We investigate the superstability of generalized derivations in non-Archimedean algebras by using a version of fixed point theorem via Cauchy functional equation.

1. Introduction

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 [1] during his talk before a Mathematical Colloquium at the University of Wisconsin in 1940.

Given a metric group G(·,ρ), a number ɛ>0, and a mapping f:G→G which satisfies the inequality ρ(f(x·y),f(x)·f(y))≤ɛ for all x,y in G, does there exist an automorphism a of G and a constant k>0, depending only on G such that ρ(a(x),f(x))≤kɛ for all x∈G?

If the answer is affirmative, we would call the equation a(x·y)=a(x)·a(y) of automorphism is stable. In 1941, Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces. In 1978, Rassias [3] generalized the theorem of Hyers 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 [4] answered the question for the case p>1, which was raised by Rassias. This new concept is known as Hyers-Ulam-Rassias or generalized Hyers-Ulam stability of functional equations [5, 6].

In 1992, Găvruţa [7] generalized the Th. M. Rassias Theorem as follows.

Suppose that (G,+) is an ablian group, X is a Banach space φ:G×G→[0,∞) which satisfies
φ̃(x,y)=12∑n=0∞2-nφ(2nx,2ny)<∞,
for all x,y∈G. If f:G→X is a mapping with
‖f(x+y)-f(x)-f(y)‖≤φ(x,y),
for all x,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 [8] 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
‖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 [9] and Miura et al. [10] proved the Ulam-Hyers stability and the Isac and Rassias-type stability of derivations [11] (see also [12, 13]); Savadkouhi et al. [14] have contributed works regarding the stability of ternary Jordan derivations. Jung and Chang [15] investigated the stability and superstability of higher derivations on rings. Recently, Ansari-Piri and Anjidani [16] discussed the superstability of generalized derivations on Banach algebras. In this paper, we investigate the superstability of generalized derivations on non-Archimedean Banach algebras by using the fixed point methods.

2. Preliminaries

In 1897, Hensel [17] has introduced a normed space which does not have the Archimedean property. It turned out that non-Archimedean spaces have many nice applications [18, 19].

A non-Archimedean field is a field 𝕂 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 (see [20, 21]).

Definition 2.1.

Let X 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:

∥x∥=0 if and only if x=0,

∥rx∥=|r|∥x∥ for all r∈𝕂 and x∈X,

∥x+y∥≤max{∥x∥,∥y∥} for all x,y∈X (the strong triangle inequality).

A sequence {xm} in a non-Archimedean space is Cauchy if and only if {xm+1-xm} converges to zero. By a complete non-Archimedean space, we mean one in which every Cauchy 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 algebra (see [22]).

Example 2.2.

Let p 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.

Definition 2.3.

Let X be a nonempty set and d:X×X→[0,∞] satisfy the following properties:

d(x,y)=0 if and only if x=y,

d(x,y)=d(y,x) (symmetry),

d(x,z)≤max{d(x,y),d(y,z)} (strong triangle in equality),

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.Definition 2.4.

Let A be a non-Archimedean algebra. An additive mapping D:A→A is said to be a ring derivation if D(xy)=D(x)y+xD(y) for all x,y∈A. An additive mapping H:A→A is said to be a generalized ring derivation if there exists a ring derivation D:A→A such that
H(xy)=xH(y)+D(x)y,
for all x,y∈A.

We need the following fixed point theorem (see [23, 24]).

Theorem 2.5 (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,
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
X*={y∈X∣d(Λkx,y)<∞},

(c) if y∈X*, then
d(y,x*)≤d(Λy,y).

3. Non-Archimedean Superstability of Generalized Derivations

Hereafter, we will assume that A is a non-Archimedean Banach algebra with unit over a non-Archimedean field 𝕂.

Theorem 3.1.

Let φ:A×A→[0,∞) be a function. Suppose that f,g:A→A are mappings such that g is additive and
‖f(x+y)-f(x)-f(y)‖≤φ(x,y),‖f(xy)-xf(y)-g(x)y‖≤φ(x,y),
for all x,y∈A. If there exists a natural number k∈𝕂 and 0<L<1,
|k|-1φ(kx,ky),|k|-1φ(kx,y),|k|-1φ(x,ky)≤Lφ(x,y),
for all x,y∈A. Then, f is a generalized ring derivation and g is a ring derivation.

Proof.

By induction on i, we prove that
‖f(ix)-if(x)‖≤max{φ(0,0),φ(x,x),φ(2x,x),…,φ((i-1)x,x)},
for all x∈A and i≥2. Let x=y in (3.1). Then,
‖f(2x)-2f(x)‖≤max{φ(0,0),φ(x,x)},n∈N0,x∈A.
This proves (3.4) for i=2. Let (3.4) holds for i=1,2,…,j. Replacing x by jx and y by x in (3.1) for each n∈ℕ0, and for all x∈A, we get
‖f((j+1)x)-f(jx)-f(x)‖≤max{φ(0,0),φ(jx,x)}.
Since
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 (3.6) that
‖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)},
for all x∈A. This proves (3.4) for all i≥2. In particular,
‖f(kx)-kf(x)‖≤ψ(x),
for all x∈A where
ψ(x)=max{φ(0,0),φ(x,x),φ(2x,x),…,φ((k-1)x,x)}(x∈A).

Let X be the set of all functions r:A→A. We define d:X×X→[0,∞] as follows:d(r,s)=inf{α>0:‖r(x)-s(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(r)(x)=k-1r(kx). Then, J is strictly contractive on X, in fact, if
‖r(x)-s(x)‖≤αψ(x),(x∈A),
then by (3.3),
‖J(r)(x)-J(s)(x)‖=|k|-1‖r(kx)-s(kx)‖≤α|k|-1ψ(kx)≤Lαψ(x),(x∈A).
It follows that
d(J(r),J(s))≤Ld(r,s)(r,s∈X).
Hence, J is a strictly contractive mapping with Lipschitz constant L. By (3.9),
‖(Jf)(x)-f(x)‖=‖k-1f(kx)-f(x)‖,|k|-1‖f(kx)-kf(x)‖≤|k|-1ψ(x)(x∈A).
This means that d(J(f),f)≤1/|k|. By Theorem 2.5, J has a unique fixed point h:A→A in the set
U={r∈X:d(r,J(f))<∞},
and for each x∈A,
h(x)=limm→∞Jm(f(x))=limk-mf(kmx).

Therefore,‖h(x+y)-h(x)-h(y)‖=limm→∞|k|-m‖f(km(x+y))-f(kmx)-f(kmy)‖≤limm→∞|k|-mmax{φ(0,0),φ(knx,kny)}≤limm→∞Lmφ(x,y)=0,
for all x,y∈A. This shows that h is additive.

Replacing x by knx in (3.2) to get‖f(knxy)-knxf(y)-g(knx)y‖≤φ(knx,y),
and so
‖f(knxy)kn-xf(y)-g(knx)kny‖≤1|k|nφ(knx,y)≤Lnφ(x,y),
for all x,y∈A and all n∈ℕ. By taking n→∞, we have
h(xy)=xf(y)+limn→∞g(knx)kny,
for all x,y∈A.

Fix m∈ℕ. By (3.21), we havexf(kmy)=h(kmxy)-limn→∞(g(knx)kn(kmy))=kmxf(y)+limn→∞(g(knkmx)kny)-kmlimn→∞(g(knx)kny)=kmxf(y)+kmlimn→∞(g(kn+mx)kn+my)-kmlimn→∞(g(knx)kny)=kmxf(y),
for all x,y∈A. Then, xf(y)=x(f(kmy)/km) for all x,y∈A and each m∈ℕ, and so by taking m→∞, we have xf(y)=xh(y). Now, we obtain h=f, since A is with unit. Replacing y by kny in (3.2), we obtain
‖f(kn(xy))-xf(kny)-kng(x)y‖≤φ(x,kny),
and hence,
‖f(knxy)kn-xf(kny)kn-g(x)y‖≤1|k|nφ(x,kny)≤Lnφ(x,y),
for all x,y∈A and each n∈ℕ. Letting n tends to infinite, we have
f(xy)=xf(y)+g(x)y.
Now, we show that g is a ring derivation. By (3.25), we get
g(xy)z=f(xyz)-xyf(z)=xf(yz)+g(x)yz-xyf(z)=(xg(y)+g(x)y)z,
for all x,y,z∈A. Therefore, we have g(xy)=xg(y)+g(x)y.

The proof of following theorem is similar to that in Theorem 3.1, hence it is omitted.

Theorem 3.2.

Let φ:A×A→[0,∞) be a function. Suppose that f,g:A→A are mappings such that g is additive and
‖f(x+y)-f(x)-f(y)‖≤φ(x,y),‖f(xy)-xf(y)-g(x)y‖≤φ(x,y),
for all x,y∈A. If there exists a natural number k∈𝕂 and 0<L<1,
|k|φ(k-1x,k-1y),|k|φ(k-1x,y),|k|φ(x,k-1y)≤Lφ(x,y),
for all x,y∈A. Then, f is a generalized ring derivation and g is a ring derivation.

The following results are immediate corollaries of Theorems 3.1 and 3.2 and Example 2.3.

Corollary 3.3.

Let A be a non-Archimedean Banach algebra over ℚp, ɛ>0, and p1,p2∈(1,∞). Suppose that f,g:A→A are mappings such that g is additive and
‖f(x+y)-f(x)-f(y)‖≤ɛ(‖x‖p1‖y‖p2),‖f(xy)-xf(y)-g(x)y‖≤ɛ(‖x‖p1‖y‖p2),
for all x,y∈A. Then, f is a generalized ring derivation and g is a ring derivation.

Corollary 3.4.

Let A be a non-Archimedean Banach algebra over ℚp, ɛ>0 and p1,p2,p1+p2∈(-∞,1). Suppose that f,g:A→A are mappings such that g is additive and
‖f(x+y)-f(x)-f(y)‖≤ɛ(‖x‖p1‖y‖p2),‖f(xy)-xf(y)-g(x)y‖≤ɛ(‖x‖p1‖y‖p2),
for all x,y∈A. Then, f is a generalized ring derivation and g is a ring derivation.

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