# Stability and Superstability of Generalized (θ, ϕ)-Derivations in Non-Archimedean Algebras: Fixed Point Theorem via the Additive Cauchy Functional Equation

Authors: M. Eshaghi Gordji; M. B. Ghaemi; G. H. Kim; Badrkhan Alizadeh
Journal: Journal of Applied Mathematics
 (2011)

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## Abstract

LetA be an algebra, and let θ, ϕ be ring automorphisms of A. An additive
mapping H:A→A is called a (θ,ϕ)-derivation if H(xy)=H(x)θ(y)+ϕ(x)H(y) for all x,y∈A. Moreover, an additive mapping F:A→A is said to be a generalized (θ,ϕ)-derivation if there exists a (θ,ϕ)-derivation H:A→A such that F(xy)=F(x)θ(y)+ϕ(x)H(y) for all x,y∈A. In this paper, we investigate the superstability of generalized (θ,ϕ)-derivations in non-Archimedean algebras by using a version of fixed point theorem via Cauchy’s functional equation.


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## Body

## 1. Introduction and Preliminaries

In 1897, Hensel [1] has introduced a normed space which does not have the Archimedean property. It turned out that non-Archimedean spaces have many nice applications [2, 3].A non-Archimedean field is a field𝕂 equipped with a function (valuation) |·| from 𝕂 into [0,∞) such that |r|=0 if and only if r=0,  |rs|=|r||s|, and |r+s|≤max{|r|,|s|} for all r,s∈𝕂. An example of a non-Archimedean valuation is the mapping |·| taking everything but 0 into 1 and |0|=0. This valuation is called trivial (see [4]).Definition 1.1.
LetX be a vector space over a scalar field 𝕂 with a non-Archimedean non-trivial valuation |·|. A function ∥·∥:X→ℝ is a non-Archimedean norm (valuation) if it satisfies the following conditions: (NA1)
∥x∥=0 if and only if x=0,(NA2)
∥rx∥=|r|∥x∥ for all r∈𝕂 and x∈X,(NA3)
∥x+y∥≤max{∥x∥,∥y∥} for all x,y∈X (the strong triangle inequality).A sequence{xm} in a non-Archimedean space is Cauchy’s if and only if {xm+1-xm} converges to zero. By a complete non-Archimedean space, we mean one in which every Cauchy’s sequence is convergent. A non-Archimedean-normed algebra is a non-Archimedean-normed space A with a linear associative multiplication, satisfying ∥xy∥≤∥x∥∥y∥ for all x,y∈A. A non-Archimedean complete normed algebra is called a non-Archimedean Banach’s algebra (see [5]).Definition 1.2.
LetX be a nonempty set, and let d:X×X→[0,∞] satisfy the following properties: (D1)
d(x,y)=0 if and only if x=y,(D2)
d(x,y)=d(y,x) (symmetry),(D3)
d(x,z)≤max{d(x,y),d(y,z)} (strong triangle inequality),for all x,y,z∈X. Then (X,d) is called a non-Archimedean generalized metric space. (X,d) is called complete if every d-Cauchy’s sequence in X is d-convergent.Definition 1.3.
LetA be a non-Archimedean algebra, and let θ, ϕ be ring automorphisms of A. An additive mapping H:A→A is called a (θ,ϕ)-derivation in case H(xy)=H(x)θ(y)+ϕ(x)H(y) holds for all x,y∈A. An additive mapping F:A→A is said to be a generalized (θ,ϕ)-derivation if there exists a (θ,ϕ)-derivation H:A→A such that
(1.1)F(xy)=F(x)θ(y)+ϕ(x)H(y)
for all x,y∈A.We need the following fixed point theorem (see [6, 7]).Theorem 1.4 (Non-Archimedean Alternative Contraction Principle).
Suppose(X,d) is a non-Archimedean generalized complete metric space and  Λ:X→X is a strictly contractive mapping; that is,
(1.2)d(Λx,Λy)≤Ld(x,y)(x,y∈X)
for some L<1. If there exists a nonnegative integer k such that d(Λk+1x,Λkx)<∞ for some x∈X, then the followings are true.(a)
The sequence{Λnx} converges to a fixed point x* of Λ.(b)
x* is a unique fixed point of Λ in
(1.3)X*={y∈X∣d(Λkx,y)<∞}.(c)
Ify∈X*, then
(1.4)d(y,x*)≤d(Λy,y).A functional equation(ξ) is superstable if every approximately solution of (ξ) is an exact solution of it.The stability of functional equations was first introduced by Ulam [8] during his talk before a mathematical colloquium at the University of Wisconsin in 1940. In 1941, Hyers [9] gave a first affirmative answer to the question of Ulam for Banach spaces. In 1978, Rassias [10] generalized the theorem of Hyers by considering the stability problem with unbounded Cauchy’s differences ∥f(x+y)-f(x)-f(y)∥≤ϵ(∥x∥p+∥y∥p),(ϵ>0,p∈[0,1)). Moreover, John Rassias [11–13] investigated the stability of some functional equations when the control function is the product of powers of norms. In 1991, Gajda [14] answered the question for the case p>1, which was raised by Rassias. This new concept is known as the Hyers-Ulam-Rassias or the generalized Hyers-Ulam stability of functional equations ([11–13, 15–35]).In 1992, Gǎvruţa [36] generalized the Rassias theorem as follows.Suppose(G,+) is an ablian group, X is a Banach space, φ:G×G→[0,∞) satisfies(1.5)φ̃(x,y)=12∑n=0∞2-nφ(2nx,2ny)<∞,for allx,y∈G. If f:G→X is a mapping with(1.6)‖f(x+y)-f(x)-f(y)‖≤φ(x,y),for allx,y∈G, then there exists a unique mapping T:G→X such that T(x+y)=T(x)+T(y) and ∥f(x)-T(x)∥≤φ̃(x,x) for all x,y∈G.In 1949, Bourgin [37] proved the following result, which is sometimes called the superstability of ring homomorphisms: suppose that A and B are Banach algebras with unit. If f:A→B is a surjective mapping such that
(1.7)‖f(x+y)-f(x)-f(y)‖≤ϵ,‖f(xy)-f(x)f(y)‖≤δ,
for some ϵ≥0, δ≥0 and for all x,y∈A, then f is a ring homomorphism.The first superstability result concerning derivations between operator algebras was obtained by Šemrl in [38]. Badora [39] proved the superstability of the functional equation f(xy)=xf(y)+f(x)y, where f is a mapping on normed algebra A with unit. Ansari-Piri and Anjidani [40] discussed the superstability of generalized derivations on Banach’s algebras. Recently, Eshaghi Gordji et al. [41] investigated the stability and superstability of higher ring derivations on non-Archimedean Banach’s algebras (see also [42]). In this paper, we investigate the superstability of generalized (θ,ϕ)-derivations on non-Archimedean Banach algebras by using the fixed point methods.


## 2. Non-Archimedean Superstability of Generalized(θ,ϕ)-Derivations

In this paper, we assume thatA is a non-Archimedean Banach’s algebra, with unit over a non-Archimedean field 𝕂, and θ, ϕ are ring automorphisms of A.Theorem 2.1.
Letφ,ψ:A×A→[0,∞) be functions. Suppose that  f:A→A is a mapping such that
(2.1)‖f(x+y)-f(x)-f(y)‖≤φ(x,y),(2.2)‖f(xy)-f(x)θ(y)-ϕ(x)g(y)‖≤ψ(x,y),
for all x,y∈A. If there exist constants K,L<1 and a natural number k∈𝕂,
(2.3)|k|-1φ(kx,ky)≤Lφ(x,y),|k|-1ψ(kx,y),|k|-1ψ(x,ky)≤Kψ(x,y),
for all x,y∈A, then f is a generalized (θ,ϕ)-derivation and g is a (θ,ϕ)-derivation.Proof.
By induction oni, we prove that for each n∈ℕ0, for all x∈A and i≥2,
(2.4)‖f(ix)-if(x)‖≤max⁡{φ(0,0),φ(x,x),φ(2x,x),…,φ((i-1)x,x)}.
Let x=y in (2.1), then
(2.5)‖f(2x)-2f(x)‖≤max⁡{φ(0,0),φ(x,x)},n∈N0,x∈A.
This proves (2.4) for i=2. Let (2.4) hold for i=1,2,…,J. Replacing x by jx and y by x in (2.1) for each n∈ℕ0, and for all x∈A, we get
(2.6)‖f((j+1)x)-f(jx)-f(x)‖≤max⁡{φ(0,0),φ(jx,x)}.
Since
(2.7)f((j+1)x)-f(jx)-f(x)=f((j+1)x)-(j+1)f(x)+(j+1)f(x)-f(jx)-f(x)=f((j+1)x)-(j+1)f(x)+jf(x)-f(jx),
for all x∈A, it follows from induction hypothesis and (2.6) that, for all x∈A,
(2.8)‖f((j+1)x)-(j+1)f(x)‖≤max⁡{‖f((j+1)x)-f(jx)-f(x)‖,‖jf(x)-f(jx)‖}≤max⁡{φ(0,0),φ(x,x),φ(2x,x),…,φ((j)x,x)}.
This proves (2.4) for all i≥2. In particular, for all x∈A,
(2.9)‖f(kx)-kf(x)‖≤Φ(x),
where
(2.10)Φ(x)=max⁡{φ(0,0),φ(x,x),φ(2x,x),…,φ((k-1)x,x)}(x∈A).
Let us define a setX of all functions r:A→A by
(2.11)X={r:A⟶A}
and introduce d on X as follows:
(2.12)d(r,s)=inf⁡{α>0:‖r(x)-s(x)‖≤αΦ(x)∀x∈A}.
It is easy to see that d defines a generalized complete metric on X. Define J:X→X by J(r)(x)=k-1r(kx). Then J is strictly contractive on X, in fact if
(2.13)‖r(x)-s(x)‖≤αΦ(x)(x∈A),
then, by (2.3),
(2.14)‖J(r)(x)-J(s)(x)‖=|k|-1‖r(kx)-s(kx)‖≤α|k|-1Φ(kx)≤LαΦ(x)(x∈A).
It follows that
(2.15)d(J(r),J(s))≤Ld(r,s)(g,h∈X).
Hence, J is strictly contractive mapping with the Lipschitz constant L. By (2.9),
(2.16)‖(Jf)(x)-f(x)‖=‖k-1f(kx)-f(x)‖,|k|-1‖f(kx)-kf(x)‖≤|k|-1Φ(x)(x∈A).
This means that d(J(f),f)≤1/|k|. By Theorem 1.4, J has a unique fixed point h:A→A in the set
(2.17)U={r∈X:d(r,J(f))<∞},
and, for each x∈A,
(2.18)h(x)=lim⁡m→∞Jm(f(x))=lim⁡k-mf(kmx).
Therefore, eachx,y∈A,
(2.19)‖h(x+y)-h(x)-h(y)‖=lim⁡m→∞|k|-m‖f(km(x+y))-f(kmx)-f(kmy)‖≤lim⁡m→∞|k|-mmax⁡{φ(0,0),φ(knx,kny)}≤lim⁡m→∞Lmφ(x,y)=0.
This shows that h is additive.
Replacingy by kny in (2.2), we get
(2.20)‖f(knxy)-f(x)θ(kny)-ϕ(x)g(kny)‖≤ψ(x,kny),
and so
(2.21)‖f(knxy)kn-f(x)θ(y)-ϕ(x)g(kny)kn‖≤1|k|nψ(x,kny)≤Knψ(x,y),
for all x,y∈A and each n∈ℕ. By taking n→∞, we have
(2.22)h(xy)=f(x)θ(y)+lim⁡n→∞ϕ(x)g(kny)kn,
for all x,y∈A.
Fixm∈ℕ. By (2.22), we have
(2.23)f(kmx)θ(y)=h(kmxy)-lim⁡n→∞ϕ(kmx)(g(kny)kn)=f(x)θ(kmy)+lim⁡n→∞ϕ(x)(g(knkmx)kn)-kmlim⁡n→∞ϕ(x)(g(knx)kn)=kmf(x)θ(y)+kmlim⁡n→∞ϕ(x)(g(kn+mx)kn+m)-kmlim⁡n→∞ϕ(x)(g(knx)kn)=kmf(x)θ(y),
for all x,y∈A. Then f(x)θ(y)=(f(kmx)/km)θ(y) for all x,y∈A and each m∈ℕ, and so, by taking m→∞, we have f(x)θ(y)=h(x)θ(x). Now we obtain h=f, since A is with unit. Replacing x by knx in (2.2), we obtain
(2.24)‖f(kn(xy))-f(knx)θ(y)-ϕ(knx)g(y)‖≤ψ(knx,y),
and; hence,
(2.25)‖f(knxy)kn-f(knx)knθ(y)-ϕ(x)g(y)‖≤1|k|nψ(knx,y)≤Knψ(x,y),
for all x,y∈A and each n∈ℕ. Sending n to infinite, we have
(2.26)f(xy)=f(x)θ(y)+ϕ(x)g(y).
By (2.26), we get
(2.27)ϕ(z)g(xy)=f(zxy)-f(z)θ(xy)=f(zx)θ(y)+ϕ(zx)g(y)-f(z)θ(xy)=[f(z)θ(x)+ϕ(z)g(x)]θ(y)+ϕ(zx)g(y)-f(z)θ(xy)=ϕ(z)[g(x)θ(y)+ϕ(x)g(y)],
for all x,y,z∈A. Therefore, we have g(xy)=g(x)θ(y)+ϕ(x)g(y).
Sincef(xy)=f(x)θ(y)+ϕ(x)g(y),  f is additive, and A is with unit, g is additive.The proof of the following theorem is similar to that in Theorem2.1; hence, it is omitted.Theorem 2.2.
Letφ,ψ:A×A→[0,∞) be functions. Suppose that  f:A→A and  g:A→A are mappings such that
(2.28)‖f(x+y)-f(x)-f(y)‖≤φ(x,y),‖f(xy)-xf(y)-g(x)y‖≤ψ(x,y),
for all x,y∈A. If there exists constants K,L<1 and a natural number k∈𝕂,
(2.29)|k|φ(k-1x,k-1y)≤Lφ(x,y),|k|ψ(k-1x,y),|k|ψ(x,k-1y)≤Kψ(x,y),
for all x,y∈A, then f is a generalized (θ,ϕ)-derivation and g is a (θ,ϕ)-derivation.In the following corollariesℚp is the field of p-adic numbers.Corollary 2.3.
LetA be a non-Archimedean Banach algebra over  ℚp,  ε>0, and let  p1,p2∈(1,∞). Suppose that
(2.30)‖f(x+y)-f(x)-f(y)‖≤ε(‖x‖p1‖y‖p2),‖f(xy)-xf(y)-g(x)y‖≤ε(‖x‖p1‖y‖p2),
for all x,y∈A. Then f is a generalized (θ,ϕ)-derivation and g is a (θ,ϕ)-derivation.Proof.
Letφ(x,y)=ψ(x,y)=ε(∥x∥p1∥y∥p2) for all x,y∈A; then
(2.31)|p|-1φ(px,py)=|p|p1+p2-1ε(‖x‖p1‖y‖p2),|p|-1φ(px,y)=|p|p1-1ε(‖x‖p1‖y‖p2),|p|-1φ(x,py)=|p|p2-1ε(‖x‖p1‖y‖p2).
Put
(2.32)L=K=max⁡{|p|p1-1,|p|p2-1,|p|p1+p2-1}=max⁡{p1-p1,p1-p2,p1-p1-p2}.
So, by Theorem 2.1, f is a generalized (θ,ϕ)-derivation and g is a (θ,ϕ)-derivation.Corollary 2.4.
LetA be a non-Archimedean Banach algebra over  ℚp,  ε>0, and let  p1,p2,p1+p2∈(-∞,1). Suppose that
(2.33)‖f(x+y)-f(x)-f(y)‖≤ε(‖x‖p1‖y‖p2),‖f(xy)-xf(y)-g(x)y‖≤ε(‖x‖p1‖y‖p2),
for all x,y∈A. Then f is a generalized (θ,ϕ)-derivation and g is a (θ,ϕ)-derivation.Proof.
Letφ(x,y)=ψ(x,y)=ε(∥x∥p1∥y∥p2) for all x,y∈A, then
(2.34)|p|φ(p-1x,p-1y)=|p|1-p1-p2ε(‖x‖p1‖y‖p2),|p|φ(p-1x,y)=|p|1-p1ε(‖x‖p1‖y‖p2),|p|φ(x,p-1y)=|p|1-p2ε(‖x‖p1‖y‖p2).
Put
(2.35)L=K=max⁡{|p|1-p1,|p|1-p2,|p|1-p1-p2}=max⁡{pp1-1,pp2-1,pp1+p2-1}.
So, by Theorem 2.2, f is a generalized (θ,ϕ)-derivation and g is a (θ,ϕ)-derivation.Similarly, we can obtain the following results.Corollary 2.5.
LetA be a non-Archimedean Banach’s algebra over  ℚp,  ε>0, δ>0, and let  p1,p2∈(1,∞). Suppose that
(2.36)‖f(x+y)-f(x)-f(y)‖≤ε(‖x‖p1+‖y‖p2),‖f(xy)-xf(y)-g(x)y‖≤δ(‖x‖p1‖y‖p2),
for all x,y∈A. Then f is a generalized (θ,ϕ)-derivation and g is a (θ,ϕ)-derivation.Corollary 2.6.
LetA be a non-Archimedean Banach’s algebra over  ℚp,  ε>0, δ>0, and let  p1,p2∈(1,∞). Suppose that
(2.37)max⁡{‖f(x+y)-f(x)-f(y)‖,‖f(xy)-xf(y)-g(x)y‖}≤εmin⁡{(‖x‖p1+‖y‖p2),‖x‖p1‖y‖p2},
for all x,y∈A. Then f is a generalized (θ,ϕ)-derivation and g is a (θ,ϕ)-derivation.Corollary 2.7.
LetA be a non-Archimedean Banach’s algebra over  ℚp, ε>0,δ>0, and let  p1,p2,p1+p2∈(-∞,1). Suppose that
(2.38)‖f(x+y)-f(x)-f(y)‖≤ε(‖x‖p1+‖y‖p2),‖f(xy)-xf(y)-g(x)y‖≤δ(‖x‖p1‖y‖p2),
for all x,y∈A. Then f is a generalized (θ,ϕ)-derivation and g is a (θ,ϕ)-derivation.



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