# Making Decisions about How to Solve Nonlinear Dynamical Systems of the Kannan Type in a New Complex Function Space of Formal Power Series

Authors: Meshayil M. Alsolmi; Arafa O. Mustafa; Mustafa M. Mohammed; Awad A. Bakery
Journal: Journal of Mathematics
 (2023)

---

## Abstract

The sufficient settings of the space generated by absolute type-weighted gamma matrices of rankp in the Nakano complex functions of the formal power series, as well as its associated prequasioperators’ ideal equipped with definite functions, are defined and explained in this paper. Assorted prequasinorms are shown to have the Fatou characteristic. Some of its geometric and topological features and the prequasioperators’ ideal that goes with it are talked about. These structures have a relationship with the fixed points of the Kannan contraction and nonexpansive mappings. By looking at real-world examples and how they are used, it is shown that there are solutions for nonlinear dynamical systems of the Kannan type.


---

## Body

## 1. Introduction

Since the book [1] on the subject came out, many researchers have looked into possible extensions to the Banach fixed point theorem (BFPT) and how it can be used. The nonlinear analysis places significant weight on the Banach contraction principle (BCP), a potent instrument for nonlinear analysis. For more interesting articles on various real-world applications of the BCP, see the following: Diening et al. [2] discussed Lebesgue and Sobolev spaces with variable exponents, Ruẑiĉka [3] gave a mathematical description of electrorheological fluids, the types of approaches to impartial probabilistic functional differential equations that are motivated by pure leaps were described by Mao et al. [4], Younis et al. [5] presented generalized contractions, Guo and Zhu [6] investigated the fixed point approach of stochastic Volterra–Levin equations with Poisson jumps, Ahmad et al. [7] provided dual partial metric type spaces, as well as converging findings, Beg [8] applied FPT to ordered uniform convexity in ordered convex metric spaces, Yang and Zhu [9] examined the kind of solutions to stochastic neutral functional differential equations of Sobolev-type, and Beg et al. [10] discussed about the polytopic fuzzy sets to multiple-attribute decision-making problems. For more interesting articles on different real-world applications of fractional calculus, see the following: Rezapour et al. [11] discussed a mathematical analysis of a system of Caputo–Fabrizio fractional differential equations for the anthrax disease model in animals, Mohammadi et al. [12] investigated a theoretical study of the Caputo–Fabrizio fractional modeling for hearing loss due to the Mumps virus with optimal control, Khan et al. [13] suggested a case study of a fractal-fractional tuberculosis model in China: existence and stability theories along with numerical simulations, Etemad et al. [14] studied some novel mathematical analyses on the fractal-fractional model of the AH1N1/09 virus and its generalized Caputo-type version, Matar et al. [15] proposed using generalized Caputo fractional derivatives to investigate the p-Laplacian nonperiodic nonlinear boundary value problem, Baleanu et al. [16] gave a new study on the mathematical modeling of the human liver with Caputo-Fabrizio fractional derivative, Tuan et al. [17] presented a mathematical model for COVID-19 transmission by using the Caputo fractional derivative, Baleanu et al. [18] used a new fractional derivative approach to analyze the model of HIV-1 infection of CD4+ T-cells, and Kannan [19] illustrated a new category of discontinuous contraction operators. For the acting of Kannan mappings on modular vector spaces, prequasi-normed spaces, and prequasi-normed spaces of nonabsolute, see [20] and [21, 22], respectively. As a generalisation of Cesàro sequence space of the variable exponent, we have defined the space generated by absolute type-weighted gamma matrices of rank rWΓr=λakrq in the Nakano complex functions of the formal power series ℓτaP, ΓrPq,τφ, where WΓr=λakrq is defined as follows:(1)λakrq=r+k−1kqkr+aa,0≤k≤a,0,k>a,where r is a positive integer, qk∈0,∞, for all k∈N, where N=0,1,2,…, and r+k−1k=r+k−1!/k!r−1!.In [23], Roopaei and Başar examined Γrτφ, where τ=p,p,… and p>1. When τ=τa∈ℜ+N, so that ℜ+N=x=xi:xi>0,for alli∈N, and τa≥1/r, for all a∈N.The spaceΓrPq,τφ is denoted by the following equation:(2)ΓrPq,τ=f∈ℂℂ:fy=∑v=0∞fv^yvandφμf<∞, for someμ>0,where ℂ is the set of complex numbers and φf=∑a=0∞∑k=0ak+r−1kqkfk^/r+aaτa. For further information on formal power series spaces and their associated behaviors, see the following: Shields [24] investigated weighted shift operators, Hedayatian [25] discussed cyclicity in the space Hpβ, Emamirad and Heshmati [26] examined chaotic weighted shifts in Bargmann spaces, and Faried et al. [27] offered s-numbers of shift mappings. In functional analysis, the mappings’ ideal hypothesis has earned much respect. The application of s-numbers is a fundamental method, see [28, 29], [30], [31], and [32].Lemma 1 (see [33]).
Ifτp>0 and yp,zp∈ℜ, where ℜ is the set of real numbers, for every p∈N, then yp+zpτp≤2K−1ypτp+zpτp, where K=max1,suppτp.Since proving several more FPTs in a particular space necessitates either increasing the space itself or expanding rapidly the self-mapping that works on it, both of these options are good. In this article, we have constructed vast spaces of solutions to a variety of nonlinear summable and difference equations. The following is an outline of the objectives of this research: In Section2, we investigate sufficient setups of ΓrPq,τ equipped with definite function φ to form prequasi-Banach (ssfps). Some topological and geometric structures of ΓrPq,τ are discussed, which are connected with the fixed point property. Section 3 explains the geometric and topological structure of the operators’ ideal generated by this function space and s-numbers. In Section 4, we look at how to configure ΓrPq,τφ with different φ and its operators ideal so that there is only one fixed point of Kannan contraction mapping (Kcm). We give some illustrative examples to clarify our results. Specifically, in Section 5, we present the requirements of this function space with definite φ for the Kannan nonexpansive mapping to have a fixed point on it, and we do this by introducing it. Numerous examples and discussions of our findings, including their practical implementations, can be found in Section 6.


## 2. Structure ofΓrPq,τ

In this section, we have examined some topological and geometric structures ofΓrPq,τ, which are connected with the fixed points of Kcm.The bounded sequence space ofℜ is indicated by ℓ∞.Theorem 1.
Ifτb∈ℜ+N∩ℓ∞, then(3)ΓrPq,τ=g∈ℂℂ:gz=∑b=0∞gb^zbandφιg<∞,for everyι>0.Proof.
Obviously,τb is a bounded sequence.
Assumeϑ is the zero function of P and(4)F=g∈ℂℂ:∃k∈Nwithgz=∑b=0kgb^zb.Definition 1 (see [34]).
A mappingh:P⟶0,∞ is called modular (m), where P is a vector space if the following setups are satisfied:(a)
Assumef∈P, then φf≥0 and φf=0⇔f=ϑ(b)
Supposef∈P and η=1, then φηf=φf(c)
Ifη∈0,1 and f,g∈P, then φηf+1−ηg≤φf+φgDefinition 2 (see [35]).
The spaceP=f∈ℂℂ:fy=∑a=0∞fa^ya is named a ssfps, when the next setups are confirmed:(1)
Supposeq∈P and pb^≤qb^, with b∈N; then, p∈P(2)
el∈P, for all l∈N, so that elz=∑b=0∞ebl^zb=zl(3)
f.∈P, if f∈P, where f.y=∑b=0∞fb/2^yb, and b/2 marks the integral part of b/2Definition 3 (see [35]).
The functionh:P⟶0,∞ is named premodular (p-m) on P, when the next parts are confirmed:(i)
Assumef∈P, then φf≥0 and φf=0⇔f=ϑ(ii)
Ifg∈P and ε∈ℂ, one has Q≥1 with φεg≤εQφg(iii)
Supposef,g∈P, one has P≥1 with φf+g≤Pφf+φg(iv)
Assumefb^≤gb^, for all b∈N, then φf≤φg(v)
One getsP0≥1 such that φf≤φf.≤P0φf(vi)
Pφ=the closure of F(vii)
One obtainsκ>0 so that φμe0≥κμφe0, for all μ∈ℂ
Also, the spacePφ is a p-mssfps. If Pφ is complete, then Pφ is said to be a premodular Banach ssfps (p-mBssfps). P-m vector spaces are a more general idea than m vector spaces. Examples of p-m vector space and m vector space are as follows.Example 1.
The spaceΓrP1/a+r−1aa=0∞,2a+3/a+2a=0∞φ, where(5)φf=infδ>0:∑a=0∞∑k=0afk^/δr+aa2a+3/a+2≤1.
An example of premodular vector space but not modular vector space is given.Example 2.
The spaceΓrP1/b+r−1bb=0∞,2b+3/b+4b=0∞φ, whereφf=∑b=0∞∑k=0bfk^/r+bb2b+3/b+4. As for every f,g∈ΓrP1/a+r−1aa=0∞,2a+3/a+4a=0∞, one has(6)φf+g2=∑a=0∞∑k=0afk^+gk^/2r+aa2a+3/a+4≤248φf+φg.Definition 4 (see [35]).
A subspacePφ of the ssfps is called a prequasi-normed ssfps (p-qNssfps), if the function h:P⟶0,∞ verifies the setups (i), (ii), and (iii) of Definition 2.4.
IfPφ is complete, then Pφ is called a p-qBssfps.Theorem 2 (see [36]).
All qNssfps are a p-qNssfps.Example 3.
The functionφf=d∑a∈N∑k=0afk^/r+aad is a qN and not a N on ΓrP1/a+r−1aa=0∞,da=0∞, for 0<d<1.Example 4.
The functionφf=∑a∈N∑k=0afk^/r+aa3a+2/a+1 is a p-qN (not a qN) on(7)ΓrP1a+r−1aa=0∞,3a+2a+1a=0∞.Example 5.
Forτa∈1,∞N, the function φf=infα>0:∑a∈N∑k=0ak+r−1kqkfk^/α/r+aaτa≤1 is a N on ΓrPq,τ.Theorem 3 (see [36]).
Every p-mssfps is a p-qNssfps.Definition 5.
(i)
SupposeδφΩ≔supφf−g:f,g∈Ω<∞, then Ω⊂ΓrPq,τφ is said to be φ-bounded(ii)
A prequasinormφ on ΓrPq,τ verifies the Fatou property (Fa-p), whenever for all gpp=1∞⊆ΓrPq,τφ under limp⟶∞φgp−g=0 and all f∈ΓrPq,τφ, one has φf−g≤suprinfp≥rφf−gp(iii)
DenoteBφh,ε≔t∈ΓrPq,τφ:φh−t≤ε for the φ-ball of radius ε≥0 and center h∈ΓrPq,τφ
AssumeI and D are the space of all increasing and decreasing sequences of real numbers, respectively.Theorem 4.
ΓrPq,τφ, where φf=∑a=0∞∑k=0ak+r−1kqkfk^/r+aaτa1/K, for all f∈ΓrPq,τ, is a p-mssfps, if the next setups are confirmed:(b1)τpp∈N∈ℓ∞∩I with τ0>1/r(b2)k+r−1kqkk=0∞∈D or k+r−1kqkk=0∞∈I∩ℓ∞ and one has C≥1 with 2k+r2k+1q2k+1≤Cz+k−1kqkProof.
(i)
Obviously,φf=0⇔f=ϑ and φf≥0.(1-i) Assumef,g∈ΓrPq,τ. One has f+gy=∑v=0∞fv^+gv^yv∈ℂ with(8)φf+g=∑a=0∞∑k=0ak+r−1kqkfk^+gk^r+aaτa1/K≤∑a=0∞∑k=0ak+r−1kqkfk^r+aaτa1/K+∑a=0∞∑k=0ak+r−1kqkgk^r+aaτa1/K=φf+φg<∞,thenf+g∈ΓrPq,τ.(ii)
One getsP≥1 with φf+g≤Pφf+φg, for all f,g∈ΓrPq,τ.(1-ii) Assumingα∈ℂ and f∈ΓrPq,τ, we obtain(9)φαf=∑a=0∞∑k=0ak+r−1kqkαfk^r+aaτa1/K≤supaατa/K∑a=0∞∑k=0ak+r−1kqkfk^r+aaτa1/K≤Qαφv<∞.Asαf∈ΓrPq,τ, by setups (1-i) and (1-ii), one can see that ΓrPq,τ is linear. Clearly, ep∈ΓrPq,τ, for all p∈N, since φep=∑a=0∞∑k=0ak+r−1kqkekp^/r+aaτa1/K=∑a=p∞p+r−1pqp/r+aaτa1/K<∞.(ii)
There isQ=max1,supaατa/K−1≥1 so that φαf≤Qαφf, for all f∈ΓrPq,τ and α∈ℂ.(2)
Assumefa≤ga, for all a∈N and g∈ΓrPq,τ. One finds(10)φf=∑a=0∞∑k=0ak+r−1kqkfk^r+aaτa1/K≤∑a=0∞∑k=0ak+r−1kqkgk^r+aaτa1/K=φg<∞,thenf∈ΓrPq,τ.(iv)
It is clear from the proof part (2)(3)
Consideringfa∈ΓrPq,τ and k+r−1kqkk=0∞∈D, we get(11)φfa/2=∑a=0∞∑k=0ak+r−1kqkfk/2^r+aaτa1/K=∑a=0∞∑k=02ak+r−1kqkfk/2^r+2a2aτ2a1/K+∑a=0∞∑k=02a+1k+r−1kqkfk/2^r+2a+12a+1τ2a+11/K≤∑a=0∞2a+r−12aq2afa^+2∑k=0ak+r−1kqkfk^r+aaτa+∑a=0∞2∑k=0afk^r+aaτa1/K,≤∑a=0∞3∑k=0ak+r−1kqkfk^r+aaτa+∑a=0∞2∑k=0afk^r+aaτa1/K=3K+2K1/Kφf,thenfa/2∈ΓrPq,τ.(v)
By using (3), one can find P0=3K+2K1/K≥1.(vi)
It is obvious thatΓrPq,τ= the closure of F.(vii)
One finds0<σ≤suppατp/K−1, for α≠0 or σ>0, for α=0 with φαe0≥σαφe0.Theorem 5.
If the conditions of Theorem4 are satisfied, then ΓrPq,τφ is a p-qBssfps, where φh=∑a=0∞∑k=0ak+r−1kqkhk^/r+aaτa1/K, for every h∈ΓrPq,τ.Proof.
According to Theorems4 and 3, the space ΓrPq,τφ is a p-qNssfp. Suppose hl=hklk=0∞ is a Cauchy sequence (CS) in ΓrPq,τφ, then for every ε∈0,1, one has l0∈N. We have for every l,m≥l0 that(12)φhl−hm=∑a=0∞∑k=0ak+r−1kqkhkl^−hkm^r+aaτa1/K<ε.
This implieshkl^−hkm^<ε. Fix k∈N, then hkm^ is a CS in ℂ. Hence, limm⟶∞hkm^−hk0^=0. One has φhl−h0<ε, for all l≥l0. As φh0≤φhl−h0+φhl<∞, then h0∈ΓrPq,τ.Theorem 6.
The functionφf=∑a=0∞∑k=0ak+r−1kqkfk^/r+aaτa1/K satisfies the Fa-p, whenever the parts of Theorem 4 are satisfied, for all f∈ΓrPq,τ.Proof.
Letgp⊆ΓrPq,τφ with limp⟶∞φgp−g=0. Since ΓrPq,τφ is a prequasi-closed space, we have g∈ΓrPq,τφ. For every f∈ΓrPq,τφ, one gets(13)φf−g=∑a=0∞∑k=0ak+r−1kqkfk^−gk^r+aaτa1/K≤∑a=0∞∑k=0ak+r−1kqkfk^−gkp^r+aaτa1/K+∑a=0∞∑k=0ak+r−1kqkgkp^−gk^r+aaτa1/K≤supminfp≥mφf−gp.Theorem 7.
The functionφf=∑a=0∞∑k=0ak+r−1kqkfk^/r+aaτa does not verify the Fa-p, for every f∈ΓrPq,τ, if the conditions of Theorem 4 are satisfied with supaτa>1.Proof.
Assumegp⊆ΓrPq,τφ so that limp⟶∞φgp−g=0. Since ΓrPq,τφ is a prequasi-closed space, one gets g∈ΓrPq,τφ. For every f∈ΓrPq,τφ, we obtain(14)φf−g=∑a=0∞∑k=0ak+r−1kqkfk^−gk^r+aaτa≤2K−1∑a=0∞∑k=0ak+r−1kqkfk^−gkp^r+aaτa+∑a=0∞∑k=0ak+r−1kqkgkp^−gk^r+aaτa≤2K−1supminfp≥mφf−gp.
We will use thatφg=∑a=0∞∑k=0ak+r−1kqkgk^/r+aaτa1/K, for every g∈ΓrPq,τ, in the next part of this section. By SPφ and BPφ, we mean the unit sphere and the unit ball of Pφ, respectively.Definition 6 (see [37]).
A functionφ∈δ2; assume for every , one has k≥2 and a>0 with(15)φ2g≤kφg+εfor eachg∈Ph,φg≤a.
Ifφ∈δ2, for every a>0 so that k≥2 depending on a, then φ∈δ2s.Theorem 8 (see [37]).
Assumeφ∈δ2s, then for every L>0 and ε>0, there exists δ>0 so that φh+t−φh<ε, h,t∈Ph, with φh≤L and φt≤δ.Theorem 9.
Suppose the conditions of Theorem4 are satisfied, then for every ε>0 and L>0 we have δ>0 so that φh+t−φh<ε, for every h,t∈ΓrPq,τφ, with φh≤L and φt≤δ.Proof.
By using Theorems4 and 8, one has φ∈δ2s.Definition 7 (see [38]).
The functionφ is strictly convex, (SC), whenever every h,t∈Pφ with φh=φt and φh+t/2=φh+φt/2, then h=t.Definition 8 (see [39]).
A sequencegp⊆Pφ, is called ε separated for some ε>0, whenever(16)sepgp=infφgp−gt:p≠t>ε.Definition 9 (see [39]).
A Banach space(BS)Pφ is named k-nearly uniformly convex (k-NUC), where k∈2,3,4,…, wheneverε>0 one finds δ∈0,1 with any gp⊆BPφ, so that sepgp≥ε one hasp1,p2,p3,…,pk∈N. This means(17)φgp1+gp2+gp3+⋯+gpkk<1−δ.
Note thatk-NUC gives reflexivity.Theorem 10.
Suppose the conditions of Theorem4 are satisfied with τ0>1, then ΓrPq,τφisk-NUC.Proof.
Assumeε∈0,1 and fn⊆BΓrPq,τφ, where fny=∑i=0∞fni^yi so that sepfn≥ε. For all m∈N, suppose fnmy=∑i=0∞fnmi^yi, where fnmi^i=0∞=0,0,0,…,fnm^,fnm+1^,…. As for all i∈N, fni^n=0∞∈ℓ∞. In view of the diagonal method, one gets fnj of fn so that fnji^ converges for all i∈N, 0≤i≤m. We obtain an increasing sequence of positive integers tm with sepfnjmj>tm≥ε. Hence, we have a sequence of positive integers vmm=0∞ with v0<v1<v2<⋯, such that(18)φKfvmm≥ε2,for every m∈N. Fix k≥2, if ε1=kp0−1−1/k−1kp0ε/4 from Theorem 9 one gets δ>0 with(19)φKf+g−φKf<ε1.
IfφKf≤1 and φKg≤δ, as φKfn≤1, for all n∈N, we have mii=0,1,2,…,k−2∈N+ such that m0<m1<m2<…<mk−2 and φKfimi≤δ. Define mk−1=mk−2+1. According to inequality (18), we have φfvmkmk≥ε/2. Suppose pi=i for 0≤i≤k−2 and pk−1=vmk−1. From inequalities (18) and (19), Jnu=uτn is convex, for all n∈N; then,(20)φKfp0+fp1+fp2+⋯+fpk−1k=∑n=0∞∑i=0ni+r−1iqifp0i^+fp1i^+⋯+fpk−1i^/kr+nrτn=∑n=0m1−1∑i=0ni+r−1iqifp0i^+fp1i^+⋯+fpk−1i^/kr+nrτn+∑n=m1∞∑i=0ni+r−1iqifp0i^+fp1i^+⋯+fpk−1i^/kr+nrτn≤∑n=0m1−1∑i=0ni+r−1iqifp0i^+fp1i^+⋯+fpk−1i^/kr+nrτn+∑n=m1∞∑i=0ni+r−1iqifp1i^+fp2i^+⋯+fpk−1i^/kr+nrτn+ε1≤∑n=0m1−11k∑j=0k−1∑i=0ni+r−1iqifpji^r+nrτn+∑n=m1m2−1∑i=0ni+r−1iqifp1i^+fp2i^+⋯+fpk−1i^/kr+nrτn+∑n=m2∞∑i=0ni+r−1iqifp1i^+fp2i^+⋯+fpk−1i^/kr+nrτn+ε1≤∑n=0m1−11k∑j=0k−1∑i=0ni+r−1iqifpji^r+nrτn+∑n=m1m2−1∑i=0ni+r−1iqifp1i^+fp2i^+⋯+fpk−1i^/kr+nrτn+∑n=m2∞∑i=0ni+r−1iqifp2i^+fp3i^+⋯+fpk−1i^/kr+nrτn+2ε1≤∑n=0m1−11k∑j=0k−1∑i=0ni+r−1iqifpji^r+nrτn+∑n=m1m2−11k∑j=1k−1∑i=0ni+r−1iqifpji^r+nrτn+∑n=m2m3−11k∑j=2k−1∑i=0ni+r−1iqifpji^r+nrτn+⋯+∑n=mk−1mk−11k∑j=k−2k−1∑i=0ni+r−1iqifpji^r+nrτn+∑n=mk∞∑i=0ni+r−1iqifpki^/kr+nrτn+k−1ε1≤hKfp0+fp1+fp2+⋯+fpk−2k+1k∑n=0mk−1∑i=0ni+r−1iqifpki^r+nrτn+∑n=mk∞∑i=0ni+r−1iqifpki^/kr+nrτn+k−1ε1≤k−1k+1k∑n=0mk−1∑i=0ni+r−1iqifpki^r+nrτn+1kτ0∑n=mk∞∑i=0ni+r−1iqifpki^r+nrτn+k−1ε1≤1−1k+1k1−∑n=mk∞∑i=0ni+r−1iqifpki^r+nrτn+1kτ0∑n=mk∞∑i=0ni+r−1iqifpki^r+nrτn+k−1ε1=1+k−1ε1−kτ0−1−1kτ0∑n=mk∞∑i=0ni+r−1iqifpki^r+nrτn≤1+k−1ε1−kτ0−1−1kτ0ε2=1−kτ0−1−1kτ0ε4.
So,ΓrPq,τφ is k-NUC.
By fixingΩ≠∅ a φ-closed, φ-bounded, and φ-convex subset of ΓrPq,τφ.Definition 10.
The spacePφ verifies the R-pr, if and only if, for every decreasing sequence Ωjj∈N of nonempty φ-closed and φ-convex subsets of Pφ with supj∈NKφf,Ωj<∞, where Kφf,Ω=infφf−g:g∈Ω, for some f∈Pφ, one has∩j∈NΩj≠∅.Theorem 11.
If the conditions of Theorem4 are satisfied with τ0>1, one has(i)
Supposef∈ΓrPq,τφ with Kφf,Ω<∞. There is a unique λ∈Ω so that Kφf,Ω=φf−λ.(ii)
ΓrPq,τφ holds the R-pr.Proof.
For (i), assumef∉Ω as Ω is φ-closed. We find C≔Kφf,Ω>0. Hence, for all i∈N, then gi∈Ω with φf−gi<C1+1/i. If gi/2 is not φ-Cauchy, one gets a gmi/2 and l0>0 with φgmi−gmj/2≥l0, for any i>j≥0. Because of(21)maxφf−gmi,φf−gmj≤C1+1mj,φgmi−gmj2≥l0≥l021+1mj,for any i>j≥0. Since τaa∈N∈ℓ∞∩I with τ0>1, one finds Vnu=uτn is strictly convex, for any n∈N. Therefore, the space ΓrPq,τφ is strictly convex,(22)φf−gmi+gmj2<C1+1mj.
Then,(23)C=Kφf,Ω<C1+1mj,for all j∈N. By putting j⟶∞, one has a contradiction. So gi/2 is φ-Cauchy. As ΓrPq,τφ is φ-complete, then gi/2φ-converges to some g. For all j∈N, one gets gi+gj/2φ-converges to g+gj/2. Since Ω is φ-closed and φ-convex, then g+gj/2∈Ω. Since g+gj/2φ-converges to 2g, then 2g∈Ω. Given Theorem 6 and assume λ=2g, as φ verifies the Fa-p, one has the following:(24)Kφf,Ω≤φf−λ≤supiinfj≥iφf−g+gj2≤supiinfj≥isupiinfp≥iφf−gp+gj2≤12supiinfp≥isupiinfp≥iφf−gp+φf−gj=Kφf,Ω.
Then,φf−λ=Kφf,Ω. Since φ is (SC), the uniqueness of λ is established. To prove (ii), if f∉Ωi0, for some i0∈N. Because of Kφf,Ωii∈N∈ℓ∞ is increasing. Put limi⟶∞Kφf,Ωi=C. Suppose C>0. Else f∈Ωi, with i∈N. By (i), one finds a unique point gi∈Ωi with Kφf,Ωi=φf−gi, for every i∈N. Clearly, gi/2φ-converges to some g∈ΓrPq,τφ. As Ωi are φ-convex, decreasing and φ-closed, one has 2g∈∩i∈NΩi.Definition 11 (see [40]).
IfPφ is a BS, then(25)WCSPφ≔infAfn:fnn=0∞⊂SPφ,Afn=A1fn,fn⟶w0,where(26)Afn=limn⟶∞supφfa−fb:a,b≥n,a≠b,A1fn=limn⟶∞infφfa−fb:a,b≥n,a≠b.Definition 12.
Pφ satisfies the φ-normal structure-property (NS-pr), if and only if, for any Ω⊆Pφ is not decreased to one point, then h∈Ω with supt∈Ωφh−t<δφΩ≔supφh−t:h,t∈Ω<∞.Theorem 12 (see [41]).
A reflexiveBSPφ with WCSPφ>1 verifies the NS-pr.Theorem 13.
If the conditions of Theorem4 are satisfied with τ0>1, thenΓrPq,τφ satisfies the φ-NS-pr.Proof.
Supposefn⊂SΓrPq,τφ is an asymptotic equidistant sequence with fn⟶w0 and ε>0. Assume v0=f0. One has i0∈N with φ∑i=i0∞v0i^ei<ε. As fn⟶0 coordinate-wise, we have n1∈N so that φ∑i=0i0−1fni^ei<ε, for n≥n1; putting v1=fn1, we have i1>i0 so that φ∑i=i1∞v0i^ei<ε. As fni⟶0 coordinate-wise, we get n2∈N so that φ∑i=0i1−1fni^ei<ε. For n≥n2, by induction, there exists a subsequence vn of fn so that(27)φ∑i=in∞vni^ei<ε,φ∑i=0in−1vn+1i^ei<ε.
Takezn=∑i=in−1in−1vni^ei for n=1,2,3,… So,(28)1≥φzn=φ∑i=0∞vni^ei−∑i=0in−1−1vni^ei−∑i=in∞vni^ei≥φ∑i=0∞vni^ei−φ∑i=1in−1−1vni^ei−φ∑i=in∞vni^ei>1−2ε.
For alln,m∈N so that n≠m, we have:(29)φvn−vm=φ∑i=0∞vni^ei−∑i=0∞vmi^ei≥φ∑i=in−1in−1vni^ei−∑i=im−1im−1vmi^ei−φ∑i=0in−1−1vni^ei−φ∑i=in∞vni^ei−φ∑i=0im−1−1vmi^ei−φ∑i=im∞vmi^ei≥φzn−zm−4ε,which gives Afn=Avn≥Azn−4ε. Take un=zn/zn, for n=1,2,3,… Then, (30)un∈SΓrPq,τφ,(31)Afn≥1−εAun−4ε.
On the other hand,φvn−vm≤φzn−zm+4ε≤φun−um+4ε, for every n,m∈N with n≠m. Hence,(32)Aun≥Afn−4ε.
Sinceε>0, by using equations (30)–(32), we get WCSΓrPq,τφ=infAun, such thatun=∑i=in−1in−1uni^ei∈SΓrPq,τφ,0=i−1<i0<…,un⟶w0 and un is asymptotic equidistant.
Fixm∈N large enough with ∑k=im−1∞b/r+krτk<ε, where b≔∑i=in−1in−1i+r−1iqiuni^. One gets, for n<m,(33)φKun−um=∑k=in−1im−1−11r+kr∑i=0ki+r−1iqiuni^τk+∑k=im−1∞1r+krb+∑i=0ki+r−1iqiumi^τk≥∑k=in−1im−1−11r+kr∑i=0ki+r−1iqiuni^τk+∑k=im−1∞1r+kr∑i=0ki+r−1iqiumi^τk=∑k=in−1∞1r+kr∑i=0ki+r−1iqiuni^τk−∑k=im−1∞br+krτk+∑k=im−1∞1r+kr∑i=0ki+r−1iqiumi^τk>1−ε+1=2−ε.that is, Anun≥2−ε1/K. Note that(34)∑k=im−1∞1r+krb+∑i=0ki+r−1iqiumi^τk1/K≤∑k=im−1∞br+krτk1/K+∑k=im−1∞1r+kr∑i=0ki+r−1iqiumi^τk1/K<ε1/K+1.
Therefore,(35)φKun−um=∑k=in−1im−1−11r+kr∑i=0ki+r−1iqiumi^τk+∑k=im−1∞1r+krb+∑i=0ki+r−1iqiumi^τk≤∑k=in−1∞1r+kr∑i=0ki+r−1iqiumi^τk+∑k=im−1∞1r+krb+∑i=0ki+r−1iqiumi^τk≤1+1+ε1/KK,for n,m∈N and n≠m. Hence, Anun≤1+1+ε1/KK1/K and, since this is true for all ε>0, we have WCSΓrPq,τφ=21/K. From Theorem 10 and Theorem 12, then the functions space ΓrPq,τφ has the φ-NS-pr.


## 3. Structure of Mappings’ Ideal

In this section, we have examined some topological and geometric structures of mappings’ ideal (MI) constructed byΓrPq,τφ, where φh=∑a=0∞∑k=0ak+r−1kqkhk^/r+aaτa1/K, for every h∈ΓrPq,τ and s− numbers, which are connected with the fixed points of Kcm.Notations 1.
(1)
Δ and Λ are two infinite dimensional BSs.Δ0 and Λ0 are arbitrary BSs.(2)
ℒΔ,Λ is the space of all bounded linear mappings from Δ into Λ. If Δ=Λ, we write ℒΔ.(3)
FΔ,Λ is the space of finite rank linear mappings from Δ into Λ.(4)
ΥΔ,Λ is the space of approximable bounded linear mappings from Δ into Λ.(5)
ℒcΔ,Λ is the space of compact bounded linear mappings from Δ into Λ.(6)
ℒ is the space of every bounded linear operators between arbitrary two BSs.(7)
Iq is the unit mapping on the t-dimensional Hilbert space ℓ2t.Definition 13 (see [42]).
An operators:ℒΔ,Λ⟶ℜ+N is called an s-number if it maps every G∈ℒΔ,Λ to a unique sdGd=0∞ that validates the following settings:(a)
G=s0G≥s1G≥s2G≥⋯≥0(b)
sp+q−1G1+G2≤spG1+sqG2, for all G1,G2∈ℒΔ,Λ and p,q∈N(c)
sqG1G2G3≤G1sqG2G3, for all G3∈ℒΔ0,Δ,G2∈ℒΔ,Λ, and G1∈ℒΛ,Λ0(d)
Ifo∈ℜ and G1∈ℒΔ,Λ, then sqoG1=osqG1(e)
Let rankG1≤b, then sbG1=0, for each G1∈ℒΔ,Λ(f)
sl<qIq=1orsl≥qIq=0
Somes-number examples are as follows:(i)
Thep-th approximation number is αpG1=infG1−G2:G2∈ℒΔ,Λand rankG2≤p(ii)
Thep-th Kolmogorov number is dpX=infdimJ≤psupf≤1infg∈JXf−gDefinition 14 (see [29]).
A sub classU of ℒ is called a MI, whenever any UΔ,Λ=U∩ℒΔ,Λ verifies the next parts:(1)
IΓ∈U, where Γ marks BS of one dimension(2)
AssumeW∈ℒΔ0,Δ,X∈UΔ,Λ, and Y∈ℒΛ,Λ0, hence YXW∈UΔ0,Λ0(3)
The spaceUΔ,Λ is linearNotations 2 (see [36]).
(36)ℐP≔ℐPΔ,Λ,whereℐPΔ,Λ≔V∈ℒΔ,Λ:fs∈P,wherefsy=∑n=0∞snVyn,ℐPα≔ℐPαΔ,Λ,whereℐPαΔ,Λ≔V∈ℒΔ,Λ::fα∈P,wherefαy=∑n=0∞αnVyn,ℐPd≔ℐPdΔ,Λ,whereℐPdΔ,Λ≔V∈ℒΔ,Λ::fd∈P,wherefdy=∑n=0∞dnVyn.Theorem 14 (see [35]).
The classℐP is a MI whenever P is an ssfps.
By Theorems4 and 14, we have the following theorem.Theorem 15.
If the conditions of Theorem4 are satisfied, then ℐΓrPq,τφ is a MI.Definition 15 (see [43]).
A functionU∈0,∞U is called a p-qN on the ideal U, if it verifies the following conditions:(1)
IfV∈UΔ,Λ,UV≥0 and UV=0, if and only if, V=0(2)
We haveQ≥1 so as to UαV≤DαUV, for all V∈UΔ,Λ and α∈ℜ(3)
There areP≥1 so that UV1+V2≤PUV1+UV2, for each V1,V2∈UΔ,Λ(4)
One can findσ≥1, when V∈ℒΔ0,Δ,X∈UΔ,Λ and Y∈ℒΛ,Λ0; hence, UYXV≤σYUXVTheorem 16 (see [44]).
EveryqN on the ideal U is a p-qN on the same ideal.Theorem 17.
If the conditions of Theorem4 are satisfied, then U is a p-qN on ℐΓrPq,τφ, so that UZ=φfs, where fs∈ΓrPq,τφ and fsy=∑n=0∞snZyn.Proof.
(1)
SupposeY∈ℐΓrPq,τφΔ,Λ, UY=φfs≥0 and UY=φfs=0, if and only if, snY=0, for every n∈N, if and only if, Y=0.(2)
One hasQ≥1 with UεX=φεfs≤QεUX, for every X∈ℐΓrPq,τφΔ,Λ and ε∈ℂ.(3)
One hasPP0≥1 so that for X1,X2∈ℐΓrPq,τφΔ,Λ, hence there are f1s,f2s∈ΓrPq,τφ with f1sy=∑n=0∞snX1yn and f2sy=∑n=0∞snX2yn. Therefore, for gsy=:∑n=0∞snX1+X2yn, we have KK0≥1 so that(37)UX1+X2=φgs≤φf1s.+f2s.≤Pφf1s.+φf2s.≤PP0UX1+UX2.(4)
One can findϱ≥1, if X∈ℒΔ0,Δ, Y∈ℐΓrPq,τφΔ,Λ and Z∈ℒΛ,Λ0, hence there is fs∈ΓrPq,τφ with fsy=∑n=0∞snYyn. Then, for gsy=:∑n=0∞snZYXyn, one has the following equation:(38)UZYX=φgs≤φXZfs≤ϱXUYZ.Theorem 18.
If the conditions of Theorem4 are satisfied, one has ℐΓrPq,τφ,U, which is a p-qB MI.Proof.
AssumeVpp∈N is a CS in ℐΓrPq,τφΔ,Λ. As ℒΔ,Λ⊇SΓrPq,τφΔ,Λ; hence, there is fsp∈ΓrPq,τφ with fspy=∑n=0∞snVpyn for every p∈N; then,(39)UVt−Vp=∑a=0∞∑k=0ak+r−1kqkskVt−Vpr+aaτa1/K≥q0Vt−Vpτ0/K,where Vpp∈N is a CS in ℒΔ,Λ. As ℒΔ,Λ is a BS, so there exists V∈ℒΔ,Λ so that limp⟶∞Vp−V=0 and since fsp∈ΓrPq,τφ, for all p∈N and ΓrPq,τφ is a p-mssfps, hence one can see(40)UV=∑a=0∞∑k=0ak+r−1kqkskVr+aaτa1/K≤∑a=0∞∑k=0ak+r−1kqksk/2V−Vpr+aaτa1/K+∑a=0∞∑k=0ak+r−1kqksk/2Vpr+aaτa1/K≤∑a=0∞∑k=0ak+r−1kqkV−Vpr+aaτa1/K+3K+2K1/K∑a=0∞∑k=0ak+r−1kqkskVpr+aaτa1/K<ε,we obtain fs∈ΓrPq,τφ; hence, V∈ℐΓrPq,τφΔ,Λ..Theorem 19.
ℐΓrPq,τφαΔ,Λ=the closure ofFΔ,Λ, if the conditions of Theorem 4 are satisfied. But the converse is not necessarily true.Proof.
Because ofep∈ΓrPq,τφ, for all p∈N and the linearity of ΓrPq,τφ. Assume Z∈FΔ,Λ and rank Z=m, for m∈N, one gets fα∈F with fαy=∑n=0m−1αnZyn, one has fα∈ΓrPq,τφ. Therefore, the closure of FΔ,Λ⊆ℐΓrPq,τφαΔ,Λ. Assume Z∈ℐΓrPq,τφαΔ,Λ, we have gα∈ΓrPq,τφ. As φgα<∞, assume ρ∈0,1, then there is a0∈N−0 with φgα−∑n=0a0−1en<ρ/2K+3ηd, for some d≥1, where η=max1,∑a=a0∞1/r+aaτa. Since αaZ is decreasing, we have(41)∑a=a0+12a0∑k=0ak+r−1kqkα2a0Zr+aaτa≤∑a=a0+12a0∑k=0ak+r−1kqkαkZr+aaτa≤∑a=a0∞∑k=0ak+r−1kqkαkZr+aaτa<ρ2K+3ηd.
Hence, there isY∈F2a0Δ,Λ so that rank Y≤2a0 and(42)∑a=2a0+13a0∑k=0ak+r−1kqkZ−Yr+aaτa≤∑a=a0+12a0∑k=0ak+r−1kqkZ−Yr+aaτa<ρ2K+3ηd.
Since the conditions of Theorem4 are satisfied, we have(43)supa=a0∞∑k=02a0−1k+r−1kqkZ−Yτa<ρ22K+2η.
Therefore, one has(44)∑a=0a0∑k=0ak+r−1kqkZ−Yr+aaτa<ρ2K+3ηd.
AsZ−Y∈ℐΓrPq,τφαΔ,Λ, hence gα∈ΓrPq,τφ, where gαy≔∑n=0∞αnZ−Yyn. In view of inequalities (43-46), one has(45)dZ,Y=φgα=∑a=03a0−1∑k=0ak+r−1kqkαkZ−Yr+aaτa+∑a=3a0∞∑k=0ak+r−1kqkαkZ−Yr+aaτa≤∑a=03a0∑k=0ak+r−1kqkZ−Yr+aaτa+∑a=a0∞∑k=0a+2a0k+r−1kqkαkZ−Yr+a+2a0a+2a0τa+2a0≤∑a=03a0∑k=0ak+r−1kqkZ−Yr+aaτa+∑a=a0∞∑k=0a+2a0k+r−1kqkαkZ−Yr+aaτa≤3∑a=0a0∑k=0ak+r−1kqkZ−Yr+aaτa+∑a=a0∞∑k=02a0−1k+r−1kqkαkZ−Y+∑k=2a0a+2a0k+r−1kqkαkZ−Yr+aaτa≤3∑a=0a0∑k=0ak+r−1kqkZ−Yr+aaτa+2K−1∑a=a0∞∑k=02a0−1k+r−1kqkαkZ−Yr+aaτa+∑a=a0∞∑k=2a0a+2a0k+r−1kqkαkZ−Yr+aaτa≤3∑a=0a0∑k=0ak+r−1kqkZ−Yr+aaτa+2K−1∑a=a0∞∑k=02a0−1k+r−1kqkαkZ−Yr+aaτa+∑a=a0∞∑k=0ak+2a0+r−1k+2a0qk+2a0αk+2a0Z−Yr+aaτa≤3∑a=0a0∑k=0ak+r−1kqkZ−Yr+aaτa+2K−1ηsupa=a0∞∑k=02a0−1k+r−1kqkZ−Yτa+∑a=a0∞∑k=0ak+r−1kqkαkZr+aaτa<ρ.
Therefore,ℐΓrPq,τφαΔ,Λ⊆the closure ofFΔ,Λ. Contrarily, one has a counter example as I2∈ℐΓrP1/k+r−1kk=0∞,0,0,2,2,2,…φαΔ,Λ, but τ0>1/r is not verified. That indicates the nonlinearity of s− type ΓrPq,τφ spaces, see open problem of Rhoades [45].Theorem 20.
Suppose the conditions of Theorem4 are satisfied with τa1<τa2, and qa2≤qa1, for all a∈N, hence(46)ℐΓrPqa1,τa1φΔ,Λ⊊ℐΓrPqa2,τa2φΔ,Λ⊊ℒΔ,Λ.Proof.
LetZ∈ℐΓrPqa1,τa1φΔ,Λ, hence gs∈ΓrPqa1,τa1φ, where gsy≔∑n=0∞snZyn. One gets(47)∑a=0∞∑k=0ak+r−1kqk2skZr+aaτa2<∑a=0∞∑k=0ak+r−1kqk1skZr+aaτa1<∞.
Then,gs∈ΓrPqa2,τa2φ, this implies Z∈ℐΓrPqa2,τa2φΔ,Λ. After, if we choose skZk=0∞ with ∑k=0ak+r−1kqk1skZ=r+aa/a+11/τa1, we have Z∈ℒΔ,Λ such that(48)∑a=0∞∑k=0ak+r−1kqk1skZr+aaτa1=∞,∑a=0∞∑k=0ak+r−1kqk2skZr+aaτa2≤∑a=0∞∑k=0ak+r−1kqk1skZr+aaτa2=∑a=0∞1a+1τa2/τa1<∞.
Then,Z∉ℐΓrPqa1,τa1φΔ,Λ and Z∈ℐΓrPqa2,τa2φΔ,Λ.
Clearly,ℐΓrPqa2,τa2φΔ,Λ⊂ℒΔ,Λ. Next, if we put skZk=0∞ with ∑k=0ak+r−1kqk2skZ=r+aa/a+11/τa2. We have Z∈ℒΔ,Λ such that Z∉ℐΓrPqa2,τa2φΔ,Λ.Theorem 21.
Assume the conditions of Theorem4 are satisfied with ∑k=0ak+r−1kqk/r+aaa=0∞∉ℓτaP, hence ℐΓrPq,τφα is minimum.Proof.
LetℐΓrPq,ταΔ,Λ=ℒΔ,Λ, one has η>0 so that UZ≤ηZ, where UZ=∑a=0∞∑k=0ak+r−1kqkαkZ/r+aaτa, for every Z∈ℒΔ,Λ. In view of Dvoretzky’s theorem [46], with p∈N, we get quotient spaces Δ/Yp and subspaces Mp of Λ which can be transformed onto ℓ2p by isomorphisms Vp and Xp with VpVp−1≤2 and XpXp−1≤2. If Ip is the identity mapping on ℓ2p, Tp is the quotient map from Δ onto Δ/Yp and Jp is the natural embedding map from Mp into Λ. Suppose mk is the Bernstein numbers [28], one has(49)1=mkIp=mkXpXp−1IpVpVp−1≤XpmkXp−1IpVpVp−1=XpmkJpXp−1IpVpVp−1≤XpdkJpXp−1IpVpVp−1=XpdkJpXp−1IpVpTpVp−1≤XpαkJpXp−1IpVpTpVp−1.for 0≤a≤p. Then, we have(50)∑a=0p∑k=0ak+r−1kqkr+aaτa≤XpVp−1τa∑a=0p∑k=0ak+r−1kqkαkJpXp−1IpVpTpr+aaτa.
Then, we obtainϱ≥1, hence(51)∑a=0p∑k=0ak+r−1kqkr+aaτa≤ϱXpVp−1∑a=0p∑k=0ak+r−1kqkαkJpXp−1IpVpTpr+aaτa≤ϱXpVp−1UJpXp−1IpVpTp≤ϱηXpVp−1JpXp−1IpVpTp≤ϱηXpVp−1JpXp−1IpVpTp=ϱηXpVp−1Xp−1IpVp≤4ϱη.
Ifp⟶∞, we have a contradiction. Hence, Δ and Λ both cannot be infinite dimensional if ℐΓrPq,ταΔ,Λ=ℒΔ,Λ. □
We can show the next theorem.Theorem 22.
If the conditions of Theorem4 are satisfied with∑k=0ak+r−1kqk/r+aaa=0∞∉ℓτaP, hence ℐΓrPq,τd is minimum.Lemma 2 (see[29]).
IfG1∈ℒΔ,Λ and G1∉ΥΔ,Λ, then G2∈ℒΔ and G3∈ℒΛ with G3G1G2Iq=Iq, for q∈N.Theorem 23 (see [29]).
In general, we have(52)FΔ⊊γΔ⊊ℒcΔ⊊ℒΔ.Theorem 24.
Let the conditions of Theorem4 be satisfied with τa1<τa2, and qa2≤qa1, for all a∈N, hence(53)ℒℐΓrPqa2,τa2φΔ,Λ,ℐΓrPqa1,τa1φΔ,Λ=γℐΓrPqa2,τa2φΔ,Λ,ℐΓrPqa1,τa1φΔ,Λ.Proof.
AssumeX∈ℒℐΓrPqa2,τa2φΔ,Λ,ℐΓrPqa1,τa1φΔ,Λ and X∉ΥℐΓrPqa2,τa2φΔ,Λ,ℐΓrPqa1,τa1φΔ,Λ. By using Lemma 2, we have Y∈ℒℐΓrPqa2,τa2φΔ,Λ and Z∈ℒℐΓrPqa1,τa1φΔ,Λ so that ZXYIb=Ib, for b∈N, then(54)IbℐΓrPqa1,τa1φΔ,Λ=∑a=0∞∑k=0ak+r−1kqk1skIbr+aaτa1≤ZXYIbℐΓrPqa2,τa2φΔ,Λ≤∑a=0∞∑k=0ak+r−1kqk2skIbr+aaτa2.
That fails Theorem20. So, X∈ΥℐΓrPqa2,τa2φΔ,Λ,ℐΓrPqa1,τa1φΔ,Λ.Corollary 1.
Let the conditions of Theorem4 be satisfied with τa1<τa2, and qa2≤qa1, for all a∈N, hence(55)ℒℐΓrPqa2,τa2φΔ,Λ,ℐΓrPqa1,τa1φΔ,Λ=ℒcℐΓrPqa2,τa2φΔ,Λ,ℐΓrPqa1,τa1φΔ,Λ.Proof.
Evidently,Υ⊂ℒc.Definition 16.
(see [29]). A BSΔ is called simple if there is a unique nontrivial closed ideal in ℒΔ.Theorem 25.
If the conditions of Theorem4 are satisfied, hence ℐΓrPq,τφ is simple.Proof.
LetX∈ℒcℐΓrPq,τφΔ,Λ and X∉ΥℐΓrPq,τφΔ,Λ. From Lemma 2, there exist Y,Z∈ℒℐΓrPq,τφΔ,Λ with ZXYIb=Ib, which gives that IℐΓrPq,τφΔ,Λ∈ℒcℐΓrPq,τφΔ,Λ. Then, ℒℐΓrPq,τφΔ,Λ=ℒcℐΓrPq,τφΔ,Λ, hence ℐΓrPq,τφ is a simple BS. □Notations 3.
(56)ℐPλ≔ℐPλΔ,Λ;ΔandΛare BSs,whereℐPλΔ,Λ≔X∈ℒΔ,Λ:fλ∈Pφ,wherefλy=∑n=0∞λnXynandX−λnXI=0,for everyn∈N.Theorem 26.
Assume the conditions of Theorem4 are satisfied with infa∑k=0ak+r−1kqk/r+aaτa>0, hence(57)ℐΓrPq,τφλΔ,Λ=ℐΓrPq,τφΔ,Λ.Proof.
LetX∈ℐΓrPq,τφλΔ,Λ, hence fλ∈ΓrPq,τφ, where fλy=∑n=0∞λnTyn and X−λnXI=0, with n∈N. We have X=λnXI, for all n∈N, so(58)snX=snλnXI=λnX.with n∈N. One gets fs∈ΓrPq,τφ, hence X∈ℐΓrPq,τφΔ,Λ. Next, suppose X∈ℐΓrPq,τφΔ,Λ. Hence, fs∈ΓrPq,τφ, one gets the following equation:(59)infa∑k=0ak+r−1kqkr+aaτa∑a=0∞saXτa≤∑a=0∞∑k=0ak+r−1kqkskXr+aaτa<∞.
Then,lima⟶∞saX=0. If X−saXI−1 exists, with a∈N. Then, X−saXI−1 exists and bounded, for all a∈N. So, lima⟶∞X−saXI−1=X−1 exists and bounded. Since ℐΓrPq,τφ,H is a prequasi-MI, we have the following equation:(60)I=XX−1∈ℐΓrPq,τφΔ,Λ⇒gs∈ΓrPq,τ⇒lima⟶∞saI=0,where gsy=∑n=0∞snIyn. This gives a contradiction, as lima⟶∞saI=1. Therefore, X−saXI=0, with a∈N, which explains X∈ℐΓrPq,τφλΔ,Λ.


## 4. Kannan Contraction Mapping

We look at how to configureΓrPq,τφ with different φ and its operators ideal so that there is a unique fixed point of Kcm in this section. We also give some illustrative examples to clarify our results.Definition 17.
A mappingV:Pφ⟶Pφ is called a Kannan φ-contraction (Kφ-c), if there is α∈0,1/2 such that φVf−Vg≤αφVf−f+φVg−g, for every f,g∈Pφ. The mapping V is named Kannan φ-nonexpansive (Kφ-NE), when α=1/2.
A functionf∈Pφ is named a fixed point of V, if Vf=f.Theorem 27.
If the conditions of Theorem4 are satisfied, and V:ΓrPq,τφ⟶ΓrPq,τφis Kφ-cm, where φf=∑a=0∞∑k=0ak+r−1kqkfk^/r+aaτa1/K,for all f∈ΓrPq,τ, then V has a unique fixed point.Proof.
Iff∈ΓrPq,τ, one has Vpf∈ΓrPq,τ. As V is a K φ-cm, we have(61)φVl+1f−Vlf≤αφVl+1f−Vlf+φVlf−Vl−1f⇒φVl+1f−Vlf≤α1−αφVlf−Vl−1f≤α1−α2φVl−1f−Vl−2f≤…≤α1−αlφVf−f.
Forl,m∈N with m>l, then(62)φVlf−Vmf≤αφVlf−Vl−1f+φVmf−Vm−1f≤αα1−αl−1+α1−αm−1φVf−f.
Then,Vlf is a CS in ΓrPq,τφ. Since ΓrPq,τφ is prequasi-BS, we obtain g∈ΓrPq,τφ with liml⟶∞Vlf=g to show that Vg=g. As φ has the Fa-p, we get(63)φVg−g≤supiinfl≥iφVl+1f−Vlf≤supiinfl≥iα1−αlφVf−f=0.
Then,Vg=g. So, g is a fixed point of V. To prove the uniqueness, suppose f,g∈ΓrPq,τφ are two not equal fixed points of V. We have(64)φf−g≤φVf−Vg≤αφVf−f+φVg−g=0.
So,f=g.Corollary 2.
Assume the settings of Theorem4 are satisfied, and V:ΓrPq,τφ⟶ΓrPq,τφis Kφ-cm, where φf=∑a=0∞∑k=0ak+r−1kqkfk^/r+aaτa1/K, for all f∈ΓrPq,τ, one has V as the unique fixed point g so that φVlf−g≤αα/1−αl−1φVf−f.Proof.
By Theorem27, we get a unique fixed point g of V. Therefore,(65)φVlf−g=φVlf−Vg≤αφVlf−Vl−1f+φVg−g=αα1−αl−1φVf−f.Example 6.
AssumeV:ΓrP1/a+r−1aa=0∞,2a+3/a+2a=0∞φ⟶ΓrP1/a+r−1aa=0∞,2a+3/a+2a=0∞φ, where φg=∑a=0∞∑k=0agk^/r+aa2a+3/a+2, for every g∈ΓrP1/a+r−1aa=0∞,2a+3/a+2a=0∞ and(66)Vg=g4,φg∈0,1,g5,φg∈1,∞.
As for eachg1,g2∈ΓrP1/a+r−1aa=0∞,2a+3/a+2a=0∞φ with φg1,φg2∈0,1, one has(67)φVg1−Vg2=φg14−g24≤1274φ3g14+φ3g24=1274φVg1−g1+φVg2−g2.
For allg1,g2∈ΓrP1/a+r−1aa=0∞,2a+3/a+2a=0∞φ with φg1,φg2∈1,∞, one has(68)φVg1−Vg2=φg15−g25≤1644φ4g15+φ4g25=1644φVg1−g1+φVg2−g2.
For allg1,g2∈ΓrP1/a+r−1aa=0∞,2a+3/a+2a=0∞φ with φg1∈0,1 and φg2∈1,∞, we get(69)φVg1−Vg2=φg14−g25≤1274φ3g14+1644φ4g25≤1274φ3g14+φ4g25=1274φVg1−g1+φVg2−g2.
So,V is K φ-c because φ satisfies the Fa-p. From Theorem 27, one has V that holds one fixed point ϑ∈ΓrP1/a+r−1aa=0∞,2a+3/a+2a=0∞φ.Definition 18.
AssumePφ is a prequasi-normed (ssfps), V:Pφ⟶Pφ and g∈Pφ. The mapping V is called φ-sequentially continuous (φ-SC) at g, if and only if, when limp⟶∞φfp−g=0, then limp⟶∞φVfp−Vg=0.Example 7.
SupposeV:ΓrP1/a+r−1aa=0∞,a+1/2a+3a=0∞φ, where φg=∑a=0∞∑k=0agk^/r+aaa+1/2a+3, for every g∈ΓrP1/a+r−1aa=0∞,a+1/2a+3a=0∞ and(70)Vg=1181+g,g0^∈0,117,117,g0^=117,118,g0^∈117,1..
V is φ-SC and discontinuous at1/17∈ΓrP1/a+r−1aa=0∞,a+1/2a+3a=0∞φ.Example 8.
AssumeV is defined as in Example 6. Supposegn⊆ΓrP1/a+r−1aa=0∞,2a+3/a+2a=0∞φ is such that limn⟶∞φgn−g0=0, where g0∈ΓrP1/a+r−1aa=0∞,2a+3/a+2a=0∞φ with φg0=1.
As the p-qNφ is continuous, we have(71)limn⟶∞φVgn−Vg0=limn⟶∞hgn4−g05=hg020>0.
So,V is not φ -SC at g0.Theorem 28.
Suppose the parts of Theorem4 are satisfied under τ0>1 and V:ΓrPq,τφ⟶ΓrPq,τφ, where φf=∑a=0∞∑k=0ak+r−1kqkfk^/r+aaτa, for all f∈ΓrPq,τ. Suppose(1)
V is Kφ-cm(2)
V is φ-SC at g∈ΓrPq,τφ(3)
There isf∈ΓrPq,τφ with Vlf has Vljf converging to g
Then,g∈ΓrPq,τφ is the unique fixed point of VProof.
Ifg is not a fixed point of V, one has Vg≠g. By parts (2) and (3), one can see(72)limlj⟶∞φVljf−g=0,limlj⟶∞φVlj+1f−Vg=0.
SinceV is K φ-c, we get(73)0<φVg−g=φVg−Vlj+1f+Vljf−g+Vlj+1f−Vljf≤22supiτi−2φVlj+1f−Vg+22supiτi−2φVljf−g+2supiτi−1αα1−αlj−1φVf−f.
Sincelj⟶∞, we obtain a contradiction. Then, g is a fixed point of V. To prove the uniqueness. Suppose g,f∈ΓrPq,τφ are two not equal fixed points of V. We get(74)φg−f≤φVg−Vf≤αφVg−g+φVf−f=0.
Hence,g=f.Example 9.
AssumeV is defined as in Example 6. Let φf=∑a=0∞∑k=0afk^/r+aa2a+3/a+2, for all f∈ΓrP1/a+r−1aa=0∞,2a+3/a+2a=0∞. Since for all f1,f2∈ΓrP1/a+r−1aa=0∞,2a+3/a+2a=0∞φ with φf1,φf2∈0,1, one gets(75)φVf1−Vf2=φf14−f24≤227φ3f14+φ3f24=227φVf1−f1+φVf2−f2.
For allf1,f2∈ΓrP1/a+r−1aa=0∞,2a+3/a+2a=0∞φ with φf1,φf2∈1,∞, one gets(76)φVf1−Vf2=φf15−f25≤14φ4f15+φ4f25=14φVf1−f1+φVf2−f2.
For allf1,f2∈ΓrP1/a+r−1aa=0∞,2a+3/a+2a=0∞φ with φf1∈0,1 and φf2∈1,∞, one gets(77)φVf1−Vf2=φf14−f25≤227φ3f14+14φ4f25≤227φ3f14+φ4f25=227φVf1−f1+φVf2−f2.
So,V is K φ -c and Vpf=f/4pφf∈0,1,f/5pφf∈1,∞.
Obviously,V is φ-SC atϑ∈ΓrP1/a+r−1aa=0∞,2a+3/a+2a=0∞φ andVpf holds Vljf converges to ϑ.By Theorem 28, the point ϑ∈ΓrP1/a+r−1aa=0∞,2a+3/a+2a=0∞φ is the only fixed point of V.
Here, the construction ofℐΓrPq,τφ,U under definite function U is presented such that there is a unique fixed point of Kcm.Definition 19.
Ap-qNU on the ideal ℐPφ verifies the Fa-p if for all Tpp∈N⊆ℐPφΔ,Λ with limp⟶∞UTp−T=0 and M∈ℐPφΔ,Λ, one gets(78)UM−T≤suppinfj≥pUM−Tj.Theorem 29.
Suppose the conditions of Theorem4 are satisfied, then ℐΓrPq,τφ,U does not hold the Fa-p.Proof.
Assuming thatTpp∈N⊆ℐΓrPq,τφΔ,Λ with limp⟶∞UTp−T=0. Since ℐΓrPq,τφ is a prequasi-closed ideal, then T∈ℐΓrPq,τφΔ,Λ. So for every M∈ℐΓrPq,τφΔ,Λ, one has(79)UM−T=∑a=0∞∑k=0ak+r−1kqkskM−Tr+aaτa1/K≤∑a=0∞∑k=0ak+r−1kqksk/2M−Tjr+aaτa1/K+∑a=0∞∑k=0ak+r−1kqksk/2Tj−Tr+aaτa1/K≤3K+2K1/Ksuppinfj≥p∑a=0∞∑k=0ak+r−1kqkskM−Tjr+aaτa1/K.Definition 20.
An operatorV:ℐPφΔ,Λ⟶ℐPφΔ,Λ is said to be U-sequentially continuous U−Sℂ at M, where M∈ℐPφΔ,Λ, if and only if, limp⟶∞UTp−M=0⇒limp⟶∞UVTp−VM=0.Example 10.
IfV:ℐΓrP1/a+r−1aa=0∞,2a+3/a+2a=0∞φΔ,Λ⟶ℐΓrP1/a+r−1aa=0∞,2a+3/a+2a=0∞φΔ,Λ, where UT=∑a=0∞∑k=0askT/r+aa2a+3/a+2, for every T∈ℐΓrP1/a+r−1aa=0∞,2a+3/a+2a=0∞φΔ,Λ and(80)VT=T6,UT∈0,1,T7,UT∈1,∞.
Obviously,V is U−Sℂ at the zero operator Θ. Let Tj⊆ℐΓrP1/a+r−1aa=0∞,2a+3/a+2a=0∞φΔ,Λ be such that limj⟶∞UTj−T0=0, where T0∈ℐΓrP1/a+r−1aa=0∞,2a+3/a+2a=0∞φΔ,Λ with UT0=1. As U is continuous, we have(81)limj⟶∞UVTj−VT0=limj⟶∞UT06−T07=UT042>0.
Hence,V is not U−Sℂ at T0.Definition 21.
An operatorW:ℐPφΔ,Λ⟶ℐPφΔ,Λ is called a Kannan U-contraction KU−c, if one has λ∈0,1/2, so that UWV−WT≤λUWV−V+UWT−T, for all V,T∈ℐPφΔ,Λ.Theorem 30.
Assume the conditions of Theorem4 are satisfied and G:ℐΓrPq,τφΔ,Λ⟶ℐΓrPq,τφΔ,Λ, where UW=∑l=0∞∑z=0lz+r−1zqzszW/r+llτl1/ℏ, for every W∈ℐΓrPq,τφΔ,Λ. The vector A∈ℐΓrPq,τφΔ,Λ is the unique fixed point of G, if the next setups are fulfilled:(a)
G is KU−c mapping(b)
G is U−Sℂ at a point A∈ℐΓrPq,τφΔ,Λ(c)
There isB∈ℐΓrPq,τφΔ,Λ such that the sequence of iterates GmB has a subsequence GmiB that converges to AProof.
Let enough setups be satisfied. AssumeA is not a fixed point of G, then GA≠A. Because of conditions (b) and (c), one has(82)limmi⟶∞UGmiB−A=0,limmi⟶∞UGmi+1B−GA=0.
AsG is K U -cm, we get(83)0<UGA−A=UGA−Gmi+1B+GmiB−A+Gmi+1B−GmiB≤3K+2K1/KUGmi+1B−GA+3K+2K2/KUGmiB−A+3K+2K2/Kλλ1−λmi−1UGB−B.
Formi⟶∞, we have a contradiction. Hence, A is a fixed point of G. To prove that the fixed point A is unique, assume we have two distinct fixed points A,D∈ℐΓrPq,τφΔ,Λ of G. Therefore, one gets(84)UA−D≤UGA−GD≤λUGA−A+UGD−D=0.
Then,A=D.Example 11.
SupposeM:ℐΓrP1/l+4l+r−1ll=0∞,2l+3/l+2l=0∞φΔ,Λ⟶ℐΓrP1/l+4l+r−1ll=0∞,2l+3/l+2l=0∞φΔ,Λ, where UA=∑l=0∞∑z=0lszA/z+4/r+ll2l+3/l+2, for each A∈ℐΓrP1/l+4l+r−1ll=0∞,2l+3/l+2l=0∞φΔ,Λ and(85)MA=A6,UA∈0,1,A7,UA∈1,∞.
Since for allA1,A2∈ℐΓrP1/l+4l+r−1ll=0∞,2l+3/l+2l=0∞φΔ,Λ with UA1,UA2∈0,1, we have(86)UMA1−MA2=UA16−A26≤21254U5A16+U5A26=21254UMA1−A1+UMA2−A2.
For allA1,A2∈ℐΓrP1/l+4l+r−1ll=0∞,2l+3/l+2l=0∞φΔ,Λ with UA1,UA2∈1,∞, one has(87)UMA1−MA2=UA17−A27≤22164U6A17+U6A27=22164UMA1−A1+UMA2−A2.
For allA1,A2∈ℐΓrP1/l+4l+r−1ll=0∞,2l+3/l+2l=0∞φΔ,Λ with UA1∈0,1 and UA2∈1,∞, one gets(88)UMA1−MA2=UA16−A27≤21254U5A16+22164U6A27≤21254UMA1−A1+UMA2−A2.
Therefore, the operatorM is KU−cm and MmA=A/6mUA∈0,1A/7mUA∈1,∞.
Evidently,M is U−SℂatΘ and MmA has a subsequence MmjA converges to Θ. According to Theorem 30, the zero operator is the only fixed point of M. Assume Aa⊆ℐΓrP1/l+4l+r−1ll=0∞,2l+3/l+2l=0∞φΔ,Λ is such that lima⟶∞UAa−A0=0, where A0∈ℐΓrP1/l+4l+r−1ll=0∞,2l+3/l+2l=0∞φΔ,Λ with UA0=1. Since the p-qNU is continuous, we have(89)lima⟶∞UMAa−MA0=lima⟶∞UA06−A07=UA042>0.
Hence,M is not U−Sℂ at A0. Therefore, the operator M is not continuous at A0.


## 5. Kannan Nonexpansive Mapping onΓrPq,τφ

We introduce the sufficient conditions ofΓrPq,τφ, where φf=∑a=0∞∑k=0ak+r−1kqkfk^/r+aaτa1/K, for every f∈ΓrPq,τ, such that the KNEM on it has a fixed point.Lemma 3.
SupposeΓrPq,τφ verifies the R-pr and the φ-qNSP. Let V:Ω⟶Ω be a Kφ-NEM. When Gt=f∈Ω:φf−Vf≤t≠∅, with t>0, put(90)Ωt=∩Bφp,j:VGt⊂Bφp,j∩Ω.
Then,Ωt≠∅, φ-convex, φ-closed subset of Ω,δφΩt≤tandVΩt⊂Ωt⊂GtProof.
SinceVGt⊂Ωt, then Ωt≠∅. As the φ-balls are φ-convex and φ-closed, then Ωt is a φ-closed and φ-convex subset of Ω. To show that Ωt⊂Gt, assume f∈Ωt. When φf−Vf=0, one has f∈Gt. Else, assume φf−Vf>0. Put(91)p=supφVg−Vf:g∈Gt.
According to the definition ofp, one gets VGt⊂BφVf,p. Therefore, Ωt⊂BφVf,p, then φf−Vf≤p. Let l>0. One has g∈Gt with p−l≤φVg−Vf. So,(92)φf−Vf−l≤p−l≤φVg−Vf≤12φf−Vf+φg−Vg≤12φf−Vf+t.
Asl is an arbitrary positive, one obtains φf−Vf≤t, then f∈Gt. Since VGt⊂Ωt, one gets VΩt⊂VGt⊂Ωt, so Ωt is V-invariant. To show that δφΩt≤t, since(93)φVf−Vg≤12φf−Vf+φg−Vg,for all f,g∈Gt, let f∈Gt. Then, VGt⊂BφVf,t. The definition of Ωt gives Ωt⊂BφVf,t. Therefore, Vf∈∩t∈ΩtBφg,t. One has φg−f≤t, for all g,f∈Ωt, so δφΩt≤t.Theorem 31.
IfΓrPq,τφ holds the φ-qNSP and the R-pr. Let V:Ω⟶Ω be a Kφ-NEM. Then V has a fixed point.Proof.
Lett0=infφf−Vf:f∈Ω and tp=t0+1/p, for all p≥1. By the definition of t0, one gets Gtp=f∈Ω:φf−Vf≤tp≠∅, with p≥1. Suppose Ωtp is defined as in Lemma 3. Clearly, Ωtp is a decreasing sequence of nonempty φ-bounded, φ-closed, and φ-convex subsets of Ω. The R-pr investigates that Ω∞=∩p≥1Ωtp≠∅. Consideringf∈Ω∞, one has φf−Vf≤tp, with p≥1. Letting p⟶∞, hence φf−Vf≤t0, then φf−Vf=t0. So, Gt0≠∅. Hence, t0=0. Else, t0>0, so V fails to have a fixed point. Suppose Ωt0 is defined in Lemma 3. Since V fails to have a fixed point and Ωt0 is V-invariant, hence Ωt0 has more than one point, then δφΩt0>0. From the φ-qNSP, one has f∈Ωt0with(94)φf−g<δφΩt0≤t0,for all g∈Ωt0. From Lemma 3, we get Ωt0⊂Gt0. From definition of Ωt0, so Vf∈Gt0⊂Ωt0. Then,(95)φf−Vf<δφΩt0≤t0,which contradicts the definition of t0. Then, t0=0 which implies that any point in Gt0 is a fixed point of V.According to Theorems11, 13 and 31, we conclude the following.Corollary 3.
If the conditions of Theorem4 are satisfied with τ0>1, and V:Ω⟶Ω is a Kφ-NEM, then V has a fixed point in Ω⊆ΓrPq,τφ.Example 12.
AssumeV:Ω⟶Ω with Vf=f/4,φf∈0,1,f/5,φf∈1,∞, where Ω=f∈ΓrP1/a+r−1aa=0∞,2a+3/a+2a=0∞φ:f0^=f1^=0 and φf=∑a=0∞∑k=0afk^/r+aa2a+3/a+2, for every f∈ΓrP1/a+r−1aa=0∞,2a+3/a+2a=0∞φ. By using Example 6,V is Kannan φ-contraction. So it is Kφ-NEM. By Corollary 3, V holds a fixed point ϑ in Ω.


## 6. Existence of Solutions of Nonlinear Difference Equations

In this section, we explore a solution inΓrPq,τφ to summable equations say (96), defined in [47], where the conditions of Theorem 4 are satisfied and φf=∑a=0∞∑k=0ak+r−1kqkfk^/r+aaτa1/K, for all f∈ΓrPq,τφ.We examine the summable equations:(96)fa^=pa^+∑m=0∞Aa,mgm,fm^.IfW:ΓrPq,τφ⟶ΓrPq,τφ is defined as follows:(97)Wfz=∑a=0∞pa^+∑m=0∞Aa,mgm,fm^za.Theorem 32.
Summable equation (96) holds an unique solution in ΓrPq,τφ, when A:N2⟶ℜ,g:N×ℂ⟶ℂ,p^:N⟶ℂ,η^:N⟶ℂ, assume there is λ∈ℜ so that supaλτa/K∈0,1/2 and for all a∈N, we have(98)∑k=0a∑m∈NAk,mgm,fm^−gm,ηm^k+r−1kqk≤λ∑k=0apk^−fk^+∑m=0∞Ak,mgm,fm^k+r−1kqk+λ∑k=0apk^−ηk^+∑m=0∞Ak,mgm,ηm^k+r−1kqk.Proof.
Let the conditions be established. Assume the mappingW:ΓrPq,τφ⟶ΓrPq,τφ is defined by equation (97). Hence,(99)φWf−Wη=∑a=0∞∑k=0a∑m∈NAk,mgm,fm^−gm,ηm^k+r−1kqkr+aaτa1/K≤supaλτa/K∑a=0∞∑k=0apk^−fk^+∑m=0∞Ak,mgm,fm^k+r−1kqkr+aaτa1/K+supaλτa/K∑a=0∞∑k=0apk^−ηk^+∑m=0∞Ak,mgm,ηm^k+r−1kqkr+aaτa1/K=supaλτa/KφWf−f+φWη−η.
By Theorem27, we have a unique solution of (96) in ΓrPq,τφ.Example 13.
ConsiderΓrP1/l+1l+r−1ll=0∞,2l+3/l+2l=0∞φ, whereφf=∑l=0∞∑z=0lfz^/z+1/r+ll2l+3/l+2, for all f∈ΓrP1/l+1l+r−1ll=0∞,2l+3/l+2l=0∞φ.
Assume the nonlinear difference equations:(100)fa^=e−3a+6i+∑m=0∞ea+mfa−2b^fa−1d^+m2+1,with i2=−1, b,d,f−2^,f−1^>0 and suppose(101)W:ΓrP1l+1l+r−1ll=0∞,2l+3l+2l=0∞φ⟶ΓrP1l+1l+r−1ll=0∞,2l+3l+2l=0∞φ.is defined by(102)Wfz=∑a=0∞e−3a+6i+∑m=0∞ea+mfa−2b^fa−1d^+m2+1za.
Evidently, there isλ∈ℜ such that supaλ2a+3/2a+4∈0,1/2 and for all a∈N, one has:(103)∑k=0a∑m=0∞ekfk−2b^fk−1d^+m2+1em−emk+r−1kqk≤λ∑k=0ae−3k+6i−fk+∑m=0∞ek+mfk−2b^fk−1d^+m2+1k+r−1kqk+λ∑k=0ae−3k+6i−ηk^+∑m=0∞ek+mηk−2b^ηk−1d^+m2+1k+r−1kqk.
According to Theorem32, the nonlinear difference (100) contains a unique solution in ΓrP1/l+1l+r−1ll=0∞,2l+3/l+2l=0∞φ.Example 14.
Assume the nonlinear difference equations (102) and W:Ω⟶Ω is defined as equation (102), where Ω=f∈ΓrP1/l+1l+r−1ll=0∞,2l+3/l+2l=0∞φ:f0^=f1^=0 and φf=∑l=0∞∑z=0lfz^/z+1/r+ll2l+3/l+2, for all f∈ΓrP1/l+1l+r−1ll=0∞,2l+3/l+2l=0∞φ. Clearly, Ω is a nonempty φ-bounded,φ-convex, andφ-closed subset ofΓrP1/l+1l+r−1ll=0∞,2l+3/l+2l=0∞φ. By using Example 13,W is Kφ-c. So it is Kφ-NEM. By Corollary 3,W has a fixed point in Ω.Theorem 33.
Consider the summable equations (96), and assume W:ΓrPq,τφ⟶ΓrPq,τφ is defined by (97),where the conditions of Theorem 4 are satisfied with τ0>1 and φf=∑a=0∞∑k=0ak+r−1kqkfk^/r+aaτa, for all f∈ΓrPq,t. The summable (96) has a unique solution Z¯∈ΓrPq,τφ, if the following conditions are satisfied:(1)
SupposeA:N2⟶ℜ,g:N×ℂ⟶ℂ,p^:N⟶ℂ,η^:N⟶ℂ, one has λ∈ℜ with 2K−1supaλτa∈0,1/2 and for all a∈N, one has(104)∑k=0a∑m∈NAk,mgm,fm^−gm,ηm^k+r−1kqk≤λ∑k=0apk^−fk^+∑m=0∞Ak,mgm,fm^k+r−1kqk+λ∑k=0apk^−ηk^+∑m=0∞Ak,mgm,ηm^k+r−1kqk.(2)
W is φ-SC at t∈ΓrPq,τφ.(3)
There isf∈ΓrPq,τφ with Wmf has Wmjf converging to t.Proof.
One has(105)φWf−Wη=∑a=0∞∑k=0a∑m∈NAk,mgm,fm^−gm,ηm^k+r−1kqkr+aaτa≤2K−1supaλτa∑a=0∞∑k=0apk^−fk^+∑m=0∞Ak,mgm,fm^k+r−1kqkr+aaτa+2K−1supaλτa∑a=0∞∑k=0apk^−ηk^+∑m=0∞Ak,mgm,ηm^k+r−1kqkr+aaτa=2K−1supaλτaφWf−f+φWη−η.
By Theorem28, one gets a unique solution t∈ΓrPq,τφ of equation (96).Example 15.
ConsiderΓrP1/l+1l+r−1ll=0∞,2l+3/l+2l=0∞φ, where φf=∑l=0∞∑z=0lfz^/z+1/r+ll2l+3/l+2, for all f∈ΓrP1/l+1l+r−1ll=0∞,2l+3/l+2l=0∞φ.
Let summable (100), W:ΓrP1/l+1l+r−1ll=0∞,2l+3/l+2l=0∞φ⟶ΓrP1/l+1l+r−1ll=0∞,2l+3/l+2l=0∞φ defined by (101). Assume W is φ-SC at t∈ΓrP1/l+1l+r−1ll=0∞,2l+3/l+2l=0∞φ, and there is f∈ΓrP1/l+1l+r−1ll=0∞,2l+3/l+2l=0∞φ with Wmf has Wmjf converging to t. Evidently, there is λ∈ℜ such that 2K−1supaλ2a+3/a+2∈0,1/2 and for all a∈N, one has(106)∑k=0a∑m=0∞ekfk−2b^fk−1d^+m2+1em−emk+r−1kqk≤λ∑k=0ae−3k+6i−fk+∑m=0∞ek+mfk−2b^fk−1d^+m2+1k+r−1kqk+λ∑k=0ae−3k+6i−ηk^+∑m=0∞ek+mηk−2b^ηk−1d^+m2+1k+r−1kqk.
By Theorem 34, the summable equations (100) have a unique solution t∈ΓrP1/l+1l+r−1ll=0∞,2l+3/l+2l=0∞φ.
It is in this part that the solution of nonlinear matrix (107) at D∈ℐΓrPq,τφΔ,Λ is investigated. Suppose the conditions of Theorem 4 are satisfied, and UG=∑l=0∞∑z=0lz+r−1zqzszG/r+llτl1/K, for every G∈ℐΓrPq,τφΔ,Λ. Consider the summable equations(107)szG=szP+∑l=0∞Az,lgl,slG.
IfW:ℐΓrPq,τφΔ,Λ⟶ℐΓrPq,τφΔ,Λ is defined as follows:(108)WG=szP+∑l=0∞Az,lgl,slGI.Theorem 34.
The summable equations (107) have a unique solution D∈ℐΓrPq,τφΔ,Λ, if the following conditions are satisfied:(a)
A:N2⟶ℜ,f:N×ℜ⟶ℜ,P∈ℒΔ,Λ,T∈ℒΔ,Λ, and for all z∈N, one has κ with supzκτz/K∈0,0.5, and(109)∑l∈NAz,lgl,slG−gl,slT≤κszP−szG+∑l∈NAz,lgl,slG+szP−szT+∑l∈NAz,lgl,slT.(b)
W is U−Sℂ at D∈ℐΓrPq,τφΔ,Λ.(c)
We haveB∈ℐΓrPq,τφΔ,Λ so that WaB has a WaiB converging to D.Proof.
IfW:ℐΓrPq,τφΔ,Λ⟶ℐΓrPq,τφΔ,Λ defined by (111). We have(110)UWG−WT=∑l=0∞∑z=0lz+r−1zqzszG−szTr+lltl1/ℏ=∑l=0∞∑z=0lz+r−1zqz∑m∈NAz,mgm,smG−gm,smTr+llτl1/k≤supzκτz/K∑l=0∞∑z=0lz+r−1zqzszP−szG+∑m∈NAz,mgm,smGr+llτl1/k+supzκτz/K∑l=0∞∑z=0lz+r−1zqzszP−szT+∑m∈NAz,mgm,smTr+llτl1/k=supzκτz/KUWG−G+UWT−T.
In view of Theorem30, we have a unique solution of (107) at D∈ℐΓrPq,τφΔ,Λ.Example 16.
Assume the classℐΓrP1/l!,2l+3/l+2φΔ,Λ, where UG=∑l=0∞∑z=0lz+r−1zszG/z!/r+ll2l+3/l+2, for all G∈ℐΓrP1/l!,2l+3/l+2φΔ,Λ. Consider the nonlinear difference equations:(111)szG=tanh2z+3+∑m=0∞tan2m+1cosh3m−zcosmbsz−2Gsinhdsz−1G+sinmz+3,where z≥2 and b,d>0 and let W:ℐΓrP1/l!,2l+3/l+2φΔ,Λ⟶ℐΓrP1/l!,2l+3/l+2φΔ,Λ be defined as follows:(112)WG=tanh2z+3+∑m=0∞tan2m+1cosh3m−zcosmbsz−2Gsinhdsz−1G+sinmz+3I.
IfW is U−Sℂ at D∈ℐΓrP1/l!,2l+3/l+2φΔ,Λ, and one obtains B∈ℐΓrP1/l!,2l+3/l+2φΔ,Λ with WaB having a WaiB converging to D, obviously,(113)∑m=0∞cosh3m−zcosmbsz−2Gsinhdsz−1G+sinmz+3tan2m+1−tan2m+1≤125tanh2z+3−szG+∑m=0∞tan2m+1cosh3m−zcosmbsz−2Gsinhdsz−1G+sinmz+3+125tanh2z+3−szT+∑m=0∞tan2m+1cosh3m−zcosmbsz−2Tsinhdsz−1T+sinmz+3.
By Theorem34, the nonlinear difference (111) has one solution D∈ℐΓrP1/l!,2l+3/l+2φΔ,Λ.


## 7. Conclusion

We defined and presented sufficient conditions of the new complex function spaceΓrPq,τφ equipped with the definite function φ to be prequasi-Banach in this paper. The Fa-p of various prequasinorms has been presented. Some topological and geometric structures in the new complex function space ΓrPq,τ and their related prequasioperator ideals, which are linked to the fixed points of the contraction and nonexpansive of the Kannan mappings are offered. We introduce many possible solutions for various nonlinear dynamical and matrix systems. Our findings on the variable exponent in the previously outlined space have bolstered several well-known theories. As a future effort, we will extend this newly constructed complex function space to its nonabsolute type and solve several fractional nonlinear dynamical and matrix systems of Kannan type in this newly constructed complex function space and their associated prequasioperator ideals.



---

*Source: 9672865-2023-01-31.xml*