# Kannan Nonexpansive Mappings on Nakano Sequence Space of Soft Reals with Some Applications

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

---

## Abstract

We developed the operators ideal in this article by extendings-soft reals and a particular space of sequences with soft real numbers. The criteria necessary for the Nakano sequence space of soft real numbers given with the definite function to be prequasi Banach and closed are investigated. This space’s (R) and normal structural features are illustrated. Fixed points have been introduced for Kannan contraction and nonexpansive mapping. Finally, we investigate whether the Kannan contraction mapping has a fixed point in the prequasi operator ideal with which it is linked. By examining some real-world instances and their applications, it is demonstrated that there exist solutions to nonlinear difference equations.


---

## Body

## 1. Introduction

The study of variable exponent Lebesgue spaces received additional impetus from the mathematical explanation of non-Newtonian fluids’ hydrodynamics (see [1, 2]). Electrorheological fluids have various applications in various fields, including military science, civil engineering, and orthopedics. Since the publication of the Banach fixed point theorem [3], there have been numerous developments in the field of mathematics. While contractions have fixed point actions, Kannan [4] illustrated a noncontinuous mapping. In Reference [5], a single attempt was made to explain Kannan operators in modular vector spaces, and this was the only one that worked. Mitrovic′ et al. [6] defined a cone bvs-metric space over Banach algebra as a generalization of metric spaces, rectangular metric spaces, b-metric spaces, rectangular b-metric spaces, v-generalized metric spaces, cone b-metric spaces over Banach algebra, and rectangular cone b-metric spaces over Banach algebra. They provided fixed point results for Banach and Kannan in cone bvs-metric spaces over Banach algebra. Debnath et al. [7] showed the existence and uniqueness of common fixed points for pairs of self-maps of the Kannan, Reich, and Chatterjea types in a complete metric space. Younis et al. [8] used concepts from graph theory and fixed point theory to provide a fixed point result for Kannan-type mappings in the context of freshly published graphical b-metric spaces. They provided suitable examples of graphs that corroborated the existing theory. They demonstrated the anticipated results by applying them to several nonlinear issues encountered in engineering and research. Younis and Singh [9] discovered adequate conditions for the existence of solutions to certain classes of Hammerstein integral equations and fractional differential equations. They extended the concept of Kannan mappings in terms of F-contraction in the context of b-metric-like spaces and provided a series of novel and nontrivial instances, as well as computer simulations, to demonstrate the established results, therefore introducing the concept in a novel way. On the other hand, several unresolved issues are offered to enthusiastic readers. More information on Kannan’s fixed point theorems can be found here (see [10–15]). The mathematics underpinnings of fuzzy set theory, which were pioneered by Zadeh [16] in 1965 and have made significant progress, are well understood in fuzzy theory. The fuzzy theory has the potential to be applied to various real-world problems. The possibility theory, for example, has been developed by several researchers, including Dubois and Prade [17] and Nahmias [18]. The contribution of probability theory, fuzzy set theory, and rough sets to the study of uncertainty is critical. Yet, these theories have some limitations as well as advantages. The theory of soft sets, developed by Molodtsov [19], was introduced as a new mathematical strategy for dealing with uncertainties to overcome these characteristics. Soft sets have been widely used in various disciplines and technologies. In particular, Maji et al. [20, 21] studied several operations on soft sets and applied their findings to decision-making problems in the literature. Several writers, including Chen [22], Pei and Miao [23], Zou and Xiao [24], and Kong et al. [25], have discovered significant characteristics of soft sets. Soft semirings, soft ideals, and idealistic soft semirings were all investigated by Feng et al. [26]. Das and Samanta developed the ideas of a soft real number and a soft real set in [27] and discussed the characteristics of each concept. These principles served as the foundation for their investigation into the concept of “soft metrics” in “[28].” (See [29, 30] for a more in-depth examination.) Based on the idea of soft elements of soft metric spaces, Abbas et al. [31] developed the concept of soft contraction mapping, which they named “soft contraction mapping.” They focused on fixed points of soft contraction maps and obtained, among other things, a soft Banach contraction principle as a result of their efforts. In their paper, Abbas et al. [32] demonstrated that every complete soft metric induces an equivalent complete usual metric. They obtained in a direct way soft metric versions of various significant fixed point theorems for metric spaces, such as the Banach contraction principle, Kannan and Meir-Keeler fixed point theorems, and Caristi theorem, Kirk’s, among other things. In [33], Chen and Lin presented an extension of the Meir and Keeler fixed point theorem to soft metric spaces, which was previously published. Many researchers working on sequence spaces and summability theory were involved in introducing fuzzy sequence spaces and studying their many characteristics. When it comes to fuzzy numbers, Nuray and Savas [34] defined and explored the Nakano sequences of fuzzy numbers, ℓFτ equipped with a definite function. The following theories use operators’ ideals: fixed point theory, Banach space geometry, normal series theory, approximation theory, and ideal transformations. For additional evidence, see [35–37]. According to Faried and Bakery [38], prequasi operator ideals are broader than quasioperator ideals. This study is aimed at introducing a certain space of soft real number sequences, abbreviated (csss), under a pre-quasi-quasi function (csss). The structure of the ideal operators has been described using this space and s-numbers. The conditions essential to generate prequasi Banach and closed (csss) ℓSτh supplied with the definite function h are investigated. This space’s (R) and normal structure properties are illustrated. Fixed points have been introduced for Kannan contraction and nonexpansive mapping. Finally, we investigate whether the Kannan contraction mapping has a fixed point in the prequasi operator ideal with which it is linked. A few real-world examples and applications demonstrate the existence of solutions to nonlinear difference equations.


## 2. Definitions and Preliminaries

Assume thatR is the set of real numbers and N is the set of nonnegative integers. We denote the collection of all nonempty bounded subsets of R by BR and E is the set of parameters.Definition 1 (see [27]).
A soft real set denoted byf~,A, or simply by f~, is a mapping f~:A⟶BR. If f~ is a single-valued mapping on A⊂E taking values in R, then f~ is called a soft element of R or a soft real number. If f~ is a single-valued mapping on A⊂E taking values in the set R+ of nonnegative real numbers, then f~ is called a nonnegative soft real number. We shall denote the set of nonnegative soft real numbers (corresponding to A) by RA∗. A constant soft real number c~ is a soft real number such that for each a∈A, we have c~a=c, where c is some real number.Definition 2 (see [39]).
For two soft real numbersf~, g~, we say that
(a)
f~≤~g~ if f~a≤~g~a, for all a∈A(b)
f~≥~g~ if f~a≥~g~a, for all a∈A(c)
f~<~g~ if f~a<~g~a, for all a∈A(d)
f~>~g~ if f~a>~g~a, for all a∈ANote that the relation≤~ is a partial order on RA. The additive identity and multiplicative identity in RA are denoted by 0~ and 1~, respectively.The arithmetic operations onRA are defined as follows:
(1)f~⊕g~λ=f~λ+g~λ:λ∈A,f~!g~λ=f~λ−g~λ:λ∈A,f~⊗g~λ=f~λg~λ:λ∈A,fgλ=f~λg~λ:λ∈Aand0∉g~λ.The absolute valuef~ of f~∈RA is defined by
(2)f~λ=f~λ:λ∈A.Letd:RA×RA⟶RA∗, where df~,g~=f~−g~ for all f~,g~∈RA. Assume md:RA×RA⟶R+ is defined by mdf~,g~=maxλ∈Adf~,g~λ.Note that(1)
RA,md is a complete metric space(2)
mdf~+k~,g~+k~=mdf~,g~ for all f~,g~,k~∈RA(3)mdf~+k~,g~+l~≤mdf~,g~+mdk~,l~.(3)
mdξf~,ξg~=ξmdf,g, for all ξ∈RDefinition 3.
A sequencef~=fj~ of soft real numbers is said to be
(a)
bounded if the setfj~:j∈N of soft real numbers is bounded; i.e., if a sequence fj~ is bounded, then there are two soft real numbers g~,l~ such that g~≤~fj~≤~l~(b)
convergent to a soft real numberf0~ if, for every ε>0, there exists n0∈N such that mdfj~,f0~<ε, for all j≥j0Byℓ∞ and ℓr, we indicate the spaces of bounded and r-absolutely summable sequences of reals. Assume ωS is the classes of all sequence spaces of soft reals. If τ=τa∈R+N, where R+N is the space of positive real sequences, we introduce Nakano sequences of soft reals such as [34] and marked it by ℓSτ=v~=va~∈ωS:hμv~<∞,forsomeμ>0, where hv~=∑a=0∞mdva~,0~τa. The space ℓSτ,., where v~=infκ>0:hv~/κ≤1 and τa≥1, for all a∈N, is a Banach space. Suppose τa∈ℓ∞, one has
(4)ℓSτ=v~=va~∈ωS:hμv~<∞,forsomeμ>0=v~=va~∈ωS:infaμτa∑a=0∞mdva~,0~τa≤∑a=0∞mdμva~,0~τa<∞,forsomeμ>0=v~=va~∈ωS:∑a=0∞mdva~,0~τa<∞=v~=va~∈ωS:hμv~<∞,foranyμ>0.Lemma 4 (see [40]).
Ifτa>0 and va,ta∈R, for all a∈N, one gets va+taτa≤2K−1vaτa+taτa, where K=max1,supaτa.


## 3. Some Properties ofℓSτ

We have investigated in this section the certain space of sequences of soft real numbers under definite function to form prequasi (csss). We present sufficient conditions ofℓSτ under definite function h to construct prequasi Banach and closed (csss). The Fatou property of different prequasi norms h on ℓSτ has been explained. We have explored the uniform convexity (UUC2), the property (R), and this space’s h-normal structure property.Definition 5.
The linear spaceU is called a certain space of sequences of soft reals (csss), when
(1)
b~qq∈N⊆U, where b~q=0~,0~,⋯,1~,0~,0~,⋯, for 1~ marks at the qth place(2)
U is solid, i.e., if Y~=Yq~∈ωS, Z~=Zq~∈U, and Yq~≤~Zq~, for all q∈N, one has Y~∈U(3)
Yq/2~q=0∞∈U, where q/2 indicates the integral part of q/2, assume Yq~q=0∞∈UDefinition 6.
A subclassUh of U is said to be a premodular (csss), if one has h∈0,∞U holds the following conditions:
(i)
SupposeY~∈U, Y~=ϑ~⇔hY~=0 with hY~≥0, where ϑ~=0~,0~,0~,(ii)
We haveQ≥1, the inequality hαY~≤QαhY~ holds, for all Y~∈U and α∈R(iii)
One hasP≥1, the inequality hY~+Z~≤PhY~+hZ~ satisfies, for all Y~,Z~∈U(iv)
WhenYq~≤~Zq~, for all q∈N, we have hYq~≤hZq~(v)
The inequalityhYq~≤hYq/2~≤P0hYq~ verifies, for some P0≥1(vi)
AssumeE is the space of finite sequences of soft real numbers, one has the closure of E=Uh(vii)
We haveσ>0 with hα~,0~,0~,0~,⋯≥σαh1~,0~,0~,0~,⋯, where α~a=α, for every a∈ADefinition 7.
IfU is a (csss). The function h∈0,∞U is said to be a prequasi norm on U, if it satisfies the following settings:
(i)
SupposeY~∈U, Y~=ϑ~⇔hY~=0 with hY~≥0, where ϑ~=0~,0~,0~,(ii)
One hasQ≥1, the inequality hαY~≤QαhY~ verifies, for all Y~∈U and α∈R(iii)
We haveP≥1, the inequality hY~+Z~≤PhY~+hZ~ satisfies, for all Y~,Z~∈UEvidently, by the last two definitions, one has the following two theorems.Theorem 8.
AssumeU is a premodular (csss), then it is prequasi normed (csss).Theorem 9.
U is a prequasi normed (csss), when it is quasinormed (csss).Definition 10.
(a)
The functionh on ℓSτ is called h-convex, when
(5)hαY~+1−αZ~≤αhY~+1−αhZ~,for all α∈0,1 and Y~,Z~∈ℓSτ(b)
Yq~q∈N⊆ℓSτh is h-convergent to Y~∈ℓSτh, if and only if, limq⟶∞hYq~−Y~=0. If the h-limit exists, then it is unique(c)
Yq~q∈N⊆ℓSτh is h-Cauchy, if limq,r⟶∞hYq~−Yr~=0(d)
Γ⊂ℓSτh is h-closed, if for every h-converges Yq~a∈N⊂Γ to Y~, one has Y~∈Γ(e)
Γ⊂ℓSτh is h-bounded, assume δhΓ=suphY~−Z~:Y~,Z~∈Γ<∞(f)
Theh-ball of radius ε≥0 and center Y~, for all Y~∈ℓSτh, is denoted by
(6)BhY~,ε=Z~∈ℓSτh:hY~−Z~≤ε.(g)
A prequasi normh on ℓSτ verifies the Fatou property, if for all sequence Zq~⊆ℓSτh with limq⟶∞hZq~−Z~=0 and every Y~∈ℓSτh, we have hY~−Z~≤suprinfq≥rhY~−Zq~Recall that the Fatou property gives theh-closedness of the h-balls. We will indicate the space of all increasing sequences of reals by I.Theorem 11.
ℓSτh, where hY=∑q=0∞mdYq~,0~τq1/K, for every Y~∈ℓSτ, is a premodular (csss), if τqq∈N∈ℓ∞∩I with τ0>0.Proof.
(i) Clearly,hY~≥0 and hY~=0⇔Y~=ϑ~.
(1-i) AssumeY~,Z~∈ℓSτ. Then,
(7)hY~+Z~=∑q=0∞mdYq~+Zq~,0~τq1/K≤∑q=0∞mdYq~,0~τq1/K+∑q=0∞mdZq~,0~τq1/K=hY~+hZ~<∞.Hence, Y~+Z~∈ℓSτ.
(ii) We haveP≥1 with hY~+Z~≤PhY~+hZ~, for every Y~,Z~∈ℓSτ.
(1-ii) Supposeα∈R and Y~∈ℓSτ, one has
(8)hαY~=∑q=0∞mdαYq~,0~τq1/K≤supqατq/K∑q=0∞mdYq~,0~τq1/K≤QαhY~<∞.
SinceαY~∈ℓSτ. By parts (1-i) and (1-ii), we have ℓSτ is linear. And b~p∈ℓSτ, for every p∈N, as hb~p=∑q=0∞mdb~p,0~τq1/K=1.
(iii) One hasQ=max1,supqατq/K−1≥1 with hαY~≤QαhY~, for every Y~∈ℓSτ and α∈R.
(2) IfYq~≤Zq~, for every q∈N and Z~∈ℓSτ. Then
(9)hY~=∑q=0∞mdYq~,0~τq1/K≤∑q=0∞mdZq~,0~τq1/K=hZ~<∞,then Y~∈ℓSτ.
(iv) Evidently, from (24).
(3) AssumeYq∈ℓSτ, one has
(10)hYq/2~=∑q=0∞mdYq/2~,0~τq1/K=∑q=0∞mdYq~,0~τ2q+∑q=0∞mdYq~,0~τ2q+11/K≤21/K∑q=0∞mdYq~,0~τq1/K=21/KhYq~,so Yq/2~∈ℓSτ. (v) From (25), there are P0=21/K≥1.
(vi) Clearly the closure ofE=ℓSτ.
(vii) One gets0<σ≤ατ0/K−1, for α≠0 or σ>0, for α=0 with
(11)α~,0~,0~,0~,⋯≥σαh1~,0~,0~,0~,⋯.Theorem 12.
Assumeτqq∈N∈ℓ∞∩I with τ0>0, one has ℓSτh which is a prequasi Banach (csss), where hY~=∑q=0∞mdYq~,0~τq1/K, for all Y~∈ℓSτ.Proof.
From Theorems11 and 8, the space ℓSτh is a prequasi normed (csss). If Yl~=Yql~q=0∞ is a Cauchy sequence in ℓSτh, then for all ε∈0,1, we have l0∈N such that for every l,m≥l0, we obtain
(12)hYl~−Ym~=∑q=0∞mdYql~−Yqm~,0~τq1/K<ε.
Therefore,mdYql~−Yqm~,0~<ε. Since RA,md is a complete metric space, so Yqm~ is a Cauchy sequence in RA, for constant q∈N. Then, limm⟶∞Yqm~=Yq0~, for fixed q∈N. So hYl~−Y0~<ε, for all l≥l0. As hY0~=hY0~−Yl~+Yl~≤hYl~−Y0~+hYl~<∞. Then, Y0~∈ℓSτ.Theorem 13.
Ifτqq∈N∈ℓ∞∩I with τ0>0, we have ℓSτh a prequasi closed (csss), where hY~=∑q=0∞mdYq~,0~τq1/K, for all Y~∈ℓSτ.Proof.
By Theorems11 and 8, the space ℓSτh is a prequasi normed (csss). When Yl~=Yql~q=0∞∈ℓSτh and liml⟶∞hYl~−Y0~=0, one has for every ε∈0,1, there is l0∈N such that for every l≥l0, one gets
(13)ε>hYl~−Y0~=∑q=0∞mdYql~−Yq0~,0~τq1/K.
Therefore,mdYql~−Yq0~,0~<ε. Since RA,md is a complete metric space, so Yql~ is a convergent sequence in RA, for constant q∈N. Then, liml⟶∞Yql~=Yq0~, for fixed q∈N. As hY0~=hY0~−Yl~+Yl~≤hYl~−Y0~+hYl~<∞. We have Y0~∈ℓSτ.Theorem 14.
The functionhY~=∑q=0∞mdYq~,0~τq1/K verifies the Fatou property, when τqq∈N∈ℓ∞∩I so that τ0>0, for every Y~∈ℓSτ.Proof.
AssumeZr~⊆ℓSτh with limr⟶∞hZr~−Z~=0. As ℓSτh is a prequasi closed space, we have Z~∈ℓSτh. For every Y~∈ℓSτh, then
(14)hY~−Z~=∑q=0∞mdYq~−Zq~,0~τq1/K≤∑q=0∞mdYq~−Zqr~,0~τq1/K+∑q=0∞mdZqr~−Zq~,0~τq1/K≤supminfr≥mhY~−Zr~.Theorem 15.
The functionhY~=∑q=0∞mdYq~,0~τq does not satisfy the Fatou property, for every Y~∈ℓSτ, if τq∈ℓ∞ and τq>1, for every q∈N.Proof.
AssumeZr~⊆ℓSτh with limr⟶∞hZr~−Z~=0. As ℓSτh is a prequasi closed space, we have Z~∈ℓSτh. For all Z~∈ℓSτh, one can see
(15)hY~−Z~=∑q=0∞mdYq~−Zq~,0~τq≤2supqτq−1∑q=0∞mdYq~−Zqr~,0~τq+∑q=0∞mdZqr~−Zq~,0~τq≤2supqτq−1supminfr≥mhY~−Zr~.Example 16.
Forτq∈1,∞N, the function hY~=infα>0:∑q∈NmdYq~/α,0~τq≤1 is a norm on ℓSτ.Example 17.
The functionhY~=∑q∈NmdYq~,0~3q+2/q+13 is a prequasi norm (not a norm) on ℓS3q+2/q+1q=0∞.Example 18.
The functionhY~=∑q∈NmdYq~,0~3q+2/q+1 is a prequasi norm (not a quasinorm) on ℓS3q+2/q+1q=0∞.Example 19.
The functionhY~=∑q∈NmdYq~,0~dd is a prequasi norm, quasi norm, and not a norm on ℓdS, for 0<d<1.Definition 20.
(1)
[41] If p>0 and q>0. Mark
(16)K2p,q=Y~,Z~:Y~,Z~∈Uh,hY~≤p,hZ~≤p,hY~−Z~2≥pq.
ForK2p,q≠∅, let
(17)K2p,q=inf1−1phY~+Z~2:Y~,Z~∈K2p,q.
SupposeK2p,q=∅, we take K2p,q=1.(2)
[41] The function h holds (UUC2) when for all r≥0 and q>0, one has β2r,q such that
(18)K2p,q>β2r,q>0,forp>r.(3)
[42] The function h is strictly convex, (SC), when for every Y~,Z~∈Uh with hY~=hZ~ and hY~+Z~/2=hY~+hZ~/2, one gets Y~=Z~Lemma 21.
(i)
[43] If t≥2 and for every f,g∈R, one has
(19)f+g2t+f−g2t≤12ft+gt.(ii)
[44] Assume 1<t≤2 and for all f,g∈R with f+g≠0, one obtains
(20)f+g2t+tt−12f−gf+g2−tf−g2t≤12ft+gt.In the next part of this section, we will use the functionh as hg~=∑p=0∞mdgp~,0~τp1/K, for all g~∈ℓSτ.Theorem 22.
Ifτqq∈N∈ℓ∞∩I so that τ0>1, one has h is (UUC2).Proof.
Supposeb>0 and a>r≥0. If f~,g~∈ℓSτh with
(21)hf~≤a,hg~≤aandhf~−g~2≥ab.
By using the definition ofh, one can see
(22)ab≤hf~−g~2=∑m=0∞mdfm~−gm~2,0~τm1/K≤2−τ0/K∑m=0∞mdfm~,0~τm1/K+∑m=0∞mdgm~,0~τm1/K=2−τ0/Khf~+hg~≤2a,then b≤2. Assume Q=x∈N:1<τx<2 and P=x∈N:τx≥2=N\Q. For all w~∈ℓSτh, one has hKw~=hPKw~+hQKw~. Therefore, hPf~−g~/2≥ab/2 or hQf~−g~/2≥ab/2. Let first hPf~−g~/2≥ab/2. In view of Lemma 21, part (i), one gets
(23)hPKf~+g~2+hPKf~−g~2≤hPKf~+hPKg~2,then
(24)hPKf~+g~2≤hPKf~+hPKg~2−ab2K.
Since(25)hQKf~+g~2≤hQKf~+hQKg~2,by summing inequalities 2 and 3, and from inequality 1, one can see
(26)hKf~+g~2≤hKf~+hKg~2−ab2K≤aK1−b2K.
This implies(27)hf~+g~2≤a1−b2K1/K.
After, assumehQf~−g~/2≥ab/2. Put B=b/4K,(28)Q1=m∈Q:mdfm~−gm~,0~≤Bmdfm~,0~+mdgm~,0~andQ2=Q\Q1.
SinceB≤1 and the power function is convex. Hence,
(29)hQ1Kf~−g~2≤∑m∈Q1Bτmmdfm~+gm~2,0~τm≤B2τ0hQ1Kf~+hQ1Kg~≤B2hQKf~+hQKg~≤B2hKf~+hKg~≤BaK.
AshQf~−g~/2≥ab/2, one has
(30)hQ2Kf~−g~2=hQKf~−g~2−hQ1Kf~−g~2≥aKb2K−b4K.
For allm∈Q2, one obtains
(31)τ0−1<τ0τ0−1≤≤τm−1τm−1−1≤τmτm−1,B<B2−τm<mdfm~−gm~,0~mdfm~,0~+mdgm~,0~2−τm.
In view of Lemma21, part (ii), one gets
(32)mdfm~+gm~2,0~τm+τ0−1B2mdfm~−gm~2,0~τm≤12mdfm~,0~τm+mdgm~,0~τm.
So(33)hQ2Kf~+g~2+τ0−1B2hQ2Kf~−g~2≤hQ2Kf~+hQ2Kg~2,then
(34)hQ2Kf~+g~2≤hQ2Kf~+hQ2Kg~2−τ0−1B2aKb2K−b4K.
As(35)hQ1Kf~+g~2≤hQ1Kf~+hQ1Kg~2,by summing inequalities 5 and 6, we have
(36)hQKf~+g~2≤hQKf~+hQKg~2−τ0−1B2aKb2K−b4K≤hQKf~+hQKg~2−τ0−12b42KaK2K−1≤hQKf~+hQKg~2−τ0−12K−1b42KaK.
As(37)hPKf~+g~2≤hPKf~+hPKg~2,by summing inequalities 7 and 8, and from inequality 1, then
(38)hKf~+g~2≤hKf~+hKg~2−τ0−12K−1b42KaK≤aK1−τ0−12K−1b42K.
So(39)hf~+g~2≤a1−τ0−12K−1b42K1/K.
Evidently,(40)1<τ0≤K<2K⇒0<τ0−12K−1<1.
From inequalities 4 and 9, and Definition20, when we take
(41)β2r,b=min1−1−b2K1/K,1−1−τ0−12K−1b42K1/K.
Therefore, we haveK2a,b>β2r,b>0, so h is (UUC2).Definition 23.
The spaceUh verifies the property (R), if and only if, for every decreasing sequence Γjj∈N of h-closed and h-convex nonempty subsets of Uh so that supj∈NKhY~,Γj<∞, for some Y~∈Uh, then ⋂j∈NΓj≠∅.By denotingΓ a nonempty h-closed and h-convex subset of ℓSτh.Theorem 24.
Supposeτqq∈N∈ℓ∞∩I so that τ0>1, we have
(i)
ifY~∈ℓSτh such that
(42)KhY~,Γ=infhY~−Z~:Z~∈Γ<∞.
One has a uniqueα~∈Γ with KhY~,Γ=hY~−α~.(ii)
ℓSτh satisfies the property (R).Proof.
To prove (i), ifY~∉Γ as Γ is h-closed, we have C≔KhY~,Γ>0. Then, for every r∈N, we have Zr~∈Γ so that hY~−Zr~<C1+1/r. Assume Zr~/2 is not h-Cauchy. There is a subsequence Zgr~/2 and l0>0 so that hZgr~−Zgj~/2≥l0, for all r>j≥0. Also, we obtain K2C1+1/r,l0/2C>α≔β2C1+1/r,l0/2C>0, for every r∈N. As
(43)maxhY~−Zgr~,hY~−Zgj~≤C1+1gj,hZgr~−Zgj~2≥l0≥C1+1gjl02C,for all r>j≥0, one has
(44)hY~−Zgr~+Zgj~2≤C1+1gj1−α.
So(45)C=KhY~,Γ≤C1+1gj1−α,for every j∈N. By choosing j⟶∞, we have
(46)0<C≤C1+1gj1−α<C.This is a contradiction. Hence, Zr~/2 is h-Cauchy. Since ℓSτh is h-complete, one has Zr~/2h-converges to some Z~. For every j∈N, we have Zr~+Zj~/2h-converges to Z~+Zj~/2. As Γ is h-closed and h-convex, we have Z~+Zj~/2∈Γ. As Z~+Zj~/2h-converges to 2Z~, one gets 2Z~∈Γ. Suppose λ~=2z~ and from Theorem 14, as h verifies the Fatou property, we get
(47)KhY~,Γ≤hY~−λ~≤supiinfj≥ihY~−Z~+Zj~2≤supiinfj≥isupiinfr≥ihY~−Zr~+Zj~2≤12supiinfr≥isupiinfr≥ihY~−Zr~+hY~−Zj~=KhY~,Γ.
SohY~−λ~=KhY~,Γ. As h is (UUC2), then it is (SC), which explains the uniqueness of λ~. To prove (ii), if Y~∉Γr0, for some r0∈N. As KhY~,Γrr∈N∈ℓ∞ is increasing, take limr⟶∞KhY~,Γr=C. If C>0, otherwise, Y~∈Γr, for every r∈N. From (i), one has one point Zr~∈Γr so that KhY~,Γr=hY~−Zr~, for all r∈N. A similar proof will show that Zr~/2h-converges to some Z~∈ℓSτh. Since Γr are h-convex, decreasing, and h-closed, we have 2Z~∈∩r∈NΓr.Definition 25.
Uh verifies the h-normal structure property, if and only if, for every nonempty h-bounded, h-convex, and h-closed subset Γ of Uh not decreased to one point, then Y~∈Γ so that
(48)supZ~∈ΓhY~−Z~<δhΓ≔suphY~−Z~:Y~,Z~∈Γ<∞.Theorem 26.
Supposeτqq∈N∈ℓ∞∩I so that τ0>1, then ℓSτh satisfies the h-normal structure property.Proof.
Theorem22 implies that h is (UUC2). Suppose Γ is a h-bounded, h-convex, and h-closed subset of ℓSτh not decreased to one point. Then, δhΓ>0. Put C=δhΓ. If Y~,Z~∈Γ with Y~≠Z~, then hY~−Z~/2=l>0. For all α~∈Γ, we have hY~−α~≤C and hZ~−α~≤C. Since Γ is h-convex, we have Y~+Z~/2∈Γ. Since
(49)hY~+Z~2−α~=hY~−α~+Z~−α~2≤C1−K2C,lC,for every α~∈Γ. We get
(50)supα~∈ΓhY~+Z~2−α~≤C1−K2C,lC<C=δhΓ.


## 4. Kannan Contraction Mapping onℓSτ

In this section, we have constructedℓSτh with distinct h so that one has a unique fixed point of Kannan contraction mapping.Definition 27.
A mappingV:Uh⟶Uh is called a Kannan h-contraction, when we have α∈0,1/2 so that hVY~−VZ~≤αhVY~−Y~+hVZ~−Z~, for every Y~,Z~∈Uh. The mapping V is said to be Kannan h-nonexpansive, if α=1/2.A vectorY~∈Uh is said to be a fixed point of V, if VY~=Y~.Theorem 28.
Supposeτqq∈N∈ℓ∞∩I so that τ0>0 and V:ℓSτh⟶ℓSτh is Kannan h-contraction mapping, where hY~=∑q=0∞mdYq~,0~τq1/K, for every Y~∈ℓSτ, then V has a unique fixed point.Proof.
LetY~∈ℓSτ, we have VpY~∈ℓSτ. Since V is a Kannan h-contraction mapping, then
(51)hVl+1Y~−VlY~≤αhVl+1Y~−VlY~+hVlY~−Vl−1Y~⟹hVl+1Y~−VlY~≤α1−αhVlY~−Vl−1Y~≤α1−α2hVl−1Y~−Vl−2Y~≤⋯≤α1−αlhVY~−Y~.
Hence, for everyl,m∈N so that m>l, then
(52)hVlY~−VmY~≤αhVlY~−Vl−1Y~+hVmY~−Vm−1Y~≤αα1−αl−1+α1−αm−1hVY~−Y~.
Therefore,VlY~ is a Cauchy sequence in ℓSτh. Since the space ℓSτh is prequasi Banach space, we have Z~∈ℓSτh so that liml⟶∞VlY~=Z~. To show that VZ~=Z~, as h holds the Fatou property, we get
(53)hVZ~−Z~≤supiinfl≥ihVl+1Y~−VlY~≤supiinfl≥iα1−αlhVY~−Y~=0,so VZ~=Z~. Hence, Z~ is a fixed point of V. To prove the uniqueness, assume Y~,Z~∈ℓSτh are two not equal fixed points of V. Then,
(54)hY~−Z~≤hVY~−VZ~≤αhVY~−Y~+hVZ~−Z~=0.
Hence,Y~=Z~.Corollary 29.
Assumeτqq∈N∈ℓ∞∩I so that τ0>1, and V:ℓSτh⟶ℓSτh is Kannan h-contraction mapping, where hY~=∑q=0∞mdYq~,0~τq1/K, for every Y~∈ℓSτ, then V has unique fixed point Z~ with hVlY~−Z~≤αα/1−αl−1hVY~−Y~.Proof.
By Theorem28, we have a unique fixed point Z~ of V. Then,
(55)hVlY~−Z~=hVlY~−VZ~≤αhVlY~−Vl−1Y~+hVZ~−Z~=αα1−αl−1hVY~−Y~.Example 30.
IfV:ℓS2q+3/q+2q=0∞h⟶ℓS2q+3/q+2q=0∞h, where hg~=∑q=0∞mdgq~,0~2q+3/q+2, for all g~∈ℓS2q+3/q+2q=0∞ and
(56)Vg~=g~4,hg~∈0,1,g~5,hg~∈1,∞.
Since for allg1~,g2~∈ℓS2q+3/q+2q=0∞h so that hg1~,hg2~∈0,1, we have
(57)hVg1~−Vg2~=hg1~4−g2~4≤1274h3g1~4+h3g2~4=1274hVg1~−g1~+hVg2~−g2~.
For everyg1~,g2~∈ℓS2q+3/q+2q=0∞h with hg1~,hg2~∈1,∞, we get
(58)hVg1~−Vg2~=hg1~5−g2~5≤1644h4g1~5+h4g2~5=1644hVg1~−g1~+hVg2~−g2~.
For eachg1~,g2~∈ℓS2q+3/q+2q=0∞h with hg1~∈0,1 and hg2~∈1,∞, one has
(59)hVg1~−Vg2~=hg1~4−g2~5≤1274h3g1~4+1644h4g2~5≤1274h3g1~4+h4g2~5=1274hVg1~−g1~+hVg2~−g2~.
Therefore,V is Kannan h-contraction. Since h holds the Fatou property, by Theorem 28, we have V that holds unique fixed point ϑ~∈ℓS2q+3/q+2q=0∞h.Definition 31.
IfUh is a prequasi normed (csss), V:Uh⟶Uh and Z~∈Uh. The mapping V is said to be h-sequentially continuous at Z~, if and only if, assume limq⟶∞hYq~−Z~=0, one has limq⟶∞hVYq~−VZ~=0.Example 32.
IfV:ℓSq+1/2q+4q=0∞h⟶ℓSq+1/2q+4q=0∞h, where hZ~=∑q=0∞mdZq~,0~q+1/2q+4, for all Z~∈ℓSq+1/2q+4q=0∞ and
(60)VZ~=118b~0+Z~,Z0~a∈0,117,117b~0,Z0~a=117,118b~0,Z0~a∈117,1.
V is obviously both h-sequentially continuous and discontinuous at 1/17b~0∈ℓSq+1/2q+4q=0∞h.Example 33.
SupposeV is defined as in Example 30. If Zn~⊆ℓS2q+3/q+2q=0∞h with limn⟶∞hZn~−Z0~=0, where Z0~∈ℓS2q+3/q+2q=0∞h so that hZ0~=1.
Since the prequasi normh is continuous, one obtains
(61)limn⟶∞hVZn~−VZ0~=limn⟶∞hZn~4−Z0~5=hZ0~20>0.
Hence,V is not h-sequentially continuous at Z0~.Theorem 34.
Assumeτqq∈N∈ℓ∞∩I so that τ0>1, and V:ℓSτh⟶ℓSτh, where hY~=∑q=0∞mdYq~,0~τq, for every Y~∈ℓSτ. If
(1)
V is Kannan h-contraction mapping(2)
V is h-sequentially continuous at Z~∈ℓSτh(3)
One hasY~∈ℓSτh so that VlY~ has VljY~ converging to Z~
Then,Z~∈ℓSτh is the only fixed point of V.Proof.
SupposeZ~ is not a fixed point of V, we have VZ~≠Z~. By using conditions (24) and (25), one has
(62)limlj⟶∞hVljY~−Z~=0,limlj⟶∞hVlj+1Y~−VZ~=0.
SinceV is Kannan h-contraction, then
(63)0<hVZ~−Z~=hVZ~−Vlj+1Y~+VljY~−Z~+Vlj+1Y~−VljY~≤22supiτi−2hVlj+1Y~−VZ~+22supiτi−2hVljY~−Z~+2supiτi−1αα1−αlj−1hVY~−Y~.
Sincelj⟶∞, this gives a contradiction. So Z~ is a fixed point of V. To prove the uniqueness, assume Z~,Y~∈ℓSτh is two not equal fixed points of V. We have
(64)hZ~−Y~≤hVZ~−VY~≤αhVZ~−Z~+hVY~−Y~=0.
Therefore,Z~=Y~.Example 35.
IfV is defined as in Example 30. Suppose hY~=∑q∈NmdYq~,0~2q+3/q+2, for every Y~∈ℓS2q+3/q+2q=0∞. As for every Y1~,Y2~∈ℓS2a+3/q+2q=0∞h so that hY1~,hY2~∈0,1, we have
(65)hVY1~−VY2~=hY1~4−Y2~4≤227h3Y1~4+h3Y2~4=227hVY1~−Y1~+hVY2~−Y2~.
For everyY1~,Y2~∈ℓS2q+3/q+2q=0∞h such that hY1~,hY2~∈1,∞, then
(66)hVY1~−VY2~=hY1~5−Y2~5≤14h4Y1~5+h4Y2~5=14hVY1~−Y1~+hVY2~−Y2~.
For everyY1~,Y2~∈ℓS2q+3/q+2q=0∞h with hY1~∈0,1 and hY2~∈1,∞, we have
(67)hVY1~−VY2~=hY1~4−Y2~5≤227h3Y1~4+14h4Y2~5≤227h3Y1~4+h4Y2~5=227hVY1~−Y1~+hVY2~−Y2~.
Then,V is Kannan h-contraction and
(68)VlY~=Y~4l,hY~∈0,1,Y~5l,hY~∈1,∞.Clearly,V is h-sequentially continuous at ϑ~∈ℓS2a+3/q+2q=0∞h and VlY~ verifies VljY~ converges to ϑ~. From Theorem 34, the element ϑ~∈ℓS2a+3/q+2q=0∞h is the only fixed point of V.


## 5. Kannan Nonexpansive Mapping onℓSτh

The enough setups ofℓSτh, where hg~=∑p=0∞mdgp~,0~τp1/K, for all g~∈ℓSτ, so that the Kannan nonexpansive mapping on it has a fixed point are presented.By lettingΓ a nonempty h-bounded, h-convex, and h-closed subset of ℓSτh.Lemma 36.
SupposeℓSτh verifies the (R) property and the h-quasinormal property. If V:Γ⟶Γ is a Kannan h-nonexpansive mapping, for t>0, put Gt=Y~∈Γ:hY~−VY~≤t≠∅. Let
(69)Γt=⋂Bhr,j:VGt⊂Bhr,j∩Γ.
Hence,Γt≠∅, h-convex, h-closed subset of Γ and VΓt⊂Γt⊂GtandδhΓt≤t.Proof.
AsVGt⊂Γt, one has Γt≠∅. Since the h-balls are h-convex and h-closed, one gets Γt is a h-closed and h-convex subset of Γ. To prove that Γt⊂Gt, let Y~∈Γt. If hY~−VY~=0, we have Y~∈Gt. Otherwise, when hY~−VY~>0, let
(70)r=suphVZ~−VY~:Z~∈Gt.
From the definition ofr, we have VGt⊂BhVY~,r. Hence, Γt⊂BhVY~,r, so hY~−VY~≤r. By taking l>0, we have Z~∈Gt so that r−l≤hVZ~−VY~. Then,
(71)hY~−VY~−l≤r−l≤hVZ~−VY~≤12hY~−VY~+hZ~−VZ~≤12hY~−VY~+t.
Sincel is an arbitrary positive, we have hY~−VY~≤t, so Y~∈Gt. As VGt⊂Γt, we have VΓt⊂VGt⊂Γt, then Γt is V-invariant. To prove that δhΓt≤t. As
(72)hVY~−VZ~≤12hY~−VY~+hZ~−VZ~,for every Y~,Z~∈Gt. If Y~∈Gt. We get VGt⊂BhVY~,t. The definition of Γt implies Γt⊂BhVY~,t. Hence, VY~∈⋂t∈ΓtBhZ~,t. Then, hZ~−Y~≤t, for all Z~,Y~∈Γt, this implies δhΓt≤t.Theorem 37.
AssumeℓSτh verifies the h-quasinormal property and the (R) property. If V:Γ⟶Γ is a Kannan h-nonexpansive mapping, so V has a fixed point.Proof.
Putt0=infhY~−VY~:Y~∈Γ and tr=t0+1/r, for all r≥1. By the definition of t0, we have Gtr=Y~∈Γ:hY~−VY~≤tr≠∅, for all r≥1. If Γtr is defined as in Lemma 36, it is obvious that Γtr is a decreasing sequence of nonempty h-bounded, h-closed, and h-convex subsets of Γ. The property (R) holds that Γ∞=⋂r≥1Γtr≠∅. Put Y~∈Γ∞, then hY~−VY~≤tr, for every r≥1. If r⟶∞, one has hY~−VY~≤t0, then hY~−VY~=t0. Hence, Gt0≠∅. Therefore, t0=0. Otherwise, t0>0 then V fails to have a fixed point. Put Γt0 as defined in Lemma 36. Since V fails to have a fixed point and Γt0 is V-invariant, so Γt0 has more than one point, then δhΓt0>0. By the h-quasinormal property, we have Y~∈Γt0 so that
(73)hY~−Z~<δhΓt0≤t0,for every Z~∈Γt0. In view of Lemma 36, one has Γt0⊂Gt0. By definition of Γt0, then VY~∈Gt0⊂Γt0. We have
(74)hY~−VY~<δhΓt0≤t0,which contradicts the definition of t0. So t0=0 which gives that any point in Gt0 is a fixed point of V.In view of Theorems24, 26, and 28, we have the following.Corollary 38.
Ifτqq∈N∈ℓ∞∩I so that τ0>1 and V:Γ⟶Γ is a Kannan h-nonexpansive mapping. One has V that holds a fixed point.Example 39.
SupposeV:Γ⟶Γ so that
(75)VY~=Y~4,hY~∈0,1,Y~5,hY~∈1,∞,where Γ=Y~∈ℓS2q+3/q+2q=0∞h:Y0~=Y1~=0~ and hY~=∑q∈NmdYq~,0~2q+3/q+2, for all Y~∈ℓS2q+3/q+2q=0∞h. From Example 35, V is Kannan h-contraction. Therefore, it is Kannan h-nonexpansive. From Corollary 38, then V has a fixed point ϑ~ in Γ.


## 6. Kannan Contraction and Structure of Operators Ideal

The structure of the operators ideal byℓSτh under definite function h, where hg~=∑p=0∞mdgp~,0~τp1/K, for all g~∈ℓSτ, and s-soft reals has been offered. Finally, we study the idea of Kannan contraction mapping in its linked prequasi operator ideal. Also, the existence of a fixed point of Kannan contraction mapping has been offered. We mark 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 indicate LΔ and FΔ.Definition 40 (see [45]).
Ans-number function is s:LΔ,Λ⟶R+N which gives all V∈LΔ,Λ a sdVd=0∞ holds the next conditions:
(a)
V=s0V≥s1V≥s2V≥⋯≥0, for every V∈LΔ,Λ(b)
sl+d−1V1+V2≤slV1+sdV2, for every 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)
SupposeV∈LΔ,Λ and γ∈R, one has sdγV=γsdV(e)
IfrankV≤d, then sdV=0, for all V∈LΔ,Λ(f)
sl≥aIa=0 or sl<aIa=1, where Ia marks the unit map on the a-dimensional Hilbert space ℓ2aDefinition 41 (see [37]).
SupposeL is the class of all bounded linear operators between any arbitrary Banach spaces. A subclass U of L is called an operator ideal, when every UΔ,Λ=U∩LΔ,Λ holds the next setups:
(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, one has YXW∈UΔ0,Λ0Notations 42.
(76)✠~U≔✠~UΔ,Λ,where✠~UΔ,Λ≔V∈LΔ,Λ:sdV~d=0∞∈U.
sdV~x=sdV, for every x∈A.Theorem 43.
IfU is a (csss), one has ✠~U an operator ideal.Proof.
(i)
SupposeV∈FΔ,Λ and rankV=n, for every n∈N, since b~i∈U, for every i∈N, and U is a linear space, then siV~i=0∞=s0V~,s1V~,⋯,sn−1V~,0~,0~,0~,⋯=∑i=0n−1siV~b~i∈U; for that V∈✠~UΔ,Λ then FΔ,Λ⊆✠~UΔ,Λ(ii)
IfV1,V2∈✠~UΔ,Λ and β1,β2∈R so by Definition 5 condition (25) one has si/2V1~i=0∞∈U and si/2V1~i=0∞∈U, as i≥2i/2, by the definition of s-numbers and siV is decreasing, we have siβ1V1+β2V2~≤s2i/2β1V1+β2V2~≤si/2β1V1+si/2β2V2~=β1si/2V1~+β2si/2V2~ for all i∈N. By Definition 5 part (2) and U is a linear space, we get siβ1V1+β2V2~i=0∞∈U; hence, β1V1+β2V2∈✠~UΔ,Λ(iii)
AssumeP∈LΔ0,Δ, T∈✠~UΔ,Λ, and R∈LΛ,Λ0, then siT~i=0∞∈U and since siRTP~≤RsiT~P, from Definition 5 parts (1) and (2), then siRTP~i=0∞∈U, then RTP∈✠~UΔ0,Λ0In view of Theorems11 and 43, we have the following theorem.Theorem 44.
Ifτqq∈N∈ℓ∞∩I so that τ0>0, then ✠~ℓSτh is an operator ideal.Definition 45 [38].
A functionH∈0,∞U is said to be a prequasi norm on the ideal U, when the next setups are verified.
(1)
IfV∈UΔ,Λ, HV≥0, and HV=0, if and only if, V=0(2)
One hasQ≥1 so as to HαV≤DαHV, for all V∈UΔ,Λ and α∈R(3)
One hasP≥1 with HV1+V2≤PHV1+HV2, for all V1,V2∈UΔ,Λ(4)
One hasσ≥1 for to if V∈LΔ0,Δ, X∈UΔ,Λ, and Y∈LΛ,Λ0, one has HYXV≤σYHXVTheorem 46 (see [38]).
H is a prequasi norm on the ideal U, whenever H is a quasinorm on the ideal U.Theorem 47.
Supposeτqq∈N∈ℓ∞∩I so that τ0>0; hence, the function H is a prequasi norm on ✠~ℓSτh, with HZ=hsqZ~q=0∞, for every Z∈✠~ℓSτhΔ,Λ.Proof.
(1)
IfX∈✠~ℓSτhΔ,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)
One hasQ≥1 with HαX=hsqαX~q=0∞≤QαHX, for every X∈✠~ℓSτhΔ,Λ and α∈R(3)
There arePP0≥1 with for X1,X2∈✠~ℓSτhΔ,Λ, we have
(77)HX1+X2=hsqX1+X2~q=0∞≤Phsq/2X1~q=0∞+hsq/2X2~q=0∞≤PP0hsqX1~q=0∞+hsqX2~q=0∞.(4)
There areρ≥1, assume X∈LΔ0,Δ, Y∈✠~ℓSτhΔ,Λ and Z∈LΛ,Λ0, one has HZYX=hsqZYX~q=0∞≤hXZsqY~q=0∞≤ρXHYZIn the next theorems, we will use the notation✠~ℓSτh,H, where HV=hsqV~q=0∞, for every V∈✠~ℓSτh.Theorem 48.
Ifτqq∈N∈ℓ∞∩I so that τ0>0, then ✠~ℓSτh,H is a prequasi Banach operator ideal.Proof.
AssumeVaa∈N is a Cauchy sequence in ✠~ℓSτhΔ,Λ. Since LΔ,Λ⊇SℓSτhΔ,Λ, we have
(78)HVr−Va=hsqVr−Va~q=0∞≥hs0Vr−Va~,0~,0~,0~,⋯≥Vr−Vaτ0/K.Then, Vaa∈N is a Cauchy sequence in LΔ,Λ. Since LΔ,Λ is a Banach space, one has V∈LΔ,Λ with lima⟶∞Va−V=0 and as sqVa~q=0∞∈ℓSτh, for every a∈N and ℓSτh is a premodular (csss). Then,
(79)HV=hsqV~q=0∞≤hsq/2V−Va~q=0∞+hsq/2~Vaq=0∞≤hVa−V1~q=0∞+21/KhsqVa~q=0∞<ε,one gets sqV~q=0∞∈ℓSτh, then V∈✠~ℓSτhΔ,Λ.Theorem 49.
Assumeτqq∈N∈ℓ∞∩I so that τ0>0, then ✠~ℓSτh,H is a prequasi closed operator ideal.Proof.
IfVa∈✠~ℓSτhΔ,Λ, for every a∈N and lima⟶∞HVa−V=0. Since LΔ,Λ⊇SℓSτhΔ,Λ, we have
(80)HVa−V=hsqVa−V~q=0∞≥hs0Va−V~,0~,0~,0~,⋯≥Va−Vτ0/K.
Hence,Vaa∈N is convergent in LΔ,Λ; i.e., lima⟶∞Va−V=0 and as sqVa~q=0∞∈ℓSτh, for every q∈N and ℓSτh is a premodular (csss). Then,
(81)HV=hsqV~q=0∞≤hsq/2V−Va~q=0∞+hsq/2~Vaq=0∞≤hVa−V1~q=0∞+21/KhsqVa~q=0∞<ε.We obtain sqV~q=0∞∈ℓSτh; hence, V∈✠~ℓSτhΔ,Λ.Definition 50.
A prequasi normH on the ideal ✠~Uh holds the Fatou property if for all Tqq∈N⊆✠~UhΔ,Λ with limq⟶∞HTq−T=0 and M∈✠~UhΔ,Λ, then
(82)HM−T≤supqinfj≥qHM−Tj.Theorem 51.
Ifτqq∈N∈ℓ∞∩I so that τ0>0, then ✠~ℓSτh,H does not satisfy the Fatou property.Proof.
LetTqq∈N⊆✠~ℓSτhΔ,Λ so that limq⟶∞HTq−T=0. As ✠~ℓSτh is a prequasi closed ideal, one has T∈✠~ℓSτhΔ,Λ. So for all M∈✠~ℓSτhΔ,Λ, then
(83)HM−T=∑q=0∞mdsqM−T~,0~τq1/K≤∑q=0∞mdsq/2M−Ti~,0~τq1/K+∑q=0∞mdsq/2Ti−T~,0~τq1/K≤21/Ksuprinfj≥r∑q=0∞mdsqM−Tj~,0~τq1/K.Definition 52.
An operatorV:✠~UhΔ,Λ⟶✠~UhΔ,Λ is called a Kannan H-contraction, if there is α∈0,1/2 so that HVT−VM≤αHVT−T+HVM−M, for every T,M∈✠~UhΔ,Λ.Definition 53.
An operatorV:✠~UhΔ,Λ⟶✠~UhΔ,Λ is called H-sequentially continuous at M, where M∈✠~UhΔ,Λ, if and only if, limr⟶∞HTr−M=0⇒limr⟶∞HVTr−VM=0.Example 54.
AssumeV:✠~ℓS2q+3/q+2q=0∞hΔ,Λ⟶✠~ℓS2q+3/q+2q=0∞hΔ,Λ, where HT=∑q=0∞mdsqT~,0~2q+3/q+2, for all T∈✠~ℓS2q+3/q+2q=0∞hΔ,Λ and
(84)VT=T6,HT∈0,1,T7,HT∈1,∞.
Clearly,V is H-sequentially continuous at the zero operator Θ∈✠~ℓS2q+3/q+2q=0∞h. Suppose Tj⊆✠~ℓS2q+3/q+2q=0∞h so that limj⟶∞HTj−T0=0, where T0∈✠~ℓS2q+3/q+2q=0∞h with HT0=1. As the prequasi norm H is continuous, we have
(85)limj⟶∞HVTj−VT0=limj⟶∞HT06−T07=HT042>0.
Hence,V is not H-sequentially continuous at T0.Theorem 55.
Ifτqq∈N∈ℓ∞∩I so that τ0>0 and V:✠~ℓSτhΔ,Λ⟶✠~ℓSτhΔ,Λ. Suppose
(i)
V is Kannan H-contraction mapping(ii)
V is H-sequentially continuous at a vector M∈✠~ℓSτhΔ,Λ(iii)
we haveG∈✠~ℓSτhΔ,Λ so that the sequence of iterates VrG has a VriG converging to MThen,M∈✠~ℓSτhΔ,Λ is the unique fixed point of V.Proof.
AssumeM is not a fixed point of V, one has VM≠M. By using conditions (ii) and (iii), one has
(86)limri⟶∞HVriG−M=0andlimri⟶∞HVri+1G−VM=0.
AsV is Kannan H-contraction, we get
(87)0<HVM−M=HVM−Vri+1G+VriG−M+Vri+1G−VriG≤21/KHVri+1G−VM+22/KHVriG−M+22/Kαα1−αri−1HVG−G.
Sinceri⟶∞, we have a contradiction. Therefore, M is a fixed point of V. To prove the uniqueness of the fixed point M, assume there are two not equal fixed points M,J∈✠~ℓSτhΔ,Λ of V. We get HM−J≤HVM−VJ≤αHVM−M+HVJ−J=0. So, M=J.Example 56.
According to Example54, as for every T1,T2∈✠~ℓS2q+3/q+2q=0∞h with HT1,HT2∈0,1, then
(88)HVT1−VT2=HT16−T26≤21254H5T16+H5T26=21254HVT1−T1+HVT2−T2.
For everyT1,T2∈✠~ℓS2q+3/q+2q=0∞h with HT1,HT2∈1,∞, then
(89)HVT1−VT2=HT17−T27≤22164H6T17+H6T27=22164HVT1−T1+HVT2−T2.
For eachT1,T2∈✠~ℓS2q+3/q+2q=0∞h with HT1∈0,1 and HT2∈1,∞, one gets
(90)HVT1−VT2=HT16−T27≤21254H5T16+22164H6T27≤21254HVT1−T1+HVT2−T2.
Therefore,V is Kannan H-contraction and
(91)VrT=T6r,HT∈0,1,T7r,HT∈1,∞.Clearly,V is H-sequentially continuous at Θ∈✠~ℓS2q+3/q+2q=0∞h and VrT has a subsequence VriT that converges to Θ. According to Theorem 55, Θ is the only fixed point of G.


## 7. Applications

In this section, some successful applications to the existence of solutions of nonlinear difference equations of soft functions are introduced.Theorem 57.
Assume the summable equations(92)Yq=Rq+∑r=0∞Dq,rmr,Yr,which are considered by many authors [46–48], and let V:ℓSτh⟶ℓSτh, where τqq∈N∈ℓ∞∩I with τ0>1 and hY~=∑j=0∞mdYj~,0~τj1/K, for all Y~∈ℓSτ, given by
(93)VYq~q∈N=Rq~+∑r=0∞Dq,rvr,Yr~q∈N.
The summable equation (92) has a unique solution in ℓSτh, when D:N2⟶R,v:N×RA⟶RA,R~:N⟶RA,Z~:N⟶RA, and for all q∈N, suppose
(94)∑r∈NDq,rvr,Yr~−vr,Zr~≤~12KRq~−Yq~+∑r=0∞Dq,rvr,Yr~+Rq~−Zq~+∑r=0∞Dq,rvr,Zr~.Proof.
We have(95)hVY~−VZ~=∑q∈NmdVYq~−VZq~,0~τq1/K=∑q∈Nmd∑r∈NDq,rvr,Yr~−vr,Zr~,0~τq1/K≤12∑q∈NmdRq~−Yq~+∑r=0∞Dq,rvr,Yr~,0~τq1/K+12∑q∈NmdRq~−Zq~+∑r=0∞Dq,rvr,Zr~,0~τq1/K=12hVY~−Y~+hVZ~−Z~.
In view of Theorem37, there is a unique solution of equation (92) in ℓSτh.Example 58.
IfℓS2q+3/q+2q=0∞h, where hY~=∑q∈NmdYq,0~2q+3/q+2, for every Y~∈ℓS2q+3/q+2q=0∞. Assume the summable equations
(96)Yq~=Rq~+∑r=0∞−1q+rYq~q2+r2+1t,so that q≥2 and t>0. Let Γ=Y~∈ℓS2q+3/q+2q=0∞h:Y0~=Y1~=0~. Clearly, Γ is a nonempty, h-convex, h-closed, and h-bounded subset of ℓS2q+3/q+2q=0∞h. Suppose V:Γ⟶Γ is defined as
(97)VYq~q≥2=Rq~+∑r=0∞−1q+rYq~q2+r2+1tq≥2.
Evidently,(98)∑r=0∞−1qYq~q2+r2+1t−1r−−1r≤~14Rq~−Yq~+∑r=0∞−1q+rYq~q2+r2+1t+Rq~−Zq~+∑r=0∞−1q+rZq~q2+r2+1t.
According to Theorem57 and Corollary 38, the summable equations (96) have a solution in Γ.Example 59.
IfℓS2q+3/q+2q=0∞h, where hY~=∑q∈NmdYq,0~2q+3/q+2, for every Y~∈ℓS2q+3/q+2q=0∞, let the nonlinear difference equations
(99)Yq~=Rq~+∑l=0∞−1q+lYq−2r~Yq−1q~+l2~+1~,so that r,q>0, Y−2~x,Y−1~x>0, for every x∈A, and suppose V:ℓS2q+3/q+2q=0∞⟶ℓS2q+3/q+2q=0∞, explained by
(100)VYqq=0∞=Rq~+∑l=0∞−1q+lYq−2r~Yq−1q~+l2~+1~q=0∞.
It is clear that(101)∑l=0∞−1qYq−2r~Yq−1q~+l2~+1~−1l−−1l≤14Rq~−Yq~+∑l=0∞−1q+lYq−2r~Yq−1q~+l2~+1~+Rq~−Zq~+∑l=0∞−1q+lZq−2r~Zq−1q~+l2~+1~.
In view of Theorem57, the nonlinear difference equations (99) have a unique solution in ℓS2q+3/q+2q=0∞.


## 8. Conclusion

The site we discussed was a “pre-quasinormed” place rather than a “quasinormed” location. In the prequasi Banach space, the concept of a fixed point of the Kannan prequasi norm contraction mapping is introduced (csss). Both (R) and the pre-quasinormal structure are supported. The occurrence of a fixed point in the Kannan nonexpansive mapping was studied in this study. A fixed point of Kannan contraction mapping in the prequasi Banach operator ideal formed by Nakano (csss) and the s-soft real numbers has also been investigated for a fixed point of Kannan contraction mapping. Finally, we have demonstrated how the results can be applied to a problem by presenting a few examples of how this has happened. Under a wide range of flexible conditions, the presence of a sequence can be established using the Nakano sequence space. Specifically, when it comes to the variable exponent in the previously described space, our key conclusions have helped to strengthen several well-established ideas.



---

*Source: 2307519-2022-05-27.xml*