# Nearly Ternary Quadratic Higher Derivations on Non-Archimedean Ternary Banach Algebras: A Fixed Point Approach

Authors: M. Eshaghi Gordji; M. B. Ghaemi; J. M. Rassias; Badrkhan Alizadeh
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We investigate the stability and superstability of ternary quadratic higher derivations in non-Archimedean ternary algebras by using a version of fixed point theorem via quadratic functional equation.


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## Body

## 1. Introduction

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. Clearly, |1|=|-1|=1 and |n|≤1 for all n∈ℕ. 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 [4–12]).Definition 1.1.
LetX be a vector space over a scalar field 𝕂 with a non-Archimedean nontrivial 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).Then, (X,∥·∥) is called a non-Archimedean normed space.We note the inequality(1.1)‖xm-xl‖≤max⁡⁡{‖xj+1-xj‖:l≤j≤m-1}(m>l),
where a sequence {xm} is Cauchy if and only if {xm+1-xm} converges to zero in a non-Archimedean space. By a complete non-Archimedean space, we mean one in which every Cauchy sequence is convergent.The most important examples of non-Archimedean spaces arep-adic numbers. A key property of p-adic numbers is that they do not satisfy the Archimedean axiom “for x,y>0, there exists n∈ℕ such that x<ny.”Letp be a prime number. For any nonzero rational number x=(a/b)pnx such that a and b are integers not divisible by p, define the p-adic absolute value |x|p:=p-nx. Then |·| is a non-Archimedean norm on ℚ. The completion of ℚ with respect to |·| is denoted by ℚp which is called the p-adic number field.Note that ifp≥3, then |2n|=1 in for each integer n.Definition 1.2.
LetX be a nonempty set and 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 sequence in X is d-convergent.Example 1.3 (see [13]).
For each nonempty setX, define
(1.2)d(x,y)={0ifx=y∞ifx≠y.
Then d is a generalized non-Archimedean metric on X.A non-Archimedean ternary algebra is a non-Archimedean vector space over a non-Archimedean field𝕂, endowed with a linear mapping, the so-called a ternary product, (x,y,z)→[xyz] of A×A×A into A such that [[xyz]tu]=[x[yzt]u]=[xy[ztu]] for all x,y,z,t,u∈A. If (A,·) is a usual binary non-Archimedean algebra, then an induced ternary multiplication can be defined by [xyz]=(x·y)·z. Hence, the non-Archimedean algebra is a natural generalization of the binary case. A normed non-Archimedean ternary algebra is a non-Archimedean ternary algebra with a norm ∥·∥ such that ∥[xyz]∥≤∥x∥∥y∥∥z∥ for x,y,z∈A. A Banach non-Archimedean ternary algebra is a normed non-Archimedean ternary algebra such that the normed non-Archimedean vector space with norm ∥·∥ is complete.The ternary algebras have been studied in nineteenth century. Their structures appeared more or less naturally in various domains of mathematical physics and data processing. The discovery of the Nambu mechanics and the progress of quantum mechanics [14], as well as work of Okubo [15] on Yang-Baxter equation gave a significant development on ternary algebras (see also [16–20]).We say that a functional equation(ξ) is stable if any function g satisfying the equation (ξ)   approximately is near to true solution of (ξ). We say that a functional equation (ξ) is superstable if every approximately solution of (ξ) is an exact solution of it (see [21]).The stability of functional equations was first introduced by Ulam [22] in 1940. In 1941, Hyers [23] gave a first affirmative answer to the question of Ulam for Banach spaces. In 1978, Rassias [24] generalized the theorem of Hyers for linear mappings by considering the stability problem with unbounded Cauchy differences ∥f(x+y)-f(x)-f(y)∥≤ϵ(∥x∥p+∥y∥p),(ϵ>0,p∈[0,1)). In 1991, Gajda [25] answered the question for the case p>1, which was raised by Rassias. This new concept is known as generalized Hyers-Ulam stability of functional equations (see [6–12, 17, 21, 25–58]). In 1982–1994, Rassias (see [59–66]) solved the Ulam problem for different mappings and for many Euler-Lagrange type quadratic mappings, by involving a product of different powers of norms. In addition, Rassias considered the mixed product sum of powers of norms control function [67].In 1949, Bourgin [41] 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(1.3)‖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.Badora [68] and Miura et al. [69] proved the Ulam-Hyers stability and the Isaac and Rassias type stability of derivations [30].The functional equation(1.4)f(x+y)+f(x-y)=2f(x)+2f(y),
is related to a symmetric biadditive function [2, 27]. It is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic equation (1.4) is said to be a quadratic mapping. It is well known that a mapping f between real vector spaces is quadratic if and only if there exists a unique symmetric bi-additive mapping B1 such that f(x)=B1(x,x) for all x. The bi-additive mapping B1 is given by B1(x,y)=(1/4)(f(x+y)-f(x-y)).The stability problem of quadratic functional equation (1.4) was proved by Skof [37] for functions f:A→B, where A is normed space and B Banach space (see also [42, 43]).Definition 1.4.
A mappingH:A→B is called a ternary quadratic homomorphism between non-Archimedean ternary algebras A,B if (1)
H is a quadratic function,(2)
H([xyz])=[H(x)H(y)H(z)], for all x,y,z∈A.For instance, letA be commutative ternary algebra, then the function f:A→A defined by f(a)=a2(a∈A) is a quadratic homomorphism.Definition 1.5.
A mappingD:A→A is called a non-Archimedean ternary quadratic derivation on ternary non-Archimedean algebra A if (1)
D is a quadratic function,(2)
D([xyz])=[D(x)y2z2]+[x2D(y)z2]+[x2y2D(z)], for all x,y,z∈A.For example, consider the algebra of2×2 matrices (1.5)A={[c1c200]:c1,c2∈C},
then it is easy to see that 𝒜 is a ternary algebra. Moreover, the function f:𝒜→𝒜 defined by (1.6)f([c1c200])=[0c2200],
is a ternary quadratic derivation.We note that ternary quadratic derivations and ternary ring derivations are different.As another example, Let𝒜 be a Banach algebra. Then we take(1.7)T=[0AA00A000],
where 𝒯 is a ternary Banach algebra equipped with the usual matrix-like operations and the following norm:(1.8)‖[0ab00c000]‖=‖a‖+‖b‖+‖c‖(a,b,c∈A).
It is known that(1.9)T*=[oA*A*00A*000]
is the dual of 𝒯 under the following norm:(1.10)‖[0fg00h000]‖=max⁡⁡{‖f‖,‖g‖,‖h‖}(f,g,h∈A*).
Let the left module action of 𝒯 on 𝒯* be trivial, and let the right module action of 𝒯 on 𝒯* is defined as follows:(1.11)〈[0fg00h000][0ab00c000],[0xy00z000]〉=g(acz),
for all f,g,h∈𝒜*,a,b,c,x,y,z∈𝒜. Then 𝒯* is a Banach 𝒯-module. Let
(1.12)[0kg00h000]∈T*.
We define D:𝒯→𝒯* by(1.13)D([0ab00c000])=[0kg00h000][00ac000000](a,b,c∈A).
Then it is easy to show that D is a ternary quadratic derivation from 𝒯 into 𝒯*.Definition 1.6.
LetA,B be two non-Archimedean ternary normed algebras over a non-Archimedean field 𝕂. For m∈ℕ∪{0}=ℕ0, a sequence H={h0,h1,…,hm} (resp., H={h0,h1,…,hn,…}) of quadratic mappings from A into B is called a ternary quadratic higher derivation of rank m (resp. infinite rank) from A into B if
(1.14)hn[xyz]=∑i+j+k=n[hi(x)hj(y)hk(z)],
holds for each n∈{0,1,…,m} (resp., n∈ℕ0) and all x,y,z∈A. The ternary quadratic higher derivation H on A is said to be strong if h0(x)=x2 for all x∈A. Of course, a ternary quadratic higher derivation of rank 0 from A into B (resp., a strong ternary quadratic higher derivation of rank 1 on A) is a ternary quadratic homomorphism (resp., a ternary quadratic derivation). So a ternary quadratic higher derivation is a generalization of both a a ternary quadratic homomorphism and a ternary quadratic derivation.Recently, Jung and Chang [70] have investigated the stability and superstability of higher derivations on rings. More recently, the first author of the present paper [46] has investigated the stability of homomorphisms and derivations on non-Archimedean Banach algebras, also Eshaghi Gordji and Alizadeh [47] by using fixed point methods, established the stability and superstability of derivations on non-Archimedean Banach algebras. In this paper, by using fixed point methods, we establish the stability and superstability of quadratic ternary higher derivations on non-Archimedean Banach ternary algebras.We need the following fixed point theorem (see [13, 71]).Theorem 1.7 (non-Archimedean alternative contraction principle).
Suppose that(X,d) is a non-Archimedean generalized complete metric space and Λ:X→X is a strictly contractive mapping, that is,
(1.15)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
(1.16)X*={y∈X∣d(Λkx,y)<∞},(c)
Ify∈X*, then
(1.17)d(y,x*)≤d(Λy,y).


## 2. Main Results

In this section,A denotes a non-Archimedean ternary normed algebra over a non-Archimedean field 𝕂, and B is a non-Archimedean Banach ternary algebra over 𝕂.Theorem 2.1.
Letφ:A×A×A→[0,∞) be a function. Suppose that F={f0,f1,…,fn,…} be a sequence of mappings from A into B such that for each n∈ℕ0,fn(0)=0,
(2.1)‖fn(x+y)+fn(x-y)-2fn(x)-2fn(y)‖≤φ(x,y,o),(2.2)‖fn[xyz]-∑i+j+t=n[fi(x)fj(y)ft(z)]‖≤φ(x,y,z),
for all x,y,z∈A. Suppose that there exist a natural number k∈𝕂 and L∈(0,1), such that
(2.3)|k|2φ(k-1x,k-1y,k-1z)≤Lφ(x,y,z),
for all x,y,z∈A. Then there exists a unique ternary quadratic higher derivation H={h0,h1,…,hn,…} of any rank from A into B such that for each n∈ℕ0,
(2.4)‖fn(x)-hn(x)‖≤Lψ(x)|k|2,
for all x∈A, where
(2.5)ψ(x)=max⁡⁡{φ(0,0,0),φ(x,x,0),φ(2x,x,0),…,φ((k-1)x,x,0)}(x∈A).Proof.
By induction oni, one can show that for each n∈ℕ0, for all x∈A and i≥2,
(2.6)‖fn(ix)-i2fn(x)‖≤max⁡⁡{φ(0,0,0),φ(x,x,0),φ(2x,x,0),…,φ((i-1)x,x,0)}.
Let x=y in (2.1), then
(2.7)‖fn(2x)-2fn(x)‖≤max⁡⁡{φ(0,0,0),φ(x,x,0)},n∈N0,x∈A.
This proves (2.6) for i=2. Let (2.6) holds for i=1,2,…J. Replacing x with jx and y with x in (2.1) for each n∈ℕ0, and for all x∈A, we get
(2.8)‖fn((j+1)x)+fn((j-1)x)-2fn(jx)-2fn(x)‖≤max⁡⁡{φ(0,0,0),φ(jx,x,0)}.
Since
(2.9)fn((j+1)x)+fn((j-1)x)-2fn(jx)-2fn(x)=fn((j+1)x)-(j+1)2fn(x)+fn((j-1)x)-(j-1)2fn(jx)-2[fn(jx)-j2fn(x)],
for all x∈A, it follows from induction hypothesis and (2.8) that for all x∈A,
(2.10)‖fn((j+1)x)-(j+1)2fn(x)‖≤max⁡⁡{(j-1)2‖fn((j+1)x)+fn((j-1)x)-2fn(jx)-2fn(x)‖,‖fn((j-1)x)-(j-1)2fn(x)‖,|2|‖j2fn(x)-fn(jx)‖}≤max⁡{φ(0,0,0),φ(x,x,0),φ(2x,x,0),…,φ((j)x,x,0)}.
This proves (2.6) for all i≥2. In particular,
(2.11)‖fn(kx)-k2fn(x)‖≤ψ(x)(x∈A).
Replacing x with k-1x in (2.11), it follows that for each x∈A,
(2.12)‖fn(x)-k2fn(k-1x)‖≤ψ(k-1x)(x∈A).
Let(2.13)X={h:A⟶B},d(g,h)=inf⁡⁡{α>0:‖g(x)-h(x)‖≤αψ(x)∀x∈A},f,g∈X.
It is easy to see that d defines a generalized non-Archimedean complete metric on X. Define J:X→X by J(h)(x)=k2h(k-1x). Then J is strictly contractive on X, in fact if
(2.14)‖g(x)-h(x)‖≤αψ(x),(x∈A),
then by (2.3),
(2.15)‖J(g)(x)-J(h)(x)‖=|k|2‖g(k-1x)-h(k-1x)‖≤α|k|2ψ(k-1x)≤Lαψ(x),(x∈A).
It follows that
(2.16)d(J(g),J(h))≤Ld(g,h)(g,h∈X).
Hence, J is strictly contractive mapping with Lipschitz constant L. By (2.12),
(2.17)‖(Jfn)(x)-fn(x)‖=‖k2fn(k-1x)-fn(x)‖≤ψ(k-1x)≤|k|-2Lψ(x)(x∈A).
This means that d(J(fn),fn)≤(L/|k|2). By Theorem 1.7, J has a unique fixed point hn:A→B in the set
(2.18)Un={gn∈X:d(gn,J(fn))<∞},
and for each x∈A,
(2.19)hn(x)=lim⁡m→∞Jm(fn(x))=lim⁡k2mfn(k-mx).
Therefore,(2.20)‖hn(x+y)+hn(x-y)-2hn(x)-2hn(y)‖=lim⁡m→∞⁡|k|2m‖fn(k-m(x+y))+fn(k-m(x-y))-2fn(k-mx)-2fn(k-my)‖≤lim⁡m→∞⁡|k|2mmax⁡⁡{φ(0,0,0),φ(k-mx,k-my,0)}≤lim⁡m→∞⁡Lmφ(x,y,0)=0,
for all x,y∈A. This shows that hn is quadratic. It follows from Theorem 1.7 that
(2.21)d(fn,hn)≤d(J(fn),fn),
that is,
(2.22)‖fn(x)-hn(x)‖≤Lψ(x)|k|2(x∈A,n∈N0).
The inequality (2.2) implies that the function Dn:A×A×A→B, defined by
(2.23)Dn(x,y,z)=fn[xyz]-∑i+j+t=n[fi(x)fj(y)ft(z)]
for each n∈ℕ0 and for all x,y,z∈A, is bounded.
By (2.3), we see that
(2.24)‖k2mDn(k-mx,k-my,k-m,k-mz)‖=|k|2m‖Dn(k-mx,k-my,k-mz)‖≤|k|2mφ(k-mx,k-my,k-mz)≤Lmφ(x,y,z),x,y∈A.
Taking the limit as m→∞, we obtain
(2.25)lim⁡m→∞k2mDn(k-mx,k-my,k-mz)=0,
for each n∈ℕ0 and for all x,y,z∈A. Now, using (2.19), (2.23), and (2.25), we have
(2.26)hn[xyz]=lim⁡m→∞k2mfn(k-m[xyz])=lim⁡r→∞k6rfn[(k-3r)xyz]=lim⁡r→∞k6rfn[(k-rx)(k-ry)(k-rz)]=lim⁡r→∞(∑i+j+t=n[k2rfi(k-rx)k2rfj(k-ry)k2rft(k-rz)]+k6rDn(k-rx,k-ry,k-rz))=∑i+j+t=n[lim⁡r→∞k2rfi(k-rx)lim⁡r→∞k2rfj(k-ry)lim⁡r→∞k2rft(k-rz)]+lim⁡r→∞k6rDn(k-rx,k-ry,k-rz)=∑i+j+t=n[hi(x)hj(y)ht(z)],
for all n∈ℕ0 and all x,y,z∈A. It follows that H={h0,h1,…,hn,…} satisfies
(2.27)hn[xyz]=∑i+j+t=n[hi(x)hj(y)htz],
for all n∈ℕ0 and all x,y,z∈A. This completes the proof of the theorem.By a same method as above, we have the following theorem.Theorem 2.2.
Letφ:A×A×A→[0,∞) be a function. Suppose that F={f0,f1,…,fn,…} is a sequence of mappings from A into B such that for each n∈ℕ0,fn(0)=0,
(2.28)‖f(x+y)+f(x-y)-2f(x)-2f(x)‖≤φ(x,y,0),‖fn([xyz])-∑i+j+t=n[fi(x)fj(y)ft(z)]‖≤φ(x,y,z),
for all x,y,z∈A. If there exist k∈𝕂 and 0<L<1 such that
(2.29)|k|-2φ(kx,ky,kz)≤Lφ(x,y,z)
for all x,y,z∈A, then there exists a unique ternary quadratic higher derivation H={h0,h1,…,hn,…} of any rank from A into B such that for each n∈ℕ0,
(2.30)‖fn(x)-hn(x)‖≤ψ(x)|k|2,
for all x∈A, where
(2.31)ψ(x)=max⁡⁡{φ(0,0,0),φ(x,x,0),φ(2x,x,0),…,φ((k-1)x,x,0)}(x∈A).Proof.
By a same reasoning of Theorem2.1, for all n∈ℕ0, and all x∈A and i≥2, we have
(2.32)‖fn(ix)-i2fn(x)‖≤max⁡⁡{φ(0,0,0),φ(x,x,0),φ(2x,x,0),…,φ((i-1)x,x,0)}.
In particular,
(2.33)‖fn(kx)-k2fn(x)‖≤ψ(x)(x∈A).
Let
(2.34)X={h:A⟶B}d(g,h)=inf⁡⁡{α>0:‖g(x)-h(x)‖≤αψ(x)∀x∈A}.
It is easy to see that d defines a generalized complete metric on X. Define J:X→X by J(h)(x)=k-2h(kx). Then J is strictly contractive on X, in fact if
(2.35)‖g(x)-h(x)‖≤αψ(x),(x∈A),
then by (2.29),
(2.36)‖J(g)(x)-J(h)(x)‖=|k|-2‖g(kx)-h(kx)‖≤α|k|-2ψ(kx)≤Lαψ(x),)(x∈A).
It follows that
(2.37)d(J(g),J(h))≤Ld(g,h)(g,h∈X).
Hence, J is strictly contractive mapping with Lipschitz constant L. By (2.33),
(2.38)‖(Jfn)(x)-fn(x)‖=‖k-2fn(kx)-fn(x)‖,|k|-2‖fn(kx)-k2fn(x)‖≤|k|-2ψ(x)(x∈A).
This means that d(J(fn),fn)≤(1/|k|2). By Theorem 1.7, J has a unique fixed point hn:A→B in the set
(2.39)Un={gn∈X:d(gn,J(fn))<∞},
and for each x∈A,
(2.40)hn(x)=lim⁡m→∞Jm(fn(x))=lim⁡k-2mfn(kmx).
Therefore,(2.41)‖hn(x+y)+hn(x-y)-2hn(x)-2hn(y)‖)=lim⁡m→∞⁡|k|-2m‖fn(km(x+y))+fn(x-y)-2fn(kmx)-2fn(kmy)‖≤lim⁡m→∞⁡|k|-2mmax⁡⁡{φ(0,0,0),φ(kx,ky,0)}≤lim⁡m→∞⁡Lmφ(x,y,0)=0,
for all x,y∈A. This shows that hn is quadratic. Again by Theorem 1.7, we have
(2.42)d(fn,hn)≤d(J(fn),fn),
that is,
(2.43)‖fn(x)-hn(x)‖≤ψ(x)|k|2(x∈A,n∈N0).
The rest of proof is similar to the proof of Theorem 2.1.In the following corollaries,ℚp is the field of p-adic numbers, where p>2 is a prime number.As a consequence of Theorem2.1, we show the Hyers-Ulam-Rassias stability of ternary quadratic higher derivations.Corollary 2.3.
Let(A,∥·∥A) be a non-Archimedean normed ternary algebra over ℚp and (B,∥·∥B) be a non-Archimedean Banach ternary algebra over ℚp. Assume that F={f0,f1,…,fn,…} is a sequence of mappings from A into B such that for each n∈ℕ0, and for all x,y∈A,
(2.44)‖fn(x+y)+fn(x-y)-2fn(x)-2fn(y)‖B≤θ(‖x‖Ar+‖y‖Ar),(2.45)‖fn[xyz]-∑i+j+t=n[fi(x)fj(y)ft(z)]‖B≤θ(‖x‖Ar+‖y‖Ar+‖z‖Ar),
for some θ>0 and r<2. Then there exists a unique ternary quadratic higher derivation H={h0,h1,…,hn,…} of any rank from A into B such that
(2.46)‖fn(x)-hn(x)‖≤2θpr‖x‖Ar(x∈A),
for all n∈ℕ0.Proof.
By (2.44), we have fn(0)=0 for all n∈ℕ0. Let φ(x,y,z)=θ(∥x∥Ar+∥y∥Ar+∥z∥Ar) for all x,y,z∈A, then
(2.47)|p|2φ(p-1x,p-1y,p-1z)=θ|p|2-r(‖x‖Ar+‖y‖Ar+‖z‖Ar)=pr-2φ(x,y,z)(x,y,z∈A).
Moreover,
(2.48)ψ(x)=max⁡⁡{φ(0,0,0),φ(x,x,0),φ(2x,x,0),…,φ((p-1)x,x,0)}=2θ‖x‖Ar(x∈A).
Put L=pr-2. By Theorem 2.1, there exists a sequence H={h0,h1,…,hn,…} with the required properties.The following corollary is similar to Corollary2.3 for the case where r>2.Corollary 2.4.
Let(A,∥·∥A) be a be a non-Archimedean normed ternary algebra over ℚp and let (B,∥·∥B) be a non-Archimedean Banach ternary algebra over ℚp. Assume that F={f0,f1,…,fn,…} is a sequence of mappings from A into B such that for each n∈ℕ0, for all x,y∈A,
(2.49)‖fn(x+y)+fn(x-y)-2fn(x)-2fn(y)‖B≤θ(‖x‖Ar+‖y‖Ar),(2.50)‖fn([xyz]-∑i+j+t=n[fi(x)fj(y)ft(z)]‖B≤θ(‖x‖Ar+‖y‖Ar+‖z‖Ar),
for some θ>0 and r>2. Then there exists a unique ternary quadratic higher derivation H={h0,h1,…,hn,…} of any rank from A into B such that
(2.51)‖fn(x)-hn(x)‖≤2θ|p|-2‖x‖Ar(x∈A),
for all n∈ℕ0.Proof.
By (2.49), fn(0)=0 for all n∈ℕ0. Let φ(x,y)=θ(∥x∥Ar+∥y∥Ar+∥z∥Ar) for all x,y,z∈A, then
(2.52)|p|-2φ(px,py,pz)=θ|p|r-2(‖x‖Ar+‖y‖Ar+‖z‖Ar)=p2-rφ(x,y,z)(x,y,z∈A).
Moreover,
(2.53)ψ(x)=max⁡⁡{φ(0,0,0),φ(x,x,0),φ(2x,x,0),…,φ((p-1)x,x,0)}=2θ‖x‖Ar(x∈A).
Put L=p2-r. By Theorem 2.2, there exists a sequence H={h0,h1,…,hn,…} with the required properties.Corollary 2.5.
Let(A,∥·∥A) be a non-Archimedean normed ternary algebra over ℚp, and let (B,∥·∥B) be a non-Archimedean Banach ternary algebra over ℚp. Assume that F={f0,f1,…,fn,…} is a sequence of mappings from A into B such that for each n∈ℕ0, and for all x,y∈A,
(2.54)‖fn(x+y)+fn(x-y)-2fn(x)-2fn(y)‖B≤max⁡⁡{‖x‖Ar,‖y‖Ar},‖fn[xyz]-∑i+j+t=n[fi(x)fj(y)ft(z)]‖B≤max⁡⁡{‖x‖Ar,‖y‖Ar,‖z‖Ar},
for some r>2. Then there exists a unique ternary quadratic higher derivation H={h0,h1,…,hn,…} of any rank from A into B such that
(2.55)‖fn(x)-hn(x)‖≤pr‖x‖Ar(x∈A),
for all n∈ℕ0.Corollary 2.6.
Let(A,∥·∥A) be a non-Archimedean normed ternary algebra over ℚp and let (B,∥·∥B) be a non-Archimedean Banach ternary algebra over ℚp. Assume that F={f0,f1,…,fn,…} is a sequence of mappings from A into B such that for each n∈ℕ0, and for all x,y∈A,
(2.56)‖fn(x+y)+fn(x-y)-2fn(x)-2fn(y)‖B≤max⁡⁡{‖x‖Ar,‖y‖Ar},‖fn[xyz]-∑i+j+t=n[fi(x)fj(y)ft(z)]‖B≤max⁡⁡{‖x‖Ar,‖y‖Ar,‖z‖Ar},
for some r<2. Then there exists a unique ternary quadratic higher derivation H={h0,h1,…,hn,…} of any rank from A into B such that
(2.57)‖fn(x)-hn(x)‖≤|p|-2‖x‖Ar(x∈A),
for all n∈ℕ0.As a consequence of Theorem2.1, we have the following superstability results for ternary quadratic higher derivations.Corollary 2.7.
Letr,s be two real numbers such that r+s<-2. Let (A,∥·∥A) be a non-Archimedean normed ternary algebra over ℚp and (B,∥·∥B) be a non-Archimedean Banach ternary algebra over ℚp. Assume that F={f0,f1,…,fn,…} is a sequence of mappings from A into B such that for each n∈ℕ0, and for all x,y∈A,
(2.58)‖fn(x+y)+fn(x-y)-2fn(x)-2fn(y)‖B≤θ(‖x‖Ar+‖y‖Ar),‖fn[xyz]-∑i+j+t=n[fi(x)fj(y)ft(z)]‖B≤θ(‖x‖Ar+‖y‖Ar)‖z‖As,
for some θ>0. Then F is a ternary quadratic higher derivation.Proof.
It follows from Theorem2.1 by putting φ(x,y,z)=θ(∥x∥Ar+∥y∥Ar)∥z∥As for all x,y,z∈A.We can prove a same result with conditionr+s>-2 by using of Theorem 2.2.Corollary 2.8.
Letr,s,t be real numbers such that r+s+t<-2. Let (A,∥·∥A) be a non-Archimedean normed ternary algebra over ℚp and (B,∥·∥B) be a non-Archimedean Banach ternary algebra over ℚp. Assume that F={f0,f1,…,fn,…} is a sequence of mappings from A into B such that for each n∈ℕ0, and for all x,y∈A,
(2.59)‖fn(x+y)+fn(x-y)-2fn(x)-2fn(y)‖B≤θ(‖x‖Ar+‖y‖Ar),‖fn[xyz]-∑i+j+t=n[fi(x)fj(y)ft(z)]‖B≤θ(‖x‖Ar‖y‖As‖z‖As),
for some θ>0. Then F is a ternary quadratic higher derivation.Proof.
It follows from Theorem2.1 by putting φ(x,y,z)=θ(∥x∥Ar∥y∥Ar∥z∥As) for all x,y,z∈A.Moreover, we can prove a same result with conditionr+s+t>-2, by applying Theorem 2.2.



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*Source: 417187-2011-11-24.xml*