# Solution and Stability of Quartic Functional Equations in Modular Spaces by Using Fatou Property

Authors: N. Uthirasamy; K. Tamilvanan; Hemant Kumar Nashine; Reny George
Journal: Journal of Function Spaces
 (2022)

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## Abstract

We propose a novel generalized quartic functional equation and investigate its Hyers–Ulam stability in modular spaces using a fixed point technique and the Fatou property in this paper.


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## Body

## 1. Introduction

The idea of functional equation stability occurs when a functional equation is replaced by an inequality that acts as a perturbation on the equation. Take into account that the topic of functional equation stability was prompted by a query posed by Ulam in 1940 [1], and Hyers response was published in [2]. By considering an unbounded Cauchy difference, Aoki [3] and Rassias [4] extended Hyers’ theorem for additive mappings and linear mappings, respectively.In [5–8], the authors examined the Hyers–Ulam-Rassias stability findings for functional equations involving many variables. They discussed the approximate solution of the septic functional equation in [9]. They established that this equation is stable in quasi-β-Banach spaces and in β,p-Banach spaces. Additionally, they established the instability of the preceding radical functional equation in a pertinent example. Khamsi investigated the notion of quasicontraction mappings in modular function spaces in the absence of the Δ2-condition, establishing the presence of fixed points and thoroughly analysing their uniqueness in [10].Kim and Tamilvanan [11] introduced a novel class of quartic functional equations and examined their Hyers–Ulam stability in fuzzy normed spaces using both the direct and fixed point methods. They demonstrated how sums and products of powers of norms may be used to manage the stability of this quartic functional equation. Wongkum et al. [12] studied the extended Ulam–Hyers–Rassias stability of quadratic functional equations extensively using fixed point theory. Their conclusions are achieved in the context of modular spaces whose modulars are lower semicontinuous but do not meet any Δ2-related requirements.Nakano established the theory of modulars on linear spaces and the accompanying theory of modular linear spaces, which was further refined by Koshi and Shimogaki [13] and Yamamuro [14] and others. Luxemburg [15], Musielak [16], and Turpin [17] and their collaborators contributed the most comprehensive elaboration of these ideas. At the moment, modulars and modular spaces theory is widely employed, most notably in the study of different Orlicz spaces [18] and interpolation theory [19, 20], both of which have several applications [16]. The significance for applications stems from the complex structure of modular function spaces, which, in addition to being Banach spaces (or F-spaces in a more generic setting), contain modular equivalents of norm or metric concepts.Notably, Nakano pioneered modular spaces with relation to order spaces in 1950 [21]. The spaces were produced using the Orlicz spaces theory, which substitutes a particular, integrated nonlinear functionality for the abstract functional that regulates the expansion of space members. They established several features of modular space in [22] and shown that every uniformly limited composition operator mapping this space onto itself must satisfy the so-called Matkowski’s requirements. In 2014, Sadeghi [23] explored the stability of functional equations in modular space using a fixed point technique.In [24], the existence and Hyers-Ulam stability of the nearly periodic solution to the fractional differential equation with impulse and fractional Brownian motion were studied under nonlocal circumstances. The research was conducted using the semigroups of operators and MÃ¶nch fixed point technique, as well as the basic theory of Hyers-Ulam stability. In [25], the authors examined at the existence and Hyers-Ulam stability of random impulsive stochastic functional differential equations with finite delays. They demonstrated that moderate solutions to the equations exist using Krasnoselskii’s fixed point approach. They then examined the Hyers-Ulam stability results under the Lipschitz condition on a bounded and closed interval. Finally, they demonstrated their findings with an example. The reader is recommended to see [26–28] and the references therein for further information on fixed point theory in modular spaces.Definition 1 (see [29]).
LetV be a vector space over Kℂorℝ. A generalized functional ρ:V⟶0,∞ is called a modular if for arbitrary u,v∈V, ρ satisfies
(a)
ρu=0 if and only if u=0(b)
ρβu=ρu for every scalar β with β=1(c)
ρβu+γv≤ρu+ρv, whenever β,γ≥0 and β+γ=1
If we replace (c) by.
(c’)ρβu+γv≤βρu+γρv, whenever β,γ≥0 and β+γ=1, then, the modular ρ is called convex. A modular ρ defines a corresponding modular space, i.e., the vector space Vρ given by:
(1)Vρ=u∈V∣ρcu⟶0asc⟶0.Definition 2 (see [29]).
IfVρ is a modular space and the sequence vn in Vρ, then
(i)
vn⟶ρv if ρvn−v⟶0 as n⟶∞(ii)
vn is known as ρ-Cauchy if ρvl−vn⟶0 as l,n⟶∞(iii)
A subsetA⊆Vρ is known as ρ-complete iff every ρ-Cauchy sequence is ρ-convergent in ADefinition 3 (see [29]).
LetVρ be a modular space and a non-empty subset A⊆Vρ. The mapping J:A⟶A is referred to as a quasicontraction, if there is k<1 satisfies
(2)ρJl−Jm≤kmaxρl−m,ρl−Jm,ρm−Jl,ρl−Jl,ρm−Jm,for any l,m∈A.Definition 4 (see [29]).
LetVρ be a modular space, a nonempty subset A⊆Vρ, and a function J:A⟶A, the J orbit around a point v is
(3)OJ≔u,Ju,J2u,⋯,the quantity
(4)YρJ≔supρp−q∣p,q∈OJ,is then related to J and is referred to as the orbital diameter of J at v. If YρJ<∞, in particular, one says that J has an orbit of v that is limited to v.Fatou property: theρ-modular will have the Fatou property iff ρv≤limm⟶∞infρvm whenever vm⟶ρv. A modular function is stated to fulfil the conditions Δ3 if there is k>0 which satisfies ρ3v≤kρv, for every v∈Vρ.In this work, we introduce new generalized quartic functional equation(5)∑1≤i≤nϕ−vi+∑j=1;i≠jnvj=n−8∑1≤i<j<k<l≤nϕvi+vj+vk+vl−n2−12n+28∑1≤i<j<k≤nϕvi+vj+vk+n3−15n2+60n−682∑1≤i<j≤nϕvi+vj+2∑1≤i<j≤nϕvi−vj+∑i=1nϕ3vi−n4−17n3+86n2−148n+5586∑i=1nϕvi,where n≥5, and investigate Hyers-Ulam stability of this quartic functional equation in modular space by using the fixed point method with the help of Fatou property.


## 2. Solution

We denoteV and W as two real vector spaces.Theorem 5.
If an even mappingϕ:V⟶W satisfies the functional equation (5) for all v1,v2,⋯,vn∈V, then, the function ϕ is quartic.Proof.
In the view of evenness, we obtainϕ−v=ϕv for all v in V. Now, setting v1=v2=⋯=vn=0 in equation (5), we have ϕ0=0. Replacing v1,v2,⋯,vn by v,0,⋯,0 in equation (5), we get
(6)ϕ3v=34ϕv,for all v∈V. Replacing v by 3v in equation (6), we have
(7)ϕ32v=342ϕv,for all v∈V. Again, replacing v by 3v in equation (7), we obtain
(8)ϕ33v=343ϕv,for all v∈V. For any nonnegative integer n≥0, we can generalize the result that
(9)ϕ3nv=34nϕv,for all v∈V. Similarly, we have
(10)ϕ3−nv=3−4nϕv,for all v∈V. Next, replacing v1,v2,⋯,vn by v1,v1,v2,0,⋯,0, we obtain
(11)ϕ2v1+v2+ϕ2v1−v2=4ϕv1−v2+4ϕv1+v2+24ϕv1−6ϕv2,for all v1,v2∈V. Hence, the function ϕ is quartic.


## 3. Hyers-Ulam Stability

We assume thatρ is a convex modular on ρ-complete modular spaces Wρ with the Fatou property that meets the Δ3-condition with 0<k≤3 in this section. Let V be a linear space as well. For convenience, we call a function ϕ:V⟶Wρ by the shorthand ϕ:V⟶Wρ:
(12)Dϕv1,v2,⋯,vn=∑1≤i≤nϕ−vi+∑j=1;i≠jnvj−n−8∑1≤i<j<k<l≤nϕvi+vj+vk+vl+n2−12n+28∑1≤i<j<k≤nϕvi+vj+vk−n3−15n2+60n−682,(13)∑1≤i<j≤nϕvi+vj−2∑1≤i<j≤nϕvi−vj−∑i=1nϕ3vi+n4−17n3+86n2−148n+5586∑i=1nϕvi,for all v1,v2,⋯,vn∈V.Theorem 6.
Letψ:Vn⟶0,+∞ be a function such that
(14)limm⟶∞134mψ3mv1,3mv2,⋯,3mvn=0,(15)ψ3v1,3v2,⋯,3vn≤34Lψv1,v2,⋯,vn,for all v1,v2,⋯,vn∈V, with L<1. If an even mapping ϕ:V⟶Wρ with ϕ0=0 and such that
(16)ρDϕv1,v2,⋯,vn≤ψv1,v2,⋯,vn,for all v1,v2,⋯,vn∈V, then, there exists a unique quartic mapping Q4:V⟶Wρ satisfying
(17)ρQ4v−ϕv≤1341−Lψv,0,⋯,0,for all v∈V.Proof.
We consider the set(18)Λ=p:V⟶Wρ,and define the function ρ¯ on Λ as follows:
(19)ρ¯p≕infθ>0:ρpv≤θψv,0,⋯,0,∀v∈V.
Now, we show thatρ¯ is a convex modular on Λ. It is easy to verify that ρ¯ satisfies the axioms (a) and (b) of a modular. Next, we will show that ρ¯ is convex, and hence, c′ is satisfied. Let ε>0 be given, then, there exist real constants θ1>0 and θ2>0 such that
(20)ρ¯p≤θ1≤ρ¯p+εandρ¯q≤θ2≤ρ¯q+ε.
Also(21)ρpv≤θ1ψv,0,⋯,0,ρqv≤θ2ψv,0,⋯,0,for all v∈V. If β+γ=1 and β,γ≥0, then, we get
(22)ρβpv+γqv≤βρpv+γρqv≤θ1β+θ2γψv,0,⋯,0,so we get
(23)ρ¯βp+γq≤βρ¯p+γρ¯q+β+γε.
This concludes thatρ¯ is convex modular on Λ. Now, we show that Λρ¯ is ρ¯-complete.
Letpn is a ρ¯-Cauchy sequence in Λρ¯ and let ε>0. Then, there exists a positive integer n0∈ℕ such that
(24)ρ¯pn−pm<ε,for all n,m≥n0. Then
(25)ρpnv−pmv≤εψv,0,⋯,0,for all v∈V and all n,m≥n0. Therefore, pnv is a ρ-Cauchy sequence in Wρ. Since Wρ is ρ-complete, so pnv is convergent in Wρ, for each v∈V. Hence, we can define a function p:V⟶Wρ by
(26)pv≔limn⟶∞pnv,for all v∈V. Since ρ satisfies the Fatou property, it follows from (25) that
(27)ρpnv−pv≤liminfm⟶∞ρpnv−pmv≤εψv,0,⋯,0,so
(28)ρ¯pn−p≤ε,for all n≥n0. Thus, pn is ρ¯-converges. Hence, Λρ¯ is ρ¯-complete.
Next, we show thatρ¯ satisfies the Fatou property. Suppose that pn is a sequence in Λρ¯ which is ρ¯-convergent to an element p∈Λρ¯.
Letε>0 be given. For each n∈ℕ, let θn be a real constant such that
(29)ρ¯pn≤θn≤ρ¯pn+ε.
So(30)ρpnv≤θnψv,0,⋯,0,for all v∈V. Since ρ satisfies the Fatou property, we get
(31)ρpv≤limn⟶∞infρpnv≤limn⟶∞infθnψv,0,⋯,0≤limn⟶∞infρ¯pn+εψv,0,⋯,0.
Thus, we obtain(32)ρ¯p≤limn⟶∞infρ¯pn+ε.
Hence,ρ¯ satisfies the Fatou property. Consider the function Ψ:Λρ¯⟶Λρ¯ by
(33)Ψpv=134p3v,for all v∈V and all p∈Λρ¯. Let p,q∈Λρ¯ and let θ∈0,1 be an arbitrary constant with ρ¯p−q<θ. From the definition of ρ¯, we obtain
(34)ρpv−qv≤θψv,0,⋯,0,for all v∈V. By inequality (15) and the above inequality, we get
(35)ρp3v34−q3v34≤134ρp3v−q3v≤134θψ3v,0,⋯,0≤θLψv,0,⋯,0,for all v∈V. Hence,
(36)ρ¯Ψp−Ψq≤Lρ¯p−q,∀p,q∈Λρ¯,i.e., Ψ is a ρ¯-contraction. Next, we show that Ψ has a bounded orbit at ϕ. Replacing v1,v2,⋯,vn by v,0,⋯,0 in (16), we get
(37)ρϕ3v−34ϕv≤ψv,0,⋯,0⇒ρϕ3v34−ϕv≤134ψv,0,⋯,0,for all v∈V. Replacing v with 3v in (37), we get
(38)ρϕ32v342−ϕ3v34≤1342ψ3v,0,⋯,0,for all v∈V. By using (37) and (38), we get
(39)ρϕ32v342−ϕv≤ρϕ32v342−ϕ3v34+ϕ3v34−ϕv≤ρϕ32v342−ϕ3v34+ρϕ3v34−ϕv≤1342ψ3v,0,⋯,0+134ψv,0,⋯,0,for all v∈V. By induction, we can easily see that
(40)ρϕ3nv34n−ϕv≤∑i=1n134iψ3i−1v,0,⋯,0≤1L34ψv,0,⋯,0∑i=1nLi≤1341−Lψv,0,⋯,0,for all v∈V. It follows from inequality (40) that
(41)ρϕ3nv34n−ϕ3mv34m≤12ρ2ϕ3nv34n−2ϕv+12ρ2ϕ3mv34m−2ϕv≤k2ρϕ3nv34n−ϕv+k2ρϕ3mv34m−ϕv≤k21341−Lψv,0,⋯,0+k21341−Lψv,0,⋯,0≤k341−Lψv,0,⋯,0,for all v∈V and all n,m∈ℕ. By the definition of ρ¯, we conclude that
(42)ρ¯Ψnϕ−Ψmϕ≤k341−L,which implies the boundedness of an orbit of Ψ at ϕ. It follows from Theorem 1.5 [29] that the sequence Ψnϕρ¯-converges to Q4∈Λρ¯. Now, by the ρ¯-contractivity of Ψ, we have
(43)ρ¯Ψnϕ−ΨQ4≤Lρ¯Ψn−1ϕ−Q4.
Passing to the limitn⟶∞ and applying the Fatou property of ρ¯, we obtain that
(44)ρ¯ΨQ4−Q4≤liminfn⟶∞ρ¯ΨQ4−Ψnϕ≤Lliminfn⟶∞ρ¯Q4−Ψn−1ϕ=0.
Therefore,Q4 is a fixed point of Ψ. Replacing v1,v2,⋯,vn by 3lv1,3lv2,⋯,3lvn in (16), we get
(45)ρDϕ3lv1,3lv2,⋯,3lvn≤ψ3lv1,3lv2,⋯,3lvn,for all v1,v2,⋯,vn∈V. Therefore
(46)ρ134lDϕ3lv1,3lv2,⋯,3lvn≤134lψ3lv1,3lv2,⋯,3lvn.
Employing the limitl⟶∞, we get
(47)DQ4v1,v2,⋯,vn=0,for all v1,v2,⋯,vn∈V. It follows from Theorem 2 that Q4 is quartic. By using (40), we get (17).
To prove the uniqueness ofQ4, let Q4′:V⟶Wρ be another quartic mapping satisfying (17). Then, Q4′ is a fixed point of Ψ.
(48)ρ¯Q4−Q4′=ρ¯ΨQ4−ΨQ4′≤Lρ¯Q4−Q4′,which implies that ρ¯Q4−Q4′=0. This proves that Q4=Q4′. Therefore, the function Q4 is unique. This completes the proof.Corollary 7.
Let a mappingψ:Vn⟶0,+∞ such that
(49)liml⟶∞134lψ3lv1,3lv2,⋯,3lvn=0,(50)ψ3v1,3v2,⋯,3vn≤L34ψv1,v2,⋯,vn,for all v1,v2,⋯,vn∈V with L<1. Suppose that a mapping ϕ:V⟶W with ϕ0=0 and such that
(51)Dϕv1,v2,⋯,vn≤ψv1,v2,⋯,vn,for all v1,v2,⋯,vn∈V, then, there exists a unique quartic mapping Q4:V⟶W satisfying
(52)Q4v−ϕv≤1341−Lψv,0,⋯,0,for all v∈V.Proof.
It is known that every normed space is modular space with the modularρv=∥v∥ and satisfies the Δ3-condition with k=3.Remark 8.
If we replaceψv1,v2,⋯,vn by α∑i=1n∥vi∥p and letting L=3p−4 in Corollary 7, we obtain the stability results for the sum of norms that
(53)Q4v−ϕv≤αvp34−3p,for all v∈V, where α and p are constants with p<4.Remark 9.
If we replaceψv1,v2,⋯,vn by α∑i=1n∥vi∥np+∏i=1n∥vi∥p and letting L=3np−4 in Corollary 7, we obtain the stability results for the sum of product of norms that
(54)Q4v−ϕv≤αvnp34−3np,for all v∈V, where α and p are constants with np<4.Theorem 10.
Letψ:Vn⟶0,+∞ be a function such that
(55)limm⟶∞34mψv13m,v23m,⋯,vn3m=0,(56)ψv13,v23,⋯,vn3≤L34ψv1,v2,⋯,vn,for all v1,v2,⋯,vn∈V with L<1. Suppose that ϕ:V⟶Wρ with ϕ0=0 and satisfies (16), then, there exists a unique quartic mapping Q4:V⟶Wρ satisfying
(57)ρQ4v−ϕv≤L341−Lψv,0,⋯,0,for all v∈V.Proof.
We consider the set(58)Λ=p:V⟶Wρ,and define the function ρ¯ on Λ as follows:
(59)ρ¯p≕infθ>0:ρpv≤θψv,0,⋯,0,∀v∈V.
Similar to the proof of Theorem6, we have
(1)
ρ¯ is a convex modular on Λ(2)
Λρ¯ is ρ¯-complete(3)
ρ¯ satisfies the Fatou property
Now, we consider the functionΨ:Λρ¯⟶Λρ¯ defined by
(60)Ψpv=34pv3,for all v∈V and all p∈Λρ¯. Let p,q∈Λρ¯ and let θ∈0,1 be an arbitrary constant with ρ¯p−q<θ. From the definition of ρ¯, we have
(61)ρpv−qv≤θψv,0,⋯,0,for all v∈V. By the assumption and the last inequality, we get
(62)ρ34pv3−34qv3≤k4ρpv3−qv3≤k4θψv3,0,⋯,0≤θLψv,0,⋯,0,v∈V.
Hence,(63)ρ¯Ψp−Ψq≤Lρ¯p−q,p,q∈Λρ¯,i.e., Ψ is a ρ¯-contraction.
Next, we prove then thatΨ has a bounded orbit at ϕ. Replacing v1,v2,⋯,vn by v,0,⋯,0 in (16), we get
(64)ρ34ϕv−ϕ3v≤ψv,0,⋯,0,for all v∈V. Replacing v with v/3 in (64), we get
(65)ρ34ϕv3−ϕv≤ψv3,0,⋯,0,for all v∈V. Replacing v with v/3 in (65), we get
(66)ρ34ϕv32−ϕv3≤ψv32,0,⋯,0,for all v∈V. By using (64), (65), and (66), we get
(67)ρ342ϕv32−ϕv≤ρ342ϕv32−34ϕv3+ρ34ϕv3−ϕv≤k4ρ34ϕv32−ϕv3+ρ34ϕv3−ϕv≤34ψv32,0,⋯,0+ψv3,0,⋯,0,for all v∈V. By induction, we can easily see that
(68)ρ34nϕv3n−ϕv≤134∑i=1n34iψv3i,0,⋯,0≤134ψv,0,⋯,0∑i=1nLi≤L341−Lψv,0,⋯,0,for all v∈V. It follows from inequality (68) that
(69)ρ34nϕv3n−34mϕv3m≤12ρ234nϕv3n−2ϕv+12ρ234mϕv3m−2ϕv≤kL341−Lψv,0,⋯,0,for all v∈V and all n,m∈ℕ. By the definition of ρ¯, we conclude that
(70)ρ¯Ψnϕ−Ψmϕ≤kL341−L,which implies the boundedness of an orbit of Ψ at ϕ. It follows from Theorem 1.5 [29] that the sequence Ψnϕρ¯-converges to Q4∈Λρ¯.
Now, by theρ¯-contractivity of Ψ, we have
(71)ρ¯Ψnϕ−ΨQ4≤Lρ¯Ψn−1ϕ−Q4.
Employing the limitn⟶∞ and applying the Fatou property of ρ¯, we obtain that
(72)ρ¯ΨQ4−Q4≤liminfn⟶∞ρ¯ΨQ4−Ψnϕ≤Lliminfn⟶∞ρ¯Q4−Ψn−1ϕ=0.
Therefore,Q4 is a fixed point of Ψ. Replacing v1,v2,⋯,vn by v1/3l,v2/3l,⋯,vn/3l in (16), we get
(73)ρDϕ3−lv1,3−lv2,⋯,3−lvn≤ψ3−lv1,3−lv2,⋯,3−lvn,for all v1,v2,⋯,vn∈V. Therefore
(74)ρ34lDϕv13l,v23l,⋯,vn3l≤k4lψv13l,v23l,⋯,vn3l.
Passing to the limitl⟶∞, we get
(75)DQ4v1,v2,⋯,vn=0,for all v1,v2,⋯,vn∈V. It follows from Theorem 2 that Q4 is quartic. By using (68), we get (57).
In order to prove the uniqueness ofQ4, consider another quartic solution Q4′:V⟶Wρ that satisfy the inequality (17). Then, Q4′ is a fixed point of Ψ.
(76)ρ¯Q4−Q4′=ρ¯ΨQ4−ΨQ4′≤Lρ¯Q4−Q4′,which implies that ρ¯Q4−Q4′=0 or Q4=Q4′. Hence, the proof is now completed.Corollary 11.
Let a mappingψ:Vn⟶0,+∞ such that
(77)liml⟶∞34lψv13l,v23l,⋯,vn3l=0,(78)ψv13,v23,⋯,vn3≤L34ψv1,v2,⋯,vn,for all v1,v2,⋯,vn∈V, with L<1. Suppose that ϕ:V⟶W with ϕ0=0 and satisfies (51), then there exists a unique quartic mapping Q4:V⟶W satisfying
(79)Q4v−ϕv≤L341−Lψv,0,⋯,0,for all v∈V.Proof.
It is known that every normed space is modular space with the modularρv=∥v∥ and satisfies the Δ3-condition with k=3.Remark 12.
If we replaceψv1,v2,⋯,vn by α∑i=1n∥vi∥p and letting L=34−p in Corollary 11, we obtain the stability results for the sum of norms that
(80)Q4v−ϕv≤αvp3p−34,for all v∈V, where α and p are constants with p>4.Remark 13.
If we replaceψv1,v2,⋯,vn by α∑i=1n∥vi∥np+∏i=1n∥vi∥p and letting L=34−np in Corollary 11, we obtain the stability results for the sum of product of norms that
(81)Q4v−ϕv≤αvnp3np−34,for all v∈V, where α and p are constants with np>4.


## 4. Counterexample

We present a counterexample to show instability of a particular condition of the equality (5) using modified example of Gajda [7].Remark 14.
If a functionϕ:ℝ⟶V satisfies the functional equation (5), then, the following assertions hold:
(1)
ϕqk/4v=qkϕv,q∈ℚ,k∈ℤ and v∈ℝ(2)
ϕv=v4ϕ1,v∈ℝ if the function ϕ is continuousExample 15.
Let a mappingϕ:ℝ⟶ℝ be defined as follows:
(82)ϕv=∑n=0∞χ3nv34n,where
(83)χv=θv4,−1<v<1,θ,else,then, the mapping ϕ:ℝ⟶ℝ satisfies
(84)Dϕv1,v2,⋯,vn≤n4−20n3+47n2−40n+540631280θ∑j=1nvj4,for all v1,v2,⋯,vn∈ℝ, but a quartic mapping Q4:ℝ⟶ℝ does not exist satisfies
(85)ϕv−Q4v≤εv4,for all v∈ℝ, where θ and ε are a constant.Proof.
It is easy to show thatϕ is bounded by 81/80θ on ℝ. If ∑j=1nvj4≥1/34 or 0, then
(86)Dϕv1,v2,⋯,vn<n4−20n3+47n2−40n+54068180θ.
Thus, (84) is valid. Next, suppose that
(87)0<∑j=1nvj4<134,then, there exists an integer m>0 satisfies
(88)134m+2≤∑j=1nvj4<134m+1.
So that34mv1<1/34,34mv2<1/34,⋯,34mvn<1/34 and
(89)3av1,3av2,⋯,3avn∑i=1n−3avi+∑j=1n3avj∑1≤i<j<k<l≤n3avi+vj+vk+vl∑1≤i<j<k≤n3avi+vj+vk∑1≤i<j≤n3avi+vj∑1≤i<j≤n3avi−vj∈−1,1,a=0,1,⋯,m−1.
Also, fora=0,1,⋯,m−1,
(90)Dχv1,v2,⋯,vn=0.
Next, by inequality (88), we obtain that
(91)Dϕv1,v2,⋯,vn≤∑a=0∞134aDχ3av1,3av2,⋯,3avn≤∑a=0∞134a∑i=1nχ−3avi+∑j=1n3avj−n−8∑1≤i<j<k<l≤nχ3avi+vj+vk+vl+n2−12n+28∑1≤i<j<k≤nχ3avi+vj+vk−n3−15n2+60n−682∑1≤i<j≤nχ3avi+vj−2∑1≤i<j≤nχ3avi−vj−∑i=1nχ3a3vi+n4−17n3+86n2−148n+5586∑i=1nχ3avi≤∑a=m∞134an4−20n3+47n2−40n+5406θ.
It follows from (88) that
(92)Dϕv1,v2,⋯,vn≤n4−20n3+47n2−40n+540631280θ∑j=1nvj4.
Thus, the functionϕ satisfies the inequality (84). Assume on a contrary that there exist a quartic solution Q4:ℝ⟶ℝ satisfying (85). For every v in ℝ, since ϕ is continuous and bounded, Q4 is limited to an open interval of origin and continuous origin.
In the view of Remark14, Q4 must be Q4v=cv4,v∈ℝ. So we obtain
(93)ϕv≤ε+cv4,v∈ℝ.
Suppose, we can choosem>0 with mθ>ε+c. If v∈0,1/3m−1, then, 3av∈0,1 for all a=0,1,⋯,m−1, we obtain
(94)ϕv=∑a=0∞χ3av34a≥∑a=0m−1θ3av434a=mθv4>ε+cv4,which contradicts.



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*Source: 5965628-2022-05-06.xml*