# Cesaŕo Summable Relative Uniform Difference Sequence of Positive Linear Functions

Authors: Awad A. Bakery; O. M. Kalthum S. K. Mohamed; Kshetrimayum Renubebeta Devi; Binod Chandra Tripathy
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this article, we introduce the class of sequence of functionsC1Δ,ru of Cesaŕo summable relative uniform difference sequence of functions. We have studied the topological properties of C1Δ,ru. We also obtain the necessary and sufficient condition to characterize the matrix classes C1Δ,ru,ℓ∞ru,C1Δ,ru,cru,P.


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## Body

## 1. Introduction

Throughout the study,ωru,C1Δ,ru,andℓ∞ru denote the classes of all relative uniform sequence space, Cesaŕo summable relative uniform difference sequence space, and bounded relative uniform sequence space, respectively.Moore in 1910 introduced the notion of uniform convergence of a sequence of functions relative to a scale function. Chittenden [1–3] gave the detailed definition of the notion as follows.Definition 1 (see [1]).
A sequencefix of single-valued, real-valued functions fix of a variable x ranging over a compact subset D of real numbers is said to be relatively uniformly convergent on D w. r. t. a scale function σx in case there exist a limiting function fx and scale function σx defined on D and for every ε, an integer n0=n0ε such that for every n≥n0 and for all x∈D,
(1)fix−fx<εσx.The notion was further discussed from various aspects by Demirci et al. [4], Demirci and Orhan [5], Devi and Tripathy [6], and many others.Example 1.
Let0<a<1 be a real number. Consider the sequence of functions fix,fi:a,1⟶R, for all i∈N defined by
(2)fix=1ix,forallx∈a,1,i∈N.This sequence of functions does not converge uniformly to 0 ona,1. However, fix converges to 0 uniformly with respect to the scale function σx defined by
(3)σx=1x,forallx∈a,1.Kizmaz [7] defined the difference sequence spaces ℓ∞Δ,cΔ,andc0Δ as follows:
(4)ZΔ=x=xi:Δxi∈Z,for Z=ℓ∞,c,c0 where Δxi=xi−xi+1,i∈N.These sequence spaces are Banach space under the norm(5)xiΔ=x1+supi∈NΔxi.The notion was further studied from different aspects by many others [8–13].The Cesaŕo sequence spaceCes∞,Cesp1<p<∞ were introduced by Shiue [14], and it has been shown that ℓ∞⊂Cesp; the inclusion is strict for 1<p<∞. Further, the Cesaŕo sequence spaces Xp and X∞ of nonabsolute type were defined by Ng and Lee [15, 16]. For a detail account of Cesaŕo difference sequence space, one may refer to [17–20].LetA=ani be an infinite matrix of real or complex numbers. Then, A transforms from the sequence space λ into the sequence space F, if Ax∈F for each sequence xi∈λ that is Ax=Anx∈F, where Anx=∑i=1∞anixi, provided that the infinite series converges for each n∈N.Matrix transformation between sequence space was studied from different aspects by many others [21–25].


## 2. Definitions and Preliminaries

Definition 2.
A sequence spaceλ is said to be solid or normal if xi∈λ implies αixi∈λ, for all αi with αi≤1, for all i∈N.Definition 3.
A sequence spaceλ is said to be monotone if it contains the canonical preimages of all its step spaces.Remark 4.
A sequence spaceλ is solid then, λ is monotone.Definition 5.
A sequence spaceλ is said to be symmetric if xi∈λ⇒xπi∈λ, for all i∈N, where π is a permutation of N, the set of natural numbers.Definition 6.
A sequence spaceλ is said to be convergence free if xi∈λ and xi=0⇒yi=0 together with yi∈λ, for all i∈N.Definition 7.
A sequence spaceλ is said to be a sequence algebra if xi.yi∈λ whenever xi and yi belongs to λ, for all i∈N.
In this article we introduce the sequence spaceC1Δ,ru of Cesaŕo summable relative uniform difference sequence of functions and it is defined as follows:
(6)C1Δ,ru=f=fix∈ωru:Δfix∈C1w.r.t.thescalefunctionσx,where Δfix∈C1 (ru) and Δfix=fix−fi+1x.


## 3. Main Results

We state the following result without proof.Theorem 8.
The sequence spaceC1Δ,ru is a normed linear space.Theorem 9.
The sequence spaceC1Δ,ru is a Banach space normed by
(7)fΔ,σ=supx≤1f1xσxx+supp≥1supx≤11p∑i=1pΔfixσxx.Proof.
Letfnx be a Cauchy sequence in C1Δ,ru where
(8)fnx=finx=f1nx,f2nx,⋯∈C1Δ,ru,foreachi∈N.Then,(9)fnx−fmxΔ,σ=supx≤1f1nx−f1mxσxx+supp≥1supx≤11p∑i=1pΔfinx−Δfimxσxx⟶0.For alln,m≥n0,
(10)fnx−fmxΔ,σ=supx≤1f1nx−f1mxσxx+supp≥1supx≤11p∑i=1pΔfinx−Δfimxσxx<ε2.f1nx is a Cauchy sequence in D w.r.t. σx for all x∈D.⇒f1nx is convergent in D w.r.t. σx for all x∈D.Letlimn⟶∞f1nx=f1x,x∈D.Similarly,limn⟶∞1/p∑i=1pΔfinx=1/p∑i=1pΔfix,x∈D.From the above equations we get,(11)limn⟶∞finx=fix,for all x∈D, for all i∈N.From (10) we have
(12)limm⟶∞f1nx−f1mxσxx=f1nx−f1xσxx<ε2,for all x∈D.Similarly,(13)limm⟶∞1p∑i=1pΔfinx−Δfimxσxx=1p∑i=1pΔfinx−Δfixσxx<ε2,for all x∈D and i∈N.Sinceε/2 is not dependent on i, we have
(14)supx≤1f1nx−f1xσxx<ε2,supp≥1supx≤11p∑i=1pΔfinx−Δfixσxx<ε2.Evidently,(15)fnx−fxΔ,σ=supx≤1f1nx−f1xσxx+supp≥1supx≤11p∑i=1pΔfinx−Δfixσxx≤ε.⇒fnx−fxΔ,σ≤ε, for all n≥n0, for all x∈D.Therefore,fnx−fx∈C1Δ,ru, for all n≥n0, for all x∈D.Then,fx=fnx−fnx−fx∈C1Δ,ru since C1Δ,ru is a linear space.Theorem 10.
The inclusionC1ru⊂C1Δ,ru strictly holds.Proof.
The proof of the theorem is obvious and the strictness of the inclusion is shown in the following example.Example 1.
Let0<a<1 be a real number and D=a,1. Consider the sequence of real valued functions fix,fi:a,1⟶R, for all i∈N, defined by
(16)fix=ix,forallx∈a,1,Δfix=fix−fi+1x=x,forallx∈a,1.
fix∈C1Δ,ru w.r.t. the scale function σx=1, for all xin D, but fix∉ℓ∞ru.
Hence, the inclusion is strict.Theorem 11.
The inclusioncΔ,ru⊂C1Δ,ru strictly holds.Proof.
The proof is obvious and the strictness of the inclusion is shown in the following example.Example 2.
Let0<a<1 be a real number and D=a,1. Consider the sequence of real valued functions fix,fi:a,1⟶R, for all i∈N, defined by
(17)fix=x,foriisodd,0,otherwise,Δfix=x,foriisodd,−x,otherwise.
We havefix∈C1Δ,ru w.r.t. the scale function σx defined by
(18)σx=1x,forallx∈a,1,but fix∉cΔ,ru. Hence, the inclusion is strict.Theorem 12.
The sequence spaceC1Δ,ru is not monotone.Proof.
The proof is shown in the following example.Example 3.
Let0<a<1 be a real number and D=a,1. Consider the sequence of real valued functions fix,fi:a,1⟶R, for all i∈N, defined by
(19)fix=ix,forallx∈a,1,Δfix=fix−fi+1x=x,forallx∈a,1.
fix∈C1Δ,ru w.r.t. the scale function defined on D by σx=1.
Letgix be the preimage of fix defined by
(20)gix=i2x,fori=k2,k∈N,0,otherwise.One cannot get a scale function that makesgix∈C1Δ,ru.HenceC1Δ,ru is not monotone.Remark 13.
The sequence spaceC1Δ,ru is not solid since C1Δ,ru is not monotone.Theorem 14.
The sequence spaceC1Δ,ru is not symmetric.Proof.
The proof of the theorem is shown with the help of the following example.Example 4.
Let us consider the sequence of functionsfix considered in Example 3. Let gix be the rearrangement sequence of functions of fix defined by
(21)gix=i+1x,fori=2j−1,j∈N,i−1x,fori=2j,j∈N.
One cannot get a scale function that makesgix∈C1Δ,ru.Hence,C1Δ,ru is not symmetric.Theorem 15.
The sequence spaceC1Δ,ru is not sequence algebra.Proof.
The proof of the theorem is shown in the following example.Example 5.
Let0<a<1 be a real number and D=a,1. Consider the sequences of real valued functions fix,fi:a,1⟶R, and gix,gi:a,1⟶R, for all i∈N, defined by
(22)fix=gix=ix,foralli∈N.fix.gix=i2x2.We get thatfix,gix∈C1Δ,ru but one cannot get a scale function that makes fix.gix∈C1Δ,ru.Hence,C1Δ,ru is not sequence algebra.Theorem 16.
The sequence spaceC1Δ,ru is not convergence free.Proof.
The proof of the theorem follows from example below.Example 6.
Let0<a<1 be a real number and D=a,1. Consider the sequence of real valued functions fix,fi:a,1⟶R, for all i∈N, defined by
(23)fix=x,forallx∈a,1,Δfix=0.Therefore,fix∈C1Δ,ru w.r.t. the constant scale function defined on D by σx=1.Let us consider another sequence of functionsgi,gi:a,1⟶R defined by
(24)gix=g1x=x;gi+1x=gm−1x+n+1x,form≥2,m,n,i∈N,C1Δ,ru=−nx,n≥2,n∈N.One cannot find a scale function that makesgix∈C1Δ,ru.Hence, the sequence spaceC1Δ,ru is not convergence free.


### 3.1. Matrix Transformation between Sequence of Functions

In this section, we give certain matrix classes between the sequence of functions.Theorem 17.
A∈C1Δ,ru,ℓ∞ru if and only if supn≥1∑i=2∞i−1ani<∞.Proof.
Letfix∈C1Δ,ru and supn≥1∑i=2∞i−1ani<∞.
(25)∑i=1∞anifixσx=−∑i=2∞i−1ani1i−1∑t=1i−1Δftxσx+f1xσx∑i=1∞ani=∑i=2∞ani∑t=1i−1Δftxσx.From the relation between dual and matrix map, we know thatcα=ℓ1. Hence,
(26)∑i=2∞ani∑t=1i−1Δftxσxconvergesabsolutelyw.r.t.σx.Forp∈N, we have, ∑i−1panifixσx=−∑i=1pani∑t=1i−1Δftxσx+f1xσx∑i=1pani.By the argument (10), we know that ∑i=1panifixσx is absolutely convergent w.r.t. scale function σx.Then,(27)∑i=1∞anifixσx≤supn∑i=2∞i−1anisupx≤1supi≥21i−1∑t=1i−1Δftxσx+f1xσxsupn∑i=2∞i−1ani.We have,∑i=1∞anifixσx<∞ since supn≥1∑i=2∞i−1ani<∞.Conversely, we know thatA is a bounded linear function from C1Δ,ru to ℓ∞ru, so we can write,
(28)∑i=1∞anifixσx=Anfσx≤supnAnfσx=Afσx∞≤AfΔ,σ.We choose a sequence of functionsfi,fi:0,1⟶R defined by
(29)fix=i−1xsgnani,for1<i≤r,0,otherwise.We getfix∈C1Δ,ru w.r.t. scale function σx=1/x with the norm fΔ,σ=1.Putting the value offix in equation (26) and letting the limit r⟶∞, we get
(30)∑i=2∞i−1ani≤A.Theorem 18.
A∈C1Δ,ru,cru,P if and only if
(i)
supn∑i=2∞i−1ani<∞(ii)
limn∑i=2∞i−1ani=−1(iii)
limnani=0 for each i(iv)
limn∑iani=0.Proof.
Let us assume that the conditions (i)-(iv) hold true and letfix∈C1Δ,ru, i.e., limi1/i∑t=1iΔftxσx=fx (say).
We know from (i) that for eachx∈D and n∈N, ∑ii−1ani converges. It follows that ∑i=2∞i−1ani1/i−1i−1∑i=2∞Δftxσx converges.
(31)∑ianifixσx=−∑i=2∞i−1ani1i−1∑t=1i−1Δftxσx−fx−fx∑ii−1ani+f1xσx∑iani.
For anyn0∈N, we have
(32)∑i=2∞i−1ani1i−1∑t=1i−1Δftxσx−fx≤supx≤1supi1i−1∑t=1i−1Δftxσx−fx∑i=2n0i−1ani+supn∑i=2∞i−1anisupx≤1supi>n01i−1∑t=1i−1Δftxσx−fxlimnsup∑i=2∞i−1ani1i−1∑t=1i−1Δftxσx−fx≤supn∑i=2∞i−1anisupx≤1supi>n01i−1∑t=1i−1Δftxσx−fx.
Letn0=0, we get ∑i=2∞i−1ani1/i−1i−1∑t=1i−1Δftxσx−fx⟶0.
Substituting this value in equation (31) and using conditions (ii) and (iv), we get
∑ianifixσx converges to fx w.r.t. the scale function σx. Conversely, let A∈C1Δ,ru,cru,P. Then, ∑ianifixσx∈cru, for all
(33)fix∈C1Δ,ru.
Condition (i) can be proceed as same as shown in Theorem17.
(ii) Letfi,fi:0,1⟶R defined by
(34)fix=ix,foralli∈N.
Then,fix∈C1Δ,ruw.r.t.σx=1/x.
SinceΔfix⟶−1, we have limn∑ii−1ani=−1.
(iii) Letfi,fi:0,1⟶R defined by
(35)fix=eix,foralli∈N.
Then,fix∈C1Δ,ruw.r.t.σx=1/x.
SinceΔfix⟶0, we have limnani=0.
(iv) Letfi,fi:0,1⟶R defined by
(36)fix=x,foralli∈N.
Then,fix∈C1Δ,ruw.r.t.σx=1/x.
SinceΔfix⟶0, we have limn∑ii−1ani=0.
The following theorem is stated without proof and can be proceeded the same as in Theorem18.Theorem 19.
A∈C1Δ,ru,c0ru if and only if
(i)
supn∑i=2∞i−1ani<∞(ii)
limn∑i=2∞i−1ani=0(iii)
limnani=0 for each i(iv)
limn∑iani=0.


## 3.1. Matrix Transformation between Sequence of Functions

In this section, we give certain matrix classes between the sequence of functions.Theorem 17.
A∈C1Δ,ru,ℓ∞ru if and only if supn≥1∑i=2∞i−1ani<∞.Proof.
Letfix∈C1Δ,ru and supn≥1∑i=2∞i−1ani<∞.
(25)∑i=1∞anifixσx=−∑i=2∞i−1ani1i−1∑t=1i−1Δftxσx+f1xσx∑i=1∞ani=∑i=2∞ani∑t=1i−1Δftxσx.From the relation between dual and matrix map, we know thatcα=ℓ1. Hence,
(26)∑i=2∞ani∑t=1i−1Δftxσxconvergesabsolutelyw.r.t.σx.Forp∈N, we have, ∑i−1panifixσx=−∑i=1pani∑t=1i−1Δftxσx+f1xσx∑i=1pani.By the argument (10), we know that ∑i=1panifixσx is absolutely convergent w.r.t. scale function σx.Then,(27)∑i=1∞anifixσx≤supn∑i=2∞i−1anisupx≤1supi≥21i−1∑t=1i−1Δftxσx+f1xσxsupn∑i=2∞i−1ani.We have,∑i=1∞anifixσx<∞ since supn≥1∑i=2∞i−1ani<∞.Conversely, we know thatA is a bounded linear function from C1Δ,ru to ℓ∞ru, so we can write,
(28)∑i=1∞anifixσx=Anfσx≤supnAnfσx=Afσx∞≤AfΔ,σ.We choose a sequence of functionsfi,fi:0,1⟶R defined by
(29)fix=i−1xsgnani,for1<i≤r,0,otherwise.We getfix∈C1Δ,ru w.r.t. scale function σx=1/x with the norm fΔ,σ=1.Putting the value offix in equation (26) and letting the limit r⟶∞, we get
(30)∑i=2∞i−1ani≤A.Theorem 18.
A∈C1Δ,ru,cru,P if and only if
(i)
supn∑i=2∞i−1ani<∞(ii)
limn∑i=2∞i−1ani=−1(iii)
limnani=0 for each i(iv)
limn∑iani=0.Proof.
Let us assume that the conditions (i)-(iv) hold true and letfix∈C1Δ,ru, i.e., limi1/i∑t=1iΔftxσx=fx (say).
We know from (i) that for eachx∈D and n∈N, ∑ii−1ani converges. It follows that ∑i=2∞i−1ani1/i−1i−1∑i=2∞Δftxσx converges.
(31)∑ianifixσx=−∑i=2∞i−1ani1i−1∑t=1i−1Δftxσx−fx−fx∑ii−1ani+f1xσx∑iani.
For anyn0∈N, we have
(32)∑i=2∞i−1ani1i−1∑t=1i−1Δftxσx−fx≤supx≤1supi1i−1∑t=1i−1Δftxσx−fx∑i=2n0i−1ani+supn∑i=2∞i−1anisupx≤1supi>n01i−1∑t=1i−1Δftxσx−fxlimnsup∑i=2∞i−1ani1i−1∑t=1i−1Δftxσx−fx≤supn∑i=2∞i−1anisupx≤1supi>n01i−1∑t=1i−1Δftxσx−fx.
Letn0=0, we get ∑i=2∞i−1ani1/i−1i−1∑t=1i−1Δftxσx−fx⟶0.
Substituting this value in equation (31) and using conditions (ii) and (iv), we get
∑ianifixσx converges to fx w.r.t. the scale function σx. Conversely, let A∈C1Δ,ru,cru,P. Then, ∑ianifixσx∈cru, for all
(33)fix∈C1Δ,ru.
Condition (i) can be proceed as same as shown in Theorem17.
(ii) Letfi,fi:0,1⟶R defined by
(34)fix=ix,foralli∈N.
Then,fix∈C1Δ,ruw.r.t.σx=1/x.
SinceΔfix⟶−1, we have limn∑ii−1ani=−1.
(iii) Letfi,fi:0,1⟶R defined by
(35)fix=eix,foralli∈N.
Then,fix∈C1Δ,ruw.r.t.σx=1/x.
SinceΔfix⟶0, we have limnani=0.
(iv) Letfi,fi:0,1⟶R defined by
(36)fix=x,foralli∈N.
Then,fix∈C1Δ,ruw.r.t.σx=1/x.
SinceΔfix⟶0, we have limn∑ii−1ani=0.
The following theorem is stated without proof and can be proceeded the same as in Theorem18.Theorem 19.
A∈C1Δ,ru,c0ru if and only if
(i)
supn∑i=2∞i−1ani<∞(ii)
limn∑i=2∞i−1ani=0(iii)
limnani=0 for each i(iv)
limn∑iani=0.


## 4. Conclusions

In this article, we studied the concept of Cesaŕo summability from the aspects of relative uniform convergence of difference sequence of positive linear functions w.r.t. a scale functionσx on a compact domain D. The class of difference sequence of functions C1Δ,ru is introduced, and its properties like solid, monotone, symmetric, sequence algebra, and convergence free are discussed. We have also further introduced characterization of matrix classes of C1Δ,ru,ℓ∞ru,C1Δ,ru,cru,P and C1Δ,ru,c0ru.



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