# Orthogonally Fixed Points and (m, m)-Hom-Derivation Equations

Authors: Abdollah Dinmohammadi
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we introduce the concept ofm-Hom-m-derivation (briefly (m, m)-Hom-derivation) equations in orthogonally Banach algebras. We use the orthogonally fixed point to investigate the hyperstability of (m, m)-Hom-derivation equations.


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## Body

## 1. Introduction and Preliminaries

The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] answered to the question of Ulam between Banach spaces. In 1978, Rassias [3] generalized for additive mappings. See [4–9] for more information.In addition, Hyers–Ulam stability of functional equations on the normed space, Banach space, and Banach algebras was given. Later, other research studies on the Hyers–Ulam stability of fractional-order systems were presented. Obloza, in 1997, was the first author who introduced the Hyers–Ulam stability for linear differential equations [10, 11]. In recent years, Hyers–Ulam stability theory of fractional differential equations has attracted great attention, for example, with the extension of the theory of fractional calculus (integral and derivative of arbitrary order) [12–16]; some researchers tried to extend the above concept for the fractional differential equations [17–20].Recently, Eshaghi Gordji et al. [21] introduced the notion of the orthogonal set which contains the notion of the orthogonality in normed space. By using the concept of orthogonal sets, Bahraini et al. [22] proved the generalization of the Diaz–Margolis [23] fixed point theorem on these sets. The study on orthogonal sets has been done by several authors (for example, see [24–26]).Definition 1 (see [21]).
LetA≠∅ and ⊥⊆A×A be a binary relation. If ⊥ satisfies(1)∃a0∀b;b⊥a0or∀b;a0⊥b,it is called an orthogonal set (or O-set), and we denote this O-set by A,⊥.
LetA,⊥ be an O-set and A,d be a generalized metric space; then, A,⊥,d is called an orthogonally generalized metric space.Definition 2 (see [21]).
LetX,⊥,d be an orthogonally metric space.(i)
A sequenceann∈ℕ is called an orthogonal sequence (briefly O-sequence) if(2)∀n;an⊥an+1or∀n;an+1⊥an.(ii)
A mappingg from X into X is ⊥−continuous in a∈X if for all O-sequences ann∈ℕ in X with an⟶a, gan⟶ga. Clearly, every continuous map is ⊥−continuous at any a∈X.(iii)
X,⊥,d is orthogonally complete (briefly O-complete) if every Cauchy O-sequence is convergent to a point in X.(iv)
A mappingg from X into X is ⊥-preserving if for all a,b∈X with a⊥b, ga⊥gb.(v)
A mappingg from X into X is called an orthogonality contraction with Lipschitz constant 0<λ<1 if(3)dga,gb≤λda,bwherea⊥b.Example 3.
LetX=0,1, and let the metric on X be Euclidean metric. For all a,b∈X, a⊥b is defined by ab≤maxa/s,b/s. Let g:X⟶X be a mapping defined by(4)ga=as,a≤1s,0,a>1s,where s=2,3,…. Since a⊥b and ab≤maxa/s,b/s, we have the following cases:(1)
Ifa=0,b≤1/s⇒ga=0,gb=b/s(2)
Ifa=0,b≤1/s⇒ga=gb=0(3)
Ifb≤1/s,a≤1/s⇒gb=b/s,fa=a/s(4)
Ifb≤1/s,a≤1/s⇒gb=b/s,ga=0
Hence, it is clear thatX is not complete, but it is easy to see that O-complete, g is not continuous and contraction on X but it is easy to see that ⊥-continuous and ⊥-λ-contraction for λ=1/s, and in the end, g is ⊥-preserving on X.Theorem 1 (see [22]).
Suppose thatX,d,⊥ is an O-complete generalized metric space. Let J:X⟶X be a ⊥-preserving, ⊥-continuous, and ⊥-λ-contraction. Let a0∈X satisfy for all b∈X, a0⊥b or for all b∈X, b⊥a0, and consider the “O-sequence of successive approximations with initial element a0 :” a0, Ja0, J2a0, …, Jna0, …. Then, either(5)dJna0,Jn+1a0=∞,∀n≥0,or there exists a positive integer n0 such that(6)dJna0,Jn+1a0<∞,for all n>n0. If the second alternative holds, then(1)
TheO-sequence of Jna0 is convergent to a fixed point a∗ of J.(2)
a∗ is the unique fixed point of J in X∗=b∈X:dJna0,b<∞.(3)
Ifb∈X, then(7)db,a∗≤11−λdb,Jb.Consider the generalized additive, quadratic, cubic, and quartic functional equation(8)fsa+b+fsa−b=sm−2fa+b+fa−b+2s2−1sm−2fa+m−21−m−226fb,where 1≤m≤4 is an integer number and s≠0,±1 is a real number.Definition 4.
LetA be an orthogonally Banach algebra, and H:A⟶A satisfying (8) and(9)Hab=HaHb,∀a,b∈Awitha⊥b,for m=1,2,3,4, is called an additive homomorphism, quadratic homomorphism, cubic homomorphism, and quartic homomorphism, respectively, or, briefly, m-homomorphisms.In 2018, Park et al. [27] defined Hom-derivation, and in the following, we defined (m, m)-Hom-derivations form=1,2,3,4 on orthogonally Banach algebra.Definition 5.
LetH:A⟶A be m-homomorphisms, m∈1,2,3,4. An m-mapping D:A⟶A is called (m, m)-Hom-derivations if D satisfies(10)Dab=DaHb+HaDb,∀a,b∈Awitha⊥b.In the present paper, we investigate the stability of orthogonally (m, m)-Hom-derivations for functional equation (8) on orthogonally Banach algebra by using the orthogonally fixed point.


## 2. Main Results

Throughout the section, letA be an orthogonally Banach algebra. Let f,h be mappings between A, for all m positive integers less then 5 and k≠0,±1, for all a,b,u,v∈A with a⊥b, and u⊥v; for simplicity, denote(11)Φmfa,b=fka+b+fka−b−km−2fa+b+fa−b−2k2−1km−2fa+m−21−m−226fb,Ψmfuv=fuhv+hufv.Suppose thatφm and ψm are mappings from A2 into 0,∞ such that(12)φmas,bs≤Lsmφma,b, (13)ψmus,vs≤Ls2mψmu,v,for all a,b,u,v∈A with a⊥b and u⊥v and for some constant 0<L<1. If we take a=b=u=v=0, φm0,0=0 and ψm0,0=0.It follows (12) and (13):(14)limn⟶∞smnφmasn,bsn=0, (15)limn⟶∞s2mnψmasn,bsn=0,for all a,b,u,v∈A with a⊥b and u⊥v and s≠0,±1.Theorem 2.
Letf,ζ:A⟶A be two mappings satisfying(16)Φmfa,b≤φma,b, (17)Φmζa,b≤φma,b, (18)ζuv−ζuζv≤ψmu,v, (19)fuv−Ψmfuv≤ψmu,v,where φm,ψma,b fulfills (12) and (13). Then, there exist unique orthogonally m-homomorphisms Hm:A⟶A and unique orthogonally (m, m)-Hom-derivations Dm:A⟶A such that(20)ζa−Hma≤12smL−1−1φma,0, (21)fa−Dma≤12smL−1−1φma,0,for all a∈A and m=1,2,3,4.Proof.
LetΔ be the set of all mappings k:A⟶A such that ka⊥smk1/sa or smk1/sa⊥ka, for all a∈A. Define dφ on Δ by(22)dφk,t=infα∈ℝ+:ka−ta≤αφma,0∀a∈A,and suppose that, for all k,t∈Δ,(23)k⊥t⟺ka⊥taorta⊥ka,for all a∈A.
Clearly,A,Δ,⊥ is an O-complete generalized metric space. Define Γ:Δ⟶Δ by Λka=smk1/sa, a∈A and m=1,2,3,4. Then,(24)Γka−Γta=smk1ka−smt1ka≤smαφmas,0≤Lαφma,0.
So, by the definition ofdφ on Δ, for k,t∈Δ with k⊥t or t⊥k and dφk,t<α, we have Γk−Γt≤αL. This shows that dφΓk,Γt≤Ldφk,t, i.e., Γ is ⊥-contraction. The function Γ is ⊥-continuous. In fact, if kn is an O-sequence in Δ which converges to k∈Δ, then for given ε>0, there exist α>0 with α<ε and n∈ℕ such that(25)kna−ka≤αφa,0.
Therefore, by the similar argument, for alla∈A, m=1,2,3,4, and n≥N, we have(26)dφΓkn,Γk≤Lα<Lε.
Clearly,Γ is ⊥-preserving.
We show that, for anyf∈Δ, we have(27)dφΓn+1f,Γnf≤∞.
In (16), put b=0. By induction, we have(28)fa−smnfasn≤Ln−1smn−1φma,0.
Then, forL∈0,1,(29)Γn+1f−Γnf=smnsfasn+1−fasn≤Lnφma,0⟶0asn⟶+∞,and then, all conditions of Theorem 3 hold. Furthermore, similarly, mapping ζ holds.
So, theO-sequence Γnf converges to the unique fixed point Dm in the set of k∈Δ:dφΓnf,k<∞, i.e.,(30)Dma=limn⟶∞Γnf=limn⟶∞smnfasn.
Also, forf∈Δ,(31)dφf,Dm≤L1−Ldφf,Γf,and by (29), df,Dm≤Lφma,0. Therefore, Dm satisfies in (21), i.e.,(32)fa−Dma≤L2sm1−Lφma,0.
Similarly,(33)Hma=limn⟶∞Λnζ=limn⟶∞smnζakn,and H satisfies in (20), i.e.,(34)ζa−Hma≤L2sm1−Lφma,0.
We claim that(1)
Hm is the unique desired orthogonally m-homomorphism which satisfies in (20)(2)
Dm is the unique desired orthogonally (m, m)-Hom-derivation which satisfies in (21)
First of all,Dm and H are two m-mappings. In fact, for all a,b∈A with a⊥b and m=1,2,3,4 and using (12), we have(35)ΦmDa,b=limn⟶∞smnΦmfasn,bsn≤limn⟶∞s2mnΦmasn,bsn=0.
So,ΦmDa,b=0. Similarly, ΦmHa,b=0. By the result in [28], Dm and Hm are orthogonally m-mappings.Proof.
of Claim (1). By using (14) and (18), we have(36)Hmuv−HmuHmv=limn⟶∞s2mnζuvs2n−ζusnζvsn≤limn⟶∞s2mnψmusn,vsn=0,for all u,v∈A with u⊥v and m=1,2,3,4.
Therefore,Hm is an orthogonally m-homomorphism.Proof.
of Claim (2). By using (14) and (19), we have(37)Dmuv−ΨmDuv=limn⟶∞s2mnfuvs2n−fusnζvsnm−ζusnmfvsn≤limn⟶∞s2mnψmusn,vsn=0,for all u,v∈A with u⊥v and m=1,2,3,4. So, Dmuv=Ψmuv for all u,v∈A with u⊥v and m=1,2,3,4. Therefore, Dm is an orthogonally (m, m)-Hom-derivation.Corollary 1.
Letθ,p∈ℝ+ with p>m, and f,ζ:A⟶A are mappings such that(38)Φmfa,b≤θap+bp, (39)Φmζa,b≤θap+bp, (40)ζuv−ζuζv≤θup+vp, (41)fuv−Ψmfuv≤θup+vp,for all a,b,u,v∈A with a⊥b and u⊥v; then, there exist unique orthogonally m-homomorphisms Hm:A⟶A and unique orthogonally (m, m)-Hom-derivations Dm:A⟶A such that(42)ζa−Hma≤θ2sp−smap, (43)fa−Dma≤θ2sp−smap.Proof.
By taking(44)φma,b≔θap+bp,ψmu,v≔θup+vp,L=sm−p,and using Theorem 2, we get the desired results.In the next theorem, we prove the hyperstability of orthogonally (m, m)-Hom-derivations for functional equation (8).Theorem 3.
Suppose there exist functionsφm,ψm such that(45)limn⟶∞smnφm0,bsn=0, (46)limn⟶∞s2mnψmukn,vsn=0,for all a,b,u,v∈A with a⊥b and u⊥v. Moreover, assume that f,ζ:A⟶A are mappings such that(47)Φmfa,b≤φm0,b, (48)Φmζa,b≤φm0,b, (49)ζuv−ζuζv≤ψmu,v, (50)fuv−Ψmuv≤ψmu,v,for all a,b,u,v∈A with a⊥b and u⊥v; then, f is an orthogonally (m, m)-Hom-derivation.Proof.
Lettingb=0 in (47), we get fsa=smfa, and by using induction on n∈ℕ, we get(51)fa=smnfasn.
Similarly, we have(52)ζa=smnζasn.
Hence, we have(53)Φmfa,b=smnΦmfasn,bsn≤smnφm0,bsn,for all a,b∈A with a⊥b and n∈ℕ. So, by n⟶∞ in (53) and using (45), we have(54)Φmfa,b=0.
It follows from (49) and (52) that(55)ζuv−ζuζv=s2mnζuvs2n−ζusnζvsn≤s2mnψmusn,vsn,for all u,v∈A with u⊥v and n∈ℕ. Hence, by n⟶∞ in (55) and using (46), we have ζuv=ζuζv for all u,v∈A with u⊥v, and it follows from (50) and (51) that(56)fuv−Ψmfuv=s2mnfuvs2n−fusnζvsnm−ζusnmfvsn≤s2mnψmusn,vsn,for all u,v∈A with u⊥v and n∈ℕ. Hence, by n⟶∞ in (56) and using (46), we have fuv=Ψmfuv. Therefore, f is an orthogonally (m, m)-Hom-derivation.Corollary 2.
Letθ,p∈ℝ+ with p>m, and f,ζ:A⟶A are mappings such that(57)Φmfa,b≤θbp,Φmga,b≤θbp,ζuv−ζuζv≤θup+vp,fuv−Ψmfuv≤θup+vp,for all a,b,u,v∈A with a⊥b and u⊥v; then, f is an (m, m)-Hom-derivation.


## 3. Conclusions

In this paper, the author introduced orthogonally (m, m)-Hom-derivation equations. Using an orthogonally fixed point theorem, it is proved that orthogonally (m, m)-Hom-derivation equations for orthogonally m-mappings on orthogonally Banach algebras can be stable and hyperstable.



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