# Some Applications of New Complex Function Space Constructed by Different Weights and Exponents

Authors: Awad A. Bakery; Elsayed A. E. Mohamed
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this article, we develop and study a new complex function space formed by varying the weights and exponents under a definite function. We investigate the geometric and topological characteristics of mapping ideals created usings-numbers and this complex function space. Also, the action of shift mappings on this complex function space has been discussed. Finally, we introduced an extension of Caristi’s fixed point theorem on it.


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## Body

## 1. Introduction

Numerous researchers are attempting to extend the Banach fixed point theorem [1] in a realistic manner. Kannan [2] recognized a subclass of mappings that execute the same fixed point operations as contractions but are not continuous. Ghoncheh [3] pioneered the study of Kannan mappings in modular vector spaces. Lebesgue spaces with variable exponents, Lr, include Nakano sequence spaces. Across the second half of the twentieth century, it was thought that these variable exponent spaces offered an adequate framework for the mathematical components of a variety of problems for which the traditional Lebesgue spaces were inadequate. Due to the importance of these areas and their consequences, they have developed a reputation as an effective instrument for resolving a wide variety of problems; presently, the study of LrΩ spaces is a developing field of research, with implications reaching across a broad range of mathematical disciplines [4]. The investigation of variable exponent Lebesgue spaces was accelerated further by the mathematical description of non-Newtonian fluid hydrodynamics [5, 6]. Non-Newtonian fluids, also known as electrorheological fluids, have a wide range of applications in a number of fields ranging from military science to civil engineering to orthopedics and beyond. Mapping ideal theory has a diverse range of applications in Banach space geometry, fixed point theory, spectral theory, and other areas of mathematics, as well as other fields of knowledge (for further information, see [7–13]). Bakery and Mohamed [14] studied the notion of a pre-quasi norm on Nakano sequence space with a variable exponent in the range 0,1. They explored the conditions under which it generates pre-quasi Banach and closed space when endowed with a particular pre-quasi norm as well as the Fatou property of various pre-quasi norms on it. Additionally, they showed the existence of a fixed point for Kannan pre-quasi norm contraction mappings on it as well as on the pre-quasi Banach operator ideal formed from this sequence space’s s-numbers. In [15], they investigated some fixed points results of Kannan non-expansive mappings on generalized Cesàro backward difference sequence space of non-absolute type.We will mark the complex and non-negative integers asℂ and ℕ=0,1,2,…, respectively. By ℂℂ, we denote the space of all complex functions with complex variable. Assuming that r=ryv∈ℕ∈0,∞ℕ, Bakery and El Dewaik [16] defined the following function space:(1)ℋwrvψ=h∈ℂℂ:hx=∑y=0∞hy^xy∈ℂ;andψϖh<∞,for someϖ>0,where(2)ψh=∑y=0∞hy^y+1ry.They studied several of the topological and geometric properties forℋwrvψ and even a pre-quasi ideal construction based on the ℋwrvψ and s-numbers. Upper bounds for s-numbers of infinite series of the weighted v-th power forward shift operator on ℋwrvψ were also introduced for some entire functions. Further, they evaluated Caristi’s fixed point theorem in ℋwrvψ. For extra information on formal power series spaces and their behaviors, see [17–20]. We denote the space of every, finite rank, approximable, and compact bounded linear mappings from a Banach space X into a Banach space Y by LX,Y, FX,Y, ΛX,Y, and LcX,Y, and if X=Y, we mark LX, FX, ΛX, and LcX, respectively. The ideal of all, finite rank, approximable, and compact mappings are denoted by L, F, Λ, and Lc. We will indicate the sequence of s-numbers, approximation numbers, and Kolmogorov numbers for any bounded linear mapping G by saGa∈ℕ, αaGa∈ℕ, and daGa∈ℕ. The mapping ideals constructed by the sequence of s-numbers, approximation numbers, and Kolmogorov numbers in sequence space V are marked by SV, SVapp, and SVKol. For any Banach spaces X and Y, we will use the following notations.Notations 1.
(see [16])(3)Sℋ≔SℋX,Y,whereSℋX,Y≔P∈LX,Y:hs∈ℋ,where,hsx=∑v=0∞svPxv∈ℂ.Sℋapp≔SℋappX,Y,whereSℋappX,Y≔P∈LX,Y:happ∈ℋ,where,happx=∑v=0∞αvPxv∈ℂ.SℋKol≔SℋKolX,Y,whereSℋKolX,Y≔P∈LX,Y:hKol∈ℋ,where,hKolx=∑v=0∞dvPxv∈ℂ.Sℋρλ≔SℋρλX,Y,whereSℋρλX,Y≔T∈LX,Y:hλ∈ℋρ,where,hλx=∑v=0∞λvPxv∈ℂandP−λvPI=0,∀v∈ℕ.
The purpose of this study is straightforward, as follows. In Section3, we introduce and investigate the complex function space ℍbv,pvρ under the definite function ρ. In Section 4, the mapping ideals constructed by s-numbers and ℍbv,pvρ are presented. We have studied their geometric and topological properties. Specifically, we explore in Section 5 the upper limits of s-numbers for infinite series of the weighted v-th power forward and backward shift mapping on ℍbv,pvρ and their applications to various entire functions. Finally, in Section 6, we present an extension of Caristi’s fixed point theorem in ℍbv,pvρ.


## 2. Definitions and Preliminaries

Letℝℕ, ℓ∞, ℓr, and c0 denote the spaces of each, bounded, r-absolutely summable, and null sequences of real numbers, respectively.Definition 1.
(see [16]). The function space ℋ=h∈ℂℂ:hy=∑v=0∞hv^yv is called a special space of formal power series (or in short ssfps), if it shows the following settings:(1)
eb∈ℋ, for all b∈ℕ, where eby=∑v=0∞evb^yv=yb.(2)
Ifg∈ℋ and hv^|≤|gv^, for all v∈ℕ, one has h∈ℋ.(3)
Supposeh∈ℋ, then h.∈ℋ, where h.y=∑b=0∞hb/2^yb and b/2 marks the integral part of b/2.Theorem 1 (see [16]).
Sℋ is a mapping ideal, when ℋ is a ssfps.We denote the space of finite formal power series byF, i.e., if h∈F, one has k∈ℕ with hy=∑v=0khv^yv. Also, θ indicates the zero function of ℋ.Definition 2.
(see [16]). A subspace ℋρ of the ssfps is said to be a pre-quasi normed ssfps, if there is a function ρ:ℋ⟶0,∞ which verifies the next conditions:(i)
Forh∈ℋ, we have ρh≥0 and h=θ⟺ρh=0.(ii)
Supposeh∈ℋ and λ∈ℂ, then there are l≥1 with ρλh≤λlρh.(iii)
Letf,g∈ℋ; then, there are K≥1 such that ρf+g≤Kρf+ρgℤ.Recall that if the spaceℋρ is complete, then ℋρ is called a pre-quasi Banach ssfps.Definition 3.
(see [16]). A subspace ℋρ of the ssfps is called a pre-modular ssfps, if there is a function ρ:ℋ⟶0,∞ which verifies the next conditions:(i)
Forh∈ℋ, we have ρh≥0 and h=θ⟺ρh=0.(ii)
Supposeh∈ℋ and λ∈ℂ, then there are l≥1 with ρλh≤λlρh.(iii)
Letf,g∈ℋ; then, there are K≥1 such that ρf+g≤Kρf+ρg.(iv)
Supposefb^|≤|gb^, for every b∈ℕ; then, ρf≤ρgF.(v)
There areK0≥1 so that ρf≤ρf.≤K0ρf.(vi)
F¯=ℋρ.(vii)
One hasξ>0 with ρλe0≥ξλρe0, where λ∈ℂ.Theorem 2 (see [16]).
Every pre-modular ssfpsℋρ is a pre-quasi normed ssfps.Definition 4.
(see [21]). A function s:LX,Y⟶0,∞ℕ is called an s-number, if the sequence sbBa=0∞, for any B∈LX,Y, satisfies the following setup:(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∈ℕ.(c)
The inequalitysaABD≤AsaB≥ holds, if D∈LX0,X, B∈LX,Y, and A∈LY,Y0; suppose that X0 and Y0 are any two Banach spaces.(d)
ForA∈LX,Y and λ∈ℝ, then saλA=λsaA.(e)
SupposerankA≤b; then, sbA=0, whenever A∈LX,Y,(f)
Assume thatIb represents the unit map on the b-dimensional Hilbert space ℓ2b; then, sr≥bIb=0 or sr<bIb=1.The following are some instances ofs-numbers:(i)
Thek-th approximation number, αkA, is presented as(4)αkA=infA−B:B∈LX,Yand rankB≤k.(ii)
Thek-th Kolmogorov number, dkA, is presented as(5)dkA=infdimY≤ksupu≤1infv∈YAu−v.Lemma 1 (see [7]).
Assume thatB∈LX,Y and B∉ΛX,Y, and we have maps D∈LX and M∈LY with MBDeb=eb, for each b∈ℕ.Definition 5.
(see [7]). A Banach space Y is named simple if LY contains one and only one non-trivial closed ideal.Theorem 3 (see [7]).
SupposeZ is a Banach space with dimZ=∞, and we have(6)FZ⫋ΛZ⫋LcZ⫋LZ.Definition 6.
(see [7]). A class U⊆L is said to be a mapping ideal if every component UX,Y=U∩LX,Y satisfies the next setups:(i)
F⊆U.(ii)
UX,Y is linear space on ℝ.(iii)
AssumeD∈LX0,X, B∈UX,Y, and A∈LY,Y0; then, ABD∈UX0,Y0.Definition 7.
(see [10]). A function g:U⟶0,∞ is called a pre-quasi norm on the ideal U if it satisfies the following setups:(1)
SupposeB∈LX,Y, gB≥0, and gB=0⟺B=0.(2)
There isM≥1 with gυA≤MυgA, for all υ∈ℂ and A∈UX,Y.(3)
One hasK≥1 so that gA1+A2≤KgA1+gA2, for every A1,A2∈UX,Y.(4)
We getC≥1 so that if A∈LX0,X, B∈UX,Y, and D∈LY,Y0, then gDBA≤CDgBA, where X0 and Y0 are normed spaces.Theorem 4 (see [10]).
Every quasi norm is a pre-quasi norm on the same ideal.With finite non-zero coordinates, we denote the space of every sequence byℱ.Theorem 5 (see [22]).
Supposes−type Vυ:=f=srT∈ℝℕ:T∈LX,Yandυf<∞. If SVυ is a mapping ideal, we have(1)
ℱ⊂s−type Vυ.(2)
AssumesrT1r=0∞∈s−type Vυ and srT2r=0∞∈s−type Vυ; then, srT1+T2r=0∞∈s−type Vυ.(3)
Ifλ∈ℂ and srTr=0∞∈s−type Vυ, then λsrTr=0∞∈s−type Vυ.(4)
Vυ is solid, i.e., if sxJx=0∞∈s−type Vυ and sxH≤sxJ, for all x∈ℕ and H,J∈LX,Y, then sxHx=0∞∈s−type Vυ.By cardG, we denote the number of elements of G.Lemma 2 (see [23]).
Supposeξii∈Ψ is a bounded family of ℝ. Hence,(7)infcardG=bsupi∉Gξi=supcardG=b+1infi∈Gξi.We will apply the next inequality [24]. For all ra,ta∈ℂℕ and qa∈0,∞ℕ, we have(8)ra+taqa≤Kraqa+taqa,where K=max1,2ϖq−1 and ϖq=max1,supaqa.Definition 8.
(see [16]). Assume ℋρ is a pre-quasi normed ssfps. A mapping Vy:ℋρ⟶ℋρ is called forward shift, if Vyh=yh, for all h∈ℋρ, where Vyhy=∑v=0∞hv^yv+1∈ℂ and ρVyh<∞.Definition 9.
(see [16]). Suppose ℋρ is a pre-quasi normed ssfps. A mapping By:ℋρ⟶ℋρ is called backward shift, if Byhy=hy−h0/y, for all h∈ℋρ, where Byhy=∑v=0∞hv+1^yv∈ℂ and ρByh<∞.Definition 10.
(see [20]). If gy=∑m=0∞amym, then Vgyhy:=∑m=0∞amVymhy.Definition 11.
(see [20]). If gy=∑m=0∞amym, then Bgyhy:=∑m=0∞amBymhy.


## 3. Pre-Modular ssfps

This section contains the space’s definitionℍbn,pnρ under the function ρ, where ρh=∑v=0∞bvhv^pv, for all h∈ℍbn,pn. We offer enough setups on ℍbn,pnρ to become pre-modular ssfps, which implies that ℍbn,pnρ is a pre-quasi Banach ssfps.Letp=pvv∈ℕ,bvv∈ℕ∈0,∞ℕ, and we define the following function space:(9)ℍbv,pv=h∈ℂℂ:hy=∑v=0∞hv^yvandργh<∞,for someγ>0.Theorem 6.
Ifpv∈ℓ∞, then(10)ℍbv,pv=h∈ℂℂ:hy=∑v=0∞hv^yvandργh<∞,for allγ>0.Proof.
(11)ℍbv,pv=h∈ℂℂ:hy=∑v=0∞hv^yvandργh<∞,for someγ>0=h∈ℂℂ:hy=∑v=0∞hv^yvand∑v=0∞γbvhv^pv<∞,for someγ>0=h∈ℂℂ:hy=∑v=0∞hv^yvandinfv|γ|pv∑v=0∞bvhv^pv<∞,for someγ>0=h∈ℂℂ:hy=∑v=0∞hv^yvand∑v=0∞bvhv^pv<∞,for anyγ>0=h∈ℂℂ:hy=∑v=0∞hv^yvandργh<∞,for anyγ>0.Hereafter, we will denote the space of all monotonic decreasing and monotonic increasing sequences of positive reals bymd↘ and mi↗, respectively.Theorem 7.
ℍbn,pn is a ssfps, if it verifies the next setups:(a1)
pn∈mi↗∩ℓ∞.(a2)
bn∈md↘, or bn∈mi↗ with C≥1 so that b2n+1≤Cbn.Proof.
(1-i) Assumef,g∈ℍbn,pn; then, fz=∑n=0∞fn^zn∈ℂand gz=∑n=0∞gn^zn∈ℂ. We have f+gz=∑n=0∞fn^+gn^zn∈ℂ. Since pn is bounded, we get(12)∑n=0∞bnfn^+bngn^pn≤K∑n=0∞bnfn^pn+∑n=0∞bngn^pn<∞,and thenf+g∈ℍbn,pn.(1-ii) Letλ∈ℂ and f∈ℍbv,pv. We have λfz=∑v=0∞λfv^zv∈ℂ. Since pv is bounded, we have(13)∑v=0∞|λbvfv^|pv≤supv|λ|pv∑v=0∞|bvfv^|pv<∞.Then,λf∈ℍbv,pv. Therefore, by using components (1-i) and (1-ii), ℍbv,pv is linear. Clearly, ek∈ℍbv,pv, for all k∈ℕ, where ekz=∑v=0∞evk^zv=zk and ∑v=0∞|bvevk^|pv=bkpk.(2)
Letfv^|≤|gv^, for all v∈ℕ and g∈ℍbv,pv. Then, gz=∑v=0∞gv^zv∈ℂ. Since bv>0, for all v∈ℕ, then(14)∑v=0∞bvfv^pv≤∑v=0∞bvgv^pv<∞.Hence,fz=∑v=0∞fv^zv∈ℂ and ρf<∞. Therefore, f∈ℍbv,pv.(3)
Letf∈ℍbv,pv, bv be an increasing sequence, and there exists C>0 such that b2v+1≤Cbv and pv is increasing. Therefore, fz=∑v=0∞fv^zv∈ℂ and ρf<∞. One has(15)ρf.=∑v=0∞bvfv/2^pv=∑v=0∞b2vfv^p2v+∑v=0∞b2v+1fv^p2v+1≤∑v=0∞b2vfv^pv+∑v=0∞b2v+1fv^pv≤max1,2supvCpvρf.
This implies thatf.z=∑v=0∞fv/2^zv∈ℂ and ρf.<∞. Hence, f.∈ℍbv,pv.Theorem 8.
Let conditions (a1) and (a2) be satisfied; then, the spaceℍbn,pnρ is a pre-modular Banach ssfps.Proof.
(i)
Evidently, for allf∈ℍbn,pn, then ρf≥0 and ρf=0⟺f=θ.(ii)
We havel=max1,supnηpn−1≥1, for all η∈ℝ∖0 and l≥1, for η=0 such that(16)ρηf=∑n=0∞ηbnfn^pn≤supnηpn∑n=0∞bnfn^pn≤lηρf,for everyf∈ℍbn,pn.(iii)
For someK=max1,2supnpn−1, we obtain(17)ρf+g=∑n=0∞bnfn^+gn^pn≤Kρf+ρg,for allf,g∈ℍbn,pn.(iv)
It is clear from the proof part (2) of Theorem7.(v)
From the proof part (3) of Theorem7, we have that K0=max1,2supnCpn≥1.(vi)
It is apparent thatF¯=ℍbn,pn.(vii)
There isζ with 0<ζ≤ηp0−1 such that ρηe0≥ζ|η|ρe0, for each η≠0 and ζ>0, if η=0.
Therefore, the spaceℍbv,pvρ is pre-modular ssfps. To show that ℍbv,pvρ is a pre-modular Banach ssfps, suppose fn is a Cauchy sequence in ℍbv,pvρ; then, for all ε∈0,1, there is n0∈ℕ such that, for all n,m≥n0, we get(18)ρfn−fm=∑v=0∞bvfvn^−fvm^pv<εϖp.
Forn,m≥n0 and v∈ℕ, we obtain(19)|fvn^−fvm^|<ε.
Hence,fvm^ is a Cauchy sequence in ℂ, for fixed v∈ℕ, so limm⟶∞fvm^=fv0^, for fixed v∈ℕ. Therefore, ρfn−f0<εϖp, for each n≥n0. Finally, to explain that f0∈ℍbv,pv, we have(20)ρf0=ρf0−fn+fn≤Kρfn−f0+ρfn<∞.
So,f0∈ℍbv,pv. This implies that ℍbv,pvρ is a pre-modular Banach ssfps.Taking into consideration Theorem2, we put forward the following theorem.Theorem 9.
Let conditions (a1) and (a2) be satisfied. Then, the spaceℍbn,pnρ is a pre-quasi Banach ssfps.Theorem 10.
Let conditions (a1) and (a2) be satisfied. Then, the spaceℍbn,pnρ is a pre-quasi closed ssfps.Proof.
Assume that the setups are verified. From Theorem9, the space ℍbn,pnρ is a pre-quasi normed ssfps. To show that ℍbn,pnρ is a pre-quasi closed ssfps, assume hmm=0∞∈ℍbn,pnρ and limm⟶∞ρhm−h0=0; then, for every ε∈0,1, there is m0∈ℕ such that for all m≥m0, one has(21)ε>ρhm−h0=∑a=0∞baham^−ha0^pa1/ϖp.
Hence, form≥m0 and a∈ℕ, we get(22)ham^−ha0^<ε.
So,ham^ is a convergent sequence in ℂ, for fixed a∈ℕ. Therefore, limm⟶∞ham^=ha0^, for fixed a∈ℕ. Finally to prove that h0∈ℍbn,pnρ, we consider(23)ρh0=ρh0−hm+hm≤Kρhm−h0+ρhm<∞,so h0∈ℍbn,pnρ. This finishes the proof.


## 4. Pre-Quasi Ideal

In this section, the mapping ideals constructed bys-numbers and ℍbn,pnρ are presented. We have studied their geometric and topological structures. We will use the notation for B∈Sℍbv,pvρ, that is, gB=ρfs, fsz=∑v=0∞svBzv∈ℂ, and ρfs=∑v=0∞bvsvBpv,for everyfs∈ℍbv,pvρ.In view of Theorems1 and 7, we conclude the next theorem.Theorem 11.
Let conditions (a1) and (a2) be satisfied. Then,Sℍbn,pn is a mapping ideal.


### 4.1. Ideal of Finite Rank Mappings

In this section, enough setups (not necessary) onℍbn,pnρ so that F is dense in Sℍbn,pnρ are investigated. This explains the non-linearity of the s−type ℍbn,pnρ spaces (Rhoades open problem [25]).Example 1.
The sequencebn=n+1/n+2n∈ℕ satisfies bn∈mi↗ and b2n+1≤Cbn, for some C≥2.Theorem 12.
FX,Y¯=Sℍbn,pnρX,Y, whenever conditions (a1) and (a2) are satisfied.Proof.
It is clear thatFX,Y¯⊂Sℍbn,pnρX,Y, since the space Sℍbn,pnρ is a mapping ideal. Currently, we substantiate that Sℍbn,pnρX,Y⊆FX,Y¯. On taking T∈Sℍbn,pnρX,Y, fs∈ℍbn,pnρ, with fsz=∑n=0∞snTzn∈ℂ. Hence, ρfs<∞, and assume ε∈0,1, so there is m∈ℕ−0 such that ρfs−∑n=0m−1en<ε/4C2, for some C≥1. While snTn∈ℕ is decreasing, we get(24)∑n=m+12mbns2mTpn≤∑n=m+12mbnsnTpn≤∑n=m∞bnsnTpn<ε4C2.
Hence, there existA∈F2mX,Y, rankA≤2m, and(25)∑n=2m+13mbnT−Apn≤∑n=m+12mbnT−Apn<ε4C2.
Sincepn is bounded,(26)∑n=0mbnT−Apn<ε4C2.
Letbn be monotonically increasing such that there exists a constant C≥1 for which b2n+1≤Cbn. Then, we have for n≥m that(27)b2m+n≤b2m+2n+1≤Cbm+n≤Cb2n≤Cb2n+1≤C2bn.
SinceT−A∈Sℍbn,pnρX,Y, then hs∈ℍbn,pnρ, where hsz:=∑n=0∞snT−Azn∈ℂ. Since pn is increasing, inequalities (2)–(5) give(28)dT,A=ρhs=∑n=03m−1bnsnT−Apn+∑n=3m∞bnsnT−Apn≤∑n=03mbnT−Apn+∑n=m∞bn+2msn+2mT−Apn+2m≤3∑n=0mbnT−Apn+C2supnpn∑n=m∞bnsnTpn<ε.
SinceI8∈Sℍn+1,1/n+1ρX,Y which gives a counter example of the converse statement, this finishes the proof.According to Theorem12, if (a1) and (a2) are fulfilled, then every compact mapping is represented by finite rank mappings; however, the reverse is not necessarily true.


### 4.2. Closed and Banach

In this part, we have investigated the sufficient conditions onℍbv,pvρ such that the pre-quasi mapping ideal Sℍbv,pvρ is Banach and closed.Theorem 13.
IfX and Y are Banach spaces and conditions (a1) and (a2) are satisfied, then the function gB=ρfs is a pre-quasi norm on Sℍbv,pvρ.Proof.
Suppose the conditions are verified, sog verifies the next setups:(1)
LetB∈Sℍbv,pvρX,Y, then we have gB=ρfs≥0, and it is clear that gB=ρfs=0, if and only if, svB=0, for all v∈N, if and only if, B=0.(2)
We havel≥1 with gλB=ρλfs≤lλρfs=lλgB, for every B∈Sℍbv,pvρX,Y and λ∈ℂ.(3)
One hasKK0≥1 for B1,B2∈Sℍbv,pvρX,Y. Therefore, f1sz=∑v=0∞svB1zv∈ℂ and f2sz=∑v=0∞svB2zv∈ℂ. Therefore, for hsz=:∑v=0∞svB1+B2zv, one can see that(29)gB1+B2=ρhs≤ρf1s.+f2s.≤Kρf1s.+ρf2s.≤KK0gB1+gB2.(4)
We haveC≥1; suppose A∈LX0,X, B∈Sℍbv,pvρX,Y, and D∈LY,Y0. Therefore, fsz=∑v=0∞svBzv∈ℂ. Then, for hsz=:∑v=0∞svDBAzv, one can see that(30)gDBA=ρhs≤ρADfs≤CAgBD.Theorem 14.
IfX and Y are Banach spaces and conditions (a1) and (a2) are satisfied, then Sℍbv,pvρ,g is a pre-quasi Banach mapping ideal.Proof.
Suppose the conditions are verified, then the functiongB=ρfs is a pre-quasi norm on Sℍbv,pvρ. Let Bm be a Cauchy sequence in Sℍbv,pvρX,Y. Therefore, fsm∈ℍbv,pvρ∈ℂ and fsmz=∑v=0∞svBmzv∈ℂ. Assume hsz=:∑v=0∞svBi−Bjzn; then, by using conditions (iv) and (vii) of Definition 3 and since LX,Y⊇Sℍbv,pvρX,Y, we get(31)gBi−Bj=ρhs≥ρs0Bi−Bje0=ρBi−Bje0≥ξBi−Bjρe0.
Thus,Bmm∈ℕ is a Cauchy sequence in LX,Y. While the space LX,Y is a Banach space, there exists B∈LX,Y with limm⟶∞Bm−B=0 and since fsm∈ℍbv,pvρ, for each m∈ℕ, using Theorem 13 and the continuity of ρ at θ, we obtain(32)gB=gB−Bm+Bm≤KK0gBm−B+gBm=KK0ρBm−B∑m=0∞em+KK0ρfsm<ε.
Thus, we havefs∈ℍbv,pvρ; then, B∈Sℍbv,pvρX,Y.Theorem 15.
IfX and Y are Banach spaces and conditions (a1) and (a2) are satisfied, then Sℍbv,pvρ,g is a pre-quasi closed mapping ideal.Proof.
Suppose the conditions are verified; then, the functiongB=ρfs is a pre-quasi norm on Sℍbv,pvρ. Assume Bm∈Sℍbv,pvρX,Y, with m∈ℕ and limm⟶∞gBm−B=0. Therefore, fsm∈ℍbv,pvρ∈ℂ and fsmz=∑v=0∞svBmzv∈ℂ. Suppose hsz=:∑v=0∞svBi−Bjzv; then, from conditions (iv) and (vii) of Definition 3 and since LX,Y⊇Sℍbv,pvρX,Y, we get(33)gB−Bj=ρhs≥ρs0B−Bje0=ρB−Bje0≥ξB−Bjρe0,and then Bmm∈ℕ is a convergent sequence in LX,Y. While the space LX,Y is a Banach space, there exists B∈LX,Y with limm⟶∞Bm−B=0 and since fsm∈ℍbv,pvρ, for each m∈ℕ, using Theorem 13 and the continuity of ρ at θ, one can see that(34)gB=gB−Bm+Bm≤KK0gBm−B+gBm=KK0ρBm−B∑m=0∞em+KK0ρfsm<ε,and we have fs∈ℍbv,pvρ; then, B∈Sℍbv,pvρX,Y.We deduce the following characteristics of thes−type ℍbv,pvρ using Theorem 5.Theorem 16.
Fors−type ℍbv,pvρ:=snT∈ℝℕ:T∈Sℍbv,pvρX,Y, the following holds:(1)
We haves−type ℍbv,pvρ⊃ℱ.(2)
IfsrT1r=0∞∈s−type ℍbv,pvρ and srT2r=0∞∈s−type ℍbv,pvρ, then srT1+T2r=0∞∈s−type ℍbn,pnρ.(3)
For allλ∈ℂ and srTr=0∞∈s−type ℍbn,pnρ, then |λ|srTr=0∞∈s−type ℍbn,pnρ.(4)
Thes−type ℍbn,pnρ is solid.


### 4.3. Smallness

We give here some inclusion relations concerning the spaceSℍbn,pnρ for different bn and pn.Theorem 17.
IfX and Y are Banach spaces with dimX=dimY=∞, 0<pn<qn, 0<an<bn, for all n∈ℕ, and setups (a1) and (a2) are satisfied, it is true that(35)Sℍbn,pnρX,Y⊂≠Sℍan,qnρX,Y⊂≠LX,Y.Proof.
SupposeT∈Sℍbn,pnρX,Y. Therefore, fs∈ℍbn,pnρ and fsz=∑n=0∞snTzn∈ℂ. One can see that(36)∑n=0∞ansnTqn<∑n=0∞bnsnTpn<∞,hence T∈Sℍan,qnρX,Y. Next, if we take T with snT=n+1−1/pn/bn, then T∉Sℍbn,pnρX,Y and T∈Sℍan,qnρX,Y. Clearly, Sℍan,qnρX,Y⊂LX,Y. By choosing T with snT=n+1−1/qn/an, then T∉Sℍan,qnρX,Y and T∈LX,Y. This finishes the proof.In this part, we investigate the setups for whichSℍbn,pnρapp is small.Theorem 18.
IfX and Y are Banach spaces with dimX=dimY=∞, assume that the conditions (a1), (a2), and bn∉ℓpn are satisfied, and hence Sℍbn,pnρapp,g is small, where gU=∑j=0∞bjαjUpj.Proof.
LetSℍbn,pnρappX,Y=LX,Y. Therefore, one gets V>0 so that gU≤VU, for every U∈LX,Y. According to Dvoretzky’s theorem [26] with r∈ℕ, there are quotient spaces X/λr and subspaces ηr of Y that mapped onto ℓ2r by isomorphisms Dr and Br with DrDr−1≤2 and BrBr−1≤2. Let Ir be the identity mapping on ℓ2r, ζr be the quotient mapping from X onto X/λr, and Jr be the natural embedding mapping from ηr into Y. Let ha, for all a∈ℕ, be the Bernstein numbers [27]; we have then(37)1=haIr=haBrBr−1IrDrDr−1≤BrhaBr−1IrDrDr−1,=BrhaJrBr−1IrDrDr−1≤BrrdaJrBr−1IrDrDr−1=BrdaJrBr−1IrDrζrDr−1≤BrrαaJrBr−1IrDrζrDr−1,for 0≤j≤r. We have l≥1 so that(38)bjpj≤BrDr−1pjbjαjJrBr−1IrDrζrpj,bjpj≤lBrbjαjJrBr−1IrDrζrpjDr−1,∑j=0rbjpj≤lBrDr−1∑j=0rbjαjJrBr−1IrDrζrpj,∑j=0rbjpj≤lBrDr−1gJrBr−1IrDrζr,∑j=0rbjpj≤lVBrDr−1JrBr−1IrDrζr,∑j=0rbjpj≤lVBrDr−1JrBr−1IrDrζr=lVBrDr−1Br−1IrDr,∑j=0rbjpj≤4lV.
Asr⟶∞, then ∑j=0∞bjpj<∞. This contradicts bn∉ℓpn. Therefore, dimX<∞ and dimY<∞. Hence, the space SHbn,pnρapp is small.By the same manner, we can easily conclude the next theorem.Theorem 19.
IfX and Y are Banach spaces with dimX=dimY=∞, assume that conditions (a1), (a2), and bn∉ℓpn are satisfied, and hence Sℍbn,pnρKol,g is small, where gU=∑j=0∞bjdjUpj.


### 4.4. Simpleness

We introduce an answer of the next question; for whichℍbn,pnρ, is the space Sℍbn,pnρ simple?Theorem 20.
Ifpn, qn verify 1≤pn<qn and 0<an<bn, for all n∈ℕ, and the setups (a1), (a2) are satisfied, then(39)LSℍan,qnρ,Sℍbn,pnρ=ΛSℍan,qnρ,Sℍbn,pnρ.Proof.
Suppose there isT∈LSℍan,qnρ,Sℍbn,pnρ. and T∉ΛSℍan,qnρ,Sℍbn,pnρ. According to Lemma 1, we can find G∈LSℍan,qnρ and B∈LSℍbn,pnρ with BTGIm=Im. For every m∈ℕ, one has(40)ImSℍbv,pvρ=∑v=0∞bvαvImpv1/suppv=∑v=0m−1bv1/suppv≤BTGImSℍav,qvρ≤∑v=0∞avαvImqv1/supqv=∑v=0m−1av1/supqv.
This contradicts Theorem17.Corollary 1.
Ifpn, qn verify 1≤pn<qn and 0<an<bn, for all n∈ℕ, and the setups (a1), (a2) are satisfied, then LSℍan,qnρ,Sℍbn,pnρ=LCSℍan,qnρ,Sℍbn,pnρ.Proof.
It is clear fromΛ⊆Lc.Theorem 21.
If setups (a1), (a2) are satisfied withp0≥1, then the space Sℍbn,pnρ is simple.Proof.
AssumeT∈LCSℍbn,pnρ and T∉ΛSℍbn,pnρ. From Lemma 1, one has G,B∈LSℍbn,pnρ so that BTGIk=Ik. We have Iℍbn,pnρ∈LCSℍbn,pnρ. Therefore, LSℍbn,pnρ=LCSℍbn,pnρ. This implies that there is one non-trivial closed ideal ΛSℍbn,pnρ in LSℍbn,pnρ.


### 4.5. Spectrum

In this part, we expound the sufficient conditions onℍbn,pnρ such that Sℍbn,pnρλ equals Sℍbn,pnρ.Theorem 22.
IfX and Y are Banach spaces with dimX=dimY=∞ and suppose setups (a1), (a2) are satisfied and infnbnpn>0, then(41)Sℍbn,pnρλX,Y=Sℍbn,pnρX,Y.Proof.
LetT∈Sℍbn,pnρλX,Y, and hence fλ∈ℍbn,pnρ, where fλz=∑n=0∞λnTzn∈ℂ with ρfλ=∑n=0∞bnλnTpn<∞, and T−λlTI=0, for all l∈ℕ. We have T=λlTI, with l∈ℕ, so slT=slλlTI=λlT, with l∈ℕ. Therefore, fs∈ℍbn,pnρ, so T∈Sℍbn,pnρX,Y.
Secondly, assumeT∈Sℍbn,pnρX,Y. Therefore, fs∈ℍbn,pnρ. Hence, we have(42)∞>∑r=0∞brsrTpr≥infrbrpr∑r=0∞srTpr.
Sinceinfrbrpr>0, then limr⟶∞srT=0. Assume T−srTI−1 exists, for every r∈ℕ. Therefore, T−srTI−1r∈ℕ∈ℓ∞. So, limr⟶∞T−srTI−1=T−1 exists and is bounded. From the pre-quasi mapping ideal of Sℍbn,pnρ,g, we obtain(43)I=TT−1∈Sℍbr,prρX,Y⇒∑r=0∞er∈ℍbr,prρ⇒∑r=0∞brpr<∞.
This contradictsinfrbrpr>0. Therefore, T−srTI=0, for every r∈ℕ. This gives T∈Sℍbn,pnρλX,Y. This provides the proof.


## 4.1. Ideal of Finite Rank Mappings

In this section, enough setups (not necessary) onℍbn,pnρ so that F is dense in Sℍbn,pnρ are investigated. This explains the non-linearity of the s−type ℍbn,pnρ spaces (Rhoades open problem [25]).Example 1.
The sequencebn=n+1/n+2n∈ℕ satisfies bn∈mi↗ and b2n+1≤Cbn, for some C≥2.Theorem 12.
FX,Y¯=Sℍbn,pnρX,Y, whenever conditions (a1) and (a2) are satisfied.Proof.
It is clear thatFX,Y¯⊂Sℍbn,pnρX,Y, since the space Sℍbn,pnρ is a mapping ideal. Currently, we substantiate that Sℍbn,pnρX,Y⊆FX,Y¯. On taking T∈Sℍbn,pnρX,Y, fs∈ℍbn,pnρ, with fsz=∑n=0∞snTzn∈ℂ. Hence, ρfs<∞, and assume ε∈0,1, so there is m∈ℕ−0 such that ρfs−∑n=0m−1en<ε/4C2, for some C≥1. While snTn∈ℕ is decreasing, we get(24)∑n=m+12mbns2mTpn≤∑n=m+12mbnsnTpn≤∑n=m∞bnsnTpn<ε4C2.
Hence, there existA∈F2mX,Y, rankA≤2m, and(25)∑n=2m+13mbnT−Apn≤∑n=m+12mbnT−Apn<ε4C2.
Sincepn is bounded,(26)∑n=0mbnT−Apn<ε4C2.
Letbn be monotonically increasing such that there exists a constant C≥1 for which b2n+1≤Cbn. Then, we have for n≥m that(27)b2m+n≤b2m+2n+1≤Cbm+n≤Cb2n≤Cb2n+1≤C2bn.
SinceT−A∈Sℍbn,pnρX,Y, then hs∈ℍbn,pnρ, where hsz:=∑n=0∞snT−Azn∈ℂ. Since pn is increasing, inequalities (2)–(5) give(28)dT,A=ρhs=∑n=03m−1bnsnT−Apn+∑n=3m∞bnsnT−Apn≤∑n=03mbnT−Apn+∑n=m∞bn+2msn+2mT−Apn+2m≤3∑n=0mbnT−Apn+C2supnpn∑n=m∞bnsnTpn<ε.
SinceI8∈Sℍn+1,1/n+1ρX,Y which gives a counter example of the converse statement, this finishes the proof.According to Theorem12, if (a1) and (a2) are fulfilled, then every compact mapping is represented by finite rank mappings; however, the reverse is not necessarily true.


## 4.2. Closed and Banach

In this part, we have investigated the sufficient conditions onℍbv,pvρ such that the pre-quasi mapping ideal Sℍbv,pvρ is Banach and closed.Theorem 13.
IfX and Y are Banach spaces and conditions (a1) and (a2) are satisfied, then the function gB=ρfs is a pre-quasi norm on Sℍbv,pvρ.Proof.
Suppose the conditions are verified, sog verifies the next setups:(1)
LetB∈Sℍbv,pvρX,Y, then we have gB=ρfs≥0, and it is clear that gB=ρfs=0, if and only if, svB=0, for all v∈N, if and only if, B=0.(2)
We havel≥1 with gλB=ρλfs≤lλρfs=lλgB, for every B∈Sℍbv,pvρX,Y and λ∈ℂ.(3)
One hasKK0≥1 for B1,B2∈Sℍbv,pvρX,Y. Therefore, f1sz=∑v=0∞svB1zv∈ℂ and f2sz=∑v=0∞svB2zv∈ℂ. Therefore, for hsz=:∑v=0∞svB1+B2zv, one can see that(29)gB1+B2=ρhs≤ρf1s.+f2s.≤Kρf1s.+ρf2s.≤KK0gB1+gB2.(4)
We haveC≥1; suppose A∈LX0,X, B∈Sℍbv,pvρX,Y, and D∈LY,Y0. Therefore, fsz=∑v=0∞svBzv∈ℂ. Then, for hsz=:∑v=0∞svDBAzv, one can see that(30)gDBA=ρhs≤ρADfs≤CAgBD.Theorem 14.
IfX and Y are Banach spaces and conditions (a1) and (a2) are satisfied, then Sℍbv,pvρ,g is a pre-quasi Banach mapping ideal.Proof.
Suppose the conditions are verified, then the functiongB=ρfs is a pre-quasi norm on Sℍbv,pvρ. Let Bm be a Cauchy sequence in Sℍbv,pvρX,Y. Therefore, fsm∈ℍbv,pvρ∈ℂ and fsmz=∑v=0∞svBmzv∈ℂ. Assume hsz=:∑v=0∞svBi−Bjzn; then, by using conditions (iv) and (vii) of Definition 3 and since LX,Y⊇Sℍbv,pvρX,Y, we get(31)gBi−Bj=ρhs≥ρs0Bi−Bje0=ρBi−Bje0≥ξBi−Bjρe0.
Thus,Bmm∈ℕ is a Cauchy sequence in LX,Y. While the space LX,Y is a Banach space, there exists B∈LX,Y with limm⟶∞Bm−B=0 and since fsm∈ℍbv,pvρ, for each m∈ℕ, using Theorem 13 and the continuity of ρ at θ, we obtain(32)gB=gB−Bm+Bm≤KK0gBm−B+gBm=KK0ρBm−B∑m=0∞em+KK0ρfsm<ε.
Thus, we havefs∈ℍbv,pvρ; then, B∈Sℍbv,pvρX,Y.Theorem 15.
IfX and Y are Banach spaces and conditions (a1) and (a2) are satisfied, then Sℍbv,pvρ,g is a pre-quasi closed mapping ideal.Proof.
Suppose the conditions are verified; then, the functiongB=ρfs is a pre-quasi norm on Sℍbv,pvρ. Assume Bm∈Sℍbv,pvρX,Y, with m∈ℕ and limm⟶∞gBm−B=0. Therefore, fsm∈ℍbv,pvρ∈ℂ and fsmz=∑v=0∞svBmzv∈ℂ. Suppose hsz=:∑v=0∞svBi−Bjzv; then, from conditions (iv) and (vii) of Definition 3 and since LX,Y⊇Sℍbv,pvρX,Y, we get(33)gB−Bj=ρhs≥ρs0B−Bje0=ρB−Bje0≥ξB−Bjρe0,and then Bmm∈ℕ is a convergent sequence in LX,Y. While the space LX,Y is a Banach space, there exists B∈LX,Y with limm⟶∞Bm−B=0 and since fsm∈ℍbv,pvρ, for each m∈ℕ, using Theorem 13 and the continuity of ρ at θ, one can see that(34)gB=gB−Bm+Bm≤KK0gBm−B+gBm=KK0ρBm−B∑m=0∞em+KK0ρfsm<ε,and we have fs∈ℍbv,pvρ; then, B∈Sℍbv,pvρX,Y.We deduce the following characteristics of thes−type ℍbv,pvρ using Theorem 5.Theorem 16.
Fors−type ℍbv,pvρ:=snT∈ℝℕ:T∈Sℍbv,pvρX,Y, the following holds:(1)
We haves−type ℍbv,pvρ⊃ℱ.(2)
IfsrT1r=0∞∈s−type ℍbv,pvρ and srT2r=0∞∈s−type ℍbv,pvρ, then srT1+T2r=0∞∈s−type ℍbn,pnρ.(3)
For allλ∈ℂ and srTr=0∞∈s−type ℍbn,pnρ, then |λ|srTr=0∞∈s−type ℍbn,pnρ.(4)
Thes−type ℍbn,pnρ is solid.


## 4.3. Smallness

We give here some inclusion relations concerning the spaceSℍbn,pnρ for different bn and pn.Theorem 17.
IfX and Y are Banach spaces with dimX=dimY=∞, 0<pn<qn, 0<an<bn, for all n∈ℕ, and setups (a1) and (a2) are satisfied, it is true that(35)Sℍbn,pnρX,Y⊂≠Sℍan,qnρX,Y⊂≠LX,Y.Proof.
SupposeT∈Sℍbn,pnρX,Y. Therefore, fs∈ℍbn,pnρ and fsz=∑n=0∞snTzn∈ℂ. One can see that(36)∑n=0∞ansnTqn<∑n=0∞bnsnTpn<∞,hence T∈Sℍan,qnρX,Y. Next, if we take T with snT=n+1−1/pn/bn, then T∉Sℍbn,pnρX,Y and T∈Sℍan,qnρX,Y. Clearly, Sℍan,qnρX,Y⊂LX,Y. By choosing T with snT=n+1−1/qn/an, then T∉Sℍan,qnρX,Y and T∈LX,Y. This finishes the proof.In this part, we investigate the setups for whichSℍbn,pnρapp is small.Theorem 18.
IfX and Y are Banach spaces with dimX=dimY=∞, assume that the conditions (a1), (a2), and bn∉ℓpn are satisfied, and hence Sℍbn,pnρapp,g is small, where gU=∑j=0∞bjαjUpj.Proof.
LetSℍbn,pnρappX,Y=LX,Y. Therefore, one gets V>0 so that gU≤VU, for every U∈LX,Y. According to Dvoretzky’s theorem [26] with r∈ℕ, there are quotient spaces X/λr and subspaces ηr of Y that mapped onto ℓ2r by isomorphisms Dr and Br with DrDr−1≤2 and BrBr−1≤2. Let Ir be the identity mapping on ℓ2r, ζr be the quotient mapping from X onto X/λr, and Jr be the natural embedding mapping from ηr into Y. Let ha, for all a∈ℕ, be the Bernstein numbers [27]; we have then(37)1=haIr=haBrBr−1IrDrDr−1≤BrhaBr−1IrDrDr−1,=BrhaJrBr−1IrDrDr−1≤BrrdaJrBr−1IrDrDr−1=BrdaJrBr−1IrDrζrDr−1≤BrrαaJrBr−1IrDrζrDr−1,for 0≤j≤r. We have l≥1 so that(38)bjpj≤BrDr−1pjbjαjJrBr−1IrDrζrpj,bjpj≤lBrbjαjJrBr−1IrDrζrpjDr−1,∑j=0rbjpj≤lBrDr−1∑j=0rbjαjJrBr−1IrDrζrpj,∑j=0rbjpj≤lBrDr−1gJrBr−1IrDrζr,∑j=0rbjpj≤lVBrDr−1JrBr−1IrDrζr,∑j=0rbjpj≤lVBrDr−1JrBr−1IrDrζr=lVBrDr−1Br−1IrDr,∑j=0rbjpj≤4lV.
Asr⟶∞, then ∑j=0∞bjpj<∞. This contradicts bn∉ℓpn. Therefore, dimX<∞ and dimY<∞. Hence, the space SHbn,pnρapp is small.By the same manner, we can easily conclude the next theorem.Theorem 19.
IfX and Y are Banach spaces with dimX=dimY=∞, assume that conditions (a1), (a2), and bn∉ℓpn are satisfied, and hence Sℍbn,pnρKol,g is small, where gU=∑j=0∞bjdjUpj.


## 4.4. Simpleness

We introduce an answer of the next question; for whichℍbn,pnρ, is the space Sℍbn,pnρ simple?Theorem 20.
Ifpn, qn verify 1≤pn<qn and 0<an<bn, for all n∈ℕ, and the setups (a1), (a2) are satisfied, then(39)LSℍan,qnρ,Sℍbn,pnρ=ΛSℍan,qnρ,Sℍbn,pnρ.Proof.
Suppose there isT∈LSℍan,qnρ,Sℍbn,pnρ. and T∉ΛSℍan,qnρ,Sℍbn,pnρ. According to Lemma 1, we can find G∈LSℍan,qnρ and B∈LSℍbn,pnρ with BTGIm=Im. For every m∈ℕ, one has(40)ImSℍbv,pvρ=∑v=0∞bvαvImpv1/suppv=∑v=0m−1bv1/suppv≤BTGImSℍav,qvρ≤∑v=0∞avαvImqv1/supqv=∑v=0m−1av1/supqv.
This contradicts Theorem17.Corollary 1.
Ifpn, qn verify 1≤pn<qn and 0<an<bn, for all n∈ℕ, and the setups (a1), (a2) are satisfied, then LSℍan,qnρ,Sℍbn,pnρ=LCSℍan,qnρ,Sℍbn,pnρ.Proof.
It is clear fromΛ⊆Lc.Theorem 21.
If setups (a1), (a2) are satisfied withp0≥1, then the space Sℍbn,pnρ is simple.Proof.
AssumeT∈LCSℍbn,pnρ and T∉ΛSℍbn,pnρ. From Lemma 1, one has G,B∈LSℍbn,pnρ so that BTGIk=Ik. We have Iℍbn,pnρ∈LCSℍbn,pnρ. Therefore, LSℍbn,pnρ=LCSℍbn,pnρ. This implies that there is one non-trivial closed ideal ΛSℍbn,pnρ in LSℍbn,pnρ.


## 4.5. Spectrum

In this part, we expound the sufficient conditions onℍbn,pnρ such that Sℍbn,pnρλ equals Sℍbn,pnρ.Theorem 22.
IfX and Y are Banach spaces with dimX=dimY=∞ and suppose setups (a1), (a2) are satisfied and infnbnpn>0, then(41)Sℍbn,pnρλX,Y=Sℍbn,pnρX,Y.Proof.
LetT∈Sℍbn,pnρλX,Y, and hence fλ∈ℍbn,pnρ, where fλz=∑n=0∞λnTzn∈ℂ with ρfλ=∑n=0∞bnλnTpn<∞, and T−λlTI=0, for all l∈ℕ. We have T=λlTI, with l∈ℕ, so slT=slλlTI=λlT, with l∈ℕ. Therefore, fs∈ℍbn,pnρ, so T∈Sℍbn,pnρX,Y.
Secondly, assumeT∈Sℍbn,pnρX,Y. Therefore, fs∈ℍbn,pnρ. Hence, we have(42)∞>∑r=0∞brsrTpr≥infrbrpr∑r=0∞srTpr.
Sinceinfrbrpr>0, then limr⟶∞srT=0. Assume T−srTI−1 exists, for every r∈ℕ. Therefore, T−srTI−1r∈ℕ∈ℓ∞. So, limr⟶∞T−srTI−1=T−1 exists and is bounded. From the pre-quasi mapping ideal of Sℍbn,pnρ,g, we obtain(43)I=TT−1∈Sℍbr,prρX,Y⇒∑r=0∞er∈ℍbr,prρ⇒∑r=0∞brpr<∞.
This contradictsinfrbrpr>0. Therefore, T−srTI=0, for every r∈ℕ. This gives T∈Sℍbn,pnρλX,Y. This provides the proof.


## 5. Application of Shift Mappings onℍbr,prρ

Specifically, we explore the upper limits ofs-numbers for infinite series of the weighted r-th power forward and backward shift mapping on ℍbr,prρ and their applications to various entire functions in this section, where ρf=∑r=0∞brfr^pr1/ϖp, for all f∈ℍbr,prρ.Theorem 23.
Let conditions (a1) and (a2) be satisfied,infnbn≥1, and suprbr+1/brpr+1/ϖp<∞; then, Vz∈Lℍbr,prρ with Vz=suprbr+1/brpr+1/ϖp.Proof.
Assume that the conditions are satisfied. Forf∈ℍbr,prρ, since pr is increasing and bounded from above with pr>0, for all r∈ℕ, then(44)ρVzf=ρzf=∑r=0∞br+1fr^pr+11/ϖp≤suprbr+1brpr+1/ϖp∑r=0∞|brfr^|pr+11/ϖp≤suprbr+1brpr+1/ϖpρf.
This givesVz∈Lℍbr,prρ with Vz≤suprbr+1/brpr+1/ϖp. Since Vz∈Lℍbr,prρ, then there is A>0 with ρVzf≤Aρf, for all f∈ℍbr,prρ. Hence, ρVzer≤Aρer, and one gets suprbr+1/brpr+1/ϖp≤Vz. This completes the proof.Theorem 24.
Let conditions (a1) and (a2) be satisfied,supnbn≥1, and suprbr/br+1pr/ϖp<∞; then, Bz∈Lℍbr,prρ with Bz=suprbr/br+1pr/ϖp.Proof.
Assume the conditions are satisfied. Forf∈ℍbr,prρ, since pr is increasing and bounded from above with pr>0, for all r∈ℕ, then(45)ρBzf=∑r=0∞brfr+1^pr1/ϖp≤suprbrbr+1pr/ϖp∑r=0∞br+1fr+1^pr1/ϖp≤suprbrbr+1pr/ϖpρf.
This givesBz∈Lℍbr,prρ with Bz≤suprbr/br+1pr/ϖp. Since Bz∈Lℍbr,prρ, then there is A>0 with ρBzf≤Aρf, for all f∈ℍbr,prρ. Hence, ρBzer≤Aρer, and one gets suprbr/br+1pr/ϖp≤Bz. This completes the proof.ByU, we denote the open unit disc in ℂ.Theorem 25.
Let conditions (a1) and (a2) be satisfied withp0≥1. If limsupbrprr=1, then every function in ℍbr,prρ is analytic on U. Furthermore, the convergence in ℍbr,prρ implies the uniform convergence on B⊆U, where B is compact.Proof.
Letlimsupbrprr=1, and h∈ℍbr,prρ. Then, hy=∑r=0∞hr^yr∈ℂ, with y∈ℂ and ρh=∑r=0∞brhr^pr1/ϖp<∞. Therefore, limsupbrhr^prr<1. This gives(46)limsuphr^prr<1limsupbrprr=1.
Aspr∈mi↗∩ℓ∞, one gets limsuphr^ry<y<1, with y∈U. Hence, hy=∑r=0∞hr^yr∈ℂ, with y∈U. Assume hky∈B, with k∈ℕ. Suppose limk⟶∞ρhk−h=0, where h∈ℍbr,prρ, and we have(47)hky−hy=∑r=0∞hrk^−hr^yr≤∑r=0∞hrk^−hr^yr≤∑n=0∞hrk^−hr^prbrpr1/ϖp=∑r=0∞yrqrbrqr1/ϖqρhk−h,∑r=0∞yrqrbrqr1/ϖq,where qr is increasing and bounded with q0≥1 and 1/pr+1/qr=1, for all r∈ℕ. Clearly, limsupr⟶∞yqr/brqr/r<1; then, ∑r=0∞yrqr/brqr<∞. So, limk⟶∞hky=hy∈B.Theorem 26.
IfVz is the forward shift mapping on ℍbr,prρ, we have(48)supcardξ=r+1infk∈ξbk+nbkpk+n/ϖp1An≤srVzn≤supcardξ=r+1infk∈ξbk+nbkpk+n/ϖp,where An=∑k=0∞bkfk^pk1/ϖp/∑k=0∞bkfk^pk+n1/ϖp.Proof.
Let cardξ=r+1 and Vznf∈ℍbr,prρ, for all f∈ℍbr,prρ, for which fy=∑k=0∞fk^yk∈ℂ with y∈ℂ and ρf=∑k=0∞bkfk^pk1/ϖp<∞. Therefore, Vznfz=∑k=0∞fk^zk+n and ρVznf=∑k=0∞bk+nfk^pk+n1/ϖp<∞.
LetPξ be a mapping on ℍbr,prρ with rank Pξ=r+1 defined by(49)Pξgz=Pξ∑k=0∞fk^zk+n=∑k∈ξ∞fk^zk+n.
SinceρPξg=∑k∈ξbk+nfk^pk+n1/ϖp≤∑k=0∞bk+nfk^pk+n1/ϖp=ρg, this gives Pξ≤1. Define a mapping Szn by Sznhz=Szn∑k∈ξfk^zk+n=∑k=0∞fk^zk, and we have(50)ρSznh=∑k=0∞bkfk^pk1/ϖp≤Un∑k∈ξbk+nfk^pk+n1/ϖp=Unρg.
This implies thatSzn≤Un, where 1≤Un=∑k=0∞bkfk^pk1/ϖp/∑k∈ξbk+nfk^pk+n1/ϖp<∞. Then, the identity mapping will be Ir+1=PξVznSzn, and from the definition of s-numbers, we have(51)srIr+1=1≤PξsrVznSzn≤srVznSzn⇒srVzn≥1Szn≥1Un=∑k∈ξbk+nfk^pk+n1/ϖp∑k=0∞bkfk^pk1/ϖp≥infk∈ξbk+nbkpk+n/ϖp1An.
Since for all cardξ=r+1, the last inequality is verified, so one can see that(52)srVzn≥supcardξ=r+1infk∈ξbk+nbkpk+n/ϖp1An.
In contrary, let cardξ=r, where ξ⊂ℕ. Define the mapping Rzn as Rznvz=Rzn∑k=0∞fk^zk=∑k∈ξfk^zk+n∑k∈ξfk^zk+n. From the definition of approximation numbers, we have(53)srVzn≤αrVzn≤Vzn−Rzn≤supfz≠0Vzn−Rznfzfz=supfz≠0∑k∉ξfk^zk+nfz≤supfz≠0∑k∉ξbk+nfk^pk+n1/ϖpfz≤supk∉ξbk+nbkpk+n/ϖp
Since for all cardξ=r, the last inequality holds and by using Lemma 2, one has(54)supcardξ=r+1infk∈ξbk+nbkpk+n/ϖp1An≤srVzn≤infcardξ=rsupk∉ξbk+nbkpk+n/ϖp=supcardξ=r+1infk∈ξbk+nbkpk+n/ϖp.
This completes the proof.Theorem 27.
IfBz is the backward shift mapping on ℍbr,prρ, then(55)supcardξ=r+1infk∈ξbkbk+npk/ϖp1Gn≤srBzn≤supcardξ=r+1infk∈ξbkbk+npk/ϖp,where Gn=∑k=0∞bkfk^pk1/ϖp/∑k∈ξbk+nfk+n^pk1/ϖp.Proof.
Assume cardξ=r+1 and Bznf∈ℍbr,prρ, for every f∈ℍbr,prρ, where fy=∑k=0∞fk^yk∈ℂ with y∈ℂ and ρf=∑k=0∞bkfk^pk1/ϖp<∞. Therefore, Bznfz=∑k=0∞fk+n^zk and ρBznf=∑k=0∞bkfk+n^pk1/ϖp<∞.
SupposePξ is a mapping on ℍbr,prρ with rank Pξ=r+1 evident by(56)Pξgz=Pξ∑k=0∞fk+n^zk=∑k∈ξ∞fk+n^zk.
AsρPξg=∑k∈ξbkfk+n^pk1/ϖp≤∑k=0∞bkfk+n^pk1/ϖp=ρg. This implies that Pξ≤1. Define a mapping Szn by Sznhz=Szn∑k∈ξfk+n^zk=∑k=0∞fk^zk, and one gets(57)ρSznh=∑k=0∞bkfk^pk1/ϖp≤Un∑k∈ξbkfk+n^pk1/ϖp=Unρh.
Therefore,Szn≤Un, where 1≤Un=∑k=0∞bkfk^pk1/ϖp/∑k∈ξbkfk+n^pk1/ϖp<∞. Hence, the identity mapping will be Ir+1=PξBznSzn, and in view of the definition of s-numbers, one has(58)srIr+1=1≤PξsrBznSzn≤srBznSzn⇒srBzn≥1Szn≥1Un=∑k∈ξbkfk+n^pk1/ϖp∑k=0∞bkfk^pk1/ϖp≥infk∈ξbkbk+npk/ϖp1Gn.
Since for every cardξ=r+1, the last inequality is confirmed, and one obtains(59)srBzn≥supcardξ=r+1infk∈ξbkbk+npk/ϖp1Gn.
In contrary, let cardξ=r, where ξ⊂ℕ. Define the mapping Rzn as Rznvz=Rzn∑k=0∞fk^zk=∑k∈ξfk+n^zk. From the definition of approximation numbers, one gets(60)srBzn≤αrBzn≤Bzn−Rzn≤sup|fz|≠0Bzn−Rznfzfz=supfz≠0∑k∉ξfk+n^zkfz≤supfz≠0∑k∉ξbkfk+n^pk1/ϖpfz≤supk∉ξbkbk+npk/ϖp.
Since for all cardξ=r, the last inequality holds, and by using Lemma 2, one has(61)supcardξ=r+1infk∈ξbkbk+npk/ϖp1Gn≤srBzn≤infcardξ=rsupk∉ξbkbk+npk/ϖp=supcardξ=r+1infk∈ξbkbk+npk/ϖp.
This finishes the proof.Theorem 28.
If conditions (a1) and (a2) are satisfied withp0≥1, let ∑m=0∞cmVzm be a shift mapping on the space ℍbr,prρ and cmm=0∞∈ℓpm/ϖp; then,(62)supj∑m=0∞cmpm+jbm+jpm+jbjpj1/ϖp≤∑m=0∞cmVzm≤supm,jbm+jbjpm+j/ϖp∑m=0∞cmpm/ϖp.Proof.
Forf∈ℍbr,prρ, we have ∑m=0∞cmVzmfz=∑j=0∞∑j=0∞cmfj^zj+m. One has(63)∑m=0∞cmVzm≥ρ∑m=0∞cmVzmejρej=∑m=0∞cmbm+jpm+jbjpj1/ϖp≥supj∑m=0∞cmpm+jbm+jpm+jbjpj11/ϖp.
Sinceρ satisfies the triangle inequality, we have(64)∑m=0∞cmVzm=supρf≠0ρ∑m=0∞cmVzmfρf≤supρf≠0∑m=0∞∑j=0∞cmfj^bm+jpm+j1/ϖp∑j=0∞fj^bjpj1/ϖp≤supm,jbm+jbjpm+j/ϖp∑m=0∞∑j=0∞cmfj^bjpm+j1/ϖp∑j=0∞fj^bjpj1/ϖp≤supm,jbm+jbjpm+j/ϖp∑m=0∞cmpm/ϖp.Theorem 29.
If conditions (a1) and (a2) are satisfied withp0≥1, let ∑j=0∞cjBzj be a shift mapping on the space ℍbr,prρ and cjj=0∞∈ℓpj/ϖp; then,(65)supk∑j=0∞cjpkbkpkbk+jpk+j1/ϖp≤∑j=0∞cjBzj≤supj,kbkbk+jpk/ϖp∑j=0∞cjpj/ϖp.Proof.
Supposef∈ℍbr,prρ, and one has ∑j=0∞cjBzjfz=∑k=0∞∑j=0∞cjfk+j^zk. We have(66)∑j=0∞cjBzj≥ρ∑j=0∞cjBzjekρek=∑j=0∞bk−jcjpk−jbkpk1/ϖp≥supk∑j=0∞cjpkbkpkbk+jpk+j1/ϖp.
Asρ verifies the triangle inequality, one can see that(67)∑j=0∞cjBzj=supρf≠0ρ∑j=0∞cjBzjfρf≤supρf≠0∑j=0∞∑k=0∞bkcjfk+j^pk1/ϖp∑k=0∞bkfk^pk1/ϖp≤supj,kbkbk+jpk/ϖp∑j=0∞∑k=0∞bk+jcjfk+j^pk1/ϖp∑k=0∞bkfk^pk1/ϖp≤supj,kbkbk+jpk/ϖp∑j=0∞cjpj/ϖp.Theorem 30.
If conditions (a1) and (a2) are satisfied withp0≥1, let ∑r=0∞crVzr be a shift mapping on ℍbr,prρ; then, the s-numbers of this mapping are given by(68)sr∑j=0∞cjVzj≤supcardξ=r+1infk∈ξsupjbj+kbkpj+k/ϖp∑j=0∞cjpj/ϖp,for allcjj=0∞∈ℓpj/ϖp.Proof.
Let cardξ=r, where ξ⊂ℕ. Define the mapping R as Rfz=R∑k=0∞fk^zk=∑k∈ξ∑j=0kcjfk−j^zk. Since the triangle inequality holds by ρ, we have(69)sr∑j=0∞cjVzj≤αr∑j=0∞cjVzj≤∑j=0∞cjVzj−R≤supρf≠0ρ∑j=0∞cjVzjf−Rfρf≤supρf≠0∑j=0∞∑k∉ξcjfk^bk+jpk+jρf≤supk∉ξ,jbj+kbkpj+k/ϖp∑j=0∞cjpj/ϖp.
As for all cardξ=r, the last inequality is verified, and one has(70)sr∑j=0∞cjVzj≤infcardξ=rsupk∉ξ,jbj+kbkpj+k/ϖp∑j=0∞cjpj/ϖp=supcardξ=r+1infk∈ξsupjbj+kbkpj+k/ϖp∑j=0∞cjpj/ϖp.
This completes the proof.Theorem 31.
If conditions (a1) and (a2) are satisfied withp0≥1, let ∑j=0∞cjBzj be a shift mapping on ℍbr,prρ; then, the s-numbers of this mapping are given by(71)sr∑j=0∞cjBzj≤supcardξ=r+1infk∈ξsupjbkbk+jpk/ϖp∑j=0∞cjpj/ϖp,for allcjj=0∞∈ℓpj/ϖp.Proof.
Let cardξ=r, where ξ⊂ℕ. Define the mapping R as Rfz=R∑k=0∞fk^zk=∑k∈ξ∑j=0kcjfk−j^zk. Since the triangle inequality holds by ρ, one gets(72)sr∑j=0∞cjBzj≤αr∑j=0∞cjBzj≤∑j=0∞cjBzj−R≤supρf≠0ρ∑j=0∞cjBzjf−Rfρf≤supρf≠0∑j=0∞∑k∉ξbkcjfk+j^pk1/ϖpρf≤supk∉ξ,jbkbk+jpk/ϖp∑j=0∞cjpj/ϖp.
As for all cardξ=r, the last inequality is verified, and one has(73)sr∑j=0∞cjBzj≤infcardξ=rsupk∉ξ,jbkbk+jpk/ϖp∑j=0∞cjpj/ϖp=supcardξ=r+1infk∈ξsupjbkbk+jpk/ϖp∑j=0∞cjpj/ϖp.
This completes the proof.The following theorems are direct actions of Theorem30 and Definition 10.Theorem 32.
If conditions (a1) and (a2) are satisfied withp0≥1, let Vez be a shift mapping on ℍbr,prρ and ez=∑r=0∞zr/r!. The upper estimation of the s-numbers of Vez is given by(74)saVez≤supcardξ=a+1infj∈ξsuprbr+jbjpr+j/ϖp∑r=0∞1r!pr/ϖp.Theorem 33.
If conditions (a1) and (a2) are satisfied withp0≥1, let Vsinz be a shift mapping on ℍbr,prρ and sinz=∑m=0∞−1mz2m+1/2m+1!. The upper estimation of the s-numbers of Vsinz is given by(75)saVsinz≤supcardξ=a+1infj∈ξsuprbr+jbjpr+j/ϖp∑r=0∞12r+1!pr/ϖp.The following theorems are direct actions of Theorem31 and Definition 11.Theorem 34.
If conditions (a1) and (a2) are satisfied withp0≥1, then the mapping Bez on ℍbr,prρ holds the following inequality:(76)srBez≤supcardξ=r+1infk∈ξsupmbkbk+mpk/ϖp∑m=0∞1m!pm/ϖp.Theorem 35.
If conditions (a1) and (a2) are satisfied withp0≥1 and the mapping Bsinz is defined on ℍbr,prρ, then the upper estimation of the s-numbers of Bsinz is given by(77)srBsinz≤supcardξ=r+1infk∈ξsupmbkbk+mpk/ϖp∑m=0∞12m+1!pm/ϖp.


## 6. Caristi’s Generalization of Fixed Point Theorem

In modular spaces, the Ekeland variational principle [28] cannot be applied because the modular does not really prove the triangle inequality. In this part, we consider an extension of Caristi’s fixed point theorem in ℍbr,prρ in light of Farkas [28].Definition 12.
(a)
The pre-quasi normed ssfpsρ on ℍbr,prρ is called ρ-convex, if ρωv+1−ωt≤ωρv+1−ωρt, for each ω∈0,1 and v,t∈ℍbn,pnρ.(b)
vaa∈ℕ⊆ℍbn,pnρ is ρ-convergent to v∈ℍbn,pnρ, if and only if, lima⟶∞ρva−v=0. If the ρ-limit exists, then it is unique.(c)
vaa∈ℕ⊆ℍbn,pnρ is ρ-Cauchy, when lima,b⟶∞ρva−vb=0.(d)
Υ⊂ℍbn,pnρ is ρ-closed, if for all ρ-converging uaa∈ℕ⊂Υ to u, and hence u∈Υ.(e)
Υ⊂ℍbn,pnρ is ρ-bounded, when δρΥ=supρv−t:v,t∈Υ<∞.(f)
Theρ-ball of radius d≥0 and center v, for every v∈ℍbn,pnρ, is defined as(78)ℬρv,d=t∈ℍbn,pnρ:ρv−t≤d.(g)
A pre-quasi normed ssfpsρ on ℍbn,pnρ satisfies the Fatou property, if for any sequence tu⊆ℍbn,pnρ with limu⟶∞ρtu−t=0 and any v∈ℍbn,pnρ,(79)ρv−t≤supminfu≥mρv−tu.Consider the fact that theρ-closedness of the ρ-balls is determined by the Fatou property.Theorem 36.
Suppose setups (a1) and (a2) are satisfied; then,ρf=∑r=0∞brfr^pr1/ϖp, for all f∈ℍbn,pnρ, holds the Fatou property.Proof.
Assume the setups are fulfilled andfi⊆ℍbn,pnρ with limi⟶∞ρfi−f=0. Since the space ℍbn,pnρ is a pre-quasi closed space, then f∈ℍbn,pnρ. Then, for any g∈ℍbn,pnρ, one can see that(80)ρg−f=∑a=0∞baga^−fa^pa1/ϖp≤∑a=0∞baga^−fai^pa1/ϖp+∑a=0∞bafai^−fa^pa1/ϖp≤supjinfi≥jρg−fi.Theorem 37.
The functionρf=∑r=0∞brfr^pr, for all f∈ℍbn,pnρ, does not satisfy the Fatou property, if setups (a1) and (a2) are satisfied with p0>1.Proof.
Let the conditions be fulfilled andfi⊆ℍbn,pnρ with limi⟶∞ρfi−f=0. Since the space ℍbn,pnρ is a pre-quasi closed space, then f∈ℍbn,pnρ. Then, for any g∈ℍbn,pnρ, we have(81)ρg−f=∑a=0∞baga^−fa^pa≤2supapa−1∑a=0∞baga^−fai^pa+∑a=0∞bafai^−fa^pa≤2supapa−1supjinfi≥jρg−fi.
Hence,ρ does not satisfy the Fatou property.Example 2.
The space of functionsℍar,qrρ is a pre-quasi normed ssfps, not quasi normed ssfps, and not a normed ssfps, where δh=∑r=0∞arhr^qr1/ϖq, for all h∈ℍar,qrρ.Example 3.
The space of functionsℍar,q, with 0<q<1, is a pre-quasi normed ssfps, quasi normed ssfps, and not a normed ssfps, where δh=∑r=0∞arhr^q1/q, for each h∈ℍar,qrδ.Example 4.
The space of functionsℍar,qr is a pre-quasi normed ssfps, a quasi normed ssfps, and a normed ssfps, where δh=infι>0:∑r=0∞arhr^/ιqr≤1, for all h∈ℍar,qrδ.Definition 13.
The functionJ:ℍbr,prδ⟶−∞,∞ is said to be lower semicontinuous at h0∈ℍbr,prδ if supV∈Vh0infh∈VJh=Jh0, for which Vh0 denotes h0’s neighborhood system.Definition 14.
The functionJ:ℍbr,prδ⟶−∞,∞ is said to be proper, when(82)DJ=f∈ℍbr,prδ:Jf<∞≠∅.Theorem 38.
IfΞ≠∅ and Ξ is a δ-closed subset of ℍbx,pxδ, with δh=∑x=0∞bxhx^px1/ϖp, for all h∈ℍbx,pxδ, and J:Ξ⟶−∞,∞ is a proper, δ-lower semicontinuous function with infh∈ΞJh>−∞, assume that λ>0, ηx⊂0,∞, and h0∈Ξ with Jh0≤infh∈ΞJh+λ. So, we have hx∈Ξ which δ-converges to few hλ, under the following conditions:(i)
δhλ−hx≤λ/2xη0, for every x∈ℕ.(ii)
Jhλ+∑x=0∞ηxδhλ−hx≤Jh0.(iii)
Whenh≠hλ, then Jhλ+∑x=0∞ηxδhλ−hx<Jh+∑x=0∞ηxδh−hx.Proof.
SetSh0=h∈Ξ:Jh+η0δh−h0≤Jh0. Since h0∈Sh0, then Sh0≠∅. As J is δ-lower semicontinuous, δ satisfies the Fatou property, and Ξ is δ-closed, we have that Sh0 is δ-closed. Select h1∈Sh0 with(83)Jh1+η0δh1−h0≤infh∈Sh0Jh+η0δh−h0+λη12η0.
Next set(84)Sh1=h∈Sh0:Jh+∑i=01ηiδh−hi≤Jh1+η0δh1−h0.
Similar toSh0, one has Sh1≠∅ and δ-closed. Suppose that we have built h0,h1,h2,…,hx and Sh0,Sh1,Sh2,…,Shx. After that, select hx+1∈Shx with(85)Jhx+1+∑i=0xηiδhx+1−hi≤infh∈ShxJh+∑i=0xηiδh−hi+ληx2xη0.
Suppose(86)Shx+1:=h∈Shx:Jh+∑i=0x+1ηiδh−hi≤Jhx+1+∑i=0xηiδhx+1−hi.
Therefore, we construct the sequenceshx and Shx by induction. For constant x∈ℕ, assume y∈Shx. One can see that(87)Jy+∑i=0xηiδy−hi≤Jhx+∑i=0x−1ηiδhx−hi,which gives(88)ηxδy−hx≤Jhx+∑i=0x−1ηiδhx−hi−Jy+∑i=0x−1ηδy−hii≤Jhx+∑i=0x−1ηiδhx−hi−infh∈Shx−1Jh+∑i=0x−1ηiδh−hi≤ληx2xη0.
SinceShx is decreasing with hx∈Shx, for each x∈ℕ, one has(89)δhx+q−hx≤λ2xη0,for each x,q∈ℕ, which gives that hx is δ− Cauchy. Since ℍbx,pxδ is δ− Banach space, hx has δ− limits hλ and ∩x∈ℕShx=hλ satisfies. As hx+1∈Shx, one has(90)Jhx+1+∑i=0xηiδhx+1−hi≤Jhx+∑i=0x−1ηiδhx−hi,which implies that Jhx+∑i=0x−1ηiδhx−hi is decreasing. After that, assume h≠hλ. So, we get r∈ℕ for which h∉Shx, for each x≥r, i.e.,(91)Jhx+∑i=0x−1ηiδhx−hi<Jh+∑i=0xηiδh−hi.
Ashλ∈Shx, with x≥r, one can see that(92)Jhλ+∑i=0xηiδhλ−hi≤Jhx+∑i=0x−1ηiδhx−hi≤Jhr+∑i=0r−1ηiδhr−hi.
Asx⟶∞ in the previous inequality, one gets(93)Jhλ+∑i=0∞ηiδhλ−hi≤Jhr+∑i=0r−1ηiδhr−hi<Jh+∑i=0rηiδh−hi≤Jh+∑i=0∞ηiδh−hi.
This implies that(94)Jhλ+∑x=0∞ηxδhλ−hx<Jh+∑x=0∞ηxδh−hx.
This finishes the proof.We discuss the concept of Caristi’s fixed point theorem inℍbx,pxδ using Theorem 38.Theorem 39.
IfΞ≠∅ and Ξ is a δ-closed subset of ℍbx,pxδ, under δh=∑x=0∞bxhx^px1/ϖp, with h∈ℍbx,pxδ, let λ>0 and ηn with 0<ν=∑x=0∞ηx<∞. U:Ξ⟶Ξ is a mapping and there is a function J:Ξ⟶−∞,∞ which is a proper and δ-lower semicontinuous under infh∈ΞJh>−∞ and(1)
δUh−g−δh−g≤δUh−h, for any h,g∈Ξ.(2)
δUh−h≤Jh−JUh, for any h∈Ξ.
Hence, there is a fixed point ofU in Ξ.Proof.
As0<ν=∑x=0∞ηx<∞, we have that J1:=νJ is proper, bounded from below, and δ-lower semicontinuous. If h∈Ξ, one has(95)νδUh−h≤J1h−J1Uh.
Asinfh∈ΞJ1h>−∞, there is h0∈Ξ with J1h0<infh∈ΞJ1h+λ. From Theorem 38, there is hx which δ-converges to few hλ∈Ξ, with(96)J1hλ+∑x=0∞ηxδhλ−hx<J1h+∑x=0∞ηxδh−hx,for all h≠hλ. Suppose that Uhλ≠hλ, and one has(97)J1hλ+∑x=0∞ηxδhλ−hx<J1Uhλ+∑x=0∞ηxδUhλ−hx,which gives(98)J1hλ−J1Uhλ<∑x=0∞ηxδUhλ−hx−∑x=0∞ηxδhλ−hx=∑x=0∞ηxδUhλ−hx−δhλ−hx.
From condition (6), one can see that(99)J1hλ−J1Uhλ<∑x=0∞ηxδUhλ−hλ=νδUhλ−hλ.
Inequality (6) gives(100)νδUhλ−hλ≤J1hλ−J1Uhλ<νδUhλ−hλ.
We have a contradiction. Hence,Uhλ=hλ. This completes the proof.



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