The stability problem for functional equations first was planed in 1940 by Ulam [1]:

Let *G*_{1} be group and *G*_{2} be a metric group with the metric *d*(·,·). Does, for any *ε >* 0, there exists *δ >* 0 such that, for any mapping *f* : *G*_{1} → *G*_{2} which satisfies *d*(*f*(*xy*), *f*(*x*)*f*(*y*)) ≤ *δ* for all *x*, *y* ∈ *G*_{1}, there exists a homomorphism *h* : *G*_{1} → *G*_{2} so that, for any *x* ∈ *G*_{1}, we have *d*(*f* (*x*), *h*(*x*)) ≤ *ε*?

In 1941, Hyers [2] answered to the Ulam's question when *G*_{1} and *G*_{2} are Banach spaces. Subsequently, the result of Hyers was generalized by Aoki [3] for additive mappings and Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias [4] has provided a lot of influences in the development of the Hyers-Ulam-Rassias stability of functional equations (for more details, see [5] where a discussion on definitions of the Hyers-Ulam stability is provided by Moszner, also [6–12]).

In this paper, we give a modification of the approach of Belaid et al. [13] in non-Archimedean spaces. Recently, Ciepliński [14] studied and proved stability of multi-additive mappings in non-Archimedean normed spaces, also see [15–22].

**Definition 1.1**. The function | · | :

*K* → ℝ is called a

*non-Archimedean valuation* or

*absolute value* over the field

*K* if it satisfies following conditions: for any

*a*,

*b* ∈

*K*,

- (1)

- (2)
|*a*| = 0 if and only if *a* = 0;

- (3)

- (4)

- (5)
there exists a member *a* _{0} ∈ *K* such that |*a* _{0}| ≠ 0, 1.

A field *K* with a non-Archimedean valuation is called a *non-Archimedean field*.

**Corollary 1.2**. |-1| = |1| = 1 *and so, for any a* ∈ *K, we have* |-*a*| = |*a*|. *Also, if* |*a*| *<* |*b*| *for any a*, *b* ∈ *K, then* |*a* + *b*| = |*b*|.

In a non-Archimedean field, the triangle inequality is satisfied and so a metric is defined. But an interesting inequality changes the usual

*Archimedean* sense of the absolute value. For any

*n* ∈ ℕ, we have |

*n* · 1| ≤ ℝ. Thus, for any

*a* ∈

*K*,

*n* ∈ ℕ and nonzero divisor

*k* ∈ ℤ of

*n*, the following inequalities hold:

**Definition 1.3**. Let

*V* be a vector space over a non-Archimedean field

*K*. A

*non-Archimedean norm* over

*V* is a function

*||* ·

*||* :

*V* → R satisfying the following conditions: for any

*α* ∈

*K* and

*u*,

*v* ∈

*V*,

- (1)
*||u||* = 0 if and only if *u* = 0;

- (2)

- (3)
*||u* + *v||* ≤ max{*||u||*, *||v||*}.

Since 0 = ||0|| = ||*v - v*|| ≤ max{||*v*||, ||-*v*||} = ||*v*|| for any *v* ∈ *V*, we have ||*v*|| ≥ 0. Any vector space *V* with a non-Archimedean norm || · || : *V* → ℝ is called a *non-Archimedean space*. If the metric *d*(*u*, *v*) = ||*u - v*|| is induced by a non-Archimedean norm || · || : *V* → ℝ on a vector space *V* which is complete, then (*V*, || · ||) is called a *complete non-Archimedean space*.

**Proposition 1.4**. ([23]) *A sequence*
*in a non-Archimedean space is a Cauchy sequence if and only if the sequence*
*converges to zero*.

Since any non-Archimedean norm satisfies the triangle inequality, any non-Archimedean norm is a continuous function from its domain to real numbers.

**Proposition 1.5**. *Let V be a normed space and E be a non-Archimedean space. Let f* : *V* → *E be a function, continuous at* 0 ∈ *V such that, for any ×* ∈ *V, f*(2*x*) = 2*f*(*x*) (*for example, additive functions*). *Then, f* = 0.

*Proof*. Since

*f*(0) = 0, for any

*ε >* 0, there exists

*δ >* 0 that, for any

*x* ∈

*V* with ||

*x*|| ≤

*δ*,

and, for any

*x* ∈

*V*, there exists

*n* ∈ ℕ that

and hence

Since this inequality holds for all *ε >* 0, it follows that, for any *x* ∈ *V*, *f*(*x*) = 0. This completes the proof.

The preceding fact is a special case of a general result for non-Archimedean spaces, that is, *every continuous function from a connected space to a non-Archimedean space is constant*. This is a consequence of *totally disconnectedness* of every non-Archimedean space (see [23]).