# Kannan Nonexpansive Mappings on Variable Exponent Function Space of Complex Variables with Some Applications

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

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## Abstract

This paper establishes the reality of a fixed point of Kannan’s prequasinorm contraction mapping on the variable exponent function space of complex variables, demonstrating that it satisfies the property (R) and possesses a prequasinormal structure. We have established the presence of a fixed point of Kannan prequasinorm nonexpansive mapping on it and Kannan prequasinorm contraction mapping in the prequasi-Banach operator ideal, created by this function space and s-numbers. Finally, we provide some applications for solutions to summable equations and illustrate instances to corroborate our findings.


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## Body

## 1. Introduction

Many mathematicians have worked on feasible extensions to the Banach fixed point theorem since the publication of book [1] on the Banach fixed point theorem. Kannan [2] approved an instance of a class of operators that perform the same fixed point operations as contractions but with a continuous flop. Ghoncheh [3] made the first attempt to characterize Kannan operators in modular vector spaces. The variable exponent Lebesgue spaces Lr contain Nakano sequence spaces. Throughout the second half of the twentieth century, it was assumed that these variable exponent spaces provided an acceptable framework for the mathematical components of several problems for which the conventional Lebesgue spaces were insufficient. Due to the relevance of these spaces and their effects, they have become a well-known and efficient tool for solving a variety of problems; nowadays, the area of LrΩ spaces is a burgeoning area of research, with ramifications extending into a wide variety of mathematical specialties [4]. The study of variable exponent Lebesgue spaces Lr received additional impetus from the mathematical description of non-Newtonian fluid hydrodynamics [5, 6]. Non-Newtonian fluids, also known as electrorheological fluids, have various applications ranging from military science to civil engineering and orthopedics. Operator ideal theory has a variety of applications in Banach space geometry, fixed point theory, spectral theory, and other branches of mathematics, among other branches of knowledge; for more details, see [7–13]. Bakery and Mohamed [14] 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 the existence of a fixed point for Kannan prequasinorm contraction mappings on it and the prequasi-Banach operator ideal generated from s-numbers belonging to this sequence space. Also, in [15], they found some fixed points results of Kannan nonexpansive mappings on generalized Cesàro backward difference sequence space of nonabsolute type. For more recent developments in contractive mappings and the existence of fixed points of nonlinear operators in various Banach spaces, Nguyen and Tram [16] examined various fixed point results with applications to involution mappings. Dehici and Redjel [17] introduced some fixed point results for nonexpansive mappings in Banach spaces. Benavides and Ramírez [18] presented some fixed points for multivalued nonexpansive mappings.We denote the set of complex numbers byℂ and ℕ0=0,1,2,… Assuming that r=ran∈ℕ0∈0,∞ℕ0, Bakery and El Dewaik [19] defined the following function space:(1)ℋwraψ=f:fz=∑y=0∞fy^zy∈ℂ,for everyz∈ℂandψϖf<∞,for someϖ>0,where(2)ψf=∑y=0∞fy^y+1ry.They developed a multitude of topological and geometric characteristics for this variable exponent weighted formal power series space, as well as the prequasi-ideal construction utilizings-number and ℋwpn. Upper bounds for s-numbers of infinite series of the weighted n-th power forward shift operator on ℋwraψ were also introduced for some entire functions. Further, they evaluated Caristi’s fixed point theorem in ℋwraψ. For extra information on formal power series spaces and their behaviors, see [20–23]. The purpose of this paper is to develop an insight into how to think about the existence of a fixed point of Kannan prequasinorm contraction mapping in the prequasi-Banach special space of formal power series, where ℋwraψ satisfies the property (R) and ℋwraψ possesses the ψ-normal structure property. It has been established that a fixed point of the Kannan prequasinorm nonexpansive mapping exists in the prequasi-Banach special space of formal power series. Additionally, we discuss the Kannan prequasinorm contraction mapping in terms of the prequasioperator ideal. The existence of a fixed point of the Kannan prequasi norm contraction mapping in the prequasi Banach operator ideal Sℋwraψ is offered, where Sℋwraψ is the class of all bounded linear mappings between any two Banach spaces with the sequence s-numbers. Finally, we discuss several applications of solutions to summable equations and illustrate our findings with some instances.


## 2. Definitions and Preliminaries

Definition 2.1 (see [19]).
The linear spaceℋ=h:hz=∑y=0∞hy^zy∈ℂ,for everyz∈ℂ is called a special space of formal power series (or in short (ssfps), if it shows the following settings:(1)
em∈ℋ, for all m∈ℕ0, where emz=∑y=0∞eym^zy=zm.(2)
Ifg∈ℋ and hy^≤gy^, for every y∈ℕ0, then h∈ℋ.(3)
Supposeh∈ℋ; then h.∈ℋ, with h.z=∑b=0∞hb/2^zb and b/2 marks the integral part of b/2.Definition 2.2 (see [19]).
A subspaceℋψ of ℋ is said to be a premodular (ssfps), if there is a function ψ:ℋ⟶0,∞ that verifies the next conditions:(i)
Forh∈ℋ, we have ψh≥0 and h=θ⇔ψh=0, where θ is the zero function of ℋ.(ii)
Supposeh∈ℋ and ϖ∈ℝ; then there is q≥1 with ψϖh≤|ϖ|qψh.(iii)
Letf,g∈ℋ; then there is A≥1 such that ψf+g≤Aψf+ψg.(iv)
Supposefb^≤gb^, for every b∈ℕ0; then ψf≤ψg.(v)
There isK0≥1 so that ψf≤ψf.≤K0ψf.(vi)
F¯=ℋψ, where F indicates the space of finite formal power series; that is, for h∈F, we have l∈ℕ0 with hz=∑y=0lhy^zy.(vii)
One hasξ>0 with ψλe0≥ξλψe0, where λ∈ℝ.It is worth noting that the continuity ofψf at θ is due to condition (ii). Condition (1) in Definition 2.1 and condition (vi) in Definition 2.2 analyze the notion that emm∈ℕ0 is a Schauder basis for ℋψ.The (ssfps)ℋψ is called a prequasinormed (ssfps) if ψ shows conditions (i)–(iii) of Definition 2.2, and if the space ℋ is complete under ψ, then ℋψ is called a prequasi-Banach (ssfps). By L, we denote the ideal of all bounded linear operators between any arbitrary Banach spaces. Also, LX,Y marks the space of all bounded linear operators from a Banach space X into a Banach space Y.Definition 2.3 (see [24]).
A functions:LX,Y⟶0,∞ℕ0 is said to be an s-number, if the sequence sbBa=0∞, for all B∈LX,Y, shows the following settings:(a)
IfB∈LX,Y, then B=s0B≥s1B≥s2B≥…≥0.(b)
sb+a−1B1+B2≤sbB1+saB2, for every B1,B2∈LX,Y,b,a∈ℕ0.(c)
The inequalitysaABD≤AsaBD holds, if D∈LX0,X,B∈LX,Y, and A∈LY,Y0, where X0 and Y0 are arbitrary Banach spaces.(d)
SupposeA∈LY,Y0 and λ∈ℝ; then saλA=λsaA.(e)
LetrankA≤b; then sbA=0, whenever A∈LY,Y0.(f)
Assume thatIλ denotes the identity mapping on the λ-dimensional Hilbert space ℓ2λ; then sr≥λIλ=0 or sr<λIλ=1.Definition 2.4.
(see [7]) A class U⊆L is said to be an operator ideal if every vector UX,Y=U∩LX,Y shows the following settings:(i)
F⊆U, where F is the ideal of all finite rank operators between any arbitrary Banach spaces.(ii)
UX,Y is linear space on ℝ.(iii)
IfD∈LX0,X,B∈UX,Y, and A∈LY,Y0, then ABD∈UX0,Y0.Definition 2.5.
(see [10]) A function g:U⟶0,∞ is called a prequasinorm on the ideal U if it shows the following settings:(1)
For eachA∈LX,Y,gA≥0 and gA=0⇔A=0.(2)
One hasM≥1 with gβA≤MβgA, for all β∈ℝ and A∈UX,Y.(3)
One hasK≥1 with gA1+A2≤KgA1+gA2, for every A1,A2∈UX,Y.(4)
There isC≥1 so that if A∈LX0,X,B∈UX,Y, and D∈LY,Y0, then gDBA≤CDgBA, where X0 and Y0 are normed spaces.Definition 2.6.
(see [19])(a)
The prequasinormed (ssfps)η on ℋwrnη is said to be η-convex, ifηεg+1−εh≤εηg+1−εηh, for all ε∈0,1 and g,h∈ℋwrnη.(b)
hyy∈ℕ0⊆ℋwryη is η-convergent to h∈ℋwrnη, if and only if limy⟶∞ηhy−h=0. If the η-limit exists, then it is unique.(c)
hyy∈ℕ0⊆ℋwryη is η-Cauchy, if limx,y⟶∞ηhx−hy=0.(d)
Y⊂ℋwryη is η-closed, if limy⟶∞ηhy−h=0, where hyy∈ℕ0⊂Y; then h∈Y.(e)
Y⊂ℋwryη is η-bounded, if δηY=supηg−h:g,h∈Y<∞.(f)
Theη-ball of radius l≥0 and center g, for all g∈ℋwryη, is defined as(3)ℬηg,l=h∈ℋwryη:ηg−h≤l.(g)
A prequasinormed (ssfps)η on ℋwry verifies the Fatou property, if for every sequence hy⊆ℋwryη with limy⟶∞ηhy−h=0 and every f∈ℋwryη, (4)ηf−g≤supjinfy≥jηf−gy.Take note that the Fatou property determined theη-balls’ closedness. By ℓ∞ and mi↗, we denote the space of real bounded sequences and the space of all monotonic increasing sequences of positive reals.Lemma 2.7.
(see [19]) The function ψf=∑r=0∞fr^/r+1pr1/K, for all f∈ℋwryη, verifies the Fatou property, when rn∈mi↗∩ℓ∞.Lemma 2.8.
(see [19]) If rn∈mi↗∩ℓ∞, then the following settings hold:(1)
The function spaceℋwryη is a prequasiclosed and Banach (ssfps), with(5)ψh=∑y=0∞hy^y+1ry,for allh∈ℋwry.(2)
The classSℋwryψ,Ψ is a prequasi-Banach and closed operator ideal, where ΨP=ψfs=∑y=0∞syP/y+1ry, where fsz=∑y=0∞syPzy∈ℂ, for every z∈ℂ.Lemma 2.9.
The following inequalities will be utilized in the continuation:(i)
[25] If b≥2 and for each f,g∈ℂ, then(6)f+g2b+f−g2b≤12fb+gb.(ii)
[26] Let 1<b≤2, and for every f,g∈ℂ with f+g≠0; then(7)f+g2b+bb−12f−gf+g2−bf−g2b≤12fb+gb.(iii)
[27] Assume by>0 and fy,gy∈ℂ, for each y∈ℕ0; then fy+gyby≤2K−1fyby+gyby, where K=max1,supyby.


## 3. Some Topological and Geometric Properties

In this section, first, we will talk about the uniform convexity (UUC 2) defined in [28] of the prequasinormed (ssfps) ℋwraψ.Definition 3.1 (see [4, 29]).
We define the prequasinormψ’s uniform convexity type behavior as follows:(1)
[30] Suppose λ>0 and β>0. Let(8)V1λ,β=f,g:f,g∈ℋwraψ,ψf≤λ,ψg≤β,ψf−g≥λβ.WhenV1λ,β≠∅, we put(9)V1λ,β=inf1−1λψf+g2:f,g∈V1λ,β.WhenV1λ,β≠∅, we put V1λ,β=1. The function ψ holds the uniform convexity (UC) if for each λ>0 and β>0, we have V1λ,β>0. Observe that, for all λ>0, then V1λ,β≠∅, for very small β>0.(2)
[28] The function ψ verifies (UUC) if for every γ≥0 and β>0, there is ζ1γ,β with(10)V1λ,β>ζ1γ,β>0,forλ>γ.(3)
[28] Suppose λ>0 and β>0. Let(11)V2λ,β=f,g:f,t∈ℋwraψ,ψf≤λ,ψg≤λ,ψf−g2≥λβ.WhenV2λ,β≠∅, we put(12)V2λ,β=inf1−1λψf+g2:f,g∈V2λ,β.WhenV2λ,β=∅, we place V2λ,β=1. The function ψ satisfies (UC 2) if for every λ>0 and β>0, one has V2λ,β>0. Observe that, for each λ>0,V2λ,β≠∅, for very small β>0.(4)
[28] The function ψ verifies (UUC 2) if for all γ≥0 and β>0, there is ζ2γ,β with(13)V2λ,β>ζ2γ,β>0,forλ>γ.(5)
[30] The function ψ is strictly convex (SC), if for all f,t∈ℋwraψ so that ψf=ψg and ψf+t/2=ψf+ψg/2, we get f=g.We will require the following comment here and in the next:ψBf=∑m∈Bfm^/m+1rm1/K, for every B⊂ℕ0 and f∈ℋwraψ. When B=∅, we put ψBf=0.Theorem 3.2.
The functionψf=∑a=0∞fa^/a+1ra1/K, for all f∈ℋwraψ, is (UUC2), if ra∈mi↗∩ℓ∞ with r0>1.Proof.
Let the condition be satisfied,b>0, and a>p≥0. Suppose f,g∈ℋwraψ so that(14)ψf≤a,ψg≤a,ψf−g2≥ab.
From the definition ofψ, we have(15)ab≤ψf−g2=∑d=0∞d+1−rdfd^−gd^2rd1K≤2−r0∑d=0∞d+1−rdfd^−gd^rd1K≤2−r0K∑d=0∞d+1−rd|fd^|rd1K+∑d=0∞d+1−rd|gd^|rd1K=2−r0Kψf+ψg≤2a,and this implies b≤2. Consequent, let Q=x∈ℕ0:1<rx<2 and P=x∈ℕ0:rx≥2=ℕ0∖Q. For every w∈ℋwraψ, we get ψKw=ψPKw+ψQKw. From the setup, one has ψPf−g/2≥ab/2 or ψQf−g/2≥ab/2. Assume first ψPf−g/2≥ab/2. By using Lemma 2.9, condition (i), we obtain(16)ψPKf+g2+ψPKf−g2≤ψPKf+ψPKg2.
This explains(17)ψPKf+g2≤ψPKf+ψPKg2−ab2K.
As(18)ψQKf+g2≤ψQKf+ψQKg2,by adding inequalities 2 and 3, and from inequality 1, we have(19)ψKf+g2≤ψKf+ψKg2−ab2K≤aK1−b2K.
This gives(20)ψf+g2≤a1−b2K1K.
Next, supposeψQf−g/2≥ab/2. Set B=b/4K,(21)Q1=d∈Q:fd^−gd^≤Bfd^+gd^andQ2=Q:Q1.
AsB≤1 and the power function is convex, (22)ψQ1Kf−g2≤∑d∈Q1Brdd+1−rdfd^+gd^2rd≤B2r0ψQ1Kf+ψQ1Kg≤B2ψQKf+ψQKg≤B2ψKf+ψKg≤BaK.
SinceψQf−g/2≥ab/2, we get(23)ψQ2Kf−g2=ψQKf−g2−ψQ1Kf−g2≥aKb2K−b4K.
For anyd∈Q2, we have(24)r0−1<r0r0−1≤⋅⋅⋅≤rd−1rd−1−1≤rdrd−1andB<B2−rd<fd^−gd^fd^+gd^2−rd.
By Lemma2.9, condition (ii), we have (25)d+1−rdfd^+gd^2rd+r0−1B2d+1−rdfd^−gd^2rd≤12d+1−rdfd^rd+d+1−rdgd^rd.
Hence,(26)ψQ2Kf+g2+r0−1B2ψQ2Kf−g2≤ψQ2Kf+ψQ2Kg2.
This investigates(27)ψQ2Kf+g2≤ψQ2Kf+ψQ2Kg2−r0−1B2aKb2K−b4K.
Since(28)ψQ1Kf+g2≤ψQ1Kf+ψQ1Kg2,by adding inequalities (27) and (28), one has(29)ψQKf+g2≤ψQKf+ψQKg2−r0−1B2aKb2K−b4K≤ψQKf+ψQKg2−r0−12b42KaK2K−1≤ψQKf+ψQKg2−r0−12K−1b42KaK.
Since(30)ψPKf+g2≤ψPKf+ψPKg2,by adding inequalities (29) and (30) and from inequality 1, we obtain(31)ψKf+g2≤ψKf+ψKg2−r0−12K−1b42KaK≤aK1−r0−12K−1b42K.
This implies(32)ψf+g2≤a1−r0−12K−1b42K1K.
It is clear that(33)1<r0≤K<2K⇒0<r0−12K−1<1.
By using inequalities 4 and 9 and Definition3.1, we put(34)ζ2p,b=min1−1−b2K1K,1−1−r0−12K−1b42K1K.
Therefore, we haveV2a,b>ζ2p,b>0, and we conclude that ψ is (UUC2).We will examine the property (R) of the prequasinormed (ssfps) ℋwraψ in this second part.Theorem 3.3.
Letra∈mi↗∩ℓ∞ with r0>1; then the next setups are satisfied:(1)
Assume thatΛ⊆ℋwraψ, Λ≠∅, ψ-closed and ψ-convex, where ψf=∑a=0∞fa^/a+1ra1/K, for all f∈ℋwraψ. Suppose f∈ℋwraψ so that(35)dψf,Λ=infψf−g:g∈Λ<∞.Hence, one has a uniqueλ∈Λ with dψf,Λ=ψf−λ.(2)
ℋwraψ satisfies the property (R). This means that, for every decreasing sequence Λxx∈ℕ0 of ψ-closed and ψ-convex nonempty subsets of ℋwraψ such that supx∈ℕ0dψf,Λx<∞, for some f∈ℋwraψ, then ∩x∈ℕ0Λx≠∅.Proof.
Suppose the setups are satisfied. To show (1), letf∉Λ as Λ is ψ-closed. Then, one has A:=dψf,Λ>0. So, for every p∈ℕ0, we have gp∈Λ with ψf−gp<A1+1/p. Assume gp/2 is not ψ-Cauchy. Therefore, one obtains a subsequence ghp/2 and b0>0 so that ψghp−ghq/2≥b0, for all p>q≥0. Furthermore, one has V2A1+1/p,b0/2A>ξ:=β2A1+1/p,b0/2A>0, for each p∈ℕ0. As(36)maxhf−ghp,hf−ghq≤A1+1hq,
and(37)ψghp−ghq2≥b0≥A1+1hqb02A.
Underp>q≥0, we get(38)ψf−ghp+ghq2≤A1+1hq1−ξ.
Therefore,(39)A=dψf,Λ≤A1+1hq1−ξ,
withq∈ℕ0. If we let q⟶∞, we get(40)0<A≤A1+1hq1−ξ<A.
We have a contradiction. Thengp/2 is ψ-Cauchy. As ℋwraψ is ψ-complete, gp/2ψ converges to some g. For all q∈ℕ0, we have the sequence gp+gq/2ψ that converges to g+gq/2. As Λ is ψ-closed and ψ-convex, one gets g+gq/2∈Λ. Surely g+gq/2ψ converges to 2g, so 2g∈Λ. For λ=2g and using Theorem 2.7, since ψ satisfies the Fatou property, we get(41)dψf,Λ≤ψf−λ≤supiinfq≥iψf−g+gq2≤supiinfq≥isupiinfp≥iψf−gp+gq2≤12supiinfq≥isupiinfp≥iψf−gp+ψf−gq=dψf,Λ.
Therefore,ψf−λ=dψf,Λ. As ψ is (UUC2), so it is SC, which implies that λ is unique. To show (2), let f∉Λp0, for some p0∈ℕ0. dψf,Λpp∈ℕ0∈ℓ∞ is increasing. Let limp⟶∞dψf,Λp=A. Suppose A>0. Else f∈Λp, for every p∈ℕ0. By using Part (1), we have one point gp∈Λp with dψf,Λp=ψf−gp, for every p∈ℕ0. A consistent proof will show that gp/2ψ converges to some g∈ℋwraψ. When Λp are ψ-convex, decreasing, and ψ-closed, we get 2g∈∩p∈ℕ0Λp.
This third part discusses the prequasinormed structure’sψ-normal structure feature (ssfps) ℋwraψ.Definition 3.4.
ℋwraψ satisfies the ψ-normal structure property if for all nonempty ψ-bounded, ψ-convex, and ψ-closed subset Λ of ℋwraψ did not decrease to one point, we have f∈Λ with(42)supg∈Λψf−g<δψΛ:=supψf−g:f,g∈Λ<∞.Theorem 3.5.
Ifra∈mi↗∩ℓ∞ with r0>1, then ℋwraψ holds the ψ-normal structure property, where ψf=∑a=0∞fa^/a+1ra1/K, for every f∈ℋwraψ.Proof.
Assume the setups are satisfied. Theorem3.2 explains that ψ is (UUC2). Let Λ be a ψ-bounded, ψ-convex, and ψ-closed subset of ℋwraψ not decreased to unique point. Hence, δψΛ>0. Let A=δψΛ. Suppose f,g∈Λ with f≠g. So ψf−g/2=b>0. For all λ∈Λ, one obtains ψf−λ≤A and ψg−λ≤A. Since Λ is ψ-convex, one has f+g/2∈Λ. Hence,(43)ψf+g2−λ=ψf−λ+g−λ2≤A1−V2A,bA.
For allλ∈Λ, (44)supλ∈Λψf+g2−λ≤A1−V2A,bA<A=δψΛ.


## 4. Kannan Contraction Mapping

In the prequasinormed space, we now develop Kannanψ-Lipschitzian mapping (ssfps). We study enough conditions on ℋwraψ with a defined prequasinorm such that Kannan prequasinorm contraction mapping has a unique fixed point.Definition 4.1.
An operatorW:ℋwraψ⟶ℋwraψ is called a Kannan ψ-Lipschitzian, if there is κ≥0, so that(45)ψWf−Wg≤κψWf−f+ψWg−g.
For allf,g∈ℋwraψ, one has the following:(1)
Ifκ∈0,1/2, then the operator W is said to be Kannan ψ-contraction.(2)
Ifκ=1/2, then the operator W is said to be Kannan ψ-nonexpansive.
A vectorg∈ℋwraψ is called a fixed point of W, when Wg=g.Theorem 4.2.
Ifra∈mi↗∩ℓ∞ and W:ℋwraψ⟶ℋwraψ is Kannan ψ-contraction mapping, where ψf=∑a=0∞fa^/a+1ra1/K, for all f∈ℋwraψ, then W has a unique fixed point.Proof.
Let the setups be satisfied. For everyf∈ℋwraψ, then Wpf∈ℋwraψ. Since W is a Kannan ψ-contraction mapping, we have(46)ψWp+1f−Wpf≤κψWp+1f−Wpf+ψWpf−Wp−1f⇒ψWp+1f−Wpf≤κ1−κψWpf−Wp−1f≤κ1−κ2ψWp−1f−Wp−2f≤⋅⋅⋅≤κ1−κpψWf−f.
Therefore, for everyp,q∈ℕ0 with q>p, then we get(47)ψWpf−Wqf≤κψWpf−Wp−1f+ψWqf−Wq−1f≤κκ1−κp−1+κ1−κq−1ψWf−f.
SoWpf is a Cauchy sequence in ℋwraψ. As the space ℋwraψ is prequasi-Banach (ssfps). Therefore, there is g∈ℋwraψ such that limp⟶∞Wpf=g. To prove that Wg=g, by Theorem 2.7, ψ holds the Fatou property, and we have(48)ψWg−g≤supiinfp≥iψWp+1f−Wpf≤supiinfp≥iκ1−κpψWf−f=0.
Hence,Wg=g. Then g is a fixed point of W. To show that the fixed point is unique, assume we have two different fixed points f,g∈ℋwraψ of W. Then, one has(49)ψf−g≤ψWb−Wt≤κψWf−f+ψWg−g=0.
Therefore,f=g.Corollary 4.3.
Letra∈mi↗∩ℓ∞ and W:ℋwraψ⟶ℋwraψ be Kannan ψ-contraction mapping, where ψf=∑a=0∞fa^/a+1ra1/K, for all f∈ℋwraψ; then W has one and only one fixed point g with ψWpf−g≤κκ/1−κp−1ψWf−f.Proof.
It is obvious, so it is omitted.Definition 4.4.
Assumeℋwraψ is a prequasinormed (ssfps) and W:ℋwraψ⟶ℋwraψ. The operator W is called ψ sequentially continuous at g∈ℋwraψ, if and only if when lima⟶∞ψfa−g=0, lima⟶∞ψWfa−Wg=0.Theorem 4.5.
Letra∈mi↗∩ℓ∞ with r0>1 and W:ℋwraψ⟶ℋwraψ, where ψf=∑a=0∞fa^/a+1ra, for all f∈ℋwraψ. The point f∈ℋwraψg∈ℋwraψ is the unique fixed point of W, if the following conditions are satisfied:(a)
W is Kannan ψ-contraction mapping.(b)
W is ψ sequentially continuous at g∈ℋwraψ.(c)
One hasf∈ℋwraψ with the sequence of iterates Wpf having a subsequence Wpif converging to g.Proof.
Suppose the settings are verified. Ifg is not a fixed point of W, then Wg≠g. By conditions (b) and (c), we have(50)limpi⟶∞ψWpif−g=0,limpi⟶∞ψWpi+1f−Wg=0.
Since the mappingW is Kannan ψ-contraction, one can see(51)0<ψWg−g=ψWg−Wpi+1f+Wpif−g+Wpi+1f−Wpif≤22supiri−2ψWpi+1f−Wg+22supiri−2ψWpif−g+2supiri−1κκ1−κpi−1ψWf−f.
Sincepi⟶∞, one has a contradiction. Hence, g is a fixed point of W. To explain that the fixed point g is unique, suppose we have two different fixed points g,b∈ℋwraψ of W. Therefore, one gets(52)ψg−b≤ψWg−Wb≤κψWg−g+ψWb−b=0.
So,g=b.Theorem 4.6.
Assumera∈0,1ℕ0 is an increasing, and W:ℋwraψ⟶ℋwraψ, where ψf=∑a=0∞fa^/a+1ra1/r0, for all f∈ℋwraψ. The point g∈ℋwraψ is the only fixed point of W, if the following conditions are satisfied:(a)
W is Kannan ψ-contraction mapping.(b)
W is ψ sequentially continuous at g∈ℋwraψ.(c)
One hasf∈ℋwraψ so that the sequence of iterates Wpf has a subsequence Wpif converging to g.Proof.
Let the conditions be verified. Ifg is not a fixed point of W, then Wg≠g. By conditions (b) and (c), we have(53)limpi⟶∞ψWpif−g=0,limpi⟶∞ψWpi+1f−Wg=0.
As the operatorW is Kannan ψ-contraction, one can see(54)0<ψWg−g=ψWg−Wpi+1f+Wpif−g+Wpi+1f−Wpif≤22r0−1−2ψWpi+1f−Wg+22r0−1−2ψWpif−g+2r0−1−1κκ1−κpi−1ψWf−f.
Sincepi⟶∞, one obtains a contradiction. Hence, g is a fixed point of W. To explain that the fixed point g is unique, assume we have two different fixed points g,b∈ℋwraψ of W. Then, one gets(55)ψg−b≤ψWg−Wb≤κψWg−g+ψWb−b=0.
So,g=b.Example 4.7.
Pick upW:ℋwa+1/2a+4a=0∞ψ⟶ℋwa+1/2a+4a=0∞ψ, where ψf=∑a∈ℕ0fa^/a+1a+1/2a+4, for all f∈ℋwa+1/2a+4a=0∞ and(56)Wf=f18,ψf∈0,1,f20,ψf∈1,∞.
For allf1,f2∈a+1/2a+4a=0∞ψ with ψf1,ψf2∈0,1, we have(57)ψWf1−Wf2=ψf118−f218≤1417ψ17f118+ψ17f218=1417ψWf1−f1+ψWf2−f2.
For allf1,f2∈ℋwa+1/2a+4a=0∞ψ with ψf1,ψf2∈1,∞, we have(58)ψWf1−Wf2=ψf120−f220≤1419ψ19f120+ψ19f220=1419ψWf1−f1+ψWf2−f2.
For allf1,f2∈ℋwa+1/2a+4a=0∞ψ with ψf1∈0,1 and ψf2∈1,∞, we have(59)ψWf1−Wf2=ψf118−f220≤1417ψ17f118+1419ψ19f220≤1417ψ17f118+ψ19f220=1417ψWf1−f1+ψWf2−f2.
Hence,W is Kannan ψ-contraction mapping. By Theorem 2.7, the function ψ satisfies the Fatou property. By Theorem 4.2, the map W has a unique fixed point θ∈ℋwa+1/2a+4a=0∞ψ.
Letfn⊆ℋwa+1/2a+4a=0∞ψ with limn⟶∞ψfn−f0=0, where f0∈ℋwa+1/2a+4a=0∞ψ and ψf0=1. Since the prequasinorm ψ is continuous, one gets(60)limn⟶∞ψWfn−Wf0=limn⟶∞ψfn18−f020=ψf0180>0.
Therefore,W is not ψ sequentially continuous at f0. Then, the map W is not continuous at f0.
Ifψf=∑a∈ℕ0fa^/a+1a+1/2a+44, for all f∈ℋwa+1/2a+4a=0∞. For all f1,f2∈ℋwa+1/2a+4a=0∞ψ with ψf1,ψf2∈0,1, one obtains(61)ψWf1−Wf2=ψf118−f218≤817ψ17f118+ψ17f218=817ψWf1−f1+ψWf2−f2.
For allf1,f2∈ℋwa+1/2a+4a=0∞ψ with ψf1,ψf2∈1,∞, we have(62)ψWf1−Wf2=ψf120−f220≤819ψ19f120+ψ19f220=819ψWf1−f1+ψWf2−f2.
For allf1,f2∈ℋwa+1/2a+4a=0∞ψ with ψf1∈0,1 and ψf2∈1,∞, we have(63)ψWf1−Wf2=ψf118−f220≤817ψ17f118+819ψ19f220≤817ψ17f118+ψ19f220=817ψWf1−f1+ψWf2−f2.
Therefore, the mapW is Kannan ψ-contraction mapping and Wpf=f/18p,ψf∈0,1,f/20p,ψf∈1,∞.
Obviously,W is ψ sequentially continuous at θ∈ℋwa+1/2a+4a=0∞ψ and Wpf has a subsequence Wpif converging to θ. By Theorem 4.5, the point θ∈ℋwa+1/2a+4a=0∞ψ is the unique fixed point of W.Example 4.8.
AssumeW:ℋwa+1/2a+4a=0∞ψ⟶ℋwa+1/2a+4a=0∞ψ, where ψf=∑a∈ℕ0fa^/a+12a+3/a+2, for all v∈ℋwa+1/2a+4a=0∞ and(64)Wf=f4,ψf∈0,1,f5,ψf∈1,∞.
For allf1,f2∈ℋwa+1/2a+4a=0∞ψ with ψf1,ψf2∈0,1, we have(65)ψWf1−Wf2=ψf14−f24≤1427ψ3f14+ψ3f24=1427ψWf1−f1+ψWf2−f2.
For allf1,f2∈ℋwa+1/2a+4a=0∞ψ with ψf1,ψf2∈1,∞, we have(66)ψWf1−Wf2=ψf15−f25≤1464ψ4f15+ψ4f25=1464ψWf1−f1+ψWf2−f2.
For allf1,f2∈ℋwa+1/2a+4a=0∞ψ with ψf1∈0,1 and ψf2∈1,∞, we have(67)ψWf1−Wf2=ψf14−f25≤1427ψ3f14+1464ψ4f25≤1427ψ3f14+ψ4f25=1427ψWf1−f1+ψWf2−f2.
Hence,W is Kannan ψ-contraction mapping. From Theorem 2.7, the function ψ satisfies the Fatou property. By Theorem 4.2, the map W has a unique fixed point θ∈ℋwa+1/2a+4a=0∞ψ.
Letfn⊆ℋwa+1/2a+4a=0∞ψ with limn⟶∞ψfn−f0=0, where f0∈ℋwa+1/2a+4a=0∞ψ and ψf0=1. Since the prequasinorm ψ is continuous, we have(68)limn⟶∞ψWfn−Wf0=limn⟶∞ψfn4−fn5=ψf020>0.
Therefore,W is not ψ sequentially continuous at f0. So, the map W is not continuous at f0.
Supposeψf=∑a∈ℕ0fa^/a+1a+1/2a+44, for all f∈ℋwa+1/2a+4a=0∞. For all f1,f2∈ℋwa+1/2a+4a=0∞ψ with ψf1,ψf2∈0,1, we have(69)ψWf1−Wf2=ψf14−f24≤227ψ3f14+ψ3f24=227ψWf1−f1+ψWf2−f2.
For allf1,f2∈ℋwa+1/2a+4a=0∞ψ with ψf1,ψf2∈1,∞, we have(70)ψWf1−Wf2=ψf15−f25≤14ψ4f15+ψ4f25=14ψWf1−f1+ψWf2−f2.
For allf1,f2∈ℋwa+1/2a+4a=0∞ψ with ψf1∈0,1 and ψf2∈1,∞, we have(71)ψWf1−Wf2=ψf14−f25≤227ψ3f14+14ψ4f25≤227ψ3f14+ψ4f25=227ψWf1−Wf1+ψWf2−Wf2.
So, the mapW is Kannan ψ-contraction mapping and Wpf=f/4p,ψf∈0,1,f/5p,ψf∈1,∞.
Obviously,W is ψ sequentially continuous at θ∈ℋwa+1/2a+4a=0∞ψ and Wpf has a subsequence Wpif converging to θ. By Theorem 4.5, the point θ∈ℋwa+1/2a+4a=0∞ψ is the unique fixed point of W.Example 4.9.
LetW:ℋwa+1/2a+4a=0∞ψ⟶ℋwa+1/2a+4a=0∞ψ, where ψf=∑a∈ℕ0fa^/a+1a+1/2a+44, for all f∈ℋwa+1/2a+4a=0∞ and(72)Wf=1181+f,f0^∈−∞,117,117,f0^=117,118,f0^∈117,∞.
For allf,g∈ℋwa+1/2a+4a=0∞ψ with f0^,g0^∈−∞,1/17, we have(73)ψWf−Wg=ψ118f−g≤817ψ17f18+ψ17g18≤817ψWf−f+ψWg−g.
For allf,g∈ℋwa+1/2a+4a=0∞ψ with f0^,g0^∈1/17,∞ and then for any ε>0, we have(74)ψWf−Wg=0≤εψWf−f+ψWg−g.For all f,g∈ℋwa+1/2a+4a=0∞ψ with f0^∈−∞,1/17 and g0^∈1/17,∞, we have(75)ψWf−Wg=ψf18≤117ψ17f18=117ψWf−f≤117ψWf−f+ψWg−g.
Hence,W is Kannan ψ-contraction mapping. Clearly, W is ψ sequentially continuous at 1/17∈ℋwa+1/2a+4a=0∞ψ, and there is f∈ℋwa+1/2a+4a=0∞ψ with f0^∈−∞,1/17 such that the sequence of iterates Wpf=∑n=1p1/18n+1/18pf has a subsequence Wpif=∑n=1p1/18n+1/18pf converging to 1/17. By Theorem 4.5, the map W has one fixed point 1/17∈ℋwa+1/2a+4a=0∞ψ. Note that W is not continuous at 1/17∈ℋwa+1/2a+4a=0∞ψ.
Ifψf=∑a∈ℕ0fa^/a+1a+1/2a+4, f∈ℋwa+1/2a+4a=0∞. For all f,g∈ℋwa+1/2a+4a=0∞ψ with f0^,g0^∈−∞,1/17, we have(76)ψWf−Wg=ψ118f−g≤1417ψ17f18+ψ17g18≤1417ψWf−f+ψWg−g.
For allf,g∈ℋwa+1/2a+4a=0∞ψ with f0^,g0^∈1/17,∞ and then for any ε>0, we have(77)ψWf−Wg=0≤εψWf−f+ψWg−g.
For allf,g∈ℋwa+1/2a+4a=0∞ψ with f0^∈−∞,1/17 and g0^∈1/17,∞, we have(78)ψWf−Wg=ψf18≤1417ψ17f18=144]17ψWf−f≤1417ψWf−f+ψWg−g.
So,W is Kannan ψ-contraction mapping. By Theorem 2.7, the function ψ satisfies the Fatou property. By Theorem 4.2, the map W has a unique fixed point 1/17∈ℋwa+1/2a+4a=0∞ψ.Example 4.10.
LetW:ℋwa+1/2a+4a=0∞ψ⟶ℋwa+1/2a+4a=0∞ψ, where ψf=∑a∈ℕ0fa^/a+1a+1/2a+4, for all v∈ℋwa+1/2a+4a=0∞ and(79)Wf=14z+f,f0^∈−∞,13,13z,f0^=13,14z,f0^∈13,∞.
For allf,g∈ℋwa+1/2a+4a=0∞ψ with f0^,g0^∈−∞,1/3, we have(80)ψWf−Wg=ψ14f−g≤227ψ3f4+ψ3g4≤227ψWf−f+ψWg−g.
For allf,g∈ℋwa+1/2a+4a=0∞ψ with f0^,g0^∈1/3,∞ and then for any ε>0, we have(81)ψWf−Wg=0≤εψWf−f+ψWg−g.
For allf,g∈ℋwa+1/2a+4a=0∞ψ with f0^∈−∞,1/3 and g0^∈1/3,∞, we have(82)ψWf−Wg=ψf4≤127ψ3f4=127ψWf−f≤127ψWf−f+ψWg−g.
Hence,W is Kannan ψ-contraction mapping. Evidently, W is ψ sequentially continuous at 1/3z∈ℋwa+1/2a+4a=0∞ψ and there is f∈ℋwa+1/2a+4a=0∞ψ with f0^∈−∞,1/3 such that the sequence of iterates Wpf=∑n=1p1/4nz+1/4pf has a subsequence Wpif=∑n=1pi1/4nz+1/4pif converging to 1/3z. By Theorem 4.5, the map W has one fixed point 1/3z∈ℋwa+1/2a+4a=0∞ψ. Note that W is not continuous at 1/3z∈ℋw2a+3/a+2a=0∞ψ.
Ifψf=∑a∈ℕ0fa^/a+12a+3/a+2, f∈ℋw2a+3/a+2a=0∞. For all f,g∈ℋw2a+3/a+2a=0∞ψ with f0^,g0^∈−∞,1/3, we have(83)ψWf−Wg=ψ14f−g≤1427ψ3f4+ψ3g4≤1427ψWf−f+ψWg−g.For all f,g∈ℋw2a+3/a+2a=0∞ψ with f0^,g0^∈1/3,∞ and then for any ε>0, we have(84)ψWf−Wg=0≤εψWf−f+ψWg−g.
For allf,g∈ℋw2a+3/a+2a=0∞ψ with f0^∈−∞,1/3 and g0^∈1/3,∞, we have(85)ψWf−Wg=ψf4≤1427ψ3f4=1427ψWf−f≤1427ψWf−f+ψWg−g.
Hence,W is Kannan ψ-contraction mapping. By Theorem 2.7, the function ψ satisfies the Fatou property. By Theorem 4.2, the map W has a unique fixed point 1/3z∈ℋw2a+3/a+2a=0∞ψ.


## 5. Kannan Nonexpansive Mapping

We examine enough conditions on the prequasinormed (ssfps)ℋwraψ for the Kannan prequasinorm nonexpansive mapping on it to have a fixed point in this section.Lemma 5.1.
Allow the prequasinormed (ssfps)ℋwraψ to validate the (R) and ψ-quasi-normal properties. If Ξ is a nonempty ψ-bounded, ψ-convex, and ψ-closed subset of ℋwraψ, suppose W:Ξ⟶Ξ is a Kannan ψ-nonexpansive mapping. For y>0, let Gy=f∈Ξ:ψf−Wf≤y≠∅. Let(86)Ξy=∩ℬψλ,γ:WGy⊂ℬψλ,γ∩Ξ.Then,Ξy is a nonempty, ψ-convex, ψ-closed subset of Ξ and(87)WΞy⊂Ξy⊂Gy,δψΞy≤y.Proof.
AsWGy⊂Ξy, this gives Ξy≠∅. As the ψ-balls are ψ-convex and ψ-closed, one has Ξy is a ψ-closed and ψ-convex subset of Ξ. To prove that Ξy⊂Gy, suppose f∈Ξy. If ψf−Wf=0, we get f∈Gy. Otherwise, suppose ψf−Wf>0. Let(88)λ=supψWw−Wf:w∈Gy.
From the definition ofλ, WGy⊂ℬψWf,λ. So, Ξy⊂ℬψWf,λ, and we have ψf−Wf≤λ. Assume η>0. Hence, there is w∈Gy so that λ−η≤ψWw−Wf. Then,(89)ψf−Wf−η≤λ−η≤ψWw−Wf≤12ψf−Wf+ψw−Ww≤12ψf−Wf+y.
Sinceη is randomly positive, we obtain ψf−Wf≤y, so we have f∈Gy. As WGy⊂Ξy, one has WΞy⊂WGy⊂Ξy; this indicates that Ξy is W-invariant, consequent to prove that δψΞy≤y, as(90)ψWf−Wg≤12ψf−Wf+ψg−Wg.
For everyf,g∈Gy, let f∈Gy. Hence, WGy⊂ℬψWf,y. The definition of Ξy provides Ξy⊂ℬψWf,y. Hence, Wf∈∩g∈Ξyℬψg,y. So, one has ψg−w≤y, for every g,w∈Ξy, this means δψΞy≤y. This completes the proof.Theorem 5.2.
Let the prequasinormed (ssfps)ℋwraψ verify the ψ-quasi-normal property and the (R) property. Assume Ξ is a nonempty, ψ-convex, ψ-closed, and ψ-bounded subset of ℋwraψ. Suppose W:Ξ⟶Ξ is a Kannan ψ-nonexpansive mapping. Hence, W has a fixed point.Proof.
Supposed0=infψf−Wf:f∈Ξ and dt=d0+1/t, for every t≥1. By using the definition of d0, we have Gdt=f∈Ξ:ψf−Wf≤dt≠∅, with t≥1. Assume Ξdt is described as in Lemma 5.1. Obviously, Ξdt is a decreasing sequence of nonempty ψ-bounded, ψ-closed, and ψ-convex subsets of Ξ. The property (R) proves that Ξ∞=∩t≥1Ξdt≠∅. Assume f∈Ξ∞; one can see ψf−Wf≤dt, for every t≥1. Suppose p⟶∞; we have ψf−Wf≤d0; this gives ψf−Wf=d0. Therefore, Gd0≠∅. We have d0=0. Else, d0>0; this gives that W fails to have a fixed point. Let Ξd0 be defined as in Lemma 5.1. As W misses to have a fixed point and Ξd0 is W-invariant, so Ξd0 has more than one point, which implies, δψΞd0>0. By the ψ-quasinormal property, there is f∈Ξd0 so that(91)ψf−g<δψΞd0≤d0.For allg∈Ξd0, by Lemma 5.1, we have Ξd0⊂Gd0. By definition of Ξd0, then Wf∈Gd0⊂Ξd0. Obviously, we have(92)ψf−Wf<δψΞd0≤d0,which contradicts the definition ofd0. So d0=0, which implies that any point in Gd0 is a fixed point of W; that is, W has a fixed point in Ξ.In view of Theorems3.3, 3.5, and 5.2, it is easy to conclude the following theorem.Theorem 5.3.
Ifra∈mi↗∩ℓ∞ with r0>1, Ξ is a nonempty, ψ-convex, ψ-closed, and ψ-bounded subset of ℋwraψ, where ψf=∑a=0∞fa^/a+1ra1/K, for every ℋwraψ, and W:Ξ⟶Ξ is a Kannan ψ-nonexpansive mapping. Then, W has a fixed point.Example 5.4.
LetW:Ξ⟶Ξ with Wf=f/4,ψf∈0,1,f/5,ψf∈1,∞,where Ξ=f∈ℋw2a+3/a+2a=0∞ψ:f0^=f1^=0 and ψf=∑a∈ℕ0fa^/a+12a+3/a+2, for all f∈ℋw2a+3/a+2a=0∞ψ. In view of example 4.8, the map W is Kannan ψ-contraction mapping. This implies Kannan ψ-nonexpansive mapping. Evidently, Ξ is a nonempty, ψ-convex, ψ-closed, and ψ-bounded subset of ℋw2a+3/a+2a=0∞ψ. By Theorem 5.3, the map W has one fixed point f=θ in Ξ.


## 6. Kannan Contraction Mappings on the Operator Ideal

We study in this section the presence of a fixed point for the Kannan prequasinorm contraction mapping in the prequasi-Banach operator ideal defined by theℋwraψ and s-numbers.Notations 6.1.
[19](93)Sℋ:=SℋX,Y;XandYare Banach Spaces,whereSℋX,Y:=P∈LX,Y:fs∈ℋ,wherefsz=∑y=0∞syPzy∈ℂ,for anyz∈ℂ.Definition 6.2.
IfX and Y are Banach spaces, a prequasinorm Ψ on the ideal Sℋwraψ, where ΨW=ψfs and fsz=∑a=0∞saWza converge for any z∈ℂ, satisfies the Fatou property if for every sequence Waa∈ℕ0⊆SℋwraψX,Y with lima⟶∞ΨWa−W=0 and any V∈SℋwraψX,Y, (94)ΨV−W≤supainfi≥aΨV−Wi.Theorem 6.3.
SupposeX and Y are Banach spaces. The prequasinorm ΨW=∑a=0∞saW/a+1ra1/K, for all W∈SℋwraψX,Y does not satisfy the Fatou property, if ra∈mi↗∩ℓ∞.Proof.
Let the condition be satisfied andWpp∈ℕ0⊆SℋwraψX,Y with limp⟶∞ΨWp−W=0. By Theorem 2.8, the space Sℋwraψ is a prequasiclosed ideal, and then W∈SℋwraψX,Y. Hence, for all V∈SℋwraψX,Y, we have(95)ΨV−W=∑a=0∞saV−Wa+1ra1K≤∑a=0∞sa/2V−Wia+1ra1K+∑a=0∞sa/2W−Wia+1ra1K≤21Ksuppinfi≥p∑a=0∞saV−Wia+1ra1K=21Ksuppinfi≥pΨV−Wi.
Hence,Ψ does not satisfy the Fatou property.Definition 6.4.
SupposeX and Y are Banach spaces. For the prequasinorm Ψ on the ideal Sℋwraψ, where ΨW=ψfs, where fsz=∑a=0∞saWza converges for any z∈ℂ, an operator G:SℋwraψX,Y⟶SℋwraψX,Y is called a Kannan Ψ-Lipschitzian, if there is κ≥0, so that(96)ΨGW−GA≤κΨGW−W+ΨGA−A,
for allW,A∈SℋwraψX,Y. An operator G is said to be(1)
KannanΨ-contraction, when κ∈0,1/2.(2)
KannanΨ-nonexpansive, when κ=1/2.Definition 6.5.
SupposeX and Y are Banach spaces. For the prequasinorm Ψ on the ideal Sℋwraψ, where ΨW=ψfs, where fsz=∑a=0∞saWza converges for any z∈ℂ, G:SℋwraψX,Y⟶SℋwraψX,Y and B∈SℋwraψX,Y. The operator G is said to be Ψ sequentially continuous at B, if and only if when limp⟶∞ΨWp−B=0, limp⟶∞ΨGWp−GB=0.Theorem 6.6.
SupposeX and Y are Banach spaces. Let ra∈mi↗∩ℓ∞ and G:SℋwraψX,Y⟶SℋwraψX,Y, where ΨW=∑a=0∞saW/a+1ra1/K, for all W∈SℋwraψX,Y. The point A∈SℋwraψX,Y is the unique fixed point of G, if the following conditions are satisfied:(a)
G is Kannan Ψ-contraction mapping.(b)
G is Ψ sequentially continuous at a point A∈SℋwraψX,Y.(c)
There isB∈SℋwraψX,Y so that the sequence of iterates GpB has a subsequence GpiB converging to A.Proof.
Let the conditions be verified. IfA is not a fixed point of G, then GA≠A. From conditions (b) and (c), we have(97)limpi⟶∞ΨGpiB−A=0,limpi⟶∞ΨGpi+1B−GA=0.
SinceG is Kannan Ψ-contraction mapping, one can see(98)0<ΨGA−A=ΨGA−Gpi+1B+GpiB−A+Gpi+1B−GpiB≤21KΨGpi+1B−GA+22KΨGpiB−A+22Kκκ1−κpi−1ΨGB−B.
Aspi⟶∞, we have a contradiction. Therefore, A is a fixed point of G. To show that the fixed point A is unique. Let us have two different fixed points A,D∈SℋwraψX,Y of G. Therefore, one has(99)ΨA−D≤ΨGA−GD≤κΨGA−A+ΨGD−D=0.
So,A=D.Example 6.7.
SupposeX and Y are Banach spaces; G:Sℋwa+1/a+2a=0∞ψX,Y⟶Sℋwa+1/a+2a=0∞ψX,Y, where ΨW=∑a=0∞saW/a+1a+1/a+2, for every W∈Sℋwa+1/a+2a=0∞ψX,Y and(100)GW=W26,ΨW∈0,1,W37,ΨW∈1,∞.
For allW1,W2∈Sℋwa+1/a+2a=0∞ψ with ΨW1,ΨW2∈0,1, we have(101)ΨGW1−GW2=ΨW126−W226≤25Ψ25W126+Ψ25W226=25ΨGW1−W1+ΨGW2−W2.
For allW1,W2∈Sℋwa+1/a+2a=0∞ψ with ΨW1,ΨW2∈1,∞, we have(102)ΨGW1−GW2=ΨW137−W237≤13Ψ36W137+Ψ36W237=13ΨGW1−W1+ΨGW2−W2.
For allW1,W2∈Sℋwa+1/a+2a=0∞ψ with ΨW1∈0,1 and ΨW2∈1,∞, we have(103)ΨGW1−GW2=ΨW126−W237≤25Ψ25W126+13Ψ36W237≤25Ψ25W126+Ψ36W237=25ΨGW1−W1+ΨGW2−W2.
Hence,G is Kannan Ψ-contraction mapping and GpW=W/26p,ΨW∈0,1,W/37p,ΨW∈0,1.
Evidently,G is Ψ sequentially continuous at the zero operator Θ∈Sℋwa+1/a+2a=0∞ψ and GpW has a subsequence GpiW converging to Θ. By Theorem 6.6, the zero operator Θ∈Sℋwa+1/a+2a=0∞ψ is the only fixed point of G. Assume Wn⊆Sℋwa+1/a+2a=0∞ψ with limn⟶∞ΨWn−W0=0, where W0∈Sℋwa+1/a+2a=0∞ψ and ΨW0=1. Since the prequasinorm Ψ is continuous, one obtains(104)limn⟶∞ΨGWn−GW0=limn⟶∞ΨWn26−W037=Ψ11W0962>0.
Therefore,G is not Ψ sequentially continuous at W0. Then, the map G is not continuous at W0.Example 6.8.
IfX and Y are Banach spaces, G:Sℋw2a+1/a+3a=0∞ψX,Y⟶Sℋw2a+1/a+3a=0∞ψX,Y, where ΨW=∑a=0∞saW/a+12a+1/a+3, for every W∈Sℋw2a+1/a+3a=0∞ψX,Y and(105)GW=W263170,ΨW∈0,1,W263171,ΨW∈1,∞.
For allW1,W2∈Sℋw2a+1/a+3a=0∞ψ with ΨW1,ΨW2∈0,1, we have(106)ΨGW1−GW2=ΨW1263170−W2263170≤26263169Ψ263169W1263170+Ψ263169W2263170=26263169ΨGW1−W1+ΨGW2−W2.
For allW1,W2∈Sℋw2a+1/a+3a=0∞ψ with ΨW1,ΨW2∈1,∞, we have(107)ΨGW1−GW2=ΨW1263171−W2263171≤26263170Ψ263170W1263171+Ψ263170W2263171=26263170ΨGW1−W1+ΨGW2−W2.
For allW1,W2∈Sℋw2a+1/a+3a=0∞ψ with ΨW1∈0,1 and ΨW1∈1,∞, we have(108)ΨGW1−GW2=ΨW1263170−W2263171≤26263169Ψ263169W1263170+26263170Ψ263170W2263171≤26263169Ψ263169W1263170+Ψ263170W2263171=26263169ΨGW1−W1+ΨGW2−W2.
Hence,G is Kannan Ψ-contraction mapping and GpW=W/263170p,ΨW∈0,1,W/263171p,ΨW∈1,∞.
Obviously,G is Ψ sequentially continuous at the zero operator Θ∈Sℋw2a+1/a+3a=0∞ψ and GpW has a subsequence GpiW converging to Θ. By Theorem 6.6, the zero operator Θ∈Sℋw2a+1/a+3a=0∞ψ is the only fixed point of G. Suppose Wn⊆Sℋw2a+1/a+3a=0∞ψ with limn⟶∞ΨWn−W0=0, where W0∈Sℋw2a+1/a+3a=0∞ψ and ΨW0=1. Since the prequasinorm Ψ is continuous, one gets(109)limn⟶∞ΨGWn−GW0=limn⟶∞ΨWn263170−W0263171=ΨW069258712070>0.
Therefore,G is not Ψ sequentially continuous at W0. Then, the map G is not continuous at W0.


## 7. Application to Nonlinear Summable Equations

Numerous authors have examined nonlinear summable equations such as (10); see [31–33]. This section is dedicated to locating a solution to (10) in ℋwraψ, where ra∈mi↗∩ℓ∞ and ψg=∑a=0∞ga^/a+1ra1/K, for every g∈ℋwraψ. Take a look at the equations that are summable:(110)ga^=pa^+∑m=0∞Aa,mfm,gm^,and assume W:ℋwraψ⟶ℋwraψ defined by(111)Wg=∑a=0∞pa^+∑m=0∞Aa,mfm,gm^za.Theorem 7.1.
The summable equation (10) has one solution in ℋwraψ, if A:ℕ02⟶ℂ, f:ℕ0×ℂ⟶ℂ, p^:ℕ0⟶ℂ, t^:ℕ0⟶ℂ, and for every a∈ℕ0, we have κ∈0,1/2, with(112)∑m∈ℕ0Aa,mfm,gm^−fm,tm^ra≤κKpa^−ga^+∑m=0∞Aa,mfm,gm^ra+pa^−ta^+∑m=0∞Aa,mfm,tm^ra.Proof.
Let the setups be verified. Consider the mappingW:ℋwraψ⟶ℋwraψ defined by (11). We have(113)ψWg−Wt=∑a∈ℕ0∑m∈ℕ0Aa,mfm,gm^−fm,tm^a+1ra1K≤κ∑a∈ℕ0pa^−ga^+∑m=0∞Aa,mfm,gm^a+1ra1K+∑a∈ℕ0pa^−ta^+∑m=0∞Aa,mfm,tm^a+1ra1K=κψWg−g+ψWt−t.
According to Theorem4.2, one obtains a unique solution of equation (10) in ℋwraψ.Example 7.2.
Assume the function spaceℋwa+1/a+2a=0∞ψ, where ψf=∑a∈ℕ0fa^/a+1a+1/a+2, for all f∈ℋwa+1/a+2a=0∞. Consider the summable equation(114)ga^=5−2a+3i+∑m=0∞−1ai+3mcosga^sinhga^+sinma+1q,where q>0 and i2=−1 and let W:ℋwa+1/a+2a=0∞ψ⟶ℋwa+1/a+2a=0∞ψ defined by(115)Wg=∑a=0∞5−2a+3i+∑m=0∞−1ai+3mcosga^sinhga^+sinma+1qza.
It is easy to see that(116)∑m=0∞−1aicosga^sinhga^+sinma+1q−13m−−13ma+1a+2≤135−2a+3i−ga^+∑m=0∞−1ai+3mcosga^sinhga^+sinma+1qa+1a+2+135−2a+3i−ta^+∑m=0∞−1ai+3mcosta^sinhta^+sinma+1qa+1a+2.
By Theorem7.1, the summable equation (114) has one solution in ℋwa+1/a+2a=0∞ψ.Example 7.3.
Given the function spaceℋw2a+1/a+3a=0∞ψ, where ψg=∑a∈ℕ0ga^/a+12a+1/a+3, for all g∈ℋw2a+1/a+3a=0∞, consider the summable equation (12). It is easy to see that(117)∑m=0∞−1aicos|ga^|sinh|ga^|+sinma+1q−13m−−13m2a+1/a+3≤195−2a+3i−ga^+∑m=0∞−1ai+3mcos|ga^|sinh|ga^|+sinma+1q2a+1/a+3+195−2a+3i−ta^+∑m=0∞−1ai+3mcos|ta^|sinh|ta^|+sinma+1q2a+1/a+3.
By Theorem7.1, the summable equation (114) has one solution in ℋwa+1/a+2a=0∞ψ.Example 7.4.
Given the function spaceℋw2a+3/a+2a=0∞ψ, where ψf=∑a∈ℕ0fa^/a+12a+3/a+2, for all f∈ℋw2a+3/a+2a=0∞, consider the summable equation (114) with a≥2 and let W:Ξ⟶Ξ, where Ξ=f∈ℋw2a+3/a+2a=0∞:f0^=f1^=0, defined by(118)Wf=∑a=2∞5−2a+3i+∑m=0∞−1ai+3mcosfa^sinhfa^+sinma+1qza.
Clearly,Ξ is a nonempty, ψ-convex, ψ-closed, and ψ-bounded subset of ℋw2a+3/a+2a=0∞ψ. It is easy to see that(119)∑m=0∞−1aicos|ga^|sinh|ga^|+sinma+1q−13m−−13m2a+3/a+2≤195−2a+3i−ga^+∑m=0∞−1ai+3mcos|ga^|sinh|ga^|+sinma+1q2a+3/a+2+195−2a+3i−ta^+∑m=0∞−1ai+3mcos|ta^|sinh|ta^|+sinma+1q2a+3/a+2.By Theorem 7.1 and Theorem 5.3, the summable equation (114) with a≥2 has a solution in Ξ.In this part, we search for a solution to nonlinear matrix (120) at D∈SℋwraψX,Y, where X and Y are Banach spaces, ra∈mi↗∩ℓ∞, and ΨG=∑a=0∞saG/a+1ra1/K, for all G∈SℋwraψX,Y. Consider the summable equation(120)saG=saP+∑m=0∞Aa,mfm,smG,and suppose W:SℋwraψX,Y⟶SℋwraψX,Y defined by(121)WG=∑a=0∞saP+∑m=0∞Aa,mfm,smGza.Theorem 7.5.
The summable equation (120) has one solution in SℋwraψX,Y, if the following conditions are satisfied:(a)
A:ℕ02⟶ℂ,f:ℕ0×0,∞⟶ℂ,P∈LX,Y,T∈LX,Y, and for every a∈ℕ0, one has κ∈0,1/2, with(122)∑m∈ℕ0Aa,mfm,smG−fm,smTra≤κKsaP−saG+∑m=0∞Aa,mfm,smGra+saP−saT+∑m=0∞Aa,mfm,smTra.(b)
W is Ψ sequentially continuous at a point D∈SℋwraψX,Y.(c)
There isB∈SℋwraψX,Y so that the sequence of iterates WpB has a subsequence WpiB converging to D.Proof.
Suppose the settings are verified. Consider the mappingW:SℋwraψX,Y⟶SℋwraψX,Y defined by (16). We have(123)ψWG−WT=∑a∈ℕ0∑m∈ℕ0Aa,mfm,smG−fm,smTa+1ra1/K≤κ∑a∈ℕ0saP−saG+∑m=0∞Aa,mfm,smGa+1ra1/K+κ∑a∈ℕ0saP−saT+∑m=0∞Aa,mfm,smTa+1ra1/K=κψWG−G+ψWT−T.
In view of Theorem6.6, one obtains a unique solution of (120) at D∈SℋwraψX,Y.Example 7.6.
Assume the function spaceℋwa+1/a+2a=0∞ψ, where ψg=∑a∈ℕ0ga^/a+1a+1/a+2, for all g∈ℋwa+1/a+2a=0∞.
Consider the nonlinear difference equation(124)ga^=e−2a+3i+∑m=0∞tan2m+1cosh3mi−acosp|ga−2^|sinhq|ga−1^|+sinma+1,where g−2^,g−1^,p,q>0,i2=−1, and let W:ℋwa+1/a+2a=0∞ψ⟶ℋwa+1/a+2a=0∞ψ defined by(125)Wg=∑a=0∞e−2a+3i+∑m=0∞tan2m+1cosh3mi−acosp|ga−2^|sinhq|ga−1^|+sinma+1za.
It is easy to see that(126)∑m=0∞cosh3mi−acosp|ga−2^|sinhq|ga−1^|+sinma+1tan2m+1−tan2m+1a+1/a+2≤15e−2a+3i−ga^+∑m=0∞tan2m+1cosh3mi−acosp|ga−2^|sinhq|ga−1^|+sinma+1a+1/a+2+15e−2a+3i−ta^+∑m=0∞tan2m+1cosh3mi−acosp|ta−2^|sinhq|ta−1^|+sinma+1a+1/a+2.
By Theorem7.1, the nonlinear difference equation (124) has one solution in ℋwa+1/a+2a=0∞ψ.Example 7.7.
Given the function spaceℋw2a+1/a+3a=0∞ψ, where ψg=∑a∈ℕ0ga^/a+1a+1/a+2, for all g∈ℋw2a+1/a+3a=0∞, consider the nonlinear difference equation (17). It is easy to see that(127)∑m=0∞cosh3mi−acosp|ga−2^|sinhq|ga−1^|+sinma+1tan2m+1−tan2m+12a+1/a+3≤125e−2a+3i−ga^+∑m=0∞tan2m+1cosh3mi−acosp|ga−2^|sinhq|ga−1^|+sinma+12a+1/a+3+125e−2a+3i−ta^+∑m=0∞tan2m+1cosh3mi−acosp|ta−2^|sinhq|ta−1^|+sinma+12a+1/a+3.
By Theorem7.1, the nonlinear difference equation (124) has one solution in ℋw2a+1/a+3a=0∞ψ.Example 7.8.
Given the function spaceℋw2a+3/a+2a=0∞ψ, where ψg=∑a∈ℕ0ga^/a+1a+1/a+2, for all g∈ℋw2a+3/a+2a=0∞, consider the nonlinear difference equation (124) with a≥1 and let W:Ξ⟶Ξ, where Ξ=g∈ℋw2a+3/a+2a=0∞:g0^=0, defined by(128)Wg=∑a=1∞e−2a+3i+∑m=0∞tan2m+1cosh3mi−acosp|ga−2^|sinhq|ga−1^|+sinma+1za.
Clearly,Ξ is a nonempty, ψ-convex, ψ-closed, and ψ-bounded subset of ℋw2a+3/a+2a=0∞ψ. It is easy to see that(129)∑m=0∞cosh3mi−acosp|ga−2^|sinhq|ga−1^|+sinma+1tan2m+1−tan2m+12a+1/a+3≤125e−2a+3i−ga^+∑m=0∞tan2m+1cosh3mi−acosp|ga−2^|sinhq|ga−1^|+sinma+12a+1/a+3+125e−2a+3i−ta^+∑m=0∞tan2m+1cosh3mi−acosp|ta−2^|sinhq|ta−1^|+sinma+12a+1/a+3.
By Theorem7.1 and Theorem 5.3, the nonlinear difference equation (124) with a≥1 has a solution in Ξ.


## 8. Conclusion

This paper studies the existence of a fixed point for Kannan’s prequasinorm contractive mappings in function spaces of complex variables. We have studied the existence of fixed points of Kannan prequasinorm nonexpansive mapping and the existence of Kannan’s prequasinorm contractive mapping in the prequasi-Banach operator ideal created by this function space ands-numbers. We have also presented some applications of summable equations. Several numerical experiments were introduced to illustrate our results. Moreover, some successful applications to the existence of solutions of nonlinear difference equations are discussed. This paper has several advantages for researchers, such as studying the fixed points of any contraction mappings on this prequasinormed function space, which is a generalization of the quasinormed function space, examining the eigenvalue problem in these new settings and noting that the closed operator ideals are certain to play an important function in the principle of Banach lattices.



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