# OrthogonallyC∗-Ternary Jordan Homomorphisms and Jordan Derivations: Solution and Stability

Authors: Vahid Keshavarz; Sedigheh Jahedi
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this work, by using some orthogonally fixed point theorem, we prove the stability and hyperstability of orthogonallyC∗-ternary Jordan homomorphisms between C∗-ternary Banach algebras and orthogonally C∗-ternary Jordan derivations of some functional equation on C∗-ternary Banach algebras.


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## Body

## 1. Introduction and Preliminaries

A classical question in the sense of a functional equation says that “when is it true that a function which approximately satisfies a functional equation must be close to an exact solution of the equation?” Ulam [1] raised the question of stability of functional equations and Hyers [2] was the first to give an affirmative answer to the question of Ulam for additive mapping between Banach spaces. In 1987, Rassias [3] proved a generalized version of the Hyers’ theorem for approximately additive maps. The study of stability problem of functional equations have been done by several authors on different spaces such as Banach, C∗-Banach algebras and modular spaces (for example see [4–13]). One of the stimulating aspects is to examine the stability of those functional equations whose general solutions exist and are useful in characterizing entropies [14].Recently, Eshaghi Gordji et al. [15] introduced the notion of the orthogonal set, which contains the notion of orthogonality in normed space. The study on orthogonal sets has been done by several authors (for example, see [16–18])Definition 1.
(see [15]). Let X≠∅ and ⊥⊆X×X be a binary relation. If there exists x0∈X such that for all y∈X,(1)y⊥x0orx0⊥y.
Then⊥ is called an orthogonally set (briefly O-set). We denote this O-set by X,⊥.
LetX,⊥ be an O-set and X,d be a generalized metric space, then X,⊥,d is called orthogonally generalized metric space.LetX,⊥,d be an orthogonally metric space.(i)
A sequencexnn∈ℕ is called orthogonally sequence (briefly O-sequence) if for any n∈ℕ,(2)xn⊥xn+1orxn+1⊥xn.(ii)
Mappingf:X⟶X is called ⊥− continuous in x∈X if for each O-sequence xnn∈ℕ in X with xn⟶x, then fxn⟶fx. Clearly, every continuous map is ⊥− continuous at any x∈X.(iii)
X,⊥,d is called orthogonally complete (briefly O-complete) if every Cauchy O-sequence is convergent to a point in X.(iv)
Mappingf:X⟶X is called ⊥-preserving if for all x,y∈X with x⊥y, then fx⊥fy.(v)
A mappingf:X⟶X is said to be orthogonally contraction (or ⊥−λ-contraction) with Lipschitz constant 0<λ<1 if(3)dfx,fy≤λdx,yifx⊥y.By using the concept of orthogonally sets, Bahraini et al. [19], proved the generalization of the Diaz and Margolis [20] fixed point theorem on these sets.Theorem 1 (see [19]).
LetX,d,⊥ be an O-complete generalized metric space. Let T:X⟶X be a ⊥-preserving, ⊥-continuous, and ⊥-λ-contraction. Let x0∈X be such that for all y∈X, x0⊥y or for all y∈X, y⊥x0, and consider the “O-sequence of successive approximations with initial element x0 ”: x0, Tx0, T2x0, …, Tnx0, …. Then, either dTnx0,Tn+1x0=∞ for all n≥0, or there exists a positive integer n0 such that dTnx0,Tn+1x0<∞ for all n>n0. If the second alternative holds, then(i)
the O-sequence ofTnx0 is convergent to a fixed point x∗ of T.(ii)
x∗ is the unique fixed point of T in X∗=y∈X:dTnx0,y<∞.(iii)
Ify∈X, then(4)dy,x∗≤11−λdy,Ty.AC∗-ternary Banach algebra A, endowed with a ternary product x,y,z⟶x,y,z of A3 into A, is a complex Banach space in which the product is ℂ-linear in the outer variables, conjugate ℂ-linear in the middle variable, and associative in the sense that \x,y,z,u,w=x,y,z,uw=x,y,z,u,w, for all x,y,z,u,w in A and satisfies x,y,z≤x.y.z,x,x,x=x3 (see [21]). If A,⋅ is a usual C∗-algebra, then an induced ternary multiplication can defined by u,v,w:=u⋅v∗⋅w. If a C∗-ternary Banach algebra A has a unital “e” such that u=u,e,e=e,e,u for all u∈A, then A with binary product u⋅v:=u,e,v and u∗:=e,u,e, is a unital C∗-algebra (see [22]).Definition 2.
Aℂ-linear mapping between C∗-ternary Banach algebras A,B; i.e. H:A⟶B, is called(1)
C∗-ternary homomorphism if(5)Hx,y,z=Hx,Hy,Hz,Hx∗=Hx∗.(2)
C∗-ternary Jordan homomorphism if(6)Hx,x,x=Hx,Hx,Hx,Hx∗=Hx∗.For allx,y,z∈A.Definition 3.
Aℂ-linear mapping D:A⟶A is called(1)
C∗-ternary derivation if(7)Dx,y,z=Dx,y,z+x,Dy,z+x,y,Dz,Dx∗=Dx∗.(2)
C∗-ternary Jordan derivation if(8)Dx,x,x=Dx,x,x+x,Dx,x+x,x,Dx,Dx∗=Dx∗.For allx,y,z∈A.
To prove main results we use the following equivalent assertions.Lemma 1 (see [23]).
Letf:A⟶B be a mapping such that(9)fy−x3+fx−3z3+f3x+3z−y3≤fx,for all x,y,z∈A. Then f is additive.Lemma 2 (see [24]).
LetA and B be two ternary Banach algebras. Let f:A⟶B be an additive mapping. Then the following assertions are equivalent:(a)
fa,a,a=fa,fa,fa, for all a∈A.(b)
fa,b,c+b,c,a+c,a,b=fa,fb,fc+fb,fc,fa+fc,fa,fb,∀a,b,c∈A.Lemma 3 (see[25]).
LetA be a ternary Banach algebras. Let f be an additive mapping from A into A. Then the following assertions are equivalent:(a)
fa,a,a=fa,a,a+a,fa,a+a,a,fa,for alla∈A(b)
fa,b,c+b,c,a+c,a,b=fa,b,c+a,fb,c+a,b,fc+fb,c,a+b,fc,a+b,c,fa+fc,a,b+c,fa,b+c,a,fb,for alla,b,c∈A.In this paper, motivated by the works of [15, 23, 26], we prove the stability of orthogonally C∗-ternary Jordan homomorphism and orthogonally C∗-ternary Jordan derivation of the functional equation(10)fty−x3+fx−3tz3+tf3x+3z−y3=tfx.On orthogonallyC∗-ternary Banach algebras, where t belongs to the set of all complex numbers eiθ with 0≤θ≤2π/n0 for some fixed positive integer number n0.


## 2. Main Results

Throughout the paper, letT1/n01 be the set of all complex numbers eiθ, where 0≤θ≤2π/n0 and n0 is a fixed positive integer number and let A,B be two C∗-ternary Banach algebras.For simplicity, denote(11)Φfx,y,z,t=fty−x3+fx−3tz3+tf3x+3z−y3−tfx,(12)Ψfx,y,z=fx,y,z+y,z,x+z,x,y−fx,fy,fz−fy,fz,fx−fz,fx,fy,where x,y,z∈A and t∈T1/n01.Suppose thatφ and ψ are two mappings from A3 into 0,∞ such that for all x,y,z∈A with x⊥y,x⊥z,y⊥z(13)φx,y,z≤L3φ3x,3y,3z,(14)ψx,y,z≤L33ψ3x,3y,3z,for some constant 0<L<1.Now, we are ready to prove the stability of orthogonallyC∗-ternary Jordan homomorphism in C∗-ternary Banach algebras.Theorem 2.
Letf:A⟶B be a mapping for which(15)Φfx,y,z,t≤φx,y,z,and(16)Ψfx,y,z≤ψx,y,z,and(17)fx∗−fx∗≤φx,x,x.For allt∈T1/n01, and x,y,z∈A with x⊥y,x⊥z,y⊥z, whose φ and ψ are defined as (13) and (14). Then there exists a unique orthogonally C∗-ternary Jordan homomorphism H:A⟶B such that(18)Hx−fx≤L1−Lφx,2x,0,for all x∈A.Proof.
LetΔ be the set of all mappings g:A⟶B such that gx⊥3g1/3x or 3g1/3x⊥gx, for all x∈A. Define dφ on Δ by(19)dφg,h=infα∈0,∞:gx−hx≤αφx,2x,0∀x∈A,and suppose that, for all g,h∈Δ, h⊥g if and only if(20)hx⊥gxorgx⊥hx.
For allx∈A.
ClearlyX,Δ,⊥ is an O-complete generalized metric space. Define Λ:Δ⟶Δ by Λgx=3g1/3x, x∈A. Then(21)Λgx−Λhx=3g13x−3h13x≤3αφx3,2x3,0≤Lαφx,2x,0.
So, by definition ofdφ on Δ, for every g,h∈Δ with g⊥h or h⊥g and dφg,h<α we have ‖Λg−Λh‖≤αL. This shows that dφΛg,Λh≤Ldφg,h, i.e. Λ is ⊥−λ-contraction. The function Λ is ⊥-continuous. In fact, if gn is an O-sequence in Δ which converges to g∈Δ, then for given ε>0, there exists α>0 with α<ε and n∈ℕ such that(22)gnx−gx≤αφx,2x,0.
For allx∈A and n∈ℕ. Therefore by the similar argument, for all x∈A and n≥N, we have(23)dφΛgn,Λg≤Lα<Lε.
Clearly,Λ is ⊥-preserving.
We show that for anyf∈Δ, we have(24)dφΛn+1f,Λnf≤∞.
In (11), put t=1, y=2x/3n−1, z=0 and x:=x/3n−1. By induction we have(25)3fx3n−fx3n≤Ln−13n−1φx,2x,0.
Then forL∈0,1,(26)Λn+1f−Λnf=3n3fx3n+1−fx3n≤Lnφx,2x,0⟶0asn⟶+∞,and then, all conditions of Theorem 1 hold.
So, the O-sequenceΛnf converges to the unique fixed point H in the set of g∈Δ:dφΛnf,g<∞, i.e.,(27)Hx=limn⟶∞Λnf=limn⟶∞3nfx3n.
Also, forf∈Δ,(28)dφf,H≤11−Ldφf,Λf,and by (26), df,H≤Lφx,2x,0. Therefore, H satisfies in (18), i.e.,(29)Hx−fx≤L1−Lφx,2x,0.
We claim thatH is the unique desired orthogonally C∗-ternary Jordan homomorphism which satisfies in (18).
First of all,H is a additive. In fact, for all t∈T1/n01, x,y,z∈A with x⊥y,x⊥z,y⊥z and using (13), we have(30)ΦHx,y,z,t=Hty−x3+Hx−3tz3+tH3x+3z−y3−tHx=limn⟶∞3nfty−x3n+1+fx−3tz3n+1+tf3x+3z−y3n+1−tfx3n≤limn⟶∞3nφx3n,y3n,z3n=0.
So by Lemma1, H is additive. By the same proof of Theorem 3 of [27], the mapping H is ℂ-linear. We show that H is unique. Let H′ be another additive mapping satisfying (18). Then, we have(31)Hx−H′x=3nHx3n−H′x3n≤3nHx3n−fx3n+3nH′x3n−fx3n≤2⋅3nφx3n,2x3n,0≤2Łnφx,2x,0.
For allx∈A. Letting n⟶∞ shows that H is unique.
Now, by using (16)(32)ΨHx,y,z=Hx,y,z+y,z,x+z,x,y−Hx,Hy,Hz−Hy,Hz,Hx−Hz,Hx,Hy=limn⟶∞33nfx3n,y3n,z3n+y3n,z3n,x3n+z3n,x3n,y3n−fx3n,fy3n,fz3n−fy3n,fz3n,fx3n−fz3n,fx3n,fy3n≤limn⟶∞33nψx3n,y3n,z3n=0,and then (14) implies that ΨHx,y,z=0 for all x,y,z∈A with x⊥y,x⊥z,y⊥z. On the other hand, by (13) and (17) we have(33)Hx∗−Hx∗=limn⟶∞3nfx∗3n−fx3n∗≤limn⟶∞3nφx3n,x3n,x3n=0.
For allx∈A. Therefore H is an orthogonally C∗-ternary Jordan homomorphism satisfying (18).In the next theorem, we prove that the self-mappingf with the same appropriate conditions which satisfied in the functional (11), can be approximated by an orthogonally C∗-ternary Jordan derivation.Denote(34)Wfx,y,z=fx,y,z+y,z,x+z,x,y−fx,y,z−x,fy,z−x,y,fz−fy,z,x−y,fz,x−y,z,fx−fz,x,y−z,fx,y−z,x,fy.Theorem 3.
Letf:A⟶A be a mapping satisfying (17) such that(35)Φfx,y,z,t≤φx,y,z,and(36)Wfx,y,z≤ψx,y,z.For allt∈T1/n01 and x,y,z∈A with x⊥y,x⊥z,y⊥z where mappings φ and ψ are satisfied in (13) and (14). Then, there exists a unique orthogonally C∗-ternary Jordan derivation D:A⟶A such that(37)Dx−fx≤L1−Lφx,2x,0,for all x∈A.Proof.
Similar to proof of Theorem2, there exists a self-mapping D on A defined by Dx:=limn⟶∞3nfx/3n satisfies (37). By using Lemma 3 and definition of Dx we have(38)WDx,y,z=Dx,y,z+y,z,x+z,x,y−Dx,y,z−x,Dy,z−x,y,Dz−Dy,z,x−y,Dz,x−y,z,Dx−Dz,x,y−z,Dx,y−z,x,Dy=limn⟶∞33nfx3n,y3n,z3n+y3n,z3n,x3n+z3n,x3n,y3n−fx3n,y3n,z3n−x3n,fy3n,z3n−x3n,y3n,fz3n−fy3n,z3n,x3n−y3n,fz3n,x3n−y3n,z3n,fx3n−fz3n,x3n,y3n−z3n,fx3n,y3n−z3n,x3n,fy3n≤limn⟶∞33nψx3n,y3n,z3n=0,for all x,y,z∈A with x⊥y,x⊥z,y⊥z. So WDx,y,z=0 for all x,y,z∈A with x⊥y,x⊥z,y⊥z. Thus, the mapping D:A⟶A is a unique orthogonally C∗-ternary Jordan derivation satisfies (37). Also, by the same argument in the proof of Theorem 2,(39)Dx∗=Dx∗.Theorems 1 and2 generalized the result of Rassias [3], whenever we define(40)φx,y,z=θxp+yp+zp,ψx,y,z=θx3p+y3p+z3p.For allθ∈ℝ+ and p≠1, in the sense of orthogonal sets.As a consequence of Theorem1, we have hyperstability of orthogonally C∗-ternary Jordan homomorphism between C∗-ternary Banach algebras.Theorem 4.
Letp≠1 and θ be nonnegative real numbers, and let f:A⟶B be a mapping such that(41)fty−x3+fx−3tz3+tf3x+3z−y3≤tfx,(42)Ψfx,y,z≤θxp+yp+zp.For allt∈T1/n01 and all x,y,z∈A with x⊥y,x⊥z,y⊥z. Then, the mapping f:A⟶B is a orthogonally C∗-ternary Jordan homomorphism.From Theorem3, we obtain hyperstability of orthogonally C∗-ternary Jordan derivation.Theorem 5.
Letp≠1 and θ be nonnegative real numbers. Let f:A⟶A be a mapping satisfies (41) such that(43)Wfx,y,z≤θxp+yp+zp.For allx,y,z∈A with x⊥y,x⊥z,y⊥z. Then the mapping f:A⟶A is an orthogonally C∗-ternary Jordan derivation.


## 3. Conclusions

In this paper, we introduced orthogonallyC∗-ternary Jordan homomorphism and C∗-ternary Jordan derivation. Using an orthogonally fixed point theorem, we proved that orthogonally C∗-ternary Jordan homomorphism and orthogonally C∗-ternary Jordan derivation of the functional (11) can be stable and hyperstable in the orthogonally C∗-ternary Banach algebras. The Hyers–Ulam stability theory has many attractions and applications in the field of fractional calculus. For farther research in this field we suggest to see the paper [28, 29].



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