# Asymptotic Behavior of Almost Quartic ⁎-Derivations on Banach ⁎-Algebras

Authors: Hark-Mahn Kim; Hwan-Yong Shin; Jinseok Park
Journal: Journal of Function Spaces
 (2019)

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## Abstract

The purpose of this paper is to obtain the stability theorems of quartic ⁎-derivations associated with the quartic functional equationf(3x-y)+f(x+y)+6f(x-y)=4f(2x-y)+4f(y)+24f(x) on Banach ⁎-algebras.


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## Body

## 1. Introduction

In 1940, Ulam [1] gave a wide ranging talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. In the next year, Hyers [2] gave a clear answer to this problem for additive mappings between Banach spaces. Then this theorem [2] was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. Since then, many mathematicians have come to deal with this problem and also there are many interesting results concerning this problem [5–8].First, we recall definition of∗-derivation.Definition 1.
LetB be a Banach ∗-algebra and let A be a Banach ∗-subalgebra of B. A C-linear mapping D:A→B is said to be derivation on A if D(ab)=D(a)b+aD(b) for all a,b∈A. Moreover, if D satisfies the additional condition D(a∗)=D(a)∗ for all a∈A, then it is called a ∗-derivation.
The stability of∗-derivations and of quadratic ∗-derivations with the Cauchy functional equation and the Jensen functional equation on Banach ∗-algebras was investigated in [9]. Jang and Park [9] proved the superstability of ∗-derivations and of quadratic ∗-derivations on C∗-algebras. The stability of ∗-derivations on Banach ∗-algebras by using fixed-point alternative was proven by Park and Bodaghi [10]. Thereafter, Yang et al. [11] obtained the stability results of cubic ∗-derivations on ∗-algebras and superstability of cubic ∗-derivations on Banach ∗-algebras which are left approximately unital. Recently, Koh and Kang [12] proved the stability of a generalized cubic ∗-derivations on Banach ∗-algebras.
In 1999, Rassias [13] treated the stability of the following quartic equation:(1)fx+2y+fx-2y+6fx=4fx+y+4fx-y+24fyfor a mapping f:X→Y where X is a linear space and Y is a Banach space. Thereafter, Lee and Chung [14] studied the general solution and stability theorem of generalized quartic functional equations in the spaces of generalized functions between real vector spaces. Kang [15] has then extended the stability theorems of the following generalized quartic functional equation: (2)fax+by+fax-by+2b2a2-b2fy=2a2a2-b2fx+ab2fx+y+fx-y,where a,b≠0,a±b≠0 in quasi-β-normed spaces. Recently, Bodaghi [16] obtained the general solution of the generalized quartic functional equation (3)fx+my+fx-my+m-12f2x=27m-9m-1fx+2m2m2-1fy+m2fx+y+fx-yfor a fixed positive integer m and proved the Hyers-Ulam stability for this quartic functional equation by the direct method and the fixed-point method on real Banach spaces and non-Archimedean spaces. For more information about the stability of quartic functional equations, we refer to [17–20].Hyer’s direct method used in [2] has been widely applied for studying the generalized Hyers-Ulam stability of various functional equations. Nevertheless, there exist also other approaches proving the Hyers-Ulam stability of functional equations. The most popular technique of proving stability of functional equations except for direct method is the fixed-point method. Although fixed-point method was used for the first time by J.A. Baker [21], most authors follow the alternative fixed-point approach [22, 23] using a theorem of Diaz and Magolis [24].In this paper, we deal with the following quartic functional equation:(4)f3x-y+fx+y+6fx-y=4f2x-y+4fy+24fxin Banach ∗-algebras. First of all, we show that (4) is equivalent to (1) and then the mapping satisfying (4) on the punctured domain at zero is quartic. In the sequel, we investigate the stability of quartic ∗-derivations associated with the given functional equation on Banach ∗-algebras by using direct method and fixed-point method, respectively.


## 2. Approximate Quartic∗-Derivations

First of all, we find out the general solution of (4) in the class of mappings between vector spaces.Lemma 2.
LetU and V be vector spaces. A mapping f:U→V satisfies the functional equation (4) if and only if the mapping f:U→V satisfies (1).Proof.
The proof is obvious by taking(x,y)≔(x-y,x) in (1) and (x,y)≔(y,x+y) in (4) on the basis of evenness of f, respectively.Throughout this section, letB be a Banach ∗-algebra and let A be a Banach ∗-subalgebra of B. For a given mapping f:A→B, we define (5)Qμfx,y=f3μx-μy+fμx+μy+6fμx-μy-4μ4f2x-y-4μ4fy-24μ4fxDfa,b=fab-a4fb-fab4for all a,b,c,x,y∈A and all μ∈T1/n01≔eiθ:0≤θ≤2π/n0,n0∈N.The following proposition provides a solution of the functional equation (4) on the punctured domain at zero.Proposition 3.
Iff:A→B is a mapping satisfying the equality Q1f(x,y)=0 for all x,y∈A-0 and f(0)=0, then Q1f(x,y)=0 for all x,y∈A and hence f is quartic.Proof.
Sincef(0)=0, it is trivial that Q1f(0,0)=0. We obtain the equalities Q1f(x,x)=2f(2x)-32f(x)=0 so f(2x)=16f(x) for x∈A-0. And we get Q1f(x,3x)-Q1f(x,-x)=6f(-2x)-6f(2x)=0; then f(-x)=f(x) for x∈A-0.
By using above properties, we can show that(6)Q1fx,0=Q1fx,2x=0,Q1f0,y=7f-y-7fy=0for x,y∈A-0. Thus, Q1f(x,y)=0 for all x,y∈A and so f is quartic.Definition 4.
A mappingδ:A→B is called a quartic homogeneous mapping if δ satisfies (1) and δ(μx)=μ4δ(x) for all x∈A and μ∈C. A quartic homogeneous mapping δ:A→B is said to be a quartic derivation if δ(ab)=a4δ(b)+δ(a)b4 for all a,b∈A. In addition, if δ(a∗)=δ(a)∗ for all a∈A, then it is called a quartic ∗-derivation.Now we present a main theorem, which is a stability of quartic functional equation (4) in Banach ∗-algebras.Theorem 5.
Letf:A→B be a mapping with f(0)=0 and let ψ:A5→[0,∞) be a function such that(7)∑i=0∞124iψ2ix,2iy,2iz,0,0<∞&limn→∞⁡128nψ0,0,0,2na,2nb=0,∑i=1∞24iψx2i,y2i,z2i,0,0<∞&limn→∞⁡28nψ0,0,0,a2n,b2n=0,respectively(8)Qμfx,y+fz∗-fz∗≤ψx,y,z,0,0,(9)Dfa,b≤ψ0,0,0,a,bfor all μ∈T1/n01 and all x,y,z,a,b∈A. Assume that the mapping t↦f(ta) from R to B is continuous for each fixed a∈A. Then there exists a unique quartic ∗-derivation δ:A→B satisfying(10)fx-δx≤132∑i=0∞124iψ2ix,2ix,0,0,0fx-δx≤132∑i=1∞24iψx2i,x2i,0,0,0,respectivelyfor all x∈A.Proof.
Takingy=x, z=0, and μ=1 in inequality (8), we get(11)116f2x-fx≤132ψx,x,0,0,0for all x∈A. By using induction, it is implied from inequality (11) that(12)124mf2mx-124nf2nx≤132∑i=nm-1ψ2ix,2ix,0,0,024i24mfx2m-24nfx2n≤132∑i=n+1m24iψx2i,x2i,0,0,0,respectivelyfor m>n≥0 and x∈A. By (7) and (12), the sequence f(2nx)/24n (24nf(x/2n),respectively) is a Cauchy sequence. So define a mapping δ:A→B by(13)limn→∞⁡f2nx24n=δxlimn→∞24nfx2n=δx,respectivelyfor all x∈A. And letting n=0 in inequality (12), we get(14)124mf2mx-fx≤132∑i=0m-1ψ2ix,2ix,0,0,024i24mfx2m-fx≤132∑i=1m24iψx2i,x2i,0,0,0,respectivelyfor m>0 and x∈A. Hence (7) and (14) show that approximate inequality (10) holds.
Next, we have to show that the mappingδ is a quartic ∗-derivation such that inequality (10) holds for all x∈A. Replacing x,y by 2ix,2iy in (8), respectively, and putting z=0, we have (15)124iQμf2ix,2iy≤ψ2ix,2iy,0,0,024i24iQμfx2i,y2i≤24iψx2i,y2i,0,0,0,respectivelyand so it follows from (7) and (13) that (16)Qμδx,y=0for all x,y∈A and μ∈T1/n01. Thus, by the same argument in the proof of Theorem 3.2 in [18], Q1δ(x,y)=0 and δ(μx)=μ4δ(x) for all x,y∈A and μ∈C, which implies that the mapping δ is quartic homogeneous by Lemma 2.
Next, replacinga,b by 2na,2nb in inequality (9), we get (17)128nDf2na,2nb≤128nψ0,0,0,2na,2nb28nDfa2n,b2n≤28nψ0,0,0,a2n,b2n,respectivelyfor all a,b∈A. By (7), we have Dδ(a,b)=0 for all a,b∈A. Letting x=y=0 and replacing z by 2nz in inequality (8), we have (18)f2nz∗24n-f2nz∗24n≤124nψ0,0,2nz,0,024nfz∗2n-24nfz2n∗≤24nψ0,0,z2n,0,0,respectivelyfor all z∈A. Also by (7), we have δ(z∗)=δ(z)∗ for all z∈A. Therefore δ is a quartic ∗-derivation.
Lastly, we should show thatδ is unique. Suppose that δ′:A→B is another quartic ∗-derivation satisfying approximate inequality (10). So (19)δx-δ′x=124nδ2nx-δ′2nx≤124nδ2nx-f2nx+f2nx-δ′2nx≤124n+1∑i=0∞124iψ2i+nx,2i+nx,0,0,0,δx-δ′x=24nδx2n-δ′x2n≤24nδx2n-fx2n+fx2n-δ′x2n≤116∑i=n+1∞24iψx2i,x2i,0,0,0,respectivelywhich tends to zero as n→∞ for all x∈A. Hence δ(x)=δ′(x) for all x∈A.Corollary 6.
Letθj,pi,qj(i=1,2,3,j=1,2) be nonnegative real constants with either 0≤pi<4,qj<8 or pi>4,qj>8(i=1,2,3,j=1,2) and let f:A→B be a mapping with f(0)=0 such that (20)Qμfx,y+fz∗-fz∗≤θ1xp1+yp2+zp3Dfa,b≤θ2aq1+bq2for all μ∈T1/n01 and x,y,z,a,b∈A. Assume that the mapping t↦f(ta) from R to B is continuous for each fixed a∈A. Then there exists a unique quartic ∗-derivation δ:A→B satisfying (21)fx-δx≤θ1xp122p1-16+θ1xp222p2-16for all x∈A.Proof.
Lettingψ(x,y,z,a,b)≔θ1(xp1+yp2+zp3)+θ2(aq1+bq2) and applying Theorem 5, we obtain the desired result.Concerning the stability of quartic homogeneous function, the following example presents that the stability of functional equationQμf(x,y)=0 in Corollary 6 with pi=4(i=1,2) does not hold.Example 7.
Letϕ:C→C be defined by(22)ϕx=x4,ifx<1,1,ifx≥1.Consider the function f:C→C defined by (23)fx=∑m=0∞α-4mϕαmxfor all x∈C, where α≥2. Then, using similar way to that in [25], f satisfies (24)Qμfx,y≤θ1x4+y4,θ1≔40α8α4-1for all x,y∈C and μ∈T1/n01, but there do not exist a quartic mapping δ:C→C and a constant γ>0 such that |f(x)-δx|≤γ|x|4 for all x∈C.However, the stability problem ofpi=4(i=1,2,3) and qj=8(j=1,2) is open in Corollary 6 concerning the stability of quartic ∗-derivations.Corollary 8.
Letθj,pi,qj(i=1,2,3,j=1,2) be nonnegative real constants with either 0≤pi<2,0≤qj<4 or pi>2,qj>4(i=1,2,3,j=1,2) and let f:A→B be a mapping with f(0)=0 such that (25)Qμfx,y+fz∗-fz∗≤θ1xp1yp2+yp2zp3+zp3xp1,Dfa,b≤θ2aq1bq2for all μ∈T1/n01 and x,y,z,a,b∈A. Assume that the mapping t↦f(ta) from R to B is continuous for each fixed a∈A. Then there exists a unique quartic ∗-derivation δ:A→B satisfying (26)fx-δx≤θ1xp1+p2216-2p1+p2for all x∈A.Proof.
Lettingψ(x,y,z,a,b)≔θ1(xp1yp2+yp2zp3+xp1zp3)+θ2(aq1bq2), as well as applying Theorem 5, we obtain the desired result.Now, we investigate the stability using the alternative fixed-point method. Before proceeding to the main result, we state the following definition and theorem which are useful for our purpose.Definition 9.
LetX be a set. A function d:X×X→[0,∞] is called a generalized metric on X if d satisfies the following: (i)
d(x,y)=0, if and only if x=y.(ii)
d(x,y)=d(y,x) for all x,y∈X.(iii)
d(x,z)≤d(x,y)+d(y,z) for all x,y,z∈X.Theorem 10 (see [24]).
Let(Ω,d) be a complete generalized metric space and let T:Ω→Ω be a mapping with Lipschitz constant L<1. Then, for each element α∈Ω, either d(Tnα,Tn+1α)=∞ for all n≥0 or there exists a natural number n0 such that (i)
d(Tnα,Tn+1α)<∞ for all n≥n0,(ii)
the sequence{Tnα} is convergent to a fixed point β∗ of T,(iii)
β∗ is the unique fixed point of T in the set Λ=β∈Ω:dTn0α,β,(iv)
d(β,β∗)≤1/1-Ld(β,Tβ) for all β∈Λ.Theorem 11.
Letf:A→B be a continuous mapping with f(0)=0 and let ψ:A5→[0,∞) be a function such that(27)Qμfx,y+fz∗-fz∗≤ψx,y,z,0,0,(28)Dfa,b≤ψ0,0,0,a,bfor all μ∈T1/n01 and a,b,x,y,z∈A. If there exist constants l1,l2∈(0,1) such that (29)ψ2x,2y,2z,0,0≤24l1ψx,y,z,0,0&ψ0,0,0,2a,2b≤28l2ψ0,0,0,a,bψx2,y2,z2,0,0≤l124ψx,y,z,0,0&ψ0,0,0,a,b≤l228ψ0,0,0,a,b,respectivelyfor all x,y,z,a,b∈A, then there exists a unique quartic ∗-derivation δ:A→B satisfying(30)fx-δx≤1321-l1ψ~x(31)fx-δx≤l1321-l1ψ~x,respectivelyfor all x∈A, where ψ~(x)=ψ(x,x,0,0,0).Proof.
First, we consider a set(32)Ω=g:A→B:g0=0and define a mapping d on Ω×Ω as follows:(33)dg1,g2≔inf⁡k∈0,∞:g1x-g2x≤kψx,x,0,0,0if there exists such constant k and d(g1,g2)≡∞, if not. Then we can easily show that d is a generalized metric on Ω and the metric space (Ω,d) is complete. We define a mapping Ψ:Ω→Ω by(34)Ψgx=124g2xΨgx=24gx2,respectivelywhere g∈Ω and for all x∈A.
Now we remark thatΨ is a strictly contractive mapping on Ω with the Lipschitz constant l1 [18].
On the other hand, lettingμ=1,y=x,z=0 in inequality (27), we get (35)116f2x-fx≤132ψx,x,0,0,016fx2-fx≤12ψx2,x2,0,0,0≤l132ψx,x,0,0,0,respectivelyfor all x∈A. This implies that (36)dΨf,f≤132dΨf,f≤l132,respectively.It follows from Theorem 10 that d(Ψnf,Ψn+1f)<∞ for all n≥0. So parts (iii) and (iv) of Theorem 10 hold on the whole Ω. Therefore, there exists a unique mapping δ:A→B such that δ is a fixed point of Ψ and (37)δx=limn→∞⁡f2nx24nδx=limn→∞⁡24nfx2n,respectivelyfor all x∈A and (38)df,δ≤11-l1dΨf,f≤1321-l1df,δ≤l11-l1dΨf,f≤l1321-l1,respectivelySo the mapping δ satisfies inequality (30) that holds for all x∈A.
Sincel1,l2∈(0,1), inequality (30) shows that(39)limi→∞⁡ψ2ix,2iy,2iz,0,024i=0&limi→∞⁡ψ0,0,0,2ia,2ib28i=0limi→∞⁡24iψx2i,y2i,z2i,0,0=0&limi→∞⁡28iψ0,0,0,a2i,b2i=0,respectivelyfor all x,y,z,a,b∈A. Replacing x,y by 2ix,2iy and putting z=0 in inequality (27), we have (40)124iQμf2ix,2iy≤ψ2ix,2iy,0,0,024i24iQμfx2i,y2i≤24iψx2i,y2i,0,0,0,respectivelyand taking the limit as i tends to infinity, we get Qμδ(x,y)=0 for all x,y∈A and all μ∈T1/n01. Also, by the same argument in the proof of Theorem 5, the mapping δ is quartic homogeneous. Next, replacing a,b by 2na,2nb in inequality (28), we get (41)128iDf2ia,2ib≤128iψ0,0,0,2ia,2ib28iDfa2i,b2i≤28iψ0,0,0,a2i,b2i,respectivelyfor all a,b∈A. By (39), we have Dδ(a,b)=0 for all a,b∈A. Letting x=y=0 and replacing z by 2nz in inequality (27), we have (42)f2nz∗24n-f2nz∗24n≤124nψ0,0,2nz,0,024nfz∗2n-24nfz2n∗≤24nψ0,0,z2n,0,0,respectivelyfor all z∈A. Also by (39), we have δ(z∗)=δ(z)∗ for all z∈A. Therefore the mapping δ is a quartic ∗-derivation.
The rest of the proof is similar to the proof of Theorem5.Corollary 12.
Letθi(i=1,2),p,q be nonnegative reals with either 0≤p<4,q<8 or p>4,q>8 and let f:A→B be a continuous mapping with f(0)=0 such that (43)Qμfx,y+fz∗-fz∗≤θ1xp+yp+zpDfa,b≤θ2aq+bqfor all μ∈T1/n01 and x,y,z,a,b∈A. Then there exists a unique quartic ∗-derivation δ on A satisfying (44)fx-δx≤θ1xp16-2pfor all x∈A.Proof.
Lettingψ(x,y,z,a,b)≔θ1(xp+yp+zp)+θ2(aq+bq) and applying Theorem 11 with l1=2-|p-4|,l2=2-|q-8|, we obtain the desired results.Corollary 13.
Letθi(i=1,2),p,q be nonnegative reals with either 0≤p<2,q<4 or p>2,q>4 and let f:A→B be a continuous mapping with f(0)=0 such that (45)Qμfx,y+fz∗-fz∗≤θ1xpyp+ypzp+zpxpDfa,b≤θ2aqbqfor all μ∈T1/n01 and x,y,z,a,b∈A. Then there exists a unique quartic ∗-derivation δ on A satisfying (46)fx-δx≤θ1x2p216-22pfor all x∈A.Proof.
Lettingψ(x,y,z,a,b)≔θ1(xpyp+ypzp+zpxp)+θ2(aqbq) and applying Theorem 11 with l1=2-2p-4,l2=2-2q-8, we obtain the desired results.



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