f1^=0.
SinceV is h-contraction. So, it is h-nonexpansive. By Corollary 59, V holds a fixed point ϑ in Γ.


## 6. Applications to Nonlinear Summable Equations

Numerous authors, for example in [47], have examined nonlinear summable equations such as (132). This section is dedicated to locating a solution to (132) in Cτ.h, where the conditions τqq∈N∈ℓ∞∩I with τ0>1 are satisfied and
(131)hf=∑q=0∞∑p=0qfp∧q+1τq1/K,for every f∈Cτ.. Take a look at the equations that are summable:
(132)ga^=ra^+∑m=0∞Aa,mfm,gm^,and assume W:Cτ.h⟶Cτ.h defined by
(133)Wgz=∑a=0∞ra^+∑m=0∞Aa,mfm,gm^za.Theorem 67.
The summable equations (132) have only one solution in Cτ.h if A:N2⟶ℂ,f:N×ℂ⟶ℂ,r^:N⟶ℂ,t^:N⟶ℂ, assume there is κ∈ℂ so that supqκτq/K∈0,1 and for every a∈N, we have
(134)∑m=0∞Aa,mfm,gm^−fm,tm^≤κga^−ta^.Proof.
Let the setups be verified. Consider the mappingW:Cτ.h⟶Cτ.h defined by (133). We have
(135)hWg−Wt=∑q=0∞∑a=0qWga∧−Wta∧q+1τq1/K=∑q=0∞∑a=0q∑m=0∞Aa,mfm,gm∧−fm,tm∧q+1τq1/K≤supqκτq/K∑q=0∞∑a=0qga∧−ta∧q+1τq1/K.According to Theorem66, one obtains a unique solution of equation (132) in Cτ.h.Example 1.
Assume the function spaceC3a+2/a+1a=0∞h, where
(136)hf=∑a=0∞∑b=0afb∧a+13a+2/a+13,
for allf∈C3a+2/a+1a=0∞.
(137)ga^=5−2a+3i+∑m=0∞−1ai+3mcosga∧sinhga∧+sinma+1q,where q>0, i2=−1 and let W:C3a+2/a+1a=0∞h⟶C3a+2/a+1a=0∞h defined by
(138)Wgz=∑a=0∞5−2a+3i+∑m=0∞−1ai+3mcosga∧sinhga∧+sinma+1qza.It is easy to see that(139)∑m=0∞−1aicosga∧sinhga∧+sinma+1q−13m−−13m≤13ga^−ta^.By Theorem67, the summable equations (137) have one solution in C3a+2/a+1a=0∞h.Example 2.
Given the function spaceC3a+2/a+1a=0∞h, where
(140)hf=∑a=0∞∑b=0afb∧a+12a+3/a+2,for all f∈C2a+3/a+2a=0∞. Consider the summable equations (137) with a≥2 and let W:Ξ⟶Ξ, where Ξ=f∈C2a+3/a+2a=0∞h:f0^=f1^=0, defined by
(141)Wfz=∑a=2∞5−2a+3i+∑m=0∞−1ai+3mcosfa∧sinhfa∧+sinma+1qza.Clearly,Ξ is a nonempty, h-convex, h-closed, and h-bounded subset of C2a+3/a+2a=0∞h. It is easy to see that
(142)∑m=0∞−1aicosga∧sinhga∧+sinma+1q−13m−−13m≤19ga^−ta^.By Theorem67 and Corollary 59, the summable equations (137) with a≥2 have a solution in Ξ.Example 3.
Assume the function spaceC3a+2/a+1a=0∞h, where
(143)hg=∑a=0∞∑b=0agb∧a+13a+2/a+13,for all g∈C3a+2/a+1a=0∞.
Consider the non-linear difference equations,(144)ga^=e−2a+3i+∑m=0∞tan2m+1cosh3mi−acospga−2^sinhqga−1^+sinma+1,where g−2^,g−1^,p,q>0, i2=−1 and let W:C3a+2/a+1a=0∞h⟶C3a+2/a+1a=0∞h defined by
(145)Wgz=∑a=0∞e−2a+3i+∑m=0∞tan2m+1cosh3mi−acospga−2^sinhqga−1^+sinma+1za.
It is easy to see that(146)∑m=0∞cosh3mi−acospga−2^sinhqga−1^+sinma+1tan2m+1−tan2m+1≤15ga^−ta^.
By Theorem67, the nonlinear difference equations (144) have one solution in C3a+2/a+1a=0∞h.Example 4.
Given the function spaceC2a+3/a+2a=0∞h, where hg=∑a=0∞∑b=0agb∧/a+12a+3/a+2, for all g∈C2a+3/a+2a=0∞. Consider the non-linear difference equations (144) with a≥1 and let W:Ξ⟶Ξ, where Ξ=g∈C2a+3/a+2a=0∞h:g0^=0, defined by
(147)Wgz=∑a=1∞e−2a+3i+∑m=0∞tan2m+1cosh3mi−acospga−2^sinhqga−1^+sinma+1za.Clearly,Ξ is a nonempty, h-convex, h-closed, and h-bounded subset of C2a+3/a+2a=0∞h. It is easy to see that
(148)∑m=0∞cosh3mi−acospga−2^sinhqga−1^+sinma+1tan2m+1−tan2m+1≤15ga^−ta^.By Theorem67 and Corollary 59, the nonlinear difference equations (144) with a≥1 have a solution in Ξ.Example 5.
The summable equations (132) have a solution in Cτ.h if
(149)K∑q=0∞∑a=0qra∧−ga∧+∑m=0∞Aa,mfm,gm∧q+1τq1/K≤ln∑q=0∞∑a=0qra∧+∑m=0∞Aa,mfm,gm∧/q+1τq∑q=0∞∑a=0qga∧/q+1τq.
Evidently, we have(150)hWg−g=∑q=0∞∑a=0qra∧−ga∧+∑m=0∞Aa,mfm,gm∧q+1τq1/K≤1Kln∑q=0∞∑a=0qra∧+∑m=0∞Aa,mfm,gm∧/q+1τq∑q=0∞∑a=0qga∧/q+1τq=lnhWg−lnhg.
By Theorem18, one gets a solution of equation (132) in Cτ.h.Example 6.
The summable equations (132) have a solution in Cτ.h, if
(151)∑q=0∞∑a=0qra∧−ga∧+∑m=0∞Aa,mfm,gm∧q+1τq1/K≤∑q=0∞∑a=0qga∧q+1τq1/K−∑q=0∞∑a=0qra∧+∑m=0∞Aa,mfm,gm∧q+1τq1/K.
Clearly, we have(152)hWg−g=∑q=0∞∑a=0qra∧−ga∧+∑m=0∞Aa,mfm,gm∧q+1τq1/K≤∑q=0∞∑a=0qga∧q+1τq1/K−∑q=0∞∑a=0qra∧+∑m=0∞Aa,mfm,gm∧q+1τq1/K=hg−hWg.
By Theorem18, one gets a solution of equation(132) in Cτ.h.
AssumeΩ is the set of all closed and bounded intervals on the real line R. For t=t1,t2 and g=g1,g2 in Ω, suppose
(153)t≤gifandonlyift1≤g1andt2≤g2.
Define a metricρ on Ω by
(154)ρt,g=maxt1−g1,t2−g2.
Matloka [48] showed that ρ is a metric on Ω, and Ω,ρ is a complete metric space.Definition 68.
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
Recall that theβ-level set of a fuzzy real number g,0<β<1 indicated by gβ is defined as
(155)gβ=y∈R:gy≥β.
The set of every upper semicontinuous, normal, convex fuzzy number, and is compact and is denoted byR0,1. The set R can be embedded in R0,1, if we define r∈R0,1 by
(156)r¯t=1,t=r,0,t≠r.
Consider the summable equations of fuzzy reals (132) and assume W:Cτ.h⟶Cτ.h defined by
(157)Wgz=∑a=0∞ρ¯ra^+∑m=0∞Aa,mfm,gm^,0¯za,where ρ¯:R0,1×R0,1⟶R+∪0 is defined by ρ¯t,g=sup0≤β≤1ρtβ,gβ. For more details about the fuzzy numbers and their properties, see Zadeh [49].Theorem 69.
The summable equations (132) have an unique solution in Cτ.h if A:N2⟶R+,f:N×R+0,1⟶R+0,1,ra^:N⟶R+0,1,ta^:N⟶R+0,1, assume there is κ∈ℂ so that supqκτq/K∈0,1 and for every a∈N, we have
(158)∑m=0∞Aa,mfm,gm^−fm,tm^≤κga^−ta^.Proof.
Let the setups be verified. Consider the mappingW:Cτ.h⟶Cτ.h defined by (157). We have
(159)hWg−Wt=∑q=0∞∑a=0qWga∧−Wta∧q+1τq1/K=∑q=0∞∑a=0qρ¯ra∧+∑m=0∞Aa,mfm,gm∧,0¯−ρ¯ra∧+∑m=0∞Aa,mfm,tm∧,0¯q+1τq1/K=∑q=0∞∑a=0q∑m=0∞Aa,mfm,gm∧−fm,tm∧q+1τq1/K≤supqκτq/K∑q=0∞∑a=0qρ¯ga∧,0¯−ρ¯ta∧,0¯q+1τq1/K=supqκτq/Khg−t.According to Theorem66, one obtains a unique solution of equation (132) in Cτ.h.


## 7. Conclusion

We discuss in this paper some topological and geometric structure ofCτ.h, the existence of Caristi’s fixed point in it, of the class ✠Cτ.h, and of the class ✠Cτ.hλ. Moreover, some geometric properties related to the fixed point theory in Cτ.h are introduced. Finally, we investigate several solutions applications to summable equations and illustrate our findings with some instances. This article has several advantages for researchers, such as studying the fixed points of any contraction mapping on this prequasispace, which is a generalization of the quasinormed spaces, a new general space of solutions for many difference equations, examining the eigenvalue problem in these new settings, and noting that the closed mappings’ ideals are certain to play an important function in the principle of Banach lattices, hence since many fixed point theorems in a particular space work by either expanding the self-mapping acting on it or expanding the space itself, as future work, we can enlarge the space Cτ.h by q-analogue or generalize the self-mapping acting on it.



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*Source: 3811326-2022-03-26.xml*