# Orthogonal Stability and Nonstability of a Generalized Quartic Functional Equation in Quasi-β-Normed Spaces

Authors: Nazek Alessa; K. Tamilvanan; K. Loganathan; T. S. Karthik; John Michael Rassias
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this work, we examine the generalized Hyers-Ulam orthogonal stability of the quartic functional equation in quasi-β-normed spaces. Moreover, we prove that this functional equation is not stable in a special condition by a counterexample.


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## Body

## 1. Introduction

In this paper,ℝ and ℂ denote sets of all real numbers and complex numbers, respectively.In the fall of 1940, Ulam [1] suggested the stability problem of functional equations concerning the stability of group homomorphisms as follows:Ulam’s question: let G1,∗, G2,⋆ be two groups and d:G2×G2→0,∞ be a metric. Given δ>0, does there exist ε>0 such that if a function g:G1→G2 satisfies the inequality
(1)dgx∗y,gx⋆gy≤δ,for all x,y∈G1, then there is a homomorphism h:G1→G2 with
(2)dgx,hx≤εforallx∈G1?In other words, under what condition does there exist a homomorphism near an approximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. In 1941, Hyers [2] gave the first affirmative answer to the question of Ulam for Banach spaces. This result was generalized by Aoki [3] for additive mappings.During the past few years, several mathematicians have published on various generalizations and applications of generalized Hyers-Ulam stability to a number of functional equations and mappings (see, for instance, [4–15]).In [16], Xu et al. obtained the general solution and investigated the Ulam stability problem for the quintic functional equation in quasi-β-normed spaces via fixed point method. This method is different from the direct method, initiated by Hyers in [2]. And also, Eskandani et al. [17, 18] obtained the general solution for the mixed additive and quadratic functional equation and a cubic functional equation and established its generalized Hyers-Ulam stability in quasi-β-normed spaces.The Ulam-type stability result for the quartic functional equation(3)Fx1+2x2+Fx1−2x2+6Fx1=4Fx1+x2+Fx1−x2+6Fx2,was first developed by Rassias [19]. Subsequently, Sahoo and Chung [20] determined the general solution of (3) without assuming any regularity conditions on the unknown function. In fact, they proved that the function f:ℝ→ℝ is a solution of (3) if and only if fx=Ax,x,x,x, where the function A:ℝ4→ℝ is symmetric and additive in each variable. Since the solution of (3) is even, we can rewrite (3) as
(4)f2x+y+f2x−y=4fx+y+4fx−y+24fx−6fy.Lee et al. [21] obtained the general solution of (4) and proved the Hyers-Ulam-Rassias stability of this equation. It is easy to show that the function fx=x4 satisfies the functional equation (4), which is called a quartic functional equation, and every solution of the quartic functional equation is said to be a quartic function. In [22] Ravi et al. have investigated the generalized Hyers-Ulam product-sum stability of functional equations and have the following theorem.Theorem 1.
Letf:E→F be a mapping which satisfies the inequality
(5)fmx+y+fmx−y−2fx+y−2fx−y−2m2−2fx+2fyF≤εxEpyEp+xE2p+yE2p,for all x,y∈E with x⊥y, where ε and p are constants with ε,p>0 and either m>1, p<1 or m<1, p>1 with m≠0, m≠±1, m≠±2, and −1≠mp−1<1. Then, the limit limn→∞m−2nfmnx exists for all x∈E, and Q:E→F is the unique orthogonally Euler-Lagrange quadratic mapping such that
(6)fx−QxF≤ε2m2−m2pxE2p,for all x∈E.In 1982, Rassias [23] provided generalizations of the Hyers-Ulam stability theorem which allows the Cauchy difference controlled by a product of different powers of norm. And then, the result of the Rassias theorem has been generalized by Gavruta [24] by replacing the unbounded Cauchy difference by a generalized control function. Also, Rassias (see [23, 25–28]) solved the Ulam problem for different mappings. In addition, Ravi et al. considered the mixed product-sum of powers of norms control function [22]. Note that the mixed product-sum function was introduced by Ravi et al. in 2008-2009 ([22, 29–31]).In this paper, we examine the generalized Hyers-Ulam orthogonal stability of the quartic functional equation as(7)ϕ∑a=1mva=∑1≤a<b<c<d≤mϕva+vb+vc+vd+−m+4∑1≤a<b<c≤mϕva+vb+vc+m2−7m+122∑1=a;a≠bmϕva+vb−∑a=1mϕ2va+−m3+9m2−26m+1206∑a=1mϕva+ϕ−va2,where m is a positive integer with ℕ−0,1,2,3,4. It is easy to see that the function ϕv=av4 is a solution of the functional equation (7).


## 2. Orthogonal Hyers-Ulam Stability

Lemma 2 (see [32].
LetE and F be real vector spaces. If the mapping ϕ:E→F satisfies the functional equation (7) for all v1,v2,⋯,vm∈E with vi⊥vj;i≠j=1,2,⋯,m, then ϕ is quartic.Remark 3.
LetEbe a linear space andϕ:ℝ→Ebe a function satisfies (7). Then, the following two assertions hold:(1)
ϕrk/4v=rkϕv for all v∈ℝ,r∈ℚ and k integers.(2)
ϕv=v4ϕ1 for all v∈ℝ if ϕ is continuous.
Here, let us considerE to be a linear space over K and F is a β,p-Banach space with p-norm ∥·∥Y.LetK be the modulus concavity of ∥·∥F.For our convenience, we use the abbreviation for a functionϕ:E→F:
(8)Δϕv1,v2,⋯,vm=ϕ∑1≤a≤nva−∑1≤a<b<c<d≤mϕva+vb+vc+vd−−m+4∑1≤a<b<c≤mϕva+vb+vc−m2−7m+122∑1=a;a≠bmϕva+vb+∑a=1mϕ2va−−m3+9m2−26m+1206∑a=1mϕva+ϕ−va2,for all v1,v2,⋯,vm∈E.Theorem 4.
Let a functionϕ:E→F which there exists ψ:Em→0,∞ such that
(9)Δϕv1,v2,⋯,vmF≤ψv1,v2,⋯,vm,v1,v2,⋯,vm∈E,with vi⊥vj,i≠j=1,2,⋯,m, and the contractively subadditive function ψ and a constant L fulfilling 21−4βL<1. Then, there exists a unique mapping Q4:E→F which is quartic such that
(10)ϕv−Q4vF≤K24βp−2Lppψv,0,⋯,0,for all v∈E.Proof.
Settingv1,v2,⋯,vm by v,0,⋯,0 in (9), we have
(11)ϕ2v−24ϕvF≤ψv,0,⋯,0,for all v∈E. Replacing v in (11) by 2mv and dividing by 24m+1β in (11) we attain
(12)ϕ2m+1v24m+1−ϕ2mv24mF≤K24m+1βψ2mv,0,⋯,0,v∈E,m>0.
We have(13)ϕ2m+1v24m+1−ϕ2iv24iFp≤∑a=imϕ2a+1v24a+1−ϕ2av24aFp≤∑a=imK24a+1βpψp2av,0,⋯,0≤Kψpv,0,⋯,024pβ∑a=im21−4βLap,for all v∈E and m≥i>0. Clearly, F is complete, the Cauchy sequence ϕ2mv/24m converges for every v∈E. Next, we define a mapping Q4:E→F by
(14)Q4v≔limm→∞ϕ2mv24m,for all v∈E. Letting i=0 and taking m→∞ in (13), we obtain (10). Next, we want to prove that Q4 is quartic. From (9) and (14) that
(15)ΔQ4v1,v2,⋯,vmFp=limm→∞Δϕ2mv1,2mv2,⋯,2mvm24mFp≤limm→∞K24mβpΔϕ2mv1,2mv2,⋯,2mvmFp≤limm→∞K24mβpψp2mv1,2mv2,⋯,2mvm≤limm→∞K2Lmp24mβpψpv1,v2,⋯,vm≤limm→∞K21−4βLmpψpv1,v2,⋯,vm=0,for all v1,v2,⋯,vm∈E with vi⊥vj,i≠j=1,2,⋯,m. Therefore, by Lemma 2, we conclude that Q4 is quartic. Next, to show that the function Q4 is unique.
Let us consider another quartic functionR4:E→F which fulfils the inequality (10) we get
(16)Q4v−R4v=limm→∞124mβϕ2mv−R42mvF≤limm→∞Kψ2mv,0,⋯,024mβ24βp−2Lpp≤limm→∞K21−4βLm24βp−2Lppψv,0,⋯,0=0.
This shows thatQ4=R4; therefore, Q4 is unique mapping. This ends the proof of the theorem.Corollary 5.
Ifβ=1 and τ be a positive real number and a function ϕ:E→F for which
(17)Δϕv1,v2,⋯,vmF≤τ,for all v1,v2,⋯,vm∈E with vi⊥vj,i≠j=1,2,⋯,m. Then, there exists Q4:E→Fwhich is a unique quartic mapping that fulfils
(18)ϕv−Q4vF≤Kτ24p−2Lpp,v∈E.The following theorem is obtained by replacing the expansive superadditive instead of the contractive subadditive in Theorem4.Theorem 6.
Let a functionϕ:E→F in which exists a mapping ψ:Em→0,∞ such that
(19)Δϕv1,v2,⋯,vmF≤ψv1,v2,⋯,vm,for all v1,v2,⋯,vm∈E with vi⊥vj,i≠j=1,2,⋯,m, and the expansively superadditive function ψ and a constant L fulfilling 24β−1L<1. Then, there exists a unique mapping Q4:E→F which is quartic which fulfils
(20)ϕv−Q4vF≤KL2p−24βLppψv,0,⋯,0,for all v∈E.With the upcoming theorems, we establish the stability of equation (7) by using an idea of Gavruta in [24].Theorem 7.
Let a mappingψ:Em→0,∞ such that
(21)limm→∞124mψ2mv1,2mv2,⋯,2mvm=0,for all v1,v2,⋯,vm∈E with vi⊥vj,i≠j=1,2,⋯,m, and
(22)ψ~Q4v≔∑a=0∞K24aβpψp2av,0,⋯,0<∞,v∈E.Ifϕ:E→F is a mapping which fulfils
(23)Δϕv1,v2,⋯,vmF≤ψv1,v2,⋯,vm,v1,v2,⋯,vm∈E,with vi⊥vj,i≠j=1,2,⋯,m, then there exists a unique mapping Q4:E→F which is quartic which satisfies
(24)ϕv−Q4vF≤K24βψ~Q4v1/p,for all v∈E.Proof.
From equation (11) in Theorem 4, we get
(25)ϕ2v−24ϕv≤ψv,0,⋯,0,v∈E.
Replacingv through 2mv in inequality (25) and dividing by 24βm+1, we obtain
(26)ϕ2m+1v24m+1−ϕ2mv24mF≤K24βm+1ψ2mv,0,⋯,0,v∈E,m>0.
Already, we know thatF is a β,p-Banach space; we obtain
(27)ϕ2m+1v24m+1−ϕ2iv24iFp≤∑a=imϕ2a+1v24a+1−ϕ2av24aFp≤K24βp∑a=im124aβpψp2av,0,⋯,0,for all v∈E with m≥i>0. From inequalities (22) and (27) that the sequence ϕ2mv/24m is Cauchy in F for every v∈E. We know that if F is complete, the sequence ϕ2mv/24m converges for every v∈E. Now, we can define a mapping Q4:E→F by
(28)Q4v≔limm→∞ϕ2mv24m,for all v∈E. Letting i=0 and taking m→∞ in (27), we obtain the result (24). The remaining proof is the same as the proof of Theorem 4.Theorem 8.
Letψ:Em→0,∞ be a mapping such that
(29)limm→∞24mψv12m,v22m,⋯,vm2m=0,v1,v2,⋯,vm∈E,with vi⊥vj,i≠j=1,2,⋯,m, and
(30)ψQ4v,0,⋯,0≔∑a=0∞24aβpψpv2a+1,0,⋯,0<∞,for all v∈E. If ϕ:E→F fulfils
(31)Δϕv1,v2,⋯,vmF≤ψv1,v2,⋯,vm,v1,v2,⋯,vm∈E,with vi⊥vj,i≠j=1,2,⋯,m. Then, there exists a unique function Q4:E→F which is quartic which fulfils
(32)ϕv−Q4vF≤K24βψ1/pv,0,⋯,0,v∈E.Proof.
From equation (11), we get
(33)ϕ2v−24ϕv≤ψv,0,⋯,0,v∈E.
Settingv by v/2m+1 in (33) and multiply by 24βm, we have
(34)24m+1ϕv2m+1−24mϕv2mF≤24βmψv2m+1,0,⋯,0,v∈E,m>0,we have
(35)24m+1ϕv2m+1−24iϕv2iFp≤∑a=im24a+1ϕv2a+1−24aϕv2aFp≤∑a=im24pβaψpv2a+1,0,⋯,0,v∈E,m≥i>0.
Then, we conclude from (42) and (34) that the sequence 24mϕv/2m is Cauchy in F for every v∈E.
AsF is complete, the sequence 24mϕv/2m converges for every v∈E. Next, we define a mapping Q4:E→F by
(36)Q4v≔limm→∞24mϕv2m,for all v∈E. Letting i=0 and taking m→∞ in (34), we obtain (32). The remaining proof is the same as the proof of Theorem 4.Corollary 9.
Lets,t be the positive real numbers such that s+t<4β or s+t>4. If a mapping ϕ:E→F satisfies the inequality
(37)Δϕv1,v2,⋯,vmF≤∏a=1mvaEs+t+∑a=1mvaEms+t,for all v1,v2,⋯,vm∈E with vi⊥vj,i≠j=1,2,⋯,m, then there exists a unique quartic mapping Q4:E→F which satisfies
(38)ϕv−Q4vF≤K∥v∥Ems+t∣24βp−2ms+tp∣p,for all v∈E.Corollary 10.
Lets,t be the positive real numbers such that s+t<4β or s+t>4. If a mapping ϕ:E→F satisfies the inequality
(39)Δϕv1,v2,⋯,vmF≤∏a=1mvaEs+t,for all v1,v2,⋯,vm∈E with vi⊥vj,i≠j=1,2,⋯,m, then the mapping ϕ:E→F is quartic.


## 3. Counterexample

Here, we proved the nonstability of equation (7) in a special condition by a counterexample which is a modified idea of Gajda [9].Example 11.
Let a mappingϕ:ℝ→ℝdefined by(40)ϕv=∑m=0∞χ2mv24m,where
(41)χv=Θv4,−1<v<1,Θ,otherwise,then the function ϕ:ℝ→ℝ fulfils
(42)Δϕv1,v2,⋯,vm≤−m3+12m2−53m+1986409615Θ∑a=1mva4,for all v1,v2,⋯,vm∈ℝ, but there does not exist a quartic mapping Q4:ℝ→ℝ such that
(43)ϕv−Q4v≤εv4,v∈ℝ,whereΘandεare constants.Proof.
Clearly,ϕ is bounded by 16/15θ on ℝ. If ∑a=1mva4≥1/24 or 0, then the left side of (29) is less than −m3+12m2−53m+198/616/15θ, and thus, (29) is true.
Next, we assume that(44)0<∑a=1mva4<124,then there exists an integer i such that
(45)124i+2≤∑a=1mva4<124i+1.So that 2i∣v1∣<1/2,2i∣v2∣<1/2,⋯,2i∣vm∣<1/2 and 2mv1,2mv2,⋯,2mvm∈−1,1 for every m=0,1,2,⋯,i−1. For m=0,1,⋯,i−1,
(46)χ∑a=1m2mva−∑1≤a<b<c<d≤mχ2mva+vb+vc+vd−−m+4∑1≤a<b<c≤mχ2mva+vb+vc−m2−7m+122∑1=a;a≠bmχ2mva+vb+∑a=1mχ2m2va−−m3+9m2−26m+1206∑a=1mχ2mva+χ−2mva2=0.
Next,(47)∣Δϕv1,v2,⋯,vm∣≤∑a=i∞124a∣Δχ2av1,2av2,⋯,2avm∣≤∑a=i∞124a∣χ∑a=1m2mva−∑1≤a<b<c<d≤mχ2mva+vb+vc+vd−−m+4∑1≤a<b<c≤mχ2mva+vb+vc−m2−7m+122∑1=a;a≠bmχ2mva+vb+∑a=1mχ2m2va−−m3+9m2−26m+1206∑a=1mχ2mva+χ−2mva2∣≤∑a=i∞124a−m3+12m2−53m+1986Θ≤−m3+12m2−53m+1986241−i15Θ.
It follows from (43) that
(48)∣Δϕv1,v2,⋯,vm∣≤−m3+12m2−53m+1986409615Θ∑a=1mva4,for all v1,v2,⋯,vm∈ℝ. Thus, ϕ satisfies (29) for all v1,v2,⋯,vm∈ℝ with vi⊥vj,i≠j=1,2,⋯,m.
Assume that there is a contrary mappingQ4:ℝ→ℝwhich is quartic which fulfils (42). We know that, for every v∈ℝ, ϕ is bounded and continuous and Q4 is bounded on any open interval containing the origin which is continuous at the origin.
In the view of Remark3, Q4 must be Q4v=av4,v∈ℝ. Thus, we have
(49)∣ϕv∣≤ε+∣a∣v4,v∈ℝ.
But we can select an integeri>0 with iΘ>ε+∣a∣. If v∈0,1/2i−1, then 2mv∈0,1 for any m=0,1,⋯,i−1, and for v, we obtain
(50)ϕv=∑m=0∞χ2mv24m≥∑m=0i−1Θ2mv424m=iΘv4>ε+∣a∣v4,which contradicts.



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