# Some Fixed Point Results of Kannan Maps on the Nakano Sequence Space

Authors: Awad A. Bakery; O. M. Kalthum S. K. Mohamed
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In the recent past, some researchers studied some fixed point results on the modular variable exponent sequence spaceℓr.ψ, where ψv=∑a=0∞1/ravara and ra≥1. They depended on their proof that the modular ψ has the Fatou property. But we have explained that this result is incorrect. Hence, in this paper, the concept of the premodular, which generalizes the modular, on the Nakano sequence space such as its variable exponent in 1,∞ and the operator ideal constructed by this sequence space and s-numbers is introduced. We construct the existence of a fixed point of Kannan contraction mapping and Kannan nonexpansive mapping acting on this space. It is interesting that several numerical experiments are presented to illustrate our results. Additionally, some successful applications to the existence of solutions of summable equations are introduced. The novelty lies in the fact that our main results have improved some well-known theorems before, which concerned the variable exponent in the aforementioned space.


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## Body

## 1. Introduction

Ideal operators and summability theorems are awfully invaluable in mathematical models and have large executions, for example, the fixed point theory, geometry of Banach spaces, normal series theory, approximation theory, and ideal transformations. For added evidence, see [1–4]. By RN, ℓ∞, ℓr, and c0, we denote the spaces of all, bounded, r-absolutely summable and null sequences of real numbers. We indicate the space of all bounded linear operators from a Banach space Z into a Banach space M by LZ,M, and if Z=M, we inscribe LZ and ed=0,0,⋯,1,0,0,⋯, while 1 displays at the dth place, for all d∈N=0,1,2,⋯.Definition 1 [5].
Ans-number function is a map detailed on LZ,M which sorts every map W∈LZ,M a nonnegative scaler sequence sdWd=0∞ overbearing that the next setting encompasses
(a)
W=s0W≥s1W≥s2W≥⋯≥0, for all W∈LZ,M(b)
sl+d−1W1+W2≤slW1+sdW2, for every W1,W2∈LZ,M, and l, d∈N(c)
Ideal property:sdVYW≤VsdYW, for every W∈LZ0,Z, Y∈LZ,M, and V∈LM,M0, where Z0 and M0 are discretionary Banach spaces(d)
ForW∈LZ,M and γ∈R, one has sdγW=γsdW(e)
Rank property: assume rankW≤d, then sdW=0, for each W∈LZ,M(f)
Norming property:sl≥aIa=0 or sl<aIa=1, where Ia mirrors the unit map on the a-dimensional Hilbert space ℓ2aThedth approximation number, established by αdW, is defined as
(1)αdW=infW−Y:Y∈LZ,MandrankY≤d.Notation 2.
The setsSA,SAZ,M, SAapp, and SAappZ,M (cf. [6]) indicate
(2)SA≔SAZ,M,where SAZ,M≔W∈LZ,M:sdWd=0∞∈A. Also, SAapp≔SAappZ,M, where SAappZ,M≔W∈LZ,M:αdWd=0∞∈A.Suppose thatr=ra∈R+N, the Nakano sequence space defined and studied in [7–9] is denoted by ℓr=v=va∈RN:ϕμv<∞,forsomeμ>0, when ϕv=∑a=0∞vara.The spaceℓr,., where v=infκ>0:ϕv/k≤1 and ra≥1, for all a∈N, is a Banach space. If ra∈ℓ∞, then,
(3)ℓr=v=va∈RN:ϕμv<∞,forsomeμ>0=v=va∈RN:infaμra∑a=0∞vara≤∑a=0∞μvara<∞,forsomeμ>0=v=va∈RN:∑a=0∞vara<∞=v=va∈RN:ϕμv<∞,foranyμ>0.The vector spacesℓr are contained in the variable exponent spaces Lr. In the second half of the twentieth century, it was assumed that these variable exponent spaces provided the proper framework for the mathematical components of numerous issues for which the classical Lebesgue spaces were insufficient. Because of the importance of these spaces and their surroundings, they have become a well-known and environmentally friendly tool in the treatment of a variety of conditions; currently, the region of LrΩ spaces is a prolific subject of research, with ramifications extending into a wide range of mathematical specialties (see [10]). The mathematical description of the hydrodynamics of non-Newtonian fluids provides an impetus for learning about variable exponent Lebesgue spaces, Lr (see [11, 12]). Applications of non-Newtonian fluids, known as electrorheological, vary from their use in army science to civil engineering and orthopedics. Faried and Bakery provided the theory of the pre-quasioperator ideal, which is more general than the quasioperator ideal, in [6]. In [7], Bakery and Abou Elmatty explained the sufficient (not necessary) setting on ℓr so that Sℓr generated a simple Banach pre-quasioperator ideal. The pre-quasioperator ideal Sℓrapp is strictly restricted to different powers. It was a small pre-quasioperator ideal. Because of the booklet of the Banach fixed point theorem [13], many mathematicians have worked on many developments. Kannan [14] 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 [15]. Bakery and Mohamed [16] explored the concept of the pre-quasinorm on the Nakano sequence space such that its variable exponent in 0,1. They explained the sufficient conditions on it, equipped with the definite pre-quasinorm to generate pre-quasi-Banach and closed space, and examined the Fatou property of different pre-quasinorms on it. Moreover, they showed the existence of a fixed point of Kannan pre-quasinorm contraction maps on it and on the pre-quasi-Banach operator ideal constructed by s-numbers which belong to this sequence space. For more details on Kannan’s fixed point theorems, see [17–24]. The aim of this paper is to examine the concept of the pre-quasinorm on ℓr with a variable exponent in 1,∞. We study the sufficient conditions on ℓr equipped with the definite pre-quasinorm to form pre-quasi-Banach and closed (sss), the existence of a fixed point of Kannan pre-quasinorm contraction mapping in the pre-quasi-Banach (sss), ℓrϕ satisfies the property (R), and ℓrϕ has the ϕ-normal structure property. The existence of a fixed point of Kannan pre-quasinorm nonexpansive mapping in the pre-quasi-Banach (sss) has been given. Finally, we examine the idea of Kannan pre-quasinorm contraction mapping in the pre-quasioperator ideal. As well, the existence of a fixed point of Kannan pre-quasinorm contraction mapping in the pre-quasi-Banach operator ideal Sℓrϕ has been introduced. Finally, some illustrative examples and applications to the existence of solutions of summable equations are given.


## 2. Definitions and Preliminaries

By0,∞A, we denote the space of all functions ϕ:A⟶0,∞. Nakano [25] introduced the concept of modular vector spaces.Definition 3.
Suppose thatA is a vector space. A function ϕ∈0,∞A is called modular if the next conditions hold
(a)
Forv∈A, v=θ⟺ϕv=0 with ϕv≥0, where θ is the zero vector of A(b)
ϕηv=ϕv holds, for all v∈A and η=1(c)
The inequalityϕαv+1−αt≤ϕv+ϕt satisfies, for all v,t∈A and α∈0,1The concept of premodular vector spaces is more general than modular vector spaces.Definition 4 [2].
The linear space of sequencesA is called a special space of sequences (sss), if
(a)
eaa∈N⊆A(b)
A is solid, i.e., assume that v=va∈RN, t=ta∈A, and va≤ta, for each a∈N, and then v∈A(c)
va/2a=0∞∈A, where a/2 indicates the integral part of a/2, in case vaa=0∞∈ADefinition 5 [6].
A subclassAϕ of A is named a premodular (sss), if there is ϕ∈0,∞A, it satisfies the next setting:
(i)
Forv∈A, v=θ⟺ϕv=0 with ϕv≥0(ii)
For someB≥1, the inequality ϕηv≤Bηϕv holds, for all v∈A and η∈R(iii)
For someJ≥1, the inequality ϕv+t≤Jϕv+ϕt satisfies, for all v,t∈A(iv)
Fora∈N and va≤ta, we have ϕva≤ϕta(v)
The inequality,ϕva≤ϕva/2≤J0ϕva includes, for some J0≥1(vi)
LetF be the space of finite sequences, then F¯=Aϕ(vii)
we haveς>0 such that ϕβ,0,0,0,⋯≥ςβϕ1,0,0,0,⋯, for all β∈R
This is an example of a premodular vector space but not a modular vector space.Example 6.
The functionϕv=∑a∈Nvaa+1/3a+44 is a premodular (not a modular) on the vector space ℓa+1/3a+4a=0∞. Since for all v,t∈ℓa+1/3a+4a=0∞, we have
(4)ϕv+t2=∑a∈Nva+ta2a+1/3a+44≤4ϕv+ϕt.Definition 7 [26].
LetA be a (sss). The function ϕ∈0,∞A is named a pre-quasinorm on A, if it provides the following setting:
(i)
Forv∈A, v=θ⟺ϕv=0 with ϕv≥0(ii)
For someB≥1, the inequality ϕηv≤Bηϕv holds, for all v∈A and η∈R(iii)
For someJ≥1, the inequality ϕv+t≤Jϕv+ϕt satisfies, for all v,t∈ATheorem 8 [26].
LetA be a premodular (sss), and then, it is pre-quasinormed (sss).Theorem 9 [26].
A is a pre-quasinormed (sss), if it is quasinormed (sss).Definition 10 [3].
Suppose thatL is the class of all bounded linear operators within any two arbitrary Banach spaces. A subclass U of L is called an operator ideal, if every element UZ,M=U∩LZ,M satisfies the next conditions:
(i)
IΓ∈U, where Γ describes the Banach space of one dimension(ii)
The spaceUZ,M is linear over R(iii)
IfW∈LZ0,Z, X∈UZ,M, and Y∈LM,M0, then, YXW∈UZ0,M0, where Z0 and M0 are normed spaces (see [27, 28])
This is the concept of the pre-quasioperator ideal which is added in general to the quasioperator ideal.Definition 11 [6].
A functionϕ∈0,∞U is called a pre-quasinorm on the ideal U if the next conditions hold:
(1)
LetW∈UZ,M, ϕW≥0, and ϕW=0, if and only if, W=0(2)
We haveD≥1 so as to ϕηW≤DηϕW, for every W∈UZ,M and η∈R(3)
We haveJ≥1 so that ϕW1+W2≤JϕW1+ϕW2, for each W1,W2∈UZ,M(4)
We haveσ≥1 if W∈LZ0,Z, X∈UZ,M, and Y∈LM,M0, and then, ϕYXW≤σYϕXWTheorem 12 [29].
Assuming thatAϕ is a pre-modular (sss), then, ϕW=ϕsaWa=0∞ is a pre-quasinorm on SAϕ.Theorem 13 [7].
LetZ and M be Banach spaces and Aϕ be a premodular (sss), and then, SAϕ,ϕ is a pre-quasi-Banach operator ideal, such that ϕW=ϕsaWa=0∞.Theorem 14 [6].
ϕ is a pre-quasinorm on the ideal U, if ϕ is a quasinorm on the ideal U.Lemma 15.
The given inequalities will be used in the sequel:(i)
Letr≥2, and for every v,t∈R [30], then(5)v+t2r+v−t2r≤12vr+tr(ii)
Assume that1<r≤2, and for all v,t∈R so that v+t≠0 [31], then(6)v+t2r+rr−12v−tv+t2−rv−t2r≤12vr+tr(iii)
Suppose thatra≥1 and va,ta∈R, for every a∈N, then, va+tara≤2K−1vara+tara, where K=supara [32]


## 3. Pre-Quasinormed (sss)

We explain the sufficient setting ofℓr equipped with a pre-quasinorm ϕ to generate pre-quasi-Banach and closed (sss). The Fatou property of a pre-quasinorm ϕ on ℓr has been given.Definition 16.
(a)
The functionϕ on ℓr is named ϕ convex, if ϕωv+1−ωt≤ωϕv+1−ωϕt, for all ω∈0,1 and v,t∈ℓr(b)
vaa∈N⊆ℓrϕ is ϕ convergent to v∈ℓrϕ, if and only if, lima⟶∞ϕva−v=0. If the ϕ limit exists, then it is unique(c)
vaa∈N⊆ℓrϕ is ϕ Cauchy, when lima,b⟶∞ϕva−vb=0(d)
Λ⊂ℓrϕ is ϕ closed, if for every ϕ-converges vaa∈N⊂Λ to v, then v∈Λ(e)
Λ⊂ℓrϕ is ϕ bounded, when δϕΛ=supϕv−t:v,t∈Λ<∞(f)
Theϕ ball of radius d≥0 and center v, for all v∈ℓrϕ, is detailed as(7)Bϕv,d=t∈ℓrϕ:ϕv−t≤d(g)
A pre-quasinormϕ on ℓr provides the Fatou property, if for all sequence ta⊆ℓrϕ with lima⟶∞ϕta−t=0 and any v∈ℓrϕ, then ϕv−t≤supjinfa≥jϕv−ta
Note that the Fatou property implies theϕ closedness of the ϕ balls.Theorem 17.
ℓrϕ, where ϕv=∑a=0∞vara1/K, for each v∈ℓr, is a premodular (sss), if raa∈N∈ℓ∞ is increasing with r0>1.Proof.
To begin with, we have to show thatℓr is a (sss):
(1)
Assumev,t∈ℓr. As ra is bounded, we get(8)ϕv+t=∑a=0∞va+tara1/K≤∑a=0∞vara1/K+∑a=0∞tara1/K=ϕv+ϕt<∞.
Hence,v+t∈ℓr.
And suppose thatη∈R and v∈ℓr. Since ra is bounded, we obtain
(9)ϕηv=∑a=0∞ηvara1/K≤supaηra/K∑a=0∞vara1/K≤Dηϕv<∞.
So,ηv∈ℓr. Therefore, by using equations (8) and (9), we have that ℓr is linear. Also, ea∈ℓr, for every a∈N, as ϕea=∑j=0∞eajrj1/K=1(2)
Supposeva≤ta, for every a∈N and t∈ℓr. We have(10)ϕv=∑a=0∞vara1/K≤∑a=0∞tara1/K=ϕt<∞.
Then,v∈ℓr.
(3)
Assuming thatva∈ℓr and ra is an increasing sequence, we have(11)ϕva/2=∑a=0∞va/2ra1/K=∑a=0∞var2a+∑a=0∞var2a+11/K≤21/K∑a=0∞vara1/K=21/Kϕva.
Then,va/2∈ℓr. As well, we prove that the functional ϕ on ℓr is a premodular:
(i)
Clearly,ϕv≥0 and ϕv=0⟺v=θ(ii)
We haveD=max1,supaηra/K−1≥1 such that ϕηv≤Dηϕv, for every v∈ℓr and η∈R(iii)
We haveJ≥1 so that ϕv+t≤Jϕv+ϕt, for every v,t∈ℓr(iv)
Evidently, from (101)(v)
From (104), we have J0=21/K≥1(vi)
Evidently,F¯=ℓr(vii)
We have0<ς≤βr0/K−1, for β≠0 or ς>0, for β=0 so that(12)ϕβ,0,0,0,⋯≥ςβϕ1,0,0,0,⋯
☐Theorem 18.
Letraa∈N∈ℓ∞ be an increase with r0>1, and then, ℓrϕ is a pre-quasi-Banach (sss), where ϕv=∑a=0∞vara1/K, for all v∈ℓr.Proof.
Suppose that the setup is satisfied. From Theorem17, the space ℓrϕ is a premodular (sss). By Theorem 8, the space ℓrϕ is a pre-quasinormed (sss). To explain that ℓrϕ is a pre-quasi-Banach (sss), suppose that vp=vapa=0∞ is a Cauchy sequence in ℓrϕ. Therefore, for all ε∈0,1, there is p0∈N so that for every p,q≥p0, we have
(13)ϕvp−vq=∑a=0∞vap−vaqra1/K<ε.
Hence, forp,q≥p0, and a∈N, we have vap−vaq<ε. Hence, vaq is a Cauchy sequence in R, for fixed a∈N, which gives limq⟶∞vaq=va0, for constant a∈N. So, ϕvp−v0<ε, for all p≥p0. Conclusively, to prove that v0∈ℓr, one has ϕv0=ϕv0−vp+vp≤ϕvp−v0+ϕvp<∞. Hence, v0∈ℓr. This gives that ℓrϕ is a pre-quasi-Banach (sss). ☐Theorem 19.
Assuming thatraa∈N∈ℓ∞ is increasing with r0>1, then, ℓrϕ is a pre-quasiclosed (sss), where ϕv=∑a=0∞vara1/K, for all v∈ℓr.Proof.
Let the setup be satisfied. From Theorem17, the space ℓrϕ is a premodular (sss). By Theorem 8, the space ℓrϕ is a pre-quasinormed (sss). To prove that ℓrϕ is a pre-quasiclosed (sss), let vp=vapa=0∞∈ℓrϕ and limp⟶∞ϕvp−v0=0; then, for each ε∈0,1, there is p0∈N such that for every p≥p0, one can see
(14)ε>ϕvp−v0=∑a=0∞vap−va0ra1/K.
Therefore, forp≥p0 and a∈N, we have vap−va0<ε. Hence, vap is a convergent sequence in R, for constant a∈N. So, limp⟶∞vap=va0, for fixed a∈N. Finally, to show that v0∈ℓr, one has
(15)ϕv0=ϕv0−vp+vp≤ϕvp−v0+ϕvp<∞.
Hence,v0∈ℓr. This implies that ℓrϕ is a pre-quasiclosed (sss). ☐Theorem 20.
The functionϕv=∑a=0∞vara1/K satisfies the Fatou property, if raa∈N∈ℓ∞ is increasing with r0>1, for every v∈ℓr.Proof.
Assume that the setup is verified andtb⊆ℓrϕ with limb⟶∞ϕtb−t=0. As the space ℓrϕ is a pre-quasiclosed space, then, t∈ℓrϕ. Hence, for all v∈ℓrϕ, we have
(16)ϕv−t=∑a=0∞va−tara1/K≤∑a=0∞va−tabra1/K+∑a=0∞tab−tara1/K≤supjinfb≥jϕv−tb.
☐Theorem 21.
The functionϕv=∑a=0∞vara does not verify the Fatou property, for every v∈ℓr, if ra∈ℓ∞ and ra>1, for each a∈N.Proof.
Assume that the setting is verified andtb⊆ℓrϕ with limb⟶∞ϕtb−t=0. As the space ℓrϕ is a pre-quasiclosed space, then, t∈ℓrϕ. Then, for all v∈ℓrϕ, one can see
(17)ϕv−t=∑a=0∞va−tara≤2supara−1∑a=0∞va−tabra+∑a=0∞tab−tara≤2supara−1supjinfb≥jϕv−tb.
Therefore,ϕ does not verify the Fatou property. ☐Similarly as Theorems17 and 19 under the conditions raa∈N∈ℓ∞ is increasing with r0>1, it is easy to prove that the space ℓr.ψ, which is studied in [33], is a pre-quasiclosed (sss), where ψv=∑a=0∞1/ravara.Theorem 22.
The functionψv=∑a=0∞1/ravara1/K satisfies the Fatou property, if raa∈N∈ℓ∞ is increasing with r0>1, for every v∈ℓr.ψ.Proof.
Assume that the setup is verified andtb⊆ℓr.ψ with limb⟶∞ψtb−t=0. As the space ℓr.ψ is a pre-quasiclosed space, then, t∈ℓr.ψ. Hence, for all v∈ℓr.ψ, we have
(18)ψv−t=∑a=0∞1rava−tara1/K≤∑a=0∞1rava−tabra1/K+∑a=0∞1ratab−tara1/K≤supjinfb≥jψv−tb.Theorem 23.
The functionψv=∑a=0∞1/ravara does not verify the Fatou property, for every v∈ℓr.ψ, if ra∈ℓ∞ and ra>1, for each a∈N.Proof.
Assume that the setting is confirmed andtb⊆ℓr.ψ with limb⟶∞ψtb−t=0. As the space ℓr.ψ is a pre-quasiclosed space, then, t∈ℓr.ψ. Then, for all v∈ℓr.ψ, one can see
(19)ψv−t=∑a=0∞1rava−tara≤2supara−1∑a=0∞1rava−tabra+∑a=0∞1ratab−tara≤2supara−1supjinfb≥jψv−tb.
Therefore,ψ does not verify the Fatou property. ☐Example 24.
The functionϕv=∑a∈Nva3a+2/a+1 is a pre-quasinorm (not a quasinorm) on ℓ3a+2/a+1a=0∞.Example 25.
The functionϕv=∑a∈Nva3a+2/a+13 is a pre-quasinorm (not a norm) on ℓ3a+2/a+1a=0∞.Example 26.
The functionϕv=∑a∈Nvarr is a pre-quasinorm, quasi norm, and not a norm on ℓr, for 0<r<1.Example 27.
Forra∈1,∞N, the function ϕv=infκ>0:∑a∈Nva/kra≤1 is a pre-quasinorm (a quasinorm and a norm) on ℓr.


## 4. Kannan Prequasiϕ Contraction Mapping

In this section, we will define Kannanϕ-Lipschitzian mapping in the pre-quasinormed (sss). We study the sufficient setting on ℓrϕ constructed with definite pre-quasinorm so that there is one and only one fixed point of Kannan pre-quasinorm contraction mapping.Definition 28.
An operatorW:Aϕ⟶Aϕ is named a Kannan ϕ-Lipschitzian, if there is ξ≥0, such that
(20)ϕWv−Wt≤ξϕWv−v+ϕWt−t,for every v,t∈Aϕ. The operator W is named
(1)
Kannanϕ contraction, if ξ∈0,1/2(2)
Kannanϕ nonexpansive, if ξ=1/2
An elementv∈Aϕ is called a fixed point of W, when Wv=v.In fact, the authors of reference [33] in Theorem 1 proved that the Kannan modular contraction mapping on a nonempty modular-closed subset of the modular space ℓr.ψ, where ψv=∑a=0∞1/ravara and ra≥1, for all a∈N, has a unique fixed point. They depended on their proof that the modular ψ has the Fatou property. But from Theorem 23, this result is incorrect. We have improved it in the next theorem.Theorem 29.
Letraa∈N∈ℓ∞ be an increase with r0>1 and W:ℓrϕ⟶ℓrϕ be Kannan ϕ contraction mapping, where ϕv=∑a=0∞vara1/K, for every v∈ℓr, so W has a unique fixed point.Proof.
Assume that the conditions are verified. For allv∈ℓr, then, Wpv∈ℓr. Since W is a Kannan ϕ contraction mapping, we have
(21)ϕWp+1v−Wpv≤ξϕWp+1v−Wpv+ϕWpv−Wp−1v⇒ϕWp+1v−Wpv≤ξ1−ξϕWpv−Wp−1v≤ξ1−ξ2ϕWp−1v−Wp−2v≤≤ξ1−ξpϕWv−v.
Therefore, for everyp,q∈N with q>p, then, we have
(22)ϕWpv−Wqv≤ξϕWpv−Wp−1v+ϕWqv−Wq−1v≤ξξ1−ξp−1+ξ1−ξq−1ϕWv−v.
Hence,Wpv is a Cauchy sequence in ℓrϕ. Since the space ℓrϕ is a pre-quasi-Banach space, so, there is t∈ℓrϕ so that limp⟶∞Wpv=t. To show that Wt=t, as ϕ has the Fatou property, we get
(23)ϕWt−t≤supiinfp≥iϕWp+1v−Wpv≤supiinfp≥iξ1−ξpϕWv−v=0.
So,Wt=t. Then, t is a fixed point of W. To prove that the fixed point is unique, assume that we have two different fixed points b,t∈ℓrϕ of W. Therefore, one can see
(24)ϕb−t≤ϕWb−Wt≤ξϕWb−b+ϕWt−t=0.
Hence,b=t. ☐Corollary 30.
Suppose thatraa∈N∈ℓ∞ is increasing with r0>1 and W:ℓrϕ⟶ℓrϕ is Kannan ϕ contraction mapping, where ϕv=∑a=0∞vara1/K, for every v∈ℓr, then W has unique fixed point b with ϕWpv−b≤ξξ/1−ξp−1ϕWv−v.Proof.
Assume that the setup is verified. By Theorem29, there is a unique fixed point b of W. Therefore, one can see
(25)ϕWpv−b=ϕWpv−Wb≤ξϕWpv−Wp−1v+ϕWb−b=ξξ1−ξp−1ϕWv−v.
☐Definition 31.
LetAϕ be a pre-quasinormed (sss), W:Aϕ⟶Aϕ and b∈Aϕ. The operator W is named ϕ sequentially continuous at b, if and only if lima⟶∞ϕva−b=0, then lima⟶∞ϕWva−Wb=0.Theorem 32.
Ifraa∈N∈ℓ∞ is increasing with r0>1 and W:ℓrϕ⟶ℓrϕ, where ϕv=∑a=0∞vara, for every v∈ℓr, the point g∈ℓrϕ is the only fixed point of W, if the next settings are verified:
(a)
W is Kannan ϕ contraction mapping(b)
W is ϕ sequentially continuous at g∈ℓrϕ(c)
We havev∈ℓrϕ such that the sequence of iterates Wpv has a subsequence Wpiv converges to gProof.
If the settings are satisfied, letg be not a fixed point of W, and then, Wg≠g. By the setups (b) and (c), one can see
(26)limpi⟶∞ϕWpiv−g=0andlimpi⟶∞ϕWpi+1v−Wg=0.
Since the operatorW is Kannan ϕ contraction, we have
(27)0<ϕWg−g=ϕWg−Wpi+1v+Wpiv−g+Wpi+1v−Wpiv≤22supiri−2ϕWpi+1v−Wg+22supiri−2ϕWpiv−g+2supiri−1ξξ1−ξpi−1ϕWv−v.
Sincepi⟶∞, we get a contradiction. Hence, g is a fixed point of W. To show that the fixed point g is unique, suppose that we have two different fixed points g,b∈ℓrϕ of W. Therefore, we have
(28)ϕg−b≤ϕWg−Wb≤ξϕWg−g+ϕWb−b=0.
So,g=b. ☐Example 33.
LetW:ℓ2a+3/a+2a=0∞ϕ⟶ℓ2a+3/a+2a=0∞ϕ, where ϕv=∑a∈Nva2a+3/a+2, for all v∈ℓ2a+3/a+2a=0∞ and
(29)Wv=v4,ϕv∈0,1,v5,ϕv∈1,∞.Since for allv1,v2∈ℓ2a+3/a+2a=0∞ϕ with ϕv1,ϕv2∈0,1, we have
(30)ϕWv1−Wv2=ϕv14−v24≤1274ϕ3v14+ϕ3v24=1274ϕWv1−v1+ϕWv2−v2.For allv1,v2∈ℓ2a+3/a+2a=0∞ϕ with ϕv1,ϕv2∈1,∞, we have
(31)ϕWv1−Wv2=ϕv15−v25≤1644ϕ4v15+ϕ4v25=1644ϕWv1−v1+ϕWv2−v2.For allv1,v2∈ℓ2a+3/a+2a=0∞ϕ with ϕv1∈0,1 and ϕv2∈1,∞, we have
(32)ϕWv1−Wv2=ϕv14−v25≤1274ϕ3v14+1644ϕ4v25≤1274ϕ3v14+ϕ4v25=1274ϕWv1−v1+ϕWv2−v2.Therefore, the mapW is Kannan ϕ contraction mapping, since ϕ satisfies the Fatou property. By Theorem 29, the map W has a unique fixed point θ∈ℓ2a+3/a+2a=0∞ϕ.Letvn⊆ℓ2a+3/a+2a=0∞ϕ be such that limn⟶∞ϕvn−v0=0, where v0∈ℓ2a+3/a+2a=0∞ϕ with ϕv0=1. Since the pre-quasinorm ϕ is continuous, we have
(33)limn⟶∞ϕWvn−Wv0=limn⟶∞ϕvn4−v05=ϕv020>0.Hence,W is not ϕ sequentially continuous at v0. So, the map W is not continuous at v0.Ifϕv=∑a∈Nva2a+3/a+2, for all v∈ℓ2a+3/a+2a=0∞. Since for all v1,v2∈ℓ2a+3/a+2a=0∞ϕ with ϕv1,ϕv2∈0,1, we have
(34)ϕWv1−Wv2=ϕv14−v24≤227ϕ3v14+ϕ3v24=227ϕWv1−v1+ϕWv2−v2.For allv1,v2∈ℓ2a+3/a+2a=0∞ϕ with ϕv1,ϕv2∈1,∞, we have
(35)ϕWv1−Wv2=ϕv15−v25≤14ϕ4v15+ϕ4v25=14ϕWv1−v1+ϕWv2−v2.For allv1,v2∈ℓ2a+3/a+2a=0∞ϕ with ϕv1∈0,1 and ϕv2∈1,∞, we have
(36)ϕWv1−Wv2=ϕv14−v25≤227ϕ3v14+14ϕ4v25≤227ϕ3v14+ϕ4v25=227ϕWv1−v1+ϕWv2−v2.Therefore, the mapW is Kannan ϕ contraction mapping and
(37)Wpv=v4p,ϕv∈0,1,v5p,ϕv∈1,∞.It is clear thatW is ϕ sequentially continuous at θ∈ℓ2a+3/a+2a=0∞ϕ and Wpv has a subsequence Wpiv that converges to θ. By Theorem 32, the point θ∈ℓ2a+3/a+2a=0∞ϕ is the only fixed point of W.Example 34.
LetWℓ2a+3/a+2a=0∞ϕ⟶ℓ2a+3/a+2a=0∞ϕ, where ϕv=∑a∈Nva2a+3/a+2, for all v∈ℓ2a+3/a+2a=0∞ϕ and
(38)Wv=14e1+v,v0∈−∞,13,13e1,v0=13,14e1,v0∈13,∞.Since for allv,t∈ℓ2a+3/a+2a=0∞ϕ with v0,t0∈−∞,1/3, we have
(39)ϕWv−Wt=ϕ14v0−t0,v1−t1,v2−t2,⋯≤227ϕ3v4+ϕ3t4≤227ϕWv−v+ϕWt−t.For allv,t∈ℓ2a+3/a+2a=0∞ϕ with v0,t0∈1/3,∞, then, for any ε>0, we have
(40)ϕWv−Wt=0≤εϕWv−v+ϕWt−t.For allv,t∈ℓ2a+3/a+2a=0∞ϕ with v0∈−∞,1/3 and t0∈1/3,∞, we have
(41)ϕWv−Wt=ϕv4≤127ϕ3v4=127ϕWv−v≤127ϕWv−v+ϕWt−t.Therefore, the mapW is Kannan ϕ contraction mapping. It is clear that W is ϕ sequentially continuous at 1/3e1∈ℓ2a+3/a+2a=0∞ϕ and there is v∈ℓ2a+3/a+2a=0∞ϕ with v0∈−∞,1/3 such that the sequence of iterates Wpv=∑n=1p1/4ne1+1/4pv has a subsequence Wpiv=∑n=1pi1/4ne1+1/4piv converges to 1/3e1. By Theorem 32, the map W has one fixed point 1/3e1∈ℓ2a+3/a+2a=0∞ϕ. Note that W is not continuous at 1/3e1∈ℓ2a+3/a+2a=0∞ϕ.Ifϕv=∑a∈Nva2a+3/a+2, for all v∈ℓ2a+3/a+2a=0∞ϕ. Since for all v,t∈ℓ2a+3/a+2a=0∞ϕ with v0,t0∈−∞,1/3, we have
(42)ϕWv−Wt=ϕ14v0−t0,v1−t1,v2−t2,⋯≤1274ϕ3v4+ϕ3t4≤1274ϕWv−v+ϕWt−t.For allv,t∈ℓ2a+3/a+2a=0∞ϕ with v0,t0∈1/3,∞, then, for any ε>0, we have
(43)ϕWv−Wt=0≤εϕWv−v+ϕWt−t.For allv,t∈ℓ2a+3/a+2a=0∞ϕ with v0∈−∞,1/3 and t0∈1/3,∞, we have
(44)ϕWv−Wt=ϕv4≤1274ϕ3v4=1274ϕWv−v≤1274ϕWv−v+ϕWt−t.Therefore, the mapW is Kannan ϕ contraction mapping. Since ϕ satisfies the Fatou property. By Theorem 29, the map W has a unique fixed point 1/3e1∈ℓ2a+3/a+2a=0∞ϕ.


## 5. Pre-Quasinormed Uniform Convexity

In this part, we investigate the uniform convexity (UUC 2) defined in [35] of the pre-quasinormed (sss) ℓrϕ.Definition 35 [10, 34].
We define the following uniform convexity-type behavior of the pre-quasinormϕ:
(1)
Assume thata>0 and b>0 [35]. Indicate that(45)ℍ1a,b=v,t:v,t∈Aϕ,ϕv≤a,ϕt≤b,ϕv−t≥ab.
Whenℍ1a,b≠∅, we put
(46)H1a,b=inf1−1aϕv+t2:v,t∈ℍ1a,b.
Whenℍ1a,b=∅, we put H1a,b=1. The function ϕ investigates the uniform convexity (UC) if for every a>0 and b>0, we have H1a,b>0. Note that for all a>0, then, ℍ1a,b≠∅, for very small b>0(2)
The functionϕ provides (UUC) if for every p≥0 and b>0, there is β1p,b based on p and b>0 so that [36](47)H1a,b>β1p,b>0,fora>p(3)
Suppose thata>0 and b>0. Indicate [36](48)ℍ2a,b=v,t:v,t∈Aϕ,ϕv≤a,ϕt≤a,ϕv−t2≥ab.
Whenℍ2a,b≠∅, we put
(49)H2a,b=inf1−1aϕv+t2:v,tv,t∈ℍ2a,b.
Whenℍ2a,b=∅, we put H2a,b=1. The function ϕ supports (UC 2) if for all a>0 and b>0, we have H2a,b>0. Observe that for each a>0, ℍ2a,b≠∅, for very small b>0(4)
The functionϕ satisfies (UUC 2) if for every p≥0 and b>0, there is β2p,b based on p and b>0 so that [36](50)H2a,b>β2p,b>0,fora>p.(5)
The functionϕ is strictly convex (SC), if for each v,t∈Aϕ so that ϕv=ϕt and ϕv+t/2=ϕv+ϕt/2, we get v=t [35]
Here and after, we will need the following notation:ϕUv=∑m∈Uvmrm1/K, for each U⊂N and v∈ℓrϕ. When U=∅, we set ϕUv=0.Theorem 36.
The pre-quasinormϕ on ℓr is (UUC 2), where ϕv=[∑d=0∞vdrd1/K, for every v∈ℓr, if raa∈N∈ℓ∞ is an increasing with r0>1.Proof.
Supposing that the setting is verified,a>p≥0 and b>0. Let v,t∈ℓr such that
(51)ϕv≤a,ϕt≤a,ϕv−t2≥ab.
From the definition ofϕ, one has
(52)ab≤ϕv−t2=∑d=0∞vd−td2rd1/K≤2−r0∑d=0∞vd−tdrd1/K≤2−r0K∑d=0∞vdrd1K+∑d=0∞tdrd1K=2−r0Kϕv+ϕt≤2a.
This implies thatb≤2. So, set P=d∈N:rd≥2 and Q=d∈N:1<rd<2=N\P. For every w∈ℓr, we have ϕKw=ϕPKw+ϕQKw. By using the setting, we obtain ϕPv−t/2≥ab/2 or ϕQv−t/2≥ab/2. Suppose first that ϕPv−t/2≥ab/2. Using Lemma 15, one can see
(53)ϕPKv+t2+ϕPKv−t2≤ϕPKv+ϕPKt2.
This gives(54)ϕPKv+t2≤ϕPKv+ϕPKt2−ab2K.
Since(55)ϕQKv+t2≤ϕQKv+ϕQKt2,one has
(56)ϕKv+t2≤ϕKv+ϕKt2−ab2K≤aK1−b2K.
This implies(57)ϕv+t2≤a1−b2K1/K.
Next, assume thatϕQv−t/2≥ab/2. Put B=b/4,(58)Q1=d∈Q:vd−td≤Bvd+td,Q2=Q\Q1.
As the power function is convex andB≤1. Hence,
(59)ϕQ1Kv−t2≤∑d∈Q1Brd∣vd∣+∣td∣2rd≤B2r0ϕQ1Kv+ϕQ1Kt≤B2ϕQKv+ϕQKt≤BaK.
AsϕQv−t/2≥ab/2, we have
(60)ϕQ2Kv−t2=ϕQKv−t2−ϕQ1Kv−t2≥aKb2K−b4K.
For alld∈Q2, one can see
(61)r0−1≤rdrd−1,B≤B2−rd≤vd−tdvd+td2−rd.
By Lemma15, we have
(62)vd+td2rd+r0−1B2vd−td2rd≤12vdrd+tdrd.
So,(63)ϕQ2Kv+t2+r0−1B2ϕQ2Kv−t2≤ϕQ2Kv+ϕQ2Kt2,this gives
(64)ϕQ2Kv+t2≤ϕQ2Kv+ϕQ2Kt2−r0−12b41+KaK2K−1.
As(65)ϕQ1Kv+t2≤ϕQ1Kv+ϕQ1Kt2,we have
(66)ϕQKv+t2≤ϕQKv+ϕQKt2−r0−12b41+KaK2K−1.
As(67)ϕPKv+t2≤ϕPKv+ϕPKt2,we get
(68)ϕv+t2≤a1−r0−12b41+K2K−11/K.
Obviously,(69)1<r0≤K<2K⇒0<r0−12K−1<1.
If we set(70)β2p,b=min1−1−b2K1/K,1−1−r0−12b41+K2K−11/K.
Hence, one hasH2a,b>β2p,b>0 and we deduce that ϕ is (UUC 2). ☐In fact, the authors of reference [37] in Proposition 3.5 proved that if infd∈Nrd>1, then, the modular space ℓr.ψ, where ψv=∑d=0∞1/rdvdrd, has the property (R). They depended on their proof that the modular ψ has the Fatou property. But from Theorem 23, this result is incorrect. Consequently, all the related results in the two references [33, 37] are incorrect. In this part, we investigate the property (R) of the pre-quasinormed (sss) ℓrϕ.Theorem 37.
Letraa∈N∈ℓ∞ be an increase with r0>1, then
(1)
The spaceℓrϕ is a pre-quasi-Banach (sss), where ϕv=∑j=0∞vjrj1/K, for every v∈ℓr(2)
Suppose thatΛ is a nonempty ϕ closed and ϕ convex subset of ℓrϕ. Assume that v∈ℓrϕ so that(71)dϕv,Λ=infϕv−tv−t:t∈Λ<∞.
Then we have a uniqueλ∈Λ so that dϕv,Λ=ϕv−λ(3)
ℓrϕ verifies the property (R), i.e., for each decreasing sequence Λjj∈N of ϕ closed and ϕ convex nonempty subsets of ℓrϕ such that supj∈Ndϕv,Λj<∞, for some v∈ℓrϕ, so we have ⋂j∈NΛj≠∅Proof.
Assume that the setting is verified. The proof of (100) comes from Theorem 18. To prove (101), suppose that v∉Λ as Λ is ϕ closed. Therefore, we have A≔dϕv,Λ>0. So, for every p∈N, there is tp∈Λ such that ϕv−tp<A1+1/p. Assume that tp/2 is not ϕ Cauchy. So there is a subsequence tfp/2 and b0>0 such that ϕtfp−tfq/2≥b0, for all p>q≥0. More, we have H2A1+1/p,b0/2A>ξ≔β2A1+1/p,b0/2A>0, for each p∈N. As
(72)maxϕv−tfp,ϕv−tfqq≤A1+1fq,ϕtfp−tfq2≥b0≥A1+1fqb02A.
For allp>q≥0, one has
(73)ϕv−tfp+tfq2≤A1+1fq1−ξ.
Hence,(74)A=dϕv,Λ≤A1+1fq1−ξ,for every q∈N. If we set q⟶∞, we have
(75)0<A≤A1+1fq1−ξ<A.
This implies a contradiction. Therefore,tp/2 is ϕ Cauchy. Since ℓrϕ is ϕ complete, hence, tp/2ϕ converges to some t. For every q∈N, we get the sequence tp+tq/2ϕ converges to t+tq/2. As Λ is ϕ closed and ϕ convex, one has t+tq/2∈Λ. Clearly, t+tq/2ϕ converges to 2t; this implies that 2t∈Λ. By setting λ=2t and using Theorem 20, as ϕ verifies the Fatou property, we have
(76)dϕv,Λ≤ϕv−λ≤supiinfq≥iϕv−t+tq2≤supiinfq≥isupiinfp≥iϕv−tp+tq2≤12supiinfq≥isupiinfp≥iϕv−tp+ϕv−tq=dϕv,Λ.
Hence,ϕv−λ=dϕv,Λ. As the function ϕ is (UUC 2), then, it is (SC), which gives the uniqueness of λ. To prove (104), suppose v∉Λp0, for some p0∈N. As dϕv,Λpp∈N∈ℓ∞ is increasing, set limp⟶∞dϕv,Λp=A, if A>0. Otherwise v∈Λp, for every p∈N. From (2), there is one point tp∈Λp such that dϕv,Λp=ϕv−tp, for all p∈N. A consistent proof will show that tp/2ϕ converges to some t∈ℓrϕ. Since Λp are ϕ convex, decreasing, and ϕ closed, we have 2t∈∩p∈NΛp. ☐In this part, we investigate theϕ normal structure property of the pre-quasinormed (sss) ℓrϕ.Definition 38.
ℓrϕ verifies the ϕ normal structure property if for every nonempty ϕ bounded, ϕ convex, and ϕ closed subset Λ of ℓrϕ that are not decreased to one point, there is v∈Λ such that
(77)supt∈Λϕv−t<δϕΛ≔supϕv−tv−t:v,t∈Λ<∞.Theorem 39.
Letraa∈N∈ℓ∞ be an increase with r0>1; then, ℓrϕ has the ϕ normal structure property, where ϕv=∑p=0∞vprp1/K, for all v∈ℓr.Proof.
Suppose that the setting is verified. Theorem36 implies that ϕ is (UUC 2). Assume that Λ is a ϕ bounded, ϕ convex, and ϕ closed subset of ℓrϕ that are not decreased to one point. Hence, δϕΛ>0. Set A=δϕΛ. Suppose that v,t∈Λ such that v≠t. Hence ϕv−t/2=b>0. For every λ∈Λ, we have ϕv−λ≤A and ϕt−λ≤A. Since Λ is ϕ convex, we have v+t/2∈Λ. So,
(78)ϕv+t2−λ=ϕv−λ+t−λ2≤A1−H2A,bA,for every λ∈Λ. Hence,
(79)supλ∈Λϕv+t2−λ≤A1−H2A,bA<A=δϕΛ.


## 6. Kannanϕ Nonexpansive Mapping on ℓrϕ

We examine here the sufficient conditions on the pre-quasi normed (sss) ℓrϕ so that the Kannan pre-quasinorm nonexpansive mapping on it has a fixed point.Lemma 40.
Let the pre-quasinormed (sss) ℓrϕ verify the (R) property and the ϕ quasinormal property. Assume that Λ is a nonempty ϕ bounded, ϕ convex, and ϕ closed subset of ℓrϕ. Suppose that W:Λ⟶Λ is a Kannan ϕ nonexpansive mapping. For a>0, assume that Ga=v∈Λ:ϕv−Wv≤a≠∅. Set
(80)Λa=⋂Bϕp,q:WGa⊂Bϕp,q∩Λ.
Then,Λa≠∅, ϕ convex, and ϕ closed subset of Λ and WΛa⊂Λa⊂GaandδϕΛa≤a.Proof.
AsWGa⊂Λa, this implies that Λa≠∅. Since the ϕ balls are ϕ convex and ϕ closed, so Λa is a ϕ closed and ϕ convex subset of Λ. To prove that Λa⊂Ga, suppose that v∈Λa. If ϕv−Wv=0, we have v∈Ga. Otherwise, let ϕv−Wv>0. Set
(81)p=supϕWw−Wv:w∈Ga.
By using the definition ofp, then, WGa⊂BϕWv,p. Hence, Λa⊂BϕWv,p; this implies that ϕv−Wv≤p. Suppose that b>0. Hence, there is w∈Ga such that p−b≤ϕWw−Wv. Hence,
(82)ϕv−Wv−b≤p−b≤ϕWw−Wv≤12ϕv−Wvv−Wv+ϕw−Ww≤12ϕv−Wv+a.
Sinceb is an arbitrary positive, we have ϕv−Wv≤a, so one has v∈Ga. As WGa⊂Λa, we have WΛa⊂WGa⊂Λa; this investigates that Λa is W invariant, consequent to prove that δϕΛa≤a. As
(83)ϕWv−Wt≤12ϕv−Wv+ϕt−Wt,for each v,t∈Ga. Assume that v∈Ga. So WGa⊂BϕWv,a. The definition of Λa implies Λa⊂BϕWv,a. So, Wv∈⋂t∈ΛaBϕt,a. Hence, we have ϕt−w≤a; for every t,w∈Λa, this gives δϕΛa≤a. This finishes the proof. ☐Theorem 41.
Picking up the pre-quasinormed (sss) ℓrϕ verifies the ϕ quasinormal property and the (R) property. Assume that Λ is a nonempty, ϕ convex, ϕ closed, and ϕ bounded subset of ℓrϕ. Suppose that W:Λ⟶Λ is a Kannan ϕ nonexpansive mapping. Then, W has a fixed point.Proof.
Seta0=infϕv−Wv:v∈Λ and ap=a0+1/p, for all p≥1. From the definition of a0, we have Gap=v∈Λ:ϕv−Wv≤ap≠∅, for all p≥1. Let Λap be studied as in Lemma 40. Obviously, Λap is a decreasing sequence of nonempty ϕ bounded, ϕ closed, and ϕ convex subsets of Λ. The property (R) gives that Λ∞=⋂p≥1Λap≠∅. Supposing that v∈Λ∞, we get ϕv−Wv≤ap, for each p≥1. Assuming that p⟶∞, one has ϕv−Wv≤a0; this implies that ϕv−Wv=a0. Hence, Ga0≠∅. We obtain a0=0. Otherwise, a0>0; this implies that W fails to have a fixed point. Suppose Λa0 as defined in Lemma 40. Since W fails to have a fixed point and Λa0 is W invariant, hence, Λa0 has more than one point; this gives, δϕΛa0>0. By the ϕ quasinormal property, there is v∈Λa0 such that
(84)ϕv−t<δϕΛa0≤a0,for each t∈Λa0. By Lemma 40, one has Λa0⊂Ga0. By definition of Λa0, then, Wv∈Ga0⊂Λa0. Evidently, this implies that
(85)ϕv−Wv<δϕΛa0≤a0.
This contradicts the definition ofa0. Hence, a0=0 which gives that any point in Ga0 is a fixed point of W, i.e., W has a fixed point in Λ. ☐Using Theorems37, 39, and 41, we have the following corollary:Corollary 42.
Letraa∈N∈ℓ∞ be an increase with r0>1. Suppose that Λ is a nonempty, ϕ convex, ϕ closed, and ϕ bounded subset of ℓrϕ, where ϕv=[∑p=0∞vprp1/k, for all v∈ℓr. Assume that W:Λ⟶Λ is a Kannan ϕ nonexpansive mapping. Then, W has a fixed point.Example 43.
Let W:Λ⟶Λ with
(86)Wv=v4,ϕv∈0,1,v5,ϕv∈1,∞,where Λ=v∈ℓ2a+3/a+2a=0∞ϕ:v0=v1=0 and ϕv=∑a∈Nva2a+3/a+2, for all v∈ℓ2a+3/a+2a=0∞ϕ. From Example 33, the map W is Kannan ϕ contraction mapping. So, it is Kannan ϕ nonexpansive mapping. Clearly, Λ is a nonempty, ϕ convex, ϕ closed, and ϕ bounded subset of ℓ2a+3/a+2a=0∞ϕ. By Corollary 42, the map W has one fixed point v=θ in Λ.


## 7. Kannan Pre-Quasicontraction on Prequasi Ideal

We study the presence of a fixed point of Kannan pre-quasinorm contraction mapping in the pre-quasi-Banach operator ideal constructed byℓrϕ and s-numbers.Theorem 44 [9].
Pick upZ and M to be Banach spaces, and raa∈N∈ℓ∞ is an increasing with r0>1, and then, Sℓrϕ,Φ, where ΦW=ϕsaWa=0∞ is a pre-quasi-Banach operator ideal.Theorem 45.
IfZ and M are Banach spaces and raa∈N∈ℓ∞ is increasing with r0>1, then, Sℓrϕ,Φ, where ΦW=ϕsaWa=0∞ is a pre-quasiclosed operator ideal.Proof.
By Theorem17, the space ℓrϕ is a premodular (sss). So, from Theorem 12, we have ΦW=ϕsaWa=0∞ which is a pre-quasinorm on Sℓrϕ. Suppose that Wq∈SℓrϕZ,M, for all q∈N and limq⟶∞ΦWq−W=0. Therefore, there is ς>0 and as LZ,M⊇SℓrϕZ,M; one has
(87)ΦWq−W=ϕsaWq−Wa=0∞≥ϕs0Wq−WWq−W,0,0,0,⋯=ϕWq−W,0,0,0,⋯≥ςWq−W.
Then,Wqq∈N is convergent in LZ,M. i.e., limq⟶∞Wq−W=0, and since saWq−Wa=0∞∈ℓrϕ, for all q∈N and ℓrϕ is a premodular (sss). Hence, one can see
(88)ΦW=ϕsaWa=0∞=ϕsaW−Wq+Wqa=0∞≤ϕsa/2W−Wqa=0∞+ϕsa/2Wqa=0∞≤ϕWq−Wa=0∞+21/KϕsaWqa=0∞<ε.
We obtainsaWa=0∞∈ℓrϕ; hence, W∈SℓrϕZ,M. ☐Definition 46.
A pre-quasinormΦ on the ideal SAϕ, where ΦW=ϕsaWa=0∞, verifies the Fatou property if for all sequence Waa∈N⊆SAϕZ,M with lima⟶∞ΦWa−W=0 and every V∈SAϕZ,M, then
(89)ΦV−W≤supainfi≥aΦV−Wi.Theorem 47.
The pre-quasinormΦW=∑a=0∞saWra1/K, for each W∈SℓrϕZ,M, does not verify the Fatou property, if raa∈N∈ℓ∞ is increasing with r0>1.Proof.
Suppose that the conditions are verified andWpp∈N⊆SℓrϕZ,M with limp⟶∞ΦWp−W=0. As the space Sℓrϕ is a pre-quasiclosed ideal; hence, W∈SℓrϕZ,M. Then, for all V∈SℓrϕZ,M, one has
(90)ΦV−W=∑a=0∞saV−Wra1/K≤∑a=0∞sa/2V−Wra1/K+∑a=0∞sa/2Wi−Wra1/K≤21/Ksuppinfi≥p∑a=0∞saV−Wra1/K.
Therefore,Φ does not verify the Fatou property. ☐Now, we explain the definition of KannanΦ-Lipschitzian mapping in the pre-quasioperator ideal.Definition 48.
For the pre-quasinormΦ on the ideal SAϕ, where ΦW=ϕsaWa=0∞. An operator G:SAϕZ,M⟶SAϕZ,M is named a Kannan Φ-Lipschitzian, if there is ξ≥0 such that
(91)ΦGW−GA≤ξΦGW−W+ΦGA−A,for every W,A∈SAϕZ,M. An operator G is named
(1)
KannanΦ contraction, if ξ∈0,1/2(2)
KannanΦ nonexpansive, if ξ=1/2Definition 49.
For the pre-quasi normΦ on the ideal SAϕ, where ΦW=ϕsaWa=0∞, G:SAϕZ,M⟶SAϕZ,M, and B∈SAϕZ,M. The operator G is named Φ sequentially continuous at B, if and only if, when limp⟶∞ΦWp−B=0, then limp⟶∞ΦGWp−GB=0.Theorem 50.
Letraa∈N∈ℓ∞ be an increase with r0>1 and G:SℓrϕZ,M⟶SℓrϕZ,M, where ΦW=∑a=0∞saWra1/K, for all W∈SℓrϕZ,M. The point A∈SℓrϕZ,M is the unique fixed point of G, if the next settings are verified:
(a)
G is Kannan Φ contraction mapping(b)
G is Φ sequentially continuous at a point A∈SℓrϕZ,M,(c)
We haveB∈SℓrϕZ,M such that the sequence of iterates GpB has a subsequence GpiB which converges to AProof.
Suppose that the settings are satisfied. IfA is not a fixed point of G, then, GA≠A. From conditions (b) and (c), one has
(92)limpi⟶∞ΦGpiB−A=0,limpi⟶∞ΦGpi+1B−GA=0.
AsG is Kannan Φ contraction mapping, we have
(93)0<ΦGA−A=ΦGA−Gpi+1B+GpiB−A+Gpi+1B−GpiB≤21/KΦGpi+1B−GA+21/KΦGpiB−A+21/Kξξ1−ξpi−1ΦGB−B.
Sincepi⟶∞, one has a contradiction. Hence, A is a fixed point of G. To prove that the fixed point A is unique, assume that we have two different fixed points A,D∈SℓrϕZ,M of G. Therefore, we have
(94)ΦA−D≤ΦGA−GD≤ξΦGA−A+ΦGD−D=0.
So,A=D. ☐Example 51.
LetZ and M be Banach spaces, G:Sℓ2a+3/a+2a=0∞ϕZ,M⟶Sℓ2a+3/a+2a=0∞ϕZ,M, where ΦW=∑a=0∞szW2a+3/a+2, for every W∈Sℓ2a+3/a+2a=0∞ϕZ,M and
(95)GW=W6,ΦW∈0,1,W7,ΦW∈1,∞.Since for allW1,W2∈Sℓ2a+3/a+2a=0∞ϕ with ΦW1,ΦW2∈0,1, we have
(96)ΦGW1−GW2=ΦW16−W26≤21254Φ5W16+Φ5W26=21254ΦGW1−W1+ΦGW2−W2.For allW1,W2∈Sℓ2a+3/a+2a=0∞ϕ with ΦW1,ΦW2∈1,∞, we have
(97)ΦGW1−GW2=ΦW17−W27≤22164Φ6W17+Φ6W27=22164ΦGW1−W1+ΦGW2−W2.For allW1,W2∈Sℓ2a+3/a+2a=0∞ϕ with ΦW1∈0,1 and ΦW2∈1,∞, we have
(98)ΦGW1−GW2=ΦW16−W27≤21254Φ5W16+22164Φ6W27≤21254ΦGW1−W1+ΦGW2−W2.Therefore, the mapW is Kannan Φ contraction mapping and GpW=W/6p,ΦW∈0,1,W/7p,ΦW∈1,∞.It is clear thatG is Φ sequentially continuous at the zero operator Θ∈Sℓ2a+3/a+2a=0∞ϕ and GpW has a subsequence GpiW which converges to Θ. By Theorem 50, the zero operator Θ∈Sℓ2a+3/a+2a=0∞ϕ is the only fixed point of G. Let Wn⊆Sℓ2a+3/a+2a=0∞ϕ be such that limn⟶∞ΦWn−W0=0, where W0∈Sℓ2a+3/a+2a=0∞ϕ with ΦW0=1. Since the pre-quasinorm Φ is continuous, we have
(99)limn⟶∞ΦGWn−GW0=limn⟶∞ΦW06−W07=ΦW042>0.Hence,G is not Φ sequentially continuous at W0. So, the map G is not continuous at W0.


## 8. Application to the Existence of Solutions of Summable Equations

Summable equations like (100) studied by Salimi et al. [38], Agarwal et al. [39], and Hussain et al. [40]. In this section, we search for a solution to (100) in ℓrϕ, where raa∈N∈ℓ∞ is increasing with r0>1 and ϕv=∑j=0∞vjrj1/K, for all v∈ℓr. Consider the summable equations
(100)va=pa+∑m=0∞Aa,mfm,vm,and letW:ℓrϕ⟶ℓrϕ defined by
(101)Wvaa∈N=pa+∑m=0∞Aa,mfm,vma∈N.Theorem 52.
The summable equation (100) has a solution in ℓrϕ; if A:N2⟶R,f:N×R⟶R,p:N⟶R, and for all a∈N, suppose that
(102)∑m∈NAa,mfm,vm−fm,tmra≤12Kpa−va+∑m=0∞Aa,mfm,vmra+pa−ta+∑m=0∞Aa,mfm,tmra.Proof.
Let the conditions be verified. Consider the mappingW:ℓrϕ⟶ℓrϕ defined by (101). We have
(103)ϕWv−Wt=∑a∈NWva−Wtara1/K=∑a∈N∑m∈NAa,mfm,vm−fm,tmra1/K≤12∑a∈Npa−va+∑m=0∞Aa,mfm,vmra1/K+∑a∈Npa−ta+∑m=0∞Aa,mfm,tmra1/K=12ϕWv−v+ϕWt−t.
Then, from Theorem41, we have a solution for equation (100) in ℓrϕ. ☐Example 53.
Given the sequence spaceℓ2a+3/a+2a=0∞ϕ, where ϕv=∑a∈Nva2a+3/a+2, for all v∈ℓ2a+3/a+2a=0∞. Consider the summable equations
(104)va=e−3a+6+∑m=0∞−1a+mvaa2+m2+1q,
witha≥2 and q>2 and let W:Λ⟶Λ, where Λ=v∈ℓ2a+3/a+2a=0∞ϕ:v0=v1=0, defined by
(105)Wvaa≥2=e−3a+6+∑m=0∞−1a+mvaa2+m2+1qa≥2.
Clearly,Λ is a nonempty, ϕ convex, ϕ closed, and ϕ bounded subset of ℓ2a+3/a+2a=0∞ϕ. It is easy to see that
(106)∑m=0∞−1avaa2+m2+1q−1m−−1m2a+3/a+2≤12e−3a+6−va+∑m=0∞−1a+mvaa2+m2+1q2a+3/a+2+e−3a+6−ta+∑m=0∞−1a+mtaa2+m2+1q2a+3/a+2.
By Theorem52, the summable equation (104) has a solution in Λ.


## 9. Conclusion

We have introduced the concept of the pre-quasinormed space, which is more general than the quasinormed space. We investigate the sufficient conditions on Nakano (sss) such as its variable exponent in 1,∞ with the known pre-quasinorm to form pre-quasi-Banach and closed (sss), the concept of a fixed point of Kannan pre-quasi-norm contraction mapping in the pre-quasi-Banach (sss), which supports the property (R), and the pre-quasinormal structure property. The existence of a fixed point of Kannan pre-quasinorm nonexpansive mapping in the pre-quasi-Banach (sss) has been examined. Also, the existence of a fixed point of Kannan pre-quasinorm contraction mapping in the pre-quasi-Banach operator ideal formed by Nakano (sss) and s-numbers has been investigated. Finally, we have explained some examples to show that the results obtained can solve a problem. The strength of this approach is that the existence results are established under flexible conditions provided by controlling the power of the Nakano sequence space. The novelty lies in the fact that our main results have improved some well-known theorems before, which concerned the variable exponent in the aforementioned space.



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