On Hyers–Ulam stability of two functional equations in non-Archimedean spaces

In this paper we prove, using the fixed point method, the generalized Hyers–Ulam stability of two functional equations in complete non-Archimedean normed spaces. One of these equations characterizes multi-Cauchy–Jensen mappings, and the other gives a characterization of multi-additive-quadratic mappings.

It is well known that among functional equations, the Cauchy equation the Jensen equation = f (x) + f (y) 2 (which is closely connected with the notion of a convex function) and the Jordan-von Neumann (quadratic) equation q(x + y) + q(x − y) = 2q(x) + 2q(y) (which is useful in some characterizations of inner product spaces) play a prominent role. A lot of information about them (in particular, about their solutions (which are said to be additive, Jensen and quadratic mappings, |rs| = |r| |s|, r, s ∈ K, |r + s| ≤ max{|r|, |s|}, r, s ∈ K.
In any non-Archimedean field we have |1| = | − 1| = 1 and |n| ≤ 1 for n ∈ N 0 . In any field K the function | · | : K → R + given by is a valuation which is called trivial, but the most important examples of non-Archimedean fields are p-adic numbers which have gained the interest of physicists for their research in some problems coming from quantum physics, p-adic strings and superstrings. Let X be a linear space over a field K with a non-Archimedean nontrivial valuation | · |. A function ∥ · ∥ : X → R + is said to be a non-Archimedean norm if it satisfies the following conditions: Then (X, ∥ · ∥) is called a non-Archimedean normed space.
In any such a space, the function d : X × X → R + , given by is a metric on X. Recall also that a sequence (x n ) n∈N of elements of a non-Archimedean normed space is Cauchy if and only if (x n+1 −x n ) n∈N converges to zero. Moreover, the addition, scalar multiplication and non-Archimedean norm are continuous mappings. The first work on the Hyers-Ulam stability of functional equations in complete non-Archimedean normed spaces (some particular cases were considered earlier; see [5] for details) is [21]. After it, a lot of papers (see, for instance, [14,27] and the references therein) on the stability of other equations in such spaces have been published.

Stability of an equation characterizing multi-Cauchy-Jensen mappings
In [2], the following characterization of multi-Cauchy-Jensen mappings was proved.
f (x 1i1 , . . . , x nin ). (2.1) In this section, we show the generalized Hyers-Ulam stability of equation (2.1) in complete non-Archimedean normed spaces (its stability in Banach spaces was proved in [2]). The proof is based on a fixed point result that can be derived from [7,Theorem 1]. To present it, we introduce the following three hypotheses: (H1) E is a nonempty set, Y is a complete non-Archimedean normed space over a non-Archimedean field of the characteristic different from 2, Now, we are in a position to present the mentioned fixed point theorem.
With this notation, we have the following result.
Vol. 18 (2016) Stability of two functional equations 437 Stability of two functional equations 5 Theorem 2.3. Suppose that V is a linear space over the rationals and let W be a complete non-Archimedean normed space over a non-Archimedean field of the characteristic different from 2. Let f : V n → W and θ : V n ×V n → R + be mappings satisfying the inequality Fix an x ∈ V n and write Then, by (2.5) and (2.6), we obtain Next, put It is easily seen that Λ has the form described in (H3) with E = V n , j = 1 and Moreover, for any ξ, µ ∈ W V n A. Bahyrycz, K. Ciepliński and J. Olko so hypothesis (H2) is also valid. Finally, using induction, one can check that for any l ∈ N 0 and x ∈ V n we have which, together with (2.3), shows that all assumptions of Theorem 2.2 are satisfied. Therefore, there exists a function F : V n → W such that and (2.4) holds. Moreover, One can now show, by induction, that for l ∈ N 0 , x 1 1 , x 1 2 ∈ V k and x 2 1 , x 2 2 ∈ V n−k . Letting l → ∞ in (2.7) and using (2.3), we obtain which means that the function F satisfies equation (2.1).
Corollary 2.4. Assume that k ≥ 1, ε > 0, V is a linear space over the rationals and W is a complete non-Archimedean normed space over a non-Archimedean field of the characteristic different from 2 such that |2| < 1.

Stability of two functional equations 439
Stability of two functional equations 7
In what follows, P stands for {−1, 1} n−k . Moreover, given f : With this notation, we have the following result.
Theorem 3.2. Let V be a commutative group, W a complete non-Archimedean normed space over a non-Archimedean field of the characteristic different from 2, f : V n → W , θ : V n × V n → R + . Assume also that for any and Then there exists a solution F : Proof. Putting (3.5) Fix an x ∈ V n and write Then, by (3.5), we obtain It is easily seen that Λ has the form described in (H3). Moreover, for any ξ, µ ∈ W V n and x ∈ V n we have so hypothesis (H2) is also valid. Finally, using induction, one can check that for any l ∈ N and x ∈ V n we have which, together with (3.3), shows that all assumptions of Theorem 2.2 are satisfied. Therefore, there exists a function F : V n → W such that and (3.4) holds. Moreover, Vol. 18 (2016) Stability of two functional equations 441 Stability of two functional equations 9 for l ∈ N, x 1 1 , x 1 2 ∈ V k and x 2 1 , x 2 2 ∈ V n−k . In order to do this, fix ) .
Next, assume that (3.6) holds for an l ∈ N.
Corollary 3.3. Assume that t ∈ R fulfills t > 2n − k, V is a normed space and W is a complete non-Archimedean normed space over a non-Archimedean field of the characteristic different from 2 such that |2| < 1. If f : V n → W satisfies the inequality Similarly, Theorem 3.2 with the control function gives the following outcome.
Corollary 3.4. Assume that t ji > 0, for i ∈ {1, 2} and j ∈ {1, . . . , n}, fulfill V is a normed space and W is a complete non-Archimedean normed space over a non-Archimedean field of the characteristic different from 2 such that |2| < 1. If f : V n → W satisfies the inequality